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60 (2014) 5
Strojniški vestnik Journal of Mechanical Engineering
Since 1955
Contents
Papers
287
Peter Avitabile, Christopher Nonis, Sergio E. Obando: System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
307
Andrea Barbarulo, Hervé Riou, Louis Kovalevsky, Pierre Ladeveze: PGDVTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems
314
Kimihiko Nakano, Matthew P. Cartmell, Honggang Hu, Rencheng Zheng: Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force
321
Diego Saba, Paola Forte, Giuseppe Vannini: Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data
331
Snehashish Chakraverty, Diptiranjan Behera: Parameter Identification of Multistorey Frame Structure from Uncertain Dynamic Data
339
Martin Česnik, Janko Slavič: Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads
349
Darryl K. Stoyko, Neil Popplewell, Arvind H. Shah: Reflection and Transmission Coefficients from Rectangular Notches in Pipes
363
Gaia Volandri, Costantino Carmignani, Francesca Di Puccio, Paola Forte: Finite Element Formulations Applied to Outer Ear Modeling
Journal of Mechanical Engineering  Strojniški vestnik
298 Fabrício César Lobato de Almeida, Michael John Brennan, Phillip Frederick Joseph, Simon Dray, Stuart Whitfield, Amarildo Tabone Paschoalini: Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
5 year 2014 volume 60 no.
Strojniški vestnik – Journal of Mechanical Engineering (SVJME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue. The international conferences selected papers are welcome for publishing as a special issue of SVJME with invited coeditor(s). Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Founding Editor Bojan Kraut University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SVJME, Aškerčeva 6, SI1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@svjme.eu, http://www.svjme.eu Print: Littera Picta, printed in 400 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok
International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).
University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
VicePresident of Publishing Council Jože Balič University of Maribor, Faculty of Mechanical Engineering, Slovenia
Cover: Reduced order system models are developed from uncoupled component modes. These reduced order system modes can be expanded using the uncoupled component modes provided they span the space of the system modes. This allows for expansion to full field from the reduced order system model. The significance is that the reduced order system model can be used for efficient dynamic response studies and the expansion allows for prediction of full field displacement and full field dynamic stressstrain. Courtesy: Structural Dynamics and Acoustic Systems Lab at UML, Massachusetts, USA
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5 Contents
Contents Strojniški vestnik  Journal of Mechanical Engineering volume 60, (2014), number 5 Ljubljana, May 2014 ISSN 00392480 Published monthly
Papers Peter Avitabile, Christopher Nonis, Sergio E. Obando: System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data 287 Fabrício César Lobato de Almeida, Michael John Brennan, Phillip Frederick Joseph, Simon Dray, Stuart Whitfield, Amarildo Tabone Paschoalini: Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes 298 Andrea Barbarulo, Hervé Riou, Louis Kovalevsky, Pierre Ladeveze: PGDVTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems 307 Kimihiko Nakano, Matthew P. Cartmell, Honggang Hu, Rencheng Zheng: Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force 314 Diego Saba, Paola Forte, Giuseppe Vannini: Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data 321 Snehashish Chakraverty, Diptiranjan Behera: Parameter Identification of Multistorey Frame Structure from Uncertain Dynamic Data 331 Martin Česnik, Janko Slavič: Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads 339 Darryl K. Stoyko, Neil Popplewell, Arvind H. Shah: Reflection and Transmission Coefficients from Rectangular Notches in Pipes 349 Gaia Volandri, Costantino Carmignani, Francesca Di Puccio, Paola Forte: Finite Element Formulations Applied to Outer Ear Modeling 363
StrojniĹĄki vestnik  Journal of Mechanical Engineering 60(2014)5 Editorial
Guest Editorial Special Issue: Recent Advances in Structural Dynamics
The International Conference on Recent Advances in Structural Dynamics has now been successfully running and hosted by the main organiser namely ISVR since 1980 and is now on its eleventh meeting. Its inception was the brainchild of Prof. Maurice Petyt, from ISVR, who continued successfully at the helm with later assistance from Howard Wolfe and Chuh Mei. The international authors who have contributed over the years reads like a whoâ€™s who list of eminent structural dynamicists, e.g. Meirovitch, Heckl, Braun, Noor, Mead, Langley, Fuller, Elishakoff, Ewins, etc. to name just a few and their work and contributions have inevitably stood the test of time. Howard Wolfe was involved in the third conference onwards due to his involvement with the United States Air Force and the topic of Acoustic Fatigue and related dynamics was actively pursued. The USA Air Force Research Office have continued their valuable support and, in all subsequent conferences, sessions have been dedicated to both numerical modelling and experimental investigation into issues relevant to supersonic aircraft. The main organisers changed over the years, as Neil Ferguson and subsequently Mike Brennan and Emiliano Rustighi lead it from ISVR with main support from Steve Rizzi (NASA Langley) and briefly Marty Ferman. Throughout the time the principles of publishing recent stateoftheart developments (theoretically, numerically and experimentally) and peer reviewed papers has been maintained. The papers presented herein are just a snapshot of the quality work that has been presented and produced. The Eleventh International Conference on Recent Advances in Structural Dynamics (RASD 2013) was organised in Pisa, Italy, whereas all of the previous meetings were held in the University of Southampton, UK. Although being still organised by ISVR, the conference has been hosted by the prestigious and historic University of Pisa, which echoes the internationalisation character and the significance of the event. As in former editions, RASD2013 has reflected the international stateoftheart of structural dynamics and dynamical systems in science and engineering practice. In fact, the conference has seen
the presentation of 127 papers by authors from over 25 countries. The papers were presented in 28 sessions covering subjects as diverse as nonlinear vibration, vibroacoustics, structural health monitoring, human structure interaction, system identification and inverse problems, earthquake engineering, modal analysis and structural modification, vibration control, civil engineering structures, numerical techniques, active vibration control, smart structures, soilstructure vibrations, reusable hypersonic platform, analytical modelling, experimental methods, fluidstructure interaction, uncertain dynamical systems, energy harvesting, stochastic dynamics, random vibrations, railway noise and vibration. A highlight of RASD2013 has also been the special session on aerostructures for reusable hypersonic platforms, followed by a moderated discussion on relevant unsolved technical challenges. There were 5 plenary lectures given by Professor Y. Suda (Dynamic Simulation and Analysis for Sustainable Transport Systems, University of Tokyo, Japan), Professor A. Metrikine (Development, evaluation and application of a wakeoscillator model for vortexinduced vibrations of marine risers, TU Delft, Netherlands), Professor J.S. Bolton (The influence of boundary conditions and internal constraints on the performance of noise control treatments: foams to metamaterials, Purdue University, USA), Dr A.M. Wickenheiser (Broadband VibrationBased Energy Harvesting: Model Reduction and Frequency UpConversion, The George Washington University, USA) and Dr S.A. Rizzi (An Overview of Virtual Acoustic Simulation of Aircraft Flyover Noise, NASA Langley Research Center, USA). Overall RASD2013 has been a fantastic occasion to exchange and disseminate ideas among esteemed colleagues scientific, technical and experimental ideas. This Special Issue is comprising nine papers; they reflect only a small part of all of the presented papers. The paper by Avitabile et al discusses system model modes developed from expansion of uncoupled component dynamic data. Almeida et al research measurement of wave attenuation in buried plastic water distribution pipes. Barbarulo et al concentrate on complex acoustical problems in midfrequency 285
range by using the variational theory of complex rays and the proper generalized decomposition. Nakano et al discuss feasibility of energy harvesting using stochastic resonance caused by axial periodic force. Saba et al review and upgrade of a bulk flow model for the analysis of honeycomb gas seals based on new high pressure experimental data. Chakraverty et al talk about parameter identification of multistorey frame structure from uncertain dynamic data. ÄŒesnik et al combine structural dynamics and vibrational
fatigue for harmonic and random loads. Stoyko et al report result of their work about reflection and transmission coefficients from rectangular notches in pipes. At the end, Volandri at al describes and explore sound propagation in the outer ear using numerical modeling. The Guest Editors would like to thank firstly to all who made the Pisa RAD2013 conference happen and secondly to all authors and reviewers that contributed to this Special Issue.
Guest Editors: Miha BolteĹžar, University of Ljubljana, Faculty of Mechanical Engineering Emiliano Rustighi, ISVR, University of Southampton Neil Ferguson, ISVR, University of Southampton
286
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 287297 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1829 Special Issue, Original Scientific Paper
Received for review: 20130809 Received revised form: 20140214 Accepted for publication: 20140328
System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data Avitabile, P. – Nonis, C. – Obando, S.E. Peter Avitabile* – Christopher Nonis – Sergio E. Obando
1 University
of Massachusetts Lowell, Department of Mechanical Engineering, Structural Dynamics and Acoustic Systems Laboratory, USA
System models are often developed from component models either as modal components or as reduced order models. The resulting system model data is only available at the reduced order space unless some expansion is performed. Of course this can be achieved using system model mapping matrices but that requires the development of the full space system model which defeats the purpose of the component synthesis approach. However, the individual uncoupled component mapping matrices can be utilized to expand to the full space of the system model provided the modes of the components span the space of the full system model. This paper shows the results of using component modes from the unconnected components as projection matrices to identify the system level full field response. Multiple analytical cases are presented to show how the selection of component modes affects the expansion results. The results show accurate system model expansion using a sufficient set of component modes that span the space of the system model. Keywords: component modeling, system modeling, expansion
0 INTRODUCTION Reduced order models are often used as component representations in a system model to reduce computation time while including appropriate structural dynamic characteristics. The model reduction is performed to lower the total number of degrees of freedom (DOF) while retaining important dynamic characteristics. The system response can then be computed with very efficient reduced order system models. Once the system level response is obtained from the reduced component system model, the full space solution is often desired for computation of the overall system dynamic stress and strain, in the individual components, at the element level. However, in order to obtain the full field response, expansion procedures are needed. Generally, to find the system model transformation matrix, the full DOF mode shapes must be computed for the full system model. This procedure is counterproductive because the purpose of model reduction is to avoid computing the full DOF model. This paper shows that the system modes can be expanded using the transformation matrices of the original uncoupled component modes and that the final system full shapes are actually not needed to predict the system level full field characteristics. To perform this proposed expansion, the system shapes are separated into component shapes and expanded on a component by component basis using the component transformation matrices obtained from the uncoupled, original component modes. The system
equivalent reduction expansion process (SEREP) [1] is used for the development of the transformation matrix because this technique exactly preserves the dynamics of the model regardless of which degrees of freedom are retained. The SEREP process used here is augmented with the variability improvement of key inaccurate node groups (VIKING) [2]. The VIKING paper [2] showed that over specifying the number of modes used in the reduction/ expansion process that span the space of the system model modes significantly improves the expansion results. Therefore, if the modes used to reduce the component span the space of the component’s shapes in the system modes, accurate results can be achieved; the error in the expanded shapes actually results from mode truncation, not from the expansion process itself. This paper shows how to obtain full field, expanded system level characteristics from the individual, uncoupled component modes of the individual components. The following sections present some background theory along with the expansion methodology proposed followed by test cases to demonstrate the technique. 1 THEORETICAL BACKGROUND Some of the pertinent equations of importance are described below relating to the modal space representation, the structural dynamic modification process and the reduction expansion process as well as correlation tools used to verify the results obtained.
*Corr. Author’s Address: University of Massachusetts Lowell, 1 University Ave, Lowell, USA, peter_avitabile@uml.edu
287
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 287297
1.1 Equations of Motion for Multiple Degree of Freedom System The general equation of motion for a multiple degree of freedom system written in matrix form is:
[ M1 ]{x} + [C1 ]{x } + [ K1 ]{x} = {F(t )}. (1)
Assuming eigensolution is:
proportional
damping,
the
[ K1 ] − λ [ M1 ] {x} = {0}. (2)
The results of the eigensolution yield the eigenvalues (natural frequencies) and eigenvectors (mode shapes). The eigenvectors are arranged in column fashion to form the modal matrix [U1]. Usually a subset of modes is included in the modal matrix to save computation time. Exclusion of modes results in truncation error which can be serious if key modes are excluded. Truncation error will be discussed in further detail in the structural dynamic modification section. The physical system can be transformed to modal space using the modal matrix as:
[ U1 ]T [ M1 ][ U1 ]{p1} + [ U1 ]T [ K1 ][ U1 ]{p1} = T = [ U1 ] {F( t )}. (3) Scaling to unit modal mass yields:
T
[ I1 ]{p1} + Ω12 {p1} = U1n {F( t )} , (4)
where [I1] is the identity matrix and Ω1 is the diagonal natural frequency matrix. More detailed information on the equation development is contained in ref. [3]. 1.2 Structural Dynamic Modification Structural dynamic modification (SDM) is a technique that uses the original mode shapes and natural
frequencies of a system to estimate the dynamic characteristics after a modification of mass and/ or stiffness is made. First, the change of mass and stiffness are transformed to modal space as shown:
∆M12 = [ U1 ] [ ∆M12 ][ U1 ] , (5)
∆K12 = [ U1 ] [ ∆K12 ][ U1 ]. (6)
T
T
The modal space mass and stiffness changes are added to the original modal space equations to give: M1 + ∆M12 {p1} +
(7) + K1 + ∆K12 {p1} = [ 0].
The eigensolution of the modified modal space model is computed and the resulting eigenvalues are the new frequencies of the system. The resulting eigenvector matrix is the [U12] matrix, which is used to transform the original modes to the new modes and is given as:
[ U 2 ] = [ U1 ][ U12 ]. (8)
The new mode shapes are [U2]. The new mode shapes are formed from linear combinations of the original mode shapes. The [U12] matrix shows how much each of the [U1] modes contributes to forming the new modes. Fig. 1 shows the formation of the new mode shapes as seen in Eq. (8). Unless [U1] includes all of the original ‘n’ system modes, there will be truncation error due to the ‘nm’ missing modes. The severity of truncation error depends on which modes are missing from [U1]. Some original system modes are more important than others for forming the new modes. A [U12] calculated using
m
Fig. 1. Structural dynamic modification, mode contribution identified using U12 [2]
288
Avitabile, P. – Nonis, C. – Obando, S.E.
\\\\\\\\
m
///////////////// nm
nm
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 287297
all the original system modes would show the correct contributions of all modes. More detailed information on SDM is contained in ref. [3]. 1.3 Physical Space System Modeling To form a physical system model, the mass and stiffness matrices of each component (A and B) are assembled in stacked form into the system mass and stiffness matrices. In physical space, these are coupled with a stiffness tie matrix; a mass tie can also be included if desired but not included in this work (see [3] for further development of the mass tie case).
as:
MA { x} + MB K A + + K TIE {x} = {F}. (9) B K
This can be cast in a modal space representation
{ } + { } K {p } + K {p } {p } + [ U ] [ ∆K ][ U ] = {0}, (10) p { } MA pA B M B p A
A
B
B
x
{x n } = x a = [T ]{x a } , (11)
d
where subscript ‘n’ signifies the full set of DOF (n DOF), ‘a’ signifies the reduced set of DOF (a DOF) and ‘d’ is the deleted DOF. The transformation matrix [T] relates the full set of DOF to the reduced set of DOF. The transformation matrix is used to reduce the mass and stiffness matrices as: M a = [ T ] [ M n ][ T ] and K a = [ T ] [ K n ][ T ]. (12) T
T
The eigensolution of these ‘a’ set mass and stiffness matrices are the modes of the reduced model. These modes can be expanded back to full space using the transformation matrix:
[ U n ] = [T ]Ua . (13)
If an optimal ‘a’ set is not selected when using methods such as Guyan condensation [4] or improved reduced system technique [5], the reduced model may not perfectly preserve the dynamics of the full space model. If system equivalent reduction expansion process (SEREP) is used, the dynamics of selected modes will be perfectly preserved regardless of the ‘a’ set selected. 1.5 System Equivalent Reduction Expansion Process (SEREP)
A
T
B
where M and K are diagonal matrices and with the mode shapes of each component stacked as:
[U] = [U
A
]
. U
[ ] B
Eq. (9) is a general equation of motion; Eq. (10) is used for the eigensolution in which the force is never included. 1.4 General Reduction/Expansion Technique Model reduction is used to reduce the number of degrees of freedom in an analytical model to reduce computing time while attempting to preserve the full DOF characteristics. The relationship between the full space and reduced space model can be written as:
The SEREP modal transformation relies on the partitioning of the modal equations representing the system DOFs relative to the modal DOFs [1]. The SEREP technique utilizes the mode shapes from a full finite element solution to map to the limited set of master DOF. SEREP is not performed to achieve efficiency in the solution but rather is intended to perform an accurate mapping matrix for the transformation. The SEREP transformation matrix is formed using a subset of modes at full space and reduced space as:
g TU = [ U n ]U a , (14)
where the modal vectors with subscripts n and a are as described in Section 1.4 and the superscript g indicates the generalized inverse. When the SEREP transformation matrix is used for model reduction/expansion as outlined in the previous section, the reduced model perfectly
System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 287297
preserves the full space dynamics of the modes in [Un] as presented in more detail in [1]. 1.6 Modal Assurance Criterion (MAC) Modal assurance criterion is a correlation tool commonly used to compare mode shapes [6]. MAC compares two vectors (ui and ej) and calculates a value from 0 to 1 that quantifies the degree of similarity between the vectors. The equation is:
MACij =
2
{e j} . (15) {u }T {u } {e }T {e } i j j i T
{u } i
A MAC value of 1 signifies perfect correlation and 0 signifies no correlation. 1.7 Pseudo Orthogonality Check Pseudo orthogonality check (POC) is a mass weighted orthogonality tool used to compare mode shapes and is given as:
POC = [ U ] [ M ][ E ]. (16) T
The POC is mass weighted. If the shapes are scaled to unit modal mass, POC ranges from 0 to 1, similar to MAC. See [7] for further information on model correlation. 2 METHODOLOGY The procedure for expanding reduced system models using limited sets of component modal information can be described in the four steps as follows. a) Reduce models of system components individually using SEREP. Connection degrees of freedom must be retained during reduction in all cases for the assembly process. The modes selected for reduction are very important for minimizing truncation error; Fig. 2 schematically shows the component reduction. b) Assemble component mass and stiffness matrices into system mass and stiffness matrices and link connection DOF; Fig. 3 schematically shows the formation of the system’s matrices.
Component A: SEREP reduction T
M aA = TuA M A T A , n u T
K aA = TuA K A T A . n u Component B: SEREP reduction T
M aB = TuB M Bn TuB , T
K aB = TuB K Bn TuB . Fig. 2. Reduce system components using SEREP
a space
MA , M sys = MB K A + K . K sys = tie B K
Fig. 3. Assemble reduced components to form system
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K A MA {x} = {0} yields: a Space λ + − K tie B B K M ω12 natural frequencies:
ω22
UA . , system mode shapes: U a = B U
Fig. 4. Perform eigensolution to calculate frequencies and mode shapes
a space
expansion
n space
UA U a = B U
TuA × U A = B B Tu × U =
UA = U B [ n ] U
↑ Tu matrices used to reduce components. Fig. 5. Expand reduced system shapes using component transformation matrices
c) Perform eigensolution on system mass and stiffness matrices to calculate frequencies and mode shapes; Fig. 4 schematically shows the solution process. d) Partition the system mode shapes into matrices of shape information for each component. Use the SEREP transformation matrices of each component to expand the component shape matrices to full space; Fig. 5 schematically shows the expansion of the reduced system matrices. The results of this expansion process are the system mode shapes at the full set of DOF whose accuracy is only affected by mode truncation error that occurred during the reduction process. This will be demonstrated by the following cases. 3 MODEL DESCRIPTION A simple two beam system model was used to illustrate the expansion technique. The beams were modeled using 2D Euler Bernoulli beam elements with only in plane motion considered. Fig. 6 shows the two beam model. The bottom beam is connected to ground at each end by a 175,126.84 N/m translational spring. The two beams are connected with two translation springs.
Fig. 6. Configuration of two beam system model
Table 1 shows the parameter used for the two beam system of Fig. 6. Table 1. Two beam system model parameters Parameter Length [m] Height [m] Width [m] Wall thickness [m] Area moment of inertia [m4] Elastic Modulus [GPa] Density [kg/m3] Nodes Beam elements Full space DOF Connector spring stiffness [N/m] Ground support spring stiffness [N/m]
System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
Top beam Bottom beam 1.27 3.56 0.038 0.038 0.076 0.076 0.0048 0.0048 2.22×107 2.22×107 69 69 2712 2712 15 29 14 28 30 56 175,126.835 175,126.84
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4 CASES STUDIED In order to evaluate the expansion methodology, two cases are presented here; other cases were also explored but are not included in this paper due to space restrictions and yield similar results supporting the accuracy and correctness of the expansion methodology presented in this paper. All cases were reduced and expanded following the procedure outlined in the methodology section. For comparison, a full space physical model and a structural dynamic modification model were also developed and compared. The SDM model is comprised of the full DOF using the same modes used for reduction. MAC and POC are used to compare mode shapes. Two cases were studied here and are described as: • Case 1: 5 modes each from component sA & B; • Case 2: 10 modes each from components A & B. These cases are described in the following sections. Fig. 7 shows the entire expansion process schematically to further describe the overall procedure. The component model modes are extracted from the full space component models (which are typically available in the design process) as shown in the upper portion of the figure. These component models are then reduced with the component transformation matrices and used to develop the reduced order system model from the reduced order component models as shown in the center portion of the figure. The individual, uncoupled component models are used to develop the transformation/expansion matrix shown on the left and right middle section of the figure; this transformation/expansion matrix already exists because it was used to reduce the components. These expansion matrices are used to expand the reduced order system model modes to obtain the full space system model modes for the assembled system model shown at the bottom of the figure. 4.1 Case 1: 5 Modes from Component A & 5 Modes from Component B Each beam was reduced to 5 translational DOF using the first 5 modes of each component beam to form the system model. The expanded results are compared to the full space physical and SDM models in Table 2 and Fig. 8. The table is broken down into the left and right sides as discussed next; the left side compares the full space physical system model results and the right side presents comparison checks to a SDM model to confirm accuracy of the results. On the left side, the full space system model lists the results obtained from a full space physical model 292
and is referred to as the reference solution. Based on the [U12] matrix generated for this case (not shown for brevity), only 6 system modes are expected to be accurate; this is due to the fact that the 5 component modes of each component only accurately span that space of the system model. The [U12] from the structural dynamic modification system modeling process identifies this and these results are anticipated. Table 2 shows that the first 6 system modes are accurately represented by the reduced component system model as evidenced by the accurate frequency prediction (<2.00% difference). The reduced order system model mode shapes are expanded to full space using the original unmodified individual component modes as described in this paper. These expanded system model modes are then checked with the full space reference model using both MAC and POC; these two columns of values on the left side of the table clearly show that the expansion of the reduced order system model using the original unmodified component modes are an accurate representation of the mode shapes. Any modes beyond the 6 system model modes are not accurately predicted because the modes of the component models do not span the space of the higher order system modes and therefore do not accurately predict those frequencies. The results on the right side of the table are presented as a check to make sure that the results are as expected. The SDM results are those obtained using the full space modal model and are compared to the expanded results; these two results must produce identical results for both the accurately predicted system modes as well as the truncated higher frequency results. The MAC and POC also substantiate this. This perfect correlation shows there is no distortion when expanding from ‘a’ space to ‘n’ space; the only source of error for the modal models is mode truncation. The results of this case clearly show that if the original unmodified component modes are sufficient to span the space of the reduced order system model to predict accurate system model frequencies, then these same component modes are also sufficient to be used as expansion matrices to obtain the full system model combined response; but this is only true for the system modes that can be accurately represented by the individual component modes. 4.2 Case 2: 10 Modes from Component A & 10 Modes from Component B In Case 1, the first 6 system modes were accurately represented by 5 component modes for each individual
Avitabile, P. – Nonis, C. – Obando, S.E.
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component. These 5 component modes are only capable of predicting the first 6 system model modes accurately. To show that truncation limits the number of accurate system modes, Case 2 uses 10 component modes for each of the individual components and based on the [U12] then the system model can only predict the first 12 system model modes accurately. Table 3 and Fig. 9 show these results. Case 2 shows that more system modes are accurately predicted due to the inclusion of more component modes to describe each reduced order component and further substantiate the results in Case 1.
4.3 Observations for Cases Studied Results of the case studies show that expanding system model modes using the component transformation matrices yields exactly the same results as a full space structural dynamic modification model, assuming the same modes are used. In the full space SDM model, mode truncation is the only source of error. There is no expansion error because the SDM is done at full space. The SDM and expansion results were shown to be equivalent. Therefore mode truncation is the only source of error and differences are not a result of the expansion
Fig. 7. Overall expansion process schematic for models studied System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
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Table 2. Case 1, expanded results compared to full Space physical and SDM model Case 1: 5 Modes each from component A & B with symmetric mounting
Mode
1 2 3 4 5 6 7 8 9 10
Full space system model comparison Frequency [Hz] Full Expanded MAC Physical model from % Diff ‘n’ space ‘a’ space 7.3 7.2 0.01 1.000 29.5 29.5 0.02 1.000 58.7 58.7 0.01 1.000 81.7 83.36 2.00 0.997 83.3 83.37 0.10 1.000 132.8 132.8 0.00 1.000 191.8 316.4 65.0 0.073 273.8 680.8 148 0.064 306.2 4461.2 1356 0.002 398.7 4977.3 1148 0.000
MAC
POC 1.000 1.000 1.000 0.998 0.999 0.999 0.253 0.252 0.058 0.000
Structural dynamic modification comparison Frequency [Hz] SDM Expanded MAC at model from % Diff ‘n’ space ‘a’ space 7.2 7.2 0.00 1.000 29.5 29.5 0.00 1.000 58.7 58.7 0.00 1.000 83.36 83.36 0.00 1.000 83.37 83.37 0.00 1.000 132.8 132.8 0.00 1.000 316.4 316.4 0.00 1.000 680.8 680.8 0.00 1.000 4461.2 4461.2 0.00 1.000 4977.3 4977.3 0.00 1.000
Expanded compared to Full space physical model
POC
MAC POC Expanded compared to Structural dynamic modification model Fig. 8. Case 1, expanded results compared to full space model and SDM
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POC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
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Table 3. Case 2, expanded results compared to full Space physical and SDM model Case 2: 10 Modes each from component A & B with symmetric mounting
Mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Full space system model comparison Frequency [Hz] Full Expanded MAC Physical model from % Diff ‘n’ space ‘a’ space 7.3 7.3 0.00 1.000 29.5 29.5 0.00 1.000 58.7 58.7 0.00 1.000 81.7 81.9 0.21 1.000 83.3 83.3 0.01 1.000 132.8 132.8 0.00 1.000 191.8 191.8 0.03 1.000 273.8 273.9 0.06 1.000 306.2 307.6 0.44 1.000 398.7 398.7 0.01 1.000 514.2 515.0 0.15 1.000 649.0 649.7 0.10 0.999 674.4 694.7 3.02 0.945 840.4 1206.6 43.6 0.000 1011.6 1837.4 81.6 0.000
technique. Truncation error is introduced when reducing the component models and forming the system model. Comparison to the full space model shows that lower order modes can be calculated accurately; higher order modes are significantly affected by mode truncation and cannot be predicted accurately, as expected. If more system modes were desired, more component modes would need to be used in the reduction of the components.
POC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 1.000 0.967 0.000 0.000
Structural dynamic modification comparison Frequency [Hz] SDM Expanded MAC at model from % Diff ‘n’ space ‘a’ space 7.3 7.3 0.00 1.000 29.5 29.5 0.00 1.000 58.7 58.7 0.00 1.000 81.9 81.9 0.00 1.000 83.3 83.3 0.00 1.000 132.8 132.8 0.00 1.000 191.8 191.8 0.00 1.000 273.9 273.9 0.00 1.000 307.6 307.6 0.00 1.000 398.7 398.7 0.00 1.000 515.0 515.0 0.00 1.000 649.7 649.7 0.00 1.000 694.7 694.7 0.00 1.000 1206.6 1206.6 0.00 1.000 1837.4 1837.4 0.00 1.000
POC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
sufficient number of modes were used to appropriately span the space for the nonlinear solution, then those very efficient models could be deployed without loss of accuracy. That previous nonlinear work coupled with this expansion methodology can accurately expand the nonlinear response to the full space of each of the individual components to provide full field response for complicated dynamic stressstrain distributions with these same very efficient reduced order nonlinear model representations. 5 CONCLUSIONS
4.4 Important Applications Results of the case studied clearly show the usefulness for development of linear system models from component data. However another very important application is for the development of nonlinear models interconnected with highly nonlinear connection elements [8] and [9]. In that work, the linear modal components are interconnected with highly nonlinear connection elements and nonlinear response was shown to be very accurately produced with highly efficient models. That work showed that as long as a
The technique proposed in this work uses the expansion matrices of uncoupled component models to expand the modes of an assembled reduced order system model. Expansion to the full space assembled system can be performed to define full space characteristics. Several reduced order system model cases were presented to demonstrate the validity of the expansion approach developed in this work. This approach works provided that the original unmodified
System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
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MAC
Expanded compared to Full space physical model
POC
MAC POC Expanded compared to Structural dynamic modification model Fig. 9. Case 2, expanded results compared to full space model and SDM
component modes retained in the reduced order model are sufficient to span the space of the final system model in order to obtain accurate system modes. The expansion matrices developed in the reduction process, which contains the same modal information as in the reduced models, are also sufficient to be used as expansion matrices to obtain full system model characteristics; but this is only true for the system modes that can be accurately represented by the individual component modes. This allows for full system level identification of modal characteristics without the need for developing the expansion matrices using the fully assembled full space system model and allows for full space prediction of important system level information such as dynamic stress and strain. 6 ACKNOWLEDGEMENTS Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA86511010009 “Development of Dynamic Response Modeling Techniques for Linear Modal 296
Components”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency. The authors are grateful for the support obtained. 7 NOMENCLATURE Symbols: Matrix [M] Analytical mass matrix [K] Analytical stiffness matrix [C] Physical damping matrix [U] Analytical modal matrix M K C
Diagonal modal mass matrix
[T] [I] [E]
Transformation matrix Identity matrix Matrix of expanded modal vectors
Diagonal modal stiffness matrix Diagonal modal damping matrix
Avitabile, P. – Nonis, C. – Obando, S.E.
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Vector {p} {F} {u} {x} {x }
Modal displacement Force Analytical mode shape Physical displacement Physical velocity
{x}
Physical acceleration
{t}
Time vector
Subscript State 1 1 2 State 2 12 State 12 Row i i j Column j n Full set of finite element DOF Reduced set of DOF a d Deleted (omitted) set of DOF U SEREP tie Stiffness tie matrix sys System AB of assembled components Superscript T Transpose g Generalized inverse k kth degree of freedom 1 Standard inverse A Component A Component B B
8 REFERENCES [1] O’Callahan, J.C., Avitabile, P., Riemer, R. (1989). System equivalent reduction expansion process. Proceedings of the 7th International Modal Analysis Conference, Las Vegas. [2] Thibault, L., Butland, A., Avitabile, P. (2012). Variability improvement of key inaccurate node groups – VIKING. Proceedings of the 13th International Modal Analysis Conference, Jacksonville. [3] Avitabile, P. (2003). Twenty years of structural dynamic modification – A review. Sound and Vibration, vol. 37, no. 1, p. 1427. [4] Guyan, R.J. (1965). Reduction of stiffness and mass matrices. AIAA Journal, vol. 3, no. 2, p. 380, DOI:10.2514/3.2874. [5] O’Callahan, J.C., (1989). A procedure for an improved reduced system (IRS) model. Proceedings of the 7th International Modal Analysis Conference, Las Vegas. [6] Allemang, R.J., Brown, D.L. (2007). A correlation coefficient for modal vector analysis. Proceedings of the 1st International Modal Analysis Conference, Orlando. [7] Avitabile, P., O’Callahan, J. (1988). Model correlation and orthogonality criteria. 6th International Modal Analysis Conference, Orlando. [8] Thibault, L., Avitabile, P., Foley, J., Wolfson, J. (2012). Equivalent reduced model technique development for nonlinear system dynamic response. Mechanical Systems and Signal Processing, vol. 36, no. 2, p. 422455, DOI:10.1016/j.ymssp.2012.07.013. [9] Marinone, T., Avitabile, P., Foley, J., Wolfson, J. (2012). Efficient computational nonlinear dynamic analysis using modal modification response technique. Mechanical Systems and Signal Processing, vol. 31, p. 6793, DOI:10.1016/j.ymssp.2012.02.011.
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 298306 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1830 Special Issue, Original Scientific Paper
Received for review: 20131019 Received revised form: 20140214 Accepted for publication: 20140401
Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
Almeida, F.C.L. – Brennan, M.J. – Joseph, P.F. – Dray, S. – Whitfield, S. – Tabone, A.P. Fabrício César Lobato de Almeida1,* – Michael John Brennan1 – Phillip Frederick Joseph2 – Simon Dray3 – Stuart Whitfield4 – Amarildo Tabone Paschoalini1 1 University
2 University
Estadual Paulista, Department of Mechanical Engineering, Brazil of Southampton, Institute of Sound and Vibration Research, United Kingdom 3 Hydrosave, United Kingdom 4 South Staffs Water, United Kingdom
Leaks in pipes are a common issue encountered in the water industry. Acoustic methods are generally successful in finding and locating leaks in metallic pipes, however, they are less effective when applied to plastic pipes. This is because leaknoise signals are heavily attenuated due to high damping in the pipewall and sound radiation into the soil. As result, high frequency leak noise does not travel long distances. To determine how far leak noise may travel in a pipe at any frequency, the attenuation of the wave responsible for leak noise propagation should be known. In this paper a new method to estimate this is described. The method is then applied to some measurements made on a bespoke pipetest rig in the UK, and the results are compared with theoretical predictions. Keywords: leak detection, wave attenuation, water industry, plastic pipes
0 INTRODUCTION Water distributions systems are susceptible to leakage, which results in a substantial wastage of water. The social and environmental effects due to leakage problems are also a matter of concern. Recently, a survey about the costs of installation/repair work of buried infrastructure in the UK has estimated that street works cost about £7bn in losses for the government annually; Social costs account for about £5.5bn and damage costs are about £1.5bn [1]. Water loss in distribution systems can typically reach between 20 and 30% of the total water production [2], but in some extreme cases this figure can rise up to 50% [3]. Leakage is the main issue responsible for such loss. Energy to supply water is one of major costs in developing countries, and it may easily consume 50% of a municipality’s budget in the developing world [4]. Hence, reducing wastage of water through leaks directy affects the cost of water distribution. Furthermore, it is estimated that between 2 and 3% of the world’s energy consumption is used to pump and treat water for urban and industrial purposes [4]. Approximately 30 to 50% of water is lost globally due to water leakage. The noise from a leak in a buried plastic pipe generally has low frequency content, occurring well below the pipe ring frequency [5]. In this frequency region, only four types of waves are, in general, responsible for most of the energy transfer in a pipe [6] and [7]. However, in buried plastic water pipes leak noise propagates in an axissymmetric wave that is predominantly a fluid wave, but is strongly coupled 298
to the pipewall, such that there is significant radial motion [5] and [7]. This strong coupling means that leak noise energy is dissipated within the pipewall and also radiates as sound into the ground [8] and [9]. The result is that the pipe acts as a lowpass filter [10], and hence measured leak noise tends to be below about 200 Hz. There are different ways of estimating the wave attenuation in pipes. Muggleton et al. [5], [8] and [11] have derived analytical expressions to predict the wavenumber in buried and submerged pipes for leak detection problems. The wavenumber is a frequency dependent complex quantity, the real part of which is related to the wavespeed, and imaginary part is related to the attenuation factor, which determines how far a wave can propagate along the pipe at any particular frequency. Hence, if the wavenumber can be predicted then the wave attenuation factor can be estimated. Measurements have also been carried out to determine the wavenumber in buried pipes [9], [12] and [13]. The aim of this paper is to investigate a new technique to estimate the wave attenuation factor insitu, in buried pipes. This technique is based on the envelope of the crosscorrelation function between two vibration signals measured on the pipe, either side of the leak. An advantage of the method is that it can also be used in the presence of a leak, which could potentially affect the estimate of wave attenuation. Following a description of the method in the next section, and some simulations in Section 2, some experiments on a bespoke test rig are reported in Section 3 to validate the proposed method.
*Corr. Author’s Address: UNESP, Av. Brasil centro, 56, Ilha Solteira, Brazil, fabricio_cesar@dem.feis.unesp.br
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1 THEORETICAL DEVELOPMENT Fig. 1 shows a schematic of a pipe with a leak between measurement Positions 1 and 2, together with actuators and sensors attached to each measurement position. The measurement positions are at convenient access points.
in which β and c are the attenuation factor and the phase velocity of the propagating wave, respectively. The frequency response function between the two accelerometer signals is given by H(ω, d1+d2). In Eq. (2) –ωβ is the imaginary part of the wavenumber Im{k}, which has units of 1/m, and the attenuation factor in the pipe is given by 20 Im{k}/ln{10}. If the actuator at Position 2 is driven at the same time as there is a leak, the crossspectral density function S21(ω) between the measurement Positions 2 and 1, is given by: ( ) S 21 (ω ) = ω 4 H * (ω , d1 ) H (ω , d 2 ) Sll (ω ) + 2
+H (ω , d1 + d 2 ) See (ω ),
Fig. 1. Schematic of a buried pipe showing the sensors and actuators mounted at measurement Positions 1 and 2 with the leak located between these positions
Two measurements are carried out. First, the actuator at Position 2 is excited, generating a wave which propages along the pipe to Position 1. The vibration of the pipe is measured using two sensors, one mounted next to the actuator at Position 2 and the other mounted at Position 1. If there is coherence between the two measured signals, the crosscorrelation function can be calculated [14], and give information about the wavespeed and attenuation factor. The actuator at Position 1 is then excited, and the measurement procedure is repeated. The wave then propagates in the opposite direction. The procedure to calculate the wave attenuation in dB/m for the section of the pipe involves some manipulation of the crosscorrelation functions, including the determination of their envelopes using the Hilbert transform [15]. Moreover, an estimation of the maximum distance that the leak noise would travel at each frequency can be calculated. In the presence of a leak two types of excitation act simultaneously on the pipe. One is the controlled excitation from an actuator, and the other is from the leak. The frequency response function (FRF) between the pressure at the leak position and the acceleration on the pipe at a distance x from the leak is given by [10]: H a (ω , x ) = −ω 2 H (ω , x), (1) where
H (ω , x ) = e −ωβ x e −iω x c , (2)
(3)
where Sll(ω) and See(ω) are the autospectral density functions of the leak and the acceleration at the actuator position, respectively, and the superscript (2) indicates that the actuator at Position 2 is being excited. ( 2) (τ ) , The corresponding crosscorrelation function R21 can be determined from the inverse Fourier transform ( 2) ( 2) (τ ) = F −1 S21 (τ ) , which F–1 of Eq. (3) to give R21 can be written as:
{
}
( 2) R21 (τ ) = Rll (τ ) ⊗ h(τ ) ⊗ δ (τ − T0 ) +
+Ree (τ ) ⊗ψ (τ ) ⊗ δ (τ − Tact ), (4)
where ⊗ denotes convolution, Rll (τ) = F–1 {Sll(ω)} is the autocorrelation of the leak signal, Ree (τ) = F–1 {See(ω)} is the autocorrelation function of the acceleration signal at the actuator excitation position, ψ (τ ) = F −1 e−ωβ ( d1+ d2 ) , h(τ ) = F −1 ω 4e −ωβ ( d1+ d2 ) , T0 = (d1 – d2) / c is the time delay due to the leak, Tact = (d1 + d2) / c is the time delay due to the actuator excitation and δ( ) is the Dirac delta function. Eq. (4) shows that the delta function δ(τ – T0) is smeared by the leak spectrum Sll(ω) and the behaviour of ω 4e −ωβ ( d1+ d2 ) , and the delta function δ(τ – Tact) is smeared by the external excitation spectrum See(ω) and the frequency characteristics of Ψ (ω ) = e −ωβ ( d1+ d2 ) . For simplicity, in the model the spectral characteristics of the leak and the excitation are considered to be white noise. The autospectral density function of the leak is thus assumed to be given by Sll(ω) = S0 and the external excitation by See(ω) = nS0, where n is the ratio of the spectral density of the leak noise to the spectral density of the external excitation. If the signals from the sensors are passed through ideal bandpass filters with frequency response G(ω), in which G(ω) = 1 for ωlow ≤ ω ≤ ωupp and zero otherwise, the crosscorrelation function becomes:
{
Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
}
{
}
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( 2) R21 (τ ) = S0 [h(τ ) ⊗ g (τ ) ⊗ δ (τ − T0 ) +
+nψ (τ ) ⊗ g(τ ) ⊗ δ (τ Tact )], (5)
where g(τ) is given by g(τ) = F–1 {G(ω)}, which can be written as:
∆ω sin(∆ωτ 2) g(τ ) = cos(ωcenτ ), (6) π ∆ωτ 2
in which Δω = ωupp – ωlow is the bandwidth of the filter and ωcen = (ωlow + ωupp) / 2 is the central frequency. The bandpass filter introduces a ripple with frequency ωcen into the crosscorrelation function. Eq. (5) shows that the delta function is further smeared by the introduction of the bandpass filter. According to Eq. (5) there are two possible peaks present in the crosscorrelation function; one is due to the leak and another is due to the external excitation. In this particular case, where the external excitation source is at Position 2, the peak in the cross correlation function related to the external excitation occurs at τ = (d1+d2) / c. However, when the external excitation is at Position 1, the peak ( 2) occurs in R21 (τ ) at τ = –(d1+d2) / c. The peak in the crosscorrelation function due to the leak occurs at τ = (d1–d2) / c irrespective of the position of the external source. If the bandpass filter is broad enough [10] such that Ψ(ω) << 1, the envelope of the cross correlation (1) ( 2) function R21 (τ ) normalised by its peak (τ ) or R21 value when τ = Tact, is given by [10]: ( 2) Env ( R21 (τ ) ) = Env ( R12(1) (τ ) ) =
βd 2
( β d ) + (τ − Tact ) 2
. (7)
The product of the normalised envelopes can be shifted so that its peak occurs at 0 rather than Tact, i.e., ( 2) Env ( R21 (τ + Tact ) ) × Env ( R12(1) (τ + Tact ) ) =
1
1 + (τ β d )
2
. (8)
It can be seen that the envelope given by Eq. (8) is a function of the attenuation factor, which is the unknown quantity to be determined, d which is known and τ which can be obtained from the measurements. Two ways of estimating the attenuation factor are proposed here. The first is based on setting Eq. (8) to be equal to 1/2, which is a measurable quantity. In this case the attenuation factor is given by:
300
β=
τ1 2 d
, (9)
where τ1/2 is the absolute value of τ when Eq. (8) is equal to 1/2. The second method is based on the area A of the envelope given in Eq. (8). If it is assumed that the envelope is symmetric then:
τ1
A = 2∫ 0
1 1 + (τ β d )
2
dτ , (10)
where τ1 is the upper limit of the integral. Evaluating the integral in Eq. (10) results in: τ1
β=
A . (11) 2d arctan (τ β d ) 0
Note that if τ1 → ∞ then arctan (τ/βd) → π / 2, so that β = A / dπ. In practice, the area A is calculated numerically from measured results and in Eq. (11) τ1 is set to a finite value to avoid noise in the data. 2 SIMULATIONS To illustrate the methodology described in the previous section, some simulations are carried out prior to presenting some experimental results in the following section. The system in Fig. 1 is considered with the parameters given in Table 1, which corresponds to one of the measurements made in the Blithfield test rig, which is described in Section 3. Table 1. Parameters used to illustrate the method to determine the wave attenuation factor
d1 d2 β c flow = ωlow / 2π fupp = ωupp / 2π n
30 m 20 m 2.9×10–4 s/m 390 m/s 20 Hz 35 Hz, 200 Hz 2×1010
Note that it is assumed arbitrarily that the amplitude of the peak due to the leak is less than half of the amplitude of the peak due to the actuator. This assumption is based on the experience that the actuator excitation has always dominated the signal at one of the sensors in the experiments carried out by the authors. Two cases are considered. In the first case, a large bandwidth is set with lower and upper limits set to 20 and 200 Hz, respectively. This represents a case where the vibration due to the actuators is effectively transmitted along the pipe, leading to good coherence over a wide frequency range. Second, a much smaller
Almeida, F.C.L. – Brennan, M.J. – Joseph, P.F. – Dray, S. – Whitfield, S. – Tabone, A.P.
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bandwidth is set with lower and upper limits set to 20 and 35 Hz, respectively. This represents a case when the distance and or damping in the pipe is large enough to significantly attenuate high frequency vibration at the sensor positions. (1) ( 2) Figs. 2a(i) and (ii) show, R21 (τ ) (τ ) and R21 normalized by their maximum values together with their envelopes calculated using the Hilbert transform, for the two different bandwidths. The largest peaks in each of the crosscorrelation functions correspond to the time delays due to wave propagation from Positions 2 to 1, and from Positions 1 to 2. Note that one of these occurs when τ is negative, and one occurs when τ is positive. The small peak in Fig. 2a(i) is due to the leak. The effect of a much reduced bandwidth on the appearance of the peak due to the leak can be seen in Fig. 2a(ii). The peak cannot be seen as it is masked by sidelobes of the peaks from the actuator excitation. This effect can be understood using Eqs. (5) and (6). When the filter bandwidth decreases, the sidelobes around the main peaks in the envelopes increase, and move away from these peaks as shown in Fig. 2a(ii).
A rule of thumb as to whether or not the peak due to the leak is masked, can be determined using an approximation for the envelopes, which is given by the modulus of the sinc function in Eq. (6). It is found that the peak due to the leak will be masked and not be present in the envelopes if T0 < Tact – 6π / Δω. To simplify the presentation of the data from the two measurements discussed above, they are combined as follows. First, R12(1) (τ ) is calculated (1) instead of R21 (τ ) , which has a peak for positive τ instead of negative τ. If the wave speed estimate is the same in each direction then this coincides with ( 2) the peak for R21 (τ ) . The product of the envelopes of these crosscorrelation functions is then calculated. This has a large peak corresponding to the time it takes for a wave to propagate from the actuator to the sensor, and the peak corresponding to the leak is much diminished. This can be seen in Figs. 2b(i) and (ii), which are normalised plots of the product ( 2) of the envelopes of R12(1) (τ ) and R21 (τ ) , for two different bandwidths. It can be seen that the effect of multiplying the two envelopes together is to accentuate the peak due to the actuators and diminish
Peak due to the leak a(i)
b(i)
( 2) Env( R21 (τ (τ )+ Tact)) × Env( R12(1)(τ(τ+) Tact))
R21
(1) R21 (τ )
( 2) R21 (τ )
a(ii)
b(ii) ( 2) (τ Env( R21 (τ )+ Tact)) × Env( R12(1)(τ(τ+) Tact))
R21
τ [s]
τ + Tact
Fig. 2. Simulations illustrating the technique to isolate the effects of the presence of a leak; the labels ‘i’ and ‘ii’ correspond to when the lower and upper limits of the bandpass filter are set at 20 to 200 Hz and 20 to 35 Hz, respectively; a(i) and a(ii), the envelope of the crosscorrelation function R21 given by Eq. (5); b(i) and b(ii), the product of the envelopes of R12(1)(τ (τ+) Tact) and R21( 2) (τ(τ )+ Tact) Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
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the peak due to the leak (although there are now two small peaks for a large bandwidth, as seen in Fig. 2b(i)). When the bandwidth is small, as in Fig 2b(ii) only a single peak due to the excitation by the actuators is evident, together with some small sidelobes. Further processing of the data to determine the wave attenuation from the product of the envelopes is discussed in the next section. 3 EXPERIMENTAL WORK The pipe rig used in this work is located at the Blithfield reservoir in Staffordshire, UK. Fig. 3 shows a schematic sideview highlighting the access points. The pipe is buried in an open field close to the South Staffs Water reservoir facility. The pipe is made from high performance polyethylene (HPPE) with Young’s modulus of 1×109 N/m2. The pipe rig is 120 m long, consisting of section lengths of 10, 20 and 30 metres, which have an outer radius of 80 mm and wall
thickness of 9.85 mm. There are 6 access points as shown in Fig. 3. The pipe extremity close to Position 1 is connected to the mains water distribution pipe, which supplies water at a pressure of about 6 bar. At the other extremity the pipe is terminated with a blank. All the access points are set in concrete to provide a rigid support for the pipe connections, while the pipe sections are buried in the ground at a depth of about 0.8 m. Fig. 4a shows one of the access points. Leak noise could be generated by opening a secondary valve fitted to a blanking piece attached to the hydrant as shown in the sketches in Figs. 4b and c. A pressure gauge was also attached to the hydrant as shown in Fig. 4c The method described Section 1 was implemented using experimental data from the Blithfield test rig. As mentioned previously, sensors and actuators were mounted at Positions 1 and 2 and the leak was induced between these positions. Details of the transducers and instrumentation are given in Tab. 2. Fig. 5 shows
Fig. 3. Schematic of the pipe rig used in the experimental work, showing the distances between the access points, and the excitation/ measurement positions and the leak position
Fig. 4. Some components of the pipe testrig; a) the pipe access point and its main valve: (1) Main valve; (2) connection point; b) sketch of the device used for generating the leak conditions: (1) plastic pipe; (2) main valve; (3) metal hydrant; (4) secondary valve; (c) pressure gauge and secondary valve attached to the hydrant: (1) secondary valve; (2) pressure gauge; (3) hydrant
Fig. 5. Photograph of the test rig at Blithfield used to validate the methods to estimate the attenuation factor of a pipe in the presence of a leak; a) instrumentation at Positions 1 and 2; b) the leak induced in the pipe
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some photographs of the experimental setup. Fig. 5a shows the instrumentation used at Positions 1 and 2, which are 50 metres apart, and Fig. 5b shows the leak. The main valve shown in Fig. 5a allowed the water contained in the buried pipe to enter the hydrant. The leak was 30 metres away from Position 1. The signals from the sensors were captured and digitised using a data acquisition system (DATS) made by PROSIG. This system is a 16 bit analog to digital converter (ADC) device type P5650, which has 8 synchronised input channels, and a sample rate of up to 100 kHz per channel. The actuator was driven using a chirp signal between 20 and 200 Hz, generated by a Hameg signal generator, reapeating every 3 seconds. The sampling frequency was 5 kHz, and a frequency resolution of 1 Hz was used in the subsequent spectral analysis. Table 2. Measurement devices used in collection of the data Device Accelerometers
Manufacturer
Type
Bruel and Kjaer
4383 and 4384
Shaker (actuator)
LDS
V201
Charge Amplifiers
Bruel and Kjaer
2635
Function Generator
Hameg
HM8130
Acquisition System
Prosig
DATS
Fig. 6 shows the powerspectral density (PSD) of accelerometermeasured signals at Positions 1 and 2 and the coherence between these signals. Data are shown for simultaneous excitation of the actuator, and with leak excitation alone. The shaker was set at its highest level or turned off with the secondary valve fully open. It can be seen in Figs. 6a and b that the leak marginally dominates the signals measured at Position 1, but the actuator strongly dominates the signal at Position 2. It can also be seen that when the actuator is turned on, the coherence between the signals at Positions 1 and 2 changes significantly due to the interaction of the different sources of vibration. Fig. 7a shows the crosscorrelation coefficient between signals at Positions 1 and 2 for three cases; (a) when the pipe is excited at Position 2 by an actuator, (b) when the leak only excites the pipe, and (c) when both the leak and actuator act simultaneously to excite the pipe. Comparing Figs. 7a and c it can be observed that the peak in the crosscorrelation coefficient reduces when the leak is turned on. This is because the leak marginally dominates the signal at Position 1 and the actuator strongly dominates the signal at Position 2 as shown in Fig. 6. Despite this, the largest peak in the crosscorrelation coefficient corresponds to the
Fig. 6. Measurements at Positions 1 and 2 with the system being excited at Position 2; the leak was induced between Positions 1 and 2; a) PSD at Position 1; b) PSD at Position 2; c) the coherence between Positions 1 and 2
Fig. 7. The crosscorrelation coefficient calculated from signals measured at Positions 1 and 2, when: a) the actuator is excited at Position 2 without any leak in the pipe; b) there is only a leak; and c) the actuator and leak acted simultaneously Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
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time delay due to the wave generated by the actuator as shown in Fig. 7c. Following the calculation of the crosscorrelation functions, their envelopes were determined as discussed in Section 1. Figures 8a and b show the results of the two techniques to estimate the attenuation factor discussed in the previous section. The experimental data is shown as a solid line and the envelopes given by Eq. (8) when the total loss is estimated by the method using Eq. (9) (red dashed line), and the other method using Eq. (11) (blue dasheddotted line). The theoretical model only predicts the experimental result at time delays close to the peak in the product of the envelopes. Thus only experimental data close to the peak is taken into account in the calculation of the attenuation factor. This is determined by setting a threshold of 0.01 for the product of the envelopes and only data greater than this value around the peak is chosen. Fig. 8a shows the estimation of the attenuation factor (envelopes) with absence of a leak. Fig. 8b shows the estimation of the attenuation factor for a case with a leak present in the pipe.
With no leak, the attenuation factor was found to be 2.8×10–4 s/m and 1.9×10–4 s/m using Eqs. (9) and (11), respectively. When the leak was induced the attenuation factor was found to be 2.5×10–4 s/m and 1.9×10–4 s/m using Eqs. (9) and (11), respectively. The attenuation factors used to estimate the envelopes in Fig. 8 are sumarized in Table 3. The difference between the attenuation factor can be observed in Figs. 8a and b as the difference in shape between the envelopes. Thus the results show that the leak only has a marginal effect on the estimation of the attenuation factor. Table 3. Attenuation factor calculated using Eq. (9) and Eq (11) for cases with and without a leak in the pipe Attenuation factor calculated using Eq. (9)
Attenuation factor calculated using Eq. (11)
No leak
2.8×10–4 s/m
1.9×10–4 s/m
Leak
2.5×10–4 s/m
1.9×10–4 s/m
The attenuation in the pipe at each frequency in dB/m is given by [10]:
a)
Cases
Attenuation = 8.67 βω.
This can be determined from a measurement by [10]:
Attenuation =
−20 ln H (ω , d1 + d 2 )
0.01
( 2) Env( R21 (τ (τ )+ Tact)) × Env( R12(1)(τ(τ+) Tact))
b)
0.01
τ + Tact Theoretical results calculated using Eq. (9) Theoretical results calculated using Eq. (11) Experimental data
Fig. 8. The normalised product of the envelopes: a) no leak; b) with leak
304
(12)
( d1 + d 2 ) ln (10 )
, (13)
Thus, if the frequency response function is calculated from the two accelerometer measurements, then Eq. (13) can be used to verify the attenuation factor determined from the methodology described in Section 1. Eq. (13) is plotted in Figs 9a(i) and b(i) using experimental data from the Blithfield test rig, when the pipe was excited at Positions 1 and 2 respectively when there was no leak. Figs. 9a(ii) and b(ii) show the corresponding results when a leak was induced. Also plotted is the attenuation calculated using the two methods discussed above, which are given by Eq. (9) and Eq. (11), using data filtered within the frequency regions shown on the graphs. The frequency bandwidth over which the analysis was conducted, was calculated using the method described in [13]. It can be seen that the area method gives a better attenuation estimate when the excitation is set at Position 1, but the half amplitude method gives a slight better attenuation estimate when the actuator is set at Position 2. Although both methods give different
Almeida, F.C.L. – Brennan, M.J. – Joseph, P.F. – Dray, S. – Whitfield, S. – Tabone, A.P.
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Fig. 9. Wave attenuation in dB/m; a(i) Actuator set at Position 1 with no leak; b(i) Actuator at Position 2 with no leak; a(ii) Actuator at Position 1 with a leak present; b(ii) Actuator set at Position 2 with a leak present
attenuation estimates, in general they give a reasonable attenuation estimate for the cases presented. It can also be seen that the presence of a leak does not interfere dramatically with the estimation of the attenuation factor.
of a leak. Experimental results from a bespoke test rig have also been presented to demonstrate the efficacy of the method.
4 CONCLUSIONS
[1] Royal, A.C.D., Atkins, P.R., Brennan, M.J., Chapman, D.N., Chen, H., Cohn, A.G., Foo, K.Y., Goddard, K.F., Hayes, R., Hao, T., Lewin, P.L., Metje, N., Muggleton, J.M., Naji, A., Orlando, G., Pennock, S.R., Redfern, M.A., Saul, A.J., Swingler, S.G., Wang, P., Rogers, C.D.F. (2011). Site assessment of multiplesensor approaches for buried utility detection. International Journal of Geophysics, Article ID 496123, DOI:10.1155/2011/496123. [2] Cheong, L.C. (1991). Unaccountedfor Water and the Economics of Leak Detection. Water Supply, vol. 9, p. IR 11 to 16.
In this paper, a method to automatically estimate the wave attenuation factor in buried water pipes, has been proposed. This method involves two measurements made by attaching an actuator to two different positions with colocated accelerometers. The technique is based on the envelope of the crosscorrelation function of the two measurements. An analytical model has been used to describe and to investigate some of the features and limitations of the method. The technique can also be used in the presence
5 REFERENCES
Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
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[3] AWWA (1987). Leaks in Water Distribution Systems – A Technical/Economic Overview, American Water Works Association, Denver. [4] Watergy: Taking Advantage of Untapped Energy and Water Efficiency Opportunities in Municipal Water Systems; from: http://pdf.usaid.gov/pdf_docs/ PNACT993.pdf, accessed on 13/02/2014. [5] Muggleton, J.M., Brennan, M.J., Pinnington, R.J. (2002). Wavenumber prediction of waves in buried pipes for water leak detection. Journal of Sound and Vibration, vol. 249, no. 5, p. 939954, DOI:10.1006/ jsvi.2001.3881. [6] Fuller, C.R., Fahy, F.J. (1982). Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. Journal of Sound and Vibration, vol. 81, no. 4, p. 501518, DOI:10.1016/0022460X(82)902930. [7] Pinnington, R.J., Briscoe, A.R. (1994). Externally applied sensor for axisymmetric waves in a fluid filled pipe. Journal of Sound and Vibration, vol. 173, no. 4, p. 503516, DOI:10.1006/jsvi.1994.1243. [8] Muggleton, J.M., Yan, J. (2013). Wavenumber prediction and measurement of axisymmetric waves in buried fluidfilled pipes: Inclusion of shear coupling at a lubricated pipe/soil interface. Journal of Sound and Vibration, vol. 332, no. 5, p. 12161230, DOI:10.1016/j. jsv.2012.10.024. [9] Muggleton, J.M., Brennan, M.J., Linford, P.W. (2004). Axisymmetric wave propagation in fluidfilled pipes:
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wavenumber measurements in vacuo and buried pipes. Journal of Sound and Vibration, vol. 270, no. 12, p. 171190, DOI:10.1016/S0022460X(03)004899. [10] Gao, Y., Brennan, M.J., Joseph, P.F., Muggleton, J.M., Hunaidi, H. (2004). A model of the correlation function of leak noise in buried plastic pipes. Journal of Sound and Vibration, vol. 227, no. 12, p. 133148, DOI:10.1016/j.jsv.2003.08.045. [11] Muggleton, J.M, Brennan, M.J. (2004). Leak noise propagation and attenuation in submerged plasticwater pipes. Journal of Sound and Vibration, vol. 278, no. 3, p. 527537, DOI:10.1016/j.jsv.2003.10.052. [12] Muggleton J.M., Gao, Y., Brennan, M.J., Pinnington, R.J. (2006). A novel sensor for measuring the acoustic pressure in buried plastic water pipes. Journal of Sound and Vibration, vol. 295, no. 35, p. 10851098, DOI:10.1016/j.jsv.2006.01.032. [13] Almeida, F.C.L. (2013). Improved Acoustic Methods for Leak Detection in Buried Plastic Water Distribution Pipes. PhD thesis, University of Southampton, Southampton. [14] Bendat, J.S., Piersol, A.G. (2000). Random Data: Analysis and Measurement Procedures. Third edition, John Wiley & Sons, Hoboken. [15] Feldman, M. (2011). Hilbert Transform Applications in Mechanical Vibration. John Wiley & Sons, Hoboken, DOI:10.1002/9781119991656.
Almeida, F.C.L. – Brennan, M.J. – Joseph, P.F. – Dray, S. – Whitfield, S. – Tabone, A.P.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 307313 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1834 Special Issue, Original Scientific Paper
Received for review: 20131016 Received revised form: 20140117 Accepted for publication: 20140401
PGDVTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems Barbarulo, A. – Riou, H. – Kovalevsky. L. – Ladeveze, P. Andrea Barbarulo1,* – Hervé Riou1 – Louis Kovalevsky2 – Pierre Ladeveze1 2 University
1 LMTCachan, ENS Cachan, France of Cambridge, Engineering department, United Kingdom
The calculation of vibrational responses of complex systems on frequency bands appears to be more and more important in engineering simulation. This is particularly true in the medium frequency regime where the solution is very sensitive to the frequency. In this work, we propose a new path to determine the frequency response of a system at many frequencies. It is based on the variational theory of complex rays (VTCR), a mid frequency dedicated numerical strategy, and the proper generalized decomposition (PGD), a model order reduction technique. The VTCR enables one to model the problem thanks to the use of waves, and the PGD expands the VTCR approximation over the frequency band through a separated variable representation. This strategy is illustrated on a 2D acoustic car cavity example. Keywords: mediumfrequency, acoustics, variational theory of complex rays, proper generalized decomposition, model order reduction
0 INTRODUCTION The prediction of the frequency responses of complex systems in frequency bands is required in many industrial applications, like car or aerospace acoustics. Since the complex matrix of the finite element is wavenumber dependent, the computation of the vibrational solution often involves the resolution of the problem at each frequency of the frequency band, then leading to a prohibitive computational cost. This is particularly true in the midfrequency regime, where the solutions are sensitive to frequency, requiring a very refined frequency discretization. The definition of advanced numerical strategies for predicting the acoustic response of complex systems in the midfrequency ranges is the subject of this work. It uses the combination of the variational theory of complex rays (VTCR) [1] and the proper generalized decomposition (PGD) [2]. The VTCR has been introduced in [3] and is dedicated to the resolution of midfrequency problems. It is a Trefftz method which uses oscillating waves to expand the field variables. It is based on an original variational formulation designed such that the approximations within the substructures are totally independent, which means that any kind of approximation can be used, even those which have a strong physical content, giving to the strategy a strong flexibility, and hence efficiency. It has already been developed for plates, [4] shells [5] acoustics in 2D [6] or 3D [1], vibration problems, and also for transient applications [7]. It distinguishes itself from the other Trefftz techniques [8] to [13] by the type of selected
shape functions and the treatment of the boundary conditions. The PGD is a model order reduction technique. Introduced in [2], it has already been successfully utilized for the resolution of multiparametric problems (problems which depend on many parameters such as the space and time problems, or the space and uncertain problems, etc.) [14] to [16]. The resolution of the vibration problem (with the VTCR) at many frequencies is such a multiparametric problem. Therefore, a combination of PGD and the VTCR is an obvious choice to handle frequency problems in medium frequency bands. The VTCR has already been extended to frequency band applications [17] and [18]. In these works, the authors proposed new algorithms for the calculation of multiple frequency solution, either by using a set of parameters to derive a discrete approximation of the frequencydependent quantities within the VTCR matrix, or by expanding the VTCR matrix into series with respect to the frequency. In this work, we propose a new path to determine the frequency response of a system at many frequencies. It is based on a combination of the VTCR, used to find the solution of the vibration problem at a fixed frequency, and the PGD, used to find the best separated variable representation of this solution over a frequency band. Power type algorithms are proposed to find it in an efficient way. A 2D numerical illustration on a car cavity is proposed to see the benefits of such an approach.
*Corr. Author’s Address: LMTCachan ENS Cachan/CNRS/Paris 6 University, PRES UniverSud Paris 61 avenue du President Wilson, F94230 Cachan, France, barbarulo@lmt.enscachan.fr
307
∑∫ ( pe − pde ) Lv (δ pe )dS + e
+ ∑ ∫δ pe ( Lv ( pe ) − vde )dS +
e Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 307313
+ ∑ ∫ ( pe − ZLv ( pe ) − hd ) Lv (δ pe )dS + e
1 THE REFERENCE ACOUSTICAL PROBLEM TO SOLVE Consider a fluid comprised in a bounded domain Ω. This fluid is characterized by its speed of sound c0 and its density ρ0. We study the steadystate vibration response of the fluid in the frequency band I = [ω0 – Δω/2; ω0 + Δω/2] where ω0 is the central frequency and Δω the frequency bandwidth. The reference problem to solve is: find the pressure p(x, ω), (x, ω) ∈ Ω×I, such that: ∆p + k 2 p = 0 over Ω × I p = pd over ∂ p Ω × I
Lv ( p ) = vd over ∂ v Ω × I p − ZLv ( p ) = hd over ∂ Z Ω × I
. (1)
+
pe − pe' = 0 along Γ e,e' . (2) Lv ( pe ) + Lv ( pe' ) = 0 along Γ e,e'
{
}
Se = pe ; ∆pe + k 2 pe = 0 over Ωe × I . (3)
Problem (Eqs. (1) and (2)) can then be formulated as: find p1 , p2, ..., pnel ∈ S1 × S2 × ... × Snel such that:
(
)
∑∫ ( pe − pde ) Lv (δ pe )dS + e
+ ∑ ∫δ pe ( Lv ( pe ) − vde )dS + e
+ ∑ ∫ ( pe − ZLv ( pe ) − hd ) Lv (δ pe )dS +
(
3 THE APPROXIMATED SOLUTION OF THE REFERENCE PROBLEM The only thing to do in order to get an approximated solution of the reference problem Eqs. (1) and (2) is to satisfy Eq. (5) in a subspace S h ⊂ S . As one can see, S is defined such that the approximations in Se can be independant of one another. As a consequence, any kind of approximated solution can be selected to span Sh, as soon as it satisfies the governing equation (first equation of Eq. (1)) in Ωe. The VTCR approximation uses propagating waves inside the acoutics domains, and consider all of them. For instance, in the 2D modeling, we have: p ( x, ω ) ∈ Se ⇔ p ( x, ω ) = ∫ X e (θ , ω ) eike x dθ , (6) where ke is the wavenumber of the vibration problem in Ωe. Corresponding 3D modeling can be found in [1]. Xe(θ,ω) corresponds to the amplitudes of the waves, and is the unknown of the problem. Different approximations of Xe(θ,ω) can be used (see [19]). If the Fourier approximation is used, we expand Xe(θ,ω) on a Fourier series in order to define the subspace Sh:
X e (θ , ω ) X eh (θ , ω ) = ∑X en (ω ) einθ . (7) As a consequence, the subspace Seh is spanned by
the shape functions φen ( x, ω ) = ∫ einθ eike x dθ and, for a
fixed frequency, X eh (ω ) are the VTCR unknowns
of the problem Eqs. (1) and (2) and are related to the amplitudes of the waves which propagates inside Ωe.
e
1 ∑ (δ pe + δ pe' ) ( Lv ( pe ) + Lv ( pe' ) )dS = 2 e,e' >e ∫
(4)
a ( p, δ p ) = l (δ p ) ∀ δ p ∈ S , (5)
308 1 Barbarulo, A. – Riou, H. – Kovalevsky. L. – Ladeveze, P. + ∑ ∫ ( pe − pe' ) ( Lv (δ pe ) − Lv (δ pe' ) )dS + 2 e,e' >e +
)
= 0 ∀ δ p1 ,..., δ pnel ∈ S1 × ... × Snel ,
where a and l are the bilinear and the linear part of Eq. (4).
2 THE VTCR FORMULATION OF THE REFERENCE PROBLEM The VTCR forumulation of an acoustic problem can be found in [6]. It necessitates the definition of the space of functions which exactly satisfy the governing equation (first equation of Eq. (1)) over the subdomains Ωe:
1 ∑ (δ pe + δ pe' ) ( Lv ( pe ) + Lv ( pe' ) )dS = 2 e,e' >e ∫
where pde, vde and hde are the boundary conditions defined in Eq. (1) but restricted to ∂Ωe. The overline designates the complex value quantity. The equivalence between Eqs. (1) and (4) car be found in references on the VTCR. Eq. (4) can be written: find p ∈ S such that:
In Eq. (1), k = (1 – iη) ω / c0 is the wave number (η is the absorption coefficient), pd is a prescribed pressure, Z is a prescribed velocity, hd a given impedance and a given function. The operator Lv is such that Lv(p) = (i/ρ0ω) (∂p/∂n). ∂pΩ, ∂vΩ and ∂zΩ are the parts of the boundary of Ω where the pressure, the velocity and the Robin conditions are prescribed. The uniqueness of the solution is ensured by a strictly positive η. If Ω is paritioned in nel nonoverlapping elements Ωe, one must also consider the additional continuity equation at the common boundary Γe,e' = Ωe ∩ Ωe' :
1 + ∑ ∫ ( pe − pe' ) ( Lv (δ pe ) − Lv (δ pe' ) )dS + 2 e,e' >e
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 307313
The substitution of Eqs. (6) and (7) into Eq. (5) leads to the frequency band matrix system : K(ω) X(ω) = F(ω),
(8)
where K(ω) and F(ω) are the projection of the bilinear and the linear forms of Eq. (5) onto the space generated by the functions φen . In the following, N will designate the size of X(ω). 4 THE COMBINATION OF THE PGD AND THE VTCR TO SOLVE THE FREQUENCY BAND PROBLEM The problem given by Eq. (8) defined on the frequency band I is a multiparametric problem: the solution X(ω) contains information related to the direction of propagation of the waves and to the frequency. Solving this problem has already be considered in previous works on the VTCR either by using a set of parameters to derive a discrete approximation of the frequencydependent quantities within the VTCR matrix or by expanding the VTCR matrix and the righthand side of the system into series with respect to the frequency [17] and [18]. Here, we propose a new path to solve this, based on a model order reduction technique through a separated variable representation of the physical data. Today, the common name, which designates a decomposition of the physical data in a separated representation of the variables, is the proper generalized decomposition (PGD) (see [20] for a general review on PGD). With such a decomposition, any physical variable can be decomposed into a separated variable decomposition: u ( x1 , ..., xR ) u M ( x1 , ..., xR ) = ∑u1m ( x1 ) × ... × umR ( xR ) ,
M being the order of the approximation. As a consequence, the strategy used to solve Eq. (8) on I is to search the solution X(ω) in the form: X (ω ) X M (ω ) = ∑X m λm (ω ) , (9)
where Xm are constant vectors in N and λm(ω) frequency functions in T, space of functions whose square integration on I is finite. Of course, neither Xm or λm(ω) are known. Then the key questions are: (i) how can we define the optimal decomposition; (ii) how can we compute it? If the solution X(ω) is known, it suffices to minimize the distance between this solution and the best approximation: M
X − ∑λm X m 2 = m =1
min
X1,..., X M ∈ N λ1,...,λM ∈T
M
X − ∑λm X m 2 , (10) m =1
according to a given norm on N and T. But here, we want to build a decomposition Eq. (8) of the solution X(ω) without knowing this solution a priori. Notice that neither λm nor Xm are uniquely defined, as any other decomposition which multiplies λm by a constant factor and divide Xm by the same factor works also. Therefore, without loss of generality, we can prescribe the normalization of Xm according to the euclidean norm on N . In order to build the best approximation, without knowing X, let us first write the problem in a variational formulation. Solving Eq. (8) on I can be written: find X(ω) ∈ N ⊗ T such that: B ( X (ω ) , Y (ω ) ) = L (Y (ω ) ) ∀Y (ω ) ∈ N ⊗ T , (11)
where, B ( X (ω ) , Y (ω ) ) = ∫Y (ω ) K (ω ) X (ω ) d ω and T
L (Y (ω ) ) = ∫Y (ω ) F (ω ) d ω T
(the superscript T stands for the complex transpose vector definition). Moreover, let us define the inner product <<.,>> on N ⊗ T by: << λ (ω ) X , γ (ω ) Y >>=
∫ ( γ (ω ) Y )
T
H (ω ) ( X λ (ω ) ) d ω , (12)
(ω ) ( H being the constant matrix where H (ω ) = Hh of the mean value of K(ω) over I and h (ω ) the frequency dependent function of the mean value of the diagonal part of K(ω)). This choice provides some optimal convergence properties [21] by preserving a relation to the initial problem to solve, and moreover ensures we have the inner product separation property , where <.,> N 1 AM ( λ X , γ Y ) = N << λ X , λ X >> − is a norm on 2 and <.,>T a norm on T. Finally,
(
)
B X M −1 +the λ Xfunctional , γ Y + L ( γAYM)(X, Y ,λ ,γ) . defined on we−introduce N × N × T × T by:
AM ( λ X , γ Y ) =
(
1 << λ X , λ X >> − 2
)
− B X M −1 + λ X , γ Y + L ( γ Y ) .
(13)
According to the Petrov Galerkin PGD approach introduced in [21], we define the best representation Eq. (9) of the solution of the problem Eq. (8) defined on I by the following minimax problem:
( λM X M , γ M YM ) ∈ arg
max
min
Y ∈ N ,γ ∈T X ∈ N ,λ∈T
AM ( λ X , γ Y ) . (14)
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Problem (Eq. (14)) can be interpreted as a pseudo eigen problem, corresponds to the definition of the best rankone separated representation of X – XM–1 and is a generalization of the proper generalized decomposition (POD) [21]. According to Eq. (14), we can see that the best separated representation makes stationary the functional AM(X, Y ,λ ,γ) with respect to X, Y, λ and γ. These stationary conditions write:
(
)
B X M −1 + λM X M , γ M Y' = L ( γ M Y' ) ∀Y' ∈ N , (15)
(
)
B X M −1 + λM X M , γ 'YM = L ( γ 'YM ) ∀γ ' ∈ T , (16) B ( λM X' , γ M YM ) =<< λM X' , λM X M >> ∀X' ∈ ,(17) N
B ( λ'X M , γ M YM ) =<< λ'X M , λM X M >> ∀λ' ∈ T . (18) Then, according to the combination of the PGD and the VTCR, the best separated variable representation Eq. (9) of the reference problem Eqs. (1) and (2) is simply the solution of Eqs. (15) to (18). 5 POWER TYPE ALGORITHM FOR THE CONSTRUCTION OF THE APPROXIMATION We have seen that the best representation Eq. (9) of problem (Eq. (8)) defined on I must verify Eqs. (15) to (18). As these equations are coupled, a natural approach to find the solution is to build solutions X, Y, λ and γ one after the other, which is the strategy adopted in power iterative algorithms. As a consequence, we can define the Algorithm 1, which looks at the solution of Eqs. (15) to (18): Algorithm 1 for m = 1 to M do Initialize λ(ω) and γ(ω) for q = 1 to Q do q Compute X m( ) and Ym( ) by using Eqs. (15) and (17) q
Normalize X m( ) and Ym( q ) and q
Compute
q λm( )
and
(
q γ m( )
(q)
by using Eqs. (16) and (18) (q)
Set ( X m , λm ) = X m , λm Compute
(q)
, the distance between
( X ( ) , λ ( ) ) and ( X ( q m
q m
if ( q ) < Q then
break end if end for 310
)
q −1) q −1 , λm( ) m
)
Set Xm = Xm–1 + λmXm Compute m , the error between KXm and F if m < M then break end if end for The first loop corresponds to the recursive construction of couples (λmXm). This loop begins with the initialization of λ(ω) and γ(ω). We always prescribe:
( KX
m −1
−F
) ( KX T
m −1
)
− F ,
as an initialization value, in order to begin the algorithm with a relation to the target problem. The second loop is the alternate construction of the pairs (Xm, λm) and (Ym, γm) and is based on a power iterative technique to find the stationary point defined by Eqs. (15) to (18). The normalization of the vectors, discussed before, is done in this loop. We introduce a stopping criteria inside the loop to see if the power iterative technique has converge or not. This criteria is based on the relative norm between the pair
( X ( ) , λ ( ) ) and ( X ( q m
q −1) q −1 , λm( ) m
q m
) , over the frequency
band. If the solution at iterationis (q) is closed to the solution at iteration (q–1), the second loop is stopped, because the power iterative technique has converged toward the stationary point of Eqs. (15) to (18). As soon as the second loop has finished, we actualize the solution (Xm, λm). Finally, we compute the error indicator m =  KXm – F  /  F  to assess the precision of the approximation Eq. (9). In practice, the alternate minimization procedure of the second loop converges very fast. As a consequence, we can then classically limit the number of iterations Q to 8 or 9. 6 NUMERICAL ILLUSTRATION Consider the closed acoustic car cavity defined on Fig. 1. This cavity is filled with a fluid close to the air (ρ0 = 1.25 kg/m3, c0 = 340 m/s and η = 0.0005). Different boundary conditions are prescribed on the boundaries of the cavity: prescribed velocity or prescribed impedance (see Fig. 1). The black zone corresponds to the measure area, where the evaluation of the acoustical energy is desired. It corresponds more or less to the zone where the hear of the driver is located. The cavity is modeled by the VTCR with 8 subcavities Ωe (see Section 1). These cavities can be
Barbarulo, A. – Riou, H. – Kovalevsky. L. – Ladeveze, P.
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seen on Fig. 1 (the continuous black lines make a separation between them). In each Ωe, the VTCR uses 2Ne + 1 shape functions φen (defined in Eq. (7)) such that the convergency criteria in [19] is respected. The considered central frequency is ω0 = 2π × 2150 rad/s and the bandwidth is Δω = 2π × 300 rad/s. The Algorithm 1 described in Section 6 is used to get an approximated solution. The parameter (see Section 5) has been retained. For the comparison, the reference solution is given by the VTCR strategy solely, with a computation at many frequencies.
band). As one can from the eyeball norm, the solutions are very closed each time. Indeed, for each considered case, the localization of the vibrational peaks and their amplitude are the same. This illustrates that the proposed Algorithm 1 is able to recover the reference solution over the whole frequency band. This is of course in agreement with the last remark on the convergence of the error indicator, whose convergence is visible on Fig. 2: this error indicator being convergent, the solution is correct over the whole frequency band.
Fig. 1. Definition of the closed acoustic car cavity considered in Section 6
Fig. 2 shows the error indicator m defined in Section 5 versus the number of PGD pairs (Xm, λm) that are selected. As one can see, this error indicator decreases, which illustrates the convergence of the Algorithm 1. As m quantifies the error in the resolution of the VTCR problem (Eq. (8)) on the whole frequency band, its convergence simply tells us that the PGDVTCR approximated solution is convergent over the whole frequency band. Fig. 3 shows a comparison between the VTCR reference solution and the PGDVTCR approximation at ω0 – Δω / 2, ω0 and ω0 + Δω / 2 (the middle and the two extreme circular frequency of the frequency
Fig. 2. The error indicator (see Section 5) versus the number of PGD pairs (Xm, λm) selected
Finally, Fig. 4 depicts the acoustical energy in the measure area defined in Fig. 1, over the whole frequency band. This energy is easy to compute from the local response, which is visible on Fig. 3. Again, the comparison of this physical quantity computed by the PGDVTCR approximation with the same quantity computed on the reference solution shows that the proposed approach is able to recover the reference solution very well, on the whole frequency band.
Fig. 3. Comparison between the VTCR reference solution and the VTCRPGD approximation (with M = 35) for the example considered in Section 6 and depicted in Fig. 1; real plot of the pressure: a) 2000 Hz, b) 2150 Hz and c) 2300 Hz PGDVTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems
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Fig. 4. Comparison between the solution given by the PGD approximation with (red color curve) and the VTCR reference solution (black color curve) for the example considered in Section 6 and described in Fig. 1
7 CONCLUSIONS We proposed here a new path for solving vibration problems on frequency bands. It is based on a combination of the VTCR and the PGD. Looking for an approximation based on a separated variable decomposition solves the VTCR problem, defined on a frequency band. The best representation of such decomposition has been defined through a minimax problem, which is a generalization of the PGD. Power type algorithms have been proposed to find the solution of the minimax problem. A 2D numerical illustration on a car cavity has been proposed to see the benefits of such an approach on complex acoustical problems. Future works will be devoted to the definition of more efficient algorithms to find a separated variable decomposition. The extension of this strategy to problems with uncertainties is also a work in progress. 8 REFERENCES [1] Kovalevsky, L., Ladevèze, P., Riou, H., Bonnet, M. (2012). The variational theory of complex rays for threedimensional helmholtz problems. Journal of Computational Acoustics, vol. 20, no. 4, DOI:10.1142/ S0218396X1250021X. [2] [2] Ladevèze, P. (1999). Nonlinear Computational Structural Mechanics  New Approaches and NonIncremental Methods of Calculation, Springer, Berlin, DOI:10.1007/9781461214328. [3] Ladevèze, P. (1996). A new computational approach for structure vibrations in the medium frequency range. Comptes Rendus de l Académie des Sciences  Series IIB  MechanicsPhysicsChemistryAstronomy, vol. 332, p. 849856.
312
[4] Rouch, P., Ladevèze, P. (2003). The variational theory of complex rays: a predictive tool for mediumfrequency vibrations. Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 2830, p. 33013315, DOI:10.1016/S00457825(03)003529. [5] Riou, H., Ladevèze, P., Rouch, P. (2004). Extension of the variational theory of complex rays to shells for medium frequency vibrations. Journal of Sound and Vibration, vol. 272, no. 12, p. 341360, DOI:10.1016/ S0022460X(03)007752. [6] Riou, H., Ladevèze, P., Sourcis, B. (2008). The multiscale VTCR approach applied to acoustics problems. Journal of Computational Acoustics, vol. 16, no. 4, p. 487505, DOI:10.1142/S0218396X08003750. [7] Ladevèze, P., Chevreuil, M. (2005). A new computational method for transient dynamics including the lowand the mediumfrequency ranges. International Journal for Numerical Methods in Engineering, vol. 64, no. 4, p. 503527, DOI:10.1002/ nme.1379. [8] Strouboulis, T., Hidajat, R. (2006). Partition of unity method for helmholtz equation: qconvergence for plane wave and waveband local bases. Applications of Mathematics, vol. 51, no. 2, p. 181204, DOI:10.1007/ s1049200600110. [9] Cessenat, O., Despres, B. (1999). Application of an ultra weak variational formulation of elliptic pdes to the twodimensional helmholtz problem. SIAM Journal on Numerical Analysis, vol. 35, no. 1, p. 255299, DOI:10.1137/S0036142995285873. [10] Monk, P., Wang, D. (1999). A leastsquares method for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, vol. 175, no. 12, p. 121136, DOI:10.1016/S00457825(98)003260. [11] Farhat, C., Harari, I., Franca, L. (2001). The discontinuous enrichment method. Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 48, p. 64556479, DOI:10.1016/S00457825(01)002328. [12] PerreyDebain, E., Trevelyan, J., Bettess, P. (2004). Wave boundary elements: a theoretical overview presenting applications in scattering of short waves. Engineering Analysis with Boundary Elements, vol. 28, no. 2, p. 131141, DOI:10.1016/S09557997(03)001279. [13] Van Genechten, B., Atak, O., Bergen, B., Deckers, E., Jonckheere, S., Lee, J.S., Maressa, A., Vergote, K., Pluymers, B., Vandepitte, D., Desmet, W. (2012). An efficient Wave Based Method for solving Helmholtz problems in three dimensional bounded domains. Engineering Analysis with Boundary Elements, vol. 36, no. 1, p. 6375, DOI:10.1016/j. enganabound.2011.07.011. [14] Ammar, A., Mokdad, B., Chinesta, F., Keunings, R. (2006). A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of NonNewtonian Fluid Mechanics,
Barbarulo, A. – Riou, H. – Kovalevsky. L. – Ladeveze, P.
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vol. 139, no. 3, p. 153176, DOI:10.1016/j. jnnfm.2006.07.007. [15] Nouy, A. (2007). A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 4548, p. 45214537, DOI:10.1016/j.cma.2007.05.016. [16] Ladevèze, P., Passieux, J.C., Néron, D. (2010). The LATIN multiscale computational method and the proper generalized decomposition. Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 2122, p. 12871296, DOI:10.1016/j.cma.2009.06.023. [17] Ladevèze, P., Rouch, P., Riou, H., Bohineust, X. (2003). Analysis of mediumfrequency vibrations in a frequency range. Journal of Computational Acoustics, vol. 11, p. 255284, DOI:10.1142/ S0218396X0300195X. [18] Ladevèze, P., Riou, H. (2005). Calculation of mediumfrequency vibrations over a wide frequency
range. Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 2729, p. 31673191, DOI:10.1016/j.cma.2004.08.009. [19] Kovalevsky, L., Ladevèze, P., Riou, H. (2012). The Fourier version of the variational theory of complex rays for mediumfrequency acoustics. Computer Methods in Applied Mechanics and Engineering, vol. 225228, p. 142153, DOI:10.1016/j.cma.2012.03.009. [20] Chinesta, F., Ladevèze, P., Cueto, E. (2011). A short review on model order reduction based on proper generalized decomposition. Archives of Computational Methods in Engineering, vol. 18, no. 4, p. 395404, DOI:10.1007/s1183101190647. [21] Nouy, A. (2010). A priori model reduction through proper generalized decomposition for solving timedependent partial differential equations. Computer Methods in Applied Mechanics and Engineering, vol. 199, no,2324, p. 16031626, DOI:10.1016/j. cma.2010.01.009.
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Received for review: 20141013 Received revised form: 20140125 Accepted for publication: 20140401
Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force Nakano, K. – Cartmell, M.P. – Hu, H. – Zheng, R. Kimihiko Nakano*,1 – Matthew P. Cartmell2 – Honggang Hu1 – Rencheng Zheng1 2 University
1 University
of Tokyo, Institute of Industrial Science, Japan of Sheffield, Department of Mechanical Engineering, United Kingdom
Stochastic resonance is a physical phenomenon where large vibration occurs when a weak sinusoidal force is applied to a bistable system. It is expected that a larger vibrational response can be produced than for a typical resonance. Then the authors utilize this system as an energy harvester, which converts energy from vibration. The energy balance between the converted energy and the energy consumed to produce the necessary weak sinusoidal force is analyzed through numerical simulations. It is shown the proposed harvester can convert more energy than a system using a typical linear system resonance. Keywords: periodic excitation, bistable system, energy harvesting, stochastic resonance
0 INTRODUCTION In recent years, the field of energy harvesting has developed strongly and has increasingly become an important application area. Energy harvesting is the process of capturing trace amounts of energy from the environment and transforming them into electrical energy, such as solar power, thermal energy, wind energy, salinity gradients, vibration energy, etc. While solar, chemical and thermal methods are sometimes viable, many researchers have recognized the abundance of vibration as a potential source of energy [1]. Vibration energy can be converted to electric energy using several types of electromechanical transducers based on electromagnetism [2], electrostatics [3] and [4] and piezoelectricity [5]. Electromechanical transducers are typically realized as linear mechanical resonators. The effectiveness of such converters is maximum when the transducer is operated at resonance, however it can be considerably suboptimal with frequencyvarying conditions and wideband vibrations [6]. If the environmental vibration frequency deviates a little from the designed frequency, the generated power decreases rapidly. Some researchers have extended output power through increasing the resonant frequency bandwidth of the harvester [7] and [8]. The harvester with a tunable resonance frequency is also considered as an effective method [9] and [10]. Given the bandwidth limitation of linear transducers, nonlinearity is also considered as a possibility for improving the effectiveness of energy harvesting [11] and [12], mainly because nonlinear systems are capable of responding over a broad frequency range. 314
The phenomenon of the stochastic resonance requires three basic ingredients: (a) an energetic activation barrier such as the double potential well of a bistable system, (b) a weak coherent input such as a periodic signal, and (c) a source of noise that is inherent to the system. Given these features, the response of the system undergoes resonancelike behaviors as a function of the noise level, hence the name stochastic resonance [13]. The original work on stochastic resonance is due to Benzi et al. [14], for explaining the periodic recurrences of the earth’s ice ages. Recently, stochastic resonance has been considered in several fields, such as signal processing [15], circuit experiments [16], and image visualization [17], etc. More especially, McInnes et al. illustrated the possibility of the application of stochastic resonance as an enhancing method for vibration energy harvesting, and demonstrated its benefit in theory [18]. For a nonlinear mechanical system, ambient vibration can be considered as source of noise which can excite a stochastically resonant system into a bistable nonlinear response. The presence of bistability makes the system capable of rapidly switching between stable states under an external nonlinear force. So, if a weak parametric excitation is applied to the system the double well potential periodically raises and lowers the potential barrier, and noiseinduced hopping between the potential wells can become synchronised with the parametric excitation, leading to stochastic resonance. On this basis, the stochastic resonance was produced in a cantilever beam where a periodic force is applied in the lateral direction to the beam [19] in the experiments. It was shown the response vibration was enhanced and
*Corr. Author’s Address: Institute of Industrial Science, The University of Tokyo, 461 Komaba, Meguroku, Tokyo, Japan, knakano@iis.utokyo.ac.jp
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the amount of the energy to be harvested increased through experiments [20]. In this study, the authors propose to apply the periodic force to the axial direction of the beam. Then dynamics of the beam is governed by a periodic excitation system. In the previous system [19] and [20], the periodic force need to be supplied from the system, however, in the proposed system, the stochastic resonance is expected to be produced in the periodic excitation system exposed to a wide bandwidth of weak ambient broadband vibration. The feasibility to cause the stochastic resonance with the periodic force in the axial direction of beam to enhance the vibration energy to be harvested is examined through numerical analysis.
Piezoelectric film
Based on the theory of stochastic resonance a bistable nonlinear vibration energy harvester is conjectured as shown in Fig. 1. The system is mainly composed of a cantilever beam with an end magnet, and another fixed magnet. It is excited by ambient vibration N(t), and there is an interaction between the elastic force of the beam and the magnetic force, generating a nonlinear response. By adjusting the distance d between the two magnets, the system can show bistability. An actuator is used to provide the source of the periodic parametric excitation. When a piezoelectric material is placed under a mechanical stress, an open circuit voltage appears across the material. Suitable piezoelectric materials for vibration energy harvesting are characterized by the large magnitude of the product of the piezoelectric voltage constant and the piezoelectric strain constant. The material can be in the form of polycrystalline ceramics, textured ceramics, thin films, and polymers. Therefore, in this paper it is considered that two piezoelectric films are fitted onto the cantilever beam to provide the nonlinear vibration energy harvester with a simple but effective mechanical to electrical energy conversion facility. Due to the configuration of the two magnets, as discussed in [19], a repulsive magnetic force FM will act between the two magnets. When FM makes the angle θ to x axis and the vertical displacement of the end magnet is yM, the vertical component, FV, can be written in terms of FM and expanded into a Taylor series, computed around yM = 0 and truncated as follows,
Magnet
θ Acos(Ωt)
S
N
yM d
N
S
N(t)
a)
yM
1 STOCHASTIC RESONANCE 1.1 Modeling of the Energy Harvester
FM yM ≈ 1/ 2 2 d yM 1 + ( d 2 ) F F 3 (1) ≈ M yM − M3 yM . d 2d
FV = FM sin θ =
y x
W(t)
xM
b) Fig. 1. An energy harvester using stochastic resonance; a) system layout, and b) simple schematic
When the periodic excitation W(t) is input to the system by the actuator in the horizontal direction, and assuming the beam is mass less, the total kinetic energy will be defined by:
Ek =
1 m[ y M 2 + ( xM − W ) 2 ], (2) 2
where xM is the lateral displacement of the end magnet. According to the thin beam theory, the potential energy of the beam whose Young’s modulus and moment of inertia are E and I, respectively, is given by: l 1 U = EI ∫ ( w′′) 2 dx, (3) 0 2 where w is the transverse displacement of the beam and w'' = d2w / dx2. Assuming yM(t) = w(l,t), where l is the length of the beam, the following relationship is derived:
3x 2 x3 w( x, t ) = v( x) w(l , t ) = 2 − 3 yM (t ), (4) 2l 2l
where v(x) is a static linear shape function of the cantilever beam. Therefore the following can be shown from Eqs. (3) and (4): l 1 U = EI ∫ (v′′) 2 dx yM 2 , (5) 2 0
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where v'' = d2f / dx2. Assuming the transverse deflection is tiny, the lateral displacement of the end magnet, xM, is given by:
3 = N (t ), (13) yM + cy M − ω 2 (1 + λ cos Ωt ) yM + byM
2 l dw xM = ∫ 1 + − 1 dx = 0 dx
=∫
to:
l
0
2
1 dw 1 l 2 . (6) dx = ∫ (v′) 2 dx yM 2 dx 2 0
Differentiating Eq. (6) with respect to time leads xM = ByM y M , (7)
l where B = ∫ (v′) 2 dx . Substituting Eq. (7) into (2), 0 the kinetic energy can be expressed as:
Ek =
1 2 1 2 2 my M + m( B 2 yM y M + W 2 − 2 ByM y MW ). (8) 2 2
According to the generalized Lagrange’s equation: ∂E d ∂Ek ∂U ( )− k + = FV , (9) dt ∂y M ∂yM ∂yM the following equation is derived: 2 2 myM (1 + B 2 yM ) − mB 2 yM y M + l + EI ∫ (v′′) 2 dx yM − mByMW = FV . (10) 0
constant periodic force, BAΩ2 can be considered to be constant. When we define λ as BAΩ2ω–2, Eq. (12) can be transformed to:
This is the governing equation for the model proposed. By neglecting the two nonlinear terms, i.e. the second and third left hand side terms, Eq. (10) can be reduced to: l F myM + ( EI ∫ (v′′) 2 dx − M ) yM − 0 d F 3 −mByM AΩ 2 cos Ωt + M3 yM = 0, (11) 2d
which is standard equation describing the system exploring stochastic resonance. 1.2 Analysis of the Conditions for Stochastic Resonance According to the three basic ingredients for stochastic resonance, a weak coherent input such as a periodic signal is required. Although the periodic excitation is too weak to make the system jump periodically from one potential well to the other, the hopping between the potential wells occur when the system disturbed by the noise is synchronized with the periodic excitation, which is the stochastic resonance. For the model proposed, the weak periodic excitation is provided by the parametric excitation. Consequently, the potential energy can be expressed as: 1 1 4 2 U ( yM , t ) = − ω 2 (1 + λ cos Ωt ) yM + byM . (14) 2 4 Fig. 2 shows the changing potential well as a function of displacement. It can be seen that the height of the potential well is modulated periodically. According to the theory of stochastic resonance, running through the frequency fulfils certain conditions, and this modulation enables the excitation N(t) to drive the system into stochastic resonance.
where W(t) = Acos(Ωt) is the periodic excitation. By adding in a viscous damping term and inputting an ambient vibration N(t), one obtains; 3 = N (t ), (12) yM + cy M − (ω 2 + BAΩ 2 cos Ωt ) yM + byM
where b=
ω2 =
l 1 FM − EI ∫ (v′′) 2 dx 0 m d
and
FM . Here, ω is corresponding to the undamped 2md 3
natural frequency of the system. Assuming W(t) is 316
Fig. 2. The potential well when ω2 = 3, λ = 0.2 and b = 1
A statistical synchronization between the noiseinduced hopping and periodic excitation will take place, when the average waiting time Tw between two
Nakano, K. – Cartmell, M.P. – Hu, H. – Zheng, R.
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noiseinduced interwell transitions is comparable with half the period of the periodic excitation Ts. The probability of a transition between the two potential wells is determined by the Kramers rate rK, and it will
be small if the potential barrier between the two wells is very large. Therefore the waiting time Tw can be defined by the Kramers rate rK as Tw = 1 / rK, and rK for the model in this paper can be expressed by:
Fig. 3. Response under ambient vibration excitation, or periodic parametric excitation: the blue line fluctuating at 10 mm and the black line fluctuating at 10 mm represent the responses under the ambient vibration excitation and the periodic parametric excitation, respectively
Fig. 4. Responses under ambient vibration excitation and parametric excitations at different frequencies for Ω; a) 0.5, b) 1.0, c) 1.5, d) 2.0, e) 2.5, and f) 3.0 rad/s Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force
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rK =
ω2 ω4 exp(− ), (15) D 2π
where D is the intensity of the ambient vibration. For example when the ambient vibration is white noise this can be defined as N (t ) = 2 D g (t ) , where g(t) is the Gauss function. Therefore the conditions for stochastic resonance defined by the frequency f of the periodic excitation in Hz and noise strength can be transformed to the following,
f =
1 rK . (16) 2
Contrary to this, when the excitation frequency in Hz f exceeds that value then stochastic resonance cannot occur. According to Eq. (16), when ω2 = 3, the range of the excitation frequency Ω for stochastic resonance where Ω = 2πf, is 1.98 to 2.12 rad/s. In order to confirm that stochastic resonance can definitely occur under this condition, the numerical simulations have been undertaken. White noise acts on the system as the ambient vibration, simultaneously the excitation with different values of Ω is also input to the system. It should be noted that it is necessary to distinguish between the effects of stochastic resonance and normal resonance. The blue line in Fig. 3 shows the displacement response of the end mass just under the ambient vibration excitation and it can be seen that the beam is unable to jump between the two potential wells. Conversely, the black line presents the displacement response for the parametric excitation on its own, operating at 1.7 Hz. This excitation alone is unable to excite the beam into bistable vibration. In Fig. 4, it can be seen that when the parametric periodic excitation and ambient vibration are input together, the system response increases. However, for different values of frequency Ω, the response is seen to change. When Ω = 0.5 rad/s, the system behaves as a single stable state for most of the simulation time. When Ω increases to 1.0 and 1.5 rad/s, the system behaves as the bistable state in a certain period. When the excitation frequency is 2.0 rad/s as shown in Fig. 4d, which is close to the theoretically predicted parametric excitation frequency for stochastic resonance, the large vibration is observed. The response is the strongest in the six cases shown in Fig. 4 and the system seems to be in the bistable state all the time. The response becomes weaker when Ω exceeds 2.0 rad/s. From these results it can be concluded that the vibration is enhanced by the stochastic resonance. 318
2 ENERGY HARVESTING For the energy harvester shown in Fig. 1, the piezoelectric film will change the damping radio of the system. According to the general model of a vibration energy harvester, the damping coefficient c is represented by ce + cm , where cm is the mechanical damping coefficient, and ce represents the electrically induced damping coefficient. The system can harvest from the energy absorbed by the electrically induced damping. The dynamics of the system can be described by rearranging Eq. (13) as follows: 2 3 yM y M + (ce + cm ) y M + byM y M −
−ω 2 (1 + λ cos Ωt ) yM y M = N (t ) y M . (17) Eq. (17) can be transformed to:
d 1 2 ω2 2 b 4 2 yM + yM ) + (ce + cm ) y M ( y M + = dt 2 2 4 = y M N (t ) + λω 2 cos(Ωt ) yM y M . (18)
Eq. (18) presents the instantaneous power input into the system, which is equal to the sum of the instantaneous power dissipated by damping and time derivative of the sum of the kinetic and 2 is potential energies. In the left hand side, 1 2 y M corresponding to the kinetic energy of the system, and 2 4 ω 2 2 yM + b 4 yM presents the potential energy. And 2 (ce + cm ) y M is the instantaneous power absorbed by 2 the damping, in which ce y M is considered to be the energy that can be harvested. In the right hand side, y M N (t ) and λω 2 cos(Ωt ) yM y are corresponding to the energy supplied to the system by the ambient vibration and the periodic excitation, respectively. As the periodic excitation is the force need be supplied to produce the stochastic resonance, the energy λω 2 cos(Ωt ) yM y M is considered to be the power consumed for the harvesting, while the energy y M N (t ) is pure energy supplied from the ambient to the harvester. When evaluating the performance of the energy harvesting, the energy λω 2 cos(Ωt ) yM y M is extracted from the energy absorbed by the electric damping 2 ce y M . Therefore the net instantaneous power which can be harvested can be given by:
P = ce y 02 − λω 2 cos(Ωt ) y0 y 0 . (19)
In order to investigate the enhancement of the energy harvesting by the stochastic resonance, a theoretical analysis with numerical simulation for the
Nakano, K. – Cartmell, M.P. – Hu, H. – Zheng, R.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 314320
Fig. 5. Net available power under different conditions: a) without periodic excitation,b) stochastic resonance state, and c) natural resonance state
available power has been carried out. Fig. 5 represents the results of numerical simulations under the condition c = 0.5 Ns/m, ce = 0.3 Ns/m, and ω2 = 3 s–2. Fig. 5 shows the net powers generated when the periodic force is not supplied, when the periodic force whose angular frequency is 2.0 rad/s, which is near the Kramers’s rate, is supplied, and when the angular frequency is 1.7 Hz, which is close to the natural frequency of the system, respectively. It is clear the system harvests the largest energy in the case of Fig. 5b. This indicates the stochastic resonance is the most effective way to generate energy from the ambient vibration. Furthermore, as we can adjust the frequency of the periodic excitation easily using the real time control unit, the stochastic resonance harvester can be operated in the best condition even if the characteristics of the ambient vibration changes. This is another advantage to the harvester using a typical resonance of the linear system. 3 CONCLUSION A nonlinear mechanical vibration harvester that fulfills the conditions for stochastic resonance has been designed. Theoretical analysis has been implemented to illustrate the possibility of stochastic resonance in a certain mechanical vibration system. Furthermore,
the feasibility to apply the stochastic resonance to the energy harvester has been investigated through numerical simulations. The effectiveness of stochastic resonance to enhance the amount of the energy to be harvested was evaluated, by comparing the harvested power with the case without the stochastic resonance. Although the power consumed in providing a periodic force reduces the total harvested power, the available power generated in stochastic resonance is obviously higher than the power generated from only ambient vibration. It is seen that stochastic resonance can act as a highly effective mean for energy harvesting from the ambient vibration. 4 REFERENCES [1] Saha, C.R., O’Donnell, T.N., McCloskey, W.P. (2008). Electromagnet generator for harvesting energy from human motion. Sensors and Actuators A: Physical, vol. 147, no. 1, p. 248253, DOI:10.1016/j.sna.2008.03.008. [2] Dai, X., Wen, Y., Li, P., Yang, J., Li, M. (2011). Energy harvesting from mechanical vibrations using multiple magnetostrictive/piezoelectric composite transducers. Sensors and Actuators A: Physical, vol. 166, no. 1, p.94101, DOI:10.1016/j.sna.2010.12.025. [3] Chandrakasan, A., Amirtharajah, R., Goodman, J., Rabiner, W. (1998). Trends in low power digital signal processing. Proceedings of the IEEE International
Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force
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Symposium on Circuits and Systems, vol. 4, p. 604607, DOI:10.1109/ISCAS.1998.699014. [4] Mitcheson, P.D., Miao, P., Stark, B.H., Yeatman, E.M., Holmes, A.S., Green, T.C. (2004). MEMS electrostatic micro power generator for low frequency operation. Sensors and Actuators A: Physical, vol. 115, no. 23, p. 523529, DOI:10.1016/j.sna.2004.04.026. [5] Yoon, S., Lee, Y., Lee, S., Lee, C. (2008). Energyharvesting Characteristics of PZT5A under gunfire shock. Materials Letters, vol. 62, no. 12, p. 36323635, DOI:10.1016/j.matlet.2008.04.042. [6] Ferrari, M., Ferrari, V., Guizzetti, M., Andò, B., Baglio, S., Trigona, C. (2009). Improved energy harvesting from wideband vibrations by nonlinear piezoelectric converters. Procedia Chemistry, Proceedings of the Eurosensors XXIII conference, vol. 1, no. 1, p. 12031206, DOI:10.1016/j.proche.2009.07.300. [7] Sari, I., Balkan, T., Kulah, H. (2008). An electromagnet micro power generator for wideband environmental vibrations. Sensors and Actuators A: Physical, vol. 145146, p. 405413, DOI:10.1016/j.sna.2007.11.021. [8] Liu, J., Fang, H., Xu, Z., Mao, X., Shen, X., Chen, D., Liao, H., Cai, B. (2008). A MEMSbased piezoelectric power generator array for vibration energy harvesting. Microelectronics Journal, vol. 39, no. 5, p. 802806, DOI:10.1016/j.mejo.2007.12.017. [9] Mansour, M.O., Arafa, M.H., Megahed, S.M. (2010). Resonator with magnetically adjustable nature frequency for vibration energy harvesting. Sensors and Actuators A: Physical, vol. 163, no. 1, p. 297303, DOI:10.1016/j.sna.2010.07.001. [10] Zhu, D., Roberts, S., Tudor, M.J., Beeby, S.P. (2010). Design and experimental characterization of a tunable vibrationbased electromagnet microgenerator. Sensors and Actuators A: Physical, vol. 158, no. 2, p. 284293, DOI:10.1016/j.sna.2010.01.002. [11] Ferrari, M., Ferrari, V., Guizzetti, M., Marioli, D. (2010). A singlemagnet nonlinear piezoelectric converter for enhanced energy harvesting from random vibrations. Procedia Engineering, vol. 5, p. 11561159, DOI;10.1016/j.proeng.2010.09.316.
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[12] Ferrari, M., Ferrari, V., Guizzetti, M., Andò, B., Baglio, S., Trigona, C. (2010). Improved energy harvesting from wideband vibrations by nonlinear piezoelectric converters. Sensors and Actuators A: Physical, vol. 162, no. 2, p. 425431, DOI:10.1016/j.sna.2010.05.022. [13] Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F. (1998). Stochastic resonance. Reviews of Modern Physics, vol. 70, no. 1, p. 223287, DOI:10.1103/ RevModPhys.70.223. [14] Benzi, R., Parisi, G., Sutera, A., Vulpiani, A. (1982). Stochastic resonance in climatic change. Tellus, vol. 34, no. 1, p. 1016, DOI:10.3402/tellusa.v34i1.10782. [15] Klamecki, B.E. (2005). Use of stochastic resonance for enhancement of lowlevel vibration signal components, Mechanical Systems and Signal Processing, vol. 19, no. 2, p. 223237, DOI:10.1016/j.ymssp.2004.03.006. [16] Harmer, G.P., Davis, B.R., Abbott, D. (2002). A review of stochastic resonance: circuits and measurement. IEEE Transactions on Instrumentation and Measurement, vol. 51, no. 2, p. 299309, DOI:10.1109/19.997828. [17] Rallabandi, V.P., Roy, P.K. (2010). Magnetic resonance image enhancement using stochastic resonance in Fourier domain. Magnetic Resonance Imaging, vol. 28, no. 9, p. 13611373, DOI:10.1016/j.mri.2010.06.014. [18] McInnes, C.R., Gorman, D.G., Cartmell, M.P. (2008). Enhanced vibrational energy harvesting using nonlinear stochastic resonance. Journal of Sound and Vibration, vol. 318, no. 45, p. 655662, DOI:10.1016/j. jsv.2008.07.017. [19] Hu, H., Nakano, K., Cartmell, M.P., Zheng, R., Ohori, M. (2012). An experimental study of stochastic resonance in a bistable mechanical system. Journal of Physics: Conference Series, vol. 382, DOI:10.1088/17426596/382/1/012024. [20] Zheng, R., Nakano, K., Hu, H., Su, D., Cartmell, M.P. (2014). An application of stochastic resonance for energy harvesting in a bistable vibrating system. Journal of Sound and Vibration, vol. 333, no. 12, p. 25682587, DOI:10.1016/j.jsv.2014.01.020.
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 321330 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1835 Special Issue, Original Scientific Paper
Received for review: 20131113 Received revised form: 20140128 Accepted for publication: 20140328
Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data Saba, D. – Forte, P. – Vannini, G. Diego Saba1,* – Paola Forte1 – Giuseppe Vannini2
1 University
of Pisa, Department of Civil and Industrial Engineering, Italy 2 GE Oil & Gas  Nuovo Pignone, Italy
The design of the gas seals used in centrifugal compressors and axial turbines requires to consider not only their aptitude to reduce leakage, but also their contribution to the overall dynamics of the machine. Honeycomb and hole pattern annular seals are often employed in compressors for the end balance piston seal or as the central balance piston seal in a backtoback arrangement. In contrast to labyrinth seals, they show a beneficial damping effect. In order to obtain an effective tool for predicting the leakage and the dynamic response of honeycomb seals, a bulk flow model has been devised in the past and, implemented in numerical codes, it is presently used in the design process. This kind of codes, however, require simplifying assumptions: in particular, one reference code available to the authors adopts the hypothesis of isothermal process. As the required level of confidence in seal design is increasing, an experimental validation and possibly some refinement are needed. In this work, the bulk flow model was reviewed and the sensitivity to different hypotheses was explored. New experimental data from a high pressure test rig were compared with the results of simulations. Keywords: gas seals, rotordynamics, honeycomb structures, vibration control
0 INTRODUCTION Since the 1960s, smoothrotor/honeycombstator annular seals have been used in process centrifugal compressors where they were employed to replace aluminium labyrinth seals consumed by the process fluid. As it turned out, honeycomb seals had significantly less leakage compared to conventional seethrough labyrinth seals for the same clearances. In a smoothrotor/honeycombstator annular seal, a honeycomb pattern of hexagonal cavities is present on the stator, see Fig. 1. Holepattern seals are very similar. The same pattern of cavities is present, but the their shape is cylindrical, see Fig. 2. It has long been known that internal seals can have a strong impact on the dynamics of a turbomachine. A destabilizing effect due to the seals was reported for the Kaybob compressor in 1975 [1] and for the Ekofisk compressor in 1976 [2].
Fig. 1. Honeycomb seal
Fig. 2. Holepattern seal
The damping capabilities of honeycomb seals were noticed and exploited later. In 1985, honeycomb seals were used to stabilize the highpressure oxygen turbopump of the space shuttle main engine [3]. Currently, honeycomb seals are often used for the balance piston seal of a high performance compressor, especially in a backtoback arrangement. When a seal is positioned half way between the bearings, where the first mode of vibration of the shaft has its maximum amplitude, its dynamic behavior becomes of greater importance. The industry would benefit from a reliable way of predicting the behavior of honeycomb and holepattern gas seals. However, despite the combined analytical and experimental effort, the predictive ability of the presently available tools is not sufficient, especially at very high pressures. In this paper, new experimental data for honeycomb gas seals are presented and compared to
*Corr. Author’s Address: University of Pisa  DICI, Largo Lucio lazzarino 1, 56100 Pisa, Italy, diego.saba@for.unipi.it
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the solutions of a simplified fluid dynamic model. In the current program the seals have been tested at higher pressures than in previous works, to the authors’ knowledge. The fluid dynamic model is based on an isothermal bulk flow model developed by Kleynhans and Childs [4] and Kleynhans [5]. Since the simulations with the isothermal model did not agree with the authors’ experimental data, a new simulation tool has been developed which adds the possibility to make different assumptions on the transport of heat and momentum. Section 1 introduces the dynamical coefficients, as a way to characterize the dynamical behavior of seals. Section 2 briefly describes the testing apparatus used to measure the dynamic coefficients. Section 3 gives a detailed description of the analytical model used in this paper. Since the aim of this work was the comparison of different variants of the bulkflow formulation, the model is described as the composition of different submodels, each one governed by its own parameters and options. Sections 4 and 5 present the results and give some concluding remarks. 1 DYNAMIC COEFFICIENTS The dynamical behavior of a rotordynamic component is characterized by means of dynamic coefficients that describe the linear relationship between the displacements of the rotor and the forces it exerts on the surrounding bodies, in this case the gas in the clearance. The relevant displacements for gas seals are the lateral displacements of the rotor. The relationship can be expressed as:
x H xx F = y H yx F
H xy x , (1) H yy y
where x , y are the Fourier transforms of the lateral displacements in two fixed orthogonal y are the Fourier transforms x , F directions and F of the corresponding forces. The frequencyresponse function Hij is a dynamic stiffness, but is called impedance in this paper, following the use of other authors in the same context [4], [6] and [7]. This choice has some advantage because the term stiffness can be used for the real part, i.e. the inphase response, without further speciﬁcation. Frequency dependent stiffness and damping coefficients are deﬁned as:
Kij = Re(Hij), Cij = Im(Hij) / ω . (2)
The terms on the diagonal Hxx, Hyy are called direct impedances, while the terms off the diagonal Hxy, Hyx are called crosscoupled impedances. Forces and displacements in Eq. (1) are represented through their Cartesian components. In other words, the displacements are decomposed into harmonic oscillations in two fixed directions. It is also possible to represent the lateral displacements of the rotor as the superposition of forward and backward circular orbits, using a new basis. The transformation rules for the change of basis are: x 1 1 xf = , (3) y −i i xb xf , xb are the components in the new basis. where The forces are decomposed accordingly into forward and backward rotating forces. The transformation rules are the same: x 1 1 F f F = . (4) y −i i F F b
and defines forward and backward impedances. The dynamic behavior is called isotropic if it is independent of the orientation of the reference system x, y. For moderate eccentricities, the behavior of a seal is usually close to isotropic. It can be shown that, for an isotropic system, the following properties hold: as:
H xx = H yy , H xy = − H yx , H fb = H bf = 0. (6) The effective stiffness and damping are deﬁned Keff = Re(Hff), Ceff = Im(Hff) / ω , (7)
and are important indicators of the performance of a seal. When the rotor axis moves along a forward precessional orbit, a positive effective stiffness indicates a centripetal reaction force, and a positive effective damping indicates a tangential stabilizing force, see Fig. 3. For an isotropic system the following relations hold:
322
In the new basis, Eq. (1) becomes: f H ff H fb F xf = , (5) b H bf H bb F xb
Keff = Kxx + ωCxy,
Saba, D. – Forte, P. – Vannini, G.
Ceff = Cxx – Kxy / ω . (8)
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Fig. 3. Effective stiffness and effective damping
Since forces and displacements are real variables, their Fourier transforms have Hermitian symmetry. The four impedance coefficients, in x / y notation, also have Hermitian symmetry, namely:
The gas used in the plant is nitrogen. The design pressure is 400 bar. The rotor is supported by active magnetic bearings (AMB). The magnetic bearings can be controlled in such a way as to impose the desired displacements to the axis of the rotor. The relative position between the shaft and each of the bearings is measured, and so is the force actuated by each bearing. The orbit imposed to the rotor axis is given by the superposition of harmonic displacements at different concurrent frequencies (multifrequency excitation).
H ij (−ω ) = H ij (ω ). (9)
On the contrary, in f / b notation the following relations hold:
H ff (−ω ) = H bb (ω ), H fb (−ω ) = H bf (ω ). (10) 2 EXPERIMENTAL DATA
The experimental results available for this work come from the ultra high pressure (UHP) test rig of GE Oil & Gas, Florence. The test seal was a smoothrotor honeycombstator seal, with convergent clearance, diameter 220 mm and length 65 mm. The ranges of values for the test parameters are given in Table 1. Table 1. Test parameters upstream pressure pressure ratio upstream temperature rotational speed preswirl ratio
pU pD / pU TU fΩ Rsw
30 to 200 bar 0.3 to 0.9 ~ 300 K 10000 rpm ~1
2.1 Testing Apparatus Only the most relevant information will be given here. A more detailed description of the testing apparatus can be found in [8]. Two identical seals are tested at once. They are mounted in the test cell in a symmetrical backtoback conﬁguration, see Fig. 4. The relevant conditions for a compressor seal are reproduced, namely the high and low pressures, respectively upstream and downstream the seals, the rotational speed of the rotor, and the swirl of the gas before entering the seals (preswirl).
Fig. 4. Test cell
Forces and displacements are measured in two conﬁgurations: with a pressure difference across the seals, and without pressure. The reason for this is the need to separate the dynamic coefficients of the seals from those of the whole system. The main contribution to the dynamic coefficients of the system, besides that of the seals, is due to the inertia of the rotor. This contribution can be computed through the displacements and the known mass distribution of the rotor. However, a direct measurement is preferred. The preswirl ratio, i.e. the ratio of the circumferential velocity component of the gas and the peripheral speed of the rotor, is generated through the swirler ring which collects the gas ﬂow from an inlet plenum and injects it toward the seals through aerodynamic nozzles evenly distributed along the circumference. The current swirler is designed to produce a high preswirl, in the range of 0.8 to 1 at 10 krpm.
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2.2 Identification Methodology The measurements of forces and displacements are sampled at a sufficiently high rate to obtain a reliable spectral decomposition in the range of the excitation frequencies. For each spectral component, Eq. (1) holds. It consists of two scalar equations, that are not enough to determine the four impedances of the general formulation. However, assuming isotropy, there are only two unknowns Hd, Hc x H F d = y − H c F
The bulk ﬂow model reduces the dimensionality of the problem, from three to two. The balance equations are written in integral form along the radial direction, and the mean values of the properties of the ﬂow are assumed to be close to the values in the bulk. The bulk ﬂow variables are thus functions of only two geometrical coordinates, axial and circumferential. In Fig. 5 the two control volumes are shown. Volume A is the clearance and volume B comprises the cavities of the honeycomb, or the holes of the hole pattern.
H c x . (11) H d y
The unknown impedances can be put in evidence by rearranging the terms: x x y H d F = . (12) y y − x H c F
The shape of the orbit of each selected frequency should not be circular or near to circular, because the matrix of coefficients in Eq. (12) would become illconditioned. Flat straightline orbits are preferred with this identiﬁcation method. To obtain the four impedances of the general case, two experiments are necessary, differing only for the excitation components and not for pressures, temperature and preswirl. Strict acceptability criteria must be met to couple two experiments. For the two coupled experiments, we can write:
1x F 1y F
2 x H xy F = 2 y H yx F
H xy x1 x2 . (13) H yy y y 2 1
which is immediate to solve. 3 SIMULATION MODEL 3.1 Bulk Flow Model The bulk ﬂow model with two control volumes, devised by Kleynhans and Childs [4] and Kleynhans [5] is an evolution of the bulk ﬂow model used by Nelson [9] for smooth annular seals. A second control volume was added to take into account the gas trapped (actually recirculating) in the cavities of the honeycomb. The effective speed of sound is slowed down by the exchange of mass between bulk ﬂow and cavities. 324
Fig. 5. Control volumes
(a ρ ),t + (a ρ uθ ),θ + (a ρ u z ), z + mass +ϕm = 0 2 momentum (a ρ uθ ),t + (a ρ uθ ),θ + (a ρ uθ u z ), z + + ap,θ + ϕ pθ = τ Rθ + τ SAθ A (a ρ u z ),t + (a ρ uθ u z ),θ + (a ρ u z2 ), z + (14) momentum + ap, z + ϕ pz = τ Rz + τ SAz (a ρ et ),t + a,t p + (a ρ ht uθ ),θ + energy +(a ρ ht u z ), z + ϕe = qR + qSA + u Rτ Rθ (b ρ B ),t − ϕm = 0 mass B (b ρ B eB ),t − ϕe = qSB energy
In the balance equations (Eq. (14)) the symbols have the meaning shown in Section 6 and commas indicate partial derivatives with respect to the variables on their right. The subscript A has been omitted from the fluid properties of the corresponding control volume. The unknowns are ρ, T, uθ, uz, ρB, TB. Any other thermodynamic quantity is derived from these using the equations of state. In this work, the ﬂuid was modeled as a perfect gas with constant heat capacity, viscosity and thermal conductivity. Radial ﬂuxes φm etc. between control volumes, friction stresses τRθ etc. and heat ﬂuxes qR etc. must be determined through additional empirical equations. Different models result from different assumptions. Two assumptions are shared by all the variants of the model. The pressure is assumed to be the
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same for the two control volumes at a given location. The temperature too is assumed to be the same for the two volumes. The assumption of thermal local equilibrium is not well supported, either theoretically or experimentally, but a more complicated model did not seem justiﬁed at this stage. Future experimental or theoretical feedback might suggest a better hypothesis. The equations are ﬁrst solved to ﬁnd the steady state, independent of t, θ. Then, a ﬁrstorder perturbation problem is set up. A precessional motion is imposed to the rotor, that entails a harmonic perturbation of the clearance, in time and along the circumferential coordinate. To complete the problem, boundary conditions must be set for the fluid at inlet and outlet. Four conditions are needed: two on the pressure, at inlet and outlet, one on the inlet swirl velocity, and one on the inlet temperature. The perturbation problem silently assumes that the flow can be described for every position z, θ and for every instant t by the same state variables as for the steady problem, namely ρ, T, uθ, uz, a. This assumpion, however, is much stronger than for the steady problem. It implies that, at the frequencies of interest, the perturbations are so slow that the vorticous flow in a honeycomb cavity differs little from the steady one identified by the same state variables.
The variants of the bulk ﬂow model implemented for this work are shown in Fig. 6. The options for the three submodels can be chosen independently.
heat exchange entrance and exit loss radial convective momentum flux
isothermal
(bt ρ ),t + (a ρ uθ ),θ + (a ρ u z ), z = 0, (a ρ uθ ),t + f p ( b ρ ),t uθ + (a ρ uθ2 ),θ + +(a ρ uθ u z ), z + ap,θ = τ Rθ + τ Sθ , (a ρ u z ),t + f p ( b ρ ),t u z + (a ρ uθ u z ),θ + +(a ρ u z2 ), z + ap, z = τ Rz + τ Sz ,
(bt ρ et ),t + (a ρ ec ),t + a,t p + (a ρ ht uθ ),θ + +(a ρ ht u z ), z = qR + qS + u Rτ Rθ , (16)
where
τ Sθ = τ SAθ + τ SBθ , τ Sz = τ SAz + τ SBz , qS = qSA + qSB . (17)
The isothermal model does not use the energy equation. 3.3 Friction Model
3.2 Variants of the Bulk Flow Model
submodel
where a nondimensional fp, chosen by the user, has been introduced to estimate the convective part of the momentum ﬂuxes. By imposing p = pB, T = TB, and using Eq. (15), we can dispose of the unknowns pB, TB and write the bulk ﬂow equations in the form:
options nonisothermal adiab. isoth. wall nq > 0 nq = 0 nq → ∞
incompressible
compressible
fp = 1
0 ≤ fp <1
Fig. 6. Variants of bulk flow model
The model described in [4] and [5] can be reproduced using the options in column two (isothermal, incompressible, fp = 1). This model has been chosen as a reference stateoftheart model. All the other options implement original submodels, to the authors’ knowledge. The radial ﬂuxes φpθ, φpz are written as:
The friction model is the same for all the variants. Friction is assumed to be isotropic: the friction coefficient is independent of ﬂow direction and the shear stresses are parallel to the relative velocity between the bulk ﬂow and each wall. The magnitude of the relative velocity is indicated in the following formulas by ur. The same friction model is used for the rotor and the stator: 1 τ = f ρ ur2 , (18) 2
f = nR m r , (19)
where Eq. (18) defines the Fanning friction factor and Eq. (19) is Blasius friction formula. Different parameters m, n are used for the stator and for the rotor. Values for the rotor can be found in the manuals. For the stator, there is no simple predictive model. Experimental values were obtained on a ﬂat plate tester [10] for some geometries and Reynolds numbers.
ϕp¸ = −τ SBθ + f pϕmuθ , ϕpz = −τ SBz + f pϕmu z , (15) Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data
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3.4 Heat Transfer Model The most important phenomenon that the heat equation and the heat transfer model should be able to describe is the amount of heat produced by friction and how much of it is transferred through the walls. The effect of the rate of heat transfer on the speed of sound is also important. The temperatures of rotor and stator are assumed to be known and equal to the temperature of the ﬂuid upstream. For the heat transfer, a simple correlation has been posited between Nusselt and Reynolds numbers Eq. (20). The heat transfer model is summarized by the formulas: q = λ (T − Taw ) , λ = Nu
κ , 2a
u2 m Nu = nq R r q , Taw = Rw T + r , (20) 2c p
where mq = 0.8 and the extreme cases for nq are 0 (adiabatic) and ∞ (isothermic wall). The adiabatic wall temperature Taw, as deﬁned by Shapiro [11], takes into account the fact that, for a compressible ﬂuid, the temperature in the bulk differs from the temperature in the viscous layer near the wall, even when the wall is adiabatic. The adiabatic wall temperature differs a little from the stagnation temperature, and their ratio Rw Eq. (20) depends mainly on Prandtl number. Since Prandtl number varies little for air, nitrogen, and other ﬂuids of interest in compressors, a ﬁxed value Rw = 0.9 was used. The correlation in Eq. (20) can be viewed as a simpliﬁcation of SiederTate correlation:
Nu = 0.072R
4/5
1/ 3
Pr (η / η w )
0.14
, (21)
where the dependence of viscosity, thermal conductivity and heat capacity on pressure and temperature have been neglected. The “isothermal wall” option is the limit case for nq → ∞, and has constant adiabatic wall temperature Taw. The bulk temperature is not constant, but decreases with velocity. 3.5 Boundary Conditions The ﬂuid, upstream, before being dragged by the movement of the rotor, can already have a circumferential velocity component, or swirl. It has been shown that the circumferential velocity affects greatly the dynamic coefficients and the stability of 326
honeycomb seals. The effective damping is decreased by a positive swirl, i.e. in the same direction of the shaft’s rotation, and is increased by a negative swirl. Two of the boundary conditions reﬂect the assumption that circumferential momentum and total enthalpy are conserved at the entrance: u Iθ = uθ , hIθ = ht . (22)
The condition on enthalpy does not apply for the isothermal model. The conditions on pressure are expressed in terms of loss coefficients, assigned by the user. The loss coefficients are the ratio of the amount of energy transformed irreversibly into heat and the kinetic energy in the conduct. For a perfect gas, assuming an adiabatic transformation, the following boundary conditions result [9]:
( = p (1 + αξ u
) /c )
pU = p 1 + αξ I u z2 / c 2 pD
2 E z
2
1/α 1/α
, , (23)
where
ξ I = (1 + k I ) / 2, ξ E = (1 − k E ) / 2, α = (γ − 1) / γ ,
c 2 = Rg T .
(24)
For the isothermal case, conditions in Eq. (23) become:
(
)
pU = p exp ξ I u z2c 2 ,
(
)
pD = p exp ξ E u z2c 2 . (25)
A formulation for incompressible ﬂuids can be chosen by the user, identical for isothermal and nonisothermal models,
pU = p + ξ I ρ u z2 ,
pD = p + ξ E ρ u z2 . (26)
This formulation was used in [4] and serves here as reference. It can be viewed as an approximation for low velocities, as in that case compressibility effects become less important. In fact, the axial Mach number at the entrance never exceeded 0.23, even in choked conditions, and no signiﬁcant differences have been found using compressible and incompressible boundary conditions. Care must be taken to recognize a choked ﬂow. The axial velocity uz increases with the axial coordinate, as in a Fanno ﬂow, even if the clearance is slightly divergent, because the expansion due to friction forces prevails. The ﬂow cannot reach the critical velocity at any point along the seal, except at the exit, where it is: Mz = 1. (27)
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When the choked condition is reached, the second boundary condition of Eq. (26) must be replaced by Eq. (27). In choked condition, the ﬂow is insensitive to the downstream pressure pD.
ms values. The choice of ms has little inﬂuence on the dynamic coefficients. Table 3. Estimate of friction coefficients ms 0.3 0.2 0.1 0
5 RESULTS The data from nine experimental tests were compared with the predictions of the model with the options and parameters of Table 2. The loss coefficients were set to kI = 0, kE = 1, meaning a reversible transformation at the entrance, and pD = p at the exit, i.e. no pressure recovery. Table 2. Simulation options for comparison with experimental data label isot momf adiab isowall
descriptive note isothermal isothermal without convective momentum flux adiabatic isothermal wall
parameter fp 1
nq 
0

1 1
0 ∞
The friction factor parameters for the stator ms, ns were not available from dedicated tests. An estimate was made using the measured mass ﬂow rates. The estimate was made by running the simulation with “isot” options and minimizing the quadratic percentage error of the mass ﬂow rates. Table 3 shows the estimated ns as a function of arbitrarily chosen ms. As can be seen from the table, the estimate of ns ﬁts quite well the experimental data for a wide range of
ns est 1.70 0.51 0.153 0.046
Rmse [%] 1.07 0.72 0.47 0.30
Experimental tests with different pressures but equal pressure ratio show a similar behavior, as expected. Fig. 7 shows experimental data in nondimensional form for different pressures but similar pressure ratios pD / PU = 0.79÷0.83. Angular frequencies and impedances are nondimensionalized respectively with ωref and Kref deﬁned by:
ωref = c / R, K ref = π RLpU / (a + b). (28)
It should be noted that the nondimensionalized frequency ω / ωref, scales with the inverse of the velocity of sound c–1, and hence with T–1/2. Different tests were run with the same excitation frequencies, but slightly different temperatures, since the ambient temperature was not controlled. The temperature shift of the nondimensionalized excitation frequencies is evident in Fig. 7. It can also be noted that this method of nondimensionalization is indeed effective in comparing different tests by similarity. The impedances Hbf, Hfb, that express the coupling between forward and backward precessions, are expected to be zero in case of isotropic dynamic
Fig. 7. Impedances in nondimensional form from four comparable experiments; the legend indicates entrance/exit pressures (bar) Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data
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behavior. The experimental data conﬁrm the isotropy, within the uncertainties of the measurements. Simulation results are shown for two cases, one with unchoked ﬂow and one with choked ﬂow, see Table 4. The Mach number in the table was computed with the isothermal model. The simulation results are displayed in Figs. 8 and 9. It can be noted that the graphs of Im(Hff) do not pass through the origin, but are negative at zero frequency. This is the effect of the positive swirl, and the reason why swirl brakes have a stabilizing effect. In both cases, the adiabatic model predicts a temperature rise up to 14 K in the bulk, due to friction.
Table 4. Cases shown in the graphics
unchoked flow choked flow
pU [bar] 64.4 48.6
pD [bar] 51.1 15.3
0.793 0.316
Mz at exit (computed) 0.23 1
The comparison between isothermal and adiabatic models is worth commenting. In the adiabatic model, the temperature should increase along the flow, because the high friction near the walls converts a sizable amount of mechanical energy into heat. The analysis has shown that this effect is partially compensated by a temperature decrease due to
Fig. 8. Unchoked case: forward and backward impedances in nondimensional form
Fig. 9. Choked case: forward and backward impedances in nondimensional form
328
pU / pD
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expansion. In fact the net result was a variation of the absolute temperature of about 5% in the worst case. Another difference is in the speed of sound, that in the adiabatic model is about 20% greater than in the isothermal model. Yet its variation seems to have little impact. The differences among the models are more evident for the choked case. This is probably due to the greater compressibility effects, which increase the sensitivity to the thermal hypotheses. The graph of the choked case shows an almost constant offset between measured and simulated Re(Hff). This is an indication that the actual clearance, during the experiment, was possibly narrower than at rest. The elastic deformations of the seal and its housing are a possible cause. Negative static stiffness can lead to instability. This kind of instability has been observed in long seals working in choked condition. In analyzing the Im(Hff) graph, it is useful to separate two kinds of discrepancies: a constant offset and difference in slope. A constant error can be ascribed to an error in swirl prediction, which, in turn, can be justified by anisotropic friction of the honeycomb, not difficult to implement. On the contrary, in order to justify a different slope in the graph a deeper revision of the model is needed, probably related to the observation reported at the end of section 3.1. 5 CONCLUSIONS To predict the dynamic behavior of honeycomb gas seals, an isothermal bulk ﬂow model is currently used. A new simulation tool has been developed which adds the possibility to make different assumptions on the transport of heat and momentum. New experimental data at higher pressures than previously available have been compared with the bulk ﬂow simulations. The results showed that the new models have no major impact on the dynamic coefficients, in the frequency range of interest, except for the effective stiffness of choked ﬂows. The fact that heat generation and transport have little effect on the temperature of the fluid and on the dynamic coefficients of the seal is a remarkable nontrivial result. Since neither the isothermal model nor the new tentative bulkflow approaches have been satisfactory in explaining the available experimental data, it is likely that a key factor has been overlooked. The assumptions underlying the perturbation equations are probably too restrictive and need a revision.
This does not necessarily mean that bulkflow approaches have to be abandoned. The ease of calculation is still a strong point in their favor and obliges to investigate unexplored model tuning parameters. 6 NOMENCLATURE γ cp / cv ratio of specific heats η dynamic viscosity θ circumferential coordinate (arc, not angle) κ thermal conductivity ρ density τ friction stresses φm radial flux of mass φp radial flux of momentum φe radial flux of energy Ω rotor angular speed angular frequency of vibration ω a radial clearance mean depth of cavities b (volume to surface ratio) bt a+b C, Cij damping matrix c cs
( dp / dT )T ( dp / dT )s
isothermal speed of sound isentropic speed of sound
e
internal energy per unit mass 1 2 (uθ + u z2 ) kinetic energy ec 2 e + ec total energy et H, Hij impedance matrix h enthalpy per unit mass ht h + ec total enthalpy K, Kij stiffness matrix k concentrated loss coefficient M z axial Mach number: uz / c for the isothermal model, uz / cs otherwise m, n coefficients of Blasius’ formula p pressure q heat flux (entering the fluid) R seal radius Rg specific gas constant Rsw uIθ / uR preswirl ratio Rr 2αρur / η Reynolds number relative to one of the walls T temperature u flow velocity uR ΩR rotor peripheral speed u r flow velocity relative to one of the walls z axial coordinate
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Subscripts θ, z component in cylindrical reference system control volumes A, B I, E entrance and exit R, S rotor and stator U, D upstream and downstream f, b component in f / b reference system unspecified component in Cartesian i, j reference system x, y component in Cartesian reference system 7 REFERENCES [1] Smith, K.J. (1975). An operation history of fractional frequency whirl. Proceedings of the 4th Turbomachinery Symposium. Texas A&M University, p. 115125. [2] Cochrane, W. (1976). New generation compressor injecting gas at Ekoﬁsk. Oil&Gas Journal, p. 6370. [3] Childs, D., Moyer, D. (1985). Vibration characteristics of the HPOTP (highpressure oxygen turbopump) of the SSME (space shuttle main engine). Journal of Engineering for Gas Turbine and Power, vol. 107, no. 1, p. 152159, DOI:10.1115/1.3239676. [4] Kleynhans, G.F., Childs, D.W. (1997). The acoustic inﬂuence of cell depth on the rotordynamic characteristics of smoothrotor/honeycombstator annular gas seals. Journal of Engineering for Gas Turbine and Power, vol. 119, no. 4, p. 949956, DOI:10.1115/1.2817079.
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[5] Kleynhans, G.F. (1996). A twocontrolvolume bulkﬂow rotordynamic analysis for smoothrotor/ honeycombstator gas annular seals. PhD Dissertation, Texas A&M University, College Station. [6] Dawson, M.P., Childs, D.W., Holt, C.G., Phillips, S.G. (2002). Measurements versus predictions for the dynamic impedance of annular gas seals – Part I: test facility and apparatus. Journal of Engineering for Gas Turbines and Power, vol. 124, no. 4, p. 958962, DOI:10.1115/1.1478075. [7] Dawson, M.P., Childs, D.W. (2002). Measurements versus predictions for the dynamic impedance of annular gas seals – Part II: smooth and honeycomb geometries. Journal of Engineering for Gas Turbines and Power, vol. 124, no. 4, p. 963970, DOI:10.1115/1.1478076. [8] Vannini, G., Cioncolini, S., Calicchio, V., Tedone, F. (2011). Development of an ultrahigh pressure rotordynamic test rig for centrifugal compressors internal seals characterization. Proceedings of the 40th Turbomachinery Symposium, College Station, p. 4659. [9] Nelson, C.C. (1985). Rotordynamic coefficients for compressible ﬂow in tapered annular seals. Journal of Tribology, vol. 107, no. 3, p. 318325, DOI:10.1115/1.3261062. [10] Ha, T.W., Childs, D.W. (1992). Frictionfactor data for ﬂatplate tests of smooth and honeycomb surfaces. Journal of Tribology, vol. 114, no. 4, p. 722729, DOI:10.1115/1.2920941. [11] Shapiro, A.H. (1953). The dynamics and thermodynamics of compressible ﬂuid ﬂow. Ronald Press Co., New York.
Saba, D. – Forte, P. – Vannini, G.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 331338 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1832 Special Issue, Original Scientific Paper
Received for review: 20131013 Received revised form: 20140116 Accepted for publication: 20140401
Parameter Identification of Multistorey Frame Structure from Uncertain Dynamic Data Chakraverty, S. – Behera, D. Snehashish Chakraverty* – Diptiranjan Behera National Institute of Technology Rourkela, India This paper investigates the identification procedure of the column stiffness of multistorey frame structures by using the prior (known) uncertain parameters and dynamic data. Uncertainties are modelled through triangular convex normalized fuzzy sets. Bounds of the identified uncertain stiffness are obtained by using a proposed fuzzy based iteration algorithm with Taylor series expansion. Example problems are solved to demonstrate the reliability and efficiency of the identification process. Keywords: stiffness matrix, mass matrix, Taylor series expansion, triangular fuzzy number, natural frequency
0 INTRODUCTION Structural dynamics problems may be categorized as direct or inverse problems. The direct problem consists of finding the response for a specified input or excitation. In the inverse problem the response is known then to develop a mathematical model of the system. The modelling problem may also be divided into two categories. In the first category the nature of the process is completely unknown. But in the second category, a considerable knowledge of the nature of the system may be available, whereas the particular values of the system parameters are unknown. In this paper the second category has been studied, where system equations are known or deducible from the physics of the system, with coefficients remaining to be estimated and modified as per the known initial dynamic characteristics. In this context various workers [1] to [3] have reviewed the state of the art of system identification in structural dynamics. Developments and various methods for studying this important field are available in literature [4]. More recent methods and practical guidelines for linear systems may be found in the work of Schoukens and Pintelon [5]. Few researchers have studied the above issues but, continuous efforts are being made to refine and develop new models for identification problems. Some representative works on the subject are available in [6] to [9]. Accordingly related works done are discussed in the subsequent paragraphs. Loh and Ton [10] have studied a system identification approach to detect changes in structural dynamic characteristics on the basis of measurements. They used the recursive instrumental variable method and extended Kalman filter algorithm for the identification procedure. Potential of using neural network to identify the internal forces of typical systems has been investigated by Chassiakos
and Masri [11]. A localized identification of many degrees of freedom structures is investigated by Zhao et al. [12] and a memorymatrix based identification methodology for structural and mechanical systems is studied by Udwadia and Proskurowski [13]. Notable studies in this field have also been done by other workers [14] to [18]. Recently various authors [19] to [22] studied different procedures for the parameter identification problems of different structures and buildings. Budipriyanto [19] addresses the application of blind source separation technique for identifying dynamic parameters of a seismicexcited multistory building from its measured response. A new technique based on second order blind identification, called the modified crosscorrelation method for the identification of the structures has been studied by Hazra et al. [20]. Rahmani and Todorovska [21] presented two new algorithms for 1D system identification of buildings during earthquakes by seismic interferometry using waveform inversion of impulse responses. Wave travel time analysis and layered shear beam models are used by Todorovska and Rahmani [22] for the system identification of buildings. Also an interesting and important review paper related to vibration based damage identification methods has been written by Fan and Qiao [23]. The objective of the structural dynamic analysis related to identification is to develop an analytical model of a structure which can be verified and adjusted by actual test results. However, this adjustment is not easy and can be done by computer with good convergence algorithm in terms of iterative cycles. As discussed above usual method of identification uses the values of the parameters initially given to the structure by an engineer. It then modifies the original parameter values as per the observed values from test by an iteration process. The parameters involved in the said problems are traditionally considered as crisp or defined exactly. But, rather than
*Corr. Author’s Address: National Institute of Technology Rourkela, Odisha, India, sne_chak@yahoo.com
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particular value, we may have only the uncertain or incomplete information about the parameters being a result of errors in measurements, observations, applying different operating conditions or it may be maintenance induced error, etc. So, for various scientific and engineering problems, it is an important issue how to deal with variables and parameters of uncertain value. Recently some effort has been made by various researchers throughout the globe to handle these uncertainties in terms of probabilistic, interval or fuzzy approach. Unfortunately, probabilistic methods may not deliver reliable results with required precision without sufficient data. Hence interval and fuzzy theory are becoming powerful tools for handling these uncertainties in recent decades. Recently, few authors [24] to [27] have studied the solution methods for fuzzy and interval system of linear equations. They have considered the system of linear equations with fuzzy and interval numbers. Also vibration analysis of structures with imprecise material properties is also done by few authors. As such papers that are related to interval and fuzzy eigenvalue problems are discussed here. An excellent paper by Chen et al. [28] who has presented a new method for calculating the upper and lower eigenvalue bound of structures with interval parameter. Uncertain bunds of eigenvalue are also studied by Friswell et al. [29]. Dimarogonas [30] discussed the vibration problem using interval analysis. Cechlarova [31] investigated the eigenvectors of the interval matrix using maxplus algebra. Recently, modal analysis of structures by using interval analysis is studied by Sim et al. [32]. Qui et al. [33] presented a paper which gives detailed analysis for exact bounds for the static response of structures with uncertainbutbounded parameters. Xia and Yu [34] studied modified interval and subinterval perturbation methods for the static response analysis of structures with interval parameters. Dynamic Analysis of structures with interval uncertainty has been explained by Modares and Mullen [35]. Fuzzy material and geometric properties have also been considered by various authors for finite element analysis. Both static and dynamic analyses of structures are excellently explained by Akpan et al. [36] using fuzzy finite element analysis. An important paper is that of Hanss et al. [37] who proposed the application of fuzzy arithmetic in the finite element analysis. Behera and Chakraverty [38, 39] investiagted various solution procedures for the static analysis of structures with fuzzy parameters. Very recently Sahoo and Chakraverty [40] presented fuzzified data based neural network modeling for health assessment 332
of multistorey shear buildings. Also soft computing methods for model updating of multistory shear buildings for simultaneous identification of mass, stiffness and damping matrices have been investigated by Khanmirza et al. [41]. In view of the above, the present study proposes a systematic mathematical model for the identification of uncertain structural parameters using the vibration characteristics consistent with the uncertain experimental data. The method first uses the values of the uncertain structural parameters (viz. as triangular fuzzy numbers) initially given to the structure by an engineer. It then modifies the original parameter values as per the observed values from test by an iteration process using Taylor series expansion. It gives uncertain fuzzy bound of modified values of the parameters to have a better estimation of structural safety. In the following sections, first preliminaries are described, followed by the mathematical modelling and identification process. Then numerical examples for two storey frame structures are described. Finally discussion and conclusions are drawn. 1 PRELIMINARIES In the following paragraph some definitions related to the present work are given. Definition 1.1 Fuzzy number [42] and [43] A fuzzy number U is convex normalised fuzzy set U of the real line R such that:
{µU ( x) : R → [0,1], ∀ x ∈ R},
where, μU is called the membership function of the fuzzy set and it is piecewise continuous. Definition 1.2 Triangular fuzzy number (TFN) A fuzzy number U is said to be triangular if: i. There exists exactly one x0 ∈ R with μU (x0) = 1 (x0 is called the mean value of U), where μU is called the membership function of the fuzzy set. ii. μU (x) is piecewise continuous. The membership function μU of an arbitrary triangular fuzzy number U = (a, b, c) may be defined as follows: x≤a 0, x−a , a≤ x≤b b − a . µU ( x) = c − x , b≤x≤c c −b 0, x≥c
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Any arbitrary triangular fuzzy number U = (a, b, c) can be represented with an ordered pair of functions through α–cut approach as:
[u (α ), u (α )] = [(b − a )α + a, − (c − b)α + c],
where α ∈[0,1]. This satisfies the following requirements: i. u(α ) is a bounded left continuous nondecreasing function over [0, 1]. ii. u (α ) is a bounded right continuous nonincreasing function over [0, 1].
[ M ]{ x} + [ K ]{x} = {0} , (1)
m 0 where [ M ] = is a 2×2 fuzzy mass matrix, 0 m
(k + k + k + k ) −(k3 + k4 ) [ K ] = 1 2 3 4 , (k3 + k4 ) −(k3 + k4 )
is a 2×2 fuzzy stiffness matrix and { X } = 2 × 1 is a fuzzy vector of displacements. Considering the simple harmonic motion, Eq. (1) can be written as a fuzzy eigenvalue problem: [ K ]{ X } = λ[ M ]{ X }. (2)
iii. u (α ) ≤ u (α ), 0 ≤ α ≤ 1.
Definition 1.3 Fuzzy arithmetic [24] and [25] As discussed above, fuzzy numbers may be transformed into an interval through α–cut approach.
By using the parametric form of fuzzy numbers, Eq. (2) will be:
So, for any arbitrary fuzzy number x = [ x(α ), x (α )], y = [ y (α ), y (α )] and scalar k, we have, x = y if and
[ K (α ), K (α )]{ X (α ), X (α )} =
= [λ (α ), λ (α )][ M (α ), M (α )]{ X (α ), X (α )}.
only if x(α ) = y (α ) and x (α ) = y (α ) . Addition: x + y = [ x(α ) + y (α ), x (α ) + y (α )]. Subtraction: x − y = [ x(α ) − y (α ), x (α ) − y (α )]. Multiplication: x × y = min (a1 , a2 , a3 , a4 ), max (a1 , a2 , a3 , a4 ) , where, a1 = x(α ) × y (α ), a2 = x(α ) × y (α ),
a3 = x (α ) × y (α ), a4 = x (α ) × y (α ),
[kx (α ), k x(α )], k < 0, and kx = [k x(α ), kx (α )], k ≥ 0. 2 MATHEMATICAL MODELLING AND METHOD OF IDENTIFICATION To investigate the present method, a twostoreyed frame structure, as shown in Fig. 1 is considered. However the general multistorey frame structure modeling may easily be extended from this example of two storey frame. This is investigated for the sake of demonstration of the procedure. The uncertain floor mass, m is assumed to be the same and the uncertain column stiffnesses k1 , k2 , k3 and k4 (as labelled in Fig. 1) are the structural parameters which are to be identified. Corresponding uncertain dynamic equation of motion in matrix form for two degrees of freedom system may be written as:
Fig. 1. Two storey frame structure
Now our aim is to solve the above fuzzy eigenvalue problem to get the lower and upper bounds of the fuzzy eigenvalues. With the above in mind, let us proceed now with the identification procedure which can handle the uncertain data. Let us assume that the uncertain structural parameters to be identified are denoted by Pi , for i = 1, 2, 3, 4. The uncertain value of the structural parameters of the prior original structure ~ˆ given initially are denoted by Pi , for i = 1, 2, 3, 4 and the corresponding fuzzy eigenvalues are symbolized ~ˆ ~ˆ as, λi ( P ) . Next the wellknown Taylor’s series expansion of the fuzzy modal parameters about the initial estimates of the parameters give;
Parameter Identification of Multistorey Frame Structure from Uncertain Dynamic Data
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{λ~( P~)} = λ~ˆ ( P~ˆ ) + [S~] {P~}− P~ˆ , (3)
where
{P} = P1, P2 , P3 , P4 T ,
~ˆ ~ˆ ~ˆ ~ˆ ~ˆ T P and = P1 , P2 , P3 , P4
[ S ] is the fuzzy eigenvalue partial derivative matrix, [∂ (λ ) / ∂ ( P )] .
Let us now denote, experimentally measured uncertain eigenvalues by {λE }. It is interesting to note here that if the values of the initial and experimental parameters are equal, then no modification is done. But if the values are different then we denote this difference by:
{δλ~} = {λ~E }− λ~ˆ . (4)
3 NUMERICAL RESULTS As mentioned earlier, the procedure is demonstrated for a two storeyed frame structure. Implementing the above procedure with the proposed iterative cycle for the revised uncertain frequencies and parameters, computer programs have been written and tested for the above problem. In the above example problem, floor masses, m = (3550, 3600, 3650) kg ,
and the column stiffnesses: k= 1 k=
3
have been taken as triangular fuzzy number. Through α–cut these may represented as: m = [50α + 3550, −50α + 3650] kg, k1 = k2 = [50α + 5350, −50α + 5450] N/m, annd k = k = [50α + 3550, −50α + 3650] N/m.
Next, let us denote the modified parameters as:
{P } = [ P 1 , P 2 , P 3 , P 4 ]T , (5)
and, in general, for n–degrees of freedom system the expression for the uncertain modified parameters from Eq. (3) can be written as: ~ ~ˆ ~ ~ P = P + Q δλ , (6)
{}
where
(
[ ]{ }
Q = S T S
)
T S .
−1
In order to have the uncertain bounds of the identified parameters with acceptable accuracy, here an iterative procedure is proposed. After finding the modified parameters from Eq. (6), these are substituted in Eq. (2) to get revised uncertain vibration characteristics viz. {λ} .
The new fuzzy eigenvalue partial derivative matrix {S} is then obtained using the current values of {P } and {λ} . From Eq. (6), the modified parameters {P t } are again found by utilizing the above values and
then the new (revised) estimates of fuzzy eigenvalues are obtained as {λ t } . If the vector norm of {λ} and {λ t } is less than some specified accuracy then the procedure is stopped and the revised parameter viz. {P t } is identified, otherwise the next iteration is to be followed.
334
(5350, 5400, 5450) N/m, k= 2 (3550, 3600, 3650) N/m, k= 4
3
4
From the prior mass and stiffness parameters, the uncertain vibration characteristics may be computed from Eq. (2) as: λ1 = (0.9314, 1, 1.0703) and λ2 = (5.8906, 6, 6.1128). Using the above sets of initial data of the fuzzy parameters with different uncertain experimental (hypothetical) test data for the frequencies, viz. λ = (0.65, 0.7, 0.75) and λ = (5.3, 5.5, 5.7) (i.e. 1E
2E
first and second experimental eigenvalues of the system) the bounds of the stiffness parameters of the structure have been identified and these are reported in Table 1. Corresponding plot for identified stiffness parameters are depicted in Figs. 2 and 3. Similarly, another set of experimental (hypothetical) fuzzy data of the natural frequencies are considered as:
λ1E = (0.88, 0.9, 0.92) and λ2 E = (5.3, 5.5, 5.7).
The identified bounds of the stiffness parameters are tabulated in Table 2 and the revised frequencies are also shown in Table 3. Corresponding plot for identified stiffness parameters are given in Figs. 4 and 5.
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 331338
Table 1. Identified lower and upper bounds of stiffness parameters Bounds of stiffness parameters [N/m]
α=0
α = 0.5
α=1
k1 = k 2
5232
5274
5316
k1 = k 2
5400.5
5358
5316
k3 = k4
3614.5
3629.8
3645
k3 = k4
3674.7
3659.9
3645
Fig. 2. Identified lower and upper bounds of stiffness parameter k1 [N/m]
Fig. 3. Identified lower and upper bounds of stiffness parameter k3 [N/m]
4 DISCUSSIONS AND CONCLUSION The present procedure systematically modifies and finally identifies the uncertain structural parameters, viz. the column stiffness for a frame structure. It uses
the prior (known) estimates of uncertain parameters and corresponding uncertain vibration characteristics. Then the algorithm estimates the bounds of present parameters utilizing the known uncertain dynamic data from some experiments. Proposed numerical
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Table 2. Identified lower and upper bounds of stiffness parameters Bounds of stiffness parameters [N/m]
α=0
α = 0.5
α=1
k1 = k 2
5286
5324
5362
k1 = k 2
5440
5401
5362
k3 = k4
3590.6
3607.7
3624.5
k3 = k4
3657.6
3641
3624.5
Fig. 4. Identified lower and upper bounds of stiffness parameter k1 [N/m]
Fig. 5. Identified lower and upper bounds of stiffness parameter k3 [N/m] Table 3. Experimental and revised lower and upper bounds of frequencies
336
α
λ1E
λ1E
λ 2E
λ 2E
λ1R
λ1R
λ 2R
λ 2R
0 0.5 1
0.88 0.89 0.9
0.92 0.91 0.9
5.3 5.4 5.5
5.7 5.6 5.5
0.9430 0.9706 0.9985
1.0552 1.0267 0.9985
5.8884 5.9476 6.0078
6.1306 6.0688 6.0078
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procedure is tested by incorporating two sets of data. The uncertainties present in the parameters are considered as triangular convex normalized fuzzy sets. It is worth mentioning that if the input data set viz. the design frequency is near to the experimental frequency data then the modified stiffness data bound has less width. This is expected as the design and experimental frequency are close means that the structure has not deteriorated much. On the other hand when the experimental data is taken a bit far from the deigned one then the estimated stiffness parameters give larger bound. These effects may be clearly seen from Tables 1 and 2. It may be noted that the accuracy of the results depends upon many factors viz. on the uncertain bound of the experimental data, initial design values of the parameters, the fuzzy computation, norm as defined etc. The present investigation may be a first of its kind to handle the identification procedure for uncertain data. Although the method has been demonstrated for a simple problem of two storey, but the method may very well be extended to higher storey frames and other structures in a similar fashion. 5 ACKNOWLEDGEMENT This work is financially supported by Board of Research in Nuclear Sciences (BRNS), DAE, and Ministry of Earth Sciences, Government of India. We would like to thank Professor Miha Boltežar and the anonymous referee for valuable comments and suggestions that have led to an improvement in both the quality and clarity of the paper. 6 REFERENCES [1] Hart, G., Yao, J. (1977). System identification in structural dynamics. Journal of Engineering Mechanics Division, vol. 103, no. 6, p. 10891104. [2] Bekey, G.A. (1970). System identification – An introduction and a survey. Simulation, vol. 15, no. 4, p. 151166, DOI:10.1177/003754977001500403 [3] Datta, A.K., Shrikhande, M., Paul, D.K. (1998). System identification of buildings – a review. Proceeding of the 11th Symposium on Earthquake Engineering, University of Roorkee, Roorkee, [4] Natke, H.G. (1982). Identification of Vibrating Structures, SpringerVerlag, Berlin, DOI:10.1007/9783709128961. [5] Schoukens, J., Pintelon, R. (1991). Identification of Linear Systems. Pergamon Press, NewYork. [6] Beck, J.L. (1978). Determining Models of Structures from Earthquake Records. Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena.
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modal data. Earthquake Engineering and Structural Mechanics, vol. 34, p. 543554, DOI:10.1002/eqe.431. [19] Budipriyanto, A. (2013). Blind source separation based dynamic parameter identification of a multistory momentresisting frame building under seismic ground motions. Procedia Engineering, vol. 54, p. 299307, DOI:10.1016/j.proeng.2013.03.027. [20] Hazra, B., Roffel, A.J., Narasimhan, S., Pandey, M.D. (2010). Modified crosscorrelation method for the blind identification of structures. Journal of Engineering Mechanics, vol. 136, no. 7, p. 889897, DOI:10.1061/ (ASCE)EM.19437889.0000133. [21] Rahmani, M., Todorovska, M.I. (2013). 1D system identification of buildings during earthquakes by seismic interferometry with waveform inversion of impulse responsesmethod and application to Millikan library. Soil Dynamics and Earthquake Engineering, vol. 47, p. 157174, DOI:10.1016/j. soildyn.2012.09.014. [22] Todorovska, M.I., Rahmani, M.T. (2013). System identification of buildings by wave travel time analysis and layered shear beam modelsSpatial resolution and accuracy. Structural Control and Health Monitoring, vol. 20, no. 5, p. 686702, DOI:10.1002/stc.1484. [23] Fan, W., Qiao, P. (2011). Vibrationbased damage identification methods: a review and comparative study. Structural Health Monitoring, vol. 10, p. 83111, DOI:10.1177/1475921710365419. [24] Behera, D., Chakraverty, S. (2012). A new method for solving real and complex fuzzy system of linear equations. Computational Mathematics and Modeling, vol. 23, no. 4, p. 507518, DOI:10.1007/s105980129152z. [25] Chakraverty, S., Behera, D. (2013). Fuzzy system of linear equations with crisp coefficients. Journal of Intelligent and Fuzzy Systems, vol. 25, no. 1, p. 201207, DOI:10.3233/IFS20120627. [26] Wang, C., Qiu, Z.P. (2013). Equivalent method for accurate solution to linear interval equations. Applied Mathematics and Mechanics, vol. 34, no. 8, p. 10311042, DOI:10.1007/s1048301317256. [27] Wang, K. (2013). Uzawa method for fuzzy linear system. Journal of Fuzzy Set Valued Analysis, vol. 2013, p. 17, DOI:10.5899/2013/jfsva00137. [28] Chen, S., Qui, Z., Song, D. (1995). A new method for computing the upper and lower bounds on frequencies of structures with interval parameters. Mechanical Research and Communications, vol. 22, p. 431439, DOI:10.1016/00936413(95)00045S. [29] Friswell, M.I., Prells, U., Penny, J.E.T. (2004). Determining uncertainty bounds for eigenvalues. Proceeding of ISMA, p. 30553064. [30] Dimarogonas, A.D. (1995). Interval analysis of vibrating systems. Journal of Sound and Vibration, vol. 183, no. 4, p. 739749, DOI:10.1006/jsvi.1995.0283. [31] Cechlarova, K. (2005). Eigenvectors of interval matrices over maxplus algebra. Discrete Applied
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Mathematics, vol. 150, no. 13, p. 215, DOI:10.1016/j. dam.2005.02.016. [32] Sim, J., Qui, Z., Wang, X. (2007). Modal analysis of structures with uncertainbutbounded parameters via interval analysis. Journal of Sound and Vibration, vol. 303, no. 12, p. 2945, DOI:10.1016/j.jsv.2006.11.038. [33] Qui, Z., Wang, X., Chen, J. (2006). Exact bounds for the static response set of structures with uncertainbutbounded parameters. International Journal of Solids and Structures, vol. 43, no. 21, p. 65746593, DOI:10.1016/j.ijsolstr.2006.01.012. [34] Xia, B., Yu, D. (2013). Modified interval and subinterval perturbation methods for the static response analysis of structures with interval parameters. Journal of Structural Engineering, DOI:10.1061/(ASCE) ST.1943541X.0000936. [35] Modares, M., Mullen, R.L. (2013). Dynamic analysis of structures with interval uncertainty. Journal of Engineering Mechanics, vol. 140, no. 4, DOI:10.1061/ (ASCE)EM.19437889.0000660. [36] Akpan, U.O., Koko, T.S., Orisamolu, I.R., Gallant, B.K. (2001). Practical fuzzy finite element analysis of structures. Finite Element Analysis and Design, vol. 38, no. 2, p. 93111, DOI:10.1016/S0168874X(01)00052X. [37] Hanss, M., Willner, K., Guidati, S. (1998). On applying fuzzy arithmetic to finite element problems. Conference of the North American Fuzzy Information Processing Society, p. 365369, DOI:10.1109/ NAFIPS.1998.715607. [38] Behera, D., Chakraverty, S. (2013). Fuzzy finite element analysis of imprecisely defined structures with fuzzy nodal force. Engineering Applications of Artificial Intelligence, vol. 26, no. 10, p. 24582466, DOI:10.1016/j.engappai.2013.07.021. [39] Behera, D., Chakraverty, S. (2013). Fuzzy analysis of structures with imprecisely defined properties. Computer Modeling in Engineering & Sciences, vol. 96, p. 317337. [40] Sahoo, D.M., Chakraverty, S. (2013). Fuzzified data based neural network modeling for health assessment of multistorey shear buildings. Advances in Artificial Neural Systems, vol. 2013, art. no. 3, p. 112, DOI:10.1155/2013/962734. [41] Khanmirza, E., Khaji, N., Majd, V.J. (2011). Model updating of multistory shear buildings for simultaneous identification of mass, stiffness and damping matrices using two different soft computing methods. Expert Systems with Applications, vol. 38, no. 5, p. 53205329, DOI:10.1016/j.eswa.2010.10.026. [42] Ross, T.J. (2007). Fuzzy Logic with Engineering Applications. Wiley Student Edition, Chichester. [43] Zimmermann, H.J. (2001). Fuzzy Set Theory and Its Application. Kluwer academic publishers, Boston, Dordrecht, London, DOI:10.1007/9789401006460.
Chakraverty, S. â€“ Behera, D.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 339348 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2013.1831 Special Issue, Original Scientific Paper
Received for review: 20131202 Received revised form: 20140130 Accepted for publication: 20140401
Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads Česnik, M. – Slavič, J. Martin Česnik – Janko Slavič*
University of Ljubljana, Faculty of Mechanical Engineering, Slovenia The presented study experimentally and theoretically researches vibrational fatigue of an aluminumalloy specimen for harmonic and random loads. The main aim of this study is to determine the influence of modal parameter changes that occur during the experimental fatigue test, on the correctness of the numerical fatigue life prediction. Firstly, the material’s fatigue parameters were obtained with harmonic base excitation of the specimen near its natural frequency. During the harmonic fatigue testing the changes in the specimen’s natural frequency and damping loss factor were monitored as the fatigue damage was accumulated at the fatigue zone. Secondly, with a validated numerical model of the specimen the stress transmissibility was obtained for the case of the randomvibration base excitation. Finally, by respecting the stress response along with the experimentally obtained material fatigue parameters the vibration fatigue life was estimated for the case of the random vibration load. The numerically predicted fatigue life was compared to the experimental results, obtained with the electrodynamic shaker. From this comparison the influence of the dampinglossfactor changes on the calculated fatigue life was clearly shown. For the case of the observed specimen, the damping loss factor included in the fatigue life estimation should be increased by more than 100 percent to give reliable prediction of the fatigue life. The presented research shows new possibilities and critical aspects in the area of accurate highcycle vibration fatigue lifeestimation of dynamic structures. Keywords: structural dynamics, vibrational fatigue, fatigue testing, harmonic excitation, random excitation, frequencydomain counting methods
0 INTRODUCTION The fatigue failure can occur even if the structure is loaded with a lowamplitude cyclic loads when the load is applied for a longer period of time. Each cycle represents a certain amount of fatigue damage that is being accumulated by the structure; when the accumulated fatigue damage at the given point on the structure reaches a certain value (usually defined as unity), the fatigue failure occurs [1]. Stress cycles that lead to the fatigue failure are a result of external loads of the structure (e.g. mechanical, temperature, vibrational load). The type of fatigue failure due to the vibrational load of the structure is known as vibrational fatigue and has been a subject of several research studies in recent years. Vibrational fatigue can be studied for harmonic or random vibration loads. In both cases the stress load leading to the fatigue failure occurs due to the structure’s own dynamic response [2] to the dynamic baseexcitation, [3] and [4]. Whenever vibrational fatigue occurs it is always accompanied with changes in the natural frequency and damping loss factor. This phenomenon can be utilized to predict the fatigue failure, [5] to [7]. However, when obtaining the fatigue parameters by applying harmonic vibration load, the changes in natural frequency and damping as a result of the increased local nonlinearities [8], lead to the varying amplitude of the stress load cycles and consequentially to the incorrect identification of the
fatigue parameters. The first part of this study deals with the experimental setup and a realtime control of an accelerated vibration fatigue test with harmonic excitation [9], that in a short time accurately obtains fatigue parameters and simultaneously monitors the changes in the natural frequency and damping loss factor. When the fatigue parameters are known one can predict the fatigue life of the structure exposed to a randomsignal vibrational load using the power spectral density of the structure’s stress response, [10] and [11]. Recently, several research studies have been done in the area of the vibrational fatigue under a random baseexcitation. Pagnacco et al. [12] optimised a thickness distribution of a plate structure to prolong the structure’s fatigue life when exposed to the random vibration. Furthermore, Paulus and Dasgupta [13] proposed a semiempirical model for fatigue life estimation for the case of the random vibration load. In their study they modeled a cantilevered beam as a singledegreeoffreedom (SDOF) lumpedparameter system and included the beam’s natural frequency shift due to the damage accumulation into the fatigue life calculation. Han et al. [14] performed the vibration fatigue analysis of a spotwelded structure with an iterative approach. Research studies relating the fatigue parameters obtained with harmonic excitation to the fatigue life under the random baseexcitation were presented by Chen et al. [15] and Yu et al. [16], where the ball grid array (BGA) solder joints were the
*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, janko.slavic@fs.unilj.si
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subject of analysis. In this study, the second part is focused on the estimation and experimental validation of the fatigue life prediction of a linear Yshaped structure under a random vibration loading, where the fatigue parameters obtained with the harmonic fatigue tests are used. Furthermore, by comparing numerical and experimental results it is clearly shown that the changes of the modal parameters must be adequately considered in order to obtain reliable fatigue life estimation. This manuscript is organized as follows. In Section 1 the theoretical background is given, concerning the specimen’s response to the base excitation and the identification of instantaneous natural frequency and damping loss factor. Additionally, some fundamentals of the fatigue life calculation in the frequency domain are given. In Section 2 the experimental setup of the fatigue test with the harmonic excitation is presented along with the obtained experimental results, needed to obtain σ – N curve of the material in question. In Section 3 the numerical and experimental results for a fatigue test with random baseexcitation are shown and compared. In Section 4 the conclusions are given. 1 THEORETICAL BACKGROUND When a dynamic system is excited in the resonant area dynamic response amplification can be observed. By taking advantage of this response amplification it is possible to achieve high stresses in the specimen by applying relatively small excitation displacements, which makes an electrodynamic shaker suitable for performing fatigue tests. The idea of the accelerated vibrational fatigue test is based on the dynamic response of the specimen; therefore, it is necessary to research the dynamic properties of the specimen before the actual fatigue test. Furthermore, the specimen’s dynamic response to the random baseexcitation represents a basis for the estimation of the fatigue life with frequencydomain counting methods.
within the shaker’s frequency range. Additionally, the excited mode shape of interest must be excitable with a translational movement in the axis of the shaker. However, by exciting the specimen in different mode shapes it is possible to achieve different stress states with a single specimen geometry.
Fig. 1. Fixed Yshaped specimen
In this research the Yshaped specimen shown in Fig. 1 is used [9]. The main three beams are arranged at 120° around the main axis and have a rectangular crosssection of 10×10 mm. The Yshaped specimen was made from aluminum alloy AS8U3 by casting and with a surface finish performed by milling. The fatigue zone was additionally fineground in order to remove any scratches that could cause the premature start of an initial crack. Two additional features are included in the Yshaped design: steel deadweights of mass 52.5 g and a round hole through the main axis. These two features are used to adjust the initial natural frequency and to position the fatigue zone on a suitable surface, respectively.
1.1 Specimen’s Dynamic Response to the Harmonic Excitation The dynamic response of the specimen depends on its modal properties; i.e., natural frequencies, mode shapes and damping loss factors. The natural frequencies and mode shapes are easily obtained using the finiteelement method and should be determined in the stage of the experiment design. For the purpose of the vibration fatigue test the natural frequency, near which the specimen is excited, must be well separated from the remaining natural frequencies and must be 340
Fig. 2. Specimen’s fourthmode shape, σeq
When performing the numerical analysis of the proposed specimen the material parameters of aluminum were used, e.g. density of ρ = 2710 kg/m3
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and Young modulus of E = 70 MPa. By evaluating the dynamic response of the Yshaped specimen the fourthmode shape at ω1 = 755 Hz (Fig. 2) was recognized as the most suitable for the nearresonance fatigue test. For the sake of simplicity the fourth natural frequency is in this manuscript denoted as ω1 rather than ω4. In order to appropriately excite the Yshaped specimen with harmonic signal and simultaneously maintain constant stress amplitude some theoretical background regarding the system’s response to harmonic base excitation must be given. Additionally, during the accelerated fatigue test the changes to the modal parameters of the specimen occur [6] and [7], which consequently alter the specimen’s response and must also be taken into the account in theoretical discussion. The following discussion is applied to the basic outline of the accelerated vibration fatigue experiment, illustrated in Fig. 3, and summarizes the methods, presented in [9]. Continuous structures, such as a Yshaped specimen, are generally described as multidegreeoffreedom (MDOF) systems; in the case of a kinematic excitation and a hysteretic damping mechanism the equilibrium equation can be written as [17]:
T + i DxT + K xT = 0. (1) Mx
Here xT denotes the total displacement vector and M , D and K are the mass, damping and stiffness matrices, respectively. For the case of kinematic excitation [4] xT can be written as:
xT (t ) = x(t ) + ι y (t ), (2)
where x(t) is a vector of relative response displacements, ι is the geometryrelated column vector and y(t) is the baseexcitation signal.
According to the Eqs. (1) and (2) the system’s relative response X a of the ath degreeof freedom in the case of the base excitation y(t) = Y sin(ωt) is deduced as [17]: X a = m a ι ω 2 Y α X a (ω ) = N
= ma ι ω 2 Y ∑
r =1
r 2 r
AX a
ω − ω 2 + iη rωr2
, (3)
where ma denotes the ath row in the mass matrix M and α X a (ω ) is the receptance function for the ath degree of freedom. Additionally, r AX a represents the massnormalized modal constant for the ath degreeoffreedom, ωr is the rth natural frequency and ηr is the hysteretic damping loss factor for the rth mode shape. Here it should be noted that since the excitation and response are single harmonics with a known frequency ω the relations x(t ) = −ω 2 ⋅ x(t ) and 2 y (t ) = −ω ⋅ y (t ) between the measured accelerations and displacements are valid. If a dynamic system is observed when excited near the pth natural frequency the ratio X a / Y can be approximated as [17]:
Xa p AX a ≈ ma ι ω 2 2 + BX a , (4) 2 2 Y ω p − ω + iη p ω p
where BX a is a constant complex quantity, in which the contribution of the remaining modes r ≠ p is accounted for. A similar deduction to that in Eq. (4) can be made for the frequency response of the measured principal stress σ 1 to the base excitation y(t). As shown by Česnik et al. [9], by comparing stress and relative displacement responses to the same base excitation, the relation:
Fig. 3. Experimental setup outline Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads
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σ1 = const. (5) X1
holds in the vicinity of a wellseparated natural frequency even when the system‘s natural frequency and damping loss factor change, as long as the constant phase shift ϕ between the excitation and response signal is maintained. This observation enables the indirect monitoring of the principal stress σ1 , using only the measured excitation and response accelerations; however, the ratio σ1 / X1 must be experimentally obtained before the accelerated fatigue test is conducted. By knowing the constant stress amplitude at the fatigue zone and a number of load cycles until fatigue failure the fatigue test of one Yshaped specimen directly gives a single point on the σ – N diagram. From this point on, the procedure for the calculation of fatigue parameters is identical to the procedure when classical fatigue testing machines are used [1]. 1.2 Simultaneous Identification of the Natural Frequency and Damping Loss Factor In order to simultaneously track changes in the natural frequency ω1 and damping loss factor η1 in the case of the nearresonant excitation a simple identification procedure is presented here, which is based on the linear singledegreeoffreedom (SDOF) assumption [17]. In Eq. (4) it was shown that the response of a baseexcited structure can be modelled with the modified receptance:
α X (ω ) = a
ω 2 p AX′
Xa a = 2 , (6) Y ω p − ω 2 +iη pω p2
where the SDOF assumption is considered with BX′ = 0 . The receptance is fully defined with known a values of the natural frequency ωp , the damping loss factor ηp and the modal constant p AX′ for the pth a mode shape. These values can be easily determined from the experimentally obtained response of the dynamic system using the circlefitting method, described in [17]. By measuring the dynamic response of the Yshaped specimen before the accelerated fatigue test, the initial receptance is obtained. As the accelerated vibration fatigue test starts, the natural frequency ω1 and the damping loss factor η1 begin to change, therefore introducing two new variables into Eq. (6) 342
that must be identified. The value of modal constant ′ is assumed constant during the fatigue test. p AX a
The identification of ω1 and η1 can be made with a numerical solution of the system of equations:
X1 = α X (ω , ω1 ,η1 ) , (7) 1 Y
φ = arg (α X (ω , ω1 ,η1 )), (8) 1
where the phase angle ϕ, relative response X1 , excitation amplitude Y and excitation frequency ω are measured during the accelerated fatigue test. 1.3 Fatigue Life Estimation for the Random BaseExcitation Contrary to the simple fatigue life estimation methods for the case of harmonic stress history [1] the estimation of fatigue life for random stress history requires a brief theoretical introduction. Damage estimation for random stress history can be done with time or frequencydomain methods. In this study only the TovoBenasciutti frequencydomain counting method [11] is used, which has shown to be very accurate for a number of different stress profiles [18]. The frequency domain counting methods are based on the frequency stress response of the excited structure. In the case of vibration excitation with a given acceleration power spectral density (PSD) matrix Saa(ω) the stress response PSD matrix is [17]:
2
Sσσ (ω ) = Hσ a (ω ) Saa (ω ), (9)
where Hσa (ω) represents the accelerationtostress transmissibility matrix of the structure. Stress response PSD is obtained from numerical model, that needs to be build and validated with the actual test specimen. Since the general stress state is multiaxial, the PSD of the equivalent stress Sσ eq (ω ) is used for the damage
intensity estimation as [19]:
Sσ (ω ) = Trace[QSσσ (ω )], (10) eq
where Q is constant matrix, defined in [19]. With known specimen‘s PSD the TovoBenasciutti method estimates the damage intensity as:
(
)
D = BTB + (1 − BTB )α 2BTB −1 α 2 DNB . (11)
Here DNB is a damage intensity for the narrow band stress profile and BTB is a factor, obtained from spectral characteristics of the stress PSD:
Česnik, M. – Slavič, J.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 339348
DNB = α 2ν p C − b
BTB =
(
(
2λ0
) ⋅ Γ 1 + b2 , (12)
(α1 − α 2 ) × (α 2 − 1) 2
b
)
× 1.112(1 + α1α 2 − (α1 + α 2 )) e 2.11α 2 + (α1 − α 2 ) . (13) In Eqs. (11) to (13) the parameters α1 , α2 , νp and λ0 are the statistical characteristics of the Sσ eq (ω ) (see [11]), b and C are the material fatigue parameters and Γ is the gamma function. The estimated fatigue life of the analyzed structure equals to Test = 1 / D . The numerical analysis of expected fatigue life is generally performed in the following manner. Firstly, the stress response PSD matrix is calculated for the given specimen geometry, boundary conditions and applied random vibration profile. Secondly, the equivalent stress PSD‘s are obtained for the nodes of interest. Thirdly, the statistical characteristics of the equivalent stress PSD‘s are calculated. Lastly, by introducing the material‘s fatigue parameters and statistical characteristics of the stress PSD into damage intensity equation (which depends on the adopted damageestimation method), the expected fatigue life for a single node is obtained.
by adjusting the excitation frequency ω. The stress amplitude is maintained constant by adjusting the excitation amplitude Y according to the indirect measurements of stress amplitude by monitoring the relative displacement X1. This is possible since Eq. (5) holds; however, the constant value of σ1 / X1 ratio needs to be determined experimentally before the fatigue test. Therefore, the calibration measurement was performed on the specimens S01 and S02, as shown in Fig. 5. The identified ratio was equal to σ1 / X1 = 3.90 MPa/μm.
Fig. 4. Calibration experiment setup
2 FATIGUE TEST WITH HARMONIC EXCITATION The first part of the experimental work concerns the harmonic fatigue test. With the series of tests presented here the fatigue parameters for the material AS8U3 are obtained along with the continuous changes of the natural frequency and damping loss factor. The experimental setup follows the testing methodology presented by Česnik et al. [9]. 2.1 Experiment Setup and Control In the presented experimental setup, the Yshaped specimen is attached with the fixation adapter to the LDS V555 electrodynamical shaker, Fig 4. To keep the system in the resonant area and to prevent the drop in the response amplitude the excitation frequency ω must be tracking the decreasing natural frequency ω1 in realtime during the accelerated fatigue test. For this case of a nearresonant singleharmonic excitation of the MDOF system the tracking of the natural frequency can be performed by maintaining the constant phase shift ϕ between the excitation y(t) and the response displacement x1(t), as explained in detail in [9]. The constant phase shift is maintained
Fig. 5. σ1 / X1 calibration measurement
2.2 Test Results The harmonic fatigue test was performed on nine Yshaped specimens, denoted with S01S09. The fatigue failure was recognized when a sudden drop of the natural frequency occurred. The instantaneous natural frequencies, identified with the method presented in Sec. 1.2, are shown in Fig. 6. The changes in the damping loss factors are shown in
Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads
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Fig. 7. In order to show the modal parameter changes throughout the fatigue life evenly for all specimens, regardless of their actual fatigue lives, a damage index is used in Figs. 6 and 7. For the case of harmonic load with constant cycle amplitude the damage is defined as Damage = n / N [1], where n and N denote a current number of cycles and a total number of cycles until failure, respectively. The summarized experimental results are listed in Table 1, where σ1 denotes the principal stress amplitude, f1 init is the initial natural frequency, min(η1) is the minimal damping loss factor and Δ f1 is the total natural frequency shift. Table 1. Experimental data of harmonic fatigue test Specimen
σ1 [MPa]
Cycles to failure
S01 S02 S03 S04 S05 S06 S07 S08 S09
95 105 105 115 115 125 125 135 135
33.0 25.0 13.5 9.71 13.1 6.87 7.98 4.87 3.24
N (×105)
f1 init
min(η1) (×104)
[Hz]
758.0 751.6 757.2 757.3 754.3 755.8 755.7 752.3 753.4
3.10 2.85 3.08 4.49 3.73 3.96 4.60 5.37 6.09
31.3 18.3 28.6 21.2 34.8 21.9 22.2 20.9 16.2
[Hz]
Δf1
Fig. 6 indicates that specimens show different frequency shifts and no value of the frequency shift can be used as a definite value for the indication of the final fatigue failure. At this point it should be mentioned that no significant temperature rise was measured during the fatigue testing; the temperature rise was within 2 °C and can be addressed to the heat conduction from the shaker‘s core to the test specimen. Additionally, from Fig. 7 it can be noticed that damping loss factor varied significantly between the tested specimens and was higher for higher loads. This can be addressed to the damping that occurs locally at the fixation of the Yshaped specimen. Therefore, in the random fatigue test the damping loss factor, used for the numerical prediction of the fatigue life, needs to be identified and considered in the numerical model separately for each tested specimen. The material’s fatigue parameters, fatigue exponent b and fatigue strength S f were calculated with least squares method. The Basquin‘s equation [1] for the aluminum alloy AS8U3 is:
σ = 688.1 N–0.170 (14)
and is illustrated in the Fig. 8 along with the σ – N experimental results.
Fig. 6. Identified natural frequencies for samples S01S09
Fig. 7. Identified damping loss factor for samples S01S09
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Fig. 8. σ – N diagram and corresponding Wöhler curve for harmonic excitation
3 EXPERIMENT WITH RANDOM EXCITATION AND VIBRATION FATIGUE ANALYSIS In this section, the second experiment is presented, where the Yshaped specimens are excited with the random acceleration profile. Here, a constant random acceleration profile is maintained at the fixation of the specimen during the whole vibration test until the fatigue failure occurs. In this way the conditions during a standard vibration test [20] are reproduced. Firstly the experimental setup and results are presented. Secondly, a numerical model for the fatigue life estimation is introduced. Finally, the comparison between experimental and predicted fatigue lives of specimens is given. 3.1 Experiment Setup and Results Experimental setup of the fatigue test with the random baseexcitation is identical to the setup, illustrated in Fig. 4, except no strain gages were used to monitor stress levels in the fatigue zone. The test was performed on the LSD V875 shaker, the acceleration profiles had constant level of the acceleration PSD in
the frequency band of 600 to 800 Hz. The selection of a narrow band random profile in the vicinity of a single natural frequency greatly reduces the influence that the remaining modes have on system’s dynamic response and damage accumulation. However, the frequency band of excitation should be wide enough to ensure the appropriate excitation of the observed modeshape even when the drop of natural frequency occurs. The excitation of the specimen was controlled with a Dactron Laser controller. Ten Yshaped specimens S11 to S20 were tested: at the start of each test the initial transmissibility was measured with very low acceleration profile to obtain initial damping, afterwards the fatigue test was performed. During the fatigue test only the transmissibility magnitude between response and control accelerometer (Fig. 4) was measured from which the changes of natural frequency with time were identified. The results of the experiment are summarized in Table 2. Table 2. Experimental data of randomexcitation fatigue test Specimen S11 S12 S13 S14 S15 S16 S17 S18 S19 S20
PSD level Time to failure ×102 [s] [(m/s2)2/Hz] 0.725 0.725 1.08 1.08 1.60 1.60 2.38 2.38 5.25 5.25
391 316 65.0 98.7 27.4 18.8 13.6 15.2 3.98 3.84
f1 init [Hz] 754 749 752 755 749 750 745 758 754 754
η1 init ×104 4.07 5.07 4.31 4.64 3.80 7.10 5.39 4.59 4.87 5.77
Δf1
[Hz] 71.7 55.6 140.4 43.7 139.7 60.8 75.6 114.3 75.9 62.6
The natural frequency shifts during the randomexcitation fatigue tests are shown in Fig. 9. From it one can observe, that significantly larger natural frequency shifts occur during the random excitation when compared to the harmonic test. During the
Fig. 9. Natural frequency drop during the randomexcitation test Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads
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harmonic test a drop of natural frequency occurred just before the crack propagated through the whole crosssection. For that case the realtime control was not sufficiently reliable and the fatigue failure was declared at this point. On the other hand, when the excitation is a random process, the fatigue crack propagates continuously until the full crosssection crack is reached. Since many authors [5] and [21] assume that frequency shift relates to the accumulated fatigue damage the fatigue failure of the specimen under random excitation was declared when a certain value of the natural frequency shift was observed. The critical natural frequency shift for the final fatigue failure at the randomexcitation test was 23.9 Hz, i.e. the average value of the total frequency shift, measured for the specimens at the harmonic tests (see Table 1). 3.2 Numerical Model The vibration fatigue life of the single tested specimen was obtained in the customly developed Vibrationfatigue plugin for the Catia analysis environment [22]. The FEM model consisted of 15600 10node tetrahedral solid elements. The size of the elements in the fatigue zone was approximately 1.5 mm, which provided reliable results of stress responses used for the fatigue life estimation. As shown in [9], the measured and numerically obtained stress distributions in the fatigue zone were in a good agreement. For the fatigue life estimation the material parameters from harmonic test along with the calculated nodal values of equivalent stress from modal analysis were used. Additionally, the initial damping loss factors (Table 2) were used to predict the Sσ (ω ) , defined in Eq. (10). eq In Fig. 10 the fatigue life estimation for specimen S12 is visualized. In order to reduce the calculation time, the fatigue life calculation was performed for the nodes in the fatigue zone, only. The comparison between the estimated and actual fatigue lives of the specimens for the case of random excitation is shown in Fig. 11. From this figure one can notice, that the numerical analysis estimated reasonably shorter fatigue lives compared to the actual fatigue lives. This deviation originates from the damping loss factor, proposed in the numerical calculation, which was measured for the intact specimen at a low excitation level. From the discussion regarding the results of the harmonic test the damping loss factor increases with the accumulated fatigue damage and with the higher excitation level (due to the type of specimen fixation). In any case, the fatigue 346
life estimation gives conservative prediction when the initial damping loss factor is used.
Fig. 10. Expected fatigue life for specimen S12
3.3 Comparison of the Experimental and Numerical Results Additionally, for the tested series of specimens, the effective damping loss factor was obtained with numerical minimization that gives a minimal sum of deviations between actual Tact and estimated Test fatigue lives:
(
∆T (η ) = ∑ log10 (Tact ,i ) − log10 (Test ,i (η )) ) . (15)
i
2
The numerical optimization was performed with the simplex method. The value of effective damping loss factor was identified as 9.594×10–4. The comparison between experiment and fatigue life estimation using the updated value of damping loss factor is shown in Fig. 12.
Fig. 11. Comparison of random excitation experiment with numerical prediction using the initial damping loss factors
From the presented comparison one can conclude, that the damping loss factor greatly influences the
Česnik, M. – Slavič, J.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 339348
fatigue life estimation. Observing the specimen S11 (marked with ▲ in Figs. 11 and 12): if the value of damping loss factor is increased by factor of 2.35 the predicted fatigue life increases by a factor of 11.3.
Fig. 12. Comparison of random excitation experiment with numerical prediction using the updated damping loss factor
The presented numerical and experimental results clearly show that uncertainties of the fatiguelife estimation rise from the inherent changes of system’s modal properties during the damage accumulation and consequential crack propagation. In order to bring the numerical fatiguelife estimation closer to the reallife vibration fatigue phenomenon, the need for further and more comprehensive research is obvious. Due to this reason, the research presented here serves as a basis for future research work. 4 CONCLUSIONS In this study an accurate examination of the vibration fatigue phenomenon is performed. The vibrational fatigue was for the case of Yshaped specimens experimentally performed with a harmonic and random baseexcitation. The comparison between experimental and numerical results was also given. Several conclusions can be drawn, which are stated in the following paragraphs. The presented experiment with the harmonic excitation has several advantages. Firstly, with the described experimental methodology it is possible to perform a fast fatigue test with a simultaneous tracking of the changes of the modal parameters. Additionally, an indirect measurement of the stress load, based only on measurements of the excitation and response acceleration, greatly shortens the specimen preparation time and additionally improves the reliability of the stress measurement during
the whole accelerated fatigue test compared to the traditional straingauge method. In its present state, the experimental setup is used for the rapid acquisition of a material’s fatigue parameters, since a highcycle accelerated fatigue test of 2×107 load cycles is achieved in approximately 7 hours. The fatigue test with random excitation diminishes the influence of the natural frequency shift on the response amplitude, as long as the acceleration profile with constant acceleration PSD is applied. On the other hand, the effect of the damping increase, which occurs with accumulated damage and at higher excitation levels, is present also in the randomexcitation testing. By comparing the experimental and numerical results the damping increase assures the conservative fatigue life prediction. Therefore, by adopting the initial damping loss factor, obtained with experimental modal analysis at low excitation levels, the fatigue life estimation is always on the conservative side. 5 REFERENCES [1] Lee, Y.L., Pan, J., Hathaway, R.B., Barkey, M.E. (2005). Fatigue Testing and Analysis: Theory and Practice. Elsevier, Oxford. [2] Kranjc, T., Slavič, J., Boltežar, M. (2013). The mass normalization of the displacement and strain mode shapes in a strain experimental modal analysis using the masschange strategy. Journal of Sound and Vibration, vol. 332, no. 26, p. 69686981, DOI:10.1016/j. jsv.2013.08.015. [3] Maia, N.M.M., Silva, J.M.M. (1997). Theoretical and Experimental Modal Analysis. Research Studies Press Ltd., Baldock, Hertfordshire. [4] Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W. (1996). Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review, Technical Report. Los Alamos, New Mexico. [5] Dirlik, T. (1985). Application of Computers in Fatigue Analysis. PhD thesis, University of Warwick, Warwick. [6] Béliveau, J.G., Vigneron, F.R., Soucy, Y., Draisey, S. (1985). Modal parameter estimation from base excitation. Journal of Sound and Vibration, vol. 107, no. 3, p. 435449, DOI:10.1016/S0022460X(86)801171. [7] Shang, D.G., Barkey, M.E., Wang, Y., Lim, T.C. (2003). Effect of fatigue damage on the dynamic response frequency of spotwelded joints. International Journal of Fatigue, vol. 25, no. 4, p. 311316, DOI:10.1016/ S01421123(02)001408. [8] Colakoglu, M., Jerina, K.L. (2003). Material damping in 6061T6511 aluminium to assess fatigue damage. Fatigue and Fracture of Engineering Materials and
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Structures, vol. 26, no. 1, p. 7984, DOI:10.1046/ j.14602695.2003.00603.x. [9] Čermelj, P., Boltežar, M. (2006). Modelling localised nonlinearities using the harmonic nonlinear super model. Journal of Sound and Vibration, vol. 298, no. 45, p. 10991112, DOI:10.1016/j.jsv.2006.06.042. [10] Česnik, M., Slavič, J., Boltežar, M. (2012). Uninterrupted and accelerated vibrational fatigue testing with simultaneous monitoring of the natural frequency and damping. Journal of Sound and Vibration, vol. 331, no. 24, p. 53705382, DOI:10.1016/j.jsv.2012.06.022. [11] Benasciutti, D. (2004). Fatigue Analysis of Random Loadings. PhD thesis, University of Ferrara, Ferrara. [12] Pagnacco, E., Lambert, S., Khalij, L., Rade, D.A. (2012). Design optimisation of linear structures subjected to dynamic random loads with respect to fatigue life. International Journal of Fatigue, vol. 43, p. 168177, DOI:10.1016/j.ijfatigue.2012.04.001. [13] Paulus, M., Dasgupta, A. (2012). Semiempirical life model of a cantilevered beam subject to random vibration. International Journal of Fatigue, vol. 45, p. 8290, DOI:10.1016/j.ijfatigue.2012.06.008. [14] Han, S.H., An, D.G., Kwak, S.J., Kang, K.W. (2013). Vibration fatigue analysis for multipoint spotwelded joints based on frequency response changes due to fatigue damage accumulation. International Journal of Fatigue, vol. 48, p. 170177, DOI:10.1016/j. ijfatigue.2012.10.017. [15] Chen, Y.S., Wang, C.S., Yang, Y.J. (2008). Combining vibration test with finite element analysis for the fatigue life estimation of PBGA components. Microelectronics
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Reliability, vol. 48, no. 4, p. 638644, DOI:10.1016/j. microrel.2007.11.006. [16] Yu, D., AlYafawi, A., Nguyen, T.T., Park, S., Chung, S. (2011). Highcycle fatigue life prediction for Pbfree BGA under random vibration loading. Microelectronics Reliability, vol. 51, no. 3, p. 649656, DOI:10.1016/j. microrel.2010.10.003. [17] Ewins, D.J. (2000). Modal Testing: Theory, Practice and Application. Research Studies Press Ltd., Baldock, Hertfordshire. [18] Mršnik, M., Slavič, J., Boltežar, M. (2013). Frequencydomain methods for a vibrationfatiguelife estimation  Application to real data. International Journal of Fatigue, vol. 47, p. 817, DOI:10.1016/j. ijfatigue.2012.07.005. [19] Pitoiset, X., Preumont, A. (2000). Spectral methods for multiaxial random fatigue analysis of metallic structures. International Journal of Fatigue, vol. 22, no. 7, p. 541550, DOI:10.1016/S01421123(00)000384. [20] US Department of the Air Force (2008). MILSTD810, Military standard environmental test methods for aerospace and ground equipment, revision G, Method 514.6: Vibration, United States Department of Defense, Arlington. [21] Wang, R.J., Shang, D.G. (2009). Fatigue life prediction based on natural frequency changes for spot welds under random loading. International Journal of Fatigue, vol. 31, no. 2, p. 361366, DOI:10.1016/j.ijfatigue. 2008.08.001. [22] Dasault systemes (2014). from http://www.3ds.com/ productsservices/catia, accessed on 2014 0409.
Česnik, M. – Slavič, J.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 349362 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1836 Special Issue, Original Scientific Paper
Received for review: 20131130 Received revised form: 20140214 Accepted for publication: 20140401
Reflection and Transmission Coefficients from Rectangular Notches in Pipes Stoyko, D.K. – Popplewell, N. – Shah, A.H. Darryl K. Stoyko1,2,* – Neil Popplewell2 – Arvind H. Shah3
2 University
1 Stress Engineering Services Canada of Manitoba, Department of Mechanical and Manufacturing Engineering, Canada 3 University of Manitoba, Department of Civil Engineering, Canada
The use of a single, nondispersive ultrasonic guided wave mode is one important approach to monitoring a structure’s health. It is advantageously nondestructive with the ability of propagating over tens of metres to detect a hidden defect. The dimensional assessment of a defect, on the other hand, requires reflection coefficients for two or more such modes. Multiple modes may be excited simultaneously by applying a short pulse to a structure’s external surface. This situation is examined here for a circular, hollow and homogeneous, isotropic pipe having negligible damping and an open rectangular notch. A finite element model is employed in a region around a notch. It is coupled to a wave function expansion in the two adjacent, effectively, semiinfinite pipes. Representative longitudinal and flexural modes are investigated for different notch dimensions. A nonaxisymmetric notch, unlike an axisymmetric notch, introduces a plethora of cross modal couplings that lead to more singularities in a reflection coefficient’s frequency dependence. There is, however, a common pattern to these distinctive singularities. It is conjectured that singularities corresponding to propagating modes may enable a notch to be detected and its dimensions determined. Keywords: pipe, notch, cutoff frequency, singularity, guided waves
0 INTRODUCTION Pipelines are used ubiquitously to transport fluids. For example, the Alberta Energy and Utilities Board (EUB) reports that a total of 377,248 km of energy related pipeline was under its jurisdiction at the end of 2005. The same source also indicates there were a total of 12,848 pipeline “incidents” between 1990 and 2005. About 95% of the reported incidents led to a pipeline leak or rupture. Hence, it is clear that extensive networks of pipelines are in widespread use and they are prone to occasional failure. Clearly a method of inspecting pipelines is required to detect and size defects and, ideally, it should be noninvasive. Guided waves are appealing because they can propagate over long distances, say tens of metres, and they are capable of rapidly interrogating entire structures, including otherwise inaccessible regions. A thorough literature review of guided wave inspection of pipes may be found in [1], so only references pertinent to the present work are given here. Early attempts of using guided waves for pipe inspection focussed on the torsional and longitudinal wave modes and considered spurious reflections as an indication of damage. More recent work has focussed also on reflections of axisymmetric pipe modes from defects. The use of flexural waves has been infrequent because “the acoustic field is much more complicated than the case of axisymmetric modes” [2]. On the other hand, the identification of spatially decaying modes, which
are introduced in a pipe by a notch and are analogous to end modes [3], has not been reported. The first objective of this paper is to demonstrate that singularities, where the term singularity is used to indicate a frequency at which a displacement response of a given guided wave mode becomes very large and behaves similarly to a resonant frequency of an undamped single degree of freedom oscillator, distinct from an unblemished pipe’s cutoff frequencies are present when a notch is introduced. These singularities are analogous, in some sense, to the end modes reported in [3]. A second objective is to describe a technique that takes advantage of these singularities to characterise the dimensions of an axisymmetric notch in a pipe. The last objective is to suggest that the extension to nonaxisymmetric and more general notch geometries is straightforward but computationally expensive. The proposed technique to detect and characterise the dimensions of an axisymmetric notch has a number of advantages. It utilizes the classical and simplest means, i.e., a radial point load acting on the pipe’s outer surface, of introducing ultrasonic energy into a pipe to simultaneously excite a number of modes. The notch may be detected by simply examining the spectral density or reflection coefficient of the response. As frequency differences are used to infer a notch’s dimensions, the need for consistent transducer coupling is reduced somewhat compared to methods that make use of amplitude changes. A single measurement can yield sufficient information to
*Corr. Author’s Address: Stress Engineering Services Canada,: #125, 12111–40th Street S.E., Calgary, Alberta, T2Z 4E6, Canada, Darryl.Stoyko@stress.com
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determine a notch’s dimensions. This attribute offers an advantage over methods that rely on the excitation of a single mode to determine a reflection coefficient, say, as at least two modes are required to uniquely determine even an axisymmetric notch’s dimensions [4]. However, because the proposed method makes use of measurements relatively close to a notch, where waves incident and scattered by the notch may interact (in the reflected field), only modest lengths of pipe are required. Consequently, the procedure can be applied only locally to a notch as waves are used whose amplitudes decay exponentially from a notch’s boundary. The technique complements, therefore, other methods that can rapidly screen long lengths of pipe. The newly discerned singularities are demonstrated to exist first by applying the hybrid semianalytical finite element (SAFE) in combination with a standard finite element procedure to axisymmetric notches in a (hollow) steel pipe. The pipe is assumed to be homogeneous, linearly elastic, isotropic, and uniformly right circular. A parametric study is undertaken subsequently in which the radial depth and axial extent of outer surface breaking, rectangular axisymmetric notches are varied independently. The results are used to illustrate how the frequency differences between the new singularities and an unblemished pipe’s cutoff frequencies can be used to detect a notch and determine its size. Solely outer surface breaking notches are considered because the simulations can be partially corroborated by existing experimental data [4]. The examples suggest that almost any set, which contains a sufficient number of accurate frequency differences, will give the same inverse solution. The modes could be selected generally but they are selected usually on the basis of ease of experimental implementation. The extension to nonaxisymmetric notches is suggested by showing that the singularities still exist and characteristic behaviours can be generalized. 1 HYBRID SAFE AND STANDARD FINITE ELEMENT PROCEDURE 1.1 Overview of Hybrid Wave FunctionStandard Finite Element Approach A hybrid wave functionstandard finite element approach employs a conventional finite element description to model the displacement field in a region completely enclosing a nonhomogeneity. The displacement field in the remainder of the waveguide is described in terms of a modal expansion of the 350
“parent” waveguide’s wave functions. (References pertinent to pipes are given in [1].) An incident wave field is generated in the parent waveguide. Waves are scattered by the nonhomogeneity and the corresponding (scattered) wave field is obtained by enforcing continuity (displacements and stresses/ nodal forces) between the finite element and wave function expansion regions. Fig. 1a shows standard orthographic views of a pipe having a nonaxisymmetric, outer surface breaking notch. The nonhomogeneity in this case is the notch. It is bounded by the planes z = 0 and z = –2zFE , which demarcate the axial extents of the finite element region. An approximate wave function expansion is used outside this region. The incident wave field is generated by the transient input excitation, shown as a radial point force in the figure, applied in the z = zL plane. The combined incident and reflected (transmitted) wave field in the parent wave guide corresponds to z ≥ 0 (z ≤ –2zFE), in the configuration shown. Note that the notched pipe shown is symmetric about the plane z = –zFE which corresponds to the finite element boundary B−. Computationally advantageous use is made of this symmetry by decomposing the input excitation into the sum of a pair of forces that are symmetric and antisymmetric about the plane z = –zFE. Then boundary conditions appropriate for symmetric and antisymmetric loadings can be applied to the finite element boundary B−. Moreover, the displacement field in the waveguide need be computed only for z ≥ 0, i.e., the reflected field, because the wave field in the transmitted field can be obtained from the reflected field by applying symmetric and antisymmetric arguments. Note that symmetry is not required to employ the hybrid wave functionstandard finite element technique. Geometries that do not possess a plane of symmetry can be accommodated by enforcing continuity conditions between the finite element region and wave guide on two cross sections. The computational effort is increased, however. Three components are required to apply the hybrid wave functionstandard finite element technique. They are: i) the wave functions of the parent waveguide, ii) a finite element description of the region enclosing the defect, and iii) a method of enforcing continuity conditions between the first two components. Each component is described briefly now. 1.2 SAFE Modelling for Pipes The SAFE formulation, detailed exhaustively in [1], provides an easily applied and accurate numerical
Stoyko, D.K. – Popplewell, N. – Shah, A.H.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 349362
Fig. 1. Illustrating; a) a nonaxisymmetrically notched pipe and b) the finite element nodal points on boundary B+ which is located at z = 0; point “O” is the origin of the cylindrical coordinate system
model by which the Green’s and wave functions of a pipe may be computed for a harmonic excitation having circular frequency ω. The frequency ω may be chosen arbitrarily so that an arbitrarily fine frequency resolution may be achieved computationally. This methodology is adapted here to compute the wave functions of a homogeneous, isotropic pipe subjected to a transient excitation. The transient excitation is decomposed into an infinite number of discrete frequency components by using a Fourier transform. Each frequency component of a point force is approximated by using a “narrow” pulse having a uniform amplitude of (2r0θ0)−1 over a circumferential distance 2r0θ0 to circumvent convergence difficulties. The r0 is the radial coordinate where the pointlike force is applied, while θ0 is the angle over which the pulse acts. This narrow pulse is represented, in turn, by employing a Fourier series, in the circumferential direction, of “ringlike” loads having separable spatial and time, t,variations. The pipe is discretized by using N layers through its thickness. (The layers are taken to have identical thicknesses here.) Each layer corresponds to a onedimensional finite element in the pipe’s radial direction for which a quadratic displacement interpolation function is assumed. A finite element approach is applied, layer by layer, in SAFE to
approximate the elastic equations of motion. The displacement is represented, like the excitation, by a Fourier series in the circumferential coordinate. The Fourier series describing the excitation and displacement are substituted into approximate equations of motion obtained from Hamilton’s principle. The result is transformed into the wavenumber domain by applying the Fourier integral transform. Then the nth circumferential harmonic (wave number) takes the form of a quadratic eigensystem which is linearized for the special case when no excitation is applied. Note that the circumferential wave numbers of a right circular pipe can take only integer values due to the requirement that the displacement field should be single valued. The resulting eigenvalues, knm, and (right) eigenvectors, R , of this eigensystem are the approximate axial φnm wave numbers and modes shapes through the thickness, respectively, for the nth circumferential harmonic. The index m is an integer value that is used to indicate the mth axial mode corresponding to the nth circumferential wave number. Modes are labelled using the standard convention described in [6]. Both the displacement (response) and excitation, for the nth circumferential harmonic, are expanded into a series of the normal modes (wave functions) of the linearized eigensystem. A displacement is obtained, for the nth
Reflection and Transmission Coefficients from Rectangular Notches in Pipes
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circumferential wavenumber and those axial cross sections having positive or negative z, by linearly superimposing the appropriate admissible 6N+3 right eigenvector solutions. Applying first the inverse Fourier transform to this sum, and then Cauchy’s residue theorem, produces the nth circumferential mode of the displacement. A linear superposition of the circumferential harmonics gives the displacement for a harmonic component of the excitation. The total displacement produced by a multifrequency excitation is found by linearly superimposing the displacements caused by each individual frequency component. See [1] for further details. 1.3 Finite Element Idealization The finite element method is a well understood tool that is in common use. Therefore, only an outline pertinent to implementing the hybrid wave functionstandard finite element technique is provided. Further pertinent details may be found in [1]. Hamilton’s principle is first applied straightforwardly in the finite element region immediately surrounding the notch. To represent a notch, elements are simply removed from the finite element mesh. See Fig. 1b. It is well understood that singularities in the stress field that occur at the notch’s corners are not described accurately by this method. However, the far field behaviour is modelled with sufficient accuracy to be meaningful. The equations of motion of the finite element region are partitioned first such that components related to the interior nodes, where no external forces are applied, can be condensed out so that only quantities on the boundaries B+ and B− remain. Then advantage is taken of the symmetry of the notched pipe about the plane z = –zFE in Fig. 1a. For (anti) symmetric loading, the displacement in the (r and θ) z direction(s), as well stress (σzz) σrz and σθz and corresponding nodal forces vanish on B−. The zero displacements on B− are condensed out and the unknown reaction forces are ignored (as they are not presently of interest). Equations are produced that contain a known dynamic stiffness matrix and, as yet, unknown finite element nodal displacements and forces on boundary B+. These unknown displacements and forces are written in terms of the unblemished pipe’s wave functions for two cases. The first case is when the notch is axisymmetric, i.e., c in Fig. 1a is 360° (2π); the second, nonaxisymmetric case is when c is less than 360° (2π). Note that a notch may have a depth, d, which is zero and represents an unblemished pipe. 352
This important case is considered in the transparency check discussed later. 1.4 Interface between the Wave Function and Finite Element Regions A single incident wave mode of unit magnitude that is incident on the plane z = 0 is considered for simplicity. The scattering caused by an arbitrary incident wave field may be constructed by appropriately scaling and superimposing the scattered wave fields calculated for all the modes present in the incident field. For the sake of discussion, let the incident wave be time harmonic with circular frequency ω and have a circumferential wave number nin, with an axial wave number knin min . Note that only modes having a nonpositive imaginary component to their axial wave number are admissible in the incident field for the configuration shown in Fig. 1a. This restriction is due to the radiation condition that requires the displacement field to remain bounded at z = ±∞. It is appropriate to use axisymmetric elements in the finite element region when the notch is axisymmetric. Then the finite element nodal points on the boundary B+, as shown in Fig. 1b, lie on the single radial line θ = 0. Note that, because the parent waveguide and the finite element region are both axisymmetric, the circumferential wave number of the scattered waves is required to be identical to that of the incident waves. See, for example, [7]. The finite element region is chosen in the axisymmetric case such that its axial boundaries correspond to those of the notch, i.e., the finite element region is bounded by the planes z = –l and z = 0. The plane of symmetry is then z = –l / 2, and –2zFE = l. This choice simultaneously simplifies the computations and reduces the number of finite elements used in the idealization. Using modal superposition, the displacements at z = 0 of the reflected wave field caused by the incident wave can be written at the finite element nodal points along the radial line θ = 0 in the form:
qs+ =
6 N +3
∑A
ms =1
R ns ms ns ms
φ
, (1)
where the sub, subscript s indicates the scattered wave field, ns = nin, Ans ms and φnRs ms are the amplitude and mode shape of the ns msth scattered mode. (Note that only modes having nonnegative, imaginary wave number components are admissible in the reflected wave field.) Continuity constraints between the wave
Stoyko, D.K. – Popplewell, N. – Shah, A.H.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 349362
function and finite element regions are applied on the nodal forces and displacements on boundary B+. The (consistent) force vector at the points along the radial line θ = 0 can be obtained, for the incident and reflected wave fields by expressing the modal stresses in terms of the displacement wave functions and integrating the product of the stresses and the finite element shape functions over the surface of each finite element. Amplitudes of the scattered waves are found straightforwardly from an invertible matrix equation that results from algebraically rearranging the equations of motion that result from applying the continuity constraints. The matrix equation is invertible because all the approximate modes are retained in the modal expansion and the number of modes is identical to the number of constraint equations. This equation must be evaluated for both the symmetric and antisymmetric components of the load, by changing the boundary conditions on B−, in order to recover their combined effects. At any location, the reflected (z ≥ 0) and transmitted (z ≤ –2zFE) wave amplitudes for the ns msth scattered mode, Rns ms and Tns ms respectively, due to a given incident mode, are given by
(
)
(
)
Rns ms = Anss ms + Anas ms / 2, (2)
and
Tns ms = Anss ms − Anas ms / 2.
(3)
The Ans ms is the ns msth scattered wave amplitude and superscript (a) s denotes the solution corresponding to the (anti) symmetric boundary conditions. Rns ms and Tns ms represent normalized reflection and transmission coefficients, respectively, because they are calculated by assuming a single incident mode having a unit amplitude. Note that the magnitudes of Rns ms and Tns ms depend on the scaling of the mode shapes; all mode shapes are scaled here to have a vector norm magnitude of unity. Moreover, the Rns ms and Tns ms represent the amplitudes of the scattered waves at the planes z = 0 and z ≤ –2zFE, respectively. The procedure for the nonaxisymmetric notch is essentially identical to before but with two major differences. First, all the circumferential wave numbers used in the modal expansion participate, in principle, in the reflected displacement field even though a single incident wave is assumed. Therefore, threedimensional elements are required now in the finite element region and clearly finite element nodal points on the boundary B+ have to be arranged around this entire boundary, as shown in Fig. 1b. Consequently
Eq. (1) becomes:
qs+ =
nmax
6 N +3
∑ ∑A
ns = nmin ms =1
R ns ms ns ms
φ
exp( jnsθ ), (4)
where nmin and nmax are the minimum and maximum circumferential wave numbers, respectively, employed in the modal expansion of the reflected field. The second difference is that, unlike before, the number of constraint equations resulting from applying the continuity conditions on boundary B+ does not generally equal the number of modes in the modal expansion so the system is not immediately invertible. Application of the principle of virtual work is applied to develop a system of equations from which the scattered waves’ amplitudes are recovered straightforwardly. Then no further modifications are necessary. Further details are available in [1]. 2 ILLUSTRATIVE EXAMPLES 2.1 Overview Having provided an overview of the hybridSAFE technique, it is applied in this section to wave scattering from two illustrative rectangular notches in an otherwise blemish free pipe. The first notch is axisymmetric, while the second is nonaxisymmetric. Both notches are outer surface breaking and they have finite radial depths and axial extents. (Note that the accuracy of the software has been checked by applying transparency and energy balance considerations [1].) The properties assumed for the unblemished pipe and excitation pulse are described first. Then an overview is given of the SAFE analysis to recover approximate wave functions for the pipe. Finally results are given for the wave scattering from the two illustrative notches. 2.2 Unblemished Pipe’s Description The unblemished pipe, whose properties are summarized in Table 1, is assumed to be uniform, hollow, right circular, homogeneous, and isotropic. These properties are representative of an unblemished, seamless, Schedule 40, 80 mm diamètre nominal (DN), steel pipe. Moreover, the properties are essentially identical to the pipe examined experimentally in [5]. Where possible, (selected) direct comparisons between the experimental data given in [5] and the simulations presented here are also given.
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2.3 Excitation’s Form The function, p(t), which describes the temporal variation of the applied force is idealized as the commonly used Gaussian modulated sine wave that has the form: t<0 0, p (t ) = 2 A exp −a ( st − τ ) sin( sω0t ), t ≥ 0, (5) where A is an amplitude, a determines the rate of decay of the pulse, s serves to “scale” time, τ centres the pulse in time, t, and ω0 sets the centre frequency of the sine wave. The constant a, s, τ, and ω0 are taken invariably to be: a = 2.29595 × 1010 s −2 ; s = 0.28;
τ = 1.4 × 10−5 s; ω0 = (5 × 105 )π rad / s. (6)
Moreover, the (body force) amplitude is always A = (μ / H) and all nondimensionalized (body) forces are given with respect to (μ / H). (In the remainder of the text, a quantity embellished with a superscript asterisk indicates that it has been nondimensionalized.) The pulse is smooth (i.e., differentiable) in both time and frequency and, with the chosen constants, has a 70 kHz centre frequency and over 99% of its energy is contained within the 35 to 107 kHz bandwidth. Therefore the Fourier integral transform of p(t),
p(ω ) , may be assumed reasonably to be contained within this finite bandwidth. The resulting forms of p(t) and  p(ω ), are illustrated in Fig. 2. Note that this excitation has been successfully employed experimentally, as in, for example, [1], [5], and [8]. Table 1. Properties assigned to the unblemished pipe Property Outer diameter, Do, [mm]
88.8
Wall thickness, H, [mm]
Mean radius, R, [mm]
Thickness to mean radius ratio, (H
Young’s modulus, E, [GPa]
Lamé constant (Shear modulus), μ Lamé constant, λ, [GPa] Ratio of Lamé constants, (λ / μ) Poisson’s ratio, ν
5.59 41.60
/ R)
0.134 216.9
(G), [GPa]
84.3 113.2 1.34 0.286
2.4 Approximate Wave Functions from SAFE In determining approximate wave functions by using SAFE, ten identically thick finite elements are used to uniformly discretize the pipe’s wall thickness, H. The circumferential angle, 2θ0, over which the spatial pulse approximates the Dirac delta function, is taken to be 0.002 radians (0.1°). Circumferential wavenumbers n, from 0 to ±16, and all the 6N + 3
Fig. 2. Applied excitation in a) time and b) frequency
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Assigned value 7932
Density, ρ, [kg m–3]
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corresponding axial modes are incorporated into the wave scattering computations. Numerical and experimental investigations of the unblemished pipe’s displacement response may be found in [1] and [8]. 2.5 Axisymmetric Notch The axisymmetric notch has the dimensional properties summarized in Table 2. Eight node, quadratic axisymmetric finite elements [7] are utilized for the finite element region around the notch. A uniform idealization is selected after successfully checking the convergence of representative reflection coefficients [1]. Ultimately, ten (five) finite elements describe the behaviour over the wall thickness in the wave function (finite element) region. Furthermore four finite elements, which together correspond to half the notch’s axial extent, are utilized axially. This selection allows longer axial notches to be represented without the need for additional axial finite elements. Note that the smallest propagating wavelength over the excitation’s bandwidth is about 3.4H, and belongs to the L(0,1) mode. Consequently, the ratio of the smallest (axial) wavelength excited to a finite element’s axial length is approximately 48, almost five times larger than the (minimum) recommended guideline given by [5] of ten elements per shortest wavelength. Table 2. Dimensions of the outer surface breaking, axisymmetric notch Property
Assigned value 2.79
Depth of notch, d, [mm]
Axial length of notch, l, [mm]
Depth to wall thickness ratio, (d
3.17
/ H) / H)
Axial length to wall thickness ratio, (l
0.500 0.568
The transient pointlike force is applied radially to the simulated pipe’s outer surface at zL* = (zL / H) = 5.1, where zL is the transmitting transducer’s axial coordinate. (Note that angles are measured relative to the idealized force’s central point of application.) The resulting radial displacement is computed on the pipe’s outer surface at θR = 0, i.e., a pure axial offset from the point load, and zR* = (zR / H) = 10.2, where zR is the axial coordinate of the receiving transducer located in the reflected field. (The transmitted and reflected fields give similar information so the former is omitted.) All pertinent positions are shown in Fig. 1. The (approximate) wave functions for the unblemished pipe were determined by using SAFE. Then the hybridSAFE technique was applied on a
mode by mode and frequency by frequency basis. A modal superposition was applied at each frequency and the inverse Fourier transform was approximated using a numerical integration scheme to recover time histories from the approximate spectral densities. Fig. 3 shows typical displacement responses predicted in time and frequency for a pipe having no notch and a pipe having the outer surface breaking, axisymmetric notch. The displacement responses are evaluated on the pipe’s outer surface where θR = 0 and zR* = (zR / H) = 10.2. A cursory examination of Fig. 3 shows that the incident and reflected waves interact and cannot be separated in time. However, the notch’s presence is discerned easily from a comparison of the corresponding spectral densities. This is because each predominant peak in the spectral density of Fig. 3b “splits” into two local maxima in Fig. 3d, one on either side of the original peak. The sharper maximum at the lower frequency also has a much larger amplitude so, for convenience, it is termed a “singularity.” Differences between the frequencies of such singularities and the nearest cutoff behaviour of the unblemished pipe, having the frequencies shown in Fig. 3b are presented in Table 3. Table 3. Frequencies that correspond to the readily identified singularities appearing in Fig. 3d; they are distinct from the unblemished pipe’s cutoff frequencies Frequency of singularity [kHz]
Difference between cutoff and singularity frequency [kHz]
m =1
43. 112
0.085
3 2 1 1 1 1 1
46.462 49.623 52.983 63.362 74.153 85.274 96.660
0.001 0.294 0.129 0.192 0.273 0.377 0.503
Circumferential wavenumber
Axial order
n±
8 ± 2 ± 4 ± 9 ±10 ±11 ±12 ±13
To help explain the frequency dependent behaviour of the notched pipe in the neighbourhood of the unblemished pipe’s cutoff frequencies, Fig. 4 shows other normalized reflection coefficients predicted for the flexural F(n,1) modes, where n equals 8 through 13, for an axisymmetric notch having (d / H) = 0.5 and (l / H) = 0.568. Each curve represents a single F(n,1) mode which is reflected into itself. (Modal conversions from F(n,1) into F(n,m), m ≠ 1, are not given for brevity. Such conversions are required to satisfy continuity and boundary conditions.) For easier comparisons, the frequency
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Fig. 3. Radial displacement predicted on the pipe’s outer surface within the reflected field where θR = 0 and zR* = (zR / H) = 10.2 for an axisymmetric notch having (d / H) = 0.5 and (l / H) = 0.568; a) and b) direct waves produced by a radial point force; c) and d) superposition of direct and reflected waves produced by a radial point force
Fig. 4. Normalized reflection coefficient caused by flexural F(n,1) modes, where n is 8 through 13 inclusive, and an axisymmetric notch having (d / H) = 0.5 and (l / H) = 0.568; the f Fc( n ,1) is the cutoff frequency of the unblemished pipe’s F(n,1) mode
axis in Fig. 4 has been normalized by the cutoff frequency of the mode in question and expanded to lower frequency ratios. It is noteworthy now that 356
each curve can be seen to possess two singularities. The singularity near a normalized frequency of 1.0 occurs, as before, at a frequency just below the
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unblemished pipe’s relevant cutoff frequency. This observation can be corroborated by noting that the normalized reflection coefficients always pass through the point (1.0,1.0) for these modes. On the other hand, the uprise around the lower normalized frequency of 0.7 always corresponds to a mode transitioning from evanescent to nonpropagating. The practical usefulness of this uprise, however, may be limited. Waves scattered from the axisymmetric notch at frequencies near this uprise decay exponentially from the notch’s vertical boundaries at a rate of about exp(–zd*). The zd* is the distance from the notch’s boundary, nondimensionalized by the unblemished pipe’s thickness, H. The decay rate in the axial direction is determined approximately based on the behaviour of the representative F(10,1) mode’s axial wavenumber found from Fig. 5. The latter figure shows that the imaginary part of the nondimensional axial wavenumber of this mode is almost one when the mode transitions from evanescent to nonpropagating. The quite large exponent suggests that the effect is very localized and likely to be masked by the propagating modes. The singularity just below the cutoff frequency of 63.553 kHz, on the other hand, is more interesting. Its effect is not so localized because the magnitude of the imaginary part of its wavenumber is much closer to zero. Indeed, Fig. 5 suggests that the
axial decay rate is about exp(–0.15zd*) at the notchinduced singularity. As a consequence of the smaller exponent, this last singularity is detectable further from the notch’s vertical boundaries. An analysis of the eigenvalues (resonant frequencies) of solely the finite element region (which can be found in [1]) indicates that the frequency of the possibly more important singularity does not correspond to a resonant frequency of the finite element region alone. It depends presumably upon the properties of both the finite element region and the parent waveguide. Moreover, the last column of Table 3 shows that the difference between the frequency of this singularity and the corresponding unblemished pipe’s cutoff frequency grows continuously as the circumferential wavenumber increases. Advantage might be taken of this trend by increasing the centre frequency of the point force to excite modes having larger circumferential wavenumbers in order to make the frequency differences easier to measure. Having examined the wave scattering from one particular notch, wave scattering by axisymmetric notches having various dimensions is considered. Fig. 6 presents the frequency difference (reduction), Δf, from a nearby cutoff frequency of the unblemished pipe caused by each notch. Results are shown for the F(10,1) mode but data for the F(11,1) and F(12,1)
Fig. 5. Real and imaginary parts of the unblemished pipe’s F(10,1) nondimensional axial wavenumber, k* = k10,1 H, as a function of frequency Reflection and Transmission Coefficients from Rectangular Notches in Pipes
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Fig. 6. Frequency differences, Δf from the unblemished pipe’s F(10,1) cutoff frequency introduced by axisymmetric notches having various dimensions
modes are similar. Frequency reductions can be seen to depend upon an axisymmetric notch’s depth and, to a less degree, its axial length. To determine these
two dimensions, constant frequency differences are projected for each of the F(10,1), F(11,1), and F(12,1) modes onto their common horizontal plane. These projections are superimposed in Fig. 7. Not surprisingly it can be seen that, due to the contours’ “Ushapes,” the depth ratio of a notch for a given axial length ratio, (l / H), and constant frequency reduction, Δf, cannot be found absolutely from any single one of the three modes. Consequently more than one mode has to be employed—a situation which is common to a reflection based procedure [4]. The intersection of the contours of two different flexural modes is usually unique. See, for example, the 200 Hz and 300 Hz contours for the F(11,1) and F(12,1) modes, respectively. The single intersection of the contours provides two coordinates which uniquely define the two dimensions of an axisymmetric notch. Interpolations are obviously needed if a frequency difference does not lie precisely on a contour line.
Fig. 7. Contour maps of constant frequency differences between the singularity produced by an axisymmetric notch and the unblemished pipe’s cutoff frequencies for the F(10,1), F(11,1), and F(12,1) modes; The solid curves correspond to contours of the F(10,1), dashed to F(11,1), and dotted to F(12,1) mode
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An example of this situation is illustrated in Fig. 7 where the frequency differences tabulated in Table 3 for the F(10,1), F(11,1), and F(12,1) modes are indicated. These differences can be used to uniquely characterise the notch’s dimensions. Fig. 7 also suggests that, for a given frequency difference, a flexural mode with a higher circumferential wavenumber has a lower position. Consequently such modes are more sensitive to smaller notches. There are instances, however, when the curves for two different modes intersect more than once. One such example seen in Fig. 7 occurs for the 400 Hz and 600 Hz contours of the F(11,1) and F(12,1) modes, respectively. In this instance the two arrowed distances from the F(10,1) mode’s 200 Hz reference contour may be used to distinguish the two intersections. Then, by interpolating linearly between the 200 and 300 Hz contours of the F(10,1) mode, a frequency difference in the F(10,1) mode of around 225 Hz would suggest a notch having (d / H) ≈ 0.73 and (l / H) ≈ 0.78. On the other hand, a frequency difference of about 260 Hz in the F(10,1) mode would imply a notch with (d / H) ≈ 0.62 and (l / H) ≈ 0.70. Clearly, however, each additional intersection requires knowledge of another mode’s frequency difference to uniquely determine a notch’s dimensions. Furthermore, an excitation such as a point force which simultaneously excites several modes becomes more advantageous as the number of required modes increases. 2.6 Nonaxisymmetric Notch The extension to nonaxisymmetric notches is considered now and the procedure for the axisymmetric notch is essentially followed. The reference nonaxisymmetric notch has dimensions identical to the axisymmetric notch considered earlier (see Table 2) with the exception that its circumferential extent is reduced to onehalf of the unblemished pipe’s circumference. Twentyseven node, brick finite elements using quadratic Lagrange interpolation polynomials in each coordinate direction [7] were employed for the finite element region around the notch. The notch was modelled again by simply removing appropriate finite elements. Consequently the far field behaviour is meaningful. Ultimately, ten identically thick finite elements described the behaviour over a full wall thickness. To reduce computer waiting time, the minimally acceptable two finite elements always represented a notch’s axial extent. However, 126 elements were deployed invariably around the pipe’s unadulterated
circumference. The radial discretization of ten elements through the unblemished pipe’s wall was selected so that the wavefunctions from the previous axisymmetric analysis could be employed. The appropriateness of the minimal axial discretization was checked by simulating the previous axisymmetric notch with the threedimensional software. The two sets of reflection and transmission coefficients were each essentially indistinguishable. The circumferential discretization was determined, after selecting the radial and axial discretizations, by considering the results from transparency tests. The number of finite elements around the unblemished pipe’s circumference was increased gradually until the reflection coefficient was less than 0.01 for all the modes propagating over some part of the excitation’s bandwidth, 35 to 107 kHz [1]. Therefore any reflection coefficient which has a magnitude greater than 0.01 for a propagating mode has an inconsequential error from this modelling component. Not surprisingly, the F(13,1) mode dictated the circumferential discretization as it has the smallest (3.6H) circumferential wavelength of the propagating modes. On the other hand, the propagating L(0,1) mode has a somewhat smaller axial wavelength of around 3.4H. Consequently, the ratio of the smallest axial wavelength of all the propagating modes to a finite element’s axial length was approximately 12—a value which is above the recommended lower bound of ten elements per shortest wavelength [5]. Similarly, the ratio of the F(13,1) mode’s (circumferential) wavelength to a finite element’s circumferential length was virtually 10. Present and previous published [5] reflection coefficients,  RL(0,2),L(0,2) , are compared in Fig. 8 for different nonaxisymmetric notches. (Note that the finite element results in [5] are extrapolated by multiplying each result from a corresponding axisymmetric notch by the percentage ratio of the part circumferential notch length to the pipe’s total circumference.) Solely the reflected L(0,2) component of the incident L(0,2) mode is considered. Note that a 50%, circumferential notch is examined in Fig. 8a, while an 11% circumferential notch is used in in Figs. 8b and d so that direct comparisons can be made to the data given in [5]. Axial extents are varied more comprehensively, however, than before. Agreement is generally reasonable and  RL(0,2),L(0,2)  is seen in Fig. 8d to grow not quite linearly with a notch’s greater axial extent. Fig. 9a is a comparable plot to Fig. 4 but with the F(11,1) mode impinging solely on the nonaxisymmetric rather than the axisymmetric
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notch. The curves labelled F(11,1) in these figure have a similar overall character. Furthermore the frequency resolution in Fig. 9b, which lies an order of magnitude finer than the commonly employed 500 Hz here, appear to produce an additional singularitylike feature. It occurs between 74.30 and 74.35 kHz; values which are again immediately below the 74.426 kHz cutoff frequency of the F(11,1) mode for the unblemished pipe. Consequently the more detailed nature of the reflections from the two notches seems little different when the F(11,1) mode is considered alone. Conversely, Fig. 9a indicates that the nonaxisymmetric, unlike the axisymmetric, notch also converts the lone incident F(11,1) mode into superimposed reflections of principally the F(n,1) modes having values of n just below eleven. On the other hand, a comparison of Figs. 4 and 9a shows that each of the individual modal contributions retains almost all the prominent features observed for the axisymmetric notch. A computationally
intensive frequency resolution, comparable to that used for F(11,1) in Fig. 9b, is still required however around singularities. Then the extension of the local procedure to dimensionalize an axisymmetric notch by using fairly small frequency differences can be explored. Furthermore, as propagating modes are also generated by nonaxisymmetric notches (through cross modal couplings), the possibility of remoter assessments could be also investigated. 3 CONCLUSIONS AND CLOSING REMARKS A hybrid SAFE and standard finite element procedure was applied to detect and characterise an open notch in an infinitely long steel pipe. Axisymmetric notches were considered first. Interactions between incident and scattered guided waves and the axisymmetric notch were shown numerically to change a radial displacement’s temporal history and introduce additional, singularitylike information in the
Fig. 8. Magnitude of the normalized reflection coefficient,  RL(0,2),L(0,2) , for different a) excitation frequencies, b) depths, c) circumferential extents, and d) axial lengths of nonaxisymmetric notches
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Fig. 9. Showing a) normalized reflection coefficients produced by the F(11,1) mode incident on the reference nonaxisymmetric notch with b) the frequency scale expanded about the F(11,1) mode’s cutoff frequency
previously unblemished pipe’s frequency response function (FRF). This information indicated the presence of a nearby notch. Moreover, frequency differences between the “singularities” of the unblemished and blemished pipes were shown to reflect an axisymmetric notch’s dimensions. A procedure by which an axisymmetric notch’s dimensions could be estimated was demonstrated by considering the frequency differences for multiple modes. It is envisioned that, in practice, the experimental and signal processing techniques described in [8] could be utilized to simultaneously excite several modes
and measure the resulting singularity frequencies. The extension to nonaxisymmetric notches was suggested. The frequency differences are seen to grow with a larger circumferential wavenumber. Advantageous use might be made of this property by increasing the centre frequency of the excitation in order to excite modes with higher circumferential wavenumbers. However, effects are localized around the notch’s axial boundaries because the frequencies of the singularities occur below nearby cutoff frequencies. This localization might be expected grow in size with a greater circumferential wavenumber because
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of the increasing frequency difference as a result of correspondingly increasing frequency differences. Because the technique applies only locally, it is unsuitable for rapidly screening large sections of pipe, unlike other complementary methods. While no new experimental data is given, preliminary (and as of yet unpublished) experiments have shown that introducing a notch in a pipe generates singularities distinct from the pipe’s cutoff frequencies. However, only “sharp,” rectangular notches are considered. The extension to notches having different geometries still needs to be examined. Notwithstanding, the hybridSAFE approach can be applied straightforwardly to any arbitrary geometry providing that a finite element mesh suitably represents a notch’s geometry. The notches considered here are larger than those of practical interest, but serve to illustrate the proposed method. It is speculated, based on the limited data obtained to date, that singularities distinct from the unblemished pipe will be generated by notchlike defects of any dimensions. It is yet to be determined, however, if the frequency differences of small notches can be measured with sufficient precision to be useful. 4 ACKNOWLEDGEMENTS All three authors acknowledge the financial support from the Natural Science and Engineering Research Council (NSERC) of Canada. The first author also wishes to acknowledge financial aid from the University of Manitoba Students’ Union (UMSU), Society of Automotive Engineers (SAE) International, University of Manitoba, Province of Manitoba, and Ms. A. Toporeck and family. The Wawanesa Mutual Insurance Company is thanked for their generous donation of goods in kind. The authors wish also to thank Dr. Joseph L. Rose, Paul Morrow
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Professor of Engineering Design and Manufacturing, of the Pennsylvania State University for the useful suggestions he made as the external examiner for the first author’s doctoral dissertation. The experimental efforts of Mr. Kazeem Adeogun are recognized. 5 REFERENCES [1] Stoyko, D. (2012). Using the Singularity Frequencies of Guided Waves to Obtain a Pipe’s Properties and Detect and Size Notches. Ph.D. Thesis, University of Manitoba, Manitoba, from http://hdl.handle. net/1993/9818. [2] Shin, H., Rose, J. (1998). Guided wave tuning principles for defect detection in tubing. Journal of Nondestructive Evaluation, vol. 17, no. 1, p. 2736, DOI:10.1023/A:1022680429232. [3] Oliver, J. (1957). Elastic wave dispersion in a cylindrical rod by a wideband shortduration pulse technique. Journal of the Acoustical Society of America, vol. 29, no. 2, p. 189195, DOI:10.1121/1.1908824. [4] Demma, A. Cawley, P., Lowe, M., Roosenbrand, A.G., Pavlakovic, B. (2004). The reflection of guided waves from notches in pipes: A guide for interpreting corrosion measurements. NDT and E International, vol. 37, no. 3, p. 167180, DOI:10.1016/j.ndteint.2003.09.004. [5] Alleyne, D., Lowe, M., Cawley, P. (1998). Reflection of guided waves from circumferential notches in pipes. Journal of Applied Mechanics, vol. 65, no. 3, p. 635641, DOI:10.1115/1.2789105. [6] Silk, M., Bainton, K. (1979). Propagation in metal tubing of ultrasonic wave modes equivalent to Lamb waves. Ultrasonics, vol. 17, no. 1, p. 1119, DOI:10.1016/0041624X(79)900064. [7] R. Cook, (1981). Concepts and Applications of Finite Element Analysis. John Wiley and Sons, New York. [8] Stoyko, D., Popplewell, N., Shah, A. (2010). Finding a pipe’s elastic and dimensional properties using ultrasonic guided wave cutoff frequencies. NDT and E International, vol. 43, no. 7, p. 568578, DOI:10.1016/j. ndteint.2010.05.013.
Stoyko, D.K. – Popplewell, N. – Shah, A.H.
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 363372 © 2014 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/svjme.2014.1837 Special Issue, Original Scientific Paper
Received for review: 20131213 Received revised form: 20140214 Accepted for publication: 20140401
Finite Element Formulations Applied to Outer Ear Modeling Volandri, G. – Carmignani, C. – Di Puccio, F. – Forte, P. Gaia Volandri* – Costantino Carmignani – Francesca Di Puccio – Paola Forte University of Pisa, Department of Civil and Industrial Engineering, Italy The work described in this paper is part of a broader research activity on the development of a virtual ear. The present study focuses on the tympanic membrane and auditory canal modeling, which are important components in sound transmission. The standard finite element method (FEM) and an alternative method (the generalized FEM), suitable for modeling sound propagation at high frequencies, were applied. Two domains (fluid and structural) for the auditory canal and the tympanic membrane, respectively, were considered in order to evaluate the coupling of the different methods and to apply a fluidstructure interaction formulation. ANSYS® software was used for solving FEM analyses, while GFEM simulations were obtained by implementing the method in Wolfram Mathematica®. Simulation results include modal response, pressure distribution in the auditory canal and displacement distribution in the tympanic membrane. The identified modal frequencies of the auditory canal agree with published data reported in the literature. The validation of such method with standard FEM simulation at increasing mesh density shows that FEM is more suitable for simulations of the human ear in the audible frequency range, although the generalized formulation could be convenient if an ear model including the whole head or the ultrasound frequency range were investigated. Keywords: finite element method, auditory canal, simulation, sound transmission
0 INTRODUCTION The work described in this paper is part of a broader research activity on the development of a model of the human hearing perception, a kind of “virtual ear”. It deals with the analysis and simulation of the vibratory behavior of the auditory apparatus in the conventionally considered audible frequency range, 20 Hz to 20 kHz. In particular, the present study is focused on the auditory canal (AC) including the tympanic membrane (TM) that represents a fundamental portion of the “normal acoustic path” formed by the outer, middle and inner ear. In the last decades, many bioengineering methods have been applied to simulate the dynamic behavior of some parts of the ear, both distributed or lumped parameter methods, such as those based on electromechanical analogy or multibody dynamics. However, for the simulation at low frequencies (<10 kHz) of sound propagation in the AC and for the mechanicalacoustic transmission through the TM, the finite element method (FEM) is the most frequently used approach [1] to [7]. The first finite element (FE) models of the TM appeared in the ‘70s [1]; since then, the FEM has been widely employed to model the ear structures due to its remarkable capability for analyzing complex geometries and the mechanical properties of anisotropic and inhomogeneous materials. Models including also, at least, the middle ear were developed by Kelly et al. [2], by Koike et al. [3], by Gan et al. [4] to [6] and Lee an Chen [7]. In the last years, hybrid FE and multibody models of the TM and middle ear were also proposed by the present authors [8] and [9].
For simulating wave propagation with a standard FE model, i.e. with polynomial element shape functions, the mesh should respect the “rule of thumb”, commonly accepted for many wave problems, that there should be at least 10 nodes per wavelength λ [10]. For sound transmission in air, λ ranges from 15.6 mm (20 kHz) to 15.6 m (20 Hz), approximately. This means that at high frequencies, conventional FEM modeling can have a high computational cost, as it requires a very dense discretization of the domain. Except for some attempts to apply domain decomposition with parallel processing, alternative methods or generalizations of conventional FEM [11] have been proposed in the literature for low wavelength acoustic problems. These methods have high accuracy and a minor need of mesh refinement with respect to standard FEM. Among these methods there are the spectral methods, as the spectral element method (SEM) or spectral finite elements (SFE), based on the fast Fourier transform (FFT) [12] or on orthogonal polynomials [13], and the wave element methods (WEM), as the ultraweak variational formulation method (UWVF) [14], the partition of unity method (PU(FE)M) [15] and the generalized finite element method (GFEM) [16] and [17]. In this paper a conventional FEM analysis, carried out on an approximated model of the system formed by the AC and the TM, is compared with the results of an alternative method: the GFEM was selected as suitable for further extension of modeling and simulation of threedimensional sound propagation problems to higher frequencies or higher dimensions of problem domain. This method was implemented in a commercial code (Wolfram Mathematica®) and applied preliminarily to simplified geometries
*Corr. Author’s Address: Department of Civil and Industrial Engineering, Largo Lazzarino 56122, Pisa, Italy, gaia.volandri@ing.unipi.it
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that approximate the anatomy of the outer ear for a comparison with standard FEM in ANSYS®. 1 OVERVIEW OF COMPARED FE METHODS The present investigation is focused on different FE formulations compared in terms both of accuracy and computational time. In this section, the theory of the adopted standard and generalized formulations is reported in brief. 1.1 Theoretical Background of the Finite Element Methods As it is wellknown, the basic idea of the FEM is to divide the continuous domain of a problem in a discrete set of elementary subdomains (elements) where the field function (e.g. displacement) can be approximated by means of simple basis functions [11]. Such basis functions are typically continuous, with piecewise continuous derivatives, and moreover can be easily integrated. In order to guarantee the global continuity of the displacement field, adjacent elements should have same values along their boundaries, thus the basis functions also support an interpolatory solution. Accordingly, the displacement u at a point x within an element e can be written as:
u e (x) = ∑ N k (x) cek , (1) k
where Nk(x) are the basis functions and cek the interpolating coefficients. Differences between the FE methods selected for comparison can be attributed mainly to the basis functions adopted in element formulations. The peculiar characteristics of standard FEM, though well known, are here reported to ease the comparison with the other method.
u e (x, t ) = N(x) ae (t ) , (3)
which introduces also the generalization to timedependent problems. It can be worth noting that in case of isoparametric elements the same basis functions are used for mapping the elements from a reference domain into the physical domain. Accordingly, the element mass and stiffness matrices are obtained from the following integrals on the element domain Ωe:
Me =
∫N
Ωe
T
ρ N dΩ, K e =
∫B
T
DB dΩ, (4)
Ωe
where ρ is the material density and B = SN, S being a differential operator for calculating strain (εe = Sue = SNae = Bae), and D an elasticity matrix [11]. FE codes calculate such integrals numerically, i.e.
∫ f (x) dΩ = ∑h f (xh ) wh ,
(5)
Ωe
in particular by means of Gaussian quadrature [11], which for polynomials gives the exact solution. Various procedures exist for the refinement of finite element solutions: the local approximation can be improved by polynomials of increasingly higher degree (p version), or, having fixed the polynomial degree p (typically p ≤ 2), by decreasing the mesh size h (h version) or increasing the mesh size h and the degree p of polynomials (hpversion) [11]. 1.1.2 Generalized Finite Element Method
u e (x) = ∑ N k (x) aek , (2)
The generalized finite element method, GFEM, is a combination of the standard FEM and the partition of unity method (PU(FE)M), aimed at introducing additional terms in the approximating function which enhance the global behaviour of the solution, also reflecting the known information about the boundary value problem [16]. Thus the standard polynomial FE (x) (or solution is enriched with special functions u handbook functions) usually nonpolynomials, that means: (x). (6) u(x, t ) = N(x)a(t ) + u
and the shape functions are typically low (first or second) order polynomials. A more compact expression can be obtained introducing the matrix form of the basis functions N(x):
However such special functions must respect the global regularity constraints, and thus are obtained by some specific (solution or problemdependent) enrichment or handbook function hf(x), multiplied by the partition of unity (PU) NPU(x), i.e.
1.1.1 Standard Finite Element Method In standard FEM, interpolating coefficients are the element nodal displacements aek (k is the node numbers), so that Eq. (1) becomes:
k
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u(x, t ) = N(x)a(t ) + N PU (x)hf(x)a* (t ), (7) where a* are nodal unknown parameters that adjust the enrichment. Since this concept is at the basis of the extended method, it is worth reminding that a partition of unity in a domain Ω is a set of functions ψi(x) such that:
∑ iψ i ( x ) = 1,
∀x ∈ Ω. (8)
Consequently any function f(x) can be reproduced by its product with the functions ψi(x). Frequently, but not necessarily, the NPU functions correspond to the standard FE shape functions N, often linear or bilinear, with a “hat” shape, defined on patches (much wider than elements) and zero everywhere else. A 2D representation of patches and elements is shown in Fig. 1. The generalization to the 3D tetrahedral case is direct.
Fig. 1. Elements and patches
The local approximability is increased due to the handbook functions, while maintaining the existing infrastructures of the FE codes. In fact, the essential boundary conditions can be imposed in GFEM exactly as in standard FEM. In [17], GFEM is applied to the solution of the problem of Helmholtz with the evaluation of special plane wave and wave band functions and functions of Vekua as handbook functions and the conclusion is that the use of plane wave functions involves a lower computational burden without substantial variations in the asymptotic accuracy, compared to more complex functions. In the 2D case the plane waves can be expressed as shown in [17]. For 3D problems, as in the simulation of the fluid domain corresponding to the auditory canal, an extension is required and the plane waves can be expressed as a direct generalization of the 2D case, detailed in [18]:
W j(i ) = eik ( x sin θl cosφm + y sin θl sin φm + z cosθl ) , (9)
with θl = l
φm = m
π p
and l = 1, 2,..., p , and similarly
2π , m = 1, 2,..., q, q
and
employed
as
handbook functions hf(x) in Eq. (7). Such a direct generalization of the 2D case provides a distribution of directions which are concentrated around the poles of an imaginary sphere. As stated in [18], it is impossible to obtain an equally spaced distribution of directions. However, the choice of the logic of distribution of directions represents an important issue since a different logic of selection of propagation directions, based on a priori knowledge about the solution, can allow reducing the number of directions required to obtain a given level of accuracy. A main issue in the implementation of GFEM is the possible linear (or almost linear) relation between the added handbook functions and the standard basis FE ones with consequent illconditioning problems [16]. As reported by many authors, an increasing number of wave directions involves a higher result accuracy with the drawback of introducing illconditioned system matrices and requiring dedicated integration techniques. Thus, it is often necessary to introduce some checks in the implementation on the condition number of the matrices and, if needed, to update the direction number assigned to each node. 1.2 FluidStructure Interaction Formulation The coupling between partial domains of the whole model represents a significant aspect of modeling that includes the debated issue of the fluid structure interaction (FSI) formulation. The fluid structure interaction at the domain interface implies that the acoustic pressure exerts a load on the structure and that the structural motion produces an effective load on the fluid. The introduction, in the system dynamics governing equations, of a coupling matrix accounting for the effective surface area and the normal to the interface area, represents a possible FSI formulation [19]. In the present study, the FSI coupling formulation described in [20] was adopted at the interface between fluid and structural domains. Such a formulation involves the building of an asymmetric element matrix for the interface elements (typically fluid elements), which have the pressure and three translational DoFs. For the interface fluid element, the structural mass, damping and stiffness matrices, as well as the fluid mass, damping and stiffness matrices,
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assume the typical FE form. Mass (MFS) and stiffness (KFS) coupling matrices are defined according to the following equations: M FS = ρ0 AT , K FS = − A, (10)
where ρ0 is the fluid density, and the coupling matrix A is obtained by the following integration on the interface surface S:
A = ∫ NnT N′T dS, (11)
S
where N and N′ represent the fluid and structural shape function matrices, respectively, and the normal to the interface is indicated with n. 2 OUTER EAR MODELS 2.1 Anatomy The auditory canal belongs to the outer portion of the ear, the tympanic membrane is instead generally included in what is called the middle ear, being at the interface of the outer and middle ear (Fig. 2).
Fig. 2. Schematic drawing of the human ear
The AC conveys the vibratory waves propagating in air to the middle ear. The morphology of the AC, though often approximated with a cylindrical geometry, typically presents two curves [21]; this shape (referred to as “S”) not only facilitates the channeling of the wave but introduces variations (typically amplifications) at some resonance frequencies. Although intersubject biological differences exist, there is a general agreement on the value of adult AC length of 25 to 32 mm. The crosssectional area ranges from 65.45 to 75.53 mm2 at the TM to 90.13 to 96.16 mm2 at the canal entrance [4] and [7]. The understanding of the peculiarities of the AC that most influence the transmission of the signal and its coupling to the middle ear are important in the design of prostheses and in reconstructive surgery. 366
The TM, located at the end side of the ear canal, forms an angle of about 140° with the upper and lower walls of the channel; such an orientation gives a useful area of about 85 mm2, greater than the orthogonal crosssection of the AC. The TM has a typical conical shape with an opening angle of 132 to 137° and a height of the cone of about 1.42 to 2 mm. The elliptical base of the cone has a vertical axis length ranging from 8.5 to 10 mm and a horizontal axis length ranging from 8 to 9 mm, with the apex (named umbo and assumed as reference point) facing the medial side [22]. The thickness of the TM is a critical parameter for modeling, since to date accurate detailed experimental measurements of the thickness distribution are not available and since it presents a high intersubject variability in terms of absolute values. Although there is a general agreement in estimating a nonuniform thickness of the membrane, in modeling an approximated single thickness value, ranging from 30 to 150 µm (with an average value of 74 µm) for the human TM is usually adopted [4], [7] and [22]. The tissue of the TM is made of a multilayer structure with fibers oriented mainly in the radial and circumferential directions. Two main regions are usually distinguished a “Pars Tensa” (PT) and a “Pars Flaccida” (PF), having different size and mechanical properties. The TM is connected on the medial side to the ossicular chain, while it is anchored along its circumference to the wall of the tympanic cavity by means of a fibrocartilaginous ring (annular ligament). 2.2 Simplified Models of the Auditory Canal and Tympanic Membrane 2.2.1 Geometry and Material Properties of the AC Two simplified models of the auditory canal were used that approximate the anatomy of an ear canal. A 22 mm long cylindrical and a pseudoanatomical geometries (shown in Fig. 3) of a human auditory canal were adopted and imported into ANSYS® environment for the FEM analysis. The pseudoanatomical geometry (including an air portion in the auricle) was extracted, through a semiautomatic segmentation algorithm, from computed tomography data provided by the Department of Otolaryngology II of Cisanello Hospital in Pisa. Size and morphology of the AC reconstructed geometry were in the literature range for adult healthy subjects (length of about 28 mm, average diameter of about 9 mm) [4].
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As regards the material properties, for the fluid contained in the AC the air medium was assumed as a compressible, inviscid fluid with uniform mean pressure and density; a 1.225 kg/m3 density value and a 340 m/s speed of sound value were, thus, adopted. The damping was not considered in this evaluation study. The AC bony wall was simulated with clamped boundary conditions and a distributed uniform harmonic sound pressure load of 90 dB SPL (corresponding to 0.632 Pa) was applied at the inlet of the AC while the outlet was not constrained. 2.2.2 Mesh Definition for the AC Model Tetrahedral elements were employed for the cylindrical and the pseudoanatomical models (Fig. 3), as it is typically adopted for complex biological geometries. The size of the elements was set by means of a convergence criterion based on the value of the first natural frequencies up to 20 kHz; the percent relative error vs. the logarithm of number of DoFs was estimated (variation less than 1%) [11].
An isotropic material model with Young’s modulus of 20 MPa was considered [1]. The Poisson’s ratio was set equal to 0.3; density was assumed equal to 1.2×103 kg/m3. Triangular shell elements with three nodes were used for the TM, when included. As concerns the boundary conditions, the TM was fully clamped at the periphery, i.e. the annular ligament was not considered, as well as the ossicular chain. 2.3 Implemented Methods The implementation and comparison of the methods was carried out in Mathematica® environment on the AC approximated geometries. Firstly, the GFEM implementation required the definition of elements and patches and the setting of the wavenumber. The partition of unity φi, and the shape functions Nk were chosen as the linear shape functions for the brick element with eight nodes of the standard FEM. (i ) In this paper special plane wave functions, W j , were implemented, whose more general expression is:
a)
b) Fig. 3. a) Cylindrical and b) anatomical geometries and mesh of a human auditory canal
2.2.3 Tympanic Membrane Model The TM was included in the model with the aim of accounting for the fluidstructure interaction. Although the TM has a peculiar shape (See Section 2.1), a flat circular geometry was adopted with the aim of facilitating the definition of the fluidstructure interface. A 0.074 mm thickness was adopted, deduced from [4].
W j(i ) = eik·r , (12)
where r is the position vector and k the wave vector. The unit vectors of the propagation directions of the planewave functions, in the 3D problems, were selected based on a priori knowledge of the solution or following the optimized “spherical covering” logic, borrowed from another method (e.g. ultra weak variational formulation, UWVF) [14]. The optimized “spherical covering” logic identifies n points on an imaginary sphere, centered in each patch node, so that the maximum distance of each node belonging to the sphere from the nearest of the n considered points is minimized. Such a criterion allows obtaining a direction distribution as homogeneous as possible. In the present study, n was chosen in the 4 to 124 range due to requirements of computational cost and the same number of directions was assigned to each vertex of the patches, although the method requires neither an equally spaced distribution of directions nor an equal number of DoFs at each node. The GFEM method was implemented for the simulation of the fluid domain while the FEM method was implemented, in Mathematica® as well, for the simulation of the structural domain (limited to the TM), when included. The ANSYS® FLUID30 and SHELL63 (Discrete Kirchoff element, DKT) formulations were adopted
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and implemented for the fluid and structural elements, respectively. The ANSYS® FLUID30 element formulation was selected as suitable for fluid/structure interaction problems and sound wave propagation applications. The element has a brick shape with eight corner nodes and four DoFs per node: three displacements (only at nodes on the interface) and pressure. The element adopts linear shape functions without extra shape functions, and a standard Gaussian quadrature for bricks (2×2×2 points). For the comparison and validation of the generalized method, the standard FEM solution at different levels of mesh refinement obtained with the commercial code ANSYS®, was used as a reference. 2.4 Comparison Tests 2.4.1 Modal and Harmonic Analyses on the AC Models Numerical freefree or constrained modal and harmonic analyses were carried out on the models, intending as constrained and freefree conditions when sound pressure load is applied or not, respectively, at the AC inlet. Pressure distribution in the AC was evaluated as main result of the harmonic analysis. In addition to a check on eigenfrequencies in a specific frequency range ([20 Hz to 20 kHz]), the modal shape correlation was quantified by the modal assurance criterion (MAC) [23]. MAC results are usually represented as a square matrix, correlating the reference modes (FEM, test, theory etc.) to verification modes. It assumes a nearly zero value in the presence of incompatible mode shapes, a unitary value in case of perfect correlation and intermediate values in case of partial correlation. 2.4.2 FSI of Auditory Canal and Tympanic Membrane Harmonic analyses were also carried out on the cylindrical geometry with circular membrane accounting for the fluidstructure interaction. The cylindrical geometry was preferred since it allows a simple definition of interfaces for the fluidstructure formulation. The material properties and the boundary conditions of the AC, except for the outlet, were set as previously described. The harmonic analysis on the implemented models provides the distribution of pressure and normal displacement in the fluid and structural domain, respectively (coupled by FSI). 368
3 RESULTS 3.1 Comparison of Methods on the AC Model In order to compare the two finite element formulations, freefree and constrained modal analyses were performed on the simplified geometries (Fig. 3) of the fluid domain. Eigenfrequencies in the 20 Hz to 20 kHz frequency range (typically the first three in the pseudoanatomical case) were compared and mode shapes correlated, using the MAC matrix. The modal frequencies obtained in the freefree and constrained modal analysis with FEM and GFEM in the cylindrical and pseudoanatomical geometries are reported and compared in Table 1. Table 1. First three modal frequencies; the results are shown for a), b) cylindrical geometries; and for c), d) pseudoanatomical geometries;a), c) for the freefree modal analysis; b), d) for the constrained modal analysis f [Hz] a)
b)
c)
d)
FEM GFEM error [%] FEM GFEM error [%] FEM GFEM error [%] FEM GFEM error [%]
1°
2°
3°
7724 6648 14 3864 3670 5 6616 5587 15 3961 3330 16
15489 13880 10 11601 11269 2.8 12022 11960 0.5 11103 11229 1.1
23257 24097 3.6 19367 19832 2.4 15258 13345 12.5 17119 17677 3.2
It is worth noting that the modal frequencies obtained with the pseudoanatomical geometry are in agreement as order of magnitude with the three resonance frequencies in the audible range (3176, 9528 and 15880 Hz) inherent to the external auditory canal, depending on its morphology and length, reported in [24]. Moreover, in the literature the first natural frequency inherent to the auditory canal is often identified in the 3 to 4 kHz range [2] and [24] confirming the estimated value. Finally, the results obtained by the two numerical approaches appear to differ majorly at low frequency where standard FEM is considered reliable and convenient while they are comparable at higher frequencies where GFEM should become more advantageous. The comparison of the mode shapes was carried out in Mathematica® environment by importing the results obtained in the different software environments for the proposed methodologies. Fig. 4 shows, for the cylindrical (a, b) and for the pseudoanatomical
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 363372
geometry (c, d), for the freefree (a, c) and constrained (b, d) modal analysis respectively, the matrix representations of the modal shape correlation of the two methods, for an equal number of nodes. The maximum/minimum values on the MAC matrix diagonal as well as the out of diagonal maximum values are shown in Fig. 4 for the four above mentioned cases.
the advantage of keeping the standard FEM mesh, even with tetrahedral elements, with nodes belonging to the element boundary.
a) a)
b)
b) c)
d)
Fig. 4. Modal shape correlation by MAC; the results are shown for the a), b) cylindrical and c), d) pseudoanatomical geometries, for the freefree a), c) and constrained b), d) modal analysis
All cases show consistency in the first three natural frequencies obtained by the various methodologies, less in the mode shapes. As an example of the harmonic analysis results, the pressure distribution at 10 kHz in the pseudoanatomical model of the AC is shown in Fig. 5. The GFEM results obtained with a coarse element mesh (144 nodes, 288 DoFs) are compared with the ANSYS® FEM results at equal and refined (3216 nodes, 3216 DoFs) mesh. The TM side is indicated in the figures. The GFEM and FEM harmonic results, in the audible frequency range agree within tight tolerances. The GFEM is more expensive from the computational point of view, since GFEM presents the difficulty of calculation in the complex field and entails problems of illconditioning and linear dependence in the phase of construction of the element matrices, for which it often requires heavy integration techniques. However, with respect to alternative methods of the literature, GFEM presents
c) Fig. 5. Harmonic analysis results at 10 kHz: pressure distribution in the pseudoanatomical geometry with a) GFEM (coarse mesh) and b) coarse mesh, and c) FEM refined mesh
In conclusion, FEM is more suitable for outer ear simulations in the 20 Hz to 20 kHz frequency range. Moreover it is available in commercial codes. However, further investigations elucidate that the GFEM can be more suitable for other applications involving higher frequencies (e.g. a 22 mm long cylindrical model at 100 kHz) or higher characteristic dimensions of the problem (e.g. a scaled/homothetic 22 cm long cylindrical model, at 10 kHz). The results of these further simulations are shown in Figs. 6 and 7, respectively, for different mesh density. The longitudinal section of the cylinder, instead of the cylindrical surface, is plotted in Figs. 6a and 7a. As one can see, the results of Fig. 6a are comparable to those of Fig. 6c which represents the
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a)
a)
b)
b)
c) Fig. 6. Harmonic analysis in a cylindrical model (22 mm long) at 100 kHz with: a) GFEM (longitudinal section) and (b, c) FEM varying the mesh density: a) 137 nodes and 166 DoFs, b) 137 nodes and DoFs, c) 28487 nodes and DoFs
FEM reference solution, obtained with a considerably higher number of degrees of freedom. Then, if Fig. 6a is compared with Fig. 6b, obtained with an equal number of degrees of freedom, the differences and the enhancement are evident. The same observations hold for Fig. 7. The use of the implemented generalized element formulations can be convenient, especially as frequency increases (and therefore the wave number) because it allows to achieve, with a coarse mesh (which does not satisfy the rule of thumb of ten nodes per wavelength conventionally accepted for standard FEM), an accuracy comparable to that obtained with a fine mesh in standard FEM formulations. These first indications suggest that the integration of these advanced techniques in a FE model of the ear could be useful if, for example, the whole head were included or if the ultrasound field were investigated. 370
c) Fig. 7. Harmonic analysis in a cylindrical model (scaled 22 cm long cylinder) at 10 kHz with: a) GFEM (longitudinal section) and b), c) FEM varying the mesh density: a) 137 nodes and 166 DoFs, b) 137 nodes and DoFs, c) 28487 nodes and DoFs
3.2 Comparison of Methods on the FSI Problem Concerning the simulation of the fluid structure problem by the combination of techniques GFEM and FEM for the fluid and structural domains, respectively, the distributions of pressure inside the AC and TM displacement at 200 Hz are compared with the results obtained with standard FEM in ANSYS® with a coarse and refined mesh, in Figs. 8 and 9, respectively. As highlighted in the previous section, the results indicate that GFEM and FEM results, in the investigated frequency range, are in agreement within acceptable tolerances. 4 CONCLUSIONS In this work some modeling aspects of the human ear canal and tympanic membrane were examined also
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Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, 363372
a) a)
b) b)
c) Fig. 8. Harmonic analysis results at 200 Hz in the FSI problem: pressure distribution in the auditory canal with: a) GFEM (coarse mesh, longitudinal section) and b) FEM (coarse mesh), c) FEM (refined mesh
considering the fluidstructural coupling that occurs between these two components of the acoustic path. Standard and generalized finite element models were implemented, tested and compared. GFEM and FEM modal and harmonic results in the 20 Hz to 20 kHz range agree within tight tolerances in all tested cases. Thus, FEM appears more suitable for simulations in the audible frequency range, assuming the typical ear size of a human being, due to its relatively limited computational burden and the availability of commercial codes. However, these preliminary results show also that the generalized finite element formulation can be convenient in shortwave acoustic problems with the aim of simulating the auditory apparatus including the whole head or investigating the ultrasound field, i.e. when the wave length is shorter than the characteristic dimension of the problem.
c) Fig. 9. Harmonic analysis results at 200 Hz in the FSI problem: normal displacement distribution in the tympanic membrane with: a) GFEM and b), FEM (coarse mesh), c) FEM (refined mesh)
The analysis results concerning the natural frequencies of the auditory canal are consistent with some published studies [2] and [24] which identify three modal frequencies in the audible range with the first modal one included in the 3 to 4 kHz range. As future developments, the implementation and integration of this method in a commercial code can be planned. The application of GFEM to a complete ear model including bone conduction in the head can be considered. Moreover GFEM can be proposed for convenient application to other fields (e.g. ultrasounds in medicine). 5 ACKNOWLEDGMENTS The support and contribution of Prof. Stefano Berrettini and Dr. Luca Bruschini of the U.O.
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Otorinolaringoiatria 2° of the Cisanello hospital in Pisa, Italy are gratefully acknowledged. 6 REFERENCES [1] Funnell, W.R.J., Laszlo, C. (1978). Modeling of the cat eardrum as a thin shell using the finiteelement method. Journal of the Acoustical Society of America, vol. 63, no. 5, p. 14611467, DOI:10.1121/1.381892. [2] Kelly, D.J., Prendergast, P.J., Blayney, A.W. (2003). The effect of prosthesis design on vibration of the reconstructed ossicular chain: a comparative finite element analysis of four prostheses. Otology & Neurotology, vol. 24, no. 1, p. 1119, DOI:10.1097/0012949220030100000004. [3] Koike, T., Wada, H., Kobayashi, T. (2002), Modeling of the human middle ear using the finiteelement method. Journal of the Acoustical Society of America, vol. 111, no. 3, p. 13061317, DOI:10.1121/1.1451073. [4] Gan, R.Z., Feng, B., Sun, Q. (2004). Threedimensional finite element modeling of human ear for sound transmission. Annals of Biomedical Engineering, vol. 32, no. 6, p. 847859, DOI:10.1023/ B:ABME.0000030260.22737.53. [5] Gan, R.Z., Sun, Q., Feng, B., Wood, M.W. (2006). Acoustic–structural coupled finite element analysis for sound transmission in human ear— Pressure distributions. Medical Engineering & Physics, vol. 28, no. 5, p. 395404, DOI:10.1016/j. medengphy.2005.07.018. [6] Gan, R.Z., Reeves, B.P., Wang, X. (2007). Modeling of sound transmission from ear canal to Cochlea. Annals of Biomedical Engineering, vol. 35, no. 12, p. 21802195, DOI:10.1007/s104390079366y. [7] Lee, C.F., Chen, P.R., Lee.W.J., Chou, Y.F., Chen, J.H., Liu, T.C. (2010). Computer aided modeling of human mastoid cavity biomechanics using finite element analysis. EURASIP Journal on Advances in Signal Processing, paper: 203037, DOI:10.1155/2010/203037. [8] Volandri, G., Di Puccio, F., Forte, P. (2012). A sensitivity study on a hybrid FE/MB human middle ear model. ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis, vol. 4, p. 217226, DOI:10.1115/ESDA201282326. [9] Volandri, G., Di Puccio, F., Forte, P., Manetti, S. (2012). Modeloriented review and multibody simulation of the ossicular chain of the human middle ear. Medical Engineering & Physics, vol. 34, no. 9, p. 13391355, DOI:10.1016/j.medengphy.2012.02.011. [10] Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P. (2005). The Finite Element Method for Fluid Dynamics. Elsvier ButterworthHeinemann, Burlington. [11] Zienkiewicz, O.C., Taylor, R.L. (2000). The Finite Element Method (5th ed.) Volume 1  The basis, Elsevier Butterworth Heinemann, Burlington.
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[12] Doyle, J.F. (1997). Wave propagation in Structures, 2nd ed., Springer Science+Business Media New York . [13] Peng, H., Meng, G., Li, F. (2009). Modeling of wave propagation in plate structures using three dimensional spectral element method for damage detection. Journal of Sound and Vibration, vol. 320, no. 46, p. 942954, DOI:10.1016/j.jsv.2008.09.005. [14] Huttunen, T., Gamallo, P., Astley, R.J. (2009). Comparison of two wave element methods for the Helmholtz problem. Communications in Numerical Methods in Engineering, vol. 25, no. 1, p. 3552, DOI:10.1002/cnm.1102. [15] Gamallo, P., Astley, R.J. (2006). The partition of unity finite element method for short wave acoustic propagation on nonuniform potential flows. International Journal for Numerical Methods in Engineering, vol. 65, no. 3, p. 425444, DOI:10.1002/ nme.1459. [16] Strouboulis, T., Babuska, I., Copps, K. (2000). The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering, vol. 181, no. 13, p. 4369, DOI:10.1016/ S00457825(99)000729. [17] Strouboulis, T., Hidajat, R., Babuska, I. (2008). The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment. Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 5, p. 364380, DOI:10.1016/j.cma.2007.05.019. [18] Laghrouche, O., Bettess, P., PerreyDebain, E., Trevelyan, J. (2003). Plane wave basis finiteelements for wave scattering in three dimensions. Communications in Numerical Methods in Engineering, vol. 19, no. 9, p. 715723, DOI:10.1002/cnm.632. [19] Fluids Analysis Guide (2005). ANSYS Release 10.0, Ansys Inc., Canonsburg. [20] Theory Reference (2004). ANSYS Release 9.0, Ansys Inc., Canonsburg. [21] Gray, H. (2000). Anatomy of the Human Body. 20th edition, Lewis, W.H. (ed.) Lea and Febiger, Philadelphia & Bartleby.com, New York, from http:// www.bartleby.com/107/ accessed on 20130101. [22] Volandri, G., Di Puccio, F., Forte, P., Carmignani, C. (2011). Biomechanics of the tympanic membrane. Journal of Biomechanics, vol. 44, no. 7, p. 12191236, DOI:10.1016/j.jbiomech.2010.12.023. [23] Ewins, D.J. (2000). Model validation: Correlation for updating. Sadhana, vol. 25, no. 3, p. 221234, DOI:10.1016/j.jbiomech.2010.12.023. [24] Vallejo, L.A., Delgado, V.M., Hidalgo, A., GilCarcedo, E., GilCarcedo, L.M., Montoya, F. (2006). Modeling of the geometry of the external auditory canal by the finite elements method. Acta Otorrinolaringológica Espaola, vol. 57, no. 2, p. 8289, DOI:10.1016/S00016519(06)786678. (in Spanish)
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Gostujoči uvodnik
SI 57
Razširjeni povzetki člankov Peter Avitabile, Christopher Nonis, Sergio E. Obando: Modalni modeli sestavljenih struktur temelječi na modalnih modelih dinamsko nevezanih podstruktur Fabrício César Lobato de Almeida, Michael John Brennan, Phillip Frederick Joseph, Simon Dray, Stuart Whitfield, Amarildo Tabone Paschoalini: Merjenje zmanjševanja valovanja v zakopani plastični cevi za distribucijo vode Andrea Barbarulo, Hervé Riou, Louis Kovalevsky, Pierre Ladeveze: PGDVTCR: PGDVTKZ: metoda za reševanje kompleksnih akustičnih sistemov v srednjem frekvenčnem območju z modelom nižjega reda Kimihiko Nakano, Matthew P. Cartmell, Honggang Hu, Rencheng Zheng: Analiza fenomena stohastične resonance pri nabiralcih energije Diego Saba, Paola Forte, Giuseppe Vannini: Pregled in nadgradnja modela kompresijskega pretoka pri analizi plinskih satnih tesnil na podlagi visokotlačnega eksperimenta Snehashish Chakraverty, Diptiranjan Behera: Identifikacija parametrov večnadstropne palične konstrukcije na podlagi negotovih dinamskih podatkov Martin Česnik, Janko Slavič: Vibracijsko utrujanje in strukturna dinamika pri harmonskih in naključnih obremenitvah Darryl K. Stoyko, Neil Popplewell, Arvind H. Shah: Koeficienti odboja in prenosa pri pravokotnih zarezah v ceveh Gaia Volandri, Costantino Carmignani, Francesca Di Puccio, Paola Forte: Formulacije končnih elementov na primeru modeliranja zunanjega ušesa Osebne vesti Doktorske disertacije, magistrska dela, diplomske naloge
SI 59 SI 60 SI 61 SI 62 SI 63 SI 64 SI 65 SI 66 SI 67 SI 68
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5 Uvodnik
Gostujoči uvodnik Tematska številka: Sodobni napredki v strukturni dinamiki Mednarodna znanstvena konferenca Sodobni napredki v strukturni dinamiki poteka v organizaciji Instituta za raziskavo hrupa in vibracij (Institute of Sound and Vibration Research  ISVR) Univerze v Southamptonu od leta 1980, leta 2013 že v enajsti ponovitvi. Nastanek konference je tesno povezan z idejnim očetom prof. Maurice Petytom iz ISVR, ki je konferenco več let vodil v sodelovanju z Howardom Wolfom ter Chuh Meiom. Eminentni znanstveniki s področja strukturne dinamike kot npr. Meirovitch, Heckl, Braun, Noor, Mead, Langley, Fuller, Elishakoff in Ewins so bili aktivni udeleženci na tej konferenci. Howard Wolfe je sodeloval pri organizaciji konference od tretje ponovitve naprej v povezavi z Zračnimi silami ZDA na področju akustičnega utrujanja v povezavi s strukturno dinamiko. Raziskovalna agencija Zračnih sil ZDA je več let finančno podpirala konferenco, tako na numeričnem kot tudi na eksperimentalnem raziskovanju na področju nadzvočnega letalstva. Z leti so se glavni organizatorji z ISVR izmenjevali, pridružili so se tudi Neil Ferguson, Mike Brennan, Emiliano Rusighi s podporo Steva Rizzija (NASA Langley) ter Martya Fermana. Izbrani članki posameznih konferenc so bili, kot pomembni izvirni znanstveni dosežki tako na teoretičnem, numeričnem ter eksperimentalnem področju strukturne dinamike, objavljeni v obliki recenziranih člankov v priznanih znanstvenih revijah. Enajsta konferenca Sodobni napredki v strukturni dinamiki (RASD 2013) je potekala v italijanski Pisi, v organizaciji prestižne tradicionalne Univerze v Pisi, medtem, ko so predhodne konference potekale v Southamptonu, v Združenem kraljestvu. RASD 2013 je podobno kot predhodne konference predstavila mednarodno sodobno znanstveno misel na področju strukturne dinamike ter dinamskih sistemov v inženirski praksi. Na konferenci je bilo predstavljenih 127 prispevkov iz več kot 25 držav. Članki so bili zbrani v 28 vsebinskih sklopov in so vsebovali nelinearna nihanja, vibroakustiko, spremljanje zdravja struktur, interakcijo človekvibracije, identifikacijo sistemov, inverzne probleme, potresno inženirstvo, modalno analizo in strukturne modifikacije, kontrolo vibracij, gradbeniške strukture, numerične tehnike,
aktivno kontrolo vibracij, pametne strukture, analitično modeliranje, eksperimentalne metode, interakcijo fluidtrdnina, nezanesljive dinamske sisteme, vibracijsko nabiranje energije, naključna nihanja, hrup in vibracije tirničnih vozil. Posebna sekcija konference je obravnavala obnovljive supersonične platforme z moderiranim razgovorom o nerešenih vprašanjih tega področja. Na konferenci so sodelovali vabljeni predavatelji, in sicer profesor Y. Suda (Dinamične simulacije ter analize modernih transportnih sistemov, Univerza iz Tokyo, Japonska), profesor A. Metrikine (Razvoj, ovrednotenje ter uporaba vibracijskih modelov pri vrtinčno vzbujenih vibracijah morskih dvižnih cevovodov, TU Delft, Nizozemska), profesor J.S. Bolton (Vpliv robnih pogojev in vmesnih omejitev pri obravnavi kontrole zvoka: od penastih do meta materialov, Univerza Purdue, ZDA), dr. A.M. Wickenheiser (Širokospektralni vibracijski nabiralci energije: Redukcija modelov in frekvenčna konverzija, Univerza George Washington, ZDA) in dr. S.A. Rizzi (Pregled virtualnih akustičnih simulacij pri preletu letal, NASA Langley Research Center, ZDA). RASD 2013 konferenca je bila edinstvena priložnost za izmenjavo in širjenje znanstvene misli med raziskovalci. Pričujoča posebna številka Strojniškega Vestnika – Journal of Mechanical Engineering vsebuje devet člankov ter tako predstavlja manjši del vseh predstavljenih člankov na konferenci. Članek avtorjev Avitabile in sodel. obravnava nihajne oblike modela na podlagi ekspanzije nevezanih dinamičnih podatkov podstruktur. Almeida in sodel. predstavljajo meritve valovnega zmanjševanja pri zakopanih plastičnih vodovodnih ceveh. Barbarulo in sodel. se osredotočajo na kompleksne akustične probleme v srednjem frekvenčnem področju z uporabo variacijske teorije posplošene dekompozicije. Nakano in sodel. raziskujejo možnost energijskih nabiralcev energije z uporabo stohastičnih resonanc na podlagi aksialne periodične sile. Saba in sodel. prikazujejo pregled in nadgradnjo modela kompresijskega pretoka pri analizi plinskih satnih tesnil na podlagi visokotlačnega eksperimenta. Chakraverty in sodel. obravnavajo identifikacijo parametrov večnadstropne stavbe na SI 57
podlagi nezanesljivih dinamskih podatkov. Česnik in sodel. združujejo strukturno dinamiko ter vibracijsko utrujanje za harmonične ter naključne obremenitve. Stoyko in sodel. predstavljajo rezultate pri odbojnih ter prenosnih koeficientih zaradi pravokotnih zarez v ceveh. Volandri in sodel. zaključujejo pregled izbranih
prispevkov z obravnavo zvočnega širjenja v zunanjem ušesu z uporabo numeričnega modeliranja. Gostujoči uredniki te posebne številke se zahvaljujejo vsem, ki so prispevali k uspehu konference RASD 2013 ter avtorjem in recenzentom objavljenih prispevkov.
Gostujoči uredniki: Miha Boltežar, Univerza v Ljubljani, Fakulteta za strojništvo Emiliano Rustighi, ISVR, Univerza v Southamptonu Neil Ferguson, ISVR, Univerza v Southamptonu
SI 58
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 59 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20130809 Prejeto popravljeno: 20140214 Odobreno za objavo: 20140328
Modalni modeli sestavljenih struktur temelječi na modalnih modelih dinamsko nevezanih podstruktur Avitabile, P. – Nonis, C. – Obando, S.E. Peter Avitabile* – Christopher Nonis – Sergio E. Obando 1 Univerza Massachusetts Lowell, Oddelek za strojništvo, Laboratorij za strukturno dinamiko in akustične sisteme, ZDA
V strukturni dinamiki so modeli sestavljenih struktur pogosto razviti na podlagi modelov posameznih komponent; pri čemer so modeli komponent lahko razviti na nivoju modalnih modelov ali pa na nivoju reduciranih strukturnih modelov. Posledično je tudi rezultirajoči model sestavljene strukture reduciran in se lahko rezultate višje stopnje pridobi zgolj s pomočjo ekspanzije modela. Ta ekspanzija je mogoča na podlagi modelnega zrcaljenja, ki povezuje polni in reducirani model; vendar pa ta korak zahteva polni model kompleksne strukture in zato sestavljanje reduciranjih sistemov izgubi smisel. Ta raziskava pokaže, da se lahko model sestavljene strukture vzpostavi na podlagi modelnega zrcaljenja polnega modela posameznih komponent; pri čemer pa morajo modalne komponente podstruktur zavzemati celotni modalni prostor sestavljene strukture. Prikazana sta dva tipična analitična primera. Prvi analitični model temelji na sistemu dveh povezanih nosilcev in modelna ekspanzija je izvedena na modalnem modelu 5 modalnih oblik; drugi analitični model pa temelji na 10 modalnih oblikah. Izkaže se, da na natančnost modelne ekspanzije bistveno vpliva napaka modalnega odreza. Študija pokaže, da je v primeru, da modalni modeli podstruktur zavzemajo celotno področje modalnega modela sestavljene strukture mogoča natančna modelna ekspanzija. Uporaba predstavljenih pristopov je zanimiva predvsem zato, ker omogoča ekspanzijo modela za polno analizo mehanskega odziva na podlagi reduciranih modalnih modelov nesklopljenih komponent. Na tak način se izračun bistveno pospeši. Neposredne aplikacije so v analizi napetostno/deformacijskega stanja polnega modela. Ključne besede: modeliranje komponet, modeliranje sestavljenih struktur, ekspanzija
*Naslov avtorja za dopisovanje: Univerza Massachusetts Lowell, 1 University Ave, Lowell, ZDA, peter_avitabile@uml.edu
SI 59
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 60 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131019 Prejeto popravljeno: 20140214 Odobreno za objavo: 20140401
Merjenje zmanjševanja valovanja v zakopani plastični cevi za distribucijo vode Fabrício César Lobato de Almeida1,* – Michael John Brennan1 – Phillip Frederick Joseph2 – Simon Dray3 – Stuart Whitfield4  Amarildo Tabone Paschoalini1 2 Univerza
1 Univerza Estadual Paulista, Oddelek za strojništvo, Brazilija v Southamptonu, Inštitut za raziskovanje zvoka in vibracij, Združeno Kraljestvo 3 Hydrosave Ltd, Združeno Kraljestvo 4 South Staffs Water, Združeno Kraljestvo
Puščanje cevi je pogost problem, ki lahko zahteva velike investicije; predvsem je to težava takrat, kadar je težko določiti mesto puščanja. Pri klasičnih kovinskih ceveh so uveljavljene metode identifikacije mesta puščanja cevi. Pri kotni identifikaciji je faktor zmanjševanja valovanja znotraj cevi bistvenega pomena. Identifikacijo pri plastičnih ceveh otežuje dejstvo, da imajo plastične cevi bistveno večje dušenje kakor kovinske in zato tudi bistveno večji faktor zmanjševanja valovanja. Dodano težavo seveda povzroča dejstvo, če je taka cev zakopana (tipično v zemljo). Z namenom identifikacije mesta puščanja, se raziskava osredotoči na zahtevno nalogo identifikacije zmanjševanja valovanja v zakopani cevi: pri tem najprej obravnava analitični model, ki pokaže omejitve pristopa in nato insitu eksperiment. Predstavljena metoda temelji na identifikaciji ovojnice križnekorelacije med dvema pospeškomeroma, ki sta nameščena na različnih mestih cevovoda. Pri analitičnem modelu in tudi pri eksperimentu je med oba pospoškomera na znani razdalji kontrolirano vpeljano tudi puščanje vode. Celotna dolžina cevovoda je 110 m in pospeškomera sta nameščena na relativno veliki razdalji 50 m. Kljub velikemu dušenju v plastični cevi je predstavljeni pristop uspešen pri identifikaciji faktorja zmanjševanja valovanja tudi v primeru puščanja vode iz cevi. Eksperimentalna potrditev na realnem cevovodnem sistemu dodatno dokazuje uporabnost pristopa. Ključne besede: identifikacija puščanja, zmanjševanje valovanja, vodna industrija, plastične cevi
SI 60
*Naslov avtorja za dopisovanje: UNESP, Av. Brasil centro, 56, Ilha Solteira, Brazilija, fabricio_cesar@dem.feis.unesp.br
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 61 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131016 Prejeto popravljeno: 20140117 Odobreno za objavo: 20140401
PGDVTKZ: metoda za reševanje kompleksnih akustičnih sistemov v srednjem frekvenčnem območju z modelom nižjega reda Barbarulo, A. – Riou, H. – Kovalevsky. L. – Ladeveze, P. Andrea Barbarulo1,* – Hervé Riou1 – Louis Kovalevsky2 – Pierre Ladeveze1 1 LMTCachan, ENS Cachan, Francija 2 Univerza
v Kembridžu, Inžinerski oddelek, Združeno kraljestvo
Prispevek predstavi razvoj napredne metode za izračun akustičnega odziva kompleksnih struktur v srednjem frekvenčnem področju. Problem se navezuje na številne industrijske aplikacije predvsem na področju avtomobilske in vesoljske akustike. Za srednje frekvenčno področje je značilno, da je odziv v veliki meri pogojen s frekvenco. To zahteva drobno frekvenčno resolucijo, kar pa je direktno povezano z dolgimi računskimi časi. Predstavljena metoda bazira na t.i. Variacijski Teoriji Kompleksnih Žarkov (VTKZ) skupaj z metodo Pravilne Generalizirane Dekompozicije (PGD). Metoda VTKZ temelji na osnovnih variacijskih enačbah, ki so zasnovane na način, da so aproksimacije znotraj posameznih domen popolnoma neodvisne od celotnega sistema. To znotraj domen omogoča aplikacijo vseh vrst aproksimacij, tudi takih z močnim fizikalnim ozadjem. S tem je dosežena velika stopnja fleksibilnosti ter učinkovitosti. Glede na ostale podobne Trefftz metode se razlikuje glede na obliko izbranih oblikovnih funkcij kot tudi z vidika obravnave robnih pogojev. Referenčni akustični problem predstavlja fluid v zaprti domeni, ki je okarakteriziran z gostoto in hitrostjo zvoka. V skladu s problematiko na tem področju je potrebno določiti odziv fluida v ustaljenem stanju na izbranem frekvenčnem območju. Reševanje akustičnega modela bazira na povezavi metod VTKZ in PGD preko dekompozicije fizikalnih spremenljivk. Uporabnost metode je prikazana na primeru izračuna akustičnega polja potniškega prostora avtomobila. V povezavi z referenčno VTKZ metodo je izvedena verifikacija predlaganega pristopa ter predstavljeni konvergenca in učinkovitost metode. Ključne besede: srednje frekvenčno območje, akustika, variacijska teorija, generalizirana dekompozicija, redukcija modela
*Naslov avtorja za dopisovanje: LMTCachan ENS Cachan/CNRS/Paris 6 University, F94230 Cachan, Francija, barbarulo@lmt.enscachan.fr
SI 61
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 62 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20141013 Prejeto popravljeno: 20140125 Odobreno za objavo: 20140401
Analiza fenomena stohastične resonance pri nabiralcih energije
Nakano, K. – Cartmell, M.P. – Hu, H. – Zheng, R. Kimihiko Nakano*,1 – Matthew P. Cartmell2 – Honggang Hu1 – Rencheng Zheng1 1 Univerza
2 Univerza
v Tokiju, Inštitut industrijske znanosti, Japonska v Sheffieldu, Oddelek za strojništvo, Združeno Kraljestvo
V delu je predstavljen fenomen stohastične resonance v povezavi z elektromehanskimi nabiralci energije. Stohastična resonanca se pojavi tedaj, ko apliciramo periodično silo na bistabilen sistem, kar povzroči vibracije z velikimi amplitudami. V primerjavi s pojavom linearne resonance je pričakovati, da lahko z omenjenim pristopom dosežemo večje amplitude vibracij ter s tem boljše izkoristke nabiralcev energije. Nabiralci energije delujejo na principu transformacije vibracij iz okolja v električno energijo. Vibracijsko energijo lahko konvertiramo v električno energijo na osnovi več tipov elektromehanskih pretvornikov, ki bazirajo na elektromagnetnem, elektrostatičnem ali piezoelektričnem principu. Elektromehanski pretvorniki so tipično v obliki mehanskih resonatorjev, katerih izkoristek je največji v območju resonance. Generirana moč nabiralcev energije je tako zelo odvisna od frekvence vzbujanja. V tej smeri predstavlja enega izmed uveljavljenih pristopov za povečanje efektivnosti prenosa nabiranja energije izraba nelinearnih fenomenov sistema. V primerjavi z linearnimi sistemi se nelinearni sistemi lahko odzovejo z višjimi amplitudami v širšem frekvenčnem območju. Predstavljen je model bistabilnega elektromehanskega resonatorja v obliki konzolno vpetega nosilca, ki je v prečni smeri kinematsko vzbujan zaradi vibracij okolja. Dodatno je sistem tudi v osni smeri vzbujan z aktuatorjem v povezavi z dvema magnetoma. Preko magnetov, ki sta nameščena ločeno na prosti konec nosilca in fiksno na podlago, je nosilec izpostavljen šibki osni periodični sili. Periodično vzbujanje sistema vodi do nelinearnega odziva oziroma do pojava t.i. stohastične resonance, kar vodi do povečanega odziva. Na osnovi numeričnih simulacij je prikazana analiza vpliva oblikovnih in obratovalnih parametrov na intenziteto pojava stohastične resonance pri nabiralcih energije. Numerične simulacije potrjujejo možnost izrabe fenomena stohastične resonance za izboljšanje učinkovitosti nabiralcev energije. Ob upoštevanju energije, potrebne za generacijo periodične sile v osni smeri nosilca, je bila kumulativna generirana energija pri vzbujanju sistema v stohastični resonanci večja, kot tedaj, ko smo vzbujali sistem le z vibracijami okolja. Predstavljene ugotovitve, ki temeljilo le na numeričnih simulacijah, je potrebno v nadaljevanju potrditi tudi na osnovi meritev. Ključne besede: periodično vzbujanje, bistabilen sistem, nabiranje energije, stohastična resonanca
SI 62
*Naslov avtorja za dopisovanje: Univerza v Tokiju, Inštitut industrijske znanosti, 461 Komaba, Meguroku, Tokijo, Japonska, knakano@iis.utokyo.ac.jp
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 63 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131113 Prejeto popravljeno: 20140128 Odobreno za objavo: 20140328
Pregled in nadgradnja modela kompresijskega pretoka pri analizi plinskih satnih tesnil na podlagi visokotlačnega eksperimenta Saba, D. – Forte, P. – Vannini, G. Diego Saba1,* – Paola Forte1 – Giuseppe Vannini2
1 Univerza
v Pizi, Oddelek za gradbeništvo in industrijski inžiniring, Italija 2 GE Oil & Gas  Nuovo Pignone, Italija
Pri zasnovi plinskih tesnil za centrifugalne kompresorje in aksialne turbine je potrebno upoštevati tako tesnilno sposobnost tesnil kot tudi vpliv tesnil na dinamski odziv stroja. Pri kompresorjih so satna tesnila ter tesnila z luknjastim vzorcem pogosto uporabljana za tesnenje ravnotežnega bobna, ki se nahaja na koncu gredi ali na sredini gredi pri simetrični konfiguraciji kompresorja. V primerjavi z labirintnimi tesnili satna tesnila izkazujejo boljše dušilne lastnosti. V preteklosti je bilo razvitih in numerično implementiranih več modelov kompresijskega pretoka, ki predstavljajo efektivno orodje za napoved puščanja in dinamskega odziva satnih tesnil ter se s pridom uporabljajo v procesu razvoja kompresorja. V obstoječih modelih je privzetih več poenostavitev; model, s katerim razpolagajo avtorji, na primer privzame hipotezo izotermalnega procesa. Z višanjem zahtevane stopnje zanesljivosti razvojnega procesa prihaja do potreb po eksperimentalni validaciji numeričnega modela ter morebitnih izboljšav letega. V članku je izdelan pregled modela kompresijskega toka iz vidika občutljivostne analize pri različnih temeljnih hipotezah. Rezultati numeričnih simulacij so primerjani z eksperimentalno pridobljenimi rezultati na visokotlačnem preizkuševališču. Ključne besede: plinska tesnila, dinamika rotorjev, satne strukture, omejevanje vibracij
*Naslov avtorja za dopisovanje: Univerza v Pizi, Oddelek za gradbeništvo in industrijski inžiniring, Largo Lucio lazzarino 1, 56100 Pisa, Italija, diego.saba@for.unipi.it
SI 63
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 64 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131013 Prejeto popravljeno: 20140116 Odobreno za objavo: 20140401
Identifikacija parametrov večnadstropne palične konstrukcije na podlagi negotovih dinamskih podatkov Chakraverty, S. – Behera, D. Snehashish Chakraverty* – Diptiranjan Behera Nacionalni inštitut za tehnologijo Rourkela, Indija
Pri strukturni dinamiki je identifikacija parametrov lahko izredno zahtevna. Masni, togostni in dušilni parametri so pogosto potrebni, npr. za validiranje numeričnih modelov. Če se masne parametre lahko določi relativno enostavno, je več težav s togostnimi in predvsem dušilnimi parametri. Ta raziskava se osredotoči predvsem na določevanje togostnih parametrov na podlagi negotovih dinamskih podatkov. Začetne vrednosti identifikacije so lahko tiste vrednosti, za katere verjamemo, da so blizu dejanskim vrednostim; te vrednosti lahko določimo na podlagi 3D modela obravnavane strukture. S spreminjanjem teh začetnih pogojev in rezultatov eksperimenta se nato na podlagi iterativnega procesa lahko približamo dejanskim vrednostim. Začetne vrednosti pa lahko niso enostavno ali enolično določene; lahko te vrednosti vključujejo večjo mero negotovosti ali temeljijo na zelo omejenih informacijah. Te negotovosti se v splošnem lahko obravnavajo glede na pričakovani raztros vrednosti, lahko se obravnavajo v določenem intervalu, lahko pa tudi s pomočjo mehke logike. Ker pa na podlagi raztrosa vrednosti lahko delamo sklepe samo, če obstaja relativno velika množica podatkov, je tako imenovana mehka logika alternativen in zelo obetaven pristop, ki se je razvil v zadnjih desetletjih. Ta raziskava predstavlja uporabo mehke logike za identifikacijo togosti na podlagi negotovih podatkov. Primer, na katerem je pristop prikazan, je numerični model večnadstropne zgradbe. Negotovosti so modelirane v obliki trikotnih konveksno normiranih mehkih množic. Meje negotovosti posameznih parametrov pa so pridobljene na podlagi predlaganega iteracijskega algoritma, ki temelji na mehkih logiki. Predstavljeni problemi kažejo relativno veliko zanesljivost in natančnost identifikacijskega procesa; ki pa seveda zelo zavisi od začetnega intervala negotovosti podatkov. Keywords: togostna matrika, masna matrika, razvoj Taylorjevih vrst, lastna frekvenca
SI 64
*Naslov avtorja za dopisovanje: Nacionalni inštitut za tehnologijo Rourkela, Odisha, Indija, sne_chak@yahoo.com
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 65 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131202 Prejeto popravljeno: 20140130 Odobreno za objavo: 20140401
Vibracijsko utrujanje in strukturna dinamika pri harmonskih in naključnih obremenitvah Česnik, M. – Slavič, J. Martin Česnik – Janko Slavič*
Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija
Pričujoča študija eksperimentalno in teoretično raziskuje fenomen vibracijskega utrujanja pri harmonskih in naključnih obremenitvah na primeru preizkušanca iz aluminijeve zlitine. Poglavitni cilj raziskave je določiti vpliv sprememb modalnih parametrov, kateri se pojavijo pri vibracijskem preizkusu, na pravilnost numerične napovedi življenjske dobe preizkušanca. V prvem koraku so bili pridobljeni materialni parametri utrujanja na podlagi eksperimenta kinematskega harmonskega vzbujanja preizkušanca v neposredni bližini lastne frekvence. Med harmonskim preizkusom se je tekom akumulacije poškodbe spremljalo tako spremembe lastne frekvence preizkušanca kot tudi spremembe pripadajočega razmernika dušenja. V drugi fazi je bila na podlagi validiranega numeričnega modela preizkušanca pridobljena frekvenčna prenosnost napetosti glede na kinematsko vzbujanje preko vpetja z naključnim signalom. V zadnjem koraku se je ob upoštevanju predhodno pridobljenih materialnih parametrov utrujanja z uporabo numeričnega modela izračunalo pričakovano življenjsko dobo preizkušanca pri kinematskem vzbujanju z naključnim signalom. Numerična napoved življenjske dobe je bila primerjana z eksperimentalnimi rezultati, pridobljenimi z elektrodinamskim stresalnikom. Iz slednje primerjave je bil identificiran velik vpliv sprememb razmernika dušenja na pravilnost numerične napovedi življenjske dobe. Za primer obravnavanega preizkušanca se je izkazalo, da je potrebno za zanesljivo numerično napoved življenjske dobe potrebno upoštevati razmernik dušenja, ki je za 100% višji od izmerjenega razmernika dušenja pri nepoškodovanem preizkušancu. Predstavljena raziskava razkriva nove možnosti in kritične vidike na področju napovedi življenjske dobe pri visokocikličnem vibracijskem utrujanju dinamskih struktur. Keywords: strukturna dinamika, vibracijsko utrujanje, preizkus utrujanja, harmonsko vzbujanje, naključno vzbujanje, števne metode v frekvenčni domeni
*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, janko.slavic@fs.unilj.si
SI 65
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 66 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131130 Prejeto popravljeno: 20140214 Odobreno za objavo: 20140401
Koeficienti odboja in prenosa pri pravokotnih zarezah v ceveh Darryl K. Stoyko1,2,* – Neil Popplewell2 – Arvind H. Shah3 2 Univerza
1 Stress Engineering Services Canada, Kanada v Manitobi, Oddelek za strojništvo in proizvodno inžinirstvo, Kanada 3 Univerza v Manitobi, Oddelek za gradbeništvo, Kanada
Uporaba posamezne, nerazpršene lastne oblike ultrazvočnega vala predstavlja pomemben pristop pri spremljanju razvoja poškodb v strukturi. Bistveno prednost tega pristopa k neporušnemu testiranju predstavlja zmožnost zaznave poškodb na razdaljah več deset metrov. Za določitev lokacije poškodbe je potrebno pridobiti koeficiente odboja za dve ali več lastnih oblik. Sočasno vzbujanje večih lastnih oblik se izvede s kratkim udarnim impulzom na zunanjo površino strukture. Opisan način je v članku uporabljen na primeru votle okrogle cevi iz homogenega in izotropnega materiala z zanemarljivim dušenjem ter odprto pravokotno zarezo. V območju zareze je uporabljen model končnih elementov, ki je sklopljen z razširitvijo valovne funkcije v dveh sosednjih polneskončnih ceveh. Tipične vzdolžne in upogibne lastne oblike valov so obravnavane pri različnih dimenzijah zareze. V primeru osno nesimetrične zareze se, za razliko od osno simetričnih zarez, pojavi množica medsebojnih sklapljanj lastnih oblik valov, kar vodi do večjega števila singularnosti pri frekvenčni odvisnosti koeficienta odboja. Značilne singularnosti se pojavljajo v pogostem vzorcu. V članku je raziskana domneva, da je na podlagi zaznanih singularnosti, ki pripadajo določenim lastnim oblikam valov, možno prepoznati poškodbo cevi ter njene dimenzije. Ključne besede: cev, zareza, frekvenca odreza, singularnost, ultrazvočni valovi
SI 66
*Naslov avtorja za dopisovanje: Stress Engineering Services Canada, #125, 12111–40th Street S.E., Calgary, Alberta, T2Z 4E6, Kanada, Darryl.Stoyko@stress.com
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 67 © 2014 Strojniški vestnik. Vse pravice pridržane. Tematska številka
Prejeto v recenzijo: 20131213 Prejeto popravljeno: 20140214 Odobreno za objavo: 20140401
Formulacije končnih elementov na primeru modeliranja zunanjega ušesa
Volandri, G. – Carmignani, C. – Di Puccio, F. – Forte, P. Gaia Volandri* – Costantino Carmignani – Francesca Di Puccio – Paola Forte Univerza v Pizi, Oddelek za gradbeništvo in industrijski inžiniring, Italija
Delo, predstaljeno v članku, je del obširnejših raziskovalnih aktivnosti na razvoju virtualnega ušesa. Obravnavana raziskava se osredotoča na modeliranje bobniča in sluhovoda, ki predstavljata pomembni komponenti pri prenosu zvoka. Za modeliranje širjenja zvoka pri visokih frekvencah sta v članku uporabljeni standardna metoda končnih elementov (MKE) ter alternativna metoda (posplošena MKE). Uporabljeni metodi sta ovrednoteni na podlagi sklapljanja dveh domen  strukturna domena za bobnič, fluidna domena za sluhovod  ter v nadaljevanju uporabljeni za formulacijo interakcije strukturafluid. Za reševanje MKE analiz je v članku uporabljen program ANSYS, simulacije na podlagi modela posplošene MKE pa so pridobljene z implementacijo lastne kode v program Wolfram Mathematica. Rezultati simulacij vključujejo dinamski odziv in porazdelitev tlaka v sluhovodu ter porazdelitev pomikov pri bobniču. Identificirane lastne frekvence sluhovoda se ujemajo z objavljenimi rezultati v literaturi. Na podlagi validacije modela z zgoščevanjem mreže je bila standardna MKE prepoznana kot ustreznejša za potrebe modeliranja človeškega ušesa v slišnem frekvenčnem območju. Posplošena formulacija MKE se lahko izkaže za ustrezno v primeru modela celotnega ušesa, ki vključuje tudi glavo, ter v primeru analize ušesa v ultrazvočnem frekvenčnem območju. Ključne besede: metoda končnih elementov, sluhovod, simulacija, prenos zvoka
*Naslov avtorja za dopisovanje: Univerza v Pizi, Oddelek za gradbeništvo in industrijski inžiniring, Largo Lazzarino 56122, Pisa, Italija, gaia.volandri@ing.unipi.it
SI 67
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 6871 Osebne objave
Doktorske disertacije, magistrska dela, diplomske naloge
DOKTORSKE DISERTACIJE Na Fakulteti za strojništvo Univerze v Ljubljani so obranili svojo doktorsko disertacijo: ● dne 8. aprila 2014 Goran MIJUŠKOVIĆ z
naslovom: »Analiza dimenzijske natančnosti mikrofrezanja grafitnih elektrod« (mentor: prof. dr. Janez Kopač); V doktorskem delu je predstavljeno mikrofrezanje grafitnih elektrod. Za določitev dimenzijskih odstopkov v procesu obdelave je bil uporabljen sistematičen pristop. Na podlagi eksperimentalnega postopka je bila izboljšana pozicijska točnost CNC obdelovalnega stroja. Razviti so bili analitični modeli za hrapavost površine in odklon orodja. Eksperimentalno so bili raziskani vplivi rezalnih parametrov in naklona površine na hrapavost površine in odklon orodja. Napovedi analitičnih in empiričnih modelov so v skladu z eksperimentalnimi rezultati odklona orodja. Za karakterizacijo procesa v področju uporabnih parametrov so bili dodatno določeni vplivi na odklon, kot so trdota grafita, naklon površine, strategije frezanja in obraba orodja. Nazadnje je bila s primerjavo eksperimentalnih parametrov in parametrov proizvajalca orodij dokazana večja produktivnost prvih; ● dne 10. aprila 2014 Marijo TELENTA z naslovom: »Vetrovna zaščita cestnega profila« (mentor: prof. dr. Jožef Duhovnik);
Naloga predstavlja eksperimentalno in numerično proučevanje ograj za zaščito pred bočnim vetrom. Eksperimentalne raziskave se izvajajo zaradi vizualizacije pretoka zraka okoli ograj in da z njimi potrdimo numerične metode. Numerične simulacije se izvajajo zato, da dobimo vpogled v mehanizme pretoka zraka, ki prispevajo k aerodinamičnim silam na vozilo, saj doslej še ni bilo numeričnih raziskav pretoka okoli geometrijsko pravilnih vetrnih ograj in objekta v njenem zavetrju. Za numerično simulacijo smo uporabili turbulentni metodi URANS in DES. Analizirali smo značilnosti vetrne sledi za ograjo pri različnih naklonov drog. Poleg tega smo proučili scenarij prehodnega bočnega vetra, pri katerem smo numerično simulirali močne sunke vetra. Nazadnje smo izvedli parametrično numerično simulacijo na avtomatiziran način za različnih naklonov drog za določen scenarij bočnega vetra; ● dne 23. aprila 2014 Gregor TAVČAR z naslovom:
»Analitičnonumeričen 3D model za popis snovnih tokov v gorivnih celicah« (mentor: izr. prof. dr. Tomaž Katrašnik, somentor: izr. prof. dr. Viktor Hacker); SI 68
Doktorsko delo predstavlja Hibriden AnalitičnoNumeričen (HAN) pristop k modeliranju gorivnih celic uporabljen za modeliranje gorivne celice z ravnimi vzporednimi kanali. Jedro principa modeliranja HAN sloni na dveh inovativnih rešitvah: (i) numeričen 1D model za tok po cevi sklopljen z analitičnim 2D modelom za transport snovi v pravokotni ravnini (ii) model za transport tekoče vode zaobjet v modelu za transport plinaste snovi. Ti dve ključni lastnosti modela HAN omogočata: (i) modeliranje obratovanja v pogojih od popolne suhosti do močne poplavljenosti; (ii) doseganje visoke natančnosti rezultatov s polno 3D razločlivostjo in (iii) doseganje kratkih računskih časov primerljivih z 1D modeli; ● dne 25. aprila 2014 Luka SELAK z naslovom: »Platforma za podporo operacijam industrijskih delovnih sistemov« (mentor: prof. dr. Alojzij Sluga);
Osnovno vprašanje proizvajalcev industrijske opreme je, kako v konkurenčnem okolju povečevati prihodke in dodano vrednost. Na drugi strani lastniki industrijske opreme stremijo k zniževanju stroškov obratovanja in vzdrževanja industrijske opreme. Odgovor na ta izziv nudi koncept produktnostoritveni sistem (PSS). PSS predvideva tesno povezovanje deležnikov, dobavitelja opreme in odjemalca ter storitveno podporo industrijskih delovnih sistemov. Obravnavan industrijski delovni sistem v tej disertaciji so hidroelektrarne (HE). Disertacija temelji na hipotezi, da je potrebno za realizacijo PSS razviti platformo za podporo operacijam industrijskih delovnih sistemov (PPOIDS). Cilj PPOIDS je doprinos k višji izkoriščenosti tehnološke opreme delovnega sistema in večji ustvarjeni vrednosti v njeni eksploataciji za vse deležnike. V okviru PPOIDS je podrobneje obravnavan koncept PSS, sistem za spremljanje stanja in diagnostiko ter spletno okolje za sodelovanje. Predstavljen je nov pristop za spremljanje delovnih sistemov, odkrivanje napak in delitev informacij ter podporo preko virtualnega diagnostičnega centra. Za odkrivanje napak in diagnostiko je predstavljen pristop, ki temelji na odkrivanju napak z metodo podpornih vektorjev (SVM). Pilotna implementacija PPOIDS je prikazana na primeru podpore obratovanju in vzdrževanju HE Arto  Blanca na reki Savi, katere ključni PSS deležniki so podjetje Litostroj Power, Hidroelektrarne na spodnji Savi in Termoelektrarna Brestanica;
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 6871
* Na Fakulteti za strojništvo Univerze v Mariboru so obranili svojo doktorsko disertacijo: ● dne 16. aprila 2014 Janko FERČEC z naslovom:
»Vpliv napetostnega stanja na mikrostrukturo ortodontske spominske zlitine NiTi« (mentorica: doc. dr. Rebeka Rudolf); V doktorski disertaciji je obravnavan problem vpliva večosnega napetostnega stanja na začetek in potek superelastičnega področja pri spominski zlitini NiTi. Za raziskovalno delo je bila uporabljena komercialno dostopna ortodontska žica NiTi z vsebnostjo 50,6 at.%Ni. V prvem delu raziskav smo izvedli karakterizacijo izbranega materiala. Določili smo transformacijske temperature Ms, Mf, As, Af, module elastičnosti, napetosti in deformacije na začetku in koncu superlastičnega področja ter analizirali mikrostrukturo v izhodnem nedeformiranem stanju. Z insitu merjenjem električnega upora med obremenjevanjem pri enoosnem nategu na napravi za simulacijo enoosnega napetostnega stanja smo potrdili uporabnost te metode za določitev prehoda v superelastično stanje oziroma za spremljanje napetostno inducirane martenzitne fazne transformacije. Na osnovi teh rezultatov smo v drugem delu raziskav razvili napravo za simulacijo večosnega napetostnega stanja z možnostjo in situ merjenja električne upornosti in mikrotrdote. Z upogibom, torzijo ter kombinacijo torzije in upogiba smo simulirali obremenitve, ki so prisotne pri ortodontskem zdravljenju. S sočasnim merjenjem električnega upora smo za ta večosna napetostna stanja določili prehod v superelastično področje. Z analizo mikrostruktur pred in po obremenitvah smo identificirali spremembe in postavili modele razvoja mikrostrukture za različna napetostna stanja; ● dne 22. aprila 2014 Matjaž FLEISINGER z naslovom: »Močno vezane računalniške simulacije s tokom gnane darrieusove turbine« (mentor: izr. prof. dr. Matej Vesenjak); Za pridobivanje energije iz vodotokov se v zadnjem času uporabljajo tudi turbine, ki so postavljene prosto v tok in predstavljajo bistveno manjši poseg v prostor kot običajne elektrarne. Postopki za preračun zmogljivosti tovrstnih turbin temeljijo na različnih aerodinamičnih modelih, medtem ko so se metode računalniške dinamike tekočin na tem področju pričele uporabljati šele pred kratkim. Tudi slednje se izvajajo z vnaprej predpisanimi parametri, ki predstavljajo predvideno ustaljeno obratovanje turbine. Za določanje obratovalne karakteristike je tako potrebno več simulacij pri različnih obratovalnih parametrih. Za poglobljeno razumevanje delovanja teh turbin so bili razviti novi pristopi računalniških simulacij,
ki zajemajo pristop s tokom gnane turbine, kar predstavlja bistveno bolj realne pogoje v simulaciji. S tem pristopom je mogoče z eno simulacijo napovedati celotno obratovalno karakteristiko za določeno hitrost toka. S takšnim pristopom so bile izvedene parametrične simulacije geometrijskih parametrov turbine. Lopatice tovrstnih turbin, posebej pri namestitvi z vodoravno osjo vrtenja, predstavljajo dolge in vitke strukture, ki so med obratovanjem izpostavljene vodnemu toku. Ta vpliva na njihovo deformacijo, slednja pa na obtekanje lopatice. Posledica tega so spremembe v zmogljivostih turbine. Z uporabo kombinacije pristopa s tokom gnane turbine in močno vezanih simulacij medsebojnega vpliva tekočine in strukture je mogoče ovrednotiti vpliv deformacije lopatic na zmogljivosti turbine. Zato je bila razvita programska rutina, ki omogoča hkratno uporabo pristopa s tokom gnane turbine in močno vezanih simulacij. Za validacijo na novo razvitih postopkov simulacije je bil izveden eksperiment s pomanjšanim modelom turbine na naravnem vodotoku, pri katerem so bili izmerjeni povesi lopatice med obratovanjem turbine. Rezultati eksperimenta in simulacij so pokazali dobro ujemanje. Primerjava rezultatov simulacij izvedenih z uveljavljenimi postopki in na novo razvitimi pristopi je pokazala na znatne razlike v delovanju turbine, zato je smiseln nadaljnji razvoj in uporaba na novo razvitih pristopov; ● dne 23. aprila 2014 Luka LEŠNIK z naslovom: »Vpliv biogoriv na proces zgorevanja v dizelskem motorju« (mentor: prof. dr. Breda KEGL); Biogoriva predstavljajo nadomestek fosilnim gorivom in lahko pripomorejo k zmanjšanju onesnaževanja okolja s toplogrednimi plini, ki so produkt zgorevanja fosilnih goriv. Najpogosteje uporabljeno biogorivo v dizelskih motorjih je biodizel, ki ga v Evropi večinoma proizvajamo iz oljne ogrščice. Pred pričetkom uporabe biodizelskega goriva v dizelskem motorju je potrebno narediti številne raziskave o njegovem vplivu na delovne karakteristike motorja, količino formiranih emisij ter na življenjsko dobo motorja. V sklopu disertacije smo za potrebe preučevanja vpliva uporabe biodizelskega goriva na proces zgorevanja v dizelskem motorju, delovne karakteristike motorja in na količino formiranih emisij uporabili numerične simulacije in meritve na realnem motorju. Za izvedbo numeričnih simulacij je bil izdelan podmodel za določitev vseh potrebnih parametrov modela zgorevanja in podmodel za določitev vseh potrebnih parametrov emisijskih modelov. Izdelava podmodelov je bila zastavljena kot inverzni problem, ki je bil rešen z uporabo optimizacijske metode. V ta SI 69
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 6871
namen smo testirali tri optimizacijske metode in za nadaljnjo delo izbrali računsko najmanj potratno LM metodo. Podmodel za določitev parametrov modela zgorevanja je bil izdelan na podlagi izmerjenega poteka tlaka v zgorevalnem prostoru motorja na treh obratovalnih režimih za čisto dizelsko in čisto biodizelsko gorivo. V podmodel za določitev parametrov modela zgorevanja smo vključili le tiste parametre modela zgorevanj, za katere smo predhodno ugotovili, da jih bomo v postopku optimizacije lahko določili. Za izdelavo podmodela za določitev parametrov emisijskih modelov so bili uporabljeni rezultati meritev emisij dušikovih oksidov NOx in ogljikovega monoksida CO za čisto dizelsko gorivo, čisto biodizelsko gorivo in njuno mešanico B50. Nova podmodela sta omogočila določitev vseh potrebnih parametrov na podlagi poznanih lastnosti goriva in obratovalnega režima motorja in sta bila uporabljena za določitev vseh potrebnih parametrov za izvedbo simulacij. Iz primerjave rezultatov je razvidno dobro ujemanje med numerično in eksperimentalno dobljenimi rezultati, ki nakazuje na to, da sta izdelana podmodela primerna za določevanje vrednosti parametrov uporabljenega modela zgorevanja in uporabljenih emisijskih modelov; ● dne 23. aprila 2014 Gregor HARIH z naslovom: »Razvoj virtualnega modela človeške roke za ergonomsko oblikovanje izdelkov« (mentor: izr. prof. dr. Bojan Dolšak); Velik del ročnih opravil je še vedno opravljen s pomočjo ročnih orodij. Pravilna zasnova ročaja orodja je tako lahko ključnega pomena za preprečevanje obolenj. Obstoječe metode načrtovanja upoštevajo valjaste ročaje in podajajo smernice za določitev optimalnih premerov za povečanje zmogljivosti in zviševanje udobja ob zmanjševanju možnosti za nastanek akutnih in kumulativnih travmatičnih obolenj. Oblika ročaja in materiali ročaja doslej niso bili podrobneje raziskani, kar bi imelo vpliv na izboljšanje ergonomije izdelka. Za premostitev omejitev glede določitve oblike ročaja, smo razvili anatomsko natančen statični virtualni model človeške roke v optimalnem krepkem oprijemu za neposredno oblikovanje ročaja orodja, ki temelji na interdisciplinarnem pristopu na osnovi medicinskega slikanja. Da bi odpravili omejitve glede pravilne določitve materiala ročaja orodja, smo uporabili metodo končnih elementov za simulacijo človeškega prsta ob oprijemu ročaja iz različnih materialov. Rezultati so pokazali, da ročaj orodja, ki temelji na razvitem virtualnem modelu človeške roke, zagotavlja bistveno večjo kontaktno površino in udobje v primerjavi s cilindričnim ročajem. Z večjo SI 70
kontaktno površino in anatomsko obliko ročaja je mogoče preprečiti prekomerne deformacije mehkega tkiva in s tem prekomerne obremenitve na roko. Numerični izračuni so pokazali, da običajni materiali ročajev orodij ne zmanjšujejo kontaktnega tlaka ob oprijemu, predlagane hiperelastične pene, ki upoštevajo nelinearno mehansko obnašanje mehkega tkiva pa lahko znatno zmanjšajo kontaktni tlak in hkrati ohranijo zadostno stopnjo stabilnosti. Rezultati tako potrjujejo domnevo, da lahko pravilna oblika in material ročaja orodja povečata učinkovitost in udobje in s tem zmanjšata tveganje za nastanek akutnih in kumulativnih travmatičnih obolenj. DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 24. aprila 2014: Gal BEVC z naslovom: »Strukturna dinamika nelinearnega sistema z uporabo metode končnih elementov« (mentor: prof. dr. Miha Boltežar, somentor: izr. prof. dr. Janko Slavič); Andraž PAPLER z naslovom: »Celovita rešitev hlajenja merilnikov debeline na napravi za valjanje tanke pločevine« (mentor: prof. dr. Alojz Poredoš); Rok STROPNIK z naslovom: »Izboljšava izkoristka fotonapetostnega sprejemnika sončne energije s pomočjo fazno spremenljive snovi« (mentor: doc. dr. Uroš Stritih, somentor: prof. dr. Vincenc Butala). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv univerzitetni diplomirani inženir strojništva: dne 24. aprila 2014: Matej VURCER z naslovom: »Analiza procesa obdelave okvirja mikrovalovne pečice s pomočjo optičnega digitalizatorja« (mentor: prof. dr. Bojan Ačko, somentor: doc. dr. Andrej Godina). * Na Fakulteti za strojništvo Univerze v Ljubljani je pridobil naziv magister inženir strojništva: dne 24. aprila 2014: Žiga SIMŠIČ z naslovom: »Vibroakustična karakterizacija vodnega ventila« (mentor: prof. dr. Miha Boltežar, somentor: izr. prof. dr. Janko Slavič).
Strojniški vestnik  Journal of Mechanical Engineering 60(2014)5, SI 6871
* Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv magister gospodarski inženir: dne 04. aprila 2014: Vid PODGORŠEK z naslovom: »Ugotavljanje vpliva zlitinskih elementov na lastnosti vermikularne litine z ekonomsko analizo nastalih stroškov« (mentorja: prof. dr. Franc Zupanič, prof. dr. Polona Tominc); dne 24. aprila 2014: Biserka FLEGAR z naslovom: »Optimizacija procesa priprave terminskih planov v podjetju Paloma d.d. Sladki Vrh« (mentorja: doc. dr. Iztok Palčič, prof. dr. Damijan Mumel). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv magister inženir strojništva: dne 23. aprila 2014:
Primož GERČAR z naslovom: »Večkriterijsko optimiranje strege pri montaži z evolucijskim pristopom« (mentor: prof. dr. Miran Brezočnik); Iztok STOPEINIG z naslovom: »Primerjava modelov termomehanskega obremenjevanja« (mentor: prof. dr. Srečko Glodež). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv diplomirani inženir strojništva (UN): dne 24. aprila 2014: Dejan BEDENIK z naslovom: »Fizikalno modeliranje turbo polnilnika« (mentor: izr. prof. dr. Bojan Dolšak, somentor: asist. dr. Jasmin Kaljun); Aleš FERLEŽ z naslovom: »Konstruiranje univerzalnega sistema za sušenje kovanih in drugih izdelkov« (mentor: izr. prof. dr. Matej Vesenjak, somentor: asist. dr. Matej Zadravec). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva (VS): dne 10. aprila 2014: Peter IVANČIČ z naslovom: »Vitka proizvodnja komponent izdelkov« (mentor: izr. prof. dr. Janez Kušar, somentor: prof. dr. Marko Starbek).
Marko KOBLAR z naslovom: »Izboljšanje mazalnih sistemov na brusilnem stroju za vroče brušenje slabov« (mentor: prof. dr. Mitjan Kalin); Jernej MUHIČ z naslovom: »Vzdrževanje smetarskih vozil v komunalnem podjetju« (mentor: prof. dr. Mitjan Kalin); Ivo PREVC z naslovom: »Zanesljivost delovanja sesalnih enot ob prisotnosti trdih delcev in kapljevin« (mentor: izr. prof. dr. Jože Tavčar, somentor: prof. dr. Jožef Duhovnik); dne 11. aprila 2014: Gregor BREŠAN z naslovom: »Delavniška izdelava derez« (mentor: doc. dr. Joško Valentinčič, somentor: doc. dr. Andrej Lebar). Tilen CIGALE z naslovom: »Konstrukcija glave rotorja helikopterja z elastičnim vpetjem lopatice« (mentor: doc. dr. Viktor Šajn); Emiljan PREGELJ z naslovom: »Meroslovne značilnosti koordinatne merilne naprave« (mentor: izr. prof. dr. Ivan Bajsić); Edo SKOČIR z naslovom: »Obdelava oblikovnih matric s potopno elektroerozijo z dodatkom električno prevodnih delcev v dielektrik« (mentor: doc. dr. Joško Valentinčič Somentor: doc. dr. Andrej Lebar); Miha ŠINKOVEC z naslovom: »Konstrukcija hitroizvozne poti vzletnopristajalne steze« (mentor: doc. dr. Viktor Šajn, somentor: pred. Miha Šorn); Anže WEICHARDT z naslovom: »Nevarnost požara v letalu zaradi baterij prenosnih elektronskih naprav« (mentor: doc. dr. Boris Jerman, somentor: izr. prof. dr. Tadej Kosel); dne 14. aprila 2014: Rok MEŽIČ z naslovom: »Snovanje in konstruiranje kosmičnika in mlina za žito« (mentor: doc. dr. Samo Zupan, somentor: prof. dr. Ivan Prebil); Tomislav RAMUŠĆAK z naslovom: »Analiza izbočevanja pločevinskih ohišij « (mentor: izr. prof. dr. Tomaž Pepelnjak). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv diplomirani inženir strojništva: dne 24. aprila 2014:
Tomaž REGORŠEK z naslovom: »Konstruiranje kabine za trikolesno vozilo« (mentor: izr. prof. dr. Stanislav Pehan, somentorica: prof. dr. Breda Kegl); Jernej ŠLAMBERGER z naslovom: »Preiskava zvarnih spojev v podjetju Palfinger d.d.« (mentor:prof. dr. Franc Zupanič, somentor: doc. dr. Tomaž Vuherer).
SI 71
Strojniški vestnik – Journal of Mechanical Engineering (SVJME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue. The international conferences selected papers are welcome for publishing as a special issue of SVJME with invited coeditor(s). Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Founding Editor Bojan Kraut
University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SVJME, Aškerčeva 6, SI1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@svjme.eu, http://www.svjme.eu Print: Littera Picta, printed in 400 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
VicePresident of Publishing Council Jože Balič
University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: Reduced order system models are developed from uncoupled component modes. These reduced order system modes can be expanded using the uncoupled component modes provided they span the space of the system modes. This allows for expansion to full field from the reduced order system model. The significance is that the reduced order system model can be used for efficient dynamic response studies and the expansion allows for prediction of full field displacement and full field dynamic stressstrain. Courtesy: Structural Dynamics and Acoustic Systems Lab at UML, Massachusetts, USA
International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue). Institutional prices include print & online access: institutional subscription price and foreign subscription €100,00 (the price of a single issue is €10,00); general public subscription and student subscription €50,00 (the price of a single issue is €5,00). Prices are exclusive of tax. Delivery is included in the price. The recipient is responsible for paying any import duties or taxes. Legal title passes to the customer on dispatch by our distributor. Single issues from current and recent volumes are available at the current singleissue price. To order the journal, please complete the form on our website. For submissions, subscriptions and all other information please visit: http://en.svjme.eu/. You can advertise on the inner and outer side of the back cover of the magazine. The authors of the published papers are invited to send photos or pictures with short explanation for cover content. We would like to thank the reviewers who have taken part in the peerreview process.
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The journal is subsidized by Slovenian Research Agency. Strojniški vestnik  Journal of Mechanical Engineering is also available on http://www.svjme.eu, where you access also to papers’ supplements, such as simulations, etc.
Instructions for Authors All manuscripts must be in English. Pages should be numbered sequentially. The maximum length of contributions is 10 pages. Longer contributions will only be accepted if authors provide justification in a cover letter. Short manuscripts should be less than 4 pages. For full instructions see the Authors Guideline section on the journal’s website: http://en.svjme.eu/. Please note that file size limit at the journal’s website is 8Mb. Announcement: The authors are kindly invited to submitt the paper through our web site: http://ojs.svjme.eu. Please note that file size limit at the journal’s website is 8Mb. The Author is also able to accompany the paper with Supplementary Files in the form of Cover Letter, data sets, research instruments, source texts, etc. The Author is able to track the submission through the editorial process  as well as participate in the copyediting and proofreading of submissions accepted for publication  by logging in, and using the username and password provided. Please provide a cover letter stating the following information about the submitted paper: 1. Paper title, list of authors and affiliations. 2. The type of your paper: original scientific paper (1.01), review scientific paper (1.02) or short scientific paper (1.03). 3. A declaration that your paper is unpublished work, not considered elsewhere for publication. 4. State the value of the paper or its practical, theoretical and scientific implications. What is new in the paper with respect to the stateoftheart in the published papers? 5. We kindly ask you to suggest at least two reviewers for your paper and give us their names and contact information (email). Every manuscript submitted to the SVJME undergoes the course of the peerreview process. THE FORMAT OF THE MANUSCRIPT The manuscript should be written in the following format:  A Title, which adequately describes the content of the manuscript.  An Abstract should not exceed 250 words. The Abstract should state the principal objectives and the scope of the investigation, as well as the methodology employed. It should summarize the results and state the principal conclusions.  6 significant key words should follow the abstract to aid indexing.  An Introduction, which should provide a review of recent literature and sufficient background information to allow the results of the article to be understood and evaluated.  A Theory or experimental methods used.  An Experimental section, which should provide details of the experimental setup and the methods used for obtaining the results.  A Results section, which should clearly and concisely present the data using figures and tables where appropriate.  A Discussion section, which should describe the relationships and generalizations shown by the results and discuss the significance of the results making comparisons with previously published work. (It may be appropriate to combine the Results and Discussion sections into a single section to improve the clarity).  Conclusions, which should present one or more conclusions that have been drawn from the results and subsequent discussion and do not duplicate the Abstract.  References, which must be cited consecutively in the text using square brackets [1] and collected together in a reference list at the end of the manuscript. Units  standard SI symbols and abbreviations should be used. Symbols for physical quantities in the text should be written in italics (e.g. v, T, n, etc.). Symbols for units that consist of letters should be in plain text (e.g. ms1, K, min, mm, etc.) Abbreviations should be spelt out in full on first appearance, e.g., variable time geometry (VTG). Meaning of symbols and units belonging to symbols should be explained in each case or quoted in a special table at the end of the manuscript before References. Figures must be cited in a consecutive numerical order in the text and referred to in both the text and the caption as Fig. 1, Fig. 2, etc. Figures should be prepared without borders and on white grounding and should be sent separately in their original formats. Pictures may be saved in resolution good enough for printing in any common format, e.g. BMP, GIF or JPG. However, graphs and line drawings should be prepared as vector images, e.g. CDR, AI. When labeling axes, physical quantities, e.g. t, v, m, etc. should be used whenever possible to minimize the need to label the axes in two languages. Multicurve graphs should have individual curves marked with a symbol. The meaning of the symbol should be explained in the figure caption. Tables should carry separate titles and must be numbered in consecutive numerical order in the text and referred to in both the text and the caption as
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http://www.svjme.eu
60 (2014) 5
Strojniški vestnik Journal of Mechanical Engineering
Since 1955
Contents
Papers
287
Peter Avitabile, Christopher Nonis, Sergio E. Obando: System Model Modes Developed from Expansion of Uncoupled Component Dynamic Data
307
Andrea Barbarulo, Hervé Riou, Louis Kovalevsky, Pierre Ladeveze: PGDVTCR: A Reduced Order Model Technique to Solve Medium Frequency Broad Band Problems on Complex Acoustical Systems
314
Kimihiko Nakano, Matthew P. Cartmell, Honggang Hu, Rencheng Zheng: Feasibility of Energy Harvesting Using Stochastic Resonance Caused by Axial Periodic Force
321
Diego Saba, Paola Forte, Giuseppe Vannini: Review and Upgrade of a Bulk Flow Model for the Analysis of Honeycomb Gas Seals Based on New High Pressure Experimental Data
331
Snehashish Chakraverty, Diptiranjan Behera: Parameter Identification of Multistorey Frame Structure from Uncertain Dynamic Data
339
Martin Česnik, Janko Slavič: Vibrational Fatigue and Structural Dynamics for Harmonic and Random Loads
349
Darryl K. Stoyko, Neil Popplewell, Arvind H. Shah: Reflection and Transmission Coefficients from Rectangular Notches in Pipes
363
Gaia Volandri, Costantino Carmignani, Francesca Di Puccio, Paola Forte: Finite Element Formulations Applied to Outer Ear Modeling
Journal of Mechanical Engineering  Strojniški vestnik
298 Fabrício César Lobato de Almeida, Michael John Brennan, Phillip Frederick Joseph, Simon Dray, Stuart Whitfield, Amarildo Tabone Paschoalini: Measurement of Wave Attenuation in Buried Plastic Water Distribution Pipes
5 year 2014 volume 60 no.
Published on May 11, 2014
The Strojniški vestnik – Journal of Mechanical Engineering publishes theoretical and practice oriented papaers, dealing with problems of mod...