Journal of Mechanical Engineering 2014 3 by Darko Svetak

http://www.sv-jme.eu

60 (2014) 3

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

147

Ignacijo Biluš, Gorazd Bombek, Marko Hočevar, Branko Širok, Tine Cenčič, Martin Petkovšek: The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

158

Siamak Pedrammehr, Mehran Mahboubkhah, Mohammad Reza Chalak Qazani, Arash Rahmani, Sajjad Pakzad: Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces

172

Georgije Bosiger, Tadej Perhavec, Janez Diaci: A Method for Optodynamic Characterization of Erbium Laser Ablation Using Piezoelectric Detection

179

Mohammad Abbasi, Ardeshir Karami Mohammadi: A Detailed Analysis of the Resonant Frequency and Sensitivity of Flexural Modes of Atomic Force Microscope Cantilevers with a Sidewall Probe Based on a Nonlocal Elasticity Theory

187

Domen Rovšček, Janko Slavič, Miha Boltežar: Tuned-Sinusoidal Method for the Operational Modal Analysis of Small and Light Structures

195

Ertugrul Selcuk Erdogan, Olcay Eksi: Prediction of Wall Thickness Distribution in Simple Thermoforming Moulds

203

Pino Koc: Sea-wave Dynamic Loading of Sailing Yacht`s Retractable Keel

Journal of Mechanical Engineering - Strojniški vestnik

Contents

3 year 2014 volume 60 no.

Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) 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 SV-JME with invited co-editor(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 SV-JME, Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.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

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: The figures present the cavitation cloud shedding in the wake of cylindrical bluff body. Laboratory cavitation station was used. The cavitation cloud dynamics was observed during different operating conditions. High speed imaging visualization of cavitation cloud structure was performed simultaneously with pressure measurements.

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 single-issue price. To order the journal, please complete the form on our website. For submissions, subscriptions and all other information please visit: http://en.sv-jme.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.

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 60, (2014), number 3 Ljubljana, March 2014 ISSN 0039-2480 Published monthly

Papers Ignacijo Biluš, Gorazd Bombek, Marko Hočevar, Branko Širok, Tine Cenčič, Martin Petkovšek: The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station Siamak Pedrammehr, Mehran Mahboubkhah, Mohammad Reza Chalak Qazani, Arash Rahmani, Sajjad Pakzad: Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces Georgije Bosiger, Tadej Perhavec, Janez Diaci: A Method for Optodynamic Characterization of Erbium Laser Ablation Using Piezoelectric Detection Mohammad Abbasi, Ardeshir Karami Mohammadi: A Detailed Analysis of the Resonant Frequency and Sensitivity of Flexural Modes of Atomic Force Microscope Cantilevers with a Sidewall Probe Based on a Nonlocal Elasticity Theory Domen Rovšček, Janko Slavič, Miha Boltežar: Tuned-Sinusoidal Method for the Operational Modal Analysis of Small and Light Structures Ertugrul Selcuk Erdogan, Olcay Eksi: Prediction of Wall Thickness Distribution in Simple Thermoforming Moulds Pino Koc: Sea-wave Dynamic Loading of Sailing Yacht`s Retractable Keel

147 158 172 179 187 195 203

Original Scientific Paper

Received for review: 2013-09-26 Received revised form: 2013-10-13 Accepted for publication: 2013-12-03

The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station Biluš, I. – Bombek, G. – Hočevar, M. – Širok, B. – Cenčič, T. – Petkovšek, M. Ignacijo Biluš1,* – Gorazd Bombek1 – Marko Hočevar2 – Branko Širok2 – Tine Cenčič2 – Martin Petkovšek2 1 University

2 University

of Maribor, Faculty of Mechanical Engineering, Slovenia of Ljubljana, Faculty of Mechanical Engineering, Slovenia

The experimental analysis of unsteady cavitating flow has been performed to compare the static pressure dynamics and cavitation cloud structure dynamics. The analysis of the unsteady cavitation flow field was performed in the wake of bluff body in the laboratory cavitation station. The pressure oscillations were measured downstream of the bluff body with a recessed installation of the pressure sensor. The cavitation cloud structure dynamics was visualized using high speed camera. Pressure and image acquisition was performed simultaneously. The results of both measurements were analyzed in low frequency and high frequency intervals. The low frequency analysis of both pressure and cloud structure oscillations was performed in the interval from 0 to 1000 Hz. The high frequency analysis of the pressure fluctuations was performed with band pass filtering from 300 to 400 kHz and amplitude demodulation. Comparison of the static pressure and cavitation cloud structures fluctuations caused by cavitation cloud shedding in the wake of bluff body showed similarity between both signals. Two distinct frequencies of flow oscillations were recognized and the influence of cavitation number on the strength of pressure and cloud structure oscillations was quantified. The amplitude demodulation method was used to show and discuss the connection between the low and high frequency pressure oscillations. Keywords: cavitation, experiment, bluff body

0 INTRODUCTION Cavitation is the formation and then immediate implosion of cavities in a liquid that are the consequence of forces acting upon the liquid. It usually occurs when a liquid is subjected to rapid changes of pressure or velocity. Cavitation can cause different undesirable effects, such as performance loss, damage by pitting and erosion, structure vibrations and noise generation in the machinery. Cavitation can take different forms, depending on hydrodynamic conditions. Stable cavities, defined as sheet cavitation, may develop in the low pressure regions attached on solid walls. If sheet cavity increases over the critical size, periodic fluctuations of cavity appear. These oscillations are accompanied with adverse pressure gradients and downstream shedding of clouds. The cavitation cloud shedding in the wake of cylindrical bluff body is the subject of present study. Many numerical and experimental studies on cavitating flows have been performed. The numerical studies, performed nowdays mainly focus either on mathematical and physical model development [1], its optimization [2] or on cavitation erosion prediction [3]. The experimental studies have longer history. Pioneer work for the case of cavitation flow visualization around a hydrofoil was carried out by Wade and Acosta [4] in 1966. The major aim of study was to measure lift and drag coefficients of the

hydrofoil with and without the presence of cavity oscillations. Several studies were performed where frequency of cavitation clouds shedding was analysed. Among them are Kjeldsen et al. [5] who classified the types and length of cavitation by a combination of angle of attack and cavitation coefficient , and compared the range of periodical shedding of the cloud cavitation to the Strouhal number. The experiments performed by George et al. [6] revealed that oscillations of cavitation clouds possess similar range of Strouhal numbers. Cavitation clouds have complex structure and behavior. Measurements performed by Kubota et al. [7] showed that the cloud cavity consisted of a large-scale vortex and cluster of small vapor bubbles located in the center of the vortex. Such complexity of cavitation clouds is reflected in periodic nature and variability of pressure oscillations. Pressure oscillations were measured using piezo-electric transmitters and their classification into local and global pulses was performed by Reisman and Brennen [8]. The development of modern visualization techniques resulted in detailed shedding phenomena and frequency content investigation performed on a case of spherical bluff body by Brandner et al. [9]. Strongly connected to this paper are studies about simultaneous pressure measurements and visualization of cavitation cloud structure using high speed imaging. Reisman et al. [10] studied shock waves in cloud cavitation using high speed visualization and

*Corr. Author’s Address: University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia, ignacijo.bilus@um.si

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pressure measurements. Langa et al. [11] performed simultaneous visualization of cavitation cloud structure and pressure fluctuations in the cavitation tunnel. Visualization of cavitation cloud structure was performed with a high speed camera while pressure fluctuations were measured by an array of sensors in a thin piezoelectric membrane. Good agreement was observed among the measured variables. Few other studies of simultaneous visualization of cavitation cloud structure exist. More often cavitation cloud structures are compared to cavitation pit damage Keil et al. [12], Van Terwisga et al. [13], Heath et al. [14]. Quantification of cavitation by visualization with several approaches was reported in the literature. Quantification methods include counting bubbles, estimation of the area or volume of cavitation and measurements of average greyness intensity of cavitation clouds. Methods also differ in the arrangement of cameras and illumination. Leppinen and Dalziel [15] reported on the case of non overlapping travelling cavitation – bubbly two phase flows using background illumination. Here the camera was used to measure the attenuation of light as it passes through a bubbly flow, and this attenuation was related to the void fraction. Makiharju et al. [16] measured the void fraction distribution in gas–liquid flows using a twodimensional X-ray densitometry system. Void fraction

was measured based on transmittance of X rays through a test specimen, and sources of measurement uncertainty such as X-ray scatter, image distortion, veiling glare and beam hardening were considered. For very low void fractions, where individual bubbles were distinguishable, void fraction was also estimated by visualization method. Cavitation bubbles border detection and size estimation using two ellipse principal axes were performed. A limited comparison of X-ray void fraction measurement with estimation based on visualization method shows a fair agreement between both methods. The X-ray measurement method was compared to visualization by Bauer et al. [17] as well. Visualization was performed by diffuse illumination from the side of the test specimen, while experimental results showed superiority of X-ray measurement. Visualization was also used by Maurus et al. [18] in the case of subcooled flow boiling to estimate void fraction. Authors arranged illumination from the side and due to the spherical shape of the bubbles, light was reflected at the phase boundary layer between vapor and liquid in diverse directions. Thus, void fraction estimation analysis included several steps to measure the number and size of bubbles. Illumination from the side using a thin light sheet was also applied by Iyer and Ceccio [19] for measuring the influence of developed cavitation on the flow of a turbulent shear layer.

Fig. 1. Experimental setup, cavitation station with cavitation test section, detail shows bluff body and pressure transducer installation

148

Biluš, I. – Bombek, G. – Hočevar, M. – Širok, B. – Cenčič, T. – Petkovšek, M.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 147-157

In the present study we investigate cavitation around a bluff body with simultaneous visualization of cavitation cloud structure and pressure measurements and compare measured variables in the area of the wake. 1 THE EXPERIMENTAL SETUP The experimental setup in Fig. 1 was used for the experiment. It consists of cavitation station with cavitation test section and measurement equipment. It is more thoroughly discussed in the following subsections. 1.1 Cavitation Station Cavitation station is shown in Fig. 1. The main elements of cavitation station are water tank, main circulation pump, vacuum pump, cavitation test section and measurement equipment for cavitation station operational point regulation. The quantity of water in the cavitation station was approximately 1 m3. It was driven by a radial circulation pump. Unprepared tap water was used for the experiments. Water quality was not measured, but we replaced the water in the cavitation station prior to measurements with fresh tap water. Operating pressure in the system was set by a vacuum pump through a three way valve at the top of the water tank. Measurement equipment for cavitation station operating point selection comprised of pressure and volume flow rate measurement equipment. Operating pressure was measured with absolute pressure transmitter ABB 2600T Series 264 VS. Pressure drop on the test section was measured with absolute pressure transmitter ABB 2600T Series 264DS. Four pressure taps were located upstream and downstream of the test section. The uncertainty of the operating pressure measurement was estimated to ±0.5% of measured value. Volumetric flow rate was measured with electromagnetic flow meter ABB COPA-XL DL43F DN 125/PN 16. The measurement uncertainty of volumetric flow rate was up to ±1% of measured value. The experiment was performed with water temperature close to ambient temperature (T = 24 °C). Water temperature was measured with Pt100 thermometer mounted inside the water tank and connected to Agilent 34970A data acquisition unit. Temperature measurement uncertainty was estimated to ±0.5 °C.

1.2 Test Section Cavitation station test section is shown in Fig. 1. The test section dimensions were 50 mm (width) × 10 mm (height) × 800 mm (length). Pressure pulsations were generated by a bluff body with diameter of 16 mm. The height of the bluff body was the same as the height of the test section. The bluff body was manufactured from hard rubber and was compressed and held in place by wall friction. No cavitation was observed near the junction of the bluff body and test section wall. The bluff body was mounted 50 mm upstream from the pressure transmitter.

Measurement equipment consisted of a fast pressure transmitter and a high speed camera.

Pressure oscillations were measured with PCB Piezotronics pressure transmitter type 111A26 with frequency range from 0.01 Hz to 400 kHz and dynamic pressure range of 34.5 bar. The transmitter was mounted to the wall of the cavitation test section as shown in Fig. 1. The pressure transmitter was mounted in an insert, which was flush mounted to the test section lower wall. The transmitter was connected to the test section through a hole 3 mm in length and 2 mm in diameter and a cavity 2 mm in length and 5 mm in diameter, as shown in Fig. 1. The transmitter was protected against cavitation erosion with the above mentioned installation installation. Power to the transmitter was supplied using PCB Piezotronics 480C02 signal conditioner. The signal conditioner has a frequency range from 0.05 to 500 kHz. Pressure was acquired with 16 bit data acquisition board national Instruments NI-USB 6351 with 1.25 MHz sample rate. The convenient roll-off of the signal conditioner and selected frequency of acquisition enabled operation without aliasing filters. The duration of acquisition was 2 s. Data was stored to the disk immediately after the acquisition. Image sequences were acquired using high speed camera Fastec Hispec4 with frequency of acquisition of 10 kHz. Image resolution was 416×272, image depth 8 bit and shutter time was 30 µs. In every operational point 10000 images were acquired in a sequence, amounting to acquisition time 1 s. For all series of images, obtained by visualization method, camera settings for brightness and contrast were constant and equal. The lens used was manual Nikkor 50 mm, aperture F1:1.2. The camera was mounted perpendicular to the cavitation station wall at a distance of 1 m. Illumination was continuously provided using 8 CREE XM-L T6 LED lights. These were mounted to the side of cavitation test section at

The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

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a distance 20 cm parallel to the cavitation station test section wall. A 70 mm cylindrical lens was attached to LED lights. It was used to focus illumination on a light sheet with thickness of approximately the same size as the height of water above the pressure transmitter. The entire distance from the bluff body to the pressure transmitter was uniformly illuminated. Such installation also prevented any unwanted reflections and provided very strong illumination required by the short shutter time. Pressure and image acquisition was synchronized with the trigger from an electronic synchronization device. The trigger started the acquisition; later acquisition timing was performed using internal clocks of the data acquisition board and the camera.

did not take into consideration the shape of cavitation clouds, which has a major influence on the cavitation cloud void fraction. While the above procedure provided useful results, other measurement methods give better estimation of void fraction. Among them is x-ray computed tomography measurement [16] or time resolved 2D X-ray densitometry [21] method. A rectangular region of interest was used as shown in Fig. 2. We used the size of ROI 65×65 pixels which corresponds to 10.4×10.4 mm. The pressure transducer hole was located in the same position as the center of ROI. All sets of images were analyzed with ROI of the same size and position.

1.3 Image Processing and Void Fraction Estimation The high speed camera acquired series of eight-bit greyscale images that were showing cavitation inside the cavitation test section. For image post-processing image with pixels can be presented as a matrix with elements. Eight-bit resolution gives 256 levels of greyness A(i, j, n), which the matrix element can occupy (0 for black pixel and 255 for white pixel) A(i, j, n) ∈ {0,1,…, 255}. Each image is thus presented as a matrix:

 A(1,1, n) … A( I ,1, n)    Image ( n ) =  M O M  . (1)  A(1, J , n) … A( I , J , n)   

For the evaluation of brightness µ of a region of interest we averaged brightness of the pixels in the region of interest (ROI) [20]:

µ (n) =

1 I ROI ⋅ J ROI

∑∑A(i, j, n). (2)

Although the exact relationship between the of cavitation clouds void fraction A and average ROI brightness µ cannot be established with single BW camera and present image acquisition configuration, we assumed that cavitation cloud void fraction A is proportional to the brightness µ in a ROI corresponding to a cavitation cloud:

A(n) ∝ µ (n). (3)

This estimation is valid under the assumption that cavitation clouds are white, uniformly illuminated and that no reflections are recorded on them. We also 150

Fig. 2. The region of interest (ROI) in the wake of bluff body

Value of estimated void fraction was normalized in every operating point, such that the void fraction value 0 represents the lowest measured ROI brightness µ and void fraction value 1 represents the highest measured ROI brightness µ. Such normalization was performed in each operating point. 2 THE RESULTS The cavitating flow characteristics were observed at three different operating conditions with different cavitation cloud dynamics. The operating conditions are related to the cavitation number value σ and are presented in Table 1. The cavitation number σ was used as follows:

σ=

pref − pv , (4) 1 ρlU ∞2 2

where pref is reference pressure in the cavitation station, pv is vapour pressure, ρl is density of water and U is water velocity. Fig. 3 shows images of the cavitation cloud cycle in the wake of bluff body for observed operating points. It is evident that cavitation number influences

Biluš, I. – Bombek, G. – Hočevar, M. – Širok, B. – Cenčič, T. – Petkovšek, M.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 147-157

OP1

t = 1815 ms

OP1

t = 1816 ms

OP1

t = 1817 ms

OP1

t = 1818 ms

OP1

t = 1819 ms

OP2

t = 1759 ms

OP2

t = 1760 ms

OP2

t = 1761 ms

OP2

t = 1762 ms

OP2

t = 1763 ms

OP3

t = 1822 ms

OP3

t = 1823 ms

OP3

t = 1824 ms

OP3

t = 1825 ms

OP3

t = 1826 ms

Fig. 3. Images of cavitation in the wake of bluff body for three operating points

Fig. 4. The estimated void fraction sample sequence in the ROI and corresponding images of the phenomenon; images a-e were recorded at times, represented by solid vertical lines. Shown is operating point OP3 The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

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the cavitation cloud shedding length and void fraction (brightness) in the wake. The phenomenon shows periodic nature for all operating points. Table 1. Selection of operating points Operational point Reference pressure [bar] Flowrate [m3/h] Cavitation number

OP1 1.50 19.8 1.31

OP2 1.44 19.6 1.26

OP3 1.40 19.5 1.22

Time series show time evolution of analyzed variables. A sample time series of void fraction A is shown in Fig. 4. In addition one period of cavitation cloud shedding is accompanied with corresponding images of the phenomenon. The rectangular frame in Fig. 4 shows the location of ROI where void fraction A was estimated for each picture in the series. Void fraction fluctuates when cavitation structure passes the ROI. In Fig. 4, subimage a shows no cavitation cloud in the ROI and void fraction is low. Images b and c show an increase of cavitation cloud in the ROI, while void fraction increases. Image d shows approximately the highest amount of cavitation in the ROI; here void

OPI 1

In the following subsection we will present time series of measured variables, corresponding power spectra and amplitude demodulation analysis.

2.1 Time Series Analysis

OPI 3

OPI 2

Fig. 5. Time signals of pressure (a, c and e) and void fraction estimation (b, d and f) for operating points OP1 to OP3

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Local pressure fluctuations and cavitation bubble collapses near the pressure transducer sensing hole contribute more to the measured pressure, while there is no equivalent of this information in the void fraction estimation. Void fraction fluctuations show only low frequency fluctuations. Estimation of void fraction at high frequencies was not possible due to low camera acquisition speed of 10 kHz and spatial averaging, introduced by a large ROI. If we analyze the influence of cavitation number on the void fraction fluctuations, we can see that in the operating point OP1 and to a lesser extent also OP2 the number of peaks decreases (comparing Figs. 7b, d and f).

OPI 1

fraction is the highest. In image e cavitation cloud left the ROI and void fraction decreased. We now compare void fraction A estimation time series with pressure p measurements time series. Both variables were simultaneously acquired and recorded at the same distance from bluff body (Fig. 2). Pressure transducer hole is located in the same position as the center of the ROI. Fig. 5 shows pressure p(t) and void fraction A(t) time series for operational points OP1 to OP3. Both variables show periodic behavior for all operating points. Pressure measurements show fluctuations of low frequency, while high frequency pressure fluctuations are imposed upon low frequency content. The low frequency pressure fluctuations correspond to cavitation cloud shedding and their passage across the ROI.

OPI 2

OPI 3

Fig. 6. Low frequency power spectra of pressure (a, c and e) and void fraction (b, d and f), unfiltered data The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

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2.2 Signals in Low Frequency Domain Fig. 6 shows the unfiltered pressure and void fraction spectra in the low frequency domain. Several peaks of fluctuations show periodic behavior of pressure and void fraction in the region bellow 1 kHz. Among them are frequency peaks at 30 and 310 Hz in the pressure spectra and void fraction spectra. Pressure and existence of void fraction are closely connected. In the case of OP1, which corresponds to the highest cavitation number, there are few significant frequencies in pressure or void fraction power spectra. As shown in Fig. 6, periodic shedding of vortices from the bluff body is responsible for frequency peaks around 310 Hz. Peaks around 310 Hz in pressure spectra and in void fraction exactly match in frequency. We assume that unsteady flow behavior inside the cavitation station is responsible for measured pressure fluctuations with peaks around 30 Hz. The presumption of flow driven fluctuations at 30 Hz seems to be confirmed with the amplitudes of peaks at 30 Hz which do not follow cavitation intensity in an individual operating point (Figs. 6c and e). Pressure fluctuations are also represented in void fraction fluctuations, which follow pressure fluctuations with the same frequency. In accordance with Eq. (4), the amplitude of peaks at 310 Hz frequency domain for both signals decreases with the intensity of cavitation. In general, Strouhal number is used to describe unsteady and oscillating flow problems. It is defined as the ratio of inertial forces due to the unsteadiness of the flow or local acceleration to the inertial forces due to changes in velocity from one point to another in the flow. f ×d St = . v In our case the flowrate was nearly the same for analyzed operating points (Table 1). According to cavitation fluctuations frequency (310 Hz), bluff body diameter (16 mm) and cavitation section surface area (5×10–4 m2) the value of Strouhal number was constant (St = 0.45) regardless of the cavitation number value. 2.3. High Frequency Analysis and Amplitude Demodulation of High Pressure Fluctuations The study of high frequency vibrations and pressure fluctuations is a known technique of cavitation analysis [18]. The frequency analysis of pressure time series was performed in the high frequency range 154

with band pass filtering ranging from 300 to 400 kHz. Due to the recessed installation of pressure sensor as explained in section 1, pressure signal was damped in comparison with flush mounting installation. We estimated that damping in the selected frequency interval was around –10 dB. The 4th order Butterworth filter was used for signal conditioning. It is known that the high frequency spectral content of pressure fluctuations cannot solely be attributed to cavitation. Other flow excitations, bearing friction, vibrations, rubbing etc. may excite high frequency pressure fluctuation. To address this problem, the amplitude demodulation technique was applied for improved signal analysis in a cavitation tunnel [22] and in a prototype turbine model by Escaler et al. [23]. The Strouhal frequency related to the natural frequency of two phase flow interacted with flow induced pulses can be determined using the amplitude demodulation technique as shown by Abbot et al. [22], Farhat et al. [24], Callenaere et al. [25] and Escaler et al. [22] and [26]. In our case, pressure pulsations were measured in a broad frequency interval. We were able to record low pressure fluctuations as shown in Fig. 6. These correspond to the frequency of cavitation clouds shedding from the bluff body (Fig. 4). We will now discuss the connection between low frequency fluctuations and amplitude demodulation of very high frequency pressure fluctuations. The cavitation cloud that travels above the pressure sensor produces very high frequency pressure fluctuations in the frequency range of several 100 kHz. These fluctuations may be a result of local cavitation bubble collapses. The latter, however, appear only when the cavitation cloud is present near the pressure transducer. We can identify low frequency pressure pulsations using amplitude demodulation technique of local cavitation bubble collapses which appear in the very high frequency range. In the frequency domain, amplitude demodulation permits identification of frequencies associated to the dynamic behavior of the cavitation clouds. In our case, this corresponds to cavitation cloud periodic shedding. In the case of cavitation in turbine machinery runner, the cavitation can be identified at a modulated frequency of a particular hydrodynamic phenomenon [23]. The amplitude demodulation procedure starts with the first step of filtering of pressure time series with high pass or band pass filter to eliminate low frequency content. In the second step Hilbert transform of pressure H(p(t)) is calculated. The fast Hilbert transform performs the discrete implementation

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of the Hilbert transform with the aid of fast Fourier transform (FFT) routines. In order to simplify, the continuous Hilbert transform is written as follows: H ( p (t )) =

1 π

∫

∞

−∞

p (τ ) dτ . (5) (t − τ )

In the third and last step, low frequency amplitude envelope of the filtered high frequency signal is created. In our case of pressure fluctuations the amplitude demodulation was calculated as: ADM (t ) = H ( p ( t ) ) + p ( t ) . (6) 2

OPI 1

Signal processing was performed using Labview fast Hilbert transform virtual instrument subroutine. The resulting filtered amplitude demodulated envelope is shown in Figs. 7 a, c and e. The spectral analysis of the amplitude demodulated pressure is shown in Figs. 7, b, d and f. Filtered and amplitude demodulated envelopes of very high frequency pressure fluctuations show periodic pulsating nature, which is connected to the cavitation cloud shedding at the bluff body. The spectral analysis in Figs. 6, b, d and f shows peaks around the frequency of 310 Hz with increasing amplitudes in the cavitation more intensive operating regimes. This statement confirms that traveling

OPI 3

OPI 2

Fig. 7. Amplitude demodulated pressure time series (a, c and e) and spectra of amplitude demodulated pressure (b, d and f) The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

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cavitating clouds produce high frequency pressure fluctuations which correspond to local cavitation bubble implosions. Using the amplitude demodulation technique, these high frequency pulsations can be transformed to periodic pulsations at lower frequencies which are associated with the dynamic behavior of cavitation clouds. In our case, the envelope pulsation frequency directly corresponds to the shedding of cavitation vortices from the bluff body. The correlation is expressive for the regimes where Von Karman structures in the wake reach the position of the pressure sensor. 3 CONCLUDING REMARKS The simultaneous study of cavitation cloud dynamics and corresponding pressure pulsations in the cavitation station was presented in the paper. Cloud cavitation takes place in the external flow and initiates periodic pressure fluctuations. Unsteady flow nature with collapsing cavitation clouds follows the vortex shedding process with Von Karman type structures in the wake of the submerged bluff body, while cavitation intensity does not affect the Strouhal number value. The present investigation shows strong interaction between pressure and cavitation cloud structure dynamics. The similarity between high speed camera and pressure transmitter signals was observed in time and frequency domain. The study shows two dominant frequency bands, connected to cavitation cloud structure passing the area of interest and cavitation dynamics within the cloud. The low frequency cavitation instabilities show a connection to high frequency dynamics inside cavitation clouds when cavities in the Von Karman structure reach the position of pressure and cavitation cloud structure sensing. For further study, much higher speed and resolution of the image acquisition system and flush mounted pressure transmitter is preferred for more detailed analysis of cavitation clouds and interaction of pressure fluctuations. 4 REFERENCES [1] Singhal, A.K., Athavale, M.M., Li H., Jiang, Y., (2002). Mathematical basis and validation of the full cavitation model. Journal of Fluids Engineering, vol. 124, no. 3 p. 617-624, DOI:10.1115/1.1486223. [2] Biluš I., Morgut M., Nobile E. (2013). Simulation of Sheet and Cloud Cavitation with Homogenous Transport Models. International Journal of Simulation Modelling, vol. 12, no. 2, p. 94-106, DOI:10.2507/ IJSIMM12(2)3.229.

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[3] Dular, M., Coutier-Delgosha, O., (2009). Numerical modelling of cavitation erosion. International Journal for Numerical Methods in Fluids, vol. 61, no. 12, p. 1388-1410, DOI:10.1002/fld.2003. [4] Wade, R.B., Acosta, A.J. (1966). Experimental observation on the flow past a plano-convex hydrofoil. Transaction of ASME, Journal of Basic Engineering, vol. 87, no. 1, p. 273-283, DOI:10.1115/1.3645829. [5] Kjeldsen, M., Arndt, R.E.A., Effertz, M. (1999). Investigation of unsteady cavitation phenomena. Proceedings of the 3rd ASME/JSME Fluids Engineering Conference, San Francisco. [6] George, D.L., Iyer, C.O., Ceccio, S.L. (2000). Measurement of the bubbly flow beneath partial attached cavities using electrical impedance probes. Journal of Fluids Engineering, vol. 122, no. 1, p. 151155, DOI:10.1115/1.483237. [7] Kubota, A., Kato, H., Yamaguchi, H., Maeda, M. (1999). Unsteady structure measurement of cloud cavitation on a foil section using conditional sampling technique. Journal of Fluids Engineering, vol. 111, no. 2, p. 204-210, DOI:10.1115/1.3243624. [8] Reisman, G.E., Brennen, C.E. (1996). Pressure pulses generated by cloud cavitation. ASME Fluids Engineering Division Conference, San Diego. [9] Brandner, P.A., Walker, G.J., Niekamp, P.N., Anderson, B. (2007). An investigation of cloud cavitation about a sphere. 16th Australasian Fluid Mechanics Conference, Crown Plaza. [10] Reisman, G. E.,Wang, Y.C., Brennen, C.E. (1998). Observations of shock waves in cloud cavitation. Journal of Fluid Mechanics, vol. 355, no. 1, p. 255283, DOI:10.1017/S0022112097007830. [11] Langa, S., Dimitrov, M., Pelz, P.F. (2012). Spatial and temporal high resolution measurement of bubble impacts. 8th International Symposium on Cavitation, Singapore, DOI:10.3850/978-981-07-2826-7_210 [12] Keil, T., Pelz, P.F., Cordes, U., Ludwig, G. (2011). Cloud cavitation and cavitation erosion in convergent divergent nozzle. WIMRC 3rd International Cavitation Forum, Coventry. [13] Van Terwisga, T.J.C., Fitzsimmons, P.A., Ziru, L., Foeth, E.J. (2009). Cavitation Erosion – A review of physical mechanisms and erosion risk models. 7th International Symposium on Cavitation, Ann Arbor. [14] Heath, D., Širok, B., Hočevar, M., & Pečnik, B. (2013). The use of the cavitation effect in the mitigation of CaCO3 deposits. Strojniški vestnik - Journal of Mechanical Engineering, vol. 59, no. 4, p. 203-215, DOI:10.5545/sv-jme.2012.732. [15] Leppinen, D.M., Dalziel, S.B. (2001). A light attenuation technique for void fraction measurement of microbubbles. Experiments in Fluids, vol. 30, no. 2, p. 214-220, DOI:10.1007/s003480000158. [16] Mäkiharju, S.A., Chang, N., Gabillet, C., Paik, B.-G., Perlin, M., Ceccio, S.L. (2013). Time resolved two dimensional X-Ray densitometry of a two phase flow downstream of a ventilated cavity. Experiments in

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Fluids, vol. 54, no. 7, p. 1-21, DOI:10.1007/s00348013-1561-z. [17] Bauer, D., Chaves, H., Arcoumanis, C. (2012). Measurements of void fraction distribution in cavitating pipe flow using x-ray CT. Measurement Science and Technology, vol. 23, no. 5, p. 10, DOI:10.1088/09570233/23/5/055302. [18] Maurus, R., Ilchenko, V., Sattelmayer, T. (2002). Study of the bubble characteristics and the local void fraction in subcooled flow boiling using digital imaging and analysing techniques. Journal of Experimental Thermal and Fluid Science, vol. 26, no. 1, p. 147-155, DOI:10.1016/S0894-1777(02)00121-8. [19] Iyer, C.O., Ceccio, S.L. (2002). The influence of developed cavitation on the flow of a turbulent shear layer. Physics of Fluids, vol. 14, no. 10, p. 3414-3431, DOI:10.1063/1.1501541. [20] Osterman, A., Hočevar, M., Širok, B., Dular, M. (2009). Characterization of incipient cavitation in axial valve by hydrophone and visualization. Experimental Thermal and Fluid Science, vol. 33, no. 4, p. 620-629, DOI:10.1016/j.expthermflusci.2008.12.008. [21] Makiharju, S., Ganesh, H., Ceccio, S. (2012). Time resolved 2D X-ray densitometry of a cavitating wedge. Bulletin of the American Physical Society, 65th Annual

Meeting of the APS Division of Fluid Dynamics, San Diego. [22] Abbot, P.A., Arndt, R.E.A., Shanahan, T.B. (1993). Modulation noise analyses of cavitating hydrofoils. ASME-FED Winter annual meeting, Bubble Noise and Cavitation Erosion in Fluid Systems, vol. 176, p. 83-94. [23] Escaler, X., Egusquiza, E., Farhat, M., Avellan, F., Coussirat, M. (2006). Detection of cavitation in hydraulic turbines. Mechanical Systems and Signal Processing, vol. 20, no.4, p. 983-1007, DOI: 10.1016/j. ymssp.2004.08.006. [24] Farhat, M., Avellan, F., Pereira, F. (1992). Pressure fluctuations downstream a leading edge cavity. La Houille Blanche, vol. 47, no. 7/8, p. 579-585, DOI:10.1051/lhb/1992061. (in French) [25] Callenaere, M., Franc, J.P., Michel, J.M., Riondet, M. (2001). The cavitation instability induced by the development of a reentrant jet. Journal of Fluid Mechanics, vol. 444, no. 1, p. 223-256, DOI:10.1017/ S0022112001005420. [26] Escaler, X., Farhat, M., Egusquiza, E., Avellan, F. (2007). Dynamics and Intensity of Erosive Partial Cavitation. Journal of Fluids Engineering, vol. 129, no. 7, p. 886-893, DOI:10.1115/1.2742748.

The Experimental Analysis of Cavitating Structure Fluctuations and Pressure Pulsations in the Cavitation Station

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Original Scientific Paper

Received for review: 2013-04-27 Received revised form: 2013-08-17 Accepted for publication: 2013-10-30

Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S. Siamak Pedrammehr1,2,* – Mehran Mahboubkhah1 – Mohammad Reza Chalak Qazani3 – Arash Rahmani4 – Sajjad Pakzad4 University of Tabriz, Faculty of Mechanical Engineering, Iran University, Faculty of Engineering and Natural Sciences, Turkey 3 Tarbiat Modares University, Department of Mechanical Engineering, Iran 4 Islamic Azad University-Ajabshir Branch, Faculty of Mechanical Engineering, Iran 1

2 Sabanci

Assuming a sinusoidal machining force, the forced vibration of a machine tools’ hexapod table in different directions is addressed in the present study. A vibration model for the hexapod table is developed and the relevant explicit equations are derived. In the vibration equation of the table, the pods are assumed as spring-damper systems and the equivalent stiffness and damping of the pods are evaluated using experimental results obtained by modal testing on one pod of the hexapod table. The results of the analytical approach have been verified by FEM simulation. The theoretical and FEM results exhibit similar trends in changes and are close to each other. The vibration of the table in different positions has been studied for rough and finish machining forces for both down and up milling. The ranges of resonance frequencies and vibration amplitudes have also been investigated. The safe functional modes of the table in terms of its upper platform’s position have subsequently been determined. Keywords: hexapod, machine tool, vibration, modal analysis, cutting force

0 INTRODUCTION In recent years, the parallel mechanism has found extensive applications as the table or spindle of machine tools. Stiffness and appropriate vibration behavior are among the primary requirements for precision machining. This requires a thorough understanding of the dynamics of the mechanism. The purpose of the present study is to partially meet this need and fill the gap existing in the literature in this respect. The control of vibration in the machine tools with the aim of improving their performance has been the focus of much research in the literature. Dohner et al. [1] have proposed a finite element method (FEM) model to analyse chatter in the spindle of a milling machine set on a hexapod platform. In their model, in order to effectively control the chatter, flexible parts were added to the borders of the spindle. However, the analytical vibration relations of the hexapod was not given. Thus, their results could not be generalized to hexapods used as a machine table. Studies to examine vibrations of hexapods with applications in vibration isolation have been carried out. Hardage and Wiens [2] presented the results of a review of a mini-modal in a Hexel Tornado 2000, where they discussed flexibility modeling using finite elements. Their investigation suggested that characteristics of resonant frequency and stiffness are dependent on the configuration of the machine. Hardage [3] has studied the structural dynamics of parallel kinematic machine 158

tools (PKMs). In another work, Wiens and Hardage [4] have developed a methodology to identify the parameters of the structural dynamics of PKMs. They derived the analytical model for the simulation of the vibration response and modal parameters for a PKM. The accuracy of the model was verified through experimental modal analysis. Ting et al. [5] have derived a dynamic model for a Stewart platformtype computer numerical control (CNC) machine by means of the Euler-Lagrange approach. The average type force model for the end milling process has also been included in the dynamic model. In their research, an appropriate estimator gain was designed for the parameter adaptation law, which is useful for the estimation of the cutting parameters. Mukherjee et al. [6] have studied the analysis of dynamics and vibration of a flexible Stewart platform. In their research, the dynamic equations were derived through the Newton-Euler approach and a dynamic stability index was developed and validated. Hong et al. [7] derived the vibration model of a parallel machine tool. In their model, the pods of the parallel mechanism were considered as spring-damper systems. They performed stability analysis through a combination of the regenerative cutting dynamics model and the vibration model. Mahboubkhah et al. [8] and [9] investigated the free vibrations and the range of natural frequencies of the machine tool’s hexapod table using different configurations and two different analytical methods. According to their work, in the first method, the mass of the pods is ignored and, in

*Corr. Author’s Address: Sabanci University, Faculty of Engineering and Natural Sciences, Orhanli, Tuzla, 34956 Istanbul, Turkey, s.pedrammehr@gmail.com

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the second one, the mass of the pods is taken into account in the calculation of the natural frequencies of the platform. Pedrammehr et al. [10] have investigated free vibrations, natural frequencies and mode shapes of the hexapod table using different configurations and two different analytical methods. They also investigated the factors influencing the dynamic features of the hexapod table. In the present study, the time variable sinusoid force with specified frequency and amplitude has been assumed to be the external force acting on the platform and the forced vibration of the moving platform has been examined analytically. The resonance frequencies and range of vibrations of the platform have been calculated. To validate the analytical method, the forced vibration of the platform is examined by FEM simulation using the harmonic analysis in ANSYS Workbench. Resonance frequencies and the vibrations of the platform in different directions have been obtained for different configurations of the mechanism. Theoretical and FEM results are close to each other, exhibiting similar trends in changes. To closely investigate the forced vibration as the result of machining on the hexapod table, milling forces are modeled analytically and forced vibrations of the platform have been studied under different conditions of roughing and finishing in both up milling and down milling operations. After calculating the resonance frequencies and the range vibrations of the platform, proper configurations of the platform are presented in order to avoid dynamic instability in different machining operations.

used. The moving platform frame {P} is attached to the geometrical center of the moving platform. The location and orientation of the moving platform frame {P}, is specified according to the base frame {W}, which is attached to the geometrical center of the stationary platform.

Fig. 1. Schematic view of the milling machinesâ&#x20AC;&#x2122; hexapod table

1 VIBRATION MODEL OF THE HEXAPOD TABLE A schematic view of the hexapod table is shown in Fig. 1. The table is installed on the workshop floor into a rigid foundation. The hexapod under investigation consists of a moving platform, a stationary platform, and six similar pods with changing lengths connecting the two aforementioned platforms to each other. The pods are connected to the upper moving platform through spherical joints and to the lower stationary platform through universal joints. The hexapod table under study was developed for a three-axis CNC milling machine. The physical specifications of the hexapod are presented in Appendix 1. The vibration model proposed by the authors for the table is illustrated in Fig. 2. To describe the motion of the upper platform as the end effecter, two coordinate systems have been

Fig. 2. Vibration model of the hexapod table

In the vibration model of the table, pods and joints are taken to be flexible. Joints are frictionless and with negligible rotational damping. It is assumed that the moving platform is rigid in order to withstand deflection under the payload. The flexible model of the manipulator is shown in Fig. 2 (only one pod is depicted). The parameters

Forced Vibration Analysis of Milling Machineâ&#x20AC;&#x2122;s Hexapod Table under Machining Forces

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shown in this figure are as follows: mp, mu, md and mb are, respectively, the total mass of the moving platform (including payload), the mass of upper part of pods, the mass of lower part of pods and the mass of the base platform. Ip is the inertia tensor of the moving platform and the payload in the base frame. lpi, lui, ldi and lbi are the displacements of the moving platform, upper and lower parts of the pods, and base along ith pod’s extensional axis (i = 1 to 6 for six pods), respectively. Csi, Cui, Cai, Cdi and Cuni are, respectively, the damping coefficients of the spherical joints, the upper part of pods, the sliding joints, the lower part of pods, and the universal joints. Ksi, Kui, Kai, Kdi and Kuni are also the stiffness coefficients of the spherical joints, the upper part of pods, the sliding joints, the lower part of pods, and the universal joints, respectively. The total damping and stiffness coefficient of the upper joints of the platform, CTi and KTi, can also be theoretically obtained by a series combination of damper’s and spring’s rules [8], respectively. Details for Ip and theoretical CTi and KTi are given in the Appendix 2. 2 EQUIVALENT STIFFNESS AND DAMPING COEFFICIENTS OF THE TABLE In the given model, the pod is linked in the bottom to the fixed platform, moves with the moving platform in the upper part and bears part of the mass of the

platform, where the elastic elements are fixed at one end and at the other end bear the vibrational forces. So, modal testing by assuming the pod as a one-end post is close to the real situation; this is carried out by linking the pod form at the bottom to the base and agitating the pod from the upper part (Fig. 3). According to Fig. 3, the modal test for one pod of the hexapod table is carried out for three situations: fully limited, semi-open and fully open. To do the experiment, a pod of the hexapod table, a piezoelectric accelerometer (type 4507, B&K Inc.) and a shaker (type 4809, B&K Inc.) are used. Considering the special boundary condition of the mechanism with fixed degrees of freedom (DOFs) in earth connection of the structure, the pod is fixed at the bottom and the shaker is adjusted and suspended from the upper part in the ball screw direction and linked through a push rod to the upper part of the platform. A force sensor is located in the upper part of the pod in series with the alternate forces exerted by oscillator. An accelerometer is also linked to the upper part of the pod in the direction of its axis. The force is exerted in a periodic random manner on the pod to agitate all vibration modes. The experiments are conducted to find the natural frequencies of length and stiffness of the pods. Thus, accelerometers are always located in the direction of longitudinal agitation of the pods and as a result, longitudinal accelerations and forces are measured.

Fig. 3. Modal analysis of the pod in a one-end-engaged situation: a) pod in the fully limited situation, b) pod in the semi-open situation, c) pod in a fully open situation

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Force exertion is carried out alternately, in the range lower than 1 N, and at a starting frequency from 0 to 6.4 kHz on the pods. These forces are adjusted such that the accelerometer can measure the responses. If the force exerted is rather high, the sensor will be agitated and moved so that it will not measure correctly. Thus, the abovementioned force should be adjusted to an optimum level. The signal of the accelerometer and the shaker force are collected and analysed using a pulse system (type 3560, B&K Inc.). Using the Fast Fourier Transform (FFT) algorithm, the frequency response functions (FRFs) of the mechanism were extracted. Using STAR software, the results were analysed for different situations of the pod. The force and acceleration curves as frequency are given for each measurement and their ratio are used to calculate the stiffness. The results of the modal test are shown in the Fig. 4 as acceleration and dynamic stiffness curves of the pods for all situations under study. Curve fitting is utilized to calculate modal parameters such as frequency, damping and mode shape [11]. In this paper, the local single degree of freedom (SDOF) category of curve fit is employed to obtain the modal parameters of pods.

Once the FRF curves are obtained, damping ratio and natural frequency can be calculated easily using star software. As the largest value of the force is located at the resonance point, static stiffness, KSt, can be calculated from the function below [12]: K St = FDyn 2 X Dynζ , (1) where FDyn denotes the exerted force, XDyn the displacement of the element in the direction of the force in dynamic situation and ζ denotes the damping ratio. Table 1. Acceleration and dynamic stiffness for all situations in modal test Situation of pod Completely closed pod Semi-open pod Completely open pod

Natural freq. [Hz]

Damping ratio ζ

Stiffness KTi Damping CTi [N/m] [Ns/m]

2546

0.033

2.50e8

1031

2466

0.037

1.46e8

726

2368

0.027

1.05e8

366

K Dyn (ω =ω ) is defined at the resonance point as the following:

K Dyn (ω =ω ) = FDyn X Dyn . (2)

Fig. 4. Acceleration and dynamic stiffness for all situations in the modal test Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces

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According to Eqs. (1) and (2), KSt is calculated as the following:

that the results for the experimental modal test are close to the results obtained by FEM.

K St = K Dyn (ω =ω ) 2ζ . (3)

3 VIBRATION EQUATION OF THE TABLE Considering the equivalent stiffness and damping forces of the moving platform, the free-body diagram of the moving platform is illustrated in Fig. 6.

Fig. 6. Free-body diagram of the moving platform

Fig. 5. Mode of vibration in a one-end-fixed situation; a) pod in a fully limited situation, b) pod in a semi-open situation, c) pod in a fully open situation

Modal parameters of one of the pods of the hexapod table, namely, natural frequency, dumping ratio and static stiffness are calculated from modal test results for all situations and are given in Table 1. In order to corroborate the results, a three dimensional model of the pod has been extracted in Solidworks, and the natural frequencies of the pod in three different conditions of completely closed, semi-open and completely open are also obtained using FEM modal analysis under ANSYS. It should be noted that the frequencies using FEM are noticeable by the mode shapes for each condition. Fig. 5 illustrates the mode shapes of vibration for a completely closed pod with a natural frequency of 2671 Hz, a semi-open pod with natural frequency of 2516 Hz and a completely open pod with a natural frequency of 2453 Hz. It is clear 162

T and  In Fig. 6, u θT are the linear and angular acceleration, respectively, of the platform expressed in frame {W}; Fmac and Mmac, the harmonic machining force and moment vectors in local coordinate frame {P} being arbitrarily exerted to the moving platform, respectively. The force and moment can be expressed in {W} by the rotation transformation, R. Details for R are given in Appendix 2. The gravity and coriolis forces are negligible in vibration analysis and have been ignored. FCi and FKi are the total stiffness and damping forces, respectively, exerted on the platform and can be obtained as follows:

FCi = CTi ∆lTi , (4)

FKi = KTi ∆lTi , (5)

l ,=and KTi ∆lTi , are the absolute velocity and inFwhich Ci = CTi ∆F Ti Ki displacement of the junction along the ith pod’s axis. Considering J–1 as an inverse Jacobian matrix l = FCi =2), CTi ∆ FKi KTi ∆lTi , can be expressed in terms (Appendix Ti ,and of the linear velocity and displacement increments of the geometric center of the moving platform in the reference coordinate frame ( u T , θ T and uT, θT), as follows:

u  ∆I = J −1   T  , (6)  θT 

u  ∆l = J −1  T  . (7)  θT 

Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S.

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Considering the free body diagram of the moving platform illustrated in Fig. 6, the force equilibrium (Newton) equation and the moment equilibrium (Euler) equation for the geometric center of the moving platform can be written as follows:

i =1

where ω is the frequency related to the sinusoid force and moment.

T + ∑ ni FCi + ∑ ni FKi = −RFmac , (8) m pu 6

i =1

θT + ∑ qi × ni FCi + ∑ qi × ni FKi = I p  = −R (M mac + r × Fmac ).

(9)

where ni and qi are respectively the unit vector along the ith pod axis and the position vector of the ith platform connection point in frame {W}. r is the position vector of the mass centre of the moving platform and payload in frame {W}. Details for r are given in Appendix 2. Substituting Eqs. (4) and (5) into Eqs. (8) and (9) and then coupling these equations yields: mp I3  0 

T  0  u u  + J − T CT J −1   T  +  I p   θT   θT 

RFmac   u  + J − T K T J −1  T  = −   . (10)  θT   R (M mac + r × Fmac ) 

where I3 is the 3×3 identity matrix; KT is a 6×6 diametric matrix whose elements are the equivalent stiffness coefficient, KTi; CT is also a 6×6 diametric matrix whose elements are the equivalent damping coefficient, CTi. The coupled vibration equation of the platform can be expressed in compact form as:

  u u  u  M V T  + CV   T  + K V  T  = FV . (11)  θT   θT   θT 

4 FORCED VIBRATION OF THE TABLE USING AN ANALYTICAL APPROACH AND FEM To investigate the forced vibration of the table, the external force on the platform has been assumed to be a sinusoid force [12]; thus, Eq. (11) can be rephrased as follows:

  u u  u  M V T  + CV   T  + K V  T  = FV sin (ωt ), (12)  θT   θT   θT 

Fig. 7. The discretized model of the manipulator

The vibration of the moving platform can, ultimately, be analysed using the above equation when its matrix coefficients are specified. The resonance frequencies and vibrations of the platform in the linear and rotational directions (i.e. uT and θT) are obtained using a programme written in MATLAB. The programme developed in the present work uses the MATLAB routine ODE45, which is based on the 4th and 5th order Runge-Kutta formulas with adaptive step-size, for solving the system of differential equations. In order to verify the results, the resonance frequencies and the vibrations of the platform are also obtained using the Full Solution Method in FEM harmonic analysis. For this purpose, a three-dimensional model of the hexapod table was developed in Solidworks. The pods are neglected in the CAD model. The model is exported to the ANSYS Workbench and the equivalent damping and stiffness of the pods (i.e. CTi and KTi) are applied instead of the pods; the necessary input data as material properties are also applied, afterwards elements of the model executed using Solid element. The model has 12589 elements and 23208 nodes. In the case of a model with a maximum payload, a cubic part with a maximum mass is modelled on the moving platform; this model has 12884 elements and 23770 nodes. Afterwards, the relevant boundary conditions are applied to the foundation connection of the manipulator and harmonic analysis was then carried out to obtain the harmonic response of the platform. The discretized model in the ANSYS Workbench with the relevant boundary conditions and external forces acting on the moving platform are shown in Fig. 7.

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in the frequency domain and in x, y and z directions. There are some lower peaks in the diagrams, and these peaks are generally due to some important resonance frequencies in other directions. For instance, the first lower peak in the figure for vibrations in the z-direction is at 112 Hz which directly shows the effect of vibrations in the x direction. The vibrations in the x direction (3.8e-3) are considerably higher than vibrations in the z direction (1.25e-5). Therefore, these vibrations may be visible in harmonic analysis of vibrations in the z direction. Table 2. Different configurations of the moving platform Configuration number 1 2 3 4 5 6

Fig. 8. Vibrations of the moving platform obtained using an analytical approach and FEM

The forced vibration of the platform has been examined using an analytical approach and FEM for a configuration of the mechanism where pods are completely open and the moving platform is in its upmost position, disoriented and without payload. In this simulation the amplitudes of the external moment have been assumed to be zero and the external force have been presumed to be Fext= [500 N 500 N 500 N]T and for the simulation. As a result of the analytical and FEM simulation, the ratio of the response amplitude to the base excitation amplitude against the base excitation frequency are shown in Fig. 8 as the amplitude of the displacements and rotations of the moving platform 164

Centre position of the moving platform Bottommost Position Middle position Upmost position

Centre position of the moving platform X, Y, Z [mm] 0, 0, 710 0, 0, 710 0, 0, 820 0, 0, 820 0, 0, 930 0, 0, 930

Mass of the platform and the payload [kg] 40.6 90.6 40.6 90.6 40.6 90.6

In general, the reason for the extra lower peaks predicted by FEM may attributed to the FEM harmonic analysis in the ANSYS Workbench program. In this analysis all the vibrational properties of the system are taken into account and a complete vibration model of the system is used to obtain resonance frequencies and displacements, whereas in the analytical model written in MATLAB, a reduced model is applied. CTi and KTi, are greatly influenced by the variation in the position and orientation of the moving platform. Total mass and inertia of the moving platform, mp and Ip, could also be affected by the impact of the variation in the weight and shape, respectively, of the dead weight installed on it. Depending on the configuration of the platform, natural and resonance frequencies will vary significantly. Therefore, six different configurations have been selected in the vicinity of the workspace to specify variations in the resonance frequencies of the moving platform (see Table 2). The results of the analytical approach and FEM for the resonance frequencies of the moving platform in different configurations are reported in Table 3. In Table 3, six resonance frequencies are presented for six linear and rotational directions. According to this table, the results of both methods are highly consistent. This can be better visualized from the comparative diagrams in Fig. 9.

Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S.

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Table 3. Resonance frequencies of the platform (in Hz) obtained using an analytical approach and FEM Configurations (Table 1) Frequency Analytical along x axis FEM Frequency Analytical along y axis FEM Frequency Analytical along z axis FEM Frequency Analytical around x axis FEM Frequency Analytical around y axis FEM Frequency Analytical around z axis FEM

1 218 211 218 213 356 358 838 729 859 732 912 856

2 146 156 146 157 314 303 557 523 573 528 604 546

3 147 147 147 148 239 246 665 619 699 621 707 634

4 98 104 98 105 211 200 431 426 444 428 469 456

5 111 112 111 113 181 187 570 557 599 597 605 602

6 75 81 75 81 160 152 369 374 380 377 403 385

The least resonance frequencies occur in the configuration in which the moving platform with the most payloads is in the upmost border of the workspace (configuration six of Table 2). This is the configuration where the length of the pods is at a maximum, i.e. they have the lowest stiffness coefficients. With a decrease in the equivalent stiffness of the pods and an increase in the mass, the natural frequencies of the table will be diminished. On

the other hand, upon reduction in the length of the pods in the lower position of the moving platform, their equivalent stiffness will be increased. This phenomenon and the decrease in the mass of the payload on the platform are two factors leading to an increase in natural frequencies. Thus, it is obvious that the factors influencing natural frequencies will also significantly influence the resonance frequencies of the moving platform. It is accepted that changes in the payload on the platform and the position (especially in direction z) will induce more changes in total mass and stiffness matrices in the vibration equation of the platform, and consequently the natural frequencies and the resonance frequencies of the moving platform will be affected. The results of both the analytical approach and FEM are obtained for the range of the maximum vibrations in different directions and in each of the configurations of Table 2 and are presented in Table 4. Another observation is that changes in the magnitude of the external force and moment do not cause any change in the resonance frequencies of the moving platform. It is obvious that these changes are directly related to the change in the amplitude of the vibrations.

Fig. 9. Comparison between the results of the analytical approach and FEM for vibrations of the platform Table 4. Range of maximum vibrations of the platform (in mm) for different configurations of the table obtained using the analytical approach and FEM

Analytical FEM

Max vibration along X 8.57e-4 to 2.94e-3 7.80e-4 to 3.78e-3

Max vibration along Y 8.67e-4 to 2.97e-3 7.53e-4 to 3.34e-3

Max vibration along Z 3.18e-5 to 1.62e-4 3.86e-5 to 2.15e-4

Max vibration along X 8.10e-5 to 2.93e-4 7.63e-5 to 3.93e-4

Max vibration along Y 6.24e-6 to 1.91e-5 5.34e-6 to 2.02e-5

Forced Vibration Analysis of Milling Machineâ&#x20AC;&#x2122;s Hexapod Table under Machining Forces

Max vibration along Z 1.64e-5 to 2.97e-4 2.06e-5 to 3.64e-4

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forces can be expressed as a function of varying uncut chip area ah(φ); yielding:

5 FORCED VIBRATION OF THE TABLE UNDER MACHINING FORCES Forced vibrations have a strong impact on the machining process when one or more of the frequencies of the cyclic shock and varying cutting force are equal or close to one or more of the natural frequencies of the machining system [13]. To this end, it seems vital to devise a comprehensive model with the capability of exactly predicting the machining process, depending on the exact modelling process of the machining forces. Thus, in order to closely examine the forced vibration as a result of the machining process on the hexapod table under study, milling forces have been modelled taking into account the range of changes in forces and different milling operations (up and down). In essence, determining the forced vibration of the platform and examining its forced response under harmonic milling forces plays a crucial role in eliminating resonance. The relationship between the direction of tool rotation and feed defines the two types of milling operations: up milling and down milling [14] and [15]. Considering the end milling process model (Fig. 10), the magnitude of forces and moments during milling are comparable. In this study, forces and moments during milling have been modelled using force equations presented in [16] using a code written in MATLAB. As mentioned in reference [16], tangential Ft(φ), radial Fr(φ) and axial Fa(φ) components of cutting

Ft(φ) = Kt ah(φ) ,

(13)

Fr(φ) = Kr ah(φ) ,

(14)

Fa(φ) = Ka ah(φ) ,

(15)

where Kt, Kr and Ka are the cutting force coefficients contributed by the shearing action in the tangential, radial and axial directions, respectively; φ is the instantaneous angle of immersion. a is the edge contact length (axial depth of cut) and h(φ) is the time dependent chip thickness variation. Chip thickness can be expressed as: h(φ) = fz sin (φ) , (16)

in which fz is the feed per tooth [mm/rev-tooth]. Considering Fig. 10, horizontal Fx (φ), normal Fy (φ) and axial Fz (φ) components of cutting forces can be derived as:

Fx (φ) = – Ft (φ) cos (φ) – Fr (φ) sin (φ), (17)

Fy (φ) = – Ft (φ) sin (φ) – Fr (φ) cos (φ), (18)

Fz (φ) = Fa (φ) , (19)

The instantaneous cutting torque, Tc (φ), can be obtained as: Tc (φ) = Ft (φ) D/2 , (20)

where D is the diameter of the milling cutter.

Fig. 10. Geometry of the milling process with cutting force components on the tooth Table 5. Range of the resonance frequencies of the platform (in Hz) for different configurations of the manipulator and different cutting conditions

Min Max

166

Resonance freq. along x

Resonance freq. along y

74 178

72 175

Resonance freq. along z 388 748

Resonance freq. around x

Resonance freq. around y

Resonance freq. around z

380 712

358 665

156 278

Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S.

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Considering the spindle speed, n [rpm], and the number of teeth on cutter, Z, tooth passing frequency, f [Hz], can be obtained as:

f = nZ / 60 .

(21)

In this study, milling forces and moments have been presumed to be the external forces and moments acting on the moving platform; therefore, the external force and moment vectors can be defined as Fmac = [Fx (φ) Fy (φ) Fz (φ)]T and Mmac = [0 0 Tc (φ)]T, respectively. In this way, forced vibrations of the platform as the result of milling forces can be examined, since the mass, damping, stiffness and force matrices in Eq. (11) are known. 6 CASE STUDY Assuming the following parameters for each condition of roughing and finishing of the workpiece (material: Ti6Al4V) with an end mill cutter with three flutes and 20 mm diameter: roughing: ae = 20 mm, a = 4 mm, fz = 0.15 mm, finishing: ae = 20 mm, a = 1 mm, fz = 0.10 mm, where ae is the radial depth of cut. Considering the spindle speed as 200 to 20000 rpm, tooth passing frequency is calculated using Eq. (21) and vibrations of the moving platform in the frequency domain of 10 to 1000 Hz, corresponding to the cutting speed of the spindle in all conditions of roughing and finishing for both up and down milling operations and for the configurations mentioned in Table 2 have been analysed. The findings of the analytical approach are given in Table 5 for the range of the resonance frequencies of the platform for different configurations of the manipulator and different cutting conditions. The range of the maximum vibration amplitudes of the moving platform as a result of the milling forces has been examined, considering the different milling operations and the different configurations of the platform mentioned in Table 2. Table 6 lists the results of the simulation.

In the present investigation, the lowest vibrations in all directions occur in the configuration in which the centre of the moving platform is located in the bottommost position of the workspace and up milling forces are applied in finishing. On the other hand, with the centre of the platform in the upper position and during roughing, more vibrations will occur. For example, the amplitude of vibrations and the resonance frequencies of these forces for the upmost position of the disoriented moving platform have been investigated against the base excitation frequency and for the minimum and maximum payload on it. The results of the simulation are illustrated in Figs. 11 and 12 for different cutting conditions. Fig. 11 illustrates the amplitude of the vibrations and resonance frequencies for the upmost position of the disoriented moving platform under up milling and for two different machining strategies of roughing and finishing. Fig. 12 also presents the results obtained by analytical simulation as the amplitude of vibrations and resonance frequencies for the upmost position of the disoriented moving platform under down milling and for two different machining conditions of roughing and finishing. Considering the results of the displacements of the moving platform as the result of different milling forces, one can infer that the range of displacement of the moving platform is proportionate to the applied force on it. Therefore, all effective parameters changing the cutting forces including cutting and geometrical parameters will influence the amplitudes of the vibrations of the moving platform. For instance, increasing the feed rate, depth of cut and number of teeth of the cutter causes an increase in cutting force parameters [17] and [18]. Thus, an increase in these factors leads to enhancement of vibrations of the platform under machining forces. It is also to be noted that increasing the spindle speed will decrease the coefficients of the cutting forces and therefore the cutting force and amplitudes of the vibrations of the platform will be decreased.

Table 6. Range of vibrations of the platform (m for displacements and rad for rotations) for different configurations of manipulator and different cutting conditions Vibration along x Roughing 9.96e-5 to 4.53e-4 Up milling Finishing 1.76e-5 to 8.03e-5 Down Roughing 1.99e-5 to 1.37e-4 milling Finishing 4.04e-6 to 2.71e-5

Vibration along y 1.21e-4 to 5.93e-4 2.48e-5 to 1.34e-4 1.84e-4 to 9.01e-4 3.03e-5 to 1.49e-4

Vibration along z 9.55e-6 to 3.71e-5 9.35e-7 to 6.87e-6 5.40e-6 to 3.48e-5 9.51e-7 to 6.25e-5

Vibration around x 9.30e-5 to 2.93e-4 1.55e-5 to 4.47e-5 1.26e-4 to 3.93e-4 2.13e-5 to 6.48e-5

Vibration around y 5.27e-6 to 2.01e-5 1.20e-6 to 3.17e-6 3.95e-5 to 1.38e-4 6.76e-6 to 2.11e-5

Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces

Vibration around z 1.64e-4 to 2.97e-3 3.12e-5 t0 6.71e-4 1.82e-4 to 3.14e-4 3.24e-5 to 6.59e-4

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Fig. 11. Vibrations of the moving platform in different directions and in different up milling operation: a) roughing, b) finishing

7 CONCLUSION AND DISCUSSIONS In the present study, forced vibrations of the hexapod table were examined using two methods, analytical and finite elements. Considering a sample sinusoid force acting on the platform, the resonance frequencies and the range of vibrations of the platform were calculated based on a programme written in MATLAB. In 168

order to verify the results of the analytical approach, harmonic analysis of the table was carried out in ANSYS Workbench under the same conditions. It was found that the results of both methods exhibit a satisfactory level of consistency. In this study, examination of the results of the analytical approach and FEM indicated that the mechanism of the case study can be calculated in terms of the characteristics

Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S.

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0 Fig. 12. Vibrations of the moving platform in different directions and in different down milling operations: a) roughing, b) finishing

of the external forces. Therefore, the resonance frequencies and the vibrations of the moving platform can be obtained for any periodic external force when the mass, damping and stiffness matrices are available. Forced vibrations of the moving platform were examined in different configurations and at different resonance frequencies and the range of platform vibrations for different machining operations were calculated in the present study. Determination of the resonance frequencies and the range of the vibrations

of the platform as a result of the machining forces is the best method for defining conditions conducive to resonance. Thus, with a careful choice of machining parameters, conditions conducive to chatter in the milling process can be avoided. According to the investigations, the lowest resonance frequencies occur in a configuration in which the moving platform is in the upmost position with the maximum payload on it. On the other hand, maximum vibrations of the moving platform are found

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for the same configuration when up milling forces are applied during the roughing process. Therefore, in order to avoid dynamic instability during the machining process in upper positions, higher cutting speed and lower cutting force are recommended and hence, upper positions of the platform will be the most proper position for high speed machining. Finishing is carried out at high cutting speeds with a lower feed rate and depth of cut. Thus, machining forces exerted on the platform will have a small magnitude under finishing conditions. It can be concluded that upper positions of the platform are the best positions for the finishing process. To produce an appropriate surface finish in upper positions of the moving platform, down milling would be the best option. Shortening the length of the pods in lower positions of the moving platform together with decreasing the payload mass are two factors contributing to the increase in the resonance frequencies of the hexapod table. Therefore, the lower positions of the platform will help the mechanism withstand high machining forces. In lower positions of the platform, the vibrations as the result of these forces would be at a minimum. Therefore, in the lower positions of the platform, machining at lower cutting speeds would be possible and these positions would be appropriate for roughing. Furthermore, in order to decrease shocks by roughing in the lower positions, up milling would be the most appropriate method. 8 APPENDIX 1 The physical specifications of the test manipulator are as follows: Radius of the moving platform 175 mm; Radius of the base 400 mm; Angular distance between two adjacent spherical joints 30˚; Angular distance between two adjacent universal joints 14˚; Minimum length of each pod 760.2 mm; Maximum length of each pod 968.9 mm; Minimum mass of moving platform together with payload 40.6 kg; Maximum mass of moving platform together with payload 90.6 kg; Maximum course in x axis = ±130 mm, in y axis = ±130 mm, and in z axis = 220 mm. 170

9 APPENDIX 2 Considering PIp as the inertia tensor of the moving platform and the payload in frame {P}, the inertia tensor of the moving platform and the payload in base frame, Ip, can be obtained using the parallel axes theorem [19] and [20], which yields:   ry2 + rz2 P  I p = R  I p + m p  −rx ry   −rx rz  

− rx ry rx2 + rz2 − ry rz

− rx rz    −ry rz   R T , rx2 + ry2   

where R is the rotation 3×3 matrix, representing the rotation of the frame {P} related to frame {W}, and can be obtained by: Cθ z Cθ y  R =  Sθ z Cθ y  − Sθ y 

− Sθ z Cθ x + Cθ z Sθ y Sθ x Cθ z Cθ x + Sθ z Sθ y Sθ x Cθ y Sθ x

Sθ z Sθ x + Cθ z Sθ y Cθ x   −Cθ z Sθ x + Sθ z Sθ y Cθ x  ,  Cθ y Cθ x 

in which Cθx = cos (θx) and Sθx = sin (θx) and the vector r = [rx ry rz]T is the position vector of the mass centre of the moving platform and the payload in frame {W} and can be obtained by: r = Rro, in which ro is the position vector of the mass centre of the moving platform and the payload in frame {P}. The inverse Jacobian matrix can be expressed as:

n1T  J −1 =   n 6T 

(q1 × n1 )T    . T (q 6 × n 6 ) 

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Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S.

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Journal of Robotic Systems, vol. 21, p. 609-623, DOI:10.1002/rob.20039. [6] Mukherjee, P., Dasgupta, B., Mallik, A.K. (2007). Dynamic stability index and vibration analysis of a flexible Stewart platform. Journal of Sound and Vibration, vol. 307, no. 3-5, p. 495-512, DOI:10.1016/j. jsv.2007.05.036. [7] Hong. D., Kim. S., Choi. W.C., Song. J-B. (2003). Analysis of machining stability for a parallel machine tool. Mechanics Based Design of Structures and Machines, vol. 31, p. 509-528, DOI:10.1081/SME120023169. [8] Mahboubkhah, M., Nategh, M.J., Esmaeilzadeh Khadem, S. (2008). Vibration analysis of machine tools’ hexapod table. The International Journal of Advanced Manufacturing Technology, vol. 38, p. 12361243, DOI:10.1007/s00170-007-1183-9. [9] Mahboubkhah, M., Nategh, M.J., Esmaeilzadeh Khadem, S. (2009). A Comprehensive Study on the Free Vibration of Machine Tools’ Hexapod Table. The International Journal of Advanced Manufacturing Technology, vol. 40, p. 1239-1251, DOI:10.1007/ s00170-008-1433-5. [10] Pedrammehr, S., Mahboubkhah, M., Khani, N. (2013). A study on vibration of Stewart platformbased machine tool table. The International Journal of Advanced Manufacturing Technology, vol. 65, p. 9911007, DOI:10.1007/s00170-012-4234-9. [11] Maia, N.M., Silva, J.M.E. (1997) Theoretical and Experimental Modal Analysis. Research Studies Press, New Delhi. [12] Tobias, S.A. (1965). Machine-Tool Vibration. Blackie and Sons Ltd., London.

[13] Cheng, K. (2009). Machining Dynamics, Fundamentals, Applications and Practices. SpringerVerlag, London. [14] Insperger, T., Mann, B.P., Stépán, G., Bayly, P.V. (2003). Stability of up-milling and down-milling, part 1: alternative analytical methods. International Journal of Machine Tools and Manufacture, vol. 43, p. 25-34, DOI:10.1016/S0890-6955(02)00159-1. [15] Long, X-H., Balachandran, B. (2007). Stability analysis for milling process. Nonlinear Dynamics, vol. 49, p. 349-359, DOI:10.1007/s11071-006-9127-8. [16] Altintas, Y. (2000). Manufacturing Automation, Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design. Cambridge University Press, UK. [17] Mann, B.P., Insperger, T., Bayly, P.V., Stépán, G. (2003). Stability of up-milling and down-milling, part 2: experimental verification. International Journal of Machine Tools and Manufacture, vol. 43, p. 35-40, DOI:10.1016/S0890-6955(02)00160-8. [18] Yangui, H., Zghal, B., Kessentini, A., Chevallier, G., Riviere, A., Haddar, M., Karral, C. (2010). Influence of Cutting and Geometrical Parameters on the Cutting Force in Milling. Scientific Research, Engineering, vol. 2, p. 751-761, DOI:10.4236/eng.2010.210097. [19] Pedrammehr, S., Mahboubkhah, M., Pakzad, S. (2011). An improved solution to the inverse dynamics of the general Stewart platform. Proceedings of IEEE International Conference on Mechatronics, no. 5971317, p. 392-397. [20] Pedrammehr, S., Mahboubkhah, M., Khani, N. (2012). Improved Dynamic Equations for the Generally Configured Stewart Platform Manipulator. Journal of Mechanical Science and Technology, vol. 26, p. 711721, DOI:10.1007/s12206-011-1231-0.

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Original Scientific Paper

Received for review: 2013-02-25 Received revised form: 2013-09-10 Accepted for publication: 2013-09-25

A Method for Optodynamic Characterization of Erbium Laser Ablation Using Piezoelectric Detection Bosiger, G. – Perhavec, T. – Diaci, J. Georgije Bosiger1,* – Tadej Perhavec1 – Janez Diaci2

2 University

1 Fotona d.d. Ljubljana, Slovenia of Ljubljana, Faculty of Mechanical Engineering, Slovenia

The paper presents a new method for characterization of Erbium laser ablation processes widely employed in various medical applications. The method is based on detection of shock waves propagating in air above the irradiated surface by means of a wideband piezoelectric sensor and analysis of the acquired signal waveforms. This sensor set-up offers the possibility for integration into an Er:YAG laser hand-piece, which opens the way to on-line process monitoring. A new model of the sensor is developed in order to take into account the relative position and orientation of the sensor and its mechanical and electrical properties. The model is verified by comparing the signal waveforms acquired at different sensor distances and orientations relative to the ablated spot with the theoretical waveforms calculated on the basis of numerical solutions of the Taylor-Sedov point explosion model and the developed sensor model. Excellent agreement is observed between the acquired and theoretical waveforms, which serves as a basis for a novel method that employs shock-wave energy released during the ablation process as a process characteristic that can be determined from the acquired signal waveforms. It is shown that shock-wave energy exhibits significantly less dependence on the position and orientation of the sensor than other waveform characteristics (time of fight, amplitude, etc.) that are currently used for the ablation process characterization. Keywords: Erbium laser, laser ablation, shock wave, piezoelectric detection, Taylor-Sedov model

0 INTRODUCTION The Er:YAG laser, with a wavelength of 2.94 μm [1], is a well-established tool in medicine and surgery, particularly in dentistry [2] and dermatology [3]. Its infrared light is absorbed strongly in water and hydroxyapatite, providing effective laser ablation of soft and hard biological tissues [4]. Numerous new medical treatments (particularly in osteotomy) would also benefit from the advantages of Er:YAG laser tissue interaction over the conventional methods (e.g. noncontact intervention, smaller heat-affected zone, absence of mechanical vibration). One of the main technical and scientific challenges yet to be solved is the development of viable and reliable systems for the on-line monitoring of the key parameters, such as the cutting depth and the type of the removed tissue [5] to [8]. Tissue ablation with the Er:YAG laser is driven by microexplosions. Absorbed laser energy is partially released in the form of nonlinear mechanical waves propagated in the ablated tissue and in the air above it. Following this, material ejection occurs. These so-called optodynamic phenomena have received considerable attention in the context of characterization of laser ablation. Various set-ups have been examined for this purpose: spatially resolved techniques, such as schlieren [9] and [10], shadowgraphy [11] to [14] or holography [15], as well as the temporally resolved ones: laser interferometer [16] to [18], laser beam 172

deflection probe [19] and [20] and capacitive or piezoelectric acoustic sensors [7], [8] and [21]. While most of these techniques represent useful research tools within controlled laboratory experiments, only a few of them exhibit the potential to be used for on-line process monitoring in real medical applications. Here, additional technical factors come into prominence: e.g. compactness, affordability, insensitivity to environmental influences, speed of response, etc. In our view, broadband piezoelectric acoustic sensors are the means to make the on-line process monitoring of Er:YAG laser ablation practicable. Fig. 1 illustrates the idea: a piezoelectric sensor is attached to the laser handpiece to detect shock waves in the air above the irradiated tissue. Considering typical conditions (geometry of the handpiece and sensor, focal length of the focusing optics, laser pulse energy, etc.) and previous research results [11] it is reasonable to assume that the shock waveform is nearly hemispherical as it impinges onto the sensor and its amplitude is decreased to the intermediate-to-weak shock level. Piezoelectric shock sensors that respond linearly to this kind of excitation are available. Existing methods of shock wave characterization using acoustic sensors rely on empirically selected signal features, such as acoustic signal energy, signal amplitude and time of flight [7], [21] and [22]. Relationships of these signal features to other influencing factors (e.g. sensor characteristics, orientation or distance from the source) are not known,

*Corr. Author’s Address: Fotona d.d., Stegne 7, SI-1210 Ljubljana, Slovenia, georgije.bosiger@fotona.com

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thus limiting the applicability of these methods to strictly controlled conditions.

wave [24]. For a Taylor-Sedov explosion, this applies well for an intermediate or weak shock over a small distance of propagation. With this assumption, the pressure transient at a given point rs on the transducer can be expressed by the pressure in a centre point r0 of the transducer, using the equation:

∆p ( t , rs ) =

 r −r  rs ∆p  t − s 0 , r0  , r0 u0  

(1)

where u0 is the shock propagation speed which is determined using the point explosion model, and where it is expressed as a function of the distance from the source P and the released energy: u0= u0(r0,E). Fig. 1. Schematic representation of a set-up for optodynamic characterization of the laser ablation processes: shock wave (SW), piezoelectric sensor (PE), Er:YAG laser ablation beam (LB)

In this paper we propose another approach that opens the way to on-line process monitoring. We develop a new model of the sensor that takes into account the relative position and orientation of the sensor as well as its mechanical and electrical properties. We verify the model by comparing signals, detected at different sensor distances and orientations relative to the ablated spot, with the theoretical waveforms determined from a numerical solution of the point explosion model [23], taking into account the model of the piezoelectric sensor. Observing excellent agreement between the theoretical and experimental waveforms, we propose a novel method that employs shock-wave energy released during the ablation process as a process characteristic that is almost independent of the exact geometrical properties of the detection set-up. 1 THEORY In this section, a new model of a piezoelectric shock sensor is presented. The finite size of the sensor and its mechanical and electrical characteristics are taken into account. Next, a numerical procedure is described that allows the determination of theoretical pressure waves at the sensor surface employing the TaylorSedov point explosion model. 1.1 Piezoelectric Sensor Model The theoretical description is simplified on the assumption that the incident shock wave, during the propagation over the piezoelectric transducer surface, can be approximated as a linear spherical acoustic

Fig. 2. Geometrical relations used in derivation of the sensor model - placement of the sensor relative to the ablation spot (P): side view (above), top view (below)

We also disregard all the effects that result due to a change of the acoustic impedance, assuming that these affect the amplitude and not the waveform. With this assumption we write:

Fa ( t ; r0 ) ∝ ∫ ∆p ( t , rs ) dAs , As

(2)

where Fa(t;r0) is the force that acts on the sensor in its axial direction, and Δp(t,rs) is the pressure transient at a point on surface As of the sensor (see Fig. 2). Vector r0 points to the sensor centre and can be expressed by the horizontal s0 and vertical h0 distance. Inserting Eq. (1) into (2) and substituting Δp with a spherical impulse disturbance δ, we get Rayleigh’s integral [25]:

h(t ; r0 ) =

⌠  r0 ⋅    ⌡ As

 r −r  δ  t − s 0 , r0  u0   dA . s rs

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(3)

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The above equation represents the response of a sensor with a finite aperture to a spherical impulse disturbance. Assuming a cylindrical sensor with a flat aperture, it is possible to express h(t;r0) in terms of the angle Θs(t;r0) determined by the intersection of the projected spherical wave onto the sensor aperture as shown on Fig. 2:

h(t ; r0 ) = 2u0 r0 Θ s (t ; r0 ).

(4)

Using the superposition principle, we find that the force acting on a sensor of finite size is proportional to the convolution of a pressure transient Δp(t,r0) in the centre of the transducer and the signal waveform Θs (t;r0) representing the impulse response of the finite sensor aperture:

Fa ( t ; r0 ) ∝ ∆p ( t , r0 ) ∗ Θ s (t ; r0 ).

(5)

In order to take into account the electrical and mechanical characteristics, we treat the sensor as a one-dimensional element, where deformation that generates electric current acts only in its axial direction. The basis for this assumption represents the design of the piezoelectric acoustic sensor, where the sensor housing and the insulation suppresses the radial excitations (see Fig. 2). Pure elastic deformations of the sensing element are assumed. With these assumptions, linear static and dynamic characteristics apply. The Piezoelectric transducer is represented with a current source and capacitor Cs in parallel [26]. Parallel resistance of the sensing element is usually large and can be neglected. Taking into account the capacitance of the cable Cc and the resistive load RL, we get a transfer function for a linear piezoelectric force sensing element, Eq. (6), where τ = RL (Cs +Cc) is the time constant, d the piezoelectric charge constant and V the measured voltage. H(s) represents the dynamic characteristics of the elastic piezoelectric structure, where natural frequencies can be determined by measuring electrical impedance on an impedance analyzer [26]. By analyzing the transfer function in Eq. (6) we can conclude that the sensor’s sensitivity in steady state is inversely proportional to the total capacitance. Due to this capacitance, the sensor behaves as a high pass RC filter with time constant τ. The sensor output is in the frequency region between 1/(2πτ) and the first natural frequency of the piezoelectric element is proportional to the force Fa. 174

V (s) d τs 1 = . Fa ( s ) cs + cc 1 + τ s H ( s )

(6)

1.2 Shock-Wave Propagation Model We employ the Taylor-Sedov point explosion model to model propagation of the spherical shock wave [23]. The model assumes that a finite amount of energy E is released instantly in an infinitesimal volume of a perfect gas. Propagation of the blast wave is described by a set of hyperbolic partial differential equations in the Euler form:

U t + F(U) r = S(U),

(7)

where U is the vector of the conserved variables, F = F(U) their fluxes and S(U) the geometric source term that results from the transformation of the Euler equations to the spherical coordinates. Subscripts denote partial derivatives with respect to the independent variables; time t and radius r. Primitive variables are the mass density ρ(r,t), fluid speed v(r,t) and pressure p(r,t). Propagation at shock wave-front r = rs(t) is governed by the Rankine-Hugoniot jump conditions. Transition to dimensionless space ξ = r/rc and time τ = t/tc = c0·t/rc coordinates is performed [23] using the characteristic radius rc = (αE / κp0)1/3 that depends on the released energy E, where c0 is the sound speed, α = 1.175 and heat capacity ratio κ = 1.4. Primitive variables are normalized with their respective values in the undisturbed gas. The solution thus becomes independent of the released energy and undisturbed gas state, therefore, numerical calculation can be performed once and scaling back into dimensional coordinates adapts it to the given conditions. In the intermediate and weak shock range, only a numerical solution of the model is obtainable. We use an explicit discrete conservative numerical scheme [27]:

U in+1 = U in −

∆t [Fi +1/ 2 − Fi −1/ 2 ] + ∆tSi , ∆r

(8)

where Δt and Δr denote time and spatial steps, respectively. The second order WAF finite volume explicit method is used for the calculation of the fluxes [27]:

Fi +1/ 2 = F ( W i +1/ 2 ) ,

where the mean average state equals: 1 W i +1/ 2 = ( Win + Win+1 ) − 2 1 N − ∑ sign (ck )ϕ k  Wi(+k1+/ 21) + Wi(+k1)/ 2  . 2 k =1

Bosiger, G. – Perhavec, T. – Diaci, J.

(9)

(10)

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 172-178

W denotes the vector of primitive variables and ck the Courant’s number. Intercell states are computed with an approximate HLLC Riemann solver [27]. The Van Leer limiter ϕ is used to prevent numerical oscillations. Time integration is performed with the forward Euler method. An analytical solution of the strong shock theory serves as an initial condition for the numerical computation. 1

resulting shock waves are generated. Water and some soft tissues form a quasi-ideal half-space whereas hard tissues usually form complicated geometry that varies from pulse to pulse during the ablation and affects the spread of the shock wave. The results obtained by ablating the water surface are relevant for the laser surgery in which the shallow holes (craters) are prepared. For laser drilling of deep holes, an appropriate shock-wave propagation model has not yet been developed. In this case the accuracy of the presented method is questionable.

∆ p/ ∆ ps

0.5

−0.5

−1 −1

−0.5

0. 5

τ − ξs

Fig. 3. Normalized theoretical waveforms retarded by the acoustic wave-front propagation time

Computation is performed over the dimensionless time interval 1 ≤ τ ≤ 35. The grid is equally spaced with 215 points for both independent variables. In the scope of this paper, only pressure is used. According to the model, the pressure transient in a given point is a function of two parameters – the distance from the source rs and the released energy E. Calculated pressure transients, normalized by the shock pressure amplitude, are shown in Fig. 3 for dimensionless distances ξs: 2 (dashed), 8, 14, 20, 26 and 32 on a time scale retarded by the time of flight of the acoustic wave-front to facilitate waveform comparison. Fig. 3 illustrates a common characteristic of spherical shock waves: the duration of the compression phase increases with distance ξs due to supersonic shock wave-front propagation while the duration of the rarefaction phase remains constant. 2 EXPERIMENT We use water as the tissue phantom for the purpose of stable and repeatable experimental validation. At the particular Erbium laser wavelength, water trapped or bonded to the tissue plays a key role in tissue ablation. Explosive expansion of laser-heated water generates strong shock waves that propagate in air above the ablated surface, followed by material ejection. The key difference between the ablation of water and the tissue is in the form of the surface on which the

Fig. 4. Experimental set-up: shock wave (SW), piezoelectric sensor (PE), photo-diode (PD), oscilloscope (OSC, pulse generator (PG), personal computer (PC)

The experimental system used to validate the above model is shown in Fig. 4. A free-running Er:YAG laser (Fidelis Plus III, Fotona) is used to irradiate the water surface as it forms a quasi-ideal half-space. The focal point of the laser exit optics is located on the water surface with a spot diameter of 0.9 mm. A signal from a pulse generator triggers the laser system, also setting the pulse duration of the laser flash lamp (45 μs). The supply voltage for laser pumping has been set to 650 V. Resulting laser pulses are short (≈2 μs), causing generation of a single shock wave rather than several that are typical for longer pulses [22]. The pulse energy was measured (Nova II, Ophir), where the mean value was 3.14 mJ with a std. deviation of 0.16 mJ. A piezoelectric sensor (CA1135, Dynasen) with a PZT-5A crystal disc of 1 mm in diameter was translated in parallel and perpendicular to the water surface using two linear stages with micrometer screws. Responses were measured at three different horizontal distances (7.25, 10.25 and 13.25 mm) and six vertical ones (10, 12, 14, 16, 18 and 20 mm). Six repetitions on each configuration were conducted. The measured signals were sampled using a digital oscilloscope (Agilent DSO6034A) with 300 MHz bandwidth and 12 bit digitization. Signal acquisition

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was triggered by a signal from an InAs photodiode (J12, Teledyne), mounted behind the back laser mirror. The signals were sent to a PC where they were saved. Room temperature and pressure were measured: T0 = 298 K, p0 = 996 mbar, using standard meteorological equipment. 3 RESULTS AND DISCUSSION We validate the described model by comparing the theoretical and measured waveforms. To enable this, the released energy is determined for every measured signal by measuring the duration of the compression phase of the measured signal. Transforming the numerical solution of the point explosion to the dimensional coordinates and using the model of the sensor, we determine a function (Eq. 11) that describes the duration of the compression waveform phase tp as a function of characteristic radius rc, given the influencing parameters: relative position of the sensor to ablation spot (determined by s0 and h0) and the sound speed c0 which is estimated from the measured room temperature. The function is formulated as a spline with 41 data points (knots) along the characteristic radius rc for values between 0.8 and 2.8 mm, whilst other parameters are held constant. At each data point a theoretical waveform (signal) is found first, from which time tp is then determined. The same procedure is repeated for other sensor positions. t p = f ( rc ; s0 , h0 , c0 ) .

(11)

0.30

0.25

0.20

0.15

176

0.15

0.10

0.05

0.00

E h [mJ]

0.30

E h [mJ]

Using measured data for tp we numerically solve the above equation for rc. Released energy for the spherical blast wave E is then determined from the

definition of the characteristic radius rc. Duration tp is determined by normalizing the signal with its peak value and searching for the time interval where the normalized amplitude exceeds the threshold value of 0.1 (10%). Signals are normalized because of simplifications of the theoretical model, where absolute signal values are unknown, Eq. (2), as the sensor is not calibrated. The results are shown in Fig. 5. Eh is the released energy for the half-space in which the shock wave forms a hemisphere (Eh = E/2). No systematic dependency of Eh on sensor positions (s0, h0, defined as shown in Figs. 2 and 4) is observed. It is to be noted that only sensor position (s0, h0) varies in this experiment, while all other parameters (especially laser energy) remain the same. Pulse-topulse variations of Eh within a sequence of repeated measurements, indicated by the error bars in Fig. 5, are mainly due to pulse-to-pulse variations of the laser pulse energy. We take the observation that Eh does not exhibit systematic dependency on sensor position as evidence that the described model correctly describes the key features of the signals and the set-up. It is of interest to note the obtained Eh value: its estimated mean value at 0.16 mJ (and std. dev. of 0.02 mJ) implies that about 5% of the incident laser pulse is converted into the energy of the shock wave. Using estimated released energies Eh, normalized theoretical signals are determined and compared to corresponding measured signal waveforms. Fig. 6 shows two limiting examples, characterized by high (53°) and low (20°) angles of incidence between the shock wave-front and the sensor aperture normal. In both examples, good agreement of the positive (compressive) phase duration is observed between the theoretical and measured waveforms. This is

0.00 12

14 10

16 12

h0 [mm]

18 14

20 16

207.25

0.00

h0 [mm]

7.25 10.25 10 12

s 0 [mm]

10.25 16 13.25 18

s 0 [mm] h0 [mm]

b) Fig. 5. Estimated energies for hemispherical explosion at a) senzor position h0 and b) sensor position s0 Bosiger, G. – Perhavec, T. – Diaci, J.

13.25

7.25

10.25

s 0 [mm]

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 172-178

expected since this parameter was used to estimate released energy. In the negative (rarefaction) phase, the measured and theoretical signals differ. These differences are the consequence of various effects that are not taken into account by the presented model (e.g. arrival of water droplets and natural vibrations of the sensor structure, diffraction of the wave, etc.).

V¯ (t)

model. Then, sensor sensitivity is calculated by dividing the peak voltage of a measured signal by the peak pressure of the corresponding theoretical signal. Analysis of the estimated sensor sensitivities shows that it varies significantly with sensor orientation (s0, h0): the estimated pressure peaks in Fig. 6 are 17 and 12 mbar for the upper and lower trace, respectively, and the corresponding sensor sensitivities are 2.4 and 5.7 μV/Pa, respectively. The observed variations of the estimated sensitivities at a fixed sensor position are less than 10%. 4 CONCLUSION

−1

V¯ (t)

−1 40

t [µ s]

b) Fig. 6. Comparison of measured and theoretical signals (dashed) for two different positions of the transducer relative to the ablation point, a) s0 = 13.25 mm, h0 = 10 mm and b) s0 = 7.25 mm, h0 = 20 mm

As a further step in model validation, we compare the theoretical and measured times of flight (TOF) of the shock wave-front. The maximum observed relative difference between a measured TOF and its theoretical counterpart is about 1%. Expressing this in terms of distance using the theoretical shock speed, we get a maximum distance uncertainty of about 0.24 mm. We attribute these deviations mainly to the uncertainties of the parameters in Eq. (11) and the measured durations tp that are used by the method for the estimation of the released energy Eh. By comparing the theoretical shock waveforms that take into account the sensor model to the respective ones that do not, we find that the former ones systematically exhibit longer duration of the compressive phase. The relative difference between the two increases with the angle of incidence from 5% at the 20° angle to 25% at the 53° angle. This is a result of the prolonged transit time of the shock wave over the finite sensor surface in the case of a larger angle of incidence. This observation presents strong evidence that the finite dimensions of the sensor in given conditions need to be taken into account. The described approach can be employed to calibrate the sensor. Peak pressures are found from the estimated energies Eh using the point explosion

We describe a method and a set-up that opens the possibilities for on-line process monitoring in Er:YAG laser ablation. We employ a piezoelectric shock sensor to detect shock waves generated in the air above the irradiated surface. Using a comprehensive sensor model and the Taylor-Sedov model of shock wave propagation, we demonstrate excellent agreement between the measured and theoretical waveforms. On this basis we propose that the released shock energy is a process characteristic that is essentially independent of the position and orientation of the sensor. We also demonstrate that this method allows peak shock wave pressure estimation and calibration of the sensor. 5 ACKNOWLEDGEMENT The operational part of this research was financed by the European Union and the European Social Fund. 6 REFERENCES [1] Bizjak, A., Nemeš, K., Možina, J. (2011). Rotatingmirror Q-switched Er:YAG laser for optodynamic studies. Strojniški vestnik – Journal of Mechanical Engineering, vol. 57, no. 1, p. 3-10, DOI:10.5545/svjme.2010.120. [2] Bader, C., Krejci, I. (2006). Indications and limitations of Er:YAG laser applications in dentistry. American Journal of Dentistry, vol. 19, no. 3, p. 178-186. [3] Hohenleutner, U., Hohenleutner, S., Bäumler, W., Landthaler, M. (1997). Fast and effective skin ablation with an Er:YAG laser: Determination of ablation rates and thermal damage zones. Lasers in Surgery and Medicine, vol. 20, no. 3, p. 242-247, DOI:10.1002/(SICI)1096-9101(1997)20:3<242::AIDLSM2>3.0.CO;2-Q. [4] Niemz, M.H. (2007). Laser-Tissue Interactions: Fundamentals and Applications, 2nd ed. Springer, Berlin, Heidelberg. [5] Burgner, J. (2010). Robot Assisted Laser Osteotomy. KIT Scientific Publishing, Karlsruhe.

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[6] Stübinger, S. (2010). Advances in bone surgery: the Er:YAG laser in oral surgery and implant dentistry. Clinical, Cosmetic and Investigational Dentistry, vol. 2, p. 47-62, DOI:10.2147/CCIDEN.S8352. [7] Nahen, K., Vogel, A. (1999). Investigations on acoustic on-line monitoring of IR laser ablation of burned skin. Lasers in Surgery and Medicine, vol. 25, no. 1, p. 69-78, DOI:10.1002/(SICI)1096-9101(1999)25:1<69::AIDLSM9>3.0.CO;2-3. [8] Rupprecht, S., Tangermann-Gerk, K., Wiltfang, J., Neukam, F.W., Schlegel A. (2004). Sensor-based laser ablation for tissue specific cutting: An experimental study. Lasers in Medical Science, vol. 19, no. 2, p. 8188, DOI:10.1007/s10103-004-0301-2. [9] Apitz, I., Vogel, A. (2005). Material ejection in nanosecond Er:YAG laser ablation of water, liver, and skin. Applied Physics A, vol. 81, no. 2, p. 329-338, DOI:10.1007/s00339-005-3213-5. [10] Nahen, K., Vogel, A. (2002) Plume dynamics and shielding by the ablation plume during Er:YAG laser ablation. Journal of Biomedical Optics, vol. 7, no. 2, p. 165-178, DOI:10.1117/1.1463047. [11] Perhavec, T., Diaci, J. (2010). A novel double-exposure shadowgraph method for observation of optodynamic shock waves using fiber-optic illumination. Strojniški vestnik - Journal of Mechanical Engineering, vol. 56, no. 7, p. 477-482. [12] Gregorčič, P., Možina, J. (2011). High-speed two-frame shadowgraphy for velocity measurements of laserinduced plasma and shock-wave evolution. Optics Letters, vol. 36, no. 15, p. 2782-2783, DOI:10.1364/ OL.36.002782. [13] Gregorčič, P., Diaci, J., Možina, J. (2012). Twodimensional measurements of laser-induced breakdown in air by high-speed two-frame shadowgraphy. Applied Physics A, p. 1-7.4. [14] Gregorcic, P., Jezersek, M., Mozina, J. (2012). Optodynamic energy-conversion efficiency during an Er:YAG-laser-pulse delivery into a liquid through different fiber-tip geometries. Journal of Biomedical Optics vol. 17, no. 7, p. 075006-1, DOI:10.1117/1. JBO.17.7.075006. [15] Amer, E., Gren, P., Sjodahl, M. (2008). Shock wave generation in laser ablation studied using pulsed digital holographic interferometry. Journal of Physics D: Applied Physics, vol. 41, p. 215502, DOI:10.1088/0022-3727/41/21/215502.

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[16] Požar, T., Možina, J. (2009). Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod. Strojniški vestnik – Journal of Mechanical Engineering, vol. 55, no. 10, p. 575-580. [17] Požar, T., Gregorčič, P., Možina, J. (2009). Optical measurements of the laser-induced ultrasonic waves on moving objects. Optics Express, vol. 17, no. 25, p. 22906-22911, DOI:10.1364/OE.17.022906. [18] Požar, T., Gregorčič, P., Možina, J. (2011). A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer. Applied Physics B: Lasers and Optics, vol. 105, no. 3, p. 575582, DOI:10.1007/s00340-011-4512-5. [19] Diaci, J., Možina, J. (1995). Multiple-pass laser beam deflection probe for detection of acoustic and weak shock waves in fluids. Review of Scientific Instruments, vol. 66, p. 4644-4648, DOI:10.1063/1.1145301. [20] Diaci, J., Mozina, J. (1994). A study of energy conversion during Nd:YAG laser ablation of metal surfaces in air by means of a laser beam deflection probe. Le Journal de Physique, vol. 4, p. 737-740. [21] Lukac, M. Grad, L., Mozina, J., Sustercic, D., Funduk, N., Skaleric, U. (1994). Optoacoustic effects during Er:YAG laser ablation in hard dental tissue, Proceedings SPIE 2327, Medical Applications of Lasers II, p. 93100. [22] Grad, L., Možina, J. (1996). Optodynamic studies of Er:YAG laser induced microexplosions in dentin. Applied Surface Science, vol. 96-98, p. 591-595, DOI:10.1016/0169-4332(95)00559-5. [23] Sedov, L.I. (1993). Similarity and Dimensional Methods in Mechanics, 10th Ed.. CRC Press, Boca Raton. [24] Kinsler, L.E., Frey, A.R. (1962). Fundamentals of Acoustics. Wiley, New York. [25] Jensen, J.A. (1999). A new calculation procedure for spatial impulse responses in ultrasound. Journal of the Acoustical Society of America, vol. 105, no. 6, p. 3266–3274, DOI:10.1121/1.424654. [26] Bentley, J.P. (2005). Principles of Measurement Systems. Pearson Education, Essex. [27] Toro, E.F. (2009). Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Dordrecht, DOI:10.1007/b79761.

Bosiger, G. – Perhavec, T. – Diaci, J.

Original Scientific Paper

Received for review: 2013-08-06 Received revised form: 2013-11-26 Accepted for publication: 2013-12-20

A Detailed Analysis of the Resonant Frequency and Sensitivity of Flexural Modes of Atomic Force Microscope Cantilevers with a Sidewall Probe Based on a Nonlocal Elasticity Theory Abbasi, M. – Mohammadi, A.K. Mohammad Abbasi* – Ardeshir Karami Mohammadi

School of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran In this paper, utilizing a nonlocal elasticity theory, the resonant frequency and sensitivities of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) are studied. This ACP comprises a horizontal cantilever and a vertical extension, and a tip located at the free end of the extension, which makes the AFM capable of topography at the sidewalls of microstructures. First, the governing differential equations of motion and boundary conditions for flexural vibration are obtained via a combination of the basic equations of nonlocal elasticity theory and Hamilton’s principle. Afterwards, a closed-form expression for the sensitivity of vibration modes is obtained using the relationship between the resonant frequency and contact stiffness between tip and sample. This analysis provide a better representation of the vibration behaviour of an AFM cantilever with a sidewall probe, for which the effects of the small scale are significant. The results of the nonlocal theory are compared to those of classic beam theory. The evaluation shows that the resonant frequency and sensitivity of the proposed ACP are size dependent, especially when the contact stiffness are high. Keywords: atomic force microscope, assembled cantilever probe, nonlocal elasticity theory, size dependent

0 INTRODUCTION Atomic-force microscopes (AFM) were originally developed to provide high-resolution images of the surface structures of both conductive and insulating samples in both air and liquid environments [1] and [2]. It is also widely used for nano-manipulation and nano-lithography in micro/nano-electromechanical systems (MEMS/NEMS) [3] and [4]. The contact between the tip of the elastic cantilever and the sample induces a dynamic interaction. The imaging rate and contrast of topographical images significantly depend on the resonant frequency and sensitivity of the AFM cantilever. Hence, investigations of the dynamic behaviour of the AFM microcantilever seem to be crucial. To date, many researchers have studied the dynamic response of the AFM cantilever [5] to [8]. Conventional AFMs consist of a microcantilever with a sharp conical or pyramidal tip located at its free end and play an important role in nano-scale surface measurements. Unfortunately, their probe tips never come in close proximity to sidewalls, regardless of how sharp and thin the tips are. Therefore, an urgent need for nano-scale surface measurements at sidewalls exists. In order to overcome the limitations of conventional AFMs, Dai et al. [9] proposed assembled cantilever probes (ACPs) for direct and non-destructive sidewall measurement of nano- and microstructures. Utilizing classic beam theory, Chang et al. [10] analysed the sensitivity and the resonant frequency an assembled cantilever probe, consisting of a horizontal AFM cantilever and a vertical extension. Kahrobaiyan et al. [11] studied the resonant

frequencies and flexural sensitivities of another form of ACPs proposed by Dai et al. [12], comprising a horizontal cantilever, a vertical extension and two tips located at the free ends of the cantilever and the extension. Beams used in MEMS and NEMS, such as AFMs, have thicknesses in the order of microns and sub-microns. The size dependency of static and dynamic behaviour of the micro-scale structures have been verified in the experimental observations [13] and [14]. For example, McFarland and Colton [15] detected a considerable difference between the stiffness values predicted by the classic beam theory and the stiffness values obtained during bending tests of polypropylene microcantilevers. In the microtorsion test of thin copper wires, Fleck et al. [16] observed that a decrease in wire diameters from 170 to 12 results in a significant increase of the torsional hardening. Lam et al. [17] observed a significant enhancement of bending rigidity caused by the beam thickness reduction in microbending tests of beams made of epoxy polymers. The lack of an internal material length scale parameter in the classic continuum models causes some deficiencies in studying small scale nanostructures. Hence, sizedependent continuum mechanics models such as the strain gradient theory [18] and [19], couple stress theory [20] and [21], modified couple stress theory [22] and nonlocal elasticity theory [23] and [24] are used. Nonlocal elasticity theory, which assumes that the stress state at a given point to be dependent on the strain states at all points in the body, have been the

*Corr. Author’s Address: School of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran, m.abbasi28@yahoo.com

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subject of much attention in nano-mechanics because of its efficiency. The nonlocal theory of elasticity has been used to investigate many fields, such as the lattice dispersion of elastic waves, wave propagation in composites, fracture mechanics, dynamic and the static analysis of carbon nanotubes and nano-rods, surface tension fluids, etc. [25] and [26]. Using beam-and-shell theories, Wang et al. [27] applied the nonlocal elasticity constitutive equations to study the vibration and buckling of carbon nanotubes. Using the nonlocal constitutive relations of Eringen [23], Reddy [28] reformulated the equations of motion of various beam theories, including the Euler–Bernoulli, Timoshenko, higher order, and Levinson beam models, using different nonlocal beam theories for bending, buckling and free vibration problems. A generalized nonlocal beam theory to derive the governing equations for different beam theories was proposed by Aydogdu [29]. Next, the effects of nonlocality and length of beams were investigated in some detail for each considered problem. Thai [30] proposed a nonlocal shear deformation beam theory for the bending, buckling, and vibration of nano-beams based on the nonlocal constitutive relations of Eringen [23]. His theory accounted for a quadratic variation of the shear strains across the thickness, and satisfied the zerotraction boundary conditions on the top and bottom surfaces of the beam without using shear correction factor. Utilizing this theory, Gheshlaghi and Mirzaei [31] studied the frequency shift of microcantileverbased sensors. They found that the normalized natural frequencies obtained for the nonlocal cantilever microbeam become smaller than those for its local counterpart . Civalek and Demir [32] developed an elastic nonlocal beam model for the bending analysis of microtubules (MTs) based on the Euler-Bernoulli beam theory. They concluded that the nonlocal continuum theory approach is superior to average (local) elasticity, especially for some boundary conditions Since ACPs’ cantilevers have thicknesses in the order of microns and sub-microns, studying their dynamic behaviour utilizing a non-classic beam theory such as nonlocal beam theory seems necessary. In this study, utilizing nonlocal elasticity theory, the sensitivity and resonant frequency of flexural vibration modes of an ACP proposed by Dai et al.[9], consisting of a horizontal cantilever and a vertical extension, are analysed. 180

1 THE NONLOCAL ELASTICITY THEORY According to the nonlocal elasticity theory proposed by Eringen [23] and [24], the stress components at a reference point x in an elastic continuum depend not only on the strain components at the same position x but also on all other points of the body. Hence, the well-known nonlocal constitutive relation in terms of a partial differential form is expressed as [23]:

(1 − µ∇ 2 )σ ij = Cijmnε mn , µ = e02 a 2 , (1)

where σij is the nonlocal stress tensor components at point x, cijmn is the fourth-order elasticity tensor, εmn is the strain tensor, μ is the nonlocal parameter, and e0 and a are the material constant and internal characteristic length, respectively, which incorporate the small-scale effect. The nonlocal parameter depends on the boundary conditions, chirality, mode shapes and type of motion [33]. Assuming that the nonlocal behaviour is negligible in the thickness direction, the nonlocal constitutive relation in Eq. (1) takes the following special relation for beams [34].

σ xx − µ

∂ 2σ xx = Eε xx , (2) ∂x 2

where E is Young’s modulus. It should be noted that for the case of μ = 0, we obtain the constitutive relations of local beam theories. 2 ANALYSIS OF THE DYNAMIC BEHAVIOUR OF ACP MICROCANTILEVERS The proposed type of AFM ACP developed in this study comprises a horizontal cantilever, a vertical extension and a tip located at the free end of the extension, which makes the AFM capable of topography at the sidewalls of microstructures. The geometrical parameters and configuration of this ACP are depicted in Fig. 1. The horizontal cantilever and vertical extension have a uniform cross section thickness b, width w, whose lengths are L and H, respectively, and the small length tip is s. Considering the ratio of the extension rigidity to cantilever rigidity, the deflection of the extension in comparison with the cantilever deflection can be neglected, and it can be assumed that the extension is rigid. Therefore, the cantilever of ACP experiences flexural vibrations during contact with the sample. The interaction between AFM tip and microstructure sidewall results in torsional vibration in the AFM probe. As shown in Fig. 1, the ACP interacts with the sample surface

Abbasi, M. – Mohammadi, A.K.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 179-186

at its tip by normal springs Kn for normal interaction and lateral springs Kl for lateral interaction. x is the coordinate along the centre of the cantilever, while w(x,t) is the vertical deflection in x-direction at time t.

1 σ ij ε ij dv, (7) 2 Ω∫

by substituting Eq. (4) into Eq. (7) and also considering Eq. (5), one obtains the following expression for the strain energy U: L

U = −∫ M

∂ 2 w ( x, t ) ∂x 2

dx. (8)

Considering the proposed ACP microcantilever depicted in Fig. 1 yields the following result for the elastic potential energy of the system: L

U = −∫ M

∂ 2 w ( x, t ) ∂x 2

1  ∂w ( L, t )  dx + K n  H  + 2  ∂x  2

∂w ( x, t )  1  + K l  w ( x, t ) − s  . 2  ∂x 

Fig. 1. Schematic diagram of an AFM cantilever micro-assembled with vertical extension and a tip located at the free end of the vertical extension for flexural vibration

Based on the flexural vibration of the ACP microcantilever depicted in Fig. 1, the components of displacement vector u are expressed as:

ux = − z

∂w( x, t ) , u y = 0, u z = w ( x, t ) , (3) ∂x

where ux, uy and uz denote the displacement along x, y and z axes, respectively. The only non-zero strains of the Euler-Bernoulli beam theory are:

ε xx = − z

∂ 2 w ( x, t ) ∂x 2

. (4)

Considering Eqs. (2) and (4) and using the moment strain relation: M = ∫ zσ xx dA, (5)

one obtains:

M −µ

∂ 2 w ( x, t ) ∂2M , (6) = − EI ∂x 2 ∂x 2

where I denotes the second moment of area about the y-axis. Then the strain energy U in a deformed isotropic linear elastic material occupying region Ω is given by:

(9)

Denoting the mass per unit length of cantilever and extension by ρ and ρe, respectively, and the extension mass and mass moment of inertia by Me and Je, and also the cantilever and extension areas by A and the mass moment of inertia of the extension by Jh, the kinetic energy of the system due to the velocity of its particles along the z-direction can be written as: 2

 ∂w ( x, t )   ∂w ( L, t )  1 L 1 ρ A  dx + M e    + ∫  2 0 2  ∂t   ∂t  2

1  ∂w ( L, t )  + J e   , 2  ∂x∂t 

(10)

1 1 2 3 where Me = ρeAH and J e = M e H = ρe AH . 3 3 Bearing in mind the aforementioned expressions for U and T, the dynamic governing equation of this ACP microcantilever as well as boundary conditions can be determined with the aid of Hamilton’s principle:

∫ (δ T − δ U + δ W ) dt = 0, (11) t1

in the above equation, We is the work done by external classic and higher-order torques which are assumed to be zero. Inserting Eqs. (9) and (10) into Eq. (11) then leads to: ∂ 2 M ( x, t ) ∂ 2 w ( x, t ) − ρA = 0, (12) 2 ∂x ∂x 2

w ( 0, t ) = 0,

∂w ( 0, t ) ∂x

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= 0, (13) 181

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 179-186

∂ 2 w ( L, t ) ∂t 2

−Je

∂M + ∂x

+ K l w ( L, t ) = 0, x=L

∂ 3 w ( L, t )

∂w ( L, t )

+ M − Kn H ∂x∂t 2 + K l sw ( L, t ) = 0,

∂x

e (14)

∂ 2W (1,τ )

M = − EI

∂x 2

+ µρ A

∂t 2

. (16)

Inserting Eq. (16) into Eqs. (12) to (15), the dynamic governing equation and the boundary conditions can be rewritten as: EI

∂ 4 w ( x, t ) ∂x 4

+ ρA

∂ 2 w ( x, t ) ∂t 2

w ( 0, t ) = 0,

µρ A

∂ w ( L, t ) 3

∂t ∂x ∂ 2 w ( L, t )

+M e

∂ 2 w ( L, t ) ∂x 2

∂t

− µρ A

+ Kn H 2

− µρ A

∂x

∂t 2 ∂x 2

= 0, (17)

= 0, (18)

∂ 3 w ( L, t ) ∂x

∂t 2

+ Je

∂ 3 w ( L, t ) ∂x∂t 2

− K l sw ( L, t ) = 0.

x w t , W= , τ= 2 L L L

ρ s µ , m= e, e= 2, L L ρ

K L3 K L3 H , β n = n , βl = l . L EI EI

∂X 4 182

∂ 2W ( X ,τ ) ∂τ 2

W ( 0,τ ) = 0,

−e

∂W ( 0,τ ) ∂X

∂ 2W (1,τ )

+ (24)

∂ W (1,τ ) 1 + mh3 + 3 ∂X ∂τ 2

− β l SW (1,τ ) = 0.

(25)

d 4V ( X )

+ eω 2

∂X 4

d 2V ( X )

V (0) =

eω 2

dV (1) dX

d 2V (1) dX

+ r2

− ω 2V ( X ) = 0, (26)

∂X 2

dV (0) = 0, (27) dX

d 3V (1) dX 3 dV (1) dX

+ rV 1 (1) = 0, (28)

+ r3V (1) = 0, (29)

where the angular frequency is denoted by ω, and r1, r2 and r3 are defined as: r1 = mhω 2 − β l , 1 r2 = β n h 2 − mh3ω 2 , (30) 3 2 r3 = eω − β l S . The solution of Eq. (26) is written as: V ( X ) = a1 sinh λ1 X + a2 cosh λ1 X + + a3 sin λ2 X + a4 cos λ2 X ,

(21)

∂τ 2 ∂X 2

∂ 2W (1,τ ) ∂τ 2

∂X

(20)

EI , ρA

∂ 4W ( X ,τ )

+ mh

∂X 3 + β lW (1,τ ) = 0,

∂τ ∂W (1,τ )

Applying this set of dimensionless variables quantities given in Eq. (21) into Eqs. (17) to (20) yields the normalized form of free vibration equation and the associated boundary conditions as: ∂ 4W ( X ,τ )

−e

∂ 3W (1,τ )

Assuming a harmonic solution for Y as W(X,τ) = V(X)eiωτ results in the following ordinary differential equation and boundary conditions:

The dimensionless variables are defined as: X=

+ βn h2

+ K l w ( L, t ) = 0, (19)

∂ 2 w ( L, t )

∂w ( L, t )

∂ 4 w ( x, t )

∂w ( 0, t ) − EI

∂X

(15)

∂ 2 w ( x, t )

∂τ 2 ∂X

−

by substituting Eq. (12) into Eq. (6), one obtains the following expression for M: ∂ 2 w ( x, t )

∂ 3W (1,τ )

= 0, (22)

= 0, (23)

(31)

where ai (i = 1, ..., 4) are some constants and λ1 and λ2 are defined as:

λ1 =

λ2 =

−eω 2 +

( eω )

+ 4ω 2

2 eω 2 +

( eω ) 2

+ 4ω 2

. (32)

Applying boundary conditions (27) to (29) into Eq. (30), the characteristics equation can be found: where

Abbasi, M. – Mohammadi, A.K.

C (ω , β ) = U1U 4 − U 2U 3 = 0, (33)

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 179-186

U1 = eω 2 λ2 ( − cosh λ1 + cos λ2 ) − λ12 λ2 cosh λ1 − −λ23 cos λ2 + r1 (

(

−λ2 sinh λ1 + sin λ2 ), λ1

(34)

U 2 = −eω 2 ( λ1 sinh λ1 + λ2 sin λ2 ) +

)

+ −λ13 sinh λ1 + λ23 sin λ2 + r1 ( − cosh λ1 + cos λ2 ) , (35) U 3 = −λ1λ2 sinh λ1 − λ22 sin λ2 +  −λ  + r2 λ2 ( − cosh λ1 + cos λ2 ) + r3  2 sinh λ1 + sin λ2  , (36)  λ1 

U 4 = −λ12 cosh λ1 − λ22 cos λ2 −

−r2 ( λ1 sinh λ1 + λ2 sin λ2 ) + r3 ( − cosh λ1 + cos λ2 ) . (37) The dimensionless sensitivity of an AFM ACP, S is defined as the differentiation of the dimensionless natural frequency with respect to the dimensionless surface contact stiffness, i.e.:

Fig. 2. The first dimensionless flexural resonant frequency for an AFM cantilever at various values of normalized nonlocal parameter

 ∂C   ∂C  ∂ω = −    . (38) ∂β l  ∂β l   ∂ω 

3 RESULTS AND DISCUSSION This section presents the results of the vibration analysis of an AFM ACP based on the nonlocal elasticity theory. The analytical expressions were obtained for the sensitivity and resonant frequency of flexural modes to indicate a better representation of the flexural behaviour of an AFM with sidewall probe where the small-scale effect are significant. To this end, we consider the geometric and material parameters of the cantilever and extension to be ρ = 2330 kg/m3, ρe = 3440 kg/m3 and h = 0.5. The default value for the normalized nonlocal parameter, e is considered to be e = 0.03 and also the lateral and normal contact stiffness are assumed as β = 0.9 βl [10], [11] and [30]. The dimensionless flexural resonant frequency of the first and second modes of the AFM cantilever based on nonlocal elasticity and classic beam theories are compared in Figs. 2 and 3 for different values of normalized nonlocal parameter. It should be noted that for the case e = 0, nonlocal elasticity theory is reduced to classic beam theory. Considering these two figures, it can be found that the resonant frequency takes at a low constant value for lower values of contact stiffness. However, by increasing the contact stiffness, the resonant frequency rises dramatically

Fig. 3. The second dimensionless flexural resonant frequency for an AFM cantilever at various values of normalized nonlocal parameter

Fig. 4. The first dimensionless flexural sensitivity for an AFM cantilever at various values of normalized nonlocal parameter

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until reaches another constant value at very high value of contact stiffness. These figures also indicate that for large values of contact stiffness, the resonant frequencies obtained by nonlocal theory are smaller than those calculated by classic beam theory. At these values of contact stiffness, an increase in normalized nonlocal parameter decreases the resonant frequency. Furthermore, it can be inferred that the resonant frequency of the second mode is more sensitive to the change in the normalized nonlocal parameter. Figs. 4 and 5 illustrate the change in the first and second flexural sensitivity modes due to the change in contact stiffness, βl and normalized nonlocal parameter, e. According to these figures, the sensitivity is maximum when the contact stiffness is low. In the first mode, an increase in contact stiffness decreases the sensitivity of the first mode, while a similar phenomenon happens in the second mode after the sensitivity experiences a minimum around the point βl = 10 and a peak approximately at the point βl = 103, respectively. Furthermore, for the higher values of contact stiffness, the sensitivities of the first mode predicted by nonlocal theory is smaller than those obtained by classic beam theory. There is a similar situation in the second mode in all ranges of contact stiffness. The flexural resonant frequency of the first mode due to the change in dimensionless contact stiffness, βl, and vertical extension length, h, is depicted in Fig. 6. It can be seen that the resonant frequency increases as the contact stiffness grows but reduces as the vertical extension enlarges. The first flexural sensitivity of the sidewall tip versus dimensionless contact stiffness, βl and various vertical extension length, h is shown in Fig. 7. From this figure, it can be concluded that an increase in vertical extension length reduces the flexural sensitivity when the contact stiffness is low. The situation is reversed for very high values of contact stiffness (βl > 100). Fig. 8 represents the first flexural sensitivity of mode 1 as a function of vertical extension length, and h at different values of normalized nonlocal parameter for βl = 103. It should be noted from Fig. 4 that the sensitivities of first mode differ from each other for different values of nonlocal parameters when the contact stiffness is high. Hence, the contact stiffness is assumed as βl = 103 in Fig. 8. From this figure, it can be inferred that the effect of change in vertical extension length h on the flexural sensitivity predicted by classic beam theory (e = 0) is significant; and a rise in the dimension length of vertical extension h increases the sensitivity dramatically. For the sensitivity calculated 184

Fig. 5. The second dimensionless flexural sensitivity for an AFM cantilever at various values of normalized nonlocal parameter

Fig. 6. The first dimensionless flexural resonant frequency for an AFM cantilever at various values of dimensionless length of the vertical extension

Fig. 7. The first dimensionless flexural sensitivity for an AFM cantilever at various values of dimensionless length of the vertical extension

Abbasi, M. – Mohammadi, A.K.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 179-186

Fig. 8. The first dimensionless flexural sensitivity for an AFM cantilever as a function of dimensionless length of the vertical extension at various normalized nonlocal parameter

by nonlocal elasticity theory, an increase in the normalized nonlocal parameter reduces the effect of vertical extension length on the sensitivity. 4 CONCLUSION In this study, the equations of motion and boundary conditions for flexural vibration of an AFM cantilever with sidewall probe have been derived utilizing nonlocal elasticity theory; analytical solutions are then presented to bring out the effect of nonlocal parameters on the resonant frequency and flexural sensitivity of the first two modes. According to the results, the sensitivities and resonant frequencies predicted by nonlocal elasticity theory are smaller than those evaluated by classic beam theory. It can also be determined that the difference between either the resonant frequency or the flexural sensitivity obtained by nonlocal elasticity theory and those calculated by classic beam theory are more significant for the second mode. Moreover, an increase in the vertical extension length diminishes the resonant frequency, while reducing the sensitivity for low values of contact stiffness and increases it for high values of contact stiffness. Furthermore, a rise in the normalized nonlocal parameter decreases the effect of vertical extension length on the sensitivity. 5 REFERENCES [1] Mazeran, P.E., Loubet, J.L. (1999). Normal and lateral modulation with a scanning force microscope,

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Abbasi, M. – Mohammadi, A.K.

Original Scientific Paper

Received for review: 2013-08-14 Received revised form: 2013-10-28 Accepted for publication: 2013-11-13

Tuned-Sinusoidal Method for the Operational Modal Analysis of Small and Light Structures Rovšček, D. – Slavič, J. – Boltežar, M. Domen Rovšček – Janko Slavič – Miha Boltežar*

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia Small and light structures have some distinctive features that intensify the difficulties when measuring their modal parameters. The mass that is added to the structure by the sensors cannot be neglected and the resonant frequencies are usually relatively high. As a result, a wide frequency range of measurements is needed. There are also difficulties with ensuring the proper excitation, so that all the measured modes are excited well and that at the same time the excitation level is not too large, which would cause a larger response of the structure than the measuring ranges of the sensors can cover. In this study an innovative method for the operational modal analysis of small and light structures is presented. The method is noncontact; therefore, there is no added mass of the sensors to the structure. The structure is acoustically excited with a pure sine signal that is tuned to each resonant frequency. A single response measurement with the laser Doppler vibrometer in individual points is needed to determine the modal parameters. A mass-change strategy is used for the mass-normalisation of the measured mode shapes. The main contribution of the presented method compared to other similar methods is that the mode shapes are better accentuated (due to the sine excitation), which can improve the results of the modal analysis on small and light structures, where the response of the structure is weak. The method is also easy to perform, because only a single response measurement is needed for each point and the excitation force does not need to be measured. The presented method gives accurate results, and this was confirmed with a comparison of the experimental and the numerical results on a sample of simple geometry. Keywords: operational modal analysis, tuned-sinusoidal method, mode-shape normalisation, small and light structures, single response, acoustic excitation

0 INTRODUCTION The rapid development of operational modal analysis (OMA) did not start until the year 2000 [1]. In the past researchers used it mostly on large structures, where the excitation measurement is difficult and therefore the ambient excitation was employed [1] to [3]. The use of OMA on small and light structures is still a subject of research. Small and light structures have some distinctive features that intensify the difficulties of measuring their modal parameters. The resonant frequencies are usually relatively high; therefore, a wide frequency range of measurement is needed. The mass that is added to the structure by the sensors cannot be neglected and there are also difficulties with ensuring the proper excitation, so that all the measured modes are well excited and that at the same time the excitation level is not too large, which would cause a larger response of the structure than the measuring range of the sensors can cover. The goal of this study was to develop an effective non-contact OMA method for small and light structures. Some non-contact methods for modal analysis were already developed [4] to [7]. In these cases either ambient or acoustic excitation was used. Parloo et al. [4] excited the structure (a 1.7-kg wooden board, constrained with clamping devices on both its far edges) using an acoustic device. The reference

response was measured with an accelerometer and the roving response with a laser Doppler vibrometer (LDV). The method involves unknown non‑contact excitation, which is suitable for small and light structures. However, the use of an accelerometer for a reference response measurement is not suitable, because the added mass of the sensor has an influence on the measured modal parameters. The main contribution of Parloo’s method [4] is that it enables the normalisation of operational mode shapes using a sensitivity analysis. Siringoringo and Fujino [5] used a LDV for the measurement of the reference response and another LDV for the measurement of the roving response. Ambient excitation was used, because the sample (a steel plate) was only fixed at one edge and the vibration transfer from the surroundings was sufficient. This method [5] works with the presumption that the input force contains equal power within the frequency range of measurement (white-noise excitation). However, if the excitation is unknown some additional peaks in the frequency spectrum of the excitation force could introduce additional peaks in the operating-deflectionshape frequency-response function (ODS FRF) that would not agree with the resonant frequencies, as mentioned by [3], [8] and [9]. Therefore, a different approach is needed. Ahmida and Ferreira [6] used a more controlled excitation. The structure was excited with two small

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, miha.boltezar@fs.uni-lj.si

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loudspeakers and the response was measured by a single LDV at an individual point of measurement. The signal that was sent to the loudspeakers for the excitation was monitored and used as an excitation signal for the experimental modal analysis (EMA). White noise excitation was used. This method is suitable for small and light structures. Ahmida and Ferreira [6] tested it on a very small structure, consisting of silicon plates. However, the method does not include the mass normalisation of the mode shapes, because the actual excitation force on the structure is not measured (only the signal that is sent to the loudspeakers is measured). It is presumed that the excitation is white noise, like the monitored signal that is sent to the loudspeakers. The mass‑normalised mode shapes are needed to determine the complete modal model of the structure; therefore, this method can still be improved. A similar method to [6] was also used by Xu and Zhu [7]. The structure was excited acoustically (with white noise) and the response was measured at individual points with an LDV. The acoustic excitation was measured by a microphone near the surface of the structure. After the measurement the EMA was performed and the results were mass‑normalised mode shapes. The measured sound excitation differs from the real excitation because it is limited to a single point

of measurement, although the sound excites the whole structure (not only one point). Therefore, the scale of the measured mode shapes could be different than the real mass‑normalised mode shapes. This measurement [7] is also sensitive to the ambient sounds and cannot be performed in a loud environment (for instance, during the operation of the machine); therefore, a different method is proposed in this study. An innovative method for the measurement of the modal parameters was developed in this paper. It is suitable for the modal analysis of small and light structures. Only one loudspeaker and one LDV are needed for the proposed method. The mass‑normalisation of the mode shapes is also enabled based on the modal sensitivity of the structure (mass-change strategy), as described in Section 1.1. The method works by tuning the acoustic excitation to each resonant frequency and exciting the structure with a pure tone (sine) excitation. When using the presented method the mode shapes are better expressed than with white-noise excitation, because all the excitation energy is concentrated at one resonant frequency, where the response is therefore also more accentuated. The structure responds as if it had only one degree of freedom (all the other modes are not excited). The more accentuated mode shapes make the method very useful for the modal analysis of

Fig. 1. Tuned-sinusoidal method for the modal analysis of small and light structures

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small and light structures, where the response of the structure is weak and it is important to highlight the measured mode shapes in order to correctly measure their amplitude. This study is organised as follows. Section 1 presents the method that is proposed in this paper. The experiment and the numerical model are described in Section 2. Section 3 presents the experimental and numerical results and their comparison. A summary of the work is given in Section 4. 1 TUNED-SINUSOIDAL METHOD The basic idea of the tuned-sinusoidal method is presented in Fig. 1. First, the response of the structure, excited by an unknown force, is measured. Different excitation techniques, such as sound excitation, impact excitation, etc., can be used, as long as the spectral density of the excitation is constant across the whole frequency range of the measurement (white‑noise excitation). The resonant frequencies ωr are on the peaks of the measured frequency spectrum. Next, the mode shapes {ψr} are measured. The structure is subjected to the acoustic excitation (with a loudspeaker) and only the response of the individual point on the structure is measured. If the structure is acoustically excited with a sine that is tuned to the resonant frequency ωr, then the structure will respond with an ODS, that can be used as an unnormalised mode shape {ψr}, as shown by Schwarz and Richardson [10]. This assumption is valid because the mode shape that corresponds to the frequency ωr prevails over all the other mode shapes in the response of the structure. Therefore, the response of the structure is almost a pure sine with a frequency ωr, as shown in Figs. 2 and 3.

at all the points of the structure need to be measured for each resonant frequency. The amplitude of the excitation is not of significant importance; it only needs to be equal for all the measurements and the response needs to be within the measurement range of the LDV. Besides the response of the structure, the signal that is sent to the loudspeaker for the excitation is also measured. The phase between the sine of the excitation and response signal is used as a phase of individual component ψir of the mode shape {ψr}. The actual phase between the excitation and the response is different from the measured phase, because the sound has to travel from the loudspeaker to the structure (the influence of the damping on the resonant frequency is neglected). However, to define the un-normalised mode shapes, only the relative relations between the phases of the individual points on the structure are needed. This is especially so when the proportional damping is presumed and the mode shapes are not complex (the phase is either 90 or –90°). The amplitude of the response sine signal defines the amplitude of the individual component ψir of the mode shape {ψr}. Therefore, the structure has to be excited and the response needs to be measured for every (ith) measurement point at all the resonant frequencies ωr to obtain the amplitude and phase of each component ψir.

Fig. 3. Amplitude spectrum of the tuned‑sinusoidal method (ωr = 692.2 Hz)

Fig. 2. Time signal of the tuned‑sinusoidal method (ωr = 692.2 Hz)

The structure needs to be excited with all the individual resonant frequencies ωr and the responses

The described tuned-sinusoidal method may sound complex in theory; however, the application of this method is relatively straightforward. The laser is pointed at every measurement point and the structure is at first excited with the lowest resonant frequency ω1. The amplitude and phase of the component ψi1 are determined with the measured response. Then, without any movements of the sensors, the excitation frequency is raised to ω2, and ψi2 is measured. This

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continues until the last resonant frequency in the frequency range of the measurement. This procedure is repeated for all the measurement points and the unnormalised mode shapes {ψr} are defined. The name tuned‑sinusoidal method is used for the described method since the acoustic sine excitation is tuned to the measured resonant frequencies. The support of the structure can be free-free or fixed. The free-free support is usually more desirable for the comparison with the numerical model. 1.1 Mass-Normalisation of the Measured Mode Shapes The measured mode shapes {ψr} are not mass normalised since the excitation force is not measured when performing the tuned-sinusoidal method. Therefore, the method presented by Parloo et al. [4] that enables the mass‑normalisation of the operational mode shapes was used. It works on the basis of the modal sensitivity of the structure. The main idea of Parloo’s method is to normalise the measured mode shapes by multiplying them with scaling factors. A known mass is added to the selected points of the structure to change the resonant frequencies. The scaling factors are calculated from these changes by using the sensitivity analysis. The term mass‑change strategy is frequently used to denote this method. Other researchers also analysed the use of modal sensitivity to mass-normalise the operational mode shapes. Lopez-Aenlle et al. [11] to [14], Fernandez et al. [15] and [16] and Coppotelli [17] gave suggestions about how to accurately normalise the mode shapes using different types of mass-change strategies. Lopez-Aenlle et al. [11] analysed the equations for the calculation of the scaling factors and developed an iterative procedure for better accuracy. In [14] and [15] Lopez-Aenlle et al. and Fernandez et al. analysed and experimentally verified the influence of the location, number and size of the added masses on the accuracy of the results. Additional instructions on how to perform accurate calculations of the scaling factors were given by Lopez-Aenlle et al. in [13]. The modal sensitivity of the rth resonant frequency ωr and the mode shape {ψr} are defined in [18], as shown in Eqs. (1) and (2), where pj represents one of the Np structural modifications.

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∂ωr2 p j , (1) j =1 ∂p j

∆ωr2 = ∑ Np

∂{φr } p j . (2) j =1 ∂p j

∆{φr } = ∑

Modal sensitivity describes the influence of the structural modifications on the modal parameters of the structure. The mass-change strategy is based on Eq. (1), as shown by Parloo et al. [4]. If the structural modifications pj are the changes of mass at different points of the structure, then they can be described by a change of the mass matrix [∆M]. A known mass [∆M] needs to be added to the structure to perform the mass-change strategy. The change of the mass matrix causes a variation of the modal parameters. If the modal parameters of the unmodified and modified structures are measured, then the scaling factors αr can be calculated as shown in Eq. (3):

(ωr2 − ωr2,M ) αr = , (3) ωr2,M {ψ r }T [∆M ]{ψ r }

where ωr denotes the rth resonant frequency of the unmodified structure and ωr,M is the rth resonant frequency of the modified structure (when the mass [∆M] is added). {ψr} is the un-normalised mode shape of the structure. A detailed derivation of Eq. (3) can be found in [4] and [11], where a presumption is made that the mode shapes do not change significantly when adding the mass to the structure ({ψr} ≈ {ψr,M}). When this presumption is valid, accurate scaling factors αr can be calculated from Eq. (3) and the mode shapes of the modified structure {ψr,M} can be used for the calculation of scaling factors instead of {ψr}, as shown by Lopez-Aenlle et al. [11]. When the presumption {ψr} ≈ {ψr,M} is not valid, it is advisable to use the Bernal projection equation [19] that gives good estimates of the scaling factor αr even in cases when the mode shapes change significantly during the normalisation procedure. The Bernal projection equation is shown in Eq. (4):

(ωr2 − ωr2,M ) Brr αr = , (4) ωr2,M {ψ r }T [∆M ]{ψ r ,M }

where Brr represents the rth diagonal element of the matrix [B]. The matrix [B] is calculated as shown in Eq. (5), where [Ψ] denotes the modal matrix of the unmodified mode shapes {ψr} and [ΨM] the modal matrix of the modified mode shapes {ψr,M}:

[ B ] = [Ψ ]−1[Ψ M ]. (5)

When the scaling factors αr are calculated, the mass-normalised mode shapes {ϕr} can be obtained

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by multiplying the un-normalised mode shapes {ψr} by αr:

{φr } = α r {ψ r }. (6)

2 EXPERIMENT AND NUMERICAL MODEL 2.1 Sample A sample with the proper geometrical and modal properties was needed to compare the experimental and numerical results. Therefore, a small steel beam with a rectangular cross-section (1.94×20.2 mm) was used (Fig. 4). The beam was 121.42 mm long and had a mass of 37 grams. The geometrical parameters are not rounded to integers, because the dimensions of the sample were measured with a calliper to ensure greater accuracy of the numerical model’s results. A total of 13 points, denoted with the numbers 0 to 12, were used for the measurement, as shown in Fig. 4. The results of the numerical model were calculated for the same 13 points. Only the bending mode shapes in the direction of the shorter side of the cross-section (1.94 mm) were measured; however, all the other mode shapes can be measured in a similar manner.

2.3 Experimental Procedure First, the response of the structure to impulse excitation was measured. The impact excitation was carried out using a small steel ball (4 mm diameter, 0.26 grams weight) that was glued to a string and swung into the structure at all the measurement points to ensure that all the mode shapes were excited well. The resonant frequencies ωr were determined from the peaks of the frequency spectrum of the response. The structure was then acoustically excited with the loudspeaker and the un-normalised mode shapes ψr were defined from the response of the structure, as described in Section 1. The experimental set‑up is shown in Fig. 5. A 6-W loudspeaker with a wide frequency range was used to excite the structure and the response of the structure was measured with a Polytec PDV-100 LDV. The frequency range of the LDV is from 0.5 Hz to 22 kHz and the sensitivity was set to 100 mm/s. For the normalisation of the mode shapes with the mass‑change strategy, six magnets, each with a mass of 0.21 grams, were added at points 1, 3, 5, 7, 9 and 11 on the structure and the resonant frequencies ωr,M of the modified structure were measured using impact excitation (with a steel ball). Then the scaling factors αr were calculated, as described in Section 1.1, and finally the normalised mode shapes ϕr were defined (ϕr = αr · ψr).

Fig. 4. The sample that was used for the measurements and to build the numerical model

2.2 Numerical Model A numerical finite-element method (FEM) model was built with commercial software (Ansys). The model was based on the geometrical and material parameters (density, modulus of elasticity, Poisson’s ratio [20]) of the sample. A numerical modal analysis was performed on the model to determine the resonant frequencies and the normalised mode shapes. Since the damping is based on the results of the measurement, it is reasonable to use only the resonant frequencies and mode shapes for the comparison with the experimental results. The results of the model are relatively accurate, because the structure is homogeneous and without any joints or other sources of non-linearity [21] and [22].

Fig. 5. Experimental set‑up for the tuned‑sinusoidal method

It was difficult to ensure a proper sine excitation above 15 kHz due to the sampling frequency limit of the analogue output module. Therefore, the measurements were limited to frequencies below 15 kHz. Six bending resonant frequencies and their respective mode shapes were measured in this frequency range. LabView was used for the acquisition and modal analysis.

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3 RESULTS OF THE NUMERICAL MODEL AND THE EXPERIMENTAL PROCEDURE

Table 1. Comparison of the resonant frequencies calculated with the numerical model (num) and measured with the tunedsinusoidal method (OMA)

Table 1 presents the experimental and numerical resonant frequencies. OMA denotes the measured resonant frequencies ωr before the mass was added to the structure for the mass-change strategy and OMA-M denotes the resonant frequencies ωr,M after the mass was added to the structure. The numerical (num) and experimental (OMA) resonant frequencies differ by less than 1.5% (∆ num), and the first four by even less than 0.3%. Therefore, it was concluded that the measurements of the resonant frequencies were accurate. After the mass is added to the structure for the mass-change strategy the resonant frequencies reduce by about 0.7 to 2.4% (∆ OMA-M). This is a relatively small change for the mass-change strategy [13]; however, it still ensures the accurate mass normalisation of the mode shapes ψr. Figs. 6 to 11 show the mass‑normalised mode shapes that were measured with the tuned‑sinusoidal method (ϕOMA) and calculated with a numerical FEM model (ϕnum). It is clear that the experimental mode shapes are in good agreement with the numerical mode shapes. This agreement proves that not only the shape of the mode shapes, but also the normalisation (the scaling factor), was accurately measured. The correlation between the experimental and the numerical mode shapes was calculated using the modal assurance criterion (MAC), which is described in detail in [18], [23] and [24]. The result of the MAC procedure is a matrix with real values between 0 and 1. The value of each element of the MAC matrix belongs to a pair of mode shapes and describes their correlation (a higher value means a better correlation). The diagonal values of the MAC matrix are equal to 1 and the non-diagonal to 0 in the ideal case (when the numerical and experimental mode shapes are in perfect correlation). The MAC results are shown in Fig. 12. All the diagonal values of the MAC matrix are close to 1 (higher than 0.96) and the non-diagonal values are close to 0 (the highest is 0.12). The MAC results prove that the mode shapes were measured well, because there is a clear correlation between the numerical and the experimental mode shapes.

Resonant freq. 1. 2. 3. 4. 5. 6.

OMA OMA-M [Hz] [Hz] 692.2 682.4 1911.1 1885.7 3752.7 3703 6209.9 6140 9272 9205 12930 12615

∆ OMA-M [%] –1.4 –1.3 –1.3 –1.1 –0.7 –2.4

Num [Hz] 691.7 1910.4 3754.5 6224.0 9335.5 13113

∆ num [%] –0.1 –0.1 +0.1 +0.2 +0.6 +1.4

Fig. 6. Comparison of the 1st experimental and numerical mode shape

Fig. 7. Comparison of the 2nd experimental and numerical mode shape

Fig. 8. Comparison of the 3rd experimental and numerical mode shape

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4 CONCLUSION

Fig. 9. Comparison of the 4th experimental and numerical mode shape

Fig. 10. Comparison of the 5th experimental and numerical mode shape

Fig. 11. Comparison of the 6th experimental and numerical mode shape

Fig. 12. MAC correlation of the experimental and the numerical mode shapes

An innovative, non‑contact method for the modal analysis of small and light structures was presented. The method is based on an acoustic sine excitation that is tuned to individual resonant frequencies. A single response measurement (with an LDV) is needed to determine the resonant frequencies and the un-normalised mode shapes. A mass‑change strategy is used to calculate the scaling factors for the normalisation of the mode shapes. The tuned‑sinusoidal method gives good results on the sample that was used in this study. The resonant frequencies are measured accurately and differ from the results of the numerical model by less than 1.5% in a wide frequency range (up to 15 kHz). The first six bending-mode shapes were also measured well, which was confirmed by the MAC analysis. The good agreement between amplitudes of the experimental and the numerical mode shapes confirms that the scaling factors for the mass‑normalisation of the measured mode shapes were correctly defined. It can, therefore, be concluded that the tuned‑sinusoidal method gives accurate results for simple small and light structures. The tuned‑sinusoidal method has some clear advantages compared to other methods used for the modal analysis of small and light structures. It is a non-contact method (the contact is only needed if the mass-normalisation is to be performed), which is very convenient, because the sensors do not add any mass to the measured structure and therefore do not affect the results of the modal analysis. The presented method is also very practical, because a single response is measured and the experimental set-up is very simple and effective. The whole structure (not just a single point) is subjected to acoustic excitation; therefore, all the measured mode shapes are excited well. An important advantage of the presented method compared to similar methods is that the mode shapes are better accentuated (due to sine excitation), which can improve measurements on small and light structures, where the response of the structure is relatively weak. The better accentuation of the mode shapes is achieved because all the excitation energy is concentrated at just a single mode and all the other modes are not excited. The method is most appropriate for lightly damped structures without closely spaced modes, because they have better distinguished individual resonant frequencies and mode shapes. Therefore, it works better on structures with not many components made of damping materials. The acoustic excitation

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makes the tuned-sinusoidal method more appropriate for smaller structures, because the excitation intensity can be too low to excite larger structures. 5 REFERENCES [1] Brincker, R, Möller, N. (2006). Operational modal analysis - a new technique to explore. Sound and Vibration, vol. 40, no. 6, p. 5-6. [2] Devriendt, C., Guillaume, P., Reynders, E, De Roeck, G. (2007). Operational modal analysis of a bridge using transmissibility measurements. Proceedings of IMAC-25: A Conference and Exposition on Structural Dynamics, p. 748-757. [3] Parloo, E., Guillaume, P., Anthonis, J., Heylen, W., Swevers, J. (2003). Modelling of sprayer boom dynamics by means of maximum likelihood identification techniques, Part 1: A comparison of input‑output and output‑only modal testing. Byosistems Engineering, vol. 85, no. 2, p. 163-171, DOI:10.1016/ S1537-5110(03)00044-8. [4] Parloo, E., Verboven, P., Guillaume, P., Van Overmeire, M. (2002). Sensitivity-based operational mode shape normalisation. Mechanical Systems and Signal Processing, vol. 16, no. 5, p. 757-767, DOI:10.1006/ mssp.2002.1498. [5] Siringoringo, D.M., Fujino, Y. (2009). Noncontact operational modal analysis of structural members by laser Doppler vibrometer. Computer-aided Civil and Infrastructure Engineering, vol. 24, no. 4, p. 249-265, DOI:10.1111/j.1467-8667.2008.00585.x. [6] Ahmida, K.M., Ferreira, L.O.S. (2004). Design and modeling of an acoustically excited doublepaddle scanner. Journal of Micromechanics and Microengineering, vol. 14, no. 10, p. 1337-1344, DOI:10.1088/0960-1317/14/10/007. [7] Xu, Y. F., Zhu, W.D. (2011). Operational modal analysis of a rectangular plate using noncontact acoustic excitation. Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Vol. 5 (Conference Proceedings of the Society for Experimental Mechanics Series). Springer, New York, p. 359-374, DOI:10.1007/978-1-4419-9428-8_30. [8] Devriendt, C., Guillaume P. (2008). Identification of modal parameters from transmissibility measurements. Journal of Sound and Vibration, vol. 314, no. 1-2, p. 343-356, DOI:10.1016/j.jsv.2007.12.022. [9] Devriendt, C., De Sitter, G., Vanlanduit, S., Guillaume, P. (2009). Operational modal analysis in the presence of harmonic excitations by the use of transmissibility measurements. Mechanical Systems and Signal Processing, vol. 23, no. 3, p. 621-635, DOI:10.1016/j. ymssp.2008.07.009. [10] Schwarz, B.J., Richardson, M.H. (2004). Measurements required for displaying operating deflection shapes. Proceedings of IMAC-22: A Conference on Structural Dynamics, p. 701-706.

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[11] Lopez-Aenlle, M., Brincker, R., Fernandez‑Canteli, A., Garcia, L.M.V. (2005). Scaling-factor estimation by the mass-change method. Proceedings of IOMAC-1, p. 5364. [12] Lopez‑Aenlle, M., Brincker, R., Fernandez‑Canteli, A. (2005). Some methods to determine scaled mode shapes in natural input modal analysis. Proceedings of IMAC-23: A Conference on Structural Dynamics, p. 141-145. [13] Lopez‑Aenlle, M., Fernandez, P., Brincker, R., Fernandez‑Canteli, A. (2010). Scaling‑factor estimation using an optimized mass-change strategy (Correction of: vol. 24, no. 5, p. 1260-1273, 2010). Mechanical Systems and Signal Processing, vol. 24, no. 8, p. 30613074. [14] Lopez-Aenlle, M., Fernandez, P., Brincker, R., Fernandez‑Canteli, A. (2007). Scaling-factor estimation using an optimized mass-change strategy, Part 1: Theory. Proceedings of IOMAC-2, p. 421-428. [15] Fernandez, P., Lopez‑Aenlle, M., Garcia, L.M.V., Brincker, R. (2007). Scaling-factor estimation using an optimized mass-change strategy, Part 2: Experimental Results. Proceedings of IOMAC-2, p. 429-436. [16] Fernandez, P., Reynolds, P., Lopez‑Aenlle, M. (2011). Scaling mode shapes in output‑only systems by a consecutive mass change method. Experimental Mechanics, vol. 51, no. 6, p. 995-1005, DOI:10.1007/ s11340-010-9400-0. [17] Coppotelli, G. (2009). On the estimate of the FRFs from operational data. Mechanical Systems and Signal Processing, vol. 23, no. 2, p. 288-299, DOI:10.1016/j. ymssp.2008.05.004. [18] Maia, N.M.M., Silva, J.M.M. (1997) Theoretical and Experimental Modal Analysis, 1st ed. Research Studies Press, John Wiley and Sons, Taunton, New York, etc. [19] Bernal, D. (2004). Modal scaling from known mass perturbations. Journal of Engineering Mechanics, vol. 130, no. 9, p. 1083-1088, DOI:10.1061/(ASCE)07339399(2004)130:9(1083). [20] Kraut, B., Puhar, J., Stropnik, J. (2003) Krautov strojniški priročnik (Kraut’s Engineering Handbook), 14th ed., Littera picta, Ljubljana. (in Slovene) [21] Čermelj, P., Boltežar, M. (2006). Modelling localised nonlinearities using the harmonic nonlinear super model. Journal of Sound and Vibration, vol. 298, no. 4-5, p. 1009-1112, DOI:10.1016/j.jsv.2006.06.042. [22] Čelič, D., Boltežar, M. (2008). Identification of the dynamic properties of joints using frequency-response functions. Journal of Sound and Vibration, vol. 317, no. 1-2, p. 158-174, DOI:10.1016/j.jsv.2008.03.009. [23] Allemang, R.J. (2003). The modal assurance criterion - Twenty years of use and abuse. Sound and Vibration, vol. 37, no. 8, p. 14-23. [24] Fotsch, D., Ewins, D.J. (2000). Application of MAC in the Frequency Domain. Proceedings of IMAC-18: A Conference on Structural Dynamics, p. 1225-1231.

Rovšček, D. – Slavič, J. – Boltežar, M.

Original Scientific Paper

Received for review: 2013-10-04 Received revised form: 2014-01-10 Accepted for publication: 2014-01-24

Prediction of Wall Thickness Distribution in Simple Thermoforming Moulds Erdogan, E.S. – Olcay Eksi, O. Ertugrul Selcuk Erdogan1 – Olcay Eksi2,*

1Trakya

2Namık

University, Engineering Faculty, Turkey Kemal University, Çorlu Engineering Faculty, Turkey

Thermoforming is widely used in manufacturing industries to produce large and labour-intensive products. Compared to other manufacturing techniques, thermoforming is an extremely efficient process that is suitable for high-efficiency mass production. In this paper, experimental thermoforming operations were carried out using a lab-scale, sheet-fed thermoformer. Carbon fibre-reinforced PP and unreinforced PS thermoplastic sheets were used in experimental thermoforming operations. The processing parameters were determined for each thermoformed material. Furthermore, a simulation of the thermoforming process was performed using LS-DynaTM software. The thickness distributions obtained from the experiments were compared with the simulation results. The results show that the parameter that most affects the wall thickness distribution is the geometry of the clamping ring. To produce thermoformed products that have a more uniform thickness distribution, the clamping tool geometry must be selected according to the geometry of the product being thermoformed. Keywords: thermoforming, wall thickness prediction, geometric element analysis (GEA), polystyrene, polypropylene

0 INTRODUCTION In thermoforming, a polymeric sheet is heated to the proper temperature, which is termed ‘the forming temperature’. This temperature depends upon the type of thermoplastic material. The polymer sheet is rubbery and elastic-plastic deformable at the forming temperature [1]. The sheet is then stretched into a female mould or over a male mould with positive or negative air pressure. The sheet is in contact with the cold mould surface for some time, which allows the polymer sheet to cool down to a temperature at which the sheet is sufficiently rigid to release from the mould. Formed sheets include the semi-finished product and the undesired trimming areas. Furthermore, thermoforming is an efficient, cost-effective manufacturing process that produces flexible, strong and durable parts. Large, highly detailed and light-weight parts can be formed economically using thermoforming. Some useful properties, such as the low tooling costs, the ease of creating aesthetically desirable finishes, and the fast adaptation to the market, make thermoforming one of the fastest growing segments in the plastics industry. Thermoforming is an integration of many techniques. Vacuum forming, pressure forming and twin-sheet forming are techniques used to produce many products with distinct external features [2]. Lieg and Giacomin [3] performed an analysis in which the method of Kershner and Giacomin was applied to triangular troughs. The analysis focused on the speed of the manufacturing process. This study of the trough issue yielded no analytical solutions for the forming time; instead, Lieg and Giacomin combined

a series expansion with a numerical solution to provide practitioners with a method to estimate the thermoforming time, the trough edge sharpness and the frozen-in stress. Ayhan and Zhang [4] investigated the effects of the process parameters, including the forming temperature, the forming air pressure and the heating time, on the wall thickness distribution of plug-assist thermoformed food containers using a multi-layered material. The resulting wall thickness data obtained for the various thermoforming parameters showed that the wall thickness was significantly affected by the forming temperature, pressure and heating time; the wall location, the container side and the interactions between the wall and the container side significantly affected the wall thickness. Rosenzweig et al. [5] studied an isothermal, one-dimensional model predicting the wall-thickness profiles of thermoformed products. The theoretical analyses of pressing into conical and truncated moulds were presented and discussed. They developed a theoretical geometric model based on eight simplifying assumptions that were independent of the material properties and the forming conditions. This model predicted the wall thickness distribution in thermoforming, which correlated with the experimental results. Azdast et al. [6] investigated the combination of free-forming and plug-assisted forming methods. During free forming, no moulds are used, and no undesired mould marks are obtained; however, the thickness distribution of the product is not desirable. In plug-assisted thermoforming, the thickness is almost uniform, but mould marks appear. A combination of the two processes is recommended in order to maximise the advantages of both processes

*Corr. Author’s Address: Namık Kemal University, Çorlu Engineering Faculty, Mechanical Engineering Department, 59860-Tekirdağ, Turkey, oeksi@nku.edu.tr

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and minimise the disadvantages. Harron et al. [7] identified the critical variables in the thermoforming process. They investigated the effects of some process parameters, such as wall thickness distribution, compressive strength, plug force and pot weight, on the final part properties. They described five factors affecting the wall thickness distribution of a thermoformed product: the sheet temperature, plug depth, plug timing, air timing and plug shape. They also found that the compressive strength of a product is directly related to the wall thickness distribution and is controlled by the same factors. Marchal et al. [8] studied the thermoforming of three different geometries (a yoghurt pot, a cellular phone housing and a safety helmet) and examined different process parameters, such as the initial temperature distribution across the sheet, the viscosity, the forming pressure and the mould surface quality. To analyse the influence of these parameters on the final thickness distribution and the final temperature distribution, the studied parameters were modified using the commercially available software Polyflow. The simulation results showed that the numerical simulation reveals all of the analytical power in the investigation of the influence of the different parameters on the quality of the final product. The numerical simulation results can be combined with the knowledge of the designers, which can reduce the time required to design a new product. Nam and Lee [9] thermoformed two distinct acrylonitrile-butadiene-styrene (ABS) polymers. The thermoforming behaviour of these polymers was compared with a numerical simulation. Hot tensile tests were performed to obtain some material parameters for the simulation. The thickness distribution obtained from the experiments was compared with the simulation results. Although Nam et al. performed a three-dimensional analysis of the thermoforming process using different ABS grades, they found that this difference does not mean that one grade is better than another. To classify these materials according to the thermoforming process parameters, a more realistic simulation is necessary for further understanding. Dong et al. [10] used the PAM-FORMTM software package to simulate the thermoforming process of a polymeric sheet. They adopted a hyper-elastic constitutive law based on the Mooney-Rivlin model in order to carry out the bubble inflation simulation and to identify the material parameters. They focused on the thickness distribution analysis and the strain states of the bubble inflation. They found that the deformation profile correlated well with the experimental results. In addition to these literature studies, there are alternative methods 196

to predict the wall thickness distribution in simple thermoforming moulds. One method is an approach that has been widely used in blow moulding and in thermoforming for more than forty years; it has been used in Germany, Poland, Russia and Japan. Geometric element analysis (GEA) [11] to [15] method follows a protocol for the stretching of an infinitely extensible membrane over a surface with a known geometry, and the polymer properties are not needed. Throne [11] and [12] studied the application of GEA to the conical thermoforming moulds. He created several equations that are used for the prediction of wall thickness distribution for deep and shallow thermoforming moulds. Kutz [13] and Crawford [14] both created calculations that can yield a useful first approximation of the dimensions of a thermoformed part. In addition, Osswald [15] studied the prediction of thickness distribution profiles in hemispherical and open cylindrical thermoforming moulds. These literature studies help scientific researchers and thermoformers to produce thermoformed packages with more uniform thickness distribution and dimensional stability. In this work, the wall thickness distributions in three thermoformed products were predicted using GEA and experimental methods. Carbon fibrereinforced, and unreinforced thermoplastic sheets were used in experimental thermoforming operations. Processing parameters were determined for each thermoformed material, and polymer sheets were formed using a lab-type thermoformer. Additionally, simulation of the thermoforming process was performed using LS-DynaTM software. The thickness distributions obtained from the experiments were compared to the simulation results. 1 MATERIALS AND METHODS Unreinforced polystyrene and carbon fibre-reinforced polypropylene sheets with different thicknesses were used in the experimental study. The thicknesses of the polystyrene sheets were 2.0 and 2.5 mm. The polystyrene used in the experiment was SABIC PS 825 E, which is a high-impact polystyrene for thermoforming. Polypropylene was extruded using a laboratory type extruder with a 55 mm screw diameter. The extruded polypropylene sheets were produced at a constant line speed of approximately 1 m/min. With the use of this laboratory extruder, thermoplastic sheets can be produced with 1 to 3 mm thicknesses and 30 mm widths. Polypropylene sheets were extruded as 5 and 15% carbon fibre-reinforced in weight in order to determine the effect of reinforcement on the final wall thickness distribution. The effect of the

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 195-202

fibre reinforcement on the thermoforming process parameters was investigated. Prepared PP sheets were extruded to be 2 mm (±0.01 mm) thick. Reinforced PP sheets were extruded using Borealis BE50-7032 Polypropylene granules. Sheets were thermoformed using a lab-scale sheet fed thermoformer that was controlled manually. Loading the sheet into the forming table, adjusting the forming temperature, opening and closing of the upper unit, setting the velocity of this unit and starting of the vacuum was entirely controlled by the researcher. Therefore, a specific cycle time was not mentioned for that process. The thermoforming unit (Yeniyurt Machinery) was not manufactured for mass production and can be used only for laboratory experiments. This unit uses only heat and a vacuum to form the sheet and can form sheets from 1 to 3 mm in thickness. The forming technique used in this experimental study is termed ‘negative forming’ or ‘vacuum forming’. In the vacuum-forming technique, female moulds are used. The mould is placed below the sheet; the sheet sags into the mould, and the part is formed down into the tool. In this study, three types of female moulds (cylindrical, conical and cubical) were used in the manufacturing of products. The sheets were cut into squares of 300×300 mm² before thermoforming. The thermoforming process parameters were determined for each material according to the manufacturer catalogue information. However, the thermoforming parameters for the extruded reinforced PP sheets were predicted through trial and error. The sheet forming temperatures for the reinforced PP sheets were modified according to the heating time. The forming temperature was controlled using twelve ceramic heaters. The heating system consists of two zones. The ceramic heaters have a 500×500 mm² heating area capacity. The first heating zone is in the centre of the complete heating system and has a 300×300 mm² heating capacity. The first heating zone was used to heat the sheets before the sheets were thermoformed. All of the dimensions were chosen for a h (height): d (diameter) ratio of 0.5. The forming temperatures were selected as 180 °C for PS, 185 °C for the 5% carbon fibre-reinforced PP and 190 °C for the 15% carbon fibre-reinforced PP sheets. Wall thickness values were measured from Point-1 (at the centre of the base of the product) to Point-2 (at the end of the radius on the rim) (Figs. 1 to 3). The wall thickness measurements were performed on at least five different products for each PS and PP sheet along a vertical cut passing through the centre. Each measurement was repeated at least three times. Measurements of the wall thicknesses were performed

using a digital micrometer (0.01 mm precision). The obtained wall thickness profiles from the experiments were compared with the results calculated using GEA for the three mould geometries. The thermoforming simulation was performed using LS-Dyna explicit software. The results for all materials were compared.

Fig. 1. Dimensions of half conical thermoformed product in mm

Fig. 2. Dimensions of half cylindrical thermoformed product in mm

Fig. 3. Dimensions of half cubical thermoformed product in mm

2 RESULTS AND DISCUSSIONS Fig. 4 shows the thickness distribution calculated by both GEA and the experimental method for polystyrene. The thickness was measured using a digital micrometer through a path that passes through the centre of the mould base. From the data in Fig. 4, the thickness distribution results generated by

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Fig. 4. The wall thickness distribution in the cylindrical thermoforming mould; polystyrene sheet 2.5 mm in thickness

Fig. 7. The wall thickness distribution obtained along a vertical cut through a symmetry axis of cubical thermoforming mould; polystyrene sheet 2 mm in thickness

Fig. 5. The wall thickness distribution in the conical thermoforming mould; polystyrene sheet 2 mm in thickness

Fig. 8. The wall thickness distribution in the conical thermoforming mould; (5% carbon fibre-reinforced sheet 2 mm in thickness

Fig. 6. The wall thickness distribution obtained along a vertical cut through a diagonal line of cubical thermoforming mould; polystyrene sheet 2 mm in thickness

Fig. 9. The wall thickness distribution in the conical thermoforming mould; 15% carbon fibre-reinforced sheet 2 mm in thickness

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the experimental method (EXP.) and the Geometric Element Analysis (GEA) do not correlate with each other. The point at which the minimum sheet thickness occurs is different for each of the abovementioned methods. In the experimental method, the minimum sheet thickness for the cylindrical mould was measured to be approximately 0.83 mm, but the minimum sheet thickness was calculated to be approximately 0.666 mm using GEA. In Fig. 5, the thickness distribution of the polystyrene product is given for comparison. The thickness values for the product are different for the experimental and numerical methods. In addition, this minimum thickness value and the point where most of the thinning occurs are closer than before for the two curves generated using GEA, finite element analysis (FEA) and the experimental method. There is slight correlation between the two thickness profile curves obtained by the experimental method and GEA in Fig. 5. In contrast, the simulation results did not have the same trend in Fig. 5. To more accurately predict the wall thickness distribution in the cubical mould, both a diagonal line and one representing the symmetry axis of the cubical mould were measured. These results are illustrated in Figs. 6 and 7; the sheet thicknesses in the centre of the product are different. The thinnest point is different between the thickness profile curves generated experimentally and using GEA. It is apparent that GEA does not accurately generate wall thickness distributions that correlate with the results measured in the experiments. To investigate the effect of fibre reinforcement on the thickness distribution, chopped carbon fibres were included during the polypropylene extrusion. Reinforcement inclusion was achieved by adding carbon fibres into the feeding hopper at 5 and 15% in weight. The resultant composite PP sheets including chopped carbon fibres were 2 mm in thickness. These PP sheets were thermoformed using only the conical

thermoforming mould. Figs. 8 and 9 represent the thickness distribution curves. There are considerable differences between the tendencies of the thickness distribution curves in both figures. The points that are the thinnest in the 5 and 15% carbon fibre-reinforced thermoformed products are not in the same location. The thickness for the thinnest area is different for the two carbon fibre-reinforced composite products. In Fig. 8, the measured thickness value is 0.9 mm on the radius that is at the base of the product. On the same radius, the minimum thicknesses calculated by GEA and FEA are 0.704 and 1.1412 mm, respectively. There is approximately a 21% difference between the actual thickness value and the thickness value obtained by GEA. FEA reveals a 26% difference compared to actual minimum thickness value. In Fig. 8, the thickness distribution curve that was obtained experimentally has a decreasing trend. In Fig. 9, the measured thickness value is 1.48 mm on the radius that is at the base of the product. At the same point, the lowest thickness values obtained using GEA and FEA are 0.704 and 1.1467 mm, respectively. There is about 52% difference between the actual thickness value and the thickness value obtained by GEA. FEA reveals a 22% difference compared to actual minimum thickness value. In Fig. 9, the variation of thickness curve obtained using the experimental method follows an increasing trend. In addition to this, in Fig. 8, there are thickness values greater than 2 mm at Arc length = 0 and 20 (at the centre of the base of the conical thermoformed product). That phenomenon occurred during the measurements because of an overlapped sheet. As a result of this, the sheet thickness was measured as being greater than it is. Taking into account that the initial thickness of the PP reinforced sheet is 2 mm, measuring the sheet thickness as 3.5 mm may be regarded as normal. Overlapping is a manufacturing problem caused by the semi-crystalline

a) b) Fig. 10. a) the wall thickness and b) resultant displacement distribution results obtained using Ls-Dyna software for a 2 mm PS sheet using the conical thermoforming mould Prediction of Wall Thickness Distribution in Simple Thermoforming Moulds

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b) c) Fig. 11. Images of the thermoformed products examined visually; a) cylindrical, b) conical and c) cubical products

structure of the PP sheet material. It was observed in all 5% carbon fibre-reinforced thermoformed specimens. Fig. 15 shows overlapped and wrinkled sheet surface at the centre of the base of the conical thermoformed product. Overlapping can occur at the base of the product or on the sidewalls. If it occurs at the base of the product as in Fig. 8, thickness variation has a decreasing trend. If overlapping occurs on the sidewalls of the product as in Fig. 9, the thickness curve has an increasing tendency. To efficiently analyse the results of the thermoforming simulation process, a quarter of a finite element model (FEM) was created. The FEM consists of 16440 nodes and 13525 shell elements; some of which are rigid (quarter of the inner surface of the conical mould), and other are deformable (quarter of the total polymeric sheet) shell elements. The wall thickness distribution result is illustrated in Fig. 10. The thickness distribution varies along the radius of the mould base. This variation is because the thermoformed product geometry is circular, whereas the clamping ring has a square geometry. To clearly show the thickness profile of the thermoformed conical, cubical and cylindrical products, a visual representation was created. The thermoformed products were placed in front of a light source, and images were taken (Fig. 11). The light areas are the thinnest and weakest parts of the thermoformed product, whereas the dark areas are the thickest and most durable. The light areas show the parts of the polymer sheet that last touched the mould surface during thermoforming. Four different points were selected to be examined using scanning electron microscope (SEM) on the composite conical thermoformed product. Fig. 12 shows the locations for these points. A cut section was taken from the centre of the chopped 200

carbon fibre-reinforced PP products. Four points were predetermined on this section. To show the fibre alignments in these points, SEM images were obtained (Figs. 13 and 14). The SEM images show that the carbon fibres are usually perpendicular to the sections in which they exist. The thickness distributions obtained using Geometric Element Analysis and

Fig. 12. Four locations where the SEM images were taken

Fig. 13. SEM images of the 5% carbon fibre-reinforced PP conical product; points 1 to 4 (magnification: 500×)

Erdogan, E.S. – Olcay Eksi, O.

StrojniĹĄki vestnik - Journal of Mechanical Engineering 60(2014)3, 195-202

Thermoforming Simulation show poor correlations with the measured data. This because carbon fibrereinforced PP products have some surface defects, which results in roughness. Furthermore, the extruder used in the production of reinforced PP sheets has a single screw. This single screw causes undesired fibre distribution in the PP material. In addition to this distribution, concentrated carbon fibre groups lead to inappropriate thickness distributions (Fig. 15).

Fig. 14. SEM images of the 15% carbon fibre-reinforced PP conical product; points 1 to 4 (magnification: 500Ă&#x2014;)

Fig. 15. 1) wrinkled and overlapped sheet surfaces; 2) unbalanced deformation of product because of heterogeneous fibre distribution; 3) undesired product thickness distribution because of the non-uniform initial sheet thickness; 4) rough and porous surface defects

3 CONCLUSIONS The experiments performed on the thermoformed products clearly show that one of the leading parameters that affects the wall thickness distribution is the geometry of the clamping ring. To produce thermoformed products that have more uniform thickness distributions, the clamping tool geometry must be selected according to the geometry of the product. It is proposed that a circular clamping tool geometry be selected for cylindrical and conical products and a rectangular one for rectangular products. That can balance the stress in all directions and provide uniform deformation characteristics, resulting in more uniform thickness distribution. Thickness distribution results have changed with the materials used in thermoforming simulation. However, there are minor differences between thickness distribution results (for example, at the same arc length value, for 5% reinforced PP thickness:1.1412 mm for 15% reinforced PP thickness 1.1467 mm). A significant difference not been found between the thickness distribution results of PS and reinforced PP sheets. In order to obtain more appropriate thickness distribution results, thermoforming simulation should be performed according to different material models that can represent polymer deformation behaviour more accurately. Furthermore, thermoforming simulation can be repeated with various polymer material models from LS-Dyna material model library. Additionally, simulations can be achieved in detail by using software packages like Ansys-Polyflow, T-SIM, etc. In this study, the geometry of the clamping tool was predicted to significantly affect the uniformity of the sidewalls and the base of the thermoformed product (Fig. 10). The amount of material in the sidewalls and base of the thermoformed product is governed by the geometry of the clamping tool. Because of the square geometry of the clamping tool, the material in the side walls stretched less along the diagonal axis of the clamping tool. This is considered to be the foremost reason for the non-uniformity of the conical and cylindrical products. As a result, the shape of the clamping ring, the homogeneity of the reinforcing fibre distribution in the matrix material, the anisotropic properties caused by the extrusion direction and the rough and porous surface defects were predicted to be the primary reasons for the nonuniform thickness distribution.

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4 ACKNOWLEDGEMENTS The authors would like to thank to EX-EN Engineering Company in Turkey for providing LsDynaTM software. The authors would also like to thank to Trakya University, Engineering Faculty in Turkey for providing Solidworks software. 5 REFERENCES [1] Throne, J.L. (1991). Guidelines for thermoforming part wall thickness. Polymer-Plastics Technology and Engineering, vol. 30, no. 7, p. 685-700, DOI:10.1080/03602559108020144. [2] Throne, J.L., Mooney, P.J. (2005). Thermoforming: growth and evolution I. Thermoforming Quarterly, vol. 24, no. 1, p. 18-20. [3] Lieg, K.L., Giacomin, A.J. (2009). Thermoforming triangular troughs. Polymer Engineering and Science, vol. 49, no. 1, p. 189-199, DOI:10.1002/pen.21239. [4] Ayhan, Z., Zhang, H. (2000). Wall thickness distribution in thermoformed food containers produced by a benco aseptic packaging machine. Polymer Engineering and Science, vol. 40, no. 1, p. 1-10, DOI:10.1002/ pen.11134. [5] Rosenzweig, N., Narkis, M., Tadmor, Z. (1979) Wall thickness distribution in thermoforming. Polymer Engineering and Science, vol. 19, no. 13, p. 946-951, DOI:10.1002/pen.760191311. [6] Azdast, T., Doniavi, A., Ahmadi, S.R., Amiri, E. (2013). Numerical and experimental analysis of wall

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thickness variation of a hemispherical PMMA sheet in thermoforming process. International Journal of Advanced Manufacturing Technologies, vol. 64, no. 1, p. 113-122, DOI:10.1007/s00170-012-4007-5. [7] Harron, G.W., Harkin-Jones, E.M.A., Martin, P.J. (2000). Influence of thermoforming parameters on final part properties. The Annual Technical Conference ANTEC 2000 Conference Proceedings, p. 3723-3727. [8] Marchal, T.M., Clemeur, N.P., Agarwal, A.K. (1998). Optimisation of the thermoforming process: a few industrial examples. The Annual Technical Conference ANTEC 1998 Conference Proceedings, p. 701-705 [9] Nam, G.J., Lee, J.W. (2001). Numerical and experimental studies of 3-dimensional thermoforming process. Journal of Reinforced Plastics and Composites, vol. 20, no. 14, p. 1182-1190. [10] Dong, Y., Lin, R.J.T., Bhattacharyya, D. (2006). Finite element simulation on thermoforming acrylic sheets using dynamic explicit method. Polymers&Polymer Composites, vol. 14, no. 3, p. 307-328. [11] Throne, J.L., (1996). Technology of Thermoforming. Hanser/Gardner Publications, Cincinnati. [12] Throne, J.L. (1987). Thermoforming. Hanser Publishers, Munich. [13] Kutz, M. (2002). Handbook of Materials Selection. John Wiley & Sons, New York, DOI:10.1002/9780470172551. [14] Crawford, R.J. (1998). Plastics Engineering. Butterworth-Heinemann, Oxford. [15] Osswald, T.A. (1998). Polymer Processing Fundamentals. Hanser/Gardner Publications, Cincinnati.

Erdogan, E.S. â&#x20AC;&#x201C; Olcay Eksi, O.

Received for review: 2013-09-04 Received revised form: 2013-10-03 Accepted for publication: 2013-10-07

Original Scientific Paper

Sea-wave Dynamic Loading of Sailing Yacht`s Retractable Keel Koc, P. Pino Koc* University of Ljubljana, Faculty of Mathematics and Physics, Slovenia A highly specialized analysis of functionality of a retractable keel lifting mechanism is presented in the paper. Because of its special design a retractable keel is vulnerable to rough sea conditions. To alleviate possible malfunctions or even damage to the keel, an analysis of lifting the keel during rough sea (WMO sea state 6) is performed. Reliable estimation of load acting on a keel is crucial for the subsequent functional or structural analysis of the keel. Aiming to obtain these loads, a sea wave analysis is performed first, where a wave train is determined by respecting statistical data regarding the required sea condition. Two wave cases are considered: head seas (waves coming from the front) and beam seas (lateral waves). A deterministic dynamic analysis on the wave-hull-keel system is then performed. The response of this system is obtained in the form of a time dependent acceleration of the centre of gravity of the vessel and in a time dependent lifting force, i.e., the force which is required for lifting the moveable part of the keel during rough sea. Finally, the required pulling force of the lifting mechanism is determined. Keywords: sailing yacht, retractable keel, wind wave, computer simulation

0 INTRODUCTION One of the tasks of a monohull sailing yacht’s keel is to compensate for yacht heeling due to the wind force. The deeper and heavier the keel is, the smaller the heeling angle or, viewed from another angle, for the same heeling angle more sails can be used. Too deep of a keel prohibits access to shallow waters and harbours. A retractable keel is the solution to this problem and still provides a good sailing ability. From a mechanical point of view, several kinematic mechanisms are possible to retract a keel. One special type of a retractable keel, used for cruising sailing yachts has been developed in recent years and is manufactured now in some shipyards worldwide [1]. It is a so-called telescopic lifting keel that is located completely outside (below) the hull. The advantage of this type of keel is that it does not influence the interior of the hull nor does it limit the cabin arrangements. The working principle of a telescopic keel is shown in Fig. 1. The keel consists of a fixed part bolted to the hull structure, a moveable part which slides along guides embedded in a fixed part, and a hydraulic cylinder for lifting and lowering the moving part. The lifting height is relatively small as compared to more common lifting keels which intrude into the hull, at the most 1 m. Since it is desirable to have a heavy keel, it is fabricated of cast iron and thick steel weldments. Although made of heavy sections which give the illusion of indestructibleness, a telescopic keel is in fact a mechanism exposed to a rough seawater environment and seafaring conditions, and is thus prone to various types of break-downs and damage.

Fig. 1. Working principle of the telescopic lifting keel

An important step in an attempt to avoid any malfunction of the telescopic keel is a reliable functional and structural analysis. 1 PROBLEM DEFINITION Basic mechanical design and a structural qualification of an ordinary, i.e., fixed keel is a well-known task, based on simple mechanical principles [2] and verified through many successful designs. When designing more complicated types of keels, lifting or canting, for example, advanced designing methods, based on finite element (FE) analysis are inevitable [3]. Also, fixed or retractable keel hydraulic design is a highly demanding task and only in the last few decades has it been thoroughly comprehended through the aid of modern computational techniques [4].

*Corr. Author’s Address: University of Ljubljana, Faculty of Mathematics and Physics, Lepi pot 11, 1000 Ljubljana, Slovenia, pino.koc@fmf.uni-lj.si

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In the structural analysis of the retractable keel, peculiarities such as sea-wave dynamic loading, extreme events (capsizing, grounding and collision), geometrical (intermittent contact between moveable and fixed parts of keel) and material (plastic deformation) nonlinearities and force variability in a hydraulic cylinder need to be considered. As such, a presentation of the whole structural analysis is too extensive; therefore, only a part of it will be presented in the paper. The response of the hydraulic cylinder during rough water loading conditions is of a special interest for the designer. It is required that the cylinder is able to retract the moving parts of the keel even in a foul weather, i.e., in our case at the sea state scale 6, according to the World Meteorological Organization (WMO). At that sea state, the Significant Wave Height (SWH or traditionally H1/3) is between 4 to 6 m. The rest of the input data, used in this work are: • moveable mass (moveable part of the keel plus ballast): app. 5,500 kg; • cylinder stroke, i.e., lifting height: 500 mm; • duration of lifting: 60 s; • yacht’s maximal speed during lifting: 1.54 m/s (3 knots); and • wave direction: Case 1: from the front (head seas) and Case 2: from the side (beam seas). The goal of analysis is to predict the minimal cylinder force which is needed to accomplish the above defined task of retracting the keel on a rough sea.

mathematical approach [6] is used in our work. It is based on the fact that by a superposition of many sinusoidal waves, a quasi irregular wave pattern, similar to that of the natural wave train, can be obtained. At the arbitrary point (x,y) on the sea surface and at the time t, the instantaneous height of the surface h, measured from the still water level is:

h ( x, y , t ) = n  2π x  2π y = ∑hAi cos  cos α i + sin α i − ωi t + ϕi  , (1) λi i =1   λi

where n is the number of superimposed sine waves, hAi, λi and αi represent the amplitude (half of the wave height H), wavelength and wave propagation direction (Fig. 2) of ith sine wave respectively. φi is the phase shift and ωi is the rotational frequency of ith wave

ωi =

2π g , (2) λi

with g as a gravitational acceleration constant.

2 SEA-WAVE ANALYSIS Note: Sea waves are primarily a consequence of the wind blowing along the sea’s surface; therefore, they are also sometimes called wind waves. In the sequel of this paper, the term sea wave will be used. Sea waves shape and floating body shape, mass and mass inertia influence the dynamic response of that body. A common approach is to evaluate wave characteristics through statistical means, using predefined wave spectra. An exact shape of the wave train cannot be retrieved from these analyses, which is not needed if a special kind of probabilistic analysis developed for naval engineering, so-called seakeeping analysis [5], is performed. In a structural analysis of the retractable keel, based on a deterministic approach of classical mechanics, a shape of the wave train, i.e., time and space dependence of the wave height needs to be known at the beginning of the analysis. Since we do not posses records of wave measurements, a pure 204

Fig. 2. Sinusoidal wave characteristics

By estimating the natural frequency of a yacht’s rolling and pitching (angular oscillation about longitudinal and lateral axis, respectively) and in concordance with [7], a combination of waves which leads to a strong motion of the yacht was found. Three sinusoidal waves (n = 3) are assembled with their individual characteristics shown in Table 1. A result of the superposition is the time dependent undulate surface of the sea. In Figs. 7 and 8 this surface is shown at several arbitrary chosen time instances. To check some statistical characteristics of artificial formed wave surface, two plots are Koc, P.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, 203-209

Fig. 3. Height of the artificial wave along a line y = 0 m and at t = 0 s

Fig. 4. Time variation of height of the artificial wave at x = 100 m and y = 0 m

given, where spatial, i.e., along a line (Fig. 3) and time (Fig. 4) dependence of artificial wave height h is displayed. From Fig. 3, a kind of periodicity can be observed. But shapes of individual waves are not repeating within 1,000 m of wave path. An average wavelength λ of the prominent waves is about 54 m. In Fig. 4, a much more regular wave pattern is seen. The period of repeating the pattern is about 60 s. A so-called zero crossing period Tz is shown in Fig. 4. An average Tz = 5.8 s is measured from the graph. The wave spectra derived from Fig. 4 would not be a smooth curve as in case of natural sea waves [5], but it would consists of three distinctive peaks of equal heights, the first at about 0.4, the second at 3 and the third at 4.8 m. Therefore, the significant wave height H1/3, defined as the average height of the waves which comprise top 33% of all waves, equals to H1/3 = 4.8 m.

than average wavelength λ will encounter crests of artificial waves at an increased rate, producing more violent movement of the vessel, which is conservative from the structural analysis point of view. 3 NUMERICAL MODEL A finite element-based computer code ABAQUS [9] is used for analysis of the yacht’s dynamic response. A mechanical scheme of the yacht’s model is given in Fig. 5.

Table 1. Sinusoidal wave data i 1 2 3

λ [m] 54 24 8

hA [m] 1.5 1.0 0.2

α [°] 0 -20 10

φ [°] 0 175 130

Characteristic wave data obtained above are compared with data at WMO scale 6, given in [8]: H1/3 = 4.6 ÷ 5.9 m, λ = 46.5 ÷ 55 m, and Tz = 8.3 ÷ 9.0 s. We can conclude that artificial wave data H1/3 and λ match data from [8], while Tz, obtained from Fig. 4, is shorter. This means that a vessel with a length smaller

Fig. 5. Mechanical scheme of wave-hull-keel system

Since the main goal of the analysis is to determine the hydraulic cylinder lifting capability, the hull, the fixed and moveable parts of the telescopic keel and the ballast are considered as rigid bodies. The total mass mh of the hull with all appendages except for the moveable parts of the keel is 20,700 kg and is applied in the centre of gravity (CG) of the hull. Masses of the

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moveable part of the keel mk and the ballast mb are 1,100 kg and 4,400 kg, respectively, and are applied in their respective centres of gravity as well. To adequately capture rotational oscillations of the yacht about three spatial axes (roll, pitch and yaw) mass moments of inertia are calculated for the hull, both parts of the keel and the ballast (Ih, Ik and Ib in Fig. 5).

•

Water pressure in the wave depends on the orbital velocity of the water particle and is smaller in the wave crest than in trough [10]. A linear variation of the pressure with the depth (vertical distance from the wave surface) is adopted in this research. This means that a higher buoyancy force is modelled on the wave crest, thus promoting pitching and heaving (i.e., translation oscillations in a vertical direction) of the yacht in case of head seas, which is conservative from a structural analysis aspect. • Slamming of the hull against the water surface is considered one of the worst dynamic loadings (except touching another hard object) [2] and therefore need to be accounted for. The slamming pressure is calculated based on a hull-to-water surface velocity difference. The water drag of the hull and the keel is neglected due to the small speed during keel lifting, as defined above. When the vessel is on the waves, damping plays a crucial role attaining a vessel’s stability and seaworthiness [10]. In our work, damping is introduced in the model through a so-called artificial damping. Defining a correct amount of damping in such cases is not a trivial task; therefore, we performed a sensitivity analysis of the yacht oscillating on still water. Three of the most important oscillating modes were examined: rolling, pitching and heaving. For each mode, critical damping was found. Damping used in a model is taken as a fraction of the critical damping and takes into account an estimated frequency of natural oscillation.

Fig. 6. FE mesh of the sailing yacht

Because the shape of the hull influences intensity and direction of static and dynamic water pressure, it is reproduced in the FE model as exactly as possible (Fig. 6). In the model, the hull is massless and is rigidly connected with mass mh. Some general data of the yacht are: hull length 18 m and draft 2.5 m (lower position of the lifting keel). The moveable part of the keel and the ballast are rigidly connected. This combined rigid body is kinematically restrained to translate along the sliding direction only (vertical direction in Fig. 5). The hydraulic cylinder inside the telescopic keel is modelled with deformable beam elements to allow extension/shortening of the cylinder and is designated as spring k in Fig. 5. Also, the stainless steel guides and reinforced polymer padding which constitute contact surfaces are modelled as deformable bodies due to their importance in the dynamic response of the lifting keel. Loads acting on the yacht are gravity and wave pressure. Since it is not allowed to be lifting the keel when the yacht is heeled due to the sails, no wind forces on the sails need to be considered in analysis. Water pressure p(x,y,z,t) on the hull is not stationary when the yacht is on the waves. To model nontypical loadings, ABAQUS offers subroutine DLOAD [9], which is used in our case to model the wave hydrodynamic phenomena: • The wave surface is modelled as described in Section 2, Eq. (1). 206

4 RESULTS OF NUMERICAL ANALYSIS 4.1 Computer Simulation of Yacht’s Behaviour on the Waves Two directions of wave incidence are considered in the work: the head seas and the beam seas. For both cases, a dynamic analysis of a yacht’s response on wave excitation was performed. In Figs. 7 and 8, an illustration of a yacht’s response is shown at three arbitrary chosen instances for the head seas case (Fig. 7) and for the beam seas (Fig. 8) simulation. Since the aim of these particular simulations is to determine the most unfavourable time sequence of yacht`s accelerations, no lifting of the keel needs to be simulated. Because of that, the moveable part of the keel can be tied on the fixed part. This means that contact calculations between a keel’s parts are deactivated, which in turn considerably speed up the simulation. Koc, P.

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Fig. 7. Yacht on head seas; a) t = 4 s; b) t = 11 s; c) t = 38 s

Fig. 8. Yacht on beam seas; a) t = 3 s; b) t = 6 s; c) t = 12 s

In Figs. 9 and 10, the results of the head seas simulation, i.e., translational and angular accelerations of the centre of gravity of yacht are displayed. It can be seen that some kind of weak periodicity appears every 12 to 14 s. Exemptions are the very first 3 s of vertical accelerations (az) and angular acceleration about the y axis, which are altering violently as compared to the rest of the graph. These abrupt changes are attributed to the imperfections of the model. Namely, at the beginning of the simulation, the hull is placed on the crest without checking the static equilibrium of gravity and buoyancy forces. Depending on the resultant force, the hull then falls down on the wave or jumps out of the crest. From diagrams, the prevailing mode of the yacht movement is also seen: heaving and pitching. In case of beam seas another intensive movement is rolling. The same acceleration histories, but for the beam seas case, are given in Figs. 11 and 12. From the above diagrams, it can be seen that at the same sea state scale 6, beam seas cause higher

accelerations (translational and angular) of the yachtâ&#x20AC;&#x2122;s centre of gravity than head seas. Therefore, it is appropriate to simulate the lifting of the keel considering the beam seaâ&#x20AC;&#x2122;s response. 4.2 Simulation of Keel Lifting Because the lifting of the keel is simulated, all contacts and friction between the fixed and moveable parts of the keel need to be considered in the simulation. This increases the analysis (CPU) time substantially. Aiming to shorten the analysis time, we decided to simulate the lifting of the keel with a higher lifting speed as prescribed by the design. The greatest effort that the hydraulic cylinder needs to give is at the beginning of the lifting, since the moveable parts of the keel are then at the greatest distance from the centre of gravity of the yacht. At that position, angular accelerations (about three space axes) of the yacht produce the biggest translational accelerations and, coupled with the shortest distance

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between upper and lower sliding surfaces (Fig. 1), the biggest friction force between the contact surfaces evolve, leading to the biggest lifting force. Therefore, it would be conservative to start the lifting of the keel at the moment when high accelerations (translational and angular) occur. By inspecting the beam seas diagrams in Figs. 11 and 12, the most intensive acceleration time sequence is determined to be between the 9th and 28th second. In the simulation of keel lifting, beam elements representing the hydraulic cylinder are manipulated such that the keel lifting is simulated within the selected time interval. In Fig. 13, the most important result of lifting simulation, i.e., the axial force in the hydraulic cylinder, is shown. At the beginning of lifting at the 9th second, the pistonâ&#x20AC;&#x2122;s rod is compressed as prescribed by the design. Namely, when the keel is in the lower position, it is pressed by the cylinder against the stop block (Fig. 1). This pressure is adequate to close all of the gaps in the polymer-stainless steel contact surface, thus achieving firm contact between the fixed and moveable parts of the keel. Also, eventual lifting of the keel due to wave acceleration is prevented. After the 9th and until the 12th second, forces in the piston gradually change from compressive to tensile. This part of the curve represents a diminishing of pre-

compressing of the contact surfaces which finally leads to separation of the contact surfaces. From this point on, the whole weight and other inertial loads, acting in the sliding direction of the moveable keel, are transmitted through the cylinder as a tensile force. After the 12th second, the dynamic response due to wave excitation can be seen. The highest tension force in the cylinder is 160Â kN. Also, it can be seen that force peaks becoming smaller in the last third of the lifting time and that the force approaches the value of the weight of the moveable part of the keel. 4.3 Discussion Determination of the wave train represents the biggest uncertainty introduced in the analysis. Namely, for the same state of the sea, which is the statistical quantity, an infinite number of deterministic wave trains can be generated. The same problem is encountered by civil engineers when evaluating earthquake effects by deterministic method such as the time integration scheme. Since the earthquake excitation is given as the frequency domain acceleration spectrum, many acceleration histories, which fit into this spectrum, can be generated. Besides repeating analyses with different acceleration histories the usual way is to

Fig. 9. Translational acceleration of yacht`s CG at head seas

Fig. 11. Translational acceleration of yacht`s CG at beam seas

Fig. 10. Angular acceleration of yacht`s CG at head seas

Fig. 12. Angular acceleration of yacht`s CG at beam seas

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Fig. 13. Piston force and stroke during lifting of the keel

perform only one analysis and then multiply the obtained results by an appropriate scaling factor. While the civil engineering community has some guidelines of how big this factor could be, we do not know for similar guidelines in naval engineering. If simply copying this scaling factor from the civil engineering field then it would be between 1.2 and 1.5. Taking the most conservative value, this means that hydraulic cylinder should be capable of pulling with a force of 1.5 × 160 kN = 240 kN. 5 CONCLUSIONS In the paper, an analysis of hydraulic cylinder capability for lifting the yacht’s keel during rough sea conditions is described. The problem is solved without using highly sophisticated computational fluid dynamics (CFD) programs. Because the yacht`s velocity relative to the water is small, a substitute method as shown in the paper could be used. The drawback for this is that several assumptions needed to be adopted - usually in the way in which the obtained solution is rather conservative (see Section 3, water pressure loading). Thus, the results are acceptable and serve as a support in the subsequent detailed designing process. 6 ACKNOWLEDGEMENT This work is fully sponsored by the Seaway group. The author wishes to express thanks to the staff of the Seaway group for their technical support during the research.

7 REFERENCES [1] Nautic Expo. Lifting keel sailboats, from: http:// www.nauticexpo.com/boat-manufacturer/lifting-keelsailboat-21398.html, accessed on 2013-09-04. [2] Larsson, L., Eliasson, R.E. (2000). Principles of Yacht Design, 2nd ed. McGraw-Hill, London, DOI:10.1038/35018068. [3] Colombo, C., Vergani, L., Modica, N., A. (2007). Numerical and experimental study of a crash absorption keel. Key Engineering Materials, vol. 348-349, p. 953-956, DOI:10.4028/www.scientific.net/KEM.348349.953. [4] Stern, F., Yang, J., Wang, Z., Sadat-Hosseini, H., Mousaviraad, M., Bhushan, S. (2012). Computational ship hydrodynamics: nowadays and way forward. Proceedings on 29th Symposium on Naval Hydrodynamics, Gothenburg. [5] Lloyd, A.R.J.M. (1989) Seakeeping: ship behaviour in rough weather. Ellis Horwood Limited, Chichester, UK. [6] Ship, behaviour on the waves (1964). Technical Encyclopedia, Volume 2, Miroslav Krleža Lexicographical Institute, Zagreb, p. 228. (in Croatian) [7] Fridsma, G. (1969). A systematic study of the roughwater performance of planning boats, Report 1275. Davidson laboratory, Stevens Institute of Technology, New Jersey. [8] Zorović, D., Mohović, R., Mohović, Đ. (2003). Towards determining the length of the wind waves of the Adriatic sea. Journal of Marine Science and Technology, vol. 50, no. 3-4, p. 145-150. (in Croatian) [9] ABAQUS/Standard, Version 6.7 (2007) User’s manual, Simulia, Providence. [10] Marchay, C.A. (1996). Seaworthiness the Forgotten Factor. Adlard Colles Nautical, London.

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Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 60, (2014), številka 3 Ljubljana, marec 2014 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki člankov Ignacijo Biluš, Gorazd Bombek, Marko Hočevar, Branko Širok, Tine Cenčič, Martin Petkovšek: Eksperimentalna analiza fluktuacije kavitacijskih struktur in tlaka v kavitacijskem kanalu Siamak Pedrammehr, Mehran Mahboubkhah, Mohammad Reza Chalak Qazani, Arash Rahmani, Sajjad Pakzad: Analiza vsiljenih vibracij šesteronožne mize rezkalnega stroja pod vplivom rezalnih sil Georgije Bosiger, Tadej Perhavec, Janez Diaci: Metoda optodinamske karakterizacije ablacije z erbijevim laserjem na osnovi piezoelektrične detekcije Mohammad Abbasi, Ardeshir Karami Mohammadi: Podrobna analiza resonančne frekvence in občutljivosti upogibnih načinov ročice mikroskopa na atomsko silo s tipalom za stranske stene s pomočjo nelokalne teorije elastičnosti Domen Rovšček, Janko Slavič, Miha Boltežar: Metoda z uglašenim sinusom za obratovalno modalno analizo majhnih in lahkih struktur Ertugrul Selcuk Erdogan, Olcay Eksi: Napovedovanje porazdelitve debeline sten pri enostavnih orodjih za termoformiranje Pino Koc: Dinamična obremenitev valovanja morja na dvižno kobilico jadrnice Osebne vesti Doktorske disertacije, magistrske naloge, diplomske naloge

SI 31 SI 32 SI 33 SI 34 SI 35 SI 36 SI 37 SI 38

Prejeto v recenzijo: 2013-09-26 Prejeto popravljeno: 2013-10-13 Odobreno za objavo: 2013-12-03

Eksperimentalna analiza fluktuacije kavitacijskih struktur in tlaka v kavitacijskem kanalu

Biluš, I. – Bombek, G. – Hočevar, M. – Širok, B. – Cenčič, T. – Petkovšek, M. Ignacijo Biluš1,* – Gorazd Bombek1 – Marko Hočevar2 – Branko Širok2 – Tine Cenčič2 – Martin Petkovšek2 1 Univerza

2 Univerza

v Mariboru, Fakulteta za strojništvo, Slovenija v Ljubljani, Fakulteta za strojništvo, Slovenija

Prispevek obravnava kavitacijo, ki se širi v obliki von Karmanovega vrtinca v tokovni brazdi za valjasto oviro. Kavitacija se pojavi zaradi lokalnega padca tlaka pod uparjalni tlak kapljevine, povzroča fluktuacije toka ter lahko privede do vibracij in obrabe hidravličnega sistema. Zaradi navedenega je poznavanje dinamike kavitacijskega toka in z njim povezanih pojavov pomembno za inženirsko prakso. Večina eksperimentalnih člankov s tega področja obravnava posamezne fizikalne veličine, povezane z dinamiko kavitacijskih struktur, medtem ko je raziskava, predstavljena v tem prispevku, veliko kompleksnejša. Namen raziskave je izvesti sočasno analizo dinamike kavitacijskega pojava z metodo vizualizacije kavitacijskih struktur in meritvijo statičnega tlaka. V prispevku so podani rezultati eksperimentalne analize nestacionarnega kavitirajočega toka v laboratorijskem kavitacijskem kanalu dimenzij 50×10×800 mm. V okviru raziskave je bila opravljena analiza in primerjava fluktuacij kavitacijskega oblaka s tlačnimi pulzacijami toka v tokovni brazdi za valjasto oviro premera 16 mm. Za vizualizacijo fluktuacij kavitacijskega oblaka je bila uporabljena hitra kamera Fastec Hispec4, za meritev tlačnih pulzacij pa v steno kanala pogreznjeno tlačno zaznavalo PCB Piezotronics 111A26. Meritve obeh signalov so bile izvedene sočasno pri različnih obratovalnih režimih, frekvenčna analiza pa je bila narejena v nizkofrekvenčnem intervalu 0 do 1000 Hz in visokofrekvenčnem intervalu 300 do 400 kHz z amplitudno demodulacijo. Rezultati analize so pokazali podobnost med signaloma v časovni in frekvenčni domeni. V obeh spektrih so prisotne oscilacije s frekvenco 30 Hz, ki predvidoma izhajajo iz tlačnih pulzacij v kavitacijskem merilnem kanalu, in oscilacije s frekvenco 310 Hz, povezane s trganjem kavitacijskega oblaka od valjaste ovire. Amplituda oscilacij sledi intenziteti kavitacije oziroma vrednosti kavitacijskega koeficienta, ki ne vpliva na vrednost Strouhalovega števila. Izvedena amplitudna demodulacija je pokazala povezavo med nizkofrekvenčnimi kavitacijskimi nestabilnostmi in visoko frekvenco fluktuacij znotraj kavitacijskega oblaka, ki se premika skozi tokovno brazdo v obliki von Karmanovega vrtinca. Za nadaljevanje raziskave je predvideno povišanje frekvence zajemanja vizualizacijskega sistema in poravnana montaža tlačnega zaznavala na steno kavitacijskega kanala. Ključne besede: kavitacija, eksperiment, tlačne pulzacije, vizualizacija, frekvenčna analiza

*Naslov avtorja za dopisovanje: Univerza v Mariboru, Fakulteta za strojništvo, Smetanova ulica 17, 2000 Maribor, Slovenija, ignacijo.bilus@um.si

SI 31

Prejeto v recenzijo: 2013-04-27 Prejeto popravljeno 2013-08-17 Odobreno za objavo: 2013-10-30

Analiza vsiljenih vibracij šesteronožne mize rezkalnega stroja pod vplivom rezalnih sil Pedrammehr, S. – Mahboubkhah, M. – Chalak Qazani, M.R. – Rahmani, A. – Pakzad, S. Siamak Pedrammehr1,2,* – Mehran Mahboubkhah1 – Mohammad Reza Chalak Qazani3 – Arash Rahmani4 – Sajjad Pakzad4 Univerza v Tabrizu, Fakulteta za strojništvo, Iran univerza, Naravoslovnotehniška fakulteta, Turčija 3 Univerza Tarbiat Modares, Oddelek za strojništvo, Iran 4 Islamska univerza Azad, Fakulteta za strojništvo, Iran 1

2 Sabancijeva

Šesteronožni mehanizmi s paralelno kinematiko od prihoda v 60. letih prejšnjega stoletja postajajo vse bolj razširjeni. Med drugim se uporabljajo v avtomobilskih in letalskih simulatorjih, za izolacijo vibracij in v napravah za nakladanje/razkladanje. Nove možnosti uporabe se pojavljajo tudi pri obdelovalnih strojih v sestavih miz in platform za vretena. Pomembna značilnost paralelnih šesteronožnih mehanizmov je njihova gibljivost, zaradi katere so zelo primerni za visokohitrostne obdelovalne stroje. Dodatna prednost šesteronožnih mehanizmov na obdelovalnih strojih je tudi šest prostostnih stopenj. Te možnosti šesteronožnih mehanizmov je mogoče uporabiti tako pri razvoju novih visokohitrostnih obdelovalnih strojev kakor tudi pri razmeroma poceni nadgradnji obstoječih ročnih strojev v napredne šestosne obdelovalne stroje. Kinematika šesteronožnih mehanizmov je dobro raziskana, medtem ko je treba njihovo dinamiko in zlasti vibracijske lastnosti še podrobneje preučiti. To še posebej velja za uporabo šesteronožnih mehanizmov v okoljih, kjer je kritičnega pomena natančnost in kakovost površine. V predstavljeni študiji je bil razvit vibracijski model šesteronožne mize in izpeljane so bile ustrezne eksplicitne enačbe. Model upošteva maso, vztrajnost, togost in dušenje različnih elementov šesteronožne mize. Mize so v enačbi vibracij sistema popisane kot sistem vzmeti in dušilk, ekvivalentna togost in dušenje pa sta bila ovrednotena eksperimentalno z modalnimi preskusi šesteronožne mize. Natančnost eksperimentalnih preskusov je bila nato validirana z modalno analizo po metodi končnih elementov (MKE). Vsiljene vibracije šesteronožne mize obdelovalnega stroja v različnih smereh so bile analizirane ob predpostavki sinusne oblike rezalne sile. Rezultati analitičnega pristopa so bili preverjeni s simulacijo po MKE. Teoretični rezultati in rezultati po MKE imajo podoben trend in se dobro ujemajo. Vibracije platforme kot končnega izvršnega člena v različnih položajih so bile analizirane za grobo in fino isto- in protismerno rezkanje. Za podrobnejšo preučitev vsiljenih vibracij šesteronožne mize pri obdelovalnem procesu so bile modelirane rezalne sile, ob upoštevanju različnih sprememb sil in operacij rezkanja. Reševanje vsiljenih vibracij platforme in preučitev vsiljenega odziva pri harmoničnih rezalnih silah je ključnega pomena za odpravo resonance. Določitev resonančnih frekvenc ter razpona vibracij platforme zaradi delovanja rezalnih sil je najboljši način za opredelitev pogojev, ki privedejo do resonance. Z natančnim izborom parametrov obdelave se je tako mogoče izogniti pojavu močnih vibracij pri procesu rezkanja. Izračunan je tudi razpon resonančnih frekvenc in amplitud vibracij. Ob analizi premikov gibljive platforme zaradi različnih rezalnih sil lahko zaključimo, da je razpon premikov platforme sorazmeren s silami, ki delujejo nanjo. Na amplitudo vibracij gibljive platforme torej vplivajo vsi tisti parametri, ki efektivno spreminjajo rezalne sile, vključno s parametri odrezavanja in geometrijo. S povečanjem podajanja, globine reza in števila zob rezkarja se torej okrepijo tudi vibracije platforme med obdelavo. Omeniti je treba tudi to, da se s povečanjem hitrosti vretena zmanjša amplituda vibracij platforme. Ob znanih resonančnih frekvencah in vibracijah platforme zaradi rezalnih sil je končno predstavljena takšna konfiguracija gibljive platforme, ki odpravlja dinamično nestabilnost pri različnih pogojih obdelave. Ključne besede: šesteronožni mehanizem, obdelovalni stroj s paralelno kinematiko, modalna analiza, rezalne sile

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*Naslov avtorja za dopisovanje: Univerza v Tabrizu, Fakulteta za strojništvo, Orhanli, Tuzla, 34956 Istanbul, Turkey, s.pedrammehr@gmail.com

Prejeto v recenzijo: 2013-02-25 Prejeto popravljeno: 2013-09-10 Odobreno za objavo: 2013-09-25

Metoda optodinamske karakterizacije ablacije z erbijevim laserjem na osnovi piezoelektrične detekcije Bosiger, G. – Perhavec, T. – Diaci, J. Georgije Bosiger1,* – Tadej Perhavec1 – Janez Diaci2 2 Univerza

1 Fotona, Slovenia v Ljubljani, Fakulteta za strojništvo, Slovenia

Laser Er:YAG z valovno dolžino 2,94 µm je uveljavljeno orodje v medicini in kirurgiji. Njegova infrardeča svetloba se močno absorbira v vodi in hidroksiapatitu, kar omogoča učinkovito lasersko ablacijo mehkih in trdih bioloških tkiv. Prednosti laserske interakcije (brezdotičnost, manjši toplotni vnos in odsotnost mehanskih vibracij) bi koristile številnim novim medicinskim posegom, a je pri tem potrebno rešiti enega izmed večjih tehničnih in znanstvenih izzivov, t.j. razvoj zanesljivega sistema za sprotni nadzor ključnih parametrov, kot sta npr. globina reza, vrsta odstranjenega tkiva ipd. Sprotno spremljanje procesa laserske ablacije bi bilo v praksi izvedljivo z uporabo piezoelektričnih akustičnih senzorjev tako, da bi zaznavali udarne valove, ki se iz mesta ablacije širijo po zraku nad biološkim tkivom in so značilni za tovrstno interakcijo. Obstoječe metode karakterizacije laserske ablacije z akustično detekcijo običajno temeljijo na empirično določenih karakteristikah akustičnih signalov. To pa jih omejuje na skrbno nadzorovane pogoje, ki jih je v praksi težko zagotoviti. V tem prispevku predstavljamo drugačen pristop, ki odpira pot k realizaciji sprotnega spremljanja laserskih medicinskih posegov. Razvijemo nov teoretični model piezoelektričnega akustičnega senzorja, ki upošteva njegov relativni položaj ter usmerjenost, kakor tudi mehanske in električne lastnosti. Širjenje udarnega vala v polprostoru opišemo s Sedov-Taylorjevim modelom točkaste eksplozije. Za potrebe validacije modela in metode realiziramo eksperiment, ki omogoča primerjavo izmerjenih signalov, detektiranih na različnih oddaljenostih in usmerjenostih senzorja glede na točko ablacije, s teoretičnimi, ki jih dobimo iz numeričnih rešitev modela točkaste eksplozije in razvitega modela senzorja. Teoretični model nato uporabimo za določitev matematične funkcije, ki podaja sproščeno energijo udarnega vala pri poljubni postavitvi piezoelektričnega senzorja na osnovi karakteristike zajetega optodinamskega signala – časa trajanja kompresijske faze. Metodo optodinamske karakterizacije eksperimentalno preverimo tako, da ocenimo sproščeno energijo na osnovi izmerjenih kompresijskih faz pri različnih relativnih položajih in orientacijah senzorja. Ocenjene sproščene energije udarnih valov kažejo bistveno manjšo sistematično odvisnost od pogojev merjenja v primerjavi z ostalimi karakteristikami, ki se običajno uporabljajo pri optodinamski karakterizaciji laserske ablacije (čas preleta, amplituda signala, ipd.). To opažanje ter dobro ujemanje teoretičnih časovnih potekov z izmerjenimi potrjujeta primernost razvitega teoretičnega modela senzorja, kot tudi predlagane metode optodinamske karakterizacije. Pridobljene ocene uporabimo za določitev izkoristka energijske pretvorbe, ki podaja delež pretvorbe energije laserskega bliska v energijo udarnega vala. Pokažemo, da lahko predstavljena metoda služi tudi za ocenjevanje maksimalnih tlakov udarnih valov ter za kalibracijo senzorja. Predstavljena spoznanja prispevka pripomorejo k uresničitvi sistemov za sprotni nadzor laserskih medicinskih in kirurških posegov. S teoretičnega vidika so optodinamski signali in s tem povezane karakteristike piezoelektričnih akustičnih senzorjev prvič obravnavane na tem področju optodinamske detekcije. Razviti teoretični model senzorja ter predstavljena metoda optodinamske karakterizacije omogočata ocenjevanje energije udarnih valov, ki je bila do sedaj mogoča le z implementacijo drugih tehnik optodinamske detekcije, npr. z uporabo laserske odklonske sonde in pri detekciji s prostorsko ločljivostjo, ki pa niso primerne za uporabo v realnih medicinskih laserskih sistemih. Ključne besede: erbijev laser, laserska ablacija, udarni val, piezoelektrična detekcija, točkasta eksplozija

*Naslov avtorja za dopisovanje: Fotona d.d., Stegne 7, SI-1210 Ljubljana, Slovenija, georgije.bosiger@fotona.com

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Prejeto v recenzijo: 2013-08-06 Prejeto popravljeno: 2013-11-26 Odobreno za objavo: 2013-12-20

Podrobna analiza resonančne frekvence in občutljivosti upogibnih načinov ročice mikroskopa na atomsko silo s tipalom za stranske stene s pomočjo nelokalne teorije elastičnosti Abbasi, M. – Mohammadi, A.K. Mohammad Abbasi* – Ardeshir Karami Mohammadi Tehniška univerza Shahrood, Fakulteta za strojništvo, Iran

Mikroskopi na atomsko silo (AFM) imajo sestavljena tipala na ročico (ACP), katerih debelina je velikostnega reda nekaj mikronov ali manj kot mikron, zato je za preučitev njihovega od velikosti odvisnega dinamičnega vedenja nujna uporaba neklasičnih teorij nosilcev, kot je npr. nelokalna teorija elastičnosti. V članku sta preučeni resonančna frekvenca in občutljivost mikroskopa na atomsko silo s sestavljenim tipalom na ročico s pomočjo nelokalne teorije elastičnosti. ACP je sestavljen iz vodoravne ročice z navpičnim podaljškom. Konica je nameščena na prostem koncu podaljška, zato lahko AFM meri topografijo stranskih sten mikrokonstrukcij. Najprej so bile s kombinacijo osnovnih enačb nelokalne teorije elastičnosti in Hamiltonovega načela določene vodilne diferencialne enačbe gibanja in robni pogoji za upogibne vibracije. Nato je bil s pomočjo zveze med resonančno frekvenco ter kontaktno togostjo med konico in vzorcem izpeljan izraz zaprte oblike za občutljivost vibracijskih načinov. Rezultat analize je boljši opis vibracijskega vedenja ročice AFM s tipalom za stranske stene, kjer so pomembni vplivi na majhnih skalah. Rezultati nelokalne teorije so primerjani s tistimi iz klasične teorije nosilcev. Izkazalo se je, da so občutljivosti in resonančne frekvence, ki jih napoveduje nelokalna teorija elastičnosti, manjše od tistih, ki jih daje klasična teorija nosilcev. Ugotovljeno je bilo tudi, da je razlika med resonančno frekvenco in upogibno občutljivostjo po nelokalni teoriji elastičnosti in po klasični teoriji nosilcev bolj signifikantna za drugi način. Resonančna frekvenca se zmanjša s povečanjem dolžine navpičnega podaljška, občutljivost pa se zmanjša pri manjših in poveča pri višjih vrednostih kontaktne togosti. Rezultati tudi kažejo, da se s povečanjem normaliziranega nelokalnega parametra zmanjša vpliv dolžine navpičnega podaljška na občutljivost. Ključne besede: mikroskop na atomsko silo, sestavljeno tipalo na ročico, nelokalna teorija elastičnosti, odvisnost od velikosti

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*Naslov avtorja za dopisovanje: Tehniška univerza Shahrood, Fakulteta za strojništvo, Shahrood, Iran, m.abbasi28@yahoo.com

Prejeto v recenzijo 2013-08-14 Prejeto popravljeno: 2013-10-28 Odobreno za objavo: 2013-11-13

Metoda z uglašenim sinusom za obratovalno modalno analizo majhnih in lahkih struktur Rovšček, D. – Slavič, J. – Boltežar, M. Domen Rovšček – Janko Slavič – Miha Boltežar*

Univerza v Ljubljani, Fakulteta za strojništvo, Slovenia

Kljub temu, da obstaja že mnogo eksperimentalnih metod za določanje modalnih parametrov (lastnih frekvenc, lastnih oblik in dušenja), se pri izvajanju teh metod pogosto pojavijo težave, ko se skuša analizirati razmeroma majhne in lahke strukture. Razlog se skriva v tem, da se pri takih strukturah z običajnimi merilnimi instrumenti (silomer, pospeškomer) na strukturo doda masa in s tem se spremenijo njene modalne karakteristike. Hkrati pa imajo take strukture pogosto zelo visoke lastne frekvence, zaradi česar je potrebna meritev v širokem frekvenčnem področju in je treba temu prilagoditi tudi izbiro senzorjev. V poimenovanju »majhne in lahke strukture« so torej zajete vse strukture, pri katerih se zaradi njihove majhne velikosti ali mase pojavijo težave pri določanju modalnih parametrov zaradi dodane mase senzorjev in visokih lastnih frekvenc. Za odpravo omenjenih težav je bila v okviru te raziskave razvita inovativna brezkontaktna metoda za obratovalno modalno analizo (OMA) majhnih in lahkih struktur iz ene same meritve odziva z laserskim merilnikom hitrosti. Metoda deluje na osnovi akustičnega sinusnega vzbujanja, ki je uglašeno na posamezne lastne frekvence strukture. Masno normiranje je bilo izvedeno z metodo masne spremembe. Uporabljeni postopek se imenuje metoda z uglašenim sinusom. Ta metoda daje dobre rezultate na uporabljenem vzorcu, saj so bile lastne frekvence zelo natančno izmerjene in se ujemajo s tistimi, ki so bile izračunane z numeričnim modelom. Poleg tega so bile tudi lastne oblike natančno določene in pravilno normirane, kar je prav tako potrdila primerjava meritev z numeričnim modelom. Metoda z uglašenim sinusom torej daje zelo dobre rezultate pri preprostih majhnih in lahkih strukturah. Slabost te metode je, da se lahko pojavijo težave pri močneje dušenih strukturah, kjer lastne frekvence niso tako izrazite in se jih zato težje določi samo iz meritev odziva na vzbujanje z belim šumom. Težave se lahko pojavijo tudi pri strukturah, ki imajo nekatere lastne frekvence zelo blizu skupaj oz. medsebojno povezane. V tem primeru izmerjene ODS (obratovalne odklonske oblike, angl. Operating Deflection Shapes) ne predstavljajo le ene lastne oblike, ampak so kombinacija več lastnih oblik. Zato nekaterih lastnih oblik v tem primeru ni mogoče določiti. Metoda z uglašenim sinusom ima tudi številne prednosti, predvsem pri majhnih in lahkih strukturah. Metoda je namreč brezkontaktna (stik je potreben le, če se normira lastne oblike z dodano maso), kar je zelo ugodno, saj masa senzorjev ne vpliva na meritev modalnih parametrov. Hkrati je tudi zelo enostavna za izvedbo, saj je potrebno le postaviti zvočnik v bližino merjene strukture in nameriti laser v merilne točke. Običajno se pri eksperimentalni modalni analizi (EMA) pojavi veliko težav, povezanih z dovajanjem vzbujevalne sile do strukture, pri tej metodi pa se jim izognemo. Ker je struktura vzbujana akustično, so bolje vzbujane tudi vse lastne oblike. Pri točkovnih vzbujevalnih silah je namreč vedno možnost, da vzbujanje poteka v vozlu lastne oblike in zato posamezne lastne oblike niso dovolj dobro izražene v rezultatih meritev. Pri majhnih in lahkih strukturah se OMA izkaže za zelo perspektiven postopek, ki bi z nadaljnjim razvojem lahko poenostavil določanje modalnih parametrov. V prihodnosti bi torej bilo smiselno nadaljevati raziskave v smeri inovativnih metod s področja OMA, ki omogočajo določanje modalnih parametrov brez stika s strukturo, pri tem pa bi bilo potrebno poskrbeti tudi za pravilno masno normiranje lastnih oblik (za verifikacijo numeričnih modelov). Z metodo, ki je predstavljena v tem prispevku, je bila OMA prilagojena za merjenje modalnih parametrov majhnih in lahkih struktur. S tem je bil odpravljen glavni vir težav pri meritvah, saj ni več treba spremljati vzbujevalne sile kakor pri klasični EMA. Razvita metoda prispeva tudi k poenostavitvi modalne analize majhnih in lahkih struktur, saj so bile dosedanje metode precej težje izvedljive in zamudnejše. Ključne besede: obratovalna modalna analiza, metoda z uglašenim sinusom, normiranje lastnih oblik, majhne in lahke strukture, enojen odziv, akustično vzbujanje

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, miha.boltezar@fs.uni-lj.si

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Prejeto v recenzijo: 2013-10-04 Prejeto popravljeno: 2014-01-10 Odobreno za objavo: 2014-01-24

Napovedovanje porazdelitve debeline sten pri enostavnih orodjih za termoformiranje Erdogan, E.S. – Olcay Eksi, O. Ertugrul Selcuk Erdogan1 – Olcay Eksi2,*

1 Univerza

2 Univerza

v Trakiji, Tehniška fakulteta, Oddelek za strojništvo, Turčija Namıka Kemala, Tehniška fakulteta v Çorluju, Oddelek za strojništvo, Turčija

Analiza po metodi geometrijskih elementov (GEA) je pristop, ki se še ni uveljavil pri napovedovanju debeline sten delov, ki nastanejo s termoformiranjem. Metoda daje porazdelitev debeline sten le za enostavne komponente, simulacije postopka termoformiranja pa zahtevajo podrobnejši popis spremenljivk procesa in nekatere parametre materiala za pripravo primernega modela po metodi končnih elementov. Simulacija kljub zahtevam daje bolj realistične rezultate kot analiza po metodi geometrijskih elementov. Cilj te študije je primerjava rezultatov analize po metodi geometrijskih elementov, analize po metodi končnih elementov in eksperimentalne analize. Polipropilen je bil ekstrudiran z laboratorijskim ekstruderjem s polžem premera 55 mm, ki omogoča izdelavo termoplastičnih folij debeline 1 do 3 mm in širine 30 mm. Ekstrudirana polipropilenska folija je bila izdelana pri konstantni hitrosti približno 1 m/min. Polipropilenska folija je bila ekstrudirana na debelino 2 mm (0,01 mm) s 5 in 15 odstotnim deležem ogljikovih vlaken za določitev vpliva ojačitve na končno porazdelitev debeline sten. Preiskan je bil vpliv ojačitvenih vlaken na parametre procesa termoformiranja. Za termoformiranje folij je bil uporabljen laboratorijski stroj z ročnim upravljanjem turškega podjetja Yeniyurt Machinery Co. Ltd. iz Istanbula. Enota ni namenjena masovni proizvodnji in primerna je le za laboratorijske poskuse. Enota za preoblikovanje uporablja samo toploto in podtlak, dela pa lahko s folijami debeline 1 do 3 mm. V eksperimentalni študiji je bila uporabljena tehnika negativnega oz. vakuumskega preoblikovanja. Pri vakuumskem termoformiranju se uporabljajo ženska orodja, ki se postavijo pod folijo, ta pa se nato oblikuje po konturi orodja. V študiji so bila za izdelavo komponent uporabljena tri različna ženska orodja (valjasto, konično in kockasto). V članku je podana korelacija med tremi različnimi metodami napovedovanja debeline sten. Navaja tudi natančnost metod napovedovanja debeline sten pri termoformiranju polimernih folij. Za termoformiranje v laboratorijski enoti so bile uporabljene folije različnih debelin iz amorfnega polistirena (PS), enostavnega za preoblikovanje, in semikristaliničnega polipropilena (PP). Profil debeline komponent je bil določen eksperimentalno. Ugotovljene porazdelitve debeline sten so bile primerjane z rezultati analize po metodi geometrijskih elementov (GEA) za tri geometrije orodja. Opravljena je bila tudi simulacija termoformiranja s programsko opremo LS-Dyna Explicit. Glavni povzročitelji neenakomerne porazdelitve debeline so oblika vpenjalnega obroča, homogenost porazdelitve ojačitvenih vlaken v matriksu, anizotropne lastnosti zaradi smeri ekstrudiranja ter grobe in porozne površinske napake. Za enakomernejšo porazdelitev debeline izdelkov je nujna izbira primerne geometrije vpenjalnega orodja, ki ustreza geometriji izdelka. Geometrija vpenjalnega orodja bi bila lahko okrogla za valjaste in konične izdelke ter pravokotna za pravokotne izdelke. Takšna oblika lahko uravnoteži napetosti v vseh smereh ter zagotovi enakomerne deformacije za enakomernejšo porazdelitev debeline. Termoformiranje se je uveljavilo v proizvodni industriji pri izdelkih, ki so veliki in zahtevajo veliko ročnega dela. V primerjavi z drugimi proizvodnimi postopki je zelo učinkovito ter je primerno za visokoučinkovito masovno proizvodnjo. Članek predstavlja inovativen pristop k napovedovanju debeline izdelkov za raziskovalce. Raziskovalci in inženirji se lahko seznanijo tudi z rezultati različnih tehnik napovedovanja debeline sten. Rezultati in ugotovitve tega dela torej podajajo novo znanje na področju postopkov termoformiranja. Ključne besede: termoformiranje, napovedovanje debeline sten, analiza po metodi geometrijskih elementov (GEA), polistiren, polipropilen, simulacija termoformiranja

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*Naslov avtorja za dopisovanje: Univerza Namıka Kemala, Tehniška fakulteta v Çorluju, Oddelek za strojništvo, 59860-Tekirdağ, Turčija, oeksi@nku.edu.tr

Prejeto v recenzijo: 2013-09-04 Prejeto popravljeno: 2013-10-03 Odobreno za objavo: 2013-10-07

Dinamična obremenitev valovanja morja na dvižno kobilico jadrnice Koc, P. Pino Koc* Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Slovenija

Kobilica na sodobnih jadrnicah je oblikovana v obliki navpično stoječega krila simetričnega prereza, ki sega globoko pod trup jadrnice. Običajno je na spodnjem koncu krila dodana še utež, prav tako hidrodinamično oblikovana, s katero zmanjšujejo nagib jadrnice, ki je posledica delovanja sile vetra na jadra. Globoko segajoča kobilica in utež je seveda primerna za jadranje, obenem pa onemogoča vplutje v plitkejše zalive in pristanišča. Z namenom ohranjanja obeh zaželjenih lastnosti, t.j., kvalitetnih jadralskih sposobnosti in dostopnosti v plitkejše vode, so razvili različne vrste dvižnih kobilic. Poseben tip dvižnih kobilic je t.i. teleskopska kobilica, ki je pritrjena na trup s spodnje strani in, za razliko od običajnih dvižnih kobilic, katerih dvižni jašek zaseda znaten del bivalnega dela v trupu jadrnice, ne sega v notranjost trupa. Oblikovanje in dimenzioniranje kobilice je zahtevno projektantsko delo, tako s področja optimizacije hidrodinamičnih razmer, kot tudi z vidika ohranjanja mehanske integritete in zagotavljanja funkcionalnosti kobilice. Ker pa je po svojem načinu delovanja teleskopska kobilica v bistvu mehanizem z masivnimi gibajočimi deli, ki mora pravilno delovati tudi pri surovih pogojih dela na morju, je projektiranje še toliko bolj zahtevno. V prispevku je opisano delo na majhnem segmentu projektiranja in sicer na ugotavljanju razmer v kobilici in dvigovalnem mehanizmu, ki nastopijo v primeru dviganja kobilice na razburkanem morju. Pri tem je bilo izbrano stanje morja 6 po 9 stopenjski WMO (World Meteorological Organization) lestvici. Deformacijsko-napetostno stanje v kobilici in dvigovalnem mehanizmu je bilo določeno z večimi numeričnimi analizami, ki so si sledile v pravilnem zaporedju, tako, da so rezultati prehodne analize služili kot vhodni podatek v naslednjo. V prvem koraku je bilo potrebno določiti zaporedje valov (wave train), ki delujejo na jadrnico. Ker nam meritve realnega valovanja niso bile na voljo, smo izdelali umetno valovanje, ki statistično ustreza stanju morja 6. Po WMO je stanje morja določeno na osnovi statistične obdelave meritev realnega valovanja in sicer je stanje morja 6 okarakterizirano z značilno višino vala (Significant Wave Height) med 4 in 6 m in značilno valovno dolžino okrog 50 m. Umetno valovanje je določeno kot vsota več sinusnih valov različnih amplitud, valovnih dolžin in smeri razprostiranja. Takšno valovanje je sicer periodično, vendar se z ustreznim sestavljanjem sinusnih valov da doseči kvazi-stohastično valovanje vsaj v krajšem časovnem intervalu, kolikor traja dviganje kobilice. V naslednjem koraku smo izvedli deterministično analizo gibanja modela trupa jadrnice na prej opisanem valovanju. Pri tem smo upoštevali bistvene dejavnike valovanja, ki vplivajo na gibanje trupa jadrnice. Rezultat analize je časovno odvisno spreminjanje pospeška in kotnega pospeška težišča jadrnice. Iz teh rezultatov smo razbrali krajši časovni interval s tako kombinacijo pospeškov, iz katere pričakujemo najneugodnejšo obremenitev za kobilico. V zadnjem koraku pa sledi podrobna analiza sestava teleskopske kobilice v kateri je simulirano tudi dviganje premičnega dela kobilice. Iz te analize smo določili silo, potrebno za dvigovanje kobilice v razburkanem morju. Opisani primer izračuna dvižne sile je bistveno drugačen od običajnih statičnih in trdnostnih analiz, s katerimi se srečuje projektant pri svojem delu. Primer prikazuje kompleksnost tovrstnih dinamičnih analiz, še posebno težave in dileme pri prehodu iz statistično določenih podatkov (stanje morja) v podatke, primerne za klasično deterministično analizo (zaporedje valov). Ključne besede: jadrnica, teleskopska dvižna kobilica, vetrni valovi, računalniška simulacija

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Lepi pot 11, 1000 Ljubljana, Slovenija, pino.koc@fmf.uni-lj.si

SI 37

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, SI 38-40 Osebne objave

Doktorske disertacije, magistrske naloge, diplomske naloge

DOKTORSKE DISERTACIJE

Na Fakulteti za strojništvo Univerze v Ljubljani je obranil svojo doktorsko disertacijo: ● dne 12. februarja 2014 Matej MIKLAVEC

Na Fakulteti za strojništvo Univerze v Mariboru je obranil svojo doktorsko disertacijo: ● dne 21. februarja 2014 Matija KOS z naslovom:

z naslovom: »Zdržljivost upogibnih nosilcev iz hibridnih kovinsko-nekovinskih materialov« (mentor: izr. prof. dr. Jernej Klemenc); V doktorskem delu je predstavljena metodologija za razvojno vrednotenje nosilnih elementov, ki so izdelani iz hibridnih kovinsko-nekovinskih materialov. S primerno oblikovanim kovinskim vložkom se doseže izboljšan spoj med kovino in polimerom z uporabo obstoječih tehnologij, in posledično višjo togost, nosilnost in varnost nosilnih elementov. V doktorski disertaciji so predstavljeni tako način izdelave oblikovnih hibridnih spojev kot tudi metodologije za numerično in eksperimentalno vrednotenje teh spojev. Metodologija je bila validirana na konkretnih primerih oblikovnih hibridnih spojev med vložkom iz jeklene pločevine S420MC in polimerom PA6GF60. Za te hibridne spoje je bila numerično in eksperimentalno določena statična natezna nosilnost ter eksperimentalno ocenjena dinamična nosilnost hibridnega spoja. Izbrana je bila tudi najprimernejša oblika hibridnega spoja za uporabo v upogibnih nosilcih; ● dne 13. februarja 2014 Goran VIŠNJIĆ z naslovom: »Geometrijsko strukturna optimizacija upogibno in prečno obremenjenega tankostenskega kompozitnega I-nosilca« (mentor: izr. prof. dr. Tadej Kosel); Strižno zaostajanje predstavlja omejitev pri optimizaciji tankostenskih kompozitnih I-nosilcev z namenom povečevanja specifične upogibne togosti in trdnosti. V raziskavi se je s pomočjo metode končnih elementov z namenom povečanja specifične upogibne trdnosti in togosti kompozitnih nosilcev raziskovalo vpliv različnih konstrukcijskih sprememb na strižno zaostajanje v pasnicah konzolno vpetega kompozitnega I-nosilca, ki je bil obremenjen z eliptično porazdeljeno prečno obremenitvijo. Na osnovi 226 različnih konfiguracij, se je analiziralo vplive razmerja med širino in debelino pasnic, strižne togosti stojine, strižne ojačitve pasnic, velikosti radija na prehodu stojina-pasnica in debeline lahkega polnila v sendviču stojine. SI 38

»Vpliv ekstremne plastične deformacije na mikrostrukturo in lastnosti kovinskih materialov« (mentor: prof. dr. Ivan Anžel); V doktorski disertaciji je obravnavana problematika vpliva ekstremne plastične deformacije na mikrostrukturne spremembe disperzijsko utrjenega bakra, s ciljem, da se združi disperzijsko in deformacijsko utrjanje in ustvari nanostrukturni kompozit z izboljšanimi lastnostmi. V okviru doktorske disertacije je bila raziskana možnost doseganja velikih globin cone notranje oksidacije (CNO). Z modificiranim Rhinesovim paketom je bila dosežena zadostna globina cone notranje oksidacije za študij obnašanja disperzijsko utrjene zlitine med ECAP (Equal channel angular pressing) postopkom. Preučeno je bilo tečenje materiala na makro nivoju pri delno notranje oksidiranem preizkušancu in homogenem referenčnem vzorcu iz modelirne mase z namenom, da se ugotovi vpliv oksidnih delcev v CNO na tečenje materiala med ECAP postopkom in določi porazdelitev deformacije in stopnjo deformacijskega utrjanja po volumnu preizkušanca. Ugotovljeno je bilo, da na makroskopskem nivoju oksidni delci ne vplivajo na tečenje materiala, zato ni razlike v tečenju cone notranje oksidacije in trdne raztopine. S primerjavo razvoja mikrostrukture med ECAP postopkom v disperzijsko utrjenem materialu in trdni raztopini je bil ovrednoten vpliv oksidnih delcev na mehanizem plastične deformacije. Ugotovljeno je bilo, da oksidni delci povečajo fragmentacijo zrn med ekstremno plastično deformacijo, kar privede do nastanka nanometrskih enakoosnih zrn, ki so obdana z amorfnim mejnim področjem. Na osnovi rezultatov je bil postavljen model plastičnega tečenja nanostrukturnega kompozita v katerem se predpostavlja, da plastično tečenje poteka z drsenjem vzdolž mej zrn.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, SI 38-40

DIPLOMSKE NALOGE

Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 25. februarja 2014: Agustin BAIXAULI BENLLOCH z naslovom: »Razvoj nadzorne plošče za projektno vodenje v proizvodnem podjetju / Development of a dashboard for project management in an industrial company« (mentor: prof. dr. Alojzij Sluga); Héctor VILLANUEVA ESCRICHE z naslovom: »Razvoj nadzorne plošče za upravljanje virov v podjetju / Development of a dashboard for managing of factory resources« (mentor: prof. dr. Alojzij Sluga); Miha BABNIK z naslovom: »Verifikacija modela toplotnega odziva aktivnega naravnega ogrevanja in hlajenja« (mentor: prof. dr. Sašo Medved).

Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva (UN): dne 29. januarja 2014: Jan LOGAR; dne 30. januarja 2014: Daniel MIKETIČ in dne 6. februarja 2014: Igor STAREŠINIČ.

* Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv magister inženir strojništva: dne 25. februarja 2014: Kristian NAGLLIC z naslovom: »Povečanje stopnje modularnosti in razširitev družine brezkrtačnih elektromotorjev« (mentor: izr. prof. dr. Jernej Klemenc); Tomaž STARMAN z naslovom: »Vpliv parametrov sušenja na lastnosti silikagela« (mentor: izr. prof. dr. Roman Šturm). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv magister inženir strojništva: dne 17. februarja 2014:

Lea BARTON z naslovom: »Zasnova in dimenzioniranje industrijskih drsnih vrat« (mentor: prof. dr. Srečko Glodež, somentor: doc. dr. Janez Kramberger); dne 27. februarja 2014:

Bogdan PLAZOVNIK z naslovom: »Upravljanje kapacitet v podjetju Metal Ravne« (mentor: doc. dr. Iztok Palčič). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister inženir oblikovanja izdelkov: dne 26. februarja 2014:

Mihael MIHELJAK z naslovom: »Oblikovanje drsalne površine na kosilnem grebenu traktorske kosilnice« (mentor: izr. prof. dr. Miran Ulbin, somentor: asist. dr. Matej Borovinšek).

* Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 5. februarja 2014: Nejc MODIC z naslovom: »Obdelovalnost nodularne in jeklene litine pri procesu obodnega frezanja glede na geometrijo rezalnega orodja« (mentor: doc. dr. Franci Pušavec, somentor: prof. dr. Janez Kopač); Luka PETRAK z naslovom: »Uporaba brezpilotnih letal za odkrivanje in nadzor divjih odlagališč na področju Ljubljanskega barja« (mentor: izr. prof. dr. Tadej Kosel); Matija SMOLIČ z naslovom: »Optimizacija parametrov krmilnika avtonomnega robota manjših dimenzij« (mentor: prof. dr. Peter Butala). * Na Fakulteti za strojništvo Univerze v Ljubljani je pridobil naziv diplomirani inženir strojništva (VS): dne 5. februarja 2014: Gregor JENIČ z naslovom: »Časovno odvisno vrednotenje kavitacijske erozije« (mentor: izr. prof. dr. Matevž Dular, somentor: prof. dr. Branko Širok). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 27. februarja 2014:

David BERDELAK z naslovom: »Snovanje novega izdelka in ustanovitev samostojne dejavnosti na trgu« (mentor: doc. dr. Iztok Palčič); Iztok DROBEŽ z naslovom: »Dimenzioniranje podporne noge za traktorske prikolice« (mentor: prof. dr. Srečko Glodež); Uroš KLEMŠE z naslovom: »Analiza sposobnosti merilnega procesa v proizvodnji paličnega mešalnika« (mentor: prof. dr. Bojan Ačko); Marko KORAT z naslovom: »Reverzibilna sklopka« (mentor: doc. dr. Aleš Belšak, somentor: izr. prof. dr. Miran Ulbin); Matjaž KUMER z naslovom: »Oblikovanje procesa izdelave plinske peči za termično obdelavo SI 39

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)3, SI 38-40

kovin« (mentor: doc. dr. Nataša Vujica Herzog, somentor: doc. dr. Janez Kramberger); Aleksander MARŠ z naslovom: »Poke Yoke sistemi v montaži komponent avtomobilskega radarskega sistema« (mentor: izr. prof. dr. Stanislav Pehan); Zdravko MRAK z naslovom: »Razvoj postopka montaže za usmerjevalec avtomobilskega žarometa« (mentor: prof. dr. Miran Brezočnik, somentor: izr. prof. dr. Borut Buchmeister); Matej POGOREVC z naslovom: »Modifikacija konstrukcije hidravlične dvižne mize« (mentor: doc. dr. Janez Kramberger);

SI 40

Gregor ROJC z naslovom: »Numerična analiza obratovalnih karakteristik aksialnega ventilatorja z lopaticami vpetimi v pesto na obodu« (mentor: doc. dr. Ignacijo Biluš, somentor: prof. dr. Brane Širok). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani gospodarski inženir (UN): dne 27. februarja 2014:

Rok KRAŠOVEC z naslovom: »Vključitev metode QFD v vrednostni management« (mentorja: doc. dr. Marjan Leber, doc. dr. Aleksandra Pisnik Korda).

Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

Image Courtesy: University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

We would like to thank the reviewers who have taken part in the peerreview process.

http://www.sv-jme.eu

60 (2014) 3

Strojniški vestnik Journal of Mechanical Engineering

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Papers

147

158

Siamak Pedrammehr, Mehran Mahboubkhah, Mohammad Reza Chalak Qazani, Arash Rahmani, Sajjad Pakzad: Forced Vibration Analysis of Milling Machine’s Hexapod Table under Machining Forces

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Georgije Bosiger, Tadej Perhavec, Janez Diaci: A Method for Optodynamic Characterization of Erbium Laser Ablation Using Piezoelectric Detection

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Domen Rovšček, Janko Slavič, Miha Boltežar: Tuned-Sinusoidal Method for the Operational Modal Analysis of Small and Light Structures

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Ertugrul Selcuk Erdogan, Olcay Eksi: Prediction of Wall Thickness Distribution in Simple Thermoforming Moulds

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Pino Koc: Sea-wave Dynamic Loading of Sailing Yacht`s Retractable Keel

Journal of Mechanical Engineering - Strojniški vestnik

Contents

3 year 2014 volume 60 no.