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http://www.sv-jme.eu

59 (2013) 5

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

281

Dunja Ravnikar, Primož Mrvar, Jožef Medved, Janez Grum: Microstructural Analysis of Laser Coating Ceramic Components TiB2 and TiC on Aluminium Alloy EN AW-6082-T651

291

Gang Cheng, Peng Xu, De-hua Yang, Hui Li, Hou-guang Liu: Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters

Hamid Reza Vosoughifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, Seyed Reza Hashemi Nezhad: Evaluation of Fluid Flow over Stepped Spillways Using the Finite Volume Method as a Novel Approach

Milosav Ognjanovic, Miroslav Milutinovic: Design for Reliability as a Key Component in Automotive Gearbox Load Capacity Identification

301 311

Jiehui Zou, Qungui Du: 323 A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems 333

Aniruddha Ghosh, Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: A Study of Thermal Behaviour during Submerged Arc Welding

Mite Tomov, Mikolaj Kuzinovski, Piotr Cichosz: A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

339

Journal of Mechanical Engineering - Strojniški vestnik

Contents

5 year 2013 volume 59

X-ray elemental map of Ti

no.

Specimen

X-ray elemental map of Al

Optical microscopy crosssection


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 Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu, http://www.sv-jme.eu Print DZS, printed in 450 copies Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia

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59 (2013) 5

Chamber of Commerce and Industry of Slovenia Metal Processing Industry Association

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Strojniški vestnik Journal of Mechanical Engineering

s

mož Mrvar, Jožef Medved, Janez Grum: alysis of Laser Coating Ceramic Components TiB2 and TiC y EN AW-6082-T651

Cover: Laser coating process of aluminum alloy with ceramics components is shown on metallographic cross sections and corresponding X-ray elemental distribution of elements.

Xu, De-hua Yang, Hui Li, Hou-guang Liu: ics of a Novel 3CPS Parallel Manipulator Based on ters

, Miroslav Milutinovic: ity as a Key Component in Automotive Gearbox Load tion

Du: oning Cube Model for Conceptual Design of Mechatronic

Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: l Behaviour during Submerged Arc Welding

aj Kuzinovski, Piotr Cichosz: of Statistic Equality of Sampling Lengths in Surface rement

Journal of Mechanical Engineering - Strojniški vestnik

ghifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, mi Nezhad: Flow over Stepped Spillways Using the Finite Volume Approach

year

no. 5 2013 59

X-ray elemental map of Ti

Specimen

X-ray elemental map of Al

volume

Optical microscopy crosssection

Image Courtesy: Laboratory for Materials Testing and Heat Treatment, Faculty of Mechanical Engineering, University of Ljubljana

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 Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. 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 President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia 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. We would like to thank the reviewers who have taken part in the peerreview process.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 59, (2013), number 5 Ljubljana, May 2013 ISSN 0039-2480 Published monthly

Papers Dunja Ravnikar, Primož Mrvar, Jožef Medved, Janez Grum: Microstructural Analysis of Laser Coated Ceramic Components TiB2 and TiC on Aluminium Alloy EN AW-6082-T651 Gang Cheng, Peng Xu, De-hua Yang, Hui Li, Hou-guang Liu: Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters Hamid Reza Vosoughifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, Seyed Reza Hashemi Nezhad: Evaluation of Fluid Flow over Stepped Spillways Using the Finite Volume Method as a Novel Approach Milosav Ognjanovic, Miroslav Milutinovic: Design for Reliability Based Methodology for Automotive Gearbox Load Capacity Identification Jiehui Zou, Qungui Du: A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems Aniruddha Ghosh, Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: A Study of Thermal Behaviour during Submerged Arc Welding Mite Tomov, Mikolaj Kuzinovski, Piotr Cichosz: A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

281 291 301 311 323 333 339


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 281-290 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.904

Original Scientific Paper

Received for review: 2012-12-07 Received revised form: 2013-02-26 Accepted for publication: 2013-03-13

Microstructural Analysis of Laser Coated Ceramic Components TiB2 and TiC on Aluminium Alloy EN AW-6082-T651 Ravnikar, D. – Mrvar, P. – Medved, J. – Grum, J. Dunja Ravnikar1 – Primož Mrvar2 – Jožef Medved2 – Janez Grum1,* 1 University

2 University

of Ljubljana, Faculty of Mechanical Engineering, Slovenia of Ljubljana, Faculty of Natural Science in Engineering, Slovenia

This paper deals with the deposition of ceramic powder coating TiB2-TiC mix with aluminium powder on the 6082-T651 aluminium alloy by means of laser coating. The resulting coating was studied by means of a microstructural and microchemical analysis. Microhardness was measured in the coating and in the substrate area under the coating. A thermodynamic analysis of the system showed the potential existence of aluminium carbide Al4C3 in the coating, whereas the EDS analysis indicated a possible occurrence of aluminium oxycarbides in the coating, containing TiB2, TiC and Al. An additional thermal analysis explained the existence of two separate exothermic peaks, on the basis of which it could be inferred that the precipitation of phase Mg2Si, which is a typical precipitate in aluminium alloy 6082, occurred. Microhardness measurements confirmed the differences in the hardness of the coating, resulting from different energy input during the coating process. Due to the thermal effect of the coating process and the rapid cooling of the coating and the coating-substrate interface, microstructural changes occurred on the substrate surface under the coating. The modified microstructure under the coating resulted in reduced hardness, which could be attributed to the use of the alloys in the precipitation-hardened state. Thermal effects of the coating process and rapid cooling contributed to the occurrence of forced dissolution with reduced microhardness. Keywords: laser coating, aluminium alloy EN AW 6082-T651, ceramic components TiC-TiB2, microstructure, microhardness

0 INTRODUCTION Aluminium alloys used in specific applications have to meet set operating requirements. Therefore, they are frequently heat-treated or hardened through cold deformation in order to obtain the desired surface properties. Surface engineering may be used to improve the required mechanical and chemical properties of different substrate materials. Žagar and Grum [1] report the improved resistance of aluminium alloys 6082-T651 and 2007-T351 to fatigue, resulting from micromechanical hardening of the surface by means of shot peening. The improved mechanical properties and resistance to corrosion of aluminium alloy 6082T651, obtained through laser shock peening, were observed by Trdan et al. [2] and [3]. Sušnik et al. [4] reported the increased hardness of the surface layer of aluminium alloy AlSi12CuNiMg, achieved via laser surface remelting. Nowadays, surface properties are frequently modified by means of laser coating, while the desired mechanical and chemical properties of the surface are obtained through the application of the coating by remelting the powder components and thin substrate layer. The application of different ceramic components may also increase the wear resistance of the surface of aluminium alloys [5] to [7]. Aluminium alloys coated with ceramic components contribute to the reduced weight of a device or a machine. The laser coating of aluminium alloys by means of metalmatrix composites is becoming ever more appealing for the application in the automotive and chemical

processing industry. Light construction materials with coatings that improve their properties may represent a significant technological advance in the field of surface engineering, as they may be the alternatively used in the design of light engines with improved efficiency [8]. The only issue with products with such coatings is related to the bonding of ceramic components with a high melting point to aluminium, which has a significantly lower melting point. Therefore, the process of preparing the surface and applying the ceramic powder components during the laser heating process is particularly important. What is also important in the application of coatings on aluminium alloys is the composition of ceramic powder components with additives that have to enable good wettability during the remelting of pre-deposited powder on the substrate. Therefore, complementary additives are added to ceramic components in view of the substrate type with the purpose of improving the coating-substrate interface under the resulting ceramic coating and the substrate surface. Katipelli et al. [5] subjected the AlMg1SiCu aluminium alloy to laser coating, whereas the surface was coated with titanium carbide to which 10% of Si was added, improving the wettability and fluidity of the resulting melt, enabling better adhesion of the coating to the substrate, and facilitating the filling of pores in the coating and in the coating-substrate interface that occur during the coating process. Chong et al. [6] coated the same aluminium alloy with a precursor of titanium carbide

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

281


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 281-290

and molybdenum; different component ratios were used. The results showed an improved wear resistance of the coating, especially with the ratio of 70% of TiC and 30% of Mo. Better adhesion of the coated layer to the substrate can be obtained by adding metal powder particles to ceramic components. These particles must have a lower melting point, such as aluminium or copper [9]. With the transition of a low-energy laser beam across the pre-deposited powder, only the component with the lower melting point is fully melted. Thus, the re-melted metal matrix with a low melting point and the respective precipitates appear in the coating, together with dispersed solid particles. Following the solidification, a metal composite coating with solid refractory particles emerges, providing the desired mechanical and chemical properties [9]. In numerous studies, aluminium alloys were coated with powder precursors of Al-Si [8] and [10] or Al-Ti [11], reinforced with TiC and TiB2 particles. Dubourg et al. [8] applied different ratios of Al-Si powder and TiC to pure aluminium. It was established that higher contents of Si and TiC increase the hardness of the coated layer. Wear tests revealed a poor wear resistance of sub-eutectic alloys with less than 12 % wt. of Si, with no TiC present. A slight increase of the wear resistance of sub-eutectic alloys was achieved with the addition of TiC. Uenishi and Kobayashi [11] applied powder precursor Al-25% Ti with the addition of TiC, TiB2 and SiC (the content of up to 40%) to pure aluminium. When TiB2 was added, it melted, resulting in a homogeneous coating of Al3Ti  /  TiB2 with finely dispersed ceramic particles. However, the addition of TiC made melting more difficult. The particles were thus dispersed in the Al3Ti matrix. The authors concluded that, at the highest available laser power density of 2.5 kW and with the addition of 40% of TiC, the Al3Ti / TiC coating with a good interface to the substrate emerges. In contrast, other authors report that the same laser power density and 40% of TiB2 do not result in good adhesion of the Al3Ti / TiB2 coating to the substrate. Anandkumar et al. [10] coated alloy Al-7 wt.% of Si with powder precursor Al-12 wt.% Si and 40 wt.% of TiB2. Increased hardness of the coated layer, i.e. 156  HV0.1, and a relatively low specific wear rate of 2.65 × 10-5 mm3/Nm was obtained. Their finding was extremely valuable, namely that during the remelting of metal and ceramic components no chemical reactions occurred between the molten aluminium and TiB2. That means that no dissolution of TiB2 took place. Similar experiments were conducted on titanium alloys, where there were also no chemical reactions between the molten titanium and TiB2 [6]. TiB2 particles remained unevenly distributed in the 282

resulting coating, which could be attributed to different powder and alloy densities [6]. The application of precursor of Al-Si and Ni, with the addition of TiC, on the Al-Si aluminium alloy also did not result in any TiC dissolution. Due to considerable differences in the powder and alloy densities, the particles remained unevenly distributed in the resulting coating [6]. 1 EXPERIMENTAL PROCEDURE 1.1 Material Selection The substrate material chosen for the study was EN AW 6082-T651 aluminium alloy; its chemical composition is given in Table 1, and mechanical properties are presented in Table 2. The first part of the designation, i.e. 6082, indicates that this alloy belongs to the 6000 series alloys, whose main alloying elements are silicon and magnesium. The second part of the designation, i.e. T651, indicates the alloy’s temper. The alloy underwent a homogenization annealing at a temperature of 525±15 °C and additional cold deformation after quenching. It was then subjected to artificial ageing at temperatures between 165 and 195 °C for 3 to 12 h [12]. The following powder precursors with aluminium were chosen for the clad material: • TiB2 with a purity of 99.5% and average particle size of 45 μm; • TiC with a purity of 99.5% and average particle size of 2 μm; • Al with a purity of 99.5% and average particle size of 45 μm. The powders were obtained from Global Tungsten & Powder Corp. Two different ratios of powder precursor were chosen: • 40 wt.% TiB2 - 40 wt.% TiC - 20 wt.% Al, • 60 wt.% TiB2 – 20 wt.% TiC - 20 wt.% Al, to determine the physical and chemical properties of the coated layer as well as of the coating-substrate interface. Aluminium was added to TiB2-TiC ceramic components to improve the adhesion of the ceramic coating to the substrate, following the laser heating and remelting process. 1.2 Specimen Preparation With the use of the water jet process, flat aluminium specimens with dimensions of 25 × 50 × 10 mm were cut from a rolled sheet with a thickness of 10  mm. Prior to powder precursor deposition, the specimen surface was sanded with SiC sandpaper with a grit

Ravnikar, D. – Mrvar, P. – Medved, J. – Grum, J.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 281-290

Table 1. Chemical composition of EN AW 6082 aluminium alloy [13] ENAW 6082

Si [wt.%] 0.7 to 1.3

Mg [wt.%] 0.6 to 1.2

Mn [wt.%] 0.4 to 1.0

Fe [wt.%] ≤ 0.5

Cr [wt.%] ≤ 0.25

Zn [wt.%] ≤ 0.2

Ti [wt.%] ≤ 0.10

Cu [wt.%] ≤ 0.10

Table 2. Mechanical properties of EN AW 6082 aluminium alloy [14] AA designation 6082

Alloy temper T651

Rm [MPa] min. 260 to 310

Rp0.2 [MPa] min. 220 to 260

A [%] min. 2 to 10

HBW 83 to 94

Table 3: Laser coating parameters Specimen designation

Type

Clad material Ratio [wt.%]

Laser traverse speed [mm/s]

Overlapping rate [%]

#1 #2 #3 #4

50 TiB2 / TiC / Al

40 / 40 / 20

60

#5

30

#6 #7 #8 #9 #10

50 TiB2 / TiC / Al

60 / 20 / 20

#11

60 30

#12

of 600 and then subjected to ultrasonic cleaning in methanol. The precursor powders were mixed with a water-based organic binder (LISI W 1583) and reducer (LISI W15833) obtained from the Warren Paint and Color Company. The prepared powder precursor with the binder and reducer was then spray-deposited on a chemically clean surface of the aluminium alloy with an estimated coating thickness of 150±25 μm. The sprayed specimens were then dried for one day at room temperature. 1.3 Laser Coating To provide good chemical bonding between the coated layer and substrate, a sufficient energy input had to be provided to re-melt the pre-deposited powder and partially also the substrate. A 3-kW IPG Photonics continuous wave fibre laser with a wavelength of 1070 nm was employed. The focus of the laser beam to the specimen surface was such as to provide a uniform spot size of 1.0  mm diameter. Regarding the desired temperature and time conditions, three different laser beam power values were chosen: 800, 1000 and 1200  W. The traverse speed of the laser beam was constant, i.e. 60  mm/s, for all laser-re-

Laser beam power [W]

Energy density [J/mm²]

800

13.33

1000

16.67

1200

20.00

800

13.33

1000

16.67

1200

20.00

800

13.33

1000

16.67

1200

20.00

800

13.33

1000

16.67

1200

20.00

melted specimens, enabling sufficient energy input of 13.33, 16.67 and 20.00 J/mm² with regard to the spot size and selected laser beam power values. Two different overlapping rates of individual laser tracks on the specimen surface were selected: 30 and 50%. The coating was applied to twelve specimens. The coating parameters are given in Table 3. 2 RESULTS AND DISCUSSION 2.1 Microstructural Analysis Supported by EDS Analysis The coated specimens for metallographic analysis were cut on a special-purpose cutting machine and mounted in Polyfast Phenolic hot mountain resin with carbon filter (Struers). Specimens were then gradually sanded with SiC sandpaper with grits from 220 to 800. The sanding and polishing process is rather demanding due to the treated elemental aluminium. Therefore, the controlled sanding conditions were chosen as follows: a set normal force of 180 N, sandpaper with a grit of 220, and a 2 min period of sanding. The period of sanding with sandpapers with grits of 500 and 800 was 1 min for each grit size. The process was followed by a 3 min polishing with a 3.0 μm diamond

Microstructural Analysis of Laser Coated Ceramic Components TiB2 and TiC on Aluminium Alloy EN AW-6082-T651

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paste and a force of 180 N, and a 1 min final polishing with a 1.0 μm paste and a force of 100 N. An aqueous solution of 2 ml HF + 3 ml HCl + 5 ml HNO3 + 90 ml H2O was used as the echant. Microstructural analysis was conducted on scanning electron microscopes NovaTM NanoSEM 230 and JEOL JSM-6500F, followed by the EDS analysis. The thickness of the coating and its porosity were determined by means of an Orthoplan optical microscope at a magnification of 100×. Fig. 1a shows an image of the cross-section of specimen #1, captured with the electron microscope at a magnification of 150×. The specimen was coated with a 40 / 40 / 20 ratio of clad material, an overlapping rate of 50% and an energy density of 13.3  J/mm². The image shows good adhesion of the coating to the substrate with no cracks and low porosity. It is of utmost importance that good adhesion of the coating to the substrate be obtained with all combinations of laser coating parameters. The method of the pre-deposition of the clad material to the substrate surface resulted in the anticipated coating thickness deviations, which,

however, have no significant effect on further results. The average thickness of the coating measured in all the coated specimens amounted to about 130 mm. The lowest average thickness of the coating was measured in specimen #10, amounting to 80.50  mm, while the highest average thickness was measured in specimen #1, amounting to 148.57 mm. The porosity rate in the coating was also determined, amounting to about 2%. TiC and TiB2 particles were relatively large, while their volume fraction in the coating was 60 to 65%. They varied in size and were distributed throughout the coating as anticipated. The region under the resulting coating is known as the laser melt zone (LMZ). Measurements of the thickness of the laser melt zone revealed that the thickness increased with greater energy input and higher overlapping rate. Due to heat transfer during the coating process and the melting of the substrate, the thickness also increased linearly through the crosssection in the direction of the coating application. The thickness of the laser melt zone, measured in the middle of the coated specimen, amounted from

Fig. 1. a) SEM image of the laser-coated layer, b) dendritic microstructure in LMZ, c) reduced number of precipitates in LMZ

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207.65 to 347.80 mm, with regard to different coating conditions. The laser melt zone shown in Fig. 1b illustrates a cellular dendritic microstructure, typical of cast aluminium alloys. These findings indicate that during the coating process, a temperature high enough to cause complete melting of the substrate surface appeared under the coating. Due to rapid local solidification, the surface transformed into a dendritic structure [15]. The size of the resulting dendrites in the laser melt zone depended on the temperature of the melt during heating, and on the cooling rate. The cooling rates gradually decreased with the depth of the laser melt zone. Therefore, a fine dendritic microstructure emerged on the substrate surface, where the cooling rate was the highest, and was transformed into a coarse structure with greater LMZ depth [15]. Due to the overheating of the substrate during the coating process, no precipitates seemed to appear in the LMZ or their number significantly decreased, as shown in Fig. 1c. A lower number of precipitates of greater sizes was observed in the heataffected zone (HAZ). In the region with a reduced number of precipitates, poorer mechanical properties may be anticipated. Due to the large differences in the melting points of titanium carbide, titanium diboride and aluminium, the added aluminium powder and then also the aluminium alloy, i.e. the substrate, began melting as the laser beam acted on the coated surface. The thickness of the re-melted layer on the surface of the substrate just beneath the coating depended mainly on the laser coating conditions. Poor conductivity of the ceramic coating resulted in the major part of the laser beam energy being consumed for the heating of the coating, whereas only a small proportion of this energy was used for the melting of aluminium. Due to the capillary effect during the remelting process, the molten aluminium filled the pores between ceramic components. A sufficiently high energy density may cause the dissolution of TiC and TiB2 particles, and the occurrence of new phases. Upon the dissolution of TiB2, a new phase AlB2 [16] may occur, whereas upon the dissolution of TiC, phase Al4C3 [5] occurs. Since TiB2 has more free energy than TiC, it is more resistant at higher temperatures. Therefore, it is more likely that phase Al4C3 will occur rather than phase AlB2. The Ti-B binary phase diagram is presented in Fig. 2. Two intermetallic compounds exist: TiB and TiB2. According to the phase diagram, there is an additional intermetallic phase formed through peritectic reaction: Ti3B4. The phase TiB2 is a congruent melting phase, and is also thermodynamically stable in the aluminium

melt. Consequently, there are no reaction products present. The transfer of B and Ti into the aluminium melt normally goes through fluoride salts (K2TiF6 and KBF4) [17]. With further cooling of such a melt, a relatively thermodynamically stable composite is normally expected. The transfer of B, Ti and C atoms can cause a local inhomogeneity or a super-saturation effect. This results in the formation of different precipitations after additional thermal treatment. Fig. 3 shows the results of a microchemical surface analysis for C, Al, Ti and B for specimen #12. The specimen was coated with a 60 /  20  /  20 ratio of clad material, an overlapping rate of 30% and an energy density of 20  J/mm². In Fig. 3, an interface between the substrate and the composite is visible.

Fig. 2. Ti-B binary phase diagram [18]

As anticipated, aluminium was found as a part of the substrate as well as a part of the composite material around facets. Carbon was found to occupy spots alongside the aluminium. It is assumed that after the decomposition of TiC, carbon bonds with aluminium and forms aluminium carbide Al4C3. Elements Ti and B were found in the regions of facets, assumed to be a part of titanium diboride. It was not possible to define which of the facets are TiC and TiB2 in our study, as was the case in the study of Du [19]. The distribution of elements in that study reveals that the concentration of titanium is found in both the rectangular and spherical types of facets. The concentration of B and C was found in rectangular and spherical facets, respectively. These results give us an indication that the rectangular facets are related to the presence of TiB2 and the spherical ones with TiC. Qualitative analysis was performed in designated measuring positions, illustrated in Figs. 4a and b. Because the specimen coating was electrically nonconductive, the surface of the coated specimens

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Fig. 3. a) SEM microphotographs and corresponding distribution of elements, b) aluminium, c) titanium, d) carbon and e) boron

was sprayed with a thin layer (7 nm) of carbon. Using EDS analysis, six elements were identified in the coating: Al, Ti, B, C, O and Si. Table 4 shows individual measuring positions in the coating with various elements, on the basis of which the presence of different phases may be inferred. Due to the prior deposition of carbon on the specimen surface, carbon wt.% values were not shown, since they would not represent the real results. It was established on the basis of a preliminary analysis that measuring positions with no facets present, i.e. in spectra 6, 7 and 9, contained carbon and aluminium, as well as oxygen and small amounts of titanium and silicon. Since Al, O and C were in proximity to each other, this could affect the occurrence of aluminium oxycarbide. Titanium present in the aluminium substrate represented the basis for the occurrence of phase Al3Ti, whereas the presence of boron could have resulted in the appearance of TiB, TiB2 and / or Ti3B4. Silicon was a part of the chemical composition of the substrate and indicated that the coating was mixed with the substrate during the coating process; some amount of silicon was transferred into the coating. Other possible elements were all under the detection limit. Thus, no other reflections were noticed, and the presence of other complex intermetallic phases, especially magnesium and compound Mg2Si, could be eliminated. Qualitative analysis of smaller facets in spectra 2, 3, 4, 5, 10, 12 and 13 indicated a greater content of 286

titanium and boron, probably confirming the existence of TiB2. The smaller part of aluminium is attributed to the background of the electron beam. It should be emphasized that titanium has a stronger signal compared to boron, because of a higher electron dissipation level (Mt = 204.38 g/mol). Table 4. EDS results given in wt.% Meas. position Spec.1 Spec.2 Spec.3 Spec.4 Spec.5 Spec.6 Spec.7 Spec.8 Spec.9 Spec.10 Spec.11 Spec.12 Spec.13

B 26.92 26.41 27.71 30.42

24.73 25.85 24.02

Al 2.91 2.28 2.11 2.17 2.86 68.46 63.78 3.20 82.90 1.91 3.57 2.40 2.01

Ti 74.16 45.46 43.83 46.87 43.12 0.69 0.69 75.33 1.13 43.15 66.36 43.21 43.85

O

Si

2.74 2.38

0.50 0.42

2.17

0.44

The results of the analysis of a smaller facet in spectrum 11 were similar to results of the analysis of bigger facets in spectra 1 and 8, where there was a higher content of titanium. In the proximity of this phase, however, there were no reaction products, such as Al3Ti or Al4C3. Thus, the smaller content of aluminium was considered to be a signal from the

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aluminium matrix. The presence of carbon in facets 1, 8 and 11 indicated the existence of TiC. In this regard, Liang et al. [20] report that the phase type may be determined on the basis of the facet shape. They agree with Du et al. [19], who claim that elongated or rectangular particles belong to TiB2, whereas the spherical particles belong to TiC. In our case, this statement could be agreed upon, except in spectrum 11, where no boron was present. Therefore, it could not be claimed that the rectangular and elongated facet in spectrum 11 confirms the presence of TiB2.

depends on the technology. The Al3Ti phase is most commonly formed through the reaction between TiC and Al, where Al4C3 is also possible. Normally, this latter carbide is formed by precipitation from the solid aluminium matrix. In the three component system T-C-Al (viewed from the aluminium richside perspective), the phase TiC decomposes to Al3Ti and Al4C3 at higher temperatures. By adding TiB and TiC to the aluminium alloy, phase equilibrium is possible as calculated in Fig. 5. The isopleth phase diagram Al-Mg is presented in Fig. 5. The equilibrium thermodynamic calculation was made for the system in which the aluminium alloy and the coating were in ratio ENÂ AW-6082 : TiB = 4 : 1.

Fig. 5. Equilibrium isopleth phase diagram of Al-Mg for ratio ENAW 6082: TiB = 4 : 1

Fig. 4. Selected and measured regions using EDS analyses of; a) specimen #4, and b) specimen #10

2.2 Thermodynamic Analysis If Ti is present in pure aluminium (or alloy) in a smaller excess, the occurrence of phase Al3Ti is possible. The method in which the phase is formed

It can be inferred from the thermodynamic calculation that (together with the standard phases, which usually occur in alloy EN AW-6082) two new phases occurred as well: TiB (MB2_C32#1) and Al4C3 (AL4C3#1). In the final equilibrium state, all of the boron will react with the titanium, and all of the carbon with the aluminium. Nevertheless, the Al4C3 phase is difficult to detect on the surface of the coating, as concluded in the research of Dubourg et al. [5]. Although the melting point of TiC is higher than the evaporation temperature of aluminium, the dissolution of TiC is possible due to a higher absorption coefficient of TiC. That is why TiC can decompose; meanwhile, aluminium powder stays in its dissolved state and causes further formation of the Al4C3 phase [5].

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2.3 Thermal Analysis In order to confirm the appearance of different phases with different ratios of ceramic components during the laser coating process, a thermal analysis was conducted. The measurement was performed under protective gas Ar with a purity degree of 5 and constant flow through the furnace. An empty corundum crucible was used as a reference. The maximum temperature reached was 550 °C using 10 K/min of heating/cooling rate. DSC (differential scanning calorimetry) was performed for two specimens with designations #3 and #9. Fig. 6 illustrates two DSC heating curves for both specimens. In both cases, two exothermic peaks were observed, which are typical for the precipitation of phases in solid state. The first thermal effect in both cases was detected at a temperature of about 215  °C, whereas the second thermal effect occurred at a slightly higher temperature, i.e. about 410  °C. The slope of the second thermal effect is much more distinctive for both samples and indicates that the process of the first thermal effect is less intensive. Because both peaks were found in a specific temperature interval, we can assume that the precipitation was gradual. Such peaks can be found in the case of precipitation of Mg2Si and Al6Mn for a particular aluminium alloy 6082. Such intense peaks can be found with relatively low concentrations of magnesium (under 0.8 wt.%).

Fig. 6. DSC heating curves

2.3 Microhardness The Vickers hardness tests were performed on a Leitz-Wetzlar tester to determine the microhardness value of the coating and in the depth under a load of 100 g and indentation time of 15 s. The measurement of the hardness of the coating was conducted at a half thickness of the coating. Sixteen iterations were performed horizontally in a randomly selected 288

position, at intervals of 500 mm. Microhardness values were also measured along the depth in increments of 50 mm to the depth of 900 mm. The greatest measuring depth was determined with regard to the depth of LMZ and HAZ. The HAZ was less prominent, since the specimen was heated at a temperature higher than precipitation only for a short period of time. The chosen method of microhardness measurement confirmed the differences in the microstructure of individual regions. Fig. 7 illustrates the profile of microhardness along the depth of the coated specimens with the energy density of 13.3 J/mm² at different overlapping rates 30 and 50% and both ratios of clad material, and with pertaining standard deviation. Due to differences in the coating thickness, the coordinate origin of the depth is placed at the coating-substrate interface. The microhardness value of the aluminium alloy prior to the application of the coating was 96.6 HV0.1. Following the coating process, this value increased on average by 66% with regard to the selected powders and laser coating parameters. The value thus amounted to around 160 HV0.1. Due to different sizes of TiC and TiB2 particles and their uneven distribution in the coating, there was a wide spread of microhardness results. Therefore, the results of measurements largely depended on the measuring position in the coating with regard to individual phase types. However, the microhardness values in the case of 30% overlapping are apparently higher than with 50% overlapping. Furthermore, the microhardness values in the case of 40 /  40 / 20 are slightly higher than 60 / 20 / 20. Measuring microhardness along the depth proved that higher heat input during the coating process results in lower microhardness values in the LMZ and HAZ. Thus, higher energy density and higher overlapping rate decreased the microhardness values in the LMZ and HAZ in the substrate under the coating surface. The average microhardness value in LMZ was 66.6 HV0.1, whereas this value amounted to 71.6 HV0.1 in the HAZ. This was a significantly lower value than the microhardness value of the aluminium alloy, i.e. 96.6 HV0.1, prior to the coating process. Lower microhardness values in the LMZ and in the HAZ may be attributed to changes in the substrate microstructure, resulting from thermal effects during the coating process and thus contributing to the process of homogenization and incomplete precipitation annealing in the LMZ and HAZ. Partial homogenization on the substrate surface could be attributed to the rapid coating process. Some of the Si underwent forced dissolution, while some of it remained in the form of Mg2Si precipitates

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Fig. 7. Profile of microhardness along the depth, following the coating process with the energy density of 13.3 J/mm2 and two ratios of clad material; a) 40 / 40 / 20, and b) 60 / 20 / 20 at different overlapping rates 30 and 50%

in the laser melt zone. After a certain period of ageing, the oversaturated solid solution started to precipitate fine sub-microscopic particles in the form of intermetallic compounds, especially within the crystal grains, resulting in the formation of internal stress on a crystalline level, and preventing the grains from moving, thus causing higher hardness values. Therefore, subsequent heat treatment by means of solution annealing and artificial ageing in a furnace made it possible to improve the microstructure, microhardness and thus also the mechanical properties of the LMZ and HAZ. DSC heating curves also indicated the existence of precipitates in the substrate, resulting from subsequent heat treatment of the coated specimens. Thus, the microhardness values in the LMZ and HAZ were higher. 3 CONCLUSIONS The research presents the laser coating of the 6082T651 aluminium alloy with different contents of ceramic components (TiB2-TiC) and the addition of aluminium (Al) in order to improve chemical bonding between the coating and the substrate at different overlapping rates and energy densities. Results and analyses conducted on different coatings applied to aluminium alloys corroborate the following conclusions: • Results of microstructural analysis confirmed that a quality coating with good adhesion to the substrate was obtained in all cases. The average coating thickness was 130 mm at an average porosity lower than 2%, corresponding to demanding applications. In all cases, good adhesion of the coating to the substrate was obtained.

In both cases of ceramic components with the addition of Al, the average coating microhardness value obtained at different combinations of laser coating conditions was 160 HV0.1, which represented a 66% higher microhardness value than that of the uncoated aluminium alloy. Measurements of microhardness of the substrate surface under the coating revealed that the heat during the remelting process affected the microstructural changes in the substrate and resulted in lower microhardness values. The average microhardness value in the LMZ was 66.6  HV0.1, whereas in the HAZ, the value was 71.6 HV0.1. The results show that at higher energy densities and higher overlapping rates, lower microhardness values were obtained in the LMZ and HAZ. Thermodynamic calculation shows that, in the final equilibrium state, all of the boron will react with the titanium, and the entire carbon with the aluminium. EDS analysis conducted in the coating revealed that a new phase, called oxycarbides, could form in the matrix. Thermal analysis, during which the coated specimens were heated to 550 °C, confirmed the existence of two separate exothermic peaks at temperatures of 215 and 410 °C. It can be inferred from the peaks in the heating curve that these two exothermic peaks were related to gradual precipitation. Since the EDS analysis did not reveal the presence of Mg and Mn in the coating, it was concluded that gradual precipitation occurred in the substrate, i.e. in the LMZ and HAZ.

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4 ACKNOWLEDGMENTS The authors gratefully acknowledge Prof. Narendra B. Dahotre and his research group from the Laboratory for Laser Materials Synthesis and Fabrication, Department of Materials Science and Engineering, University of North Texas, Denton, Texas, USA, for carrying out the laser coating. 5 REFERENCES [1] Žagar, S., Grum, J. (2011). Surface Integrity after mechanical hardening of various Aluminium alloys. Strojniški vestnik – Journal of Mechanical Enginerring, vol. 57, no. 4, p. 334-344, DOI:10.5545/ sv-jme.2010.092, DOI:10.5545/sv-jme.2010.092. [2] Trdan, U., Grum, J. (2012). Evaluation of corrosion resistance of AA6082-T651 aluminium alloy after laser shock peening by means of cyclic polarisation and ElS methods. Corrosion Science, vol. 59, p. 324-333, DOI:10.1016/j.corsci.2012.03.019. [3] Trdan, U., Porro, J.A., Oca-a, J.L., Grum, J. (2012). Laser shock peening without absorbent coating (LSPwC) effect on 3D surface topography and mechanical properties of 6082-T651 Al alloy. Surface & Coatings Technology, vol. 208, p. 109-116, DOI:10.1016/j.surfcoat.2012.08.048. [4] Sušnik, J., Šturm, R., Grum, J. (2012). Influence of laser surface remelting on Al-Si alloy properties. Strojniški vestnik – Journal of Mechanical Engineering, vol. 58, no. 10, p. 614-620, DOI:10.5545/sv-jme.2012.696. [5] Katipelli, L.R., Agarwal, A., Dahotre, N.B. (2000). Laser surface engineered TiC coating on 6061 Al alloy - microstructure and wear. Applied Surface Science, vol. 153, no. 2-3, p. 65-78, DOI:10.1016/S01694332(99)00368-2. [6] Chong, P.H., Man, H.C., Yue, T.M. (2002). Laser fabrication of Mo-TiC MMC on AA6061 aluminum alloy surface. Surface and Coating Technology, vol. 154, no. 2-3 p. 268-275, DOI:10.1016/S02578972(01)01719-4. [7] Anandkumar, R., Almeida, A., Colaco, R., Vilar, R. Ocelik, V., De Hosson, J. Th. M. (2007). Microstructure and wear studies of laser clad Al-Si/SiC(p) composite coatings. Surface & Coatings Technology, vol. 201, no. 24, p. 9497-9505, DOI:10.1016/j.surfcoat.2007.04.003. [8] Dubourg, L., Ursescu, D., Hlawka, F., Cornet, A. (2005). Laser cladding of MMC coatings on aluminium substrate: influence of composition and microstructure

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on mechanical properties. Wear, vol. 258, no. 11-12, p.1745-1754, DOI:10.1016/j.surfcoat.2007.04.003. [9] Čekada, M., Panjan, P. (2004). Laser methods for surface protection. Vakuumist, vol. 24, no. 4, p. 4-10. (In Slovene) [10] Anandkumar, R., Almeida, A., Vilar, R. (2011). Wear behavior of Al-12Si/TiB2 coatings produced by laser clading. Surface and Coatings Technology, vol. 205, no. 13-14, p. 3824-3832, DOI:10.1016/j. surfcoat.2011.01.048. [11] Uenishi, K., Kobayashi, K.F. (1999). Formation of surface layer based on Al3Ti on aluminum by laser cladding and its compatibility with ceramics. Intermetalics, vol. 7, no.5, p. 553-559, DOI:10.1016/ S0966-9795(98)00071-5. [12] Brandes, E.A., Brook, G.B. (1998). Smithells Light Metals Handbook. Butterworth-Heinemann, Oxford. [13] European Aluminum Association and the Matter Project, from http://aluminum.matter.org.uk/aluselect/, accessed on 2012-04-16. [14] BS EN 485-2:2004. (2004). Aluminium and aluminium alloys - Sheet. strip and plate - Part 2: Mechanical properties. British standard Institution, London. [15] Kadolkar, P., Dahotre, N.B. (2002). Variation of structure with input energy during laser surface engineering of ceramic coatings on aluminum alloys. Applied Surface Science, vol. 199, no. 1-4, p. 222-233, DOI:10.1016/S0169-4332(02)00799-7. [16] Xu, J., Li, Z., Zhu, W., Liu, Z., Liu, W. (2007). Investigation on microstructural characterization of in situ TiB/Al metal matrix composite by laser cladding. Materials Science and Engineering A, vol. 477, no. 1-2, p. 307-313, DOI:10.1016/j.msea.2006.10.057. [17] Lakshmi, S., Lu, L., Gupta, M. (1998). In situ preparation of TiB2 reinforced Al based composites. Journal of Materials Processsing Technology, vol. 73, no. 1-3, p. 160-166, DOI:10.1016/S09240136(97)00225-2. [18] Murray, J.L., Liao, P.K., Spear, K.E. (1987). The Ti–B (titanium–boron) system. Murray, J.L. (ed.), Phase Diagrams of Binary Titanium Alloys. ASM International, Metals Park, p. 33-38. [19] Du, B., Zou, Z., Wang, X., Li, Q. (2007). In situ synthesis of TiC-TiB2 reinforced FeCrSiB composite coating by laser cladding. Surface Review and Letters, vol. 14, p. 315-319, DOI:10.1142/S0218625X07009414. [20] Liang, Y., Han, Z., Zhang, Z., Li, X., Ren, L. (2012). Effect of Cu content in Cu-Ti-B4C system on fabricating TiC/TiB2 particulates locally reinforced steel matrix composites. Materials and Design, vol. 40, p. 64-69, DOI:10.1016/j.matdes.2012.03.023.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 291-300 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.727

Original Scientific Paper

Received for review: 2012-08-06 Received revised form: 2012-10-11 Accepted for publication: 2013-02-14

Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters Cheng, G. – Xu, P. – Yang, D. – Li, H. – Liu, H. Gang Cheng1,* – Peng Xu1 – De-hua Yang2 – Hui Li2 – Hou-guang Liu1

1 China

University of Mining and Technology, College of Mechanical and Electrical Engineering, China of Sciences, National Astronomical Observatories, Nanjing Institute of Astronomical Optics and Technology, China

2 Chinese Academy

In order to adjust the poses of the segment mirrors and give the correct surface shape to a large aperture telescope, an active adjusting platform for segment mirror with a novel 3CPS parallel manipulator as core module is proposed. The platform has 6 degree-of-freedoms (DOFs) including three translational freedoms and three rotational freedoms. Its kinematics are analysed systematically. By means of the Rodrigues parameters method, the formulae for solving the inverse/forward displacement, the inverse/forward velocity, and the inverse/ forward acceleration kinematics are derived. A numerical simulation of the kinematics model is then carried out combining the topological structure characteristics of the manipulator. The correctness of the kinematics model is verified by an experiment in which the pose of moving platform is measured using a photogrammetric method. Keywords: active adjusting platform, 3CPS parallel manipulator, Rodrigues parameters, kinematics, photogrammetry

0 INTRODUCTION Astronomy seeks the detection of more distant and dim celestial bodies. Large aperture optical systems are significant in astronomy research for their increased light gathering capability and angular resolution in the object space [1]. Therefore, the development of telescopes with large apertures and high imaging quality is necessary. However, as the size of the telescopes increases, they become increasingly sensitive to external disturbances such as thermal gradients, gravity, and wind, and also to internal disturbances from support equipment such as pumps, cryocoolers, and fans [2]. In order to decrease these influences, most large aperture telescopes in the future will be segmented [3], such as the US/Japanese Thirty Meter Telescope and the European-ELT 42m telescope projects. For these telescopes, the large aperture primary mirrors are spliced by many small aperture and thin segment mirrors. The advantages of this approach, where the mass and size of the primary mirror are no longer the main factors affecting the observation results, are obvious. However, comparing the telescope with one entire primary mirror, the main problems with this approach are the position and orientation disorders of the segment mirrors. The development of an active adjusting mechanism for a large number of mirrors with multiple degrees of freedom is urgent. In recent years, parallel manipulators have seen growing applications in robotics, machine tools, positioning systems, measurement devices, and so on [4] to [8]. The classic 6 DOFs parallel manipulator (Stewart platform) has many characteristics, such as

simple structure, concise principle, flexible function, high accuracy, and high stiffness. It has also been used in the field of astronomical telescope and instruments [9] to [11]. However, the six driving legs of the Stewart platform have strong coupling movements and the distance between the moving platform and fixed base is usually large. These shortcomings, as well as the structure type of the Stewart platform, are not suitable for making large-scale active adjustments in compact spaces. In order to overcome the defects mentioned above, an active adjusting platform prototype of a segment mirror with a novel 3CPS parallel manipulator as the core module is proposed. The parallel manipulator is composed of a moving platform, a fixed base and three CPS chains, where the notation CPS denotes the kinematic chain made up of a cylindrical joint, a prismatic joint, and a spherical joint in series. Kinematic analysis is a common basis of dynamic analysis and control system design. The kinematics of parallel manipulators includes inverse kinematics and forward kinematics. Numerous researchers have made contributions to this. Cheng et al. [12] studied the inverse/forward displacement, velocity, and acceleration kinematics of a 3SPS+1PS bionic parallel test platform by means of the unit quaternion method. Lu et al. [13] studied kinematics, statics, and workspaces of a 3R1T 4 DOFs and a 1R3T 4 DOFs parallel manipulators, comprehensively. Gallardo et al. [14] studied the kinematics of modular spatial hyper-redundant manipulators formed from RPS-type limbs based on screw theory and recursive method. Cui et al. [15] analysed the kinematics of a TAU parallel manipulator based on a D-H model and solved

*Corr. Author’s Address: China University of Mining and Technology, College of Mechanical and Electrical Engineering, 221008, Xuzhou, China, chg@cumt.edu.cn

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its forward kinematics in closed forms by a Jacobian approximation method. Varedi et al. [16] analysed the kinematics of an offset 3UPU translational parallel manipulator by the homotopy continuation method, which alleviates the drawbacks of traditional numerical techniques, namely: the acquirement of good initial guess values, the problem of convergence, and computing time. The main research methods of kinematics can be divided into analytical and numerical ones. For the parallel manipulators whose structures are not complex, the analytical method can solve the kinematics competently and solution procedures can be fully automated. For parallel manipulators with complex structures, the analytical method is always inadequate and the obtained solutions can be too complex to subsequently analyse. The computational accuracy and speed of the numerical method depend on the complexity of mechanisms and algorithms themselves whose flexibility and portability is quite often poor. Due to the simple mechanical structure, the analytical method is adopted to analyse the kinematics in this paper. The rest of the paper is organized as follows: In section 1, the structure of the active adjusting platform prototype is described and the reference systems of the manipulator are established. In section 2, the inverse/ forward displacement, the inverse/forward velocity, and the inverse/forward acceleration kinematics of the manipulator are studied based on the Rodrigues parameters. In section 3, a numerical simulation of the kinematics analysis is conducted, and lastly the numerical results are validated by experiments.

legs and the corresponding sides of triangular B are equal. Three equal-length short legs are connected to the horizontal legs and the lengths were unchanged. In the following analysis, the lengths of the short legs can add to the lengths of the vertical legs. The short legs are connected to m with sphere joints (Si, i = 1, 2, 3). The distances of mount points in m are equal to each other. The relative coordinate system {m} is attached to m at point o. The y-axis of {m} passes through point a2, the z-axis is perpendicular to m pointing upward. The x, y and z axis follow the RHR. Three spherical joints are installed on m and the distance from mount point to point o is denoted as e. When the length of the vertical legs are equal, namely: hi = h0 (i = 1, 2, 3) and the length of the horizontal legs are equal, namely: li = l0 (i = 1, 2, 3), the manipulator is at the equilibrium position.

a)

1 DESCRIPTION OF 3CPS PARALLEL MANIPULATOR The active adjusting platform for the segment mirror considered in this paper is shown in Fig. 1a and the topological structure of its core module, a 3CPS parallel manipulator, is shown in Fig. 1b. Taking the 3CPS parallel manipulator as an analysis object, reference systems for kinematic analysis are established. The absolute coordinate system {B} is fixed on B at point O. The Y-axis of {B} passes through point A2, the Z-axis is perpendicular to B pointing to m. The X-axis can be determined by the other two axes following the right-hand-rule (RHR). Three vertical legs with cylindrical joints (Ci, i =1, 2, 3) are installed symmetrically about point O on B. Every mount point is equidistant from point O and the distance denotes as E. Three horizontal legs with prismatic joints (Pi, i = 1, 2, 3) are fixed on the end points of three vertical legs (Di, i = 1, 2, 3). They rotate around Z-axis in the same direction and the angles between the horizontal 292

b) Fig. 1. The 3CPS parallel manipulator; a) a prototype of the active adjusting platform, and b) the topological structure of the active adjusting platform

This prototype has partial DOF-decoupling motion characteristics. The rotations around the X-axis, Y-axis and the translations along the Z-axis are driven by the three vertical leg hi (i = 1, 2, 3),

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while the translations along the X-axis, the Y-axis and the rotations around Z-axis are driven by the three horizontal legs li (i = 1, 2, 3), respectively. When the manipulator is at the equilibrium position, the moving platform is parallel to the fixed base. Let θ be the rotational angle around the Z-axis between the moving platform and the fixed base. According to the cosine formula, the formula for solving θ is expressed as:

θ = arc cos(

2

2

2 0

E +e −l ). (1) 2 Ee

Using a Kutzbach Grübler equation [17], the DOF of 3CPS parallel manipulator is calculated as: g

F = 6 ( n − g − 1) + ∑ f i = 6, (2)

where, λ02 = 1 + Φ12 + Φ22 + Φ32 and (xl, xm, xn, yl, ym, yn, zl, zm, zn) are the nine orientation parameters of m in {B}. By rotation transformation, the absolute coordinates aiB of the point ai (i = 1, 2, 3) can be derived from Eqs. (3) and (4). The absolute coordinates DiB of the point Di (i = 1, 2, 3) can be expressed as follows: DiB = AiB + RaiB , (5)

where, R = [0 0 0; 0 0 0; 0 0 1]. According to the formula of distance between two spatial points, the length of the vertical leg hi (i = 1, 2, 3) and the horizontal leg li (i = 1, 2, 3) can be derived as follows:

i =1

where, F is the DOF of the manipulator, n is the number of components, g is the number of kinematics pairs, and fi is the degree of freedom of the ith kinematics pair. It can be seen clearly that the parallel manipulator has 6 DOFs including three translational freedoms and three rotational freedoms. 2 MANIPULATOR KINEMATICS ANALYSES

 X Ai   X ai   xai      m B A =  YAi  , ai =  yai  , ai =  Yai  ,  Z Ai   Z ai   zai  B i

Xo  o B =  Yo  , aiB = Do aim + o B , (3)  Z o 

where, oB is a vector of point o on m in {B}, (Xo, Yo, Zo) are the components of oB, Do is a rotation transformation matrix from {m} to {B} based on Rodrigues parameters and it can be written as follows [18]:  xl yl zl  Do =  xm ym zm  =  xn yn zn  =

(D

B i

− AiB )

li =

(a

B i

− DiB ) ( aiB − DiB ). (6)

T

(D

B i

− AiB ) ,

T

The unit vector ςi along the vertical legs hi, the unit vector δi along the horizontal legs li and the vector ei of line oai can be expressed as:

2.1 Inverse/Forward Displacement Analysis Before analysing the kinematics of 3CPS manipulator, the coordinates of the points Ai (i = 1, 2, 3) in {B} and the coordinates of the points ai (i = 1, 2, 3) in {m} and {B} must be determined. They are expressed as:

hi =

ςi =

DiB − AiB a B − DiB , ´i = i , ei = aiB − o B . (7) hi li

The unit vector ξi is the tangent vector of the horizontal legs li rotating around vertical legs hi and can be expressed as follow:

ξi = δi × ςi . (8)

When given three Rodrigues parameters Φi (i = 1, 2, 3) and oB, the inverse displacement parameters (hi, li, ςi, δi, ei, i = 1, 2, 3) can be solved from Eqs. (6) and (7). When given the lengths of the six input driving legs hi (i = 1, 2, 3) and li (i = 1, 2, 3), from Eq. (6), the position and orientation parameters of moving platform can be obtained by solving a nonlinear kinematic equations system. 2.2 Inverse/forward Velocity and Jacobian Matrix Let V be the general velocity of m at point o and v be the linear velocity, while ω is the angular velocity of m at point o and vi is a linear velocity of m at point ai. Let vh and vl be the input velocities of vertical legs and horizontal legs, respectively. This can be expressed as follows:

1 + Φ12 − Φ22 − Φ32 2 (ΦΦ  2 (ΦΦ 1 3 + Φ2 ) 1 2 − Φ3 ) 1  2 (Φ2Φ1 + Φ3 ) 1 − Φ12 + Φ22 − Φ32 2 (Φ2Φ3 − Φ1 )  , (4) 2  λ0  2 Φ Φ − Φ 2 (Φ3Φ2 + Φ1 ) 1 − Φ12 − Φ22 + Φ32   ( 3 1 2) Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters

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 v1   ω1  v  V =   , v = v2  , ω = ω2  , ω  6×1  v3  ω3   vh1   vl1    vi = v + ω × ei , vh = vh 2  , vl = vl 2  . (9)  vh 3   vl 3 

Let uo, vo and wo be the unit vectors of x, y and z axes of the moving platform in {B}, respectively. These expressions can be obtained by rotation transformation from {m} to {B}:

1  0 0     uo = Do 0  , vo = Do 1  , wo = Do 0  . (10) 0  0  1 

The angular velocity ω of m at point o can be expressed as follows:

ω = uoω1 + voω2 + woω3 . (11)

The ω1, ω2 and ω3 are the components of ω and can be obtained by dot-multiplying Eq. (11) with (vo×wo), (uo×wo) and (uo×vo) at both sides, respectively:

ω1 =

( u × wo ) ⋅ ω (v o × wo ) ⋅ ω , , ω2 = o (uo × wo ) ⋅ vo (vo × wo ) ⋅ uo

ω3 =

(uo × vo ) ⋅ ω . (uo × vo ) ⋅ wo

(12)

According to the geometrical characteristics of the manipulator, the linear velocity vhi along vertical legs hi (i = 1, 2, 3) and the linear velocity vli along horizontal legs li (i = 1, 2, 3) can be obtained from Eq. (9) and can be expressed as follows:

vhi = vi ⋅ ςi = ςi T 

( ei × ςi )

vli = vi ⋅ δ i = δ iT 

( ei × δ i )

T

T

 V , (13a)   V . (13b) 

By combining Eq. (13a) and Eq. (13b), the formulae for solving the inverse/forward velocities can be obtained and expressed as follows:

294

ς1T  vh1   v  ς2T  h2   T  vh 3  ς3   = JV , J =  T  vl1  δ 1  vl 2   T   δ 2  vl 3   T δ 3

( e1 × ς1 )   T ( e2 × ς2 )   T ( e3 × ς3 )  , (14) T  ( e1 × δ 1 )  T ( e2 × δ 2 )  T ( e3 × δ 3 )  6×6 T

where, J is a velocity Jacobian matrix. From Eqs. (8) and (9), the formulae for solving the tangential velocities of the horizontal legs at point ai when rotating around the vertical legs are expressed as: vti = vi ⋅ ξ i = ξ iT 

( ei × ξ i )

T

 V . (15) 

From Eqs. (6) and (15), the angular velocities ωCi of the vertical legs are derived as shown below:

ωCi =

vti , (i = 1, 2, 3). (16) li

2.3 Inverse/forward Acceleration and Hessian Matrix First, a skew symmetric matrix is briefly introduced. Suppose two vectors μ and ν and a skew symmetric matrix S(μ) for μ:  0  µx  ν x       µ =  µ y  , ν = ν y  , S ( µ ) =  µ z −µ y  µ z  ν z  

−µz 0 µx

µy   − µ x  . (17) 0 

Eq. (17) satisfies the following relationships [19]:

µ ×ν = S ( µ ) ⋅ν = − S (ν ) ⋅ µ , S ( µ ) = −S ( µ ) , T

− S ( µ ) + S ( µ ) ⋅ S ( µ ) = I 3×3 , (18) 2

T

where, I3×3 is an order 3 unit matrix. Let A be the general acceleration of m at point o, where a and ε are the linear and angular acceleration of m at point o, respectively. Let ah and al be the input accelerations of the vertical legs and the horizontal legs, respectively. They can be expressed as follows:

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 291-300

 ax  ε x  a    A=   , a =  a y  , ε = ε y  , ε   az  ε z   ah1   al1  ah =  ah 2  , al =  al 2  . (19)  ah 3   al 3 

Similarly, the equation can be obtained as:

( ei × δ i )

( ei × ςi )

( ei × ςi )

 A + 01×3   (i = 1, 2, 3), T

T

V ,  (20a)

where,

( ei × ςi )

T

T

=  − S (ςi ) (ω × ei )  =

T

( e × δ ) i

i

01×3  0  1×3

T

=  S ( ei ) δi  = −δiT S ( ei ) .

(

( ei × δ i )

ei × δi

δ T  i Hi =

)

T

01×3 

( ei × δ i )

+ δiT 

( e × δ i

i

T

+ ei × δi

)

T

 V. (21a) 

The derivative of the unit vector δi along the horizontal legs li can be expressed as: v vδ 1 δi = li − li i =  − S (δ i ) 2 li li li where,

(

vliδ i = δ iT 

(

S (δ i ) 2 S (ei )  V , (21b)

( ei × δ i )

T

)

V δ i = 

)

= δ iδ iT −δ iδ iT S ( ei )  V . (21c)

From Eq. (21b), the equation can be shown as:  − S (δ i ) 1 01×3  = V T  2 li  − S ( ei ) S (δ i ) 2

δiT

03×3   . (21d) 03×3 

03×3  , S ( ei ) S (δ i )  S (δ i ) S ( ei ) 2

S ( ei ) S (δ i )

2

  . (21f) S ( ei ) 

1

( e × δ

− S (δ i )

i

2

 li  − S (e ) S (δ

)

+ ei × δi

i

T

 =V TH , i 

  , l S ( e ) S (δ ) + S (e ) S (δ ) S (e )  6×6 (i = 4, 5, 6). (21g) 2

S (δ i ) S ( ei )

i

)

2

2

i

i

i

i

i

i

ain = JA + V T HV , A = J −1 ( ain − V T HV ) , ain = [ ah1

 A+ 

0  = V T  3×3  03×3

By substituting Eqs. (20c) and (21g) into Eqs. (20a) and (21a) respectively, an inverse acceleration ain, a forward acceleration A, and a Hessian matrix H are derived as:

By differentiation of Eq. (13b) with respect to time, the acceleration ali along the ith horizontal leg is derived as below: ali = δ iT 

T

  = 1 V T 03×3  li 03×3

i

0 0  T ( ei × ςi )  = V T 03×3 S e 3×S3 ς  = V T H i , ( i ) ( i )  3×3 03×3 03×3  (20c) Hi =   , (i = 1, 2, 3). 03×3 S ( ei ) S (ςi )  6×6

(21e)

Integrating Eqs. (21d) and (21f), we get the following equation:

T

From Eq. (20b), we obtain:

T

From Eq. (21e), the equation is deduced as:

= ( S ( ei ) S ( ei ) ω ) = ω T S ( ei ) S ( ei ) . (20b)

T

=  − S (δ i ) (ω × ei )  = = ( S (δ i ) S ( ei ) ω ) = ω T S ( ei ) S (δ i ) ,

By differentiation of Eq. (13a) with respect to time, the acceleration ahi along the ith vertical leg is derived as below:

ahi = ςi T 

T

H = [ H1

ah 2 H2

ah 3

al1

al 2

al 3 ] ,

H3

H4

H5

H 6 ] . (22)

T

T

At this point, the kinematic model of the 3CPS parallel manipulator is established. 3 NUMERICAL SIMULATIONS AND EXPERIMENTAL VERIFICATION 3.1 Numerical Simulations The designed structural parameters of the developed active adjusting platform are noted in Table 1 and the set variations of the six independent pose parameters of the moving platform (Xo, Yo, Zo, α, β, γ) are shown in Fig. 2. The corresponding Rodrigues parameters Φ1, Φ2 and Φ3 vary continuously with time. Following these variations, a numerical simulation of the inverse kinematics is implemented using the Matlab program at 0.01 s intervals.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 291-300

a)

b) Fig. 2. Variations in the six pose parameters; a) position variations (Xo, Yo, Zo), and b) orientation variations (α, β, γ)

a)

a)

b) Fig. 3. Length variations in the driving legs; a) length variations in the vertical legs, and b) length variations in the horizontal legs

b) Fig. 4. Velocity variations in the driving legs; a) velocity variations in the vertical legs, and b) velocity variations in the horizontal legs

The length variations in the vertical legs and in the horizontal legs are shown in Fig. 3. The length ranges of the active legs are all within the design limits. Combined with the structural parameters, the driving legs have no structural interference during movement. The length variations of active legs approximate to 296

the simple harmonic curves and this characteristic benefits the control of the moving platform. The velocity variations of the active legs according to Fig. 3 are shown in Fig. 4, and the corresponding acceleration variations are shown in Fig. 5. Fig. 4a and Fig. 5a indicate that the vertical

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a)

b) Fig. 5. Acceleration variations in the driving legs; a) acceleration variations in the vertical legs, and b) acceleration variations in the horizontal legs

legs h1 and h2 have approximately the same velocity and acceleration variation ranges, while that of leg h3 is slightly smaller than the other two. Fig. 5 indicates that all three horizontal legs have large variation ranges of acceleration compared to the vertical legs. The large variation ranges in the acceleration increase the demand for the dynamic performance of the driving motors, especially for that of the horizontal legs. The curves of velocities and accelerations vary smoothly and have no abrupt change points. Therefore, there is no rigid shock in the motion of the moving platform. This is important for the dynamic behaviour of the moving platform.

orientations of the moving platform in this paper [21] and [22]. Single camera solutions are of interest in restricted applications, such as the restrictions in terms of costs, synchronisation demands or spatial observation conditions.

Table 1. Main parameters of 3CPS parallel manipulator Structural parameters of the manipulator Installation Radius of B (E) Installation Radius of m (e) Minimum length of vertical legs Maximum length of vertical legs Minimum length of horizontal legs Maximum length of horizontal legs

Value [mm] 250 200 170 210 30 250

3.2 Experimental Verification Photogrammetric measurement based on computer vision and image processing technology has become an excellent measurement method. It has been widely used in the industrial and scientific fields, such as in industrial inspection, reverse engineering, and robot vision systems [20]. In order to verify the correctness of the theoretical kinematics model, the photogrammetric pose measurement method with a single camera is used to measure the positions and

Fig. 6. Test environment for photogrammetric measurement

The test environment for photogrammetric measurement is shown in Fig. 6. The moving platform of the 3CPS parallel manipulator is made of a hexagonal glass plate, which is covered by a black and white grid pattern. The pose is measured by using a NIKON D80 camera after calibrating using free software calib-for-matlab. Considering the image size and processing speed, a medium image resolution (2896Ă—1944 pixels) is used in the experiment. In addition, the computer is used to control the camera shutter and the ambient temperature is monitored to ensure the stability of the measuring process.

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The process for testing the kinematic performance as well as concrete steps for photogrammetry to measure a specific pose are listed as follows: a) connect all the components of the developed parallel manipulator as shown in Fig. 1; b) post the grid pattern, align with the center of glass plate, and check the motion of the motors; c) connect the camera to the computer, sample several different poses of the grid pattern and calibrate camera; d) obtain the pose of the moving platform at a fixed time according to the designed trajectory and sampled frequency; e) substitute the pose into the theoretical kinematic model and calculate the corresponding leg length; f) control the motor according to the calculated leg length, use a height gauge, and vernier caliper to measure the length of the legs and shoot the grid paper at the fixed pose; g) calculate the poses of the moving platform by using the calibrated parameters of the camera.

a)

process. The purpose of the experiment is to verify the theoretical kinematics model in this paper. Considering the efficiency and cost required for the measurement, it does not make sense to test one by one and cover the whole range of calculations. To compare the experimental results with the designed trajectories intuitively, only the errors of position and orientation between two groups from 0 to 0.1 s with a sampling time at 0.01 s are shown in Fig.7. Therefore, there are 11 groups of data in the error sequence. Errors exist between the designed trajectories and the experimental results. Fig. 7 shows that the position error is about 0.2 mm and the orientation error is about 0.3°. An initial analysis of causes of the errors is as follows: 1) Errors of measurement tool: The grid pattern is printed using an inkjet printer and the paper is not flat due to the infiltration of ink. The sizes of the grid are also different. After measurement, the average size of the grid is 40.07×40.23 mm. However, a nominal size 40.00×40.00 mm is used in the simulation process. There are also differences between the coordinate system of the grid pattern and the moving platform due to the placement error. 2) Differences between the physical prototype and the simulation model: Manufacturing and assembly errors, which are inevitable, exist in the physical prototype, while nominal ideal size is used in the simulation process. Moreover, the control system has no automatic feedback. 3) Influences due to the environment: During the experiment, changes in ambient temperature, vibration and even light have a great impact on camera calibration and measurement accuracy. All the errors mentioned above will cause pose errors of the moving platform in the photogrammetric measurement. However, considering the errors of the measurement tools, the differences between the physical prototype and the simulation model, as well as environmental influences, the comparison results in Fig. 7 show that the theoretical value and the experimental results agree very well in a generally manner and verify the correctness of the kinematics model. 4 CONCLUSIONS

b)

Fig. 7. Error between designed trajectories and experimental results; a) error of position, and b) error of orientation

In order to get a smooth curve, we selected sampling time at 0.01 s. Thus, we obtain 100 groups of data within a period of 1 s during the simulation 298

A prototype of the 3CPS active adjusting platform for a segment mirror with 6 DOFs has been established. This prototype has partial DOF-decoupling motion characteristics and the mobility properties of the manipulator are analysed. The kinematics of the manipulator is studied in depth. The derived

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 291-300

final formulae for solving the inverse/forward displacement, velocity, and acceleration of the 6 DOFs parallel active adjusting platform are quite simple. According to the topological structure, the characteristics of the manipulator and the designed trajectories of the moving platform, the lengths, velocities, and accelerations of the six driving legs are constructed based on inverse kinematics by numerical simulation. Meanwhile, the analytic results of this parallel manipulator are verified by the experiments in which the pose of moving platform is measured using a photogrammetric method. Though errors exist between the designed trajectories and the experimental results, the experimental results prove the correctness of the kinematics model overall. 5 NOMENCLATURE {m} {B} Do hi (i=1,2,3) li (i=1,2,3) O o ai (i=1,2,3) Ai (i=1,2,3) Di (i=1,2,3) e E ς (i=1,2,3) δi (i=1,2,3) uo, vo, wo vhi (i=1,2,3) vli (i=1,2,3) V J ωCi(i=1,2,3) ahi (i=1,2,3) ali (i=1,2,3) A H

relative coordinate system o-xyz fixed on m absolute coordinate system O-XYZ fixed on B rotation transformation matrix from {m} to {B} active vertical leg and its length active horizontal leg and its length center point of B center point of m attachment points of m with surrounding legs attachment points of B with surrounding legs end point of the vertical legs distance from ai to o distance from Ai to O unit vector of hi unit vector of li unit vectors of x, y and z velocity of active vertical leg hi velocity of active horizontal leg li the forward general velocity, V=[v ω]T Jacobian matrix angular velocities of vertical legs acceleration of active vertical leg hi acceleration of active horizontal leg li the forward general acceleration, A = [a ε]T Hessian matrix

6 ACKNOWLEDGMENTS Financial support for this work, provided by the National Natural Science Foundation of China (Grant No. 51275512), the Fundamental Research Funds for the Central Universities (Grant No. 2012LWB36), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, is gratefully acknowledged. 7 REFERENCES [1] Chang, J., Cao, J., Cheng, D., Feng, S., Jiang, H. (2011). The system design and assemble for the high resolution telescope system with segmented aperture. Optik-International Journal for Light and Electron Optics, vol. 122, no. 18, p. 1628-1632, DOI:10.1016/j. ijleo.2010.10.015. [2] Preumont, A., Bastaits, R., Rodrigues, G. (2009). Scale effects in active optics of large segmented mirrors. Mechatronics, vol. 19, no. 8, p. 1286-1293, DOI:10.1016/j.mechatronics.2009.08.005. [3] Shore, P., Cunninghamb, C., DeBra, D., Evans, C., Hough, J., Gilmozzi, R., Kunzmann, H., Morantz, P., Tonnellier, X. (2010). Precision engineering for astronomy and gravity science. CIRP Annals Manufacturing Technology, vol. 59, no. 2, p. 694-716, DOI:10.1016/j.cirp.2010.05.003. [4] Cheng, G., Yu, J., Ge, S., Zhang, S. (2011). Workspace analysis of 3SPS+1PS bionic parallel test platform for hip joint simulator. Proceeding of the Institution of Mechanical Part C - Journal of Mechanical Engineering Science, vol. 225, no. 9, p. 2216-2231, DOI:10.1177/0954406211404864. [5] Cheng, G., Gu, W., Yu, J., Tang, P. (2011). Overall structure calibration of 3-UCR parallel manipulator based on quaternion method. Strojniški vestnik Journal of Mechanical Engineering, vol. 57, no. 10, p. 719-729, DOI:10.5545/sv-jme.2010.167. [6] Harib, K., Sharif Ullah, A., Hammami, A. (2007). A hexapod-based machine tool with hybrid structure: Kinematic analysis and trajectory planning. International Journal of Machine Tools and Manufacture, vol. 47, no. 9, p. 1426-1432, DOI:10.1016/j.ijmachtools.2006.09.021. [7] Terrier, M., Dugas, A., Hascoet, J. (2004). Qualification of parallel kinematics machines in high-speed milling on free form surfaces. International Journal of Machine Tools and Manufacture, vol. 44, no. 7-8, p. 865-877, DOI:10.1016/j.ijmachtools.2003.11.003. [8] Chanala, H., Duca, E., Raya, P., Hascoe, J. (2007). A new approach for the geometrical calibration of parallel kinematics machines tools based on the machining of a dedicated part. International Journal of Machine Tools and Manufacture, vol. 47, no. 7-8, p. 1151-1163, DOI:10.1016/j.ijmachtools.2006.09.006.

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[9] Riccardi, A., Xompero, M., Zanotti, D., Busoni, L. (2008). The adaptive secondary mirror for the Large Binocular Telescope: results of acceptance laboratory test. Proceedings of the SPIE, vol. 6, p. 7015-7037, DOI:10.1117/12.790527. [10] Zago, L., Genequand, P., Moerschell, J. (1998). Extremely compact secondary mirror unit for the SOFIA telescope capable of 6-degree-of-freedom alignment plus chopping. Proceedings of the SPIE, vol. 3352, p. 666-674, DOI:10.1117/12.319276. [11] Schipani, P., Marty, L. (2006). Stewart platform kinematics and secondary mirror aberration control. Proceedings of the SPIE, vol. 6273, p. 1026-1237, DOI:10.1117/12.670537. [12] Cheng, G., Yu, J., Gu, W. (2012). Kinematic analysis of 3SPS+1PS bionic parallel test platform for hip joint simulator based on unit quaternion. Robotics and Computer Integrated Manufacturing, vol. 28, no. 2, p. 257-264, DOI:10.1016/j.rcim.2011.09.007. [13] Lu, Y., Shi, Y., Huang, Z., Yu, J., Li, S., Tian, X. (2009). Kinematics/statics of a 4-DOF over-constrained parallel manipulator with 3 legs. Mechanism and Machine Theory, vol. 44, no. 8, p. 1497-1506, DOI:10.1016/j. mechmachtheory.2008.12.001. [14] Gallardoa, J., Lessoa, R., Rico, J., Alici, G. (2011). The kinematics of modular spatial hyper-redundant manipulators formed from RPS-type limbs. Robotics and Autonomous Systems, vol. 59, no. 1, p. 12-21, DOI:10.1016/j.robot.2010.09.005. [15] Cui, H., Zhu, Z., Gan, Z., Brogardh, T. (2005). Kinematic analysis and error modeling of TAU

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parallel robot. Robotics and Computer Integrated Manufacturing, vol. 21, no. 6, p. 497-505, DOI:10.1016/j.rcim.2004.07.018. [16] Varedi, S., Daniali, H., Ganji, D. (2009). Kinematics of an offset 3-UPU translational parallel manipulator by the homotopy continuation method. Nonlinear Analysis: Real World Applications, vol. 10, no. 3, p. 1767-1774, DOI:10.1016/j.nonrwa.2008.02.014. [17] Zhang, Y., Mu, D. (2010). New concept and new theory of mobility calculation for multi-loop mechanisms. Science China Technology Science, vol. 53, no. 6, p. 1598-1604, DOI:10.1007/s11431-010-3100-y. [18] Zhou, J., Miao, Y., Wang, M. (2004). Attitude representation using Rodrigues parameter. Journal of Astronautics, vol. 25, no. 5, p. 514-519. (in Chinese) [19] Craig, J. (2005). Introduction to robotics: mechanics and control, 3rd ed. Prentice Hall/Pearson Press, New York. [20] Papa, G., Torkar, D. (2009) Visual control of an industrial robot manipulator: accuracy estimation. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 12, p. 781-787. [21] Li, H., Yang, D., Zhai, C. (2010). Research on the pose measurement of a 6-DOF platform using a single camera. Optical Technique, vol. 36, no. 3, p. 344-349. (in Chinese) [22] Luhmann, T. (2009). Precision potential of photogrammetric 6 DOF pose estimation with a single camera. ISPRS Journal of Photogrammetry and Remote Sensing, vol. 64, no. 3, p. 275-284, DOI:10.1016/j. isprsjprs.2009.01.002.

Cheng, G. – Xu, P. – Yang, D. – Li, H. – Liu, H.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 301-310 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.669

Original Scientific Paper

Received for review: 2012-06-18 Received revised form: 2012-11-23 Accepted for publication: 2013-01-17

Evaluation of Fluid Flow over Stepped Spillways Using the Finite Volume Method as a Novel Approach

Vosoughifar, H.R. – Dolatshah, A. – Sadat Shokouhi, S.K. – Hashemi Nezhad, S.R. Hamid Reza Vosoughifar1,* – Azam Dolatshah2 – Seyed Kazem Sadat Shokouhi1 – Seyed Reza Hashemi Nezhad1 1 Islamic Azad

University, South Tehran Branch, Department of Civil Engineering, Iran University, Dezful Branch, Department of Civil Engineering, Iran

2 Islamic Azad

This research deals with the development of a Computational Fluid Dynamics (CFD) code called V-Flow using MATLAB for the two-dimensional modelling of unsteady flow over stepped spillways. V-Flow can be coupled with GAMBIT software and different spillway geometries can be modelled using voronoi mesh elements. The governing equations of flow over stepped spillways were discretized using the Finite Volume Method (FVM). The Power-Law scheme, implicit time approximation, Gauss–Seidel method, and SIMPLE algorithm were used for the discretization procedure. The flow was considered to be a laminar fluid flow with no turbulence model. The V-Flow model was validated against velocity vectors, streamlines, static pressures, dynamic pressures, and total pressures over the spillway obtained from the FLUENT model application. The experimental model simulated using both V-Flow and FLUENT for validation according to Gonzalez’s experiments. A comparison between the results obtained from the application of the different models showed good agreement according to the mass imbalance, which serves as both a useful indicator of the convergence of the fluid flow solution and as an accuracy criterion to compare the V-Flow and FLUENT results. Keywords: stepped spillways, finite volume method, discretization, voronoi mesh, numerical modelling, roller compacted concrete

0 INTRODUCTION A stepped spillway is a hydraulic structure, which is an integrated part of the dam, that allows the safe passage of overtopping flows. Due to increased air entrainment (i.e. aeration flow), stepped spillways dissipate the energy of the flow. This means that they reduce the level of energy in dissipater structures downstream from the spillway and lower the risks of cavitation. In recent years, the development of the Roller Compacted Concrete (RCC) technique has improved stepped spillways due to the low-cost and relatively high-speed construction this method enables. Most studies investigating the flow over stepped spillways have been undertaken utilizing both laboratory experiments on scaled models and analytical and numerical approaches. For example, Degoutte et al. [1] suggested that the energy dissipation is higher in the jet flow than the skimming flow. They also discovered that there are two types of jet flow: one with fully developed hydraulic jumps and the other one with partially developed hydraulics jumps. Gonzalez and Chanson conducted an experimental study to gain a better understanding of the flow properties in stepped chutes with slopes typical of embankment dams. Their work yielded a new design procedure including some key issues not foreseen in prior studies [2]. Pfister and Hager presented visual observations made with a high-speed camera and air concentration measurements in the vicinity of

the pseudo-bottom air inception point on a stepped model spillway [3]. Felder and Chanson conducted a physical study in a moderate slope-stepped chute and tested five configurations. The results were compared in terms of flow patterns, energy dissipation, and flow resistance [4]. Chanson and Felder also studied the two-phase gas-liquid flow properties of highvelocity open channel flows in a large-size channel experimentally. The results demonstrated high levels of turbulence in the high-speed, highly turbulent freesurface flows [5]. Barani et al. [6] used the feasible direction method and a wooden physical model to determine the optimum slope and step height of a stepped spillway and to calculate the dissipated energy of the flow. Si-ying et al. [7] studied the effect of aerator type on countering cavitation damage in the stepped chute at the Murum Hydraulic Power Station. They optimized the shape of the aerator and observed the aeration effects by hydraulic modelling. With the advent of high-performance computers and the development of robust Computational Fluid Dynamic (CFD) software, the development of complementary analytical tools for resolving the intricacies of the flow pattern have become feasible, for examples see Chen et al. [8], Tabbara et al. [9], and Naderi Rad and Teimouri[10]. Carvalho and Martins [11] also investigated hydraulic jumps on the steps of stepped spillways analytically, physically, and numerically. They designed a conceptual prototype using classic hydraulic formulae. A large-scale model was adapted and an experimental study was conducted

*Corr. Author’s Address: Islamic Azad University, South Tehran Branch, Tehran, Iran, vosoughifar@azad.ac.ir

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to examine the similarity of hydraulic jumps at each step, minimizing the hydraulic jump length, and maximizing the discharge per unit width. A numerical model based on the Two-Dimensional (2D) Reynoldsaveraged Navier-Stokes (RANS) equations was used for evaluating velocities and pressures and for characterizing the hydraulic forces on the baffles and sills. Bombardelli et al. [12] presented and discussed the results of a comprehensive study addressing the non-aerated region of the skimming flow in steep stepped spillways. They used a relatively large physical model of the spillway to acquire data on flow velocities and water levels. Numerical simulations using a commercial code were then performed to reproduce those experimental conditions. In addition, Hanbay et al. [13] created two intelligent models to predict flow conditions and aeration efficiency in stepped cascades using critical flow depth, step height, and channel slope. Salmasi [14] developed a procedure using an Artificial Neural Network (ANN) for calculating the energy dissipation of flow over a stepped spillway chute. The work was based on a wide variety of experimental facilities in large-size models. In this paper, the advantages of the CFD software are considered and a program written to investigate the flow pattern problems. In general, the Finite Volume Method (FVM) and voronoi mesh grid are used to solve the governing equations and to estimate the flow characteristics over stepped spillways, especially the velocity vectors, streamlines, static pressures, dynamic pressures, and total pressures over steps. 1 RESEARCH METHODOLOGY 1.1 Governing Equations

1.2 Numerical Modelling Algorithm

The 2D governing equations of a Newtonian, viscous, incompressible, unsteady state and turbulent flow can be described via the Continuity and NavierStokes equations, Eqs. (1) and (2), respectively [15], whereas the last term of the Navier-Stokes equation is eliminated when describing the laminar flow state. → →

∇ .V = 0, (1)

ρ

    ∂ϕ + ρV ⋅∇ϕ = −∇P + ρ gi + µ∇ 2ϕ − ρ∇ ⋅ u ′v′. (2) ∂t

  where V , ρ, ϕ, t, P, gi , μ, u' and v’ are the velocity vector, water density, velocity vector components in the x or y direction (ϕ = u, v), time, pressure, gravity acceleration in the x or y direction (i = x, y), 302

dynamic viscosity and fluctuating values of velocity vector components in the x or y direction (ϕ' = u', v') respectively. An additional term called Reynolds stresses − ρ ∇ ⋅ u ′v′ ) in the right side of Eq. (2) arises from the ( time-averaged equations for turbulent flow compared to laminar flow. This is due to an averaging operation in which it is assumed that there are random fluctuations about the mean value. It means that ‘ φ = φ + φ' and P = P + P′ , while φ and P are the mean values of the velocity vector components in the x or y direction ( φ = u , v ) and the pressure, respectively. In addition, ϕ' and P' are fluctuating values of the velocity vector components and pressure around their mean values, respectively. However, as mentioned before, if the last term is ignored, the Navier-Stokes equation represents the laminar flow state. Many turbulence models such as the k-ε and Renormalization Group (RNG) models employ the concept of a turbulent viscosity to express the turbulent stresses. The result is that the time-averaged equations for turbulent flow have the same appearance at the equations for laminar flow, except that the laminar exchange coefficient such as viscosity is replaced by the effective exchange coefficient. However, only laminar flow was considered in this research, i.e. no turbulent model was taken into account. Instead, attempts have been made to link the pressure and velocity fields somehow and to correct them during the solution procedure for the discrete governing equations using the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm [16]. The main features of this process are explained in the following section.

In Finite Volume Method (FVM), the integration domain is covered with control volumes. Each control volume surrounds a node, which lies on a grid mesh. Mesh grids are generally divided into structured and unstructured meshes. Due to its generality, the FVM can handle any type of mesh, structured as well as unstructured. Unstructured grid methods employ an arbitrary collection of elements to fill the domain. For instance, the voronoi and Delaunay are types of unstructured meshes. Dirichlet [17] used a special form of the voronoi tessellation in his study of positive quadratic forms. Voronoi later published a generalization of this concept that would apply to higher dimensions and so introduced the concept in its modern form. A Delaunay triangulation is created

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by several algorithms such as the Watson-Bowyer and Step-By-Step Algorithms [18]. In this paper, the studied domain was discretized using the FVM for solving the discrete governing equations and the unstructured voronoi mesh grid established using the MATLAB software [19]. In this approach, the domain was divided into several distinct control volumes without any overlapping. By integrating the governing differential equations over every control volume, an algebraic equations system was created for every single control volume, and each equation links the parameter investigated in a control volume node to different numbers of adjacent nodes, consequently the parameter was computed in each node [20]. In order to solve discrete equations the parameter investigated in each node was computed considering its discrete equation and newest adjacent nodes’ values. The solution procedure can be expressed as follow: • Initial guess for the investigated parameter (i.e. ϕ) in each node at initial conditions. • Calculation of the value in an investigated node considering its discrete equation. • Performing previous step for all nodes in the studied domain; one cycle is performed by repetition of this step. • Verification of the convergence clause. If this clause is satisfied, the computing will end, otherwise the computation will be repeated from the second step. The Riemann boundary condition was also utilized for computing values in boundaries, because the exact values of the boundary conditions were not distinct. Therefore, we can assume that a layer which is near the boundary layer, where ∂ / ∂x = 0 and ∂ / ∂y = 0, is defined and that the calculated value for the boundary adjacent nodes is transformed into the related boundary nodes. This procedure will continue until the difference in results converges to zero.

where; anb = D f A(|

Ff

|) + [[0, − Ff ]], (5)

Df

a0" = ∑ ( anb + F ) f , (6)

f

 Df =   

µA

( xnb − x0 ) + ( ynb − y0 ) 2

2

  ,  (7) f

Ff = ( ρ vε A ) f . (8)

The general discrete governing equation of the unsteady state flow problem was also discretized using implicit time approximation as shown in Eq. (9).  a0ϕ0 = ∑ ( anbϕ nb ) f − ∑ PA f

f

( )

f

 ρ ∆ϑ + ρ g ∆ϑ + 0 ϕ00 , (9) ∆t

where;

 F  anb = D f A  f  + [[0, − Ff ]]. (10)  Df   

and are defined in Eqs. (7) and (8), respectively. The two general discrete equations for evaluating the velocity components in two directions (u, v) can be determined from Eq. (9), where the velocity components (u, v) need to be inserted into the equation instead of ϕ. Furthermore, other changes due to different coordinate directions (x, y) should be also considered.

1.3 Discretization of Governing Equations Discretization of the Continuity and Navier-Stokes equations was performed utilizing the FVM. Fig. 1 illustrates the 2D voronoi mesh grid used for describing these equations. The discrete Continuity and the unsteady state Navier-Stocks equations were discretized as shown in Eqs. (3) and (4), respectively.

∑( F )

f

f

= 0, (3)

 a0"ϕ0 = ∑ ( anbϕ nb ) f − ∑ PA f

f

( )

f

Fig. 1. The 2D schematic voronoi mesh cell used to describe the discretization of the governing equations

 + ρ gi ∆ϑ , (4)

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1.4 Pressure and Velocity Correction Equations and Mass Imbalance Once the two general discrete velocity equations have been derived, we can attempt to solve the different terms and parameters of these equations. The two obtained equations, which evaluate the velocity components in two directions, should satisfy the continuity equation. Thus, the continuity equation will be obtained as shown in Eq. (11). Fig. 2 shows the velocity components of the 2D schematic voronoi cell used to describe the discretization of the governing equations. Ax2 ρ 2 g y ∆y f ρ A2f f Ff = ( ρ vε A ) f = P − P + . (11) ( 0 1) af af

( )

ρ A2f , (13) af

( anb′ ) f =

′ ) f , (14) a0′ = ∑ ( anb f ,0

b = −∑ Ff* . (15) f ,0

The mass source, b, serves as a useful indicator of the convergence of the fluid flow solution. If the face flow rates, F*, satisfy the discrete continuity equation, b will be zero. Indeed the b term is a measure of the resulting mass imbalance. The pressure correction equation corrects the pressure and velocity fields to ensure that the resulting field annihilates this mass imbalance [21] and [16]. In this research, the mass imbalance was considered to be a factor indicating the accuracy of the results. The results were finally studied and compared with each other according to the mass imbalance factor at the end of the modelling practice. 1.5 Numerical Model

Fig. 2. Velocity components of the 2D schematic voronoi cell for describing the discretization of the governing equations

With regard to the SIMPLE algorithm, let u* and v* be the discrete u and v fields resulting from a solution of the discrete u and v momentum equations. In addition, let P* represent the discrete pressure field that is used in the solution of the momentum equations. If the pressure field p* is only a guess or a prevailing iterate, the discrete u* and v* obtained by solving the momentum equations will not, in general, satisfy the discrete continuity equation. So therefore we should let p = p* + p'. In addition, to know how the velocity components respond to this change in pressure, we should let u = u* + u' and v = v* + u'. These corrected values satisfy Eq. (11). Finally, Eq. (12) can be obtained for the evaluation of the pressure correction.

′ pnb ′ ) f + b, (12) a0′ p0′ = ∑ ( anb f ,0

304

In this study, a finite volume program called V-Flow was developed using the MATLAB programming for 2D modelling of unsteady flows over stepped spillways. V-Flow can discretize the domain using the unstructured voronoi mesh grid with different geometries, boundary conditions, and arbitrary vertices. V-Flow can be connected to the GAMBIT software [22] to model the arbitrary and complicated geometry. The study domain was modelled in GAMBIT using triangular elements. Then, the coordinates of the created nodes (not the created triangular mesh grid) were exported from GAMBIT and imported into V-Flow. V-Flow created the Delaunay triangulation based on the coordination of the grid. Then, it created the voronoi mesh grid according to the relationship between the Delaunay triangulation and voronoi mesh generation. Generally, the Power-Law [16] and fully implicit schemes were used for discretization of the governing algebraic equations. Although it is somewhat more complicated than other schemes, the Power-Law scheme expressions are not particularly expensive to compute and they provide an extremely good representation of the exponential behaviour. However, when the general discrete equation is related as Eq. (4) and anb is

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described as Eq. (5), then A( | Ff / Df | ) can be written as Eq. (16) in terms of the Power-Law scheme.

A(|

Ff Df

|) = [[0,(1 − 0.1 |

Ff Df

|)5 ]]. (16)

In addition, the iterative solution Gauss–Seidel method was employed to solve the algebraic equations set resulting from the discretization. The SIMPLE algorithm was also used for the pressure correction equation. The flow was considered to be a laminar fluid flow and no turbulent model was applied. Consequently, V-Flow can evaluate the velocity and pressure field in both directions and it can display the flow pattern over stepped spillways. On the other hand, the FLUENT CFD software [23] was implemented as a reference in order to evaluate the precision of V-Flow. FLUENT also uses the FVM to solve the governing equations. It provides physical models on different types of unstructured 2D/3D meshes. Its modelling and grid generation is conducted using GAMBIT. Here the SIMPLE algorithm, Power-Law scheme, and the segregated solver in the laminar flow state were used to investigate the verification model compared with the V-Flow model. The results obtained from both V-Flow and FLUENT were then compared with each other, especially where the mass imbalance was concerned. 1.6 Case Study To test the precision of the V-Flow code developed in this study, a physical model of a large stepped spillway with flat horizontal steps was considered for simulation of the flow properties using the V-Flow code and FLUENT software. This broadcrested stepped spillway physical model was created by Gonzalez [24] at the hydraulics laboratory of University of Queensland, Australia. The model is 3.15 m long, 0.9 m high and 1 m wide. It has 9 steps and each step is 0.1 m high and 0.35 m long. The slope angle of the chute is 15.94°, the length of the crest is 0.617 m from the upstream face of the spillway to the tip of the first step, and the total length of the model is 4 m including upstream and downstream in addition to the spillway. In this paper, the broad-crested profile has an angle steepness of 15.94° which is equal to the chute slope angle, because the V-Flow was prepared to model the non-free surface flow. Therefore, it was essential to change the crest of the model.

1.7 Boundary Conditions Boundary conditions comprise four boundaries. The first boundary condition is the area beneath the steps made from soil. This is a non-slip wall boundary condition. The second one is the inlet flow boundary just at the left face of the spillway. This boundary is defined as the velocity inlet boundary condition. The total velocity, which is parallel to the slopping crest, is considered to be 2 m/s and the water depth is assumed to be 0.208 m. The third boundary condition is the air pressure boundary condition, which is a line parallel to the chute at top of the flow region over the spillway. The pressure was defined as Pa. The fourth one is the outflow boundary condition just at the right face of the spillway and both the velocity components and pressure are undefined. 1.8 Convergence Criterion In statistics, the Mean Absolute Error (MAE) is a quantity used to measure how close predictions are to the eventual outcomes. It is a common measure of forecast error in time series analysis. The Mean Absolute Error is given by Eq. (17).

MAE =

1 n ∑ fi − ei , (17) n i =1

where fi is the prediction and ei is the true value. In this paper, MAE has been considered equal to 10–6 in order to lead to convergence for both V-Flow and FLUENT. 2 RESULTS 2.1 Prepared Code A large stepped spillway physical case study was modelled using GAMBIT software. The coordination nodes of the triangular mesh grid were imported into the V-Flow code. A Delaunay triangulation was produced based on these nodes and a voronoi mesh grid was consequently created utilizing the Delaunay triangulation. Fig.3 illustrates the Delaunay triangulation and voronoi mesh generation in V-Flow. The model was run using the V-Flow code and Figs. 4 to 8 show the results of the analysis procedure, which include velocity vectors, streamlines, static pressure, dynamic pressure and total pressure contours over the spillway.

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Fig. 3. The Delaunay triangulation and voronoi mesh generation created by V-Flow

Fig. 4. Velocity vectors over the stepped spillway obtained using V-Flow

Fig. 5. Streamlines and static pressures over the stepped spillway obtained using V-Flow

Fig. 6. Static pressure contours over the stepped spillway obtained using V-Flow

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Fig. 7. Dynamic pressure contours over the stepped spillway obtained using V-Flow

Fig. 8. Total pressure contours over the stepped spillway obtained using V-Flow

Fig. 9. Velocity vectors over the stepped spillway obtained using FLUENT

Fig. 10. Streamlines over the stepped spillway obtained using FLUENT Evaluation of Fluid Flow over Stepped Spillways Using the Finite Volume Method as a Novel Approach

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2.2 CFD Software for Validation

2.3 Comparison

The large stepped spillway physical case study was simulated using FLUENT software for validation of the V-Flow code. The GAMBIT software was used for triangular mesh generation which is similar to the Delaunay triangulation for V-Flow. Figs. 9 to 13 show velocity vectors, streamlines, static pressure, dynamic pressure and total pressure contours over the spillway.

Nodes, mesh grids, velocity vectors, streamlines, static pressures, dynamic pressures, and total pressures over the spillway are shown in Figs.3 to 13 for both V-Flow and FLUENT. Total pressure contours indicate a decline in pressure of 0.1043 to 0.1008 MPa from the fluid surface to the tip and vertical face of steps. During the solution procedure, as the obtained

Fig. 11. Static pressure contours over the stepped spillway obtained using FLUENT

Fig. 12. Dynamic pressure contours over the stepped spillway obtained using FLUENT

Fig. 13. Total pressure contours over the stepped spillway obtained using FLUENT

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responses become closer to final responses, the mass imbalance at each control volume will be at its lowest. So, the lowest mass imbalance leads to achieving the best result. Table 1 shows the mass imbalance values for both V-Flow and FLUENT. As it can be seen, the relative mass imbalance on average for an individual cell is 0.0000213 or 0.0000329 for V-Flow or FLUENT, respectively. The difference between these two codes is about 0.0000129, and V-Flow provides a mean mass imbalance less than that of FLUENT. A small difference is observed between the maximum and minimum mass imbalance values of V-Flow compared to FLUENT. The sum total of the mass imbalance at all control volumes can be used as a criterion to determine whether the continuity equation is satisfied. This relative sum total is 0.04331 and 0.0626 in the laminar flow state for V-Flow and FLUENT, respectively. Finally, Table 2 shows a comparison between some other properties of V-Flow and FLUENT, such as mesh type, number of elements, and iterations. Table 1. Comparison between mass imbalance values Mass Imbalance (b)

V-Flow FLUENT

Minimum

Maximum

Mean

-0.00997 -0.02157

0.00571 0.00961

0.0000213 0.0000324

Sum Total for all cells 0.04331 0.0626

Table 2. Comparison between some basic properties of V-Flow and FLUENT Mesh grid type Number of elements [-] Maximum time [s] Time step[s] Iterations after several time steps [-] Mean Absolute Error (MAE)

V-Flow Voronoi 2032 11 0.0001 600 10–6

FLUENT Triangular 3692 14 0.000178 3000 10–6

3 CONCLUSIONS The FVM with the Power-Law scheme and implicit time approximation can discretize the governing equations of the 2D unsteady laminar incompressible flow problem over stepped spillways to prepare a CFD numerical code named V-Flow using MATLAB programming. V-Flow can be coupled with the GAMBIT software and used to model different geometries with voronoi mesh elements. According to the results, velocity vectors and streamlines illustrate re-circulating vortexes at the corner of steps in both V-Flow and FLUENT, as

they were observed in experiments performed by Gonzalez [24]. This is an important property as a result of establishing the skimming flow regime over stepped spillways, which can be simulated using V-Flow precisely. On the other hand, V-Flow was prepared using a no turbulence model, whereas the recirculating vortices clearly appeared at the corner of steps. An important property of V-Flow is to reduce the procedure solution. Furthermore, the results demonstrate that the static pressure significantly increased where the streamlines are in contact with the horizontal face of steps. Static pressure contours show an increase in pressure from the fluid surface to the bottom. Dynamic pressure contours show that the pressure is higher in regions where the velocity is high. Increasing the total pressure at the tip and vertical face of steps raises the risk of cavitation in this area compared to others. Despite the fact that the number of elements used by the V-Flow code is less than that used by FLUENT, the V-Flow results are very precise. This may be due to the efficiency of the voronoi mesh grid in modelling the geometry and to the discretized equations. The node that is being studied in the voronoi mesh grid is at the centre of the cell and it is considered to be the central node of the computational control volume. This is not the case for the triangular mesh grid used by FLUENT due to the placement of the nodes at the tip of the triangular control volumes. It is essential to transfer the nodes that are on boundaries from the tip of triangles to the middle of the sides using some approximations. Therefore, voronoi elements can model the geometries better than triangular elements. The mass imbalance criterion can be used to investigate the precision of the results and the convergence during the iterative solution. Both V-Flow and FLUENT give precise results due to a very small mass imbalance and similar output results due to a very small mean mass imbalance difference. A small difference was observed between the maximum and minimum mass imbalance values of V-Flow compared with FLUENT. This means that V-Flow can make a more uniform mass balance than FLUENT. The sum total of the mass imbalance of V-Flow is less than that of FLUENT. This may indicate that the velocity field obtained using V-Flow satisfies the continuity equation approximately three times better than the velocity field obtained using FLUENT. 4 REFERENCES [1] Degoutte, G., Peyras, L., Royet, P. (1992). Disscussion of skimming flow in stepped spillways by Rajaratnam.

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ASCE Journal of Hydraulic Engineering, vol. 118, no. 1, p. 111-114., DOI:10.1061/(ASCE)07339429(1992)118:1(111.2). [2] Gonzalez, C.A., Chanson, H. (2007). Hydraulic design of stepped spillways and downstream energy dissipaters for embankment dams. Dam Engineering, vol. 17, no. 4, p. 223-244. [3] Pfister, M., Hager, W.H. (2011). Self-entrainment of air on stepped spillways. International Journal of Multiphase Flow, vol. 37, no. 2, p. 99-107, DOI:10.1016/j.ijmultiphaseflow.2010.10.007. [4] Felder, S., Chanson, H. (2011). Energy dissipation down a stepped spillway with non-uniform step heights. Journal of Hydraulic Engineering, vol. 137, no. 11, p. 1543-1548, DOI:10.1061/(ASCE)HY.19437900.0000455. [5] Chanson, H., Felder, S. (2010). Turbulence measurements in air-water self-aerated flows: basic analysis and results. 7th International Conference on Multiphase Flow, Tampa. [6] Barani, G.A., Rahnama, M.B., Bagheri, H. (2005). Optimization of stepped spillway dimensions and investigation of flow energy dissipation over a physical model. Journal of Applied Sciences, vol. 5, no. 5, p. 878-882, DOI:10.3923/jas.2005.878.882. [7] Si-ying, W., Dong-mei, H., Cai-huan, W. (2012). Aerator of Stepped Chute in Murum Hydropower Station. Procedia Engineering, vol. 28, p. 803-807, DOI:10.1016/j.proeng.2012.01.813. [8] Chen, Q., Dai, G., Liu, H. (2002). Volume of fluid model for turbulence numerical simulation of stepped spillway overflow. ASCE Journal of Hydraulic Engineering, ,vol. 128, no. 7, p. 683-688, DOI:10.1061/ (ASCE)0733-9429(2002)128:7(683). [9] Tabbara, M., Chatila, J., Awwad, R. (2005). Computational simulation of flow over stepped spillways. Computer & Structures, vol. 83, no. 27, p. 2215-2224, DOI:10.1016/j.compstruc.2005.04.005. [10] Naderi Rad, I., Teimouri, M. (2010). An investigation of flow energy dissipation in simple stepped spillways by numerical model. European Journal of Scientific Research, vol. 45, no. 4, p. 544-553. [11] Carvalho, R., Martins, R. (2009). Stepped spillway with hydraulic jumps: application of a numerical model to a scale model of a conceptual prototype. ASCE Journal

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of Hydraulic Engineering, vol. 135, no. 7, p. 615-619, DOI:10.1061/(ASCE)HY.1943-7900.0000042. [12] Bombardelli, F.A., Meireles, I., Matos, J. (2011). Laboratory measurements and multi-block numerical simulations of the mean flow and turbulence in the non-aerated skimming flow region of steep stepped spillways. Environmental Fluid Mechanics, vol. 11, no. 3, p. 263-288, DOI:10.1007/s10652-010-9188-6. [13] Hanbay, D., Baylar, A., Batan, M. (2009). Prediction of aeration efficiency on stepped cascades by using least square support vector machines. Express System whit Application, vol. 36, no. 3, p. 4248-4252, DOI:10.1016/j.eswa.2008.03.003. [14] Salmasi, F. (2010). An artificial neural network (ANN) for hydraulics of flow on stepped chutes. European Journal of Scientific Research, vol. 145, no. 3, p. 450457. [15] Shaughnessy, E.J., Katz, I.M., Schaffer, J.P. (2005). Introduction to Fluid Mechanics. Oxford University Press, Oxford. [16] Patankar, S.V. (1980). Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York. [17] Dirichlet, G.L. (1850). Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Journal für die Reine und Angewandte Mathematik, vol. 40, p. 209-227, DOI:10.1515/ crll.1850.40.209. (in German) [18] Delaunay, B. (1934). On empty sphere; A memory of Georges Voronoi. Bulletin of the Academy of Sciences of the U.S.S.R., vol. 7, no. 6, p. 793-800. [19] MATLAB. (2011). MATLAB R2011a (7.12), User’s Guide, Mathworks, San Francisco. [20] Fletcher, C.A.J. (1991). Computational Techniques for Fluid Dynamics, Volume 2, 2nd ed., Springer Verlag, New York. [21] Murthy, J.Y. (2002). Numerical Methods in Heat, Mass, and Momentum Transfer. School of Mechanical Engineering, Purdue University, Purdue. [22] Gambit. (2007). Version 2.4.6 User’s Guide, Fluent, Lebanon, New Hampshire. [23] Fluent. (2009). Fluent User’s Guide. Ansys Inc., Lebanon. [24] Gonzalez, C.A. (2005). An Experimental Study of FreeSurface Aeration on Embankment Stepped Chutes. Ph.D. thesis, Department of Civil Engineering, The University of Queensland, Brisbane.

Vosoughifar, H.R. – Dolatshah, A. – Sadat Shokouhi, S.K. – Hashemi Nezhad, S.R.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 311-322 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.769

Original Scientific Paper

Received for review: 2012-05-21 Received revised form: 2012-10-16 Accepted for publication: 2012-11-29

Design for Reliability Based Methodology for Automotive Gearbox Load Capacity Identification Ognjanovic, M. – Milutinovic, M. Milosav Ognjanovic1 – Miroslav Milutinovic2,*

1 University

2 University

of Belgrade, Faculty of Mechanical Engineering, Serbia of East Sarajevo, Faculty of Mechanical Engineering, Bosnia and Herzegovina

Robust and Axiomatic design, a property-based approach in design, is applied and integrated into a new methodology for developing Functional Requirements (FR) or Design Parameters (DP). The reliability of the design structure and the elementary reliability of the design components are used as a functional requirement of the automotive gearbox, in relation to the service life and operation conditions, and also as a design constraint in analytical relationships. For this purpose, elementary reliability and allowable stress are defined in a specific way. The automotive gearbox, operating under varying and random operation conditions, is used as a case study. The same design structure has to operate under different operation conditions. In these circumstances, the carrying capacity as a functional requirement is related to the operation conditions and operation regime. The model presented in this paper, as well as a computer program, enable identification of this carrying capacity. This paper discusses an interdisciplinary and multi-methodological integrated approach to the presented task. Experimental data regarding the failure probability of gearbox components and the probability of operation conditions processing, the decomposition of gearbox structure, and the elementary reliability treatment as a component of design property are only some of the methods applied. Keywords: robust design, axiomatic design, gearbox, reliability

0 INTRODUCTION The functional requirements of design structures, for example, the load capacity, reliability, service life, etc. are strongly related to operation conditions that are, as a rule, random and very often uncertain. The basic idea is to apply a set of methods in Engineering Design in order to develop a methodology for load capacity identification with regard to operation conditions. An automotive gearbox has been taken for the case study, because it represents a design structure with very variable operation conditions. The reliability of structures such as gearboxes is one of the main indicators of quality and an important functional requirement. In order to fulfil this objective, various methods and approaches are taken into consideration, such as Robust and Axiomatic Design, Property-based design, Design for X, and various studies in the fields of automotive gearboxes, gears and gear failures, the reliability of gear transmission components, etc. Today, Axiomatic design provides a basis for analytic presentation of the relationship between functional requirements and design parameters. The Axiomatic principles developed by Suh, continue to perfectly for various applications [1], but this methodology is also suitable for combining with the Robust Design approach. On the other hand, the Property-based design approach together with the system structure and functional requirements or quality indicators deduction provides possibilities for arranging specific procedures and approaches for the design parameters or the carrying capacity

identification for design structures such as automotive gearboxes. Robust Design is a verified methodology, which provides good results on the first attempt. It also provides good design results (the relationship between functional requirements and design parameters) under variable operation conditions, and is applied for solving problems in many kinds of specific design cases. For horizontal axis wind turbines, the Robust design is carried out using the Taguchi method [2] and multi-objective optimization of the design parameters. The robustness of uncertain design variables can be provided by optimization using evolutionary algorithms [3]. The optimization is performed by analysing whether the solutions are truly robust and how robust optimal solutions differ from the performance maximizing solutions. Further perfection of robustness is provided for quality variables that vary over time [4]. Using the expectation of maximal quality loss over a time period, this work suggests the possibility of quantifying the robustness for timedependent quality characteristics. The reliability of technical systems such as automotive gearboxes is a typical time-dependent quality indicator that has to be treated in a similar way. In addition, the desired elementary reliability at the end of the service life of a design component can be applied as a design constraint for the design parameters definition [5] and [6]. The reliability of technical systems has been studied for a long time, basically for the sake of system maintenance, and has been oriented in two directions. The first direction is a continuation of the research for

*Corr. Author’s Address: University of East Sarajevo, Faculty of Mechanical Engineering, Vuka Karadzica 30,71123 East Sarajevo, B&H, m.milutinovic82@gmail.com

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maintenance necessities, such as reliability prediction [7], short-term reliability analysis of power generating units [8] in order to identify the actual reliability level of the power-generating unit, or the application of Natvig measures in the reliability identification of repairable systems [9]. The other direction is oriented towards the necessity for product development and design. This can be used for direct applications such as design optimization [10] in conditions of nonGaussian distribution, identification of reliability decrease in the course of service life [11] or investment analysis for increasing reliability of the technical system [12]. The decomposition of a technical system into components and the decomposition of their functional requirements and indicators of quality is an important step in both the reliability analysis and the design parameters definition. The decomposition of the system and relationships into a transformation matrix enables application of the axiomatic approach or determines the reliability of a certain component analysis. Very complex structures such as a vehicle [13] can be decomposed down to the level of design modules (engine, transmission, break system, etc.) and then into the level of their components (gears, bearings, coupling, etc.). The decoupling of the transformation matrixes is the main task of this process and the results are applicable in solving various design problems, such as stochastic problems in reliability [14], design parameters, functional requirements, and the quality indicators definition. Decomposition is also used in gear teeth wear analysis [15] and in the design components’ failure propagation [16]. The Design for X (DFX) [17] is oriented towards a certain aspect of product analysis or synthesis, which is the opposite of holistic or integrated approaches to product development and design. The property-based approach in design also takes into consideration a chosen property and provides the possibility of solving a certain problem in design [18]. The ”Design for X“ was the proposed implementation platform used in rapidly configuring the optimal design solution based on the company’s requirements in the case of refrigerators [19]. Further analysis continues with the property-based approach, where the elementary reliability of the structure component is the property used as a design constraint in the load capacity and design parameters identification. This is a continuation of articles [20] and [21]. The established relationships can then be used for direct or reverse engineering such as [22]. In order to connect automotive gearboxes as in case study [23], the results for the torque measurements of these structures [24] 312

and [25], as well as torque based vehicle speed control [26], are used. This paper offers a new and specific approach based on the integration of axiomatic and robust methodology directed towards load capacity identification of the existing design structure or the design parameters definition in the design process of the new mechanical structure. Reliability plays multiple roles, as a: functional requirement, design property, design constraint, indicator of quality, and indicator of design behaviour. Compared to other design optimisation approaches, the advantage of this particular one is the possibility of controlling the probabilistic variation in operation conditions and the fulfilment of design robustness, i.e. reaching design parameters or functional requirement (load capacity) insensitivity to the variation in operation conditions. 1 PROBLEM DEFINITION Basically, the main task of the design process is to create a relationahip between the product function, product structure, and product behaviour. The product function explains the effects of the system and creates a correlation between the input conditions and output effects. Product behaviour represents the interaction of the function with the environment and how the product fulfils its function. This is a result of the properties and characteristics of the assemblies and parts in relation to the environment and system use. In order to fulfil the functional requirements and to obtain the desired system behaviour, it is necessary to create a satisfactory design structure combining components (mechanical, electrical, information, etc.) in corresponding assemblies and system. One of the models for product structure creation is the V-model established by VDI-2206 as the “Design methodology of mechatronic systems“ which contains a synthesis of the system in order to obtain a design structure based on functional requirements, and an analysis aimed at integrating the system behaviour. This means a synchronized process of design structure processing and a functional requirements’ transformation into design structure behaviour. For this approach, known as Property-based design in [18], a monitoring system is established to provide the desired system behaviour starting from the defined functional requirements (Fig. 1). The reliability of technical systems is an important indicator of the system’s quality. In the clarification of the tasks design stage, reliability is one of the main functional requirements, and, in the operation process of a technical system, reliability is one of

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Fig. 1. V- Model of design process addapted for the property based product development with reliability as functional requirement, design property and as an indicator of system behaviour

the main indicators of quality and system behaviour. By decomposing the desired reliability of the system (functional requirement) to the design component level, the elementary reliability becomes the design property of the component (Fig. 1). According to [18], design properties of the design components are the result of the parts properties (intensive and extensive) and parts characteristics. These characteristics are the physical and chemical description of the material, geometrical (shape, dimensions,.etc.) and structural (joints and parts) interactions. Elementary reliability as a design property is a specific term in this paper, characterized by the following: (a) In the reliability system hierarchy it is connected to the technical system’s basic component. (b) As there is more than one possible failure in one component, the elementary reliability is connected to the top probable failure only. (c) This is a complex probability that contains the probability of operation conditions and the failure probability for certain operation conditions (see the case of gears in Eq. (1)). Reliability is one of the functional requirements of the system. An automotive gearbox is chosen as a case study for the presented approach, because it is complex in the functional, structural, and behavioural

regard. A set of functional requirements contains the load capacity, number of speed levels, reliability level, volume and mass value, vibration and noise level, handling suitability, etc. Load capacity is the main indicator of gearbox design and as such is the main aim of this paper and the developed approach. For the purpose of applying the axiomatic and robust design approach in further processing, the abovementioned group of functional requirements is divided into two groups. In order to simplify the axiomatic relationship, the load capacity, volume, and number of transmission stages are used as functional requirements. The rest of the functional requirements, such as reliability, vibration and noise level, are used for the design constraints definition in order to fulfil the robustness of the design process. The section on carrying (load) capacity identification will discuss this in more detail. For a given technical system (for example, the automotive gearbox), functional requirements can also be used as design constraints (Fig. 2) for the purpose of applying both the axiomatic and robust design. Members of matrix A are directly related to constraints related to operation conditions and selected functional requirements. In this case, the elementary reliability of gearbox components is the design constraint

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that provides robustness to the design process. The vibration and noise of a gearbox can be used as a design constraint in order to harmonize the interaction of design parameters, but this article does not address this aspect of the design.

Fig. 2. The relationship of functional requirements FR, design parameters DP, operation conditions, and constraints in the axiomatic and robust design of automotive gearboxes

couplings. The output and input shafts are coaxial and all gear pairs have the same centre distances.

Fig. 3. The automotive gearbox function structure

2 STRUCTURING OF THE SYSTEM The product model, i.e. the technical system model, displays chosen characteristics or properties in a form suitable for the desired analysis. The model also represents the structure and relationship of these characteristics and properties. The planned automotive gearbox analysis requires a few models, such as a functional one, then a model of the design structure, a model of the parts and assembly geometry, models of reliability of the system, and of elementary reliability, etc. A model of gearbox function is presented in Fig. 3. This is a gearbox with six speeds and an additional one for reverse motion. Input speed is transformed into slow speed and then for every output speed transforms back to a corresponding speed level. The last, sixth speed, is equal to input speed without transformation. Output speed for reverse motion is additionally reduced and the direction is reversed. The transformation of input power carries out the function marked by 0 and then for the I to V speed levels an additional transformation is carried out for the functions marked by 1 to 5 and also for the reverse motion by the function labelled R. The maximal speed level VI is the input speed without transformation. The control and choice of speed level is provided by the steering system with couplings c1 to c3 and cR, in interaction with the vehicle driver. The function structure transformed into organ structure (conceptual design) can be presented as shown in Fig.4. This structure consists of three shafts: the input, output and middle shaft. Gear pairs j = 0, 1, 2, …, R are located on these shafts together with the 314

Fig. 4. Organ structure (conceptual design) of the automotive gearbox

The structure of design components and assemblies is presented in Fig. 5. This is a 3D geometric model of a complete gearbox and its component representatives, the gear pair, coupling, shaft, bearing, eagling, etc. This structure allows one to create a reliability structure model. Compared to other reliability models, the model for axiomatic and robust design has to be created according to the following procedure and limitations. The technical system, i.e. the automotive gearbox, has to have the ability to be decomposed into components that can be completely replaced in case of damage. These are (Fig. 5): seven gear pairs, three couplings, five bearings, and two seals. In Fig. 6, a separate elementary reliability is recognized for each of these as a design property of the component (sub-assembly). The steering (st) sub-system is a specific assembly and Rst is avoided in this analysis of reliability. Total reliability R is the product of component reliabilities, i.e. elementary reliabilities Rgj, Rbj, Rcj, Rsej.

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Fig. 5. 3D model of automotive gearbox and its components used for load measurement and for case study

by wear or by teeth fracture, couplings by the damage of movable parts, and by the damage of the parts for the fixing position, etc. When there is a possibility of more than one failure in a component, the reliability of the component is equal to the lowest reliability of all elementary reliabilities for every failure separately. For example, the elementary reliability of the gear pair is equal to the elementary reliability with regard to teeth wear; the reliability of coupling is equal to the reliability with regard to the damage to the fixed position parts, etc. In this way, the total reliability of the system (gearbox) is decomposed to the level of probable damage to the component (sub-assembly) and presented in Fig. 6. This is a serial reliability structure composed of the above-mentioned gearbox component reliabilities. 3 A MODEL FOR THE GEAR PAIR ELEMENTARY RELIABILITY CALCULATION

Fig. 6. Model of automotive gearbox reliability

The reliability of design components can also be complex. One component can show two or more types of damage. The teeth of gear pairs can be damaged

Elementary unreliability is a complex probability that includes the probability of certain operation conditions (load, stress, etc.) as well as the failure probability of these operation conditions. Elementary unreliability is, in fact, the product of these two probabilities. For example, gear teeth damage can occur if the gear pair is exposed to the load (stress),

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which can be identified by the probability of stress σHi as pi(σHi) = nΣi / nΣ (nΣi is stress cycles number of stress σHi and nΣ, total stress cycles number in the course of service life). A low level of stress does not produce damage, i.e. it is necessary to consider failure probability in relation to the stress level PFi (σHi). If the gear pair in the course of service life is exposed to stress levels i = 1, 2, …, k, the gear pair elementary unreliability for gear pair j is integral, that is, a sum of complex probabilities throughout the entire service life, i.e.:

the identified load capacity and conditions closer to highways will increase the load capacity. These operation conditions are the customer’s requirement presented in the form of combination of various exploitation conditions (load spectrum). This is the base for load capacity identification for these (desired) operation conditions.

k

Fpj = ∑ pi (σ Hi ) PFi (σ Hi ). (1) i

Elementary reliability against this certain failure is Rgj = 1 – Fpj. Failure is possible if two conditions are fulfilled: first, that the stress exists and second, that this stress is able to produce damage. If neither of these conditions exist, failure unreliability is close to zero. Therefore, it is necessary to identify the probabilities of both operation conditions, i.e. the operation regime and the failure probability. 3.1 Gearbox Operation Regime A standard procedure for identifying the operation regime does not exist. A combination of measurements under usage conditions and statistical estimations make it possible to determine the operation regime. VDI norms define load spectrum parameters only. In order to identify the operation regime of a chosen type of gearbox, extensive experimental research has been carried out. The first stage of research was processed by applying the method of interview. About 30 drivers with extensive experience driving trucks answered various questions in a specially prepared questionnaire. Fig. 7 presents a diagram that shows the participation of every gearbox speed in the course of vehicle use under various conditions. In mountainous or hilly areas, lower level gearbox speeds predominate in the course of service life, while in flatter areas the middle levels of speed are more common. Under highway conditions, high speeds dominate. Since the vehicles (trucks and buses) do not operate under only one of these conditions, a combination is common. One of the possible combinations is presented in Fig. 7. This combination includes equal participation of mountain, flat ground, and highway conditions, for the purpose of the case study in the next analysis. The robustness of the presented approach is in the various combinations, which affects the value of the identified gearbox load capacity. For instance, the conditions closer to mountain conditions will reduce 316

Fig. 7. Participation of gearbox speeds in truck drives under various operation conditions

The second stage of the gearbox operation regime identification is the measurement of output torque under the chosen operation conditions for every gearbox speed. The measurement was carried out using strain gauge transducers at the output shaft of the gearbox. A transducer was connected to the software for data processing using a telemetric transmitter. Table 1 lists the maximum values of the measured torques during the operation. Table 1. Maximal output torques measured under operation conditions Gearbox speeds I II III IV V VI R

Maximal output torque Toutp-max [Nm] 4824 4630 1987 820 710 550 4900

By combining statistical samples identified by measurement, interviews and assessment, the arrangement of the total statistical load spectrum of every gearbox stage (speed level) is obtained. The load spectrum shows the participation of gear pair torque in a one million (106) gear teeth mesh (Fig. 8). Except for stage VI, gear pair 0 (Figs. 3 and 4) and a corresponding gear pair 1 for stage I, gear pair 2 in stage II, etc. participate in all other stages. Using load spectrums for each transmission stage I, II, III, etc.

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by a corresponding transformation, the load spectrum of each gear pair is arranged and presented in Fig. 8. The ratio xi = Ti / T1 is equal to 0 to 1, where T1 is the maximal torque of the pinion of the gear pair, T10 for gear pair 0, T11 for gear pair 1, T12 for gear pair 2, etc. Torque Ti is a variable torque. Maximal torque T1 for the pinion of each gear pair is obtained by recalculation (deduction) of the gearbox maximal output torque presented in Table 1.

Fig. 8. Load spectrums for the pinions of each gear pair designed based on the measured operation load of the chosen gearbox for gear pairs j = 0, 1, 2, …, R.

4 LOAD CAPACITY IDENTIFICATION According to Fig. 2, the robust and axiomatic design of automotive gearboxes is provided by minimizing the necessary information and eliminating the design parameters sensitivity related to operation conditions. Minimum information is obtained by functional requirements and design parameters number minimization. Design parameters connected to the gearbox geometry are parametrically connected with the major parameters such as gear diameters, gear width, gear centre distance, etc. In this way, the number of design parameters responsible for functional requirements is significantly reduced. Other non-geometric DPs are found in transformation matrix A. The number of functional requirements (FR) is also reduced and contains only the major ones, such as gearbox load capacity, speed levels and service life. Operation conditions and level of reliability, together with material characteristics and design properties of the gearbox, are also included in the calculation of the matrix A members.

3.2 Failure Probability of Gears Fig. 9 shows the range of failure probability (PF) dissipation for gear teeth flanks wear. This range of dissipation is obtained by laboratory testing and defined by ISO 6336 and DIN 3990 for gears made of alloyed steel (15CrNi6) with carbonised teeth flanks. For each stress cycle number nΣi, the Weibull function of failure probability is:

PF (σ H ) = 1 − e

σ  − H   η 

β

. (2)

Parameters of Weibull function η and β define the bound values from the Fig. 9 diagram for the corresponding stress cycle number nΣi i.e. σH0.1 and σH0.9. These two values replaced in Eq. (2) together with the corresponding failure probability PF(σH) = 0.1 and PF(σH) = 0.9 give equations for calculating variables β and η:

β=

log ( −ln 0.1 −ln 0.9 ) σ H 0.1 , η= . (3) β log (σ H 0.9 σ H 0.1 ) −ln (1 − 0.1)

Fig. 9. The range of wear probability distribution of carbonised gear teeth flanks

4.1 Elementary Reliability of the Gear Pair The value that brings into correlation the operation conditions, material characteristics, and design properties randomness is the design allowable stress. The allowable stress also provides robustness to the design process because this value includes possible variations in operation conditions and randomness of characteristics and properties. In order to present its effect, Fig. 10 gives the relationship between the operation load (stress) spectrum and failure probability of gear teeth flanks. Possible variations of operation conditions and failure probability are included in the elementary reliability calculation

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according to Eq. (1). It is possible to carry out a reverse calculation of the design allowable stress level for the desired elementary reliability by using various approaches. One of them is an iterative calculation of reliability with varying operation stresses σHi (keeping the same ratio between them in the spectrum). When the calculated reliability is close to the desired one, the maximum stress in the spectrum represents the design allowable stress σHdes = σH1.

Fig. 10. Operation stress and failure probability range relationship in the elementary reliability calculation of a given gear pair and the design allowable stress σHdes definition

Fig. 10 shows that the failure probability and gear pair reliability are strongly correlated with the stress cycle number nΣi in the course of gearbox service life. For the purpose of σHdes identification for a certain gear pair, the transformation of load spectrum in Fig. 8 for the corresponding gear pair is necessary. The transformation contains a few steps. The first one is the calculation of Hertzian stress σH according to Eq. (4), i.e. the transformation of the maximum pinion torque T1 into maximal stress σH1. The next step is the transformation of the continual logarithmic spectrum line into a few levels σHi (Fig. 10). The third step is the calculation of the total number of stress cycles nΣ and dividing of this number into the cycles number of participating stress levels nΣi. In order to be comparable, Fig. 8 shows all load spectrums presented in the same scale of 106 cycles and with a relative torque value of xi = Ti / T1max = 0 ~ 1. For calculation purposes, Fig. 10 shows the relative scales transformed into the absolute. For a given gear pair for a vehicle service life of one million kilometres, the number of pinion revolutions nΣ is calculated and then divided into nΣI, i.e. nΣ = ΣnΣi. The ratio between σHi levels and nΣi has to be maintained as shown in the load spectrums ratios of torques in Fig. 8.

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4.2 Decomposition of the Reliability System According to the reliability system presented in Fig. 6, the total reliability of the gearbox is R = Rg·Rc·Rb·Rse·Rst. The reliability of the gears contains the elementary reliability of all gear pairs against teeth flanks wear, i.e. Rg = Rg0·Rg1·Rg2·Rg3· ·Rg4·Rg5·RgR and includes j = 7 components. The reliability of couplings Rc = Rc1·Rc2·Rc3 also contains c = 3 components. Engagement of a gear set for reverse motion R is carried out without the coupling, using middle gear axial movement, carried out by the steering system. The reliability of the bearings is Rb = Rb1·Rb2·Rb3·Rb4·Rb5, it contains b = 5 elementary reliabilities, and the reliability of the seals Rse = Rse1·Rse2, se = 2. The total number of elementary reliabilities, including the reliability of the steering assembly, is r = g+c+b+se+st = 7+3+5+2+1 = 18. The desired reliability, R, as a functional requirement after the desired service life has to be decomposed to the level of elementary reliabilities Rr (Fig. 6). The simplest way to decompose it is in the form Rr = R1/r. The advantage of this calculation is the development of a design structure that is composed of components with the same level of reliabilities. The service life resources of all components will disappear at the same time. On the other hand, various combinations are possible. 4.3 Load Capacity Calculation The calculation of load capacity implies that all design parameters, design properties of components, and characteristics of materials and parts are known and defined. The load capacity of the existing automotive gearbox is the result of load capacities for every speed in relation to the load spectrum. For a certain gear pair this relationship is: 2T1 u + 1 K ≤ σ Hdes , b ⋅ d12 u (4)  2 d12 u  T1 =  σ Hdes  ⋅ b = a ⋅ b. 2 KZ u 2 +1  

σH = Z

In the above formulas T1 is the torque at the pinion (lower gear in gear pair), d1 is the pitch diameter of the pinion, b is the gear pair width, u = z2 / z1 is the transmission ratio, where z1 is teeth number of the pinion, and z2 teeth number of the gear. The load distribution at the gear teeth contacts and the load dynamics effect are included in value K, and the

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contact conditions are given by value Z, all according to ISO 6336 and DIN 3990 standards concerning helical and spur gear calculations. The value ajj in Nm/mm denotes the member in the axiomatic matrix A and connects the functional requirement T1j and the gear pair width design parameter bj (in mm). The design allowable stress σHdesj is calculated in relation to the elementary reliability for a corresponding gear pair j and all the material and operation characteristics including variation and randomness. For all gear pairs in the gearbox this relationship takes the form: T10   a00 T   0  11   T12   0    T13  =  0 T   0  14   T15   0    T1R   0

0 a11 0 0 0 0

0 0 a22 0 0 0

0 0 0 a33 0 0

0

0

0

0 0 0 0 a44 0 0

0 0 0 0 0 a55 0

  b0   b   1    b2      b3  . (5)  b   4  0   b5    aRR  bR  0 0 0 0 0

Computer Program Description The design parameters definition (DPD) is the result of satisfying functional requirements (FR) according to the presented methodology. Furthermore, the design parameters have to be determined with regard to numerous additional limitations, such as volume minimization, suspension, vibration and noise level minimization and insulation, cost minimization, modularity provision, i.e. the automotive gearbox is compatible with the design module in various vehicles, etc. For this purpose, axiomatic and robust methodology should be combined with interactive decision-making and iterative repetition of these calculations in order to obtain maximal harmony of the DP with the desired functional requirements and design constraints. The computer program DRAG has been developed for that purpose.

These are torques at the pinions of the gear pairs. The next step is the recalculation of these torques at the input torque. When the load capacity of the gearbox is identified, the maximum input torque can be obtained. These data are suitable for analysis of the design parameters correlation and optimality under the chosen operation conditions. 4.4

Load capacity and DP Harmonization

The design parameters definition of gear transmission units for common application is presented in [5] and geometric parameterization of gears is processed. One of the points of this paper is successive calculation of the transformation matrix members in order to decouple them, i.e. decoupling the transformation matrix according to the axiomatic rule. In this way, the problem of complex interrelation between design parameters (DP) of gears, shafts and bearings is solved. Eq. (5) represents the relationship between the DP and FR of independent gear pairs that operate separately when engaged by a corresponding coupling, and this is the reason why transformation matrix A is simply decoupled. The calculation of other DPs such as coupling dimensions, shafts dimensions, bearing dimensions or load capacity is made possible by applying the successive calculation procedure presented in [5].

Fig. 11. Modules of DRAG software for automotive gearbox optimization

Fig. 11 shows the DRAG (Design for Reliability of Automotive Gearboxes) program structure. The program consists of five modules with tasks that require communication with a designer in order to be solved. The first of these is the module defining the gear flank allowable stress (design Hertzian stress σHdes). This module contains a complex calculation of the elementary reliability of gears based on the operation load (operation stress) spectrum and possible failure probability (Fig. 10). The calculation iterates until the calculated elementary reliability

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becomes equal to the elementary reliability obtained by the deduction of the desired total gearbox reliability at the end of service life. The maximal stress level in the last iteration is σHdes. The input data for these calculations are the operation load spectrum, the gear pair service life, and the range of gear flank failure probability distribution. The load spectrum is the result of various combinations of operation conditions of the automotive gearbox and service life. The failure probability range is the result of extensive laboratory testing or data presented in DIN 3990 or ISO 6336. An interactive variation of operation conditions and the desired reliability produces a variation of σHdes, which provides robustness for the further calculated DP or FR. The second module shown in Fig. 11 is the load capacity identification of the automotive gearbox. The central relationship in this module is the matrix (axiomatic) Eq. (5) which defines the individual load capacity of gear pairs in the automotive gearbox. The first stage in this process is the calculation of values of matrix members a, which correlated with the already identified design stress σHdes, gear pair dimensions and other influences presented by Eq. (4). The calculated load capacities T1i of individual gear pairs are not the load capacity of the complete gearbox. These represent the basis for calculating the total value of input Tinput and output torque Toutput for every level of gearbox speed. The minimum value of the input torque represents the load capacity of the gearbox. The optimal design of the gearbox with design parameters harmonised with the operation conditions has similar values of input torque for every speed level. For the purpose of the existing gearbox redesign or new design development, it is necessary to define the new design parameters (DP) for the desired load capacity and for other functional requirements (FR). This is shown in the third software module in Fig. 11. The calculation process is reversed in the second module. The input torque (gearbox load capacity) is decomposed to the level of the individual gear pair and then follows the inverse calculation according to Eq. (5). This step has two possibilities. If the centre distance and diameters of the gear pairs are fixed, the results of the calculation are the gear widths of individual gear pairs. The second possibility is the calculation of the necessary gear diameters with a defined relationship between the gear diameter and gear width. Using the results of the gear diameters calculation, it is necessary to decide upon the same centre distance of all gear pairs and then repeat the calculation in order to change the gear pair widths if necessary. According to the presented procedure, 320

this module is very interactive and needs numerous decisions and input data for further calculation. Experimental Data and Computer Program Verification The last module for gear calculation given in Fig. 11 is the module for interactive optimization. Calculations contained in the previous two modules, as a rule, give no harmonised design parameters or load capacities of individual gear pairs. It is possible to obtain the optimal design structure or full design utility if the load capacities and design parameters of the individual design components are harmonised between each other and with the operation conditions. The gearbox load capacity (input torque) calculated based on the load capacity of individual gear pairs, can be a variable value. Harmonization means calculation of the design parameters, e.g. the gear width, in order to obtain the equilibrium of input torques, which is calculated based on the load capacities of individual gear pairs. This is also a harmonization of design parameters with operation conditions, i.e. with the functional requirements that operation conditions produce. The case study is carried out using the existing FAMOS six-speed gearbox which is mounted in trucks FAP 1620, 1921, 2021, TAM-170 and in bus transmission FAP 537.3. Table 2 presents the calculated load capacities (input torques) for the design parameters of this gearbox and Figs. 7 and 8 show the operation conditions. The load capacities of individual gear pairs are also presented. There is a great variation in input torque Tinput. The load capacity of individual gear pairs is not harmonised with the load spectrums for the operation conditions used in the calculation (middle conditions). Table 2. Load capacity of individual gear pairs for DP not harmonized with operation conditions Gear pair 0 1 2 3 4 5 R

Gear width b [mm] 35 44 32 31 27.5 32.5 57

Gear pair load capacity T1 [Nm] 1000 1200 1100 1367 950 600 1250

Gearbox load capacity Tinput [Nm] 1000 746 683 849 659 825 776

Table 3 shows the result, i.e. the new values of the gear widths of individual gear pairs, which is harmonized to the operation conditions for nominal input torques 750 and 850 Nm. For this purpose, the middle operation regime and the total reliability

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of gearbox 0.92 (for individual gear pair 0.99) after one million kilometres of service life were used. Gears are made from steel 15CrNi6 with carbonised flanks. A lower level of reliability and easier operation conditions can provide a higher level of gearbox load capacity. The DRAG software allows for the analysis of data varying trends at an intermediary point between modules. These variations are the result of the total operation conditions decomposition to the level of individual gearbox component operation conditions. This, however, requires a separate analysis of correlations between operation conditions and the design allowable stresses, FR and DP. Table 3. Gearbox DP harmonized with operation conditions for load capacity 750 and 850 Nm Gearbox load capacity for all Gearbox load capacity for all speeds Tinput = 850 Nm speeds Tinput = 750 Nm Gear pair Gear width Gear pair load Gear width Gear pair load b [mm] capacity T1 [Nm] b [mm] capacity T1 [Nm] 0 26 750 30 850 1 44 1206 50 1367 2 35 1206 39 1367 3 27 1206 31 1367 4 31 1081 35 1225 5 29 545 33 618 R 55 1206 62 1367

The presented procedure provides a harmonisation of design parameters with operation conditions. As it is impossible to adapt the design parameters to individual customer needs, the procedure is available for reverse application. Design parameters fall in the middle of the range of their variation. Individual customers may need gearboxes for various operation conditions. The load capacity can be calculated for each of them. For difficult conditions the load capacity has a lower value and for easier conditions this value is higher. The software module for the bearings calculation (Fig. 11) is not directly coupled with modules for the calculation and design parameters harmonisation of gear pairs. The harmonisation of gear parameters, coupling and shafts parameters (dimensions) is followed by the identification of the individual load spectrum for each bearing according to operation conditions. These spectrums present the basis for the service life calculation of each of the five bearings or the opposite, for the individual bearing DP definition. The constraints for this calculation are the available space, elementary reliability, etc. Software interactivity allows users to make key decisions, such as choosing the combination of operating conditions, choosing the characteristics of components (material,

thermal treatment, technology processing, etc.), as well as selection of component design parameters, automotive gearbox service life, and reliability level at the end of service life, etc. The DRAG software is a tool for automotive gearbox harmonisations with the desired operating conditions, i.e. the load capacity and design parameters by applying the robust design and property-based design methodology. 5 CONCLUSIONS The main objective of this paper is to develop a procedure that can correlate the functional requirements, random operation conditions and other limitations accordingly to the design parameters and provide a model for the design of a certain mechanical structure. The following contributions have been presented: • Integration of robust and axiomatic design methodology and design-based properties. • Reliability is used as the design requirement, the design components property, and as the quality and design behaviour indicator. therefore, a specific approach for defining elementary reliability is established. This reliability is a complex probability made up of operation conditions (load) probability and the failure probability of components exposed to a certain load level. • As a case study, the automotive gearbox is decomposed into components and the system of reliability is structured to the level of elementary reliability, which represents a design property of automotive components and a design constraint. • Random operation conditions of the gearbox are included by using load spectrums, which are designed incorporating the results of the torque measurement, the users’ interview method and the assessment. Load spectrums and gearbox speed participation in the course of service life provide robustness to the approach. • The computer program for load capacity and design parameter calculation of automotive gearboxes for random operation conditions is developed. The DRAG program is based on the developed model, which provides robustness to the results. 6 ACKNOWLEDGMENT This article is a contribution to the Ministry of Education and Science of Serbia funded project TR 035006.

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7 REFERENCES [1] Suh, N. (2007). Ergonomics, axiomatic design and complexity theory. Theoretical Issues in Ergonomics Science, vol. 8, no. 2, p. 101-121, DOI:10.1080/14639220601092509. [2] Hu, Y., Rao, S. (2011). Robust Design of Horizontal Axis Wind Turbines Using Taguchi Method. Journal of Mechanical Design, vol. 133, no. 11, paper: 111009. [3] Saha, A., Ray, T. (2011). Practical Robust Design Optimization Using Evolutionary Algorithms. Journal of Mechanical Design, vol. 133, no. 10, paper 101012. [4] Du, X. (2012). Toward Time-Dependent Robustness Metrics. Journal of Mechanical Design, vol. 134, no. 1, paper: 011004. [5] Ognjanovic, M., Benur, M. (2011). Experimental research for robust design of power transmission components. Meccanica, vol. 46, p. 699-710, DOI:10.1007/s11012-010-9331-y. [6] Dersjo, T., Olsson, M. (2011). Reliability based design optimization using a single constraint approximation poin. Journal of Mechanical Design, vol. 133, no. 3, paper: 031006. [7] Moura, M., Zio, E., Lins, I., Droguett, E. (2011). Failure and reliability prediction by support vector machines regression of time series data. Reliability Engineering and System Safety, vol. 96, p.1527-1534, DOI:10.1016/j.ress.2011.06.006. [8] Lisnianski, A., Elmakias, D., Laredo, D., Haim, H.B. (2012). A multi-state Markov model for a shortterm reliability analysis of a power generating unit. Reliability Engineering and System Safety, vol. 98, p.16, DOI:10.1016/j.ress.2011.10.008. [9] Natvig, B., Huseby, A., Reistadbakk, M. (2011). Measures of component importance in repairable multistate systems - a numericalstudy. Reliability Engineering and System Safety, vol. 96, p. 1680-1690, DOI:10.1016/j.ress.2011.07.006. [10] Noh, Y., Choi, K., Lee, I. (2011). Reliability-based design optimization with confidence level for nonGaussian distributions using bootstrap method. Journal of Mechanical Design, vol. 133, no. 9, paper: 091001. [11] Jiang, R., Murthy, P. (2011). A study of Weibull shape parameter: Properties and significance. Reliability Engineering and System Safety, vol. 96, p. 1619-1626, DOI:10.1016/j.ress.2011.09.003. [12] Murthy, P., Rausand, M., Virtanen, S. (2009). Investment in new product reliability. Reliability Engineering and System Safety, vol. 94, no. 10, p. 1593-1600, DOI:10.1016/j.ress.2009.02.031. [13] Li, S. (2011). A matrix-based clustering approach for the decomposition of design problems. Research in Engineering Design, vol. 22, p. 263-278, DOI:10.1007/ s00163-011-0111-z. [14] Minguez, R., Conejo, A., Garcia-Bertrand, R. (2011). Reliability and decomposition techniques to solve certain class of stochastic programming problems.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 323-332 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.757

Original Scientific Paper

Received for review: 2012-08-27 Received revised form: 2013-02-14 Accepted for publication: 2013-02-18

A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems Zou, J. – Du, Q. Jiehui Zou* – Qungui Du

South China University of Technology, School of Mechanical and Automotive Engineering, China In order to improve the conceptual design of mechatronic systems, a refined functional representation aimed at functional reasoning is presented in this paper. Based on the refined functional representation, we proposed a cube model for functional reasoning. We compared the cube model with the systematic model through an illustration. The cube model can be regarded as an improvement of the systematic model. In addition, the application scope of the cube model is discussed. The proposed cube model can be applied to design other systems (except holonomic mechatronic systems). Illustrational comparison and discussion showed that the proposed cube model was clear and easy to use for designing various technical systems. Keywords: cube model, refined functional representation, functional reasoning, design process, conceptual design, engineering design

0 INTRODUCTION In the initial design period, the trial and error method is generally used. However, modern technology systems design must be guided by modern design theory [1] to [4]. Conceptual design is the first stage of engineering design and includes three kinds of processes: functional reasoning, concept solving, and solution synthesizing. Functional reasoning in particular is increasingly regarded as an important technique in engineering [5] to [7]. The essence of functional reasoning is the reasoning process from overall functions to all levels of sub-functions through several sets of nested functional decompositions, in which each functional decomposition generates the next level of sub-functions from a function. Historically, there are influential functional reasoning approaches or models for conceptual design, such as Freeman and Newells' model [8], the Zigzag model by Suh [9], the Scheme for functional reasoning [10] and [11], the Function logic approach [12], Gero's FBS-model [13], the Function-behaviorstate model [14], Function-to-form mapping [15], and the Function-oriented theoretical framework [16]. Each of these models provides a framework to show the reasoning process of the whole. Nevertheless, when these models are applied in actual functional decomposition, a common question naturally arises: how are the lower level sub-functions generated? We believe that the problem is caused by the unclear relationship between the sub-function and the function. Moreover, Garbacz has pointed out that the semantics of the relationship that “x is a sub-function of y” is still unclear [17]. So, functional reasoning largely depends on the inspiration and experience of the designers rather than knowledge.

The systematic model proposed by Pahl and Beitz differs from other functional reasoning approaches. In the model, the relationships between the subfunction and function can be described by flows. Thus the relationships are relatively clear [18]. In the systematic model, the overall function of a technical system is represented by a black-box operation dealing with the flows of materials, energies, and signals at first. This overall function is then progressively expanded into combinations of sub-functions. This combination is called function-structure. This process of simplification is continued until the sub-functions of the function-structure are so simple that each subfunction can be provided by the corresponding scheme (concept solution). However, this model is not perfect either. Firstly, it is difficult to extract (determine) and comprehensively describe all the input and output flows according to the demands of the customers. Secondly, when both the overall function description and sub-function description are included in the three sorts of flows, function decomposition has no definite target. Thus, the same puzzle arises: how are the lower level sub-functions generated? If the three sorts of flows are separately analyzed, it is only necessary to trace one sort of flow separately during the function decomposition. In this way, the target becomes clear and functional reasoning is simple. Thus, defects in the systematic model are avoided. Based on this strategy, we refine the functional representation and propose how to describe the functions with one sort of flow and then provide three functional reasoning rules. Based on the function description and reasoning rules, we propose a new functional reasoning model named the cube model. The cube model can be regarded as an improved systematic model. An illustrational comparison and discussion showed that the proposed cube model was

*Corr. Author’s Address: South China University of Technology, School of Mechanical and Automotive Engineering, Guangzhou, China, zoujiehui@yahoo.cn

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clear and easy to use for designing various technical systems. The paper’s structure was inspired by the work of Chakrabarti and Bligh [10].

flows). The rectangular frame indicates one function block.

1 A REFINED FUNCTIONAL REPRESENTATION The essence of conceptual design is to describe the design problem as an intended function and find a physical solution to provide it. Therefore, function is crucial to conceptual design and functional representation is a precondition of the functional reasoning model. However, there is still no consensus about the meaning of the term “function” itself in engineering [7] and [19] to [22]. 1.1 Refining the Flows It is well-known that there are three sorts of flows: material flows, energy flows, and information flows. Pahl and Beitz believed that the function could be described by changes in the energy, material and information flows (hereafter respectively E_flow, M_flow, and I_flow for short). Thus, this functional representation comprises three sorts of flows, as shown in Fig. 1. A double-headed arrow indicates that flow number is uncertain (0, 1 or more flows). Different flow types are indicated by different lines. Heavy line, fine line, and fine dotted line respectively indicate energy, material, and information flows. The rectangular frame indicates one function block.

Fig. 1. A functional representation of Pahl and Beitz

Even when we accept the functional representation of Pahl and Beitz, we have to bear in mind that some flows of a system have no direct relationship with its function. So, we must modify it slightly in order to rationalize the representation of the functions. Therefore, we refine the flows, and define the function as the change in only one sort of flow (energy, material or information flow). That is to say, in our refined version, there is only one kind of flow of the function. These kinds of flows are called target flow and the other flows are discriminatingly called condition flows. The refined functional representation is shown in Fig. 2. The double-headed arrow indicates that flow number is uncertain (0, 1 or more flows). The hollow line indicates the target flow of uncertain kinds of flow (probably material, energy or information 324

Fig. 2. The refined functional representation

Here, a functional block is a physical entity with a proper function. It can be regarded as a flow processor, which may be a whole technical system, subsystem, device, piece of equipment, instrument, or one part of these. There are four reasons to refine the functional representation: Firstly, the refined function definition is more able to reflect the core demand of customers and the main intention of designers (Umeda, et al. represent function as an association of the designer's intention [14] and Ullman considers function to be a human abstraction of behavior often implying intention [23]). For instance, the function of a water heater is to increase the temperature of input water. The contrast of the input cold water and the output hot water reflects the demand of customers and the main intention of designers. The other flows of energy and information are not directly related to the demand of customers and the intention of designers. Secondly, the refined function definition is more able to show the duty of a functional block (Hubka, et al. regard function as the duty of a technical system to deliver specified effects at its input [24]). The duty of a water heater is to change the input cold water to output hot water and the input of electric energy and the output of losing thermal energy are not its duty. Thirdly, we believe that the condition flows serve to change the target flows. For example, in a water heater, the energy and information flows serve to change the material flows (water). Fourthly, the main objective is to facilitate functional reasoning, which will be discussed in Section 2. 1.2 Types of Change We consider a function to be a change in target flows. The changes refer to the number of target flows or the attributes of one target flow. According to the numbers of flows, all of the refined functions are divided into four parts: singleinput/single-output, single-input/multiple-output, multiple-input/single output, and multiple-input/ multiple-output.

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In the functions of single-input/single-output, the change does not refer to the number of flows but the attributes of the flow. All the attributes of a flow can be classified into two categories: qualitative attributes and quantitative attributes. Generally, the value region of a qualitative attribute is discrete (for instance, the modality of a material is a qualitative attribute, it has three values: solid state, liquid state, and gaseous state) and the value region of a quantitative attribute is indiscrete (for instance, the temperature of a material is a quantitative attribute and its value region are indiscrete). In a word, the refined function representation comprises only one sort of flow, such as target flows. The changes include three kinds of elements: the number of flows, the qualitative attributes of a flow, and the quantitative attributes of a flow. 2 THE FUNCTIONAL REASONING CUBE MODEL In this section, based on the proposed functional representation, a new functional reasoning model named the cube model is proposed. 2.1 The Three Functional Reasoning Rules If we consider a technical system as a flow processor, its functional reasoning is tracing the flow change over time, such as searching for middle flows or the middle status points of a flow. The core contribution of this refined functional representation is separating the three sorts of flows. Based on the refined functional representation, three rules are presented to guide the functional reasoning. Thus, the process of functional reasoning will become regular and clear immediately. The rules are as follows. Rule 1: take one kind of flow into account for reasoning and then consider other kinds of flows; Rule 2: take the change in flow number into account and then consider the change in flow attributes; Rule 3: take the change in flow qualitative attributes into account and then consider the change in flow quantitative attributes. 2.2 The Two Kinds of Reasoning Subprocesses According to Rule 1, functional reasoning as a whole can be separated into three functional decompositions. The decompositions need only to trace the change in one kind of flow (target flows). The bridge between two functional decompositions is the condition flow, which is determined by concept solving. The condition

flows are the target flows of the next decomposition. That is to say, the whole process of functional reasoning includes two kinds of sub-processes. One is the functional decomposition by tracing target flows and the other is the determination of condition flows through concept solving. 2.2.1 Functional Decomposition by Tracing the Change in Target Flows Rule 2 and Rule 3 can be used in this sub-process to guide the functional reasoning. The essence of tracing the change in the flow number is to search for and add the middle flows (new flows, e.g. Target flow 5 in Fig. 3). And the essence of tracing the change in a flow attribute is to search for and add the middle status points of a flow (new status of a flow, e.g. Target flow 6 in Fig. 3), as shown in Figure 3. The single-headed arrow indicates one flow and the double-headed arrow indicates that the flow number is uncertain (0, 1 or more flows). A hollow line indicates the target flow of uncertain kinds (probably material, energy or information flows). The rectangular frame indicates one function block.

Fig. 3. Functional decomposition by tracing the change in target flows

2.2.2 Determining Condition Flows through Concept Solving This sub-process is inspired by both Freeman & Newells' model [8] and the Zigzag model [9]. The common characteristic of the two models is that the functional decomposition and the concept solving are performed alternately. The concept solving aims to provide the functions that are generated by functional decomposition, and the concept solutions are the basis of the next round of functional decomposition. From the functional reasoning point of view, the concept

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solving (F-S) is inserted into the sequential two rounds of functional decomposition (F-Fs) as F-S-Fs. The goal of this sub-process is to determine condition flows through concept solving. The condition flows are the target flows of the next functional decomposition, so this sub-process is a precursor to the next functional decomposition, as shown in Fig. 4.

Fig. 4. Determining condition flows through concept solving

2.3 A Holonomic Mechatronic System The proposed functional reasoning model can be applied to design various technical systems, but a holonomic mechatronic system is presumed to be the application object. Other kinds of systems will be discussed in Section 4. Here, a holonomic mechatronic system is particularly defined as a processor of three sorts of flows and its target flows are the material flows. A holonomic mechatronic system consists of three functional subsystems including the executing subsystem, driving subsystem, and control subsystem. Their duties are treating material, transforming energy, and real-time control, respectively. Their interrelationships can be described by the different flows, as shown in Fig. 5.

(1) Target flows of the executing subsystem are the material flows, see arrows 1 and 2; (2) The change in the materials depends on the energy flows, see arrow 3; (3) Target flows of the driving subsystem are the energy flows, see arrow 3 and 4; (4) The executing subsystem and the driving subsystem should work under the control of the control subsystem, see arrows 5 and 6; (5) Target flows of the control subsystem are the information flows. The input is from the exterior or interior of the other two subsystems, except for time (it is ubiquitous), see arrows 7, 8 and 9; (6) The work of the control subsystem requires energy flows from the driving subsystem, see the arrow 10. In addition, the target flows of the three subsystems are material, energy, and information flows, so they are also respectively named the material subsystem, energy subsystem, and information subsystem (hereafter, respectively, M_subsystem, E_ subsystem and I_subsystem for short). 2.4 Description and Definition of the Cube Model The cube model is composed of six planes (Fig. 6). The three visible planes (front, left, and top) refer to the three function-structures of the three subsystems and belong to the functional domain, and the other three invisible planes (back, right, and bottom) refer to the concept solutions of the three subsystems and belong to the physical domain. The unfolded graph of the cube and the definitions for the six planes are shown in Fig. 6.

Fig. 5. Relationships among the described subsystems using the various kinds of flows Fig. 6. The unfolded cube and the definitions of the six planes

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2.5 The Building Process of the Cube Model 2.5.1 The Whole Building Sequence The building process of the cube model is a functional reasoning process, so the proposed functional reasoning model is also called the cube model. The whole building sequence of the cube applied to design a holonomic mechatronic system is shown in Fig. 7.

Step 5: Building the top plane (function-structure of I_subsystem) through functional decomposition by the I_flows (the input and output I_flows are decided by the control strategy and they are the condition flows of the M_subsystem & E_subsystem, which are determined by the concept solutions in the back & right planes, see arrows 2 and 3). Step 6: Building the bottom plane (concept solutions of I_subsystem, I_Ss) through concept solving to provide the information functions (I_Fs) in the top plane. Step 7: Adjusting the left and right planes by adding the E_flows into the E_subsystem (the E_ flows are the condition flows of the I_subsystem and come from the bottom plane, see arrow 4). The whole building sequence can be simply expressed as: Front-Back-Left-Right-Top-BottomLeft-Right. 2.5.2 The Detailed Building Process

Fig. 7. The sketch of building the cube

The cube model can be built in seven steps according to the relationships among the three subsystems (see section 2.3) and the two reasoning sub-processes (see section 2.2). Step 1: Building the front plane (functionstructure of M_subsystem) through functional decomposition by the M_flows (the M_flows are extracted from the customer requirements). Step 2: Building the back plane (concept solutions of the M_subsystem, M_Ss) through concept solving to provide the material functions (M_Fs) in the front plane. Step 3: Building the left plane (function-structure of E_subsystem) through functional decomposition by the E_flows (the input E_flows are decided by the working environment, and the output E_flows are condition flows of the M_subsystem which are determined by the concept solutions in the back plane, see arrow 1). Step 4: Building the right plane (concept solutions of E_subsystem, E_Ss) through concept solving to provide the energy functions (E_Fs) in the left plane.

The building processes of the front (left, top) plane and the back (right, bottom) plane are inseparable and iterative. If one concept solution can not provide the function in the front plane, function decomposition and concept solving will continue until all the functions in the front plane (left or top) are provided by the concept solutions of the back (right or bottom) plane. The detailed reasoning processes and reasoning results are shown in Fig. 8.

Fig. 8. Flow chart of detailed functional reasoning and results

A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems

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3 AN ILLUSTRATIONAL COMPARISON In this section, the systematic model and the cube model are applied to design an integrated road mending machine (IRMM). 3.1 The Intended Design Object As shown in Fig. 9, the conventional road mending method contains four stages and each stage requires a special tool. An integrated road mending machine (IRMM) is regarded as the design object for performing the entire mending task according to new mending methods.

Step 2: Functional reasoning Even if all the input-output flows in the overall function are determined, too many targets hamper functional decomposition. In this way, we must consider the decomposition of three kinds of flows and the docking of three kinds of flows after the decomposition. 3.3 Functional Reasoning with the Cube Model The target flows of the IRMM system are material flows and its essential function is to generate the mixtures and transport the mixtures to the appointed place. The whole building sequence is shown in Fig. 7 (Front-Back-Left-Right-Top-Bottom-Left-Right). Step 1: Building the front plane The goal of this step is to generate the functionstructure of the M_subsystem through functional decomposition. Fig. 11 shows the process of functional decomposition by tracing the changes of M_flows according to Rule 2 and Rule 3.

Fig. 9. Functional requirements of the IRMM

3.2 Functional Reasoning using the Systematic Model In the systematic model, we will first represent the overall function as a black-boxed operation on the flows of materials, energies, and signals. Then, this overall function will be decomposed into the combined sub-functions. Step 1: Ascertaining the overall function According to the new mending method, the input M_flows include sand, air, and asphalt, and the output mixtures include pure air, air & asphalt, air & sand & asphalt, and air & sand. But we cannot entirely ascertain the input-output E_flows and I_flows, as shown in Fig. 10.

Fig. 10. Overall function of the IRMM

328

Fig. 11. Process of functional decomposition by M_flows

The final result of functional decomposition by the M_flows is shown in Fig. 12.

Fig. 12. Function-structure of the M_subsystem

In addition, this process supports innovative design by various functional decompositions and the following steps 3 and 5 are similar. Zou, J. – Du, Q.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 323-332

Step 2: Building the back plane The concept solutions of the M_subsystem (M_ S1 to M_S7) are diagramed to provide the M_Fs (Fig. 13). In addition, this process supports innovative design by generating different concept solutions to provide a function (e.g., M_S1(a) and M_S1(b) are the different concept solutions to provide M_F1, see Fig. 13.) Fig. 14. Function of the E_subsystem

Fig. 15. Function-structure of the E_subsystem

Fig. 13. Concept solutions of M_subsystem

Step 3: Building the left plane Most M_Fs require energy support. The required energy is the output E_flows of the E_subsystem. The input E_flows are decided by the working environment. As the IRMM is involved in working outdoors, neither electric power nor solar energy is the ideal input E_flows. An alternative program is the chemical energy from diesel oil. The function of the E_system is shown in Fig. 14. The result of functional decomposition by the E_ flows (function-structure of E_subsystem) is shown in Fig. 15. Step 4: Building the right plane The concept solutions of the E_subsystem (E_S1 to E_S6) are to provide the E_flows. The solutions are described in the shadow ellipses (Fig. 16).

Fig. 16. Concept solutions of the E_subsystem

Fig. 17. Function of the I_subsystem

A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems

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E_subsystem, or operator. The function of the I_ subsystem is shown in Fig. 17. The result of functional decomposition by I_ flows (function-structure of I_subsystem) is shown in Fig. 18. Step 6: Building the bottom plane The concept solutions of the I_subsystem (I_S1 to I_S11) provide the I_flows. The solutions are described in the shadow ellipse (Fig. 19). Step 7: Adjusting the left and the right plane The required E_flow of the I_subsystem is lowvoltage DC, which can be directly provided by E_F4 (see Fig. 15). The entire function-structure of the system is composed of three function-structures of the subsystems in the front, left, and top planes. According to the plane definitions of the cube model, the final unfolded cube model is shown in Fig. 20. The concept solutions of the three subsystems are in the back, right, and bottom planes, respectively. The entire concept solution of the system will be generated by concept synthesizing (not covered in this paper).

Fig. 18. Function-structure of the I_subsystem

3.4 Improvements to the Systematic Model The cube model can be regarded as an improvement of the systematic model. The obvious improvements are shown in Table 1. The cube model is characterized by clear target flows, three reasoning rules, simple function structure, and a clear reasoning foundation in physical knowledge.

Fig. 19. Concept solutions of the I_subsystem

Table 1. Differences between the systematic model and the cube model Improvements Functional representation Processes Results

Fig. 20. Complete functional reasoning results of the IRMM using the cube model

Step 5: Building the top plane According to the control strategy, asphalt temperature, compressed air pressure, and the content of the mixture should be controlled by the I_subsystem. The input and output I_flows contain these parameters and come from the M_subsystem, 330

Systematic model Change in three sorts of flows Unclear

a whole functionstructure Reasoning foundation inspiration and experience

Cube model Change in target flows Clear process with three reasoning rules Three simpler function-structures physical knowledge

4 DISCUSSION OF THE SCOPE OF APPLICATION OF THE CUBE MODEL The proposed cube model can be used to design other sorts of systems (except holonomic mechatronic systems). The difference lies in the building sequence of the cube, as shown in Table 2.

Zou, J. – Du, Q.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 323-332

Front-Back-Left-Right

With the cube model proposed in the paper, functional reasoning is converted into tracing the change in one kind of flow. In this way, the cube model can comprehensively utilize physical knowledge in functional reasoning, providing the opportunity for the realization of computer-aided concept design and computer automated concept design.

Top-Bottom-Left-Right

6 ACKNOWLEDGEMENTS

Left-Right-Top-BottomLeft-Right

The authors are grateful to Guangdong Province Project Foundation of Science & Technology, China (No. 2003C101003), Loux John for revising the paper, and the anonymous referees for their helpful comments and suggestions.

Table 2. The different building sequences for various technical systems Flows

System

Instances Vehicle Modern (M+E+I) RoadMechanical machine Traditional Bicycle (M+E) Mechanical Handcart Information Mobile-phone (I+E) processor Radio AutoModern energy (E+I) changer transformation Servo-motor Traditional Engine (E) energy Motor transformation

Sequence Front-Back-Left-RightTop-Bottom -Left-Right

Left-Right

• M: M_flows E: E_flows I: I_flows. • The first letter in bracket denotes the kind of target flows of the whole technical system.

5 CONCLUSION Pahl and Beitz believe that function can be represented by the changes in material, energy, and information flows. However, we believe that only one sort of flow in the input-output flows of a function block can reflect the requirements of the customers, intentions of the designers, and duties of technical systems. We call this kind of flow the target flow. As conditional flows, other sorts of flows serve to change the target flows. Based on the above ideas, we proposed one new refined functional representation, the function description of target flows. According to functional representation, we also proposed a new functional reasoning model: the cube model. In the cube model, function reasoning is divided into three rounds of functional decomposition according to the kind of flow. In the decomposition, it is only necessary to trace the change of one kind of flow. During two rounds of functional decomposition, the condition flow obtained through solution solving is used as the connection bridge. All the functional decomposition and function solving may be described as one cube. The cube model can be considered as an improved systematic model. In addition to the advantages of the systematic model, the cube model also has the following features: clear reasoning based on physical knowledge, simple results, and a wide scope of application in the design of various technical systems. In addition, the functional reasoning method used in current concept design is lacking in definite rules and largely depends on the experience and inspiration of the designers. The lack of reasoning rules hampers the usage of computer tools in functional reasoning.

7 REFERENCES [1] Preston, B. (2009). Philosophical theories of artifact function. Meijers, A. (ed). Philosophy of Technology and Engineering Sciences. North Holland, Amsterdam, p. 213-233. [2] Popovic, V., Vasic, B., Petrovic, M., Mitic, S. (2011). System Approach to Vehicle Suspension System Control in CAE Environment. Strojniški vestnik Journal of Mechanical Engineering, vol. 57, no. 2, p. 100-109, DOI:10.5545/sv-jme.2009.018. [3] Otto, K.N., Wood, K.L. (2000). Product Design: Techniques in Reverse Engineering and New Product Development. Prentice Hall, Upper Saddle River. [4] Iacob, R., Popescu, D., Mitrouchev, P. (2012). Assembly/disassembly analysis and modeling techniques: A review. Strojniški vestnik - Journal of Mechanical Engineering, vol. 58, no. 11, p. 653-664, DOI:10.5545/sv-jme.2011.183. [5] Pahl, G., Wallace, K. (2002). Using the Concept of Functions to Help Synthesize Solutions. Chakrabarti, A. (ed.), Engineering Design Synthesis. Springer, London, p. 109-119. [6] Hirtz, J., Stone, R.B., McAdams, D.A., Szykman, S., Wood, K.L. (2002). A Functional Basis for Engineering Design: Reconciling and Evolving Previous Efforts. Research in Engineering Design, vol. 13, no. 2, p. 6582. [7] van Eck, D., McAdams, D.A., Vermaas, P.E. (2007). Functional decomposition in engineering: A survey. International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, file no. 34232, p. 1-10. [8] Freeman, P, Newell, A. (1971). A model for functional reasoning in design. Proceedings of the 2nd International Joint Conference on Artificial Intelligence, p. 621-633. [9] Suh, N.P. (2000). Axiomatic Design: Advances and Applications. MIT Press, Cambridge. [10] Chakrabarti, A., Bligh, T.B. (2001). A scheme for functional reasoning in conceptual design. Design

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Studies, vol. 22, no. 6, p. 493-517, DOI:10.1016/ S0142-694X(01)00008-4. [11] Liu, Y.C., Bligh, T., Chakrabarti, A. (2003). Towards an 'ideal' approach for concept generation. Design Studies, vol. 24, no. 4, p. 341-355, DOI:10.1016/S0142694X(03)00003-6. [12] Sturges, R.H., Shaughnessy, K.O., Kilani, M.I. (1996). Computational model for conceptual design based on extended function logic. Artificial Intelligence for Engineering Design, Analysis, and Manufacturing, vol. 10, no. 4, p. 255-274, DOI:10.1017/ S089006040000161X. [13] Gero, J.S., Kannengiesser, U. (2004). The situated function-behaviour-structure framework. Design Studies, vol. 25, no. 4 p. 373-391, DOI:10.1016/j. destud.2003.10.010. [14] Umeda, Y., Ishii, M., Youshioka, M., Shimomura, Y., Tomiyama, T. (1996). Supporting conceptual design based on the function-behavior-state modeler. Artificial Intelligence for Engineering Design, Analysis, and Manufacturing, vol. 10, no. 4, p. 275-288, DOI:10.1017/S0890060400001621. [15] Roy, U., Pramanik, N., Sudarsan, R., Sriram, R.D, Lyons, K.W. (2001). Function-to-form mapping: model representation and application in design synthesis. Computer-Aided Design, vol. 33, no. 10, p. 699-719, DOI:10.1016/S0010-4485(00)00100-7. [16] Yong, X., Zou, H.J. (2007). A function-oriented theoretical framework for mechatronic system design. Strojniški vestnik - Journal of Mechanical Engineering, vol. 53, no. 4, p. 241-252.

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[17] Garbacz, P. (2007) A formal model of functional decomposition. ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, file no. 99097, p. 1-10. [18] Pahl, G., Beitz, W. (1996). Engineering Design: a Systematic Approach. SpringerVerlag, London. [19] Chittaro, L., Kumar, A.N. (1998). Reasoning about function and its applications to engineering. Artificial Intelligence in Engineering, vol. 12, no. 4, p. 331-336, DOI:10.1016/S0954-1810(97)10008-5. [20] Deng, Y.M. (2002). Function and behavior representation in conceptual mechanical design. Artificial Intelligence for Engineering Design, Analysis, and Manufacturing, vol. 16, no. 5, p. 343362, DOI:10.1017/S0890060402165024. [21] Far, B.H., Elamy, A.H. (2005). Functional reasoning theories: problems and perspectives. Artificial Intelligence for Engineering Design, Analysis, and Manufacturing, vol. 19, no. 2, p. 75-88, DOI:10.1017/ S0890060405050080. [22] Zadnik, Z., Karakasic, M., Kljajin, M., Duhovnik, J. (2009). Function and functionality in the conceptual design process. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 7-8, p. 455-471. [23] Ullman, DG. (2002). Toward the ideal mechanical engineering design support system. Research in Engineering Design, vol. 13, no. 2, p. 55-64. [24] Hubka, V., Eder, W.E. (1988). Theory of Technical Systems: A Total Concept Theory for Engineering Design. Springer Verlag, Berlin, DOI:10.1007/978-3642-52121-8.

Zou, J. – Du, Q.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 333-338 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.775

Original Scientific Paper

Received for review: 2012-09-03 Received revised form: 2013-02-18 Accepted for publication: 2013-03-29

A Study of Thermal Behaviour during Submerged Arc Welding Ghosh, A. – Barman, N. – Chattopadhyay, H. – Hloch, S. Aniruddha Ghosh1,* – Nilkanta Barman2 – Himadri Chattopadhyay2 – Sergej Hloch3

1 Government

College of Engineering & Textile Technology, Depertment of Mechanical Engineering, India 2 Jadavpur University, Department of Mechanical Engineering, India 3 Technical University of Košice, Faculty of Manufacturing Technologies, Slovak Republic

The present study reports on thermal behavior during submerged arc welding considering an oval shaped heat source. The welding process is represented by the energy conservation equation where the moving electrode is incorporated considering an oval shaped Gaussian heat source. The governing equation is solved based on the Green function solution and subsequently considered the effect of heat in the weld pool and convection after welding. The prediction agrees well with the experiments and it was found that an oval shape is a better approximation for the heat source. Keywords: submerged arc welding, oval shape Gaussian heat source, parabolic weld pool, convective heat loss, lumped system

0 INTRODUCTION In arc welding processes, a moving heat source is generally applied for joining plates. The process includes heating as well as cooling of the welded plate whose temperature changes over time and space. Control of the temperature distribution is important for maintaining the quality of the welded plate, which depends on many process parameters (current, voltage, travel speed, etc.). Hence, a detailed study of the distribution of temperature using input process parameters is essential. In the literature, two methods are primarily used for modelling the transient temperature distribution during the welding of plates: numerical [1] to [3] and analytical [4] to [6] methods. In the present work, an analytical solution is used to predict the transient temperature distribution during arc welding of two steel plates. In the last few decades, many researchers have reported on the transient temperature behavior during welding of plates considering a 2-D heat source. A classical solution for the traisient temperature field is Rosenthal‘s solution [5], which deals with a semiinfinite body subjected to instant point, line, and 2-D surface heat sources. Eagar and Tsai [6] considered a 2-D bell shape for the heat source and predicted the temperature distribution in the heat source region. Jeong and Cho [7] introduced a 2-D Gaussian heat source for determining the temperature field of a finite thick plate. However, the 2-D model of the heat source is unable to predict the penetration effect. Thus, Goldak et al. [1] first introduced a 3-D double ellipsoidal moving heat source for predicting the thermal behaviour during welding. Nguyen et al. [8] presented an analytical solution for the transient temperature field of a semi infinite body subjected to a 3-D dynamic heat source. Winczek [9] described an

analytical solution for the transient temperature field of a semi infinite body caused by a volumetric heat source of Gaussian distribution with different motions. Winczek [10] also provided an analytical solution that considers temperature increments caused by liquid metal and heat radiation of the moving electrode during welding. Li and Lu [11] predicted the temperature distribution considering a hybrid heat source model, the combination of bell shape and ellipsoidal heat sources, during submerged arc welding. The literature shows that mostly ellipsoidal and bell shape heat sources are studied by the researchers. The literature also shows that the shape of the heat source depends on the input process parameters and welding process. Consideration of an oval shape heat source provides an appropriate temperature distribution, particularly in submerged arc welding,. In the present work, therfore, the transient thermal behaviour during submerged arc welding is predicted analytically considering a moving oval heat source. Finally, the analytical prediction is compared with experiments. 1 DESCRIPTION OF THE PHYSICAL PROBLEM The present work considers the joining of two steel plates (30×15×2 cm) using submerged arc welding. The chemical composition of the steel is described in Table 1. A V-groove of angle 60° was cut on the work pieces along the X-axis. The plates were then firmly fixed to a base plate where a root opening of 0.1 cm was provided to join the plates, keeping the electrode positive and perpendicular to the plates. Fig. 1 shows a schematic of the system where a copper-coated electrode in coil form with a 0.315 cm diameter travels along the X-axis. The electrode is connected to a constant voltage, retifyer type power source with a 1200 A capacity. During welding, a basic fluoride

*Corr. Author’s Address: Government College of Engineering & Textile Technology, Department of Mechanical Engineering, India, agmech74@gmail.com

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Table 1. Chemical composition of the C-Mn steel work piece [%] C 0.18

Sn 0.36

Mn 1.58

P 0.023

S 0.027

type granular flux is used. During processing, the temperature at different positions in the welded plate is recorded (as shown in Fig. 1) using two infrared thermometers (OMEGA SCOPE OS524E, 2482 °C, accuracy of 0.2 °C) at positions P1 and P2.

Cr 0.06

Ni 0.03

Mo 0.01

A=

Cu 0.04

Al 0.05

2 abc 1 × ( m2 /( 4 a )) × Q0 , (3) π 3/ 2 e

where Q0 (Q0 = I × V × ŋ) is the total heat input. V, I and ŋ are the welding voltage, current and arc efficiency, respectively. In the present case, the arc efficiency is considered to be 1 [6]. It was found from the experimental data that the shape of a weld pool is oval on the X-Y plane (z = 0). The corresponding shape of the weld pool, based on the experimenatal data, is shown in Fig. 3. The equation of the oval weld pool was found to be: x2 y2 + × e0.3 x = 1. (4) 2 1.3 1.132 The equation for an oval shape heat source, based on Eq. (2), is considered to be:

Fig. 1. Schematic of submerged arc welding; the temperature of points (P1 and P2) was recorded

2 MATHEMATICAL MODEL In the present study, the heat transfer behaviour in the welded plate is representated by the transient heat conduction equation. The corresponding governing equation is given as:

k ∇ 2T + q = ρ C p

∂T , (1) ∂t

where T = T(x, y, z, t) is the temperature at a point (x, y, z) at time t and q is a 3-D Gaussian oval heat source. A schematic of the heat source is shown in Fig. 2. During welding, the heat source moves along the x-axis with a constant velocity (v). The heat generated by the heat source at a point (x, y, z) at any instant t is considered to be:

q ( x, y, z ) = Ae − ( ax

2

+ ( by 2 + cz 2 )×emx )

, (2)

where A is maximum heat density and a, b, c and m are the heat source parameters. The maximum heat density (A) is found to be: 334

( ax + (by 2

2

)

)

+ cz 2 × e mx = 1, (5)

where a is a semi major axis, b is a semi minor axis, and c is another semi-principal axis of an ellipsoid (ax2 + by2 + cz2 = 1). Hence, in the present work, m is considered to be 0.3. The constants (a, b and c) for the oval shape bead geometry are taken from Goldak and Akhlaghi [12] as:

a=

ln(20) , L2

b=

ln(20) , (7) B2

and c =

(6)

ln(20) , (8) C2

where B, C and L are the half of the bead width, penetration of weld pool, and half of the major axis of oval shape (L ≈ 1.15 B), respectively. B and C were determined experimentally. The corresponding values of B and C are given in Table 2. Initial and Boundary Conditions Intially, the heat source is located at: x = 0 at t = 0 ,

T(x, y, z, 0) = T0,

Ghosh, A. – Barman, N. – Chattopadhyay, H. – Hloch, S.

T(±∞, ±∞, ±∞, t) = T0

(9) (10)


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 333-338

Table 2. The values for the bead parameters (B ,C/P and R) Voltage [V] 35

Current [A] 450

Travel speed (v) [cm/min] 30

Bead width (B) [mm] 22.66

Penetration (C / P) [mm] 7.78

Maximum reinforcement height (R) [mm] 1.94

Fig. 2. Schematic of an oval heat source during welding with weld pool

Fig. 3. The shape of the heat source (oval) based on experimental data

3 TEMPERATURE INCREMENT CAUSED BY HEAT RADIATION OF THE MOVING ELECTRODE

For an infinite medium, the temperature increment at point (x, y, z) at an instant t is given as:

A general solution [5] of the governing Eq. (1) based on the Green function for an instantaneous unit heat source is:

G ( x − x ', y − y ', z − z ', t − t ') = [− 1 = e 3/ 2 ρ c p [4απ ( t − t ')]

( x − x ' )2 + ( y − y ' )2 + ( z − z ' )2 ] 4α ( t −t ')

, (11)

where α (α = k / ρcp) is the thermal diffusivity and t' is the electrode travel time. Eq. (11) gives the temperature increment at a point (x, y, z) at an instant t for the unit heat source applied at point (x', y', z') at an instant t'. Due to the linearity of Eq. (11), the temperature variation induced at point (x, y, z) at time t by the instantaneous heat source q (x', y', z', t') applied at (x', y', z') at time t' is: q (x', y', z', t') G (x – x', y – y', z – z', t – t'). (12)

∆T ( x, y, z , t ) = ∫

t

0

q ( x' , y' , z' , t' )G ( x − x' , y − y' , z − z' , t − t' )dt.

∫ ∫ ∫

q ( x' , y' , z' , t' ) ×

×G ( x − x' , y − y' , z − z' , t − t' )dx dy' dz ′ dt' . (13)

In the present work, the temperature distribution considering an oval shape heat source was found to be: ∆T ( x, y, z , t ) = ∫

t

0

×

Q0 1 × × 2 3 / 2 ρ C π  4πα ( t − t' )  3/ 2 p  

2 2 abc × 1e m / ( 4a ) × I x × I y × I z dt ′, 3/ 2 π

(14)

where Ix, Iy and Iz are calculated as Eqs. (15) to (17). ∞

Iz = ∫ e

(−

( z − z' )2 ) 4α ( t −t' )

−∞

It is assumed that the heat is continuously generated at point (x', y', z') from t' = 0, thus the temperature increment at point (x, y, z) at time t is given as:

t

0 −∝ −∝ −∝

tπα (t − t' )

=

4cα f ( x' )(t − t' ) + 1 ∞

Iy = ∫ e

(−

( y − y' )2 ) 4α ( t −t' )

−∞

× e − ( cz' 

=

× e − ( cy' 

tπα (t − t' ) 4bα e mx' (t − t' ) + 1

A Study of Thermal Behaviour during Submerged Arc Welding

×e

×e

2

) emx'

2

 dz' = 

cemx' z 2 4 cf ( x' )α ( t −t' ) +1

) emx'

, (15)

 dz' = 

bemx' y 2 4 bemx'α ( t −t' ) +1

, (16) 335


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 333-338

Ix = ∫ e

(−

( x − x' )2 ) 4α ( t −t' )

−∞

 ( x − x' )2   4α t 

−  qv dTwb = e 3/ 2 C p ρ (4πα t )

× e − ( ax' )  × I y × I z dx' =   2

 tπα (t − t' ) = × mx  (4bα (t − t' )e + 1)(4cα (t − t' )e mx + 1) ×e

−(

bz 2 e− mx by 2 emx − ) 4 bα ( t −t' ) emx +1 4 cα ( t −t' ) emx +1

 2 × e − ax  x +  

 tπα (t − t' ) × + − mx  (4bα (t − t' )e + 1)(4cα (t − t' )e − mx + 1) ×e

−(

bz 2 e− mx by 2 e− mx − ) 4 bα ( t −t' ) emx +1 4 cα ( t −t' ) emx +1

×e

− ax 2 −

4 x2 4α ( t −t' )

1 1 ∆T ( x, y, z , t ) = ∫ × × 3/ 2 3/ 2 0 2 8 ρ C pπ [πα (t − t' ) ] t

1 abc × m2 /( 4 a ) × Q0 × I xv × I y × I z dt' . 3/ 2 π e

×e

 ( y − y' )2  −   4α t   

×e

 ( z − z' ) 2  −   4α t   

dz' dy' dx',

where qv is a volumetric heat source at point (x', y', z') due to the molten weldpool. The corresponding total increment in temperature (ΔTwb) for deposition of the liquid melt is: ∆Twb ( x, y, z , t ) = ∫

t

∆l / 2

B/2

∫ ∫

4 Ry 2 / B 2 + P + R

0 − ∆l / 2 − B / 2 4 Ry 2 / B 2

  x. (17)  

As this work considers a moving electrode during welding, the corresponding effect is incorporated in the Ix term as Ixv˝= f (x–vt'). Eq. (18) thus becomes:

×

×

×e

 ( x − x' )2  −   4α t   

×e

 ( y − y' )2  −   4α t   

×e

 ( z − z' )2  −   4α t   

qv × C p ρ (4πα t )3/ 2

dz' dy' dx' dt ,

(22)

where, qv is the volumetric heat source rate. Hence, the total increment in the temperature during welding is given as: T(x, y, z, t) – T∞ = ΔT(x, y, z, t) + Twb(x, y, z, t), (23) where T(x, y, z, t) is the variable temperature of the welded plate and T∞ is the initial temperature of plate.

(18)

4 EFFECT OF MOLTEN METAL ON THE TEMPERATURE INCREMENT The present work also considers the effect of molten liquid (weldpool) on the temperature increment. A schematic of a parabolic shaped weldpool is shown in Fig. 4 where a local coordinate is considered at the top of the weldpool. The top surface of the plate is shown by a dotted line in Fig. 4. The weld reinforcement (the part of weldpool above the top surface of plate) is given as: z=

4R 2 y , (19) B2

where R is the maximum reinforcement height. In the case of penetration depth, the equation of the parabola is given as: z=

4R 2 y + P + R, (20) B2

Hence, the total amount of heat delivered by the weldpool is:

Q=∫

∆l / 2

B/2

∫ ∫

4 Ry 2 / B 2 + P + R

− ∆l / 2 − B / 2 4 Ry 2 / B 2

qv dz'dy'dx'. (21)

Fig. 4. Geometry of a weldpool

5 CHANGE IN TEMPERATURE DUE TO CONVECTIVE HEAT LOSS To incoporate the heat loss due to convective heat transfer after welding, the welded plate is divided into a number of cubical segments (10×10×10 mm). It was found that the Biot number (Bi) for each segment is 0.003. Hence, based on the lumped capacitance model, the temperature, T(x, y, z, ti), at time (ti – t) after welding is given as:

T ( x, y, z , ti ) − T∞ = e −τ (ti −t ) , (24) T ( x, y, z , t ) − T∞

where τ is the time constant (τ = h /(ρLcCp)). Lc is the characteristic length of the welded plate.

The temperature increment is given as: 336

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 333-338

source. Eq. (23) represents the variation in temperature during welding and Eq. (24) represents the variation in temperature after welding. Fig. 5 shows the temperature variation at two different points in the plate over time. The point P1 (2.5 cm, 2 cm, 0) is near the weldpool and the point P2 (2.5 cm, 10 cm, 0) is far from the weldpool. It was found that as the electrode moves toward the points, the temperature of the points increases. Fig. 5a shows that the temperature is very high (~2000 °C) when the electrode comes near to point P1, which is a molten state. On the other hand, the maximum temperature at point P2 is about 475 °C. After welding, the temperature of both the points decreases with time as heat is transferred from the plate surfaces by convection. Subsequently, the predicted temperature is compared with the experimental measurements. A good agreement was found with the experiments. The present prediction is also compared for different shapes of heat sources. Fig. 6 shows the variation in temperature at point P1 for different shapes of the heat sources. It was found that the prediction considering an oval shape for the heat source provides a better approximation with the experiments.

a)

7 CONCLUSIONS b)

Fig. 5. Variation in temperature with time considering an oval shape heat source at: a) point P1 (2.5 cm, 1 cm, 0) and b) point P2 (2.5 cm, 10 cm, 0 cm)

The present work considers an analytical solution to temperature variation during submerged arc welding considering an oval shaped Gaussian heat source. The solution considers the effect of electrode movement and heat release from the weld pool during welding, as well as convective heat transfer from the plate surface after welding. The present prediction agrees well with the experiments. It was found that the oval shape is a better approximation for the heat source and that taking convective heat loss into account provides good accuracy after welding. 8 NOMENCLATURE

Fig. 6. Variation in temperature over time for different shape heat sources at point P1 (2.5 cm, 1 cm, 0 cm)

6 RESULTS AND DISCUSSION The present work predicts the temperature variation during submergerd arc welding based on an analytical solution considering an oval shape Gussian heat

ρ Cp k h T q t, t', ti x, y, z x', y, z' a, b, c, m

Density [kg/m3] Heat capacity [W/(kg·K)] Thermal conductivity [W/(m·K)] Convective heat transfer coefficient [W/(m2·K)] Temperature [°C] Heat density [W/m3] Time [s] Position where heat source is applied Point where temperature is measured Heat source parameters

A Study of Thermal Behaviour during Submerged Arc Welding

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Q0 I V ŋ T (x, y, z, t)

Heat input [W] Current [A] Voltage [V] Arc efficiency Temperature of welded plate at (x, y, z) point at time t Weld bead parameters L, B, C Penetration C/P ΔTwb (x, y, z, t) Change of temperature due to application of oval heat source [°C] ΔToa (x, y, z, t) Change of temperature due to heat transfer from molten electrode [°C] 7 REFERENCES [1] Goldak, J., Chakraborty, A., Bibby, M. (1984). A new finite element model for welding heat sources. Metallurgical Transactions B, vol. 15, no. 2, p. 299305, DOI:10.1007/BF02667333. [2] Gue, M., Goldak, J.A. (1994). Steady-state formulation for stress and distortion of welds. Journal of Engineering for Industry, vol. 116, p. 467-474, DOI:10.1115/1.2902130. [3] Younise, B., Rakin, M., Medjo, B. (2010). Numerical analysis of constraint effect on ductile tearing in strength mismatched welded CCT specimens using micromechanical approach. Tehnički vjesnik - Technical Gazette, vol. 17, no. 4, p. 411-418. [4] Ghosh, A., Chattopadhyaya, S., Hloch, S. (2012). Prediction of weld bead parameters, transient temperature distribution & HAZ width of submerged

338

arc welded structural steel plates. Tehnicki Vjesnik Technical Gazette, vol. 19, no. 3, p. 617-620. [5] Fachinotti, V.D., Cardona, A. (2008). Semi-analytical solution of the thermal field induced by a moving double-ellipsoidal welding heat source in a semiinfinite body. Mecanica Computacional, vol. 27, p. 1519-1530. [6] Eager, T.W., Tsai, N.S. (1983). Temperature fields produced by traveling distributed heat sources. Welding Journal, vol. 62, no. 12, p. 346-355. [7] Jeong, S.K., Cho, H.S. (1997). An analytical solution to predict the transient temperature distribution in fillet arc welds, Welding Journal, vol. 76, no. 6, p. 223-232. [8] Nguyen, N.T., Ohta, A., Suzuki, N., Maeda, Y. (1999). Analytical solutions for transient temperature of semiinfinite body subjected to 3-D moving heat source. Welding Journal, p. 265- 274. [9] Winczek, J. (2010). Analytical solution to transient temperature field in a half infinite body caused by moving volumetric heat source. International Journal of Heat Mass Transfer, vol. 53, p. 5774-5781, DOI:10.1016/j.ijheatmasstransfer.2010.07.065. [10] Winczek, J. (2011). A new approach to modelling of temperature field in surface steel elements. International Journal of Heat and Mass Transfer, vol. 54, no. 13, p. 4702-4709, DOI:10.1016/j. ijheatmasstransfer.2011.06.007. [11] Li, P., Lu, H. (2012). Hybrid heat source model designing and parameter prediction on tandem submerged arc welding. International Journal of Advanced Manufacturing Technology, vol. 62, Issue 5-8, p. 577-585, DOI:10.1007/s00170-011-3829-x. [12] Goldak, J.A., Akhlaghi, M. (2005). Computational Welding Mechanics. Springer, New York,NY 10013, USA, p. 71-115.

Ghosh, A. – Barman, N. – Chattopadhyay, H. – Hloch, S.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348 © 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.606

Review Scientific Paper

Received for review: 2012-05-23 Received revised form: 2013-02-11 Accepted for publication: 2013-03-29

A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement Tomov, M. – Kuzinovski, M. – Cichosz, P. Mite Tomov1,* – Mikolaj Kuzinovski1 – Piotr Cichosz2

1 “Ss.

Cyril and Methodius” University in Skopje, Faculty of Mechanical Engineering, Macedonia University of Technology, Institute of Production Engineering and Automation, Poland

2 Wroclaw

This paper presents the procedure for obtaining the primary profile, the waviness profile, and the roughness profile when measuring surface roughness, based on a set of recommendations from several ISO standards. An analysis was conducted and key insights into the procedure are given. In addition, a proposal to introduce a new parameter of statistical equality of sampling lengths when measuring surface roughness is included and the possible benefits from introducing the new parameter are stated. Keywords: parameter of statistical equality of sampling lengths, surface roughness, roughness profile, waviness profile, primary profile, mean value, standard deviation

0 INTRODUCTION Accelerated growth in the applications of surface metrology has led to significant changes in the approach to and study of surface metrology. However, the essence of surface metrology is that it must enable full control of surface constitution and provide an understanding of the characteristics that are important from a functional point of view. In [1] and [2] there is a detailed and systemic description of the genesis of the surface metrology discipline, as well as the expected future development directions. The development of surface metrology, presented in [2], is shifting away from profile towards aerial characterization, away from stochastic towards structural surfaces, and away from simple shapes towards complex freeform geometries. On the other hand, the current product surface roughness prediction models, at least for conventional machining, are based on a planar projection of the tool geometry (regardless of whether the geometry is defined or not) onto the surface of the work piece [3] and [4]. Hence the conclusion that studies of the roughness profile and the roughness parameters will continue to remain important, but that the emphasis will shift towards ways to determine them more accurately. This paper examines the conditions needed for a more accurate determination of the roughness profile and the roughness parameters by means of a new parameter: the statistical equality of sampling lengths.

1 PROCEDURE FOR OBTAINING THE ROUGHNESS PROFILE WHEN MEASURING SURFACE ROUGHNESS The procedure for obtaining the roughness profile, and thereby the roughness parameters, has evolved together with improvements in measuring devices. In order to realiably compare the measured values of the roughness parameters, the procedures and recommendations used to measure and obtain the roughness profile are usually included in the national standards on a local level or in the international standards on global level, and these are usually harmonized with one another. The paper provides an overview of the current ISO standards’ recommendations on obtaining roughness profiles using contact skidless instruments. This provides the procedure for obtaining the roughness profile illustrated in Fig. 1. Before the measurement process is started, the section of the surface that will be measured should be determined. The reference system is placed so that the x-axis runs perpendicular to the process traces (the lay). The intersection between the plane oriented along the x-axis and the surface irregularities determine the surface profile. The measurement can be made using any available measuring instrument that uses a stylus and a skidless probe. As the stylus traces the surface profile, the traced profile is determined by mechanical filtration, as shown in Position 4 in Fig. 1. The digitalized form of the traced profile facilitates the determination of the total profile, as shown in Position 5, which contains the nominal form, the roughness profile, and the waviness profile. The next step is to remove the nominal form from the total profile. If a flat surface is measured then, according to [5], the

*Corr. Author’s Address: “Ss. Cyril and Methodius” University, Faculty of Mechanical Engineering, Karpoš II. bb, 1000 Skopje, Macedonia, mitetomov@yahoo.com

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Real surface

1

ISO 4287:1997

2 z

y x

and other.

Traced profile

6

Nominal form (removed)

µm

5 4

ISO 3274:1996 ISO 4287:1997

5

Total profile

µm

20

3

15

2 1

10 5

0

0.5

1

1.5

2

2.5

3

3.5

0

4 mm

0

0.5

1

1.5

2

2.5

3

3.5

4 mm

8

Mean line for the primary profile

Primary profile

µm

1

Application of the profil filter λs

7

Noise

µm

0 -1

2

-2 0

0 .5

1

1 .5

2

2 .5

3

3 .5

1

4 mm

ISO 4287:1997; ISO 3274:1996; ISO 11562:1996; ISO 16610 series….

0

9

-1 -2 0

0.5

1

1.5

2

2.5

3

3.5 m m

Application of the profil filter λc 3 2

Application of the profil filter λf

11

1 0 -1

3.0

-2

2.0

-3 0

0.5

1

1.5

2

2.5

3

3.5

4

1.0

ISO 4287:1997

0.0

10 µm

ISO 4287:1997; ISO 3274:1996; ISO 11562:1996; ISO 16610 series….

0

ISO 3274:1996

4

Measujhgvvhhhbrem ent process

ISO 3274:1996

3 Measurement process

Mean line for the roughness profile

Roughness profile

-1.0 -2.0 -3.0 0

0.5

1

1.5

2

2.5

Waviness profile

3

3.5

4

ISO 4287:1997; ISO 3274:1996; ISO 11562:1996; ISO 16610 series….

ISO 3274:1996

ISO 4288:1996 ISO 3274:1996

Surface profile

1 0 -1 -2 0

0.5

1

1.5

2

2.5

3

3.5 mm

12 Roughness paremeter Ra, Rq, Rt, Rp, Rv…..

Fig. 1. Procedure for obtaining the primary profile, the roughness profile, and the waviness profile

most frequently used method is the least squares mean line method (Position 6). The process of software filtering using the profile filters λs, λc and λf begins from Position 7 onwards and the application of the λs profile filter facilitates the removal of the elements with small amplitudes and large frequencies (for example: noise) from the total profile. The application 340

of the λs profile filter, according to [6], always comes after the removal of the nominal form. The primary profile, shown in Position 8, is obtained after the removal of the nominal form, as well as the waves with small amplitudes and large frequencies, from the total profile. The primary profile is described using P parameters. When the λc profile filter is applied to

Tomov, M. – Kuzinovski, M. – Cichosz, P.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

the primary profile, the mean line for the roughness profile is obtained, and the roughness profile (Position 10 in Fig. 1) is obtained when the values of the mean line coordinates are subtracted from the values of the primary profile coordinates. The roughness profile is described using R parameters (roughness parameters). When the components with long waveforms are removed from the mean line obtained by application of the λc profile filter, the waviness profile is obtained (Position 11 in Fig. 1). The waviness profile is described using W parameters. 2 THE NEED FOR INTRODUCING A NEW PARAMETER The current definitions, mathematical formulations, and graphic interpretations of the parameters used to describe the primary profile and the roughness and waviness profiles are based on the M system, which requires the determination of the referent mean line in order to express and determine the parameters. The procedure presented in Fig. 1 clearly shows that the determination of the referent mean line for the purposes of determining the roughness profile is done through a filtering process using the λc profile filter. A great deal of research in the field of surface metrology is dedicated to studying the filtering as well as the metrological characteristics of the profile filters used. The primary objective that the profile filters need to fulfill is to have the determined mean line pass through the center of the irregularities comprising the profile through which the mean line passes. In practice, the most frequently used filter is the Gaussian filter. Its metrological characteristics for an open profile are standardized in [7] and [8]. Studies on the application of this profile filter have identified some disadvantages to its use [9] to [11]. Namely, the filter mean line determined using the Gaussian filter can be distorted at the ends of the profile as a result of the openness of the primary profile, which is not the case when applying the filter on closed profiles. Another disadvantage worth mentioning is its sensitivity to the deep grooves of the primary profiles, which leads to a false characterisation of the roughness profile in close proximity to the groove. An attempt has been made to remedy these disadvantages of the Gaussian profile filter with the introduction of ISO 13565-1:1996 [12], ISO 1661021:2011 [13], and ISO/TS 16610-28:2010 [14]. ISO 13565-1:1996 proposes a special filtering method using the Gaussian filter on primary profiles with deep grooves, ISO 16610-21:2011 introduces several types of new Gaussian filters, and ISO/TS 1661028:2010 presents a method for removing the possible

distortions at the end of the filter mean line. However, [9] and [10] emphasize that these deficiencies appear only when the primary profile is highly wavy, i.e. when the profile features sudden changes in the forms of the irregularities along its length. Additional momentum in the research activities related to filtration in surface metrology has been provided by the International Standardization Organization (ISO) which, according to [1], issued a document (а resolution) designated as ISO/TC 213 in 1996 and established a working group. The primary aim of this group is filtration research. The initiative to form such a group came from the needs of the industry as well as from the frequent debates about the disadvantages of the most frequently used filters at the time, i.e. the Gaussian and the 2RC filter. The results of the work of this group have facilitated the establishment of a filter development framework (including the standards and technical specifications that will be developed in the future), but on a solely mathematical basis. The documents derived from the work of the working group are presented as technical specifications (ISO/TS 16610 series). The new profile filters currently introduced by the ISO/TS 16610 series include: Gaussian filters [13], Gaussian regression filters [15], Spline filters [16] and [17], Spline wavelets [18], Disk and horizontal linesegment filters [19], Scale space techniques [20], and Motif filters (under development). During a real measuring process, the operator (the metrologist), shown in Position 9 in Fig. 1, faces a situation where he/she has to select an appropriate λc profile filter with metrological characteristics depending on the characteristics and the shape of the primary profile. However, the operator assesses the state (shape and character) of the primary profile qualitatively (high level of waviness, low level of waviness, significant or insignificant end distortions, etc.). This qualitative assessment of the primary profile is based on the experience and the conviction of the operator and can lead to an inappropriate selection of a profile filter type. The in-depth analysis regarding the filtering process and the use of profile filters suggests that if the irregularities that comprise the total profile are evenly and uniformly accumulated around the mean line used to determine the primary profile, then the probability of drawing mean lines using any existing λc profile filter that will not fulfill the requirement to pass through the middle is minimal. Accordingly, the authors of this paper conclude that a qualitative assessment of the total or the primary profile should be replaced by a parameter that will

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provide information about the irregularities in the total or the primary profile around the mean line. Therefore, the introduction of the new parameter will help link the state of the profiles obtained from the topography with the methodology for obtaining the parameter values, as shown in Fig. 1, irrespective of whether they are parameters of the primary, waviness or the roughness profile, with an ultimate view to determining these parameters as accurately as possible. 3 A NEW PARAMETER FOR THE STATISTICAL EQUALITY OF SAMPLING LENGTHS Our proposal is to have the new parameter determined after the drawing of the mean line for determining the primary profile, position 8 of Fig. 1, and the information that it should provide about the shape (state) of the profile should take into account the accumulation of the statistical characteristics around one value (the mean value). If the surveyed profile obtained from the surface is considered as a set of points in a reference system, it is clear that the magnitudes that have to be included in the determination of the new parameters are the basic

z

Mean value of primary profile

distribution measures, i.e. mean arithmetic value, variation (dispersion), and the standard deviation. To determine whether a function is stationary and ergodic or not, the mean value (statistical and temporal) for a given part of the function (of an exact size) should be compared to other parts of the function of the same size [21]. Henceforth, any further research will make use only of the methodology for comparing parts of the function, regardless of whether the profile is a stationary and an ergodic function. If this methodology is applied to the primary profile represented by discrete values in a reference system, then the statistical characteristics of the irregularities of a segment of the profile need to be compared to other segments of irregularities along the profile. The segments (parts) whose basic statistical characteristics will be compared to each other are considered to be equal to the sampling length lr. This value has already been standardized in [5]. This allows a direct connection to the software filtration, where the profile filter size is equal to the sampling length. For the purposes of deriving the mathematical formulation of the new parameter, we will consider two theoretical cases of scattering the irregularities around the mean line.

Approximately equal standard deviations

Primary profile

x Different mean value

Sampling length

Standard deviation of primary profile

Fig. 2. Case one: different mean values and approximately equal standard deviations

z

Primary profile

Different standard deviations

x

Sampling length

Approximately equal mean value

Mean value of primary profile

Fig. 3. Case two: approximately equal mean values and different standard deviatios

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

Case one: The points used to represent the primary profile within the sampling lengths have different mean values and approximately equal standard deviations, as shown in Fig. 2. Case two: The points used to represent the primary profile within the sampling lengths have approximately equal mean values and different standard deviations, as shown in Fig. 3. The primary profiles in Figs. 2 and 3 are presented on a Cartesian right-oriented reference system, where the horizontal longitudinal axis is marked with x, while the vertical axis is the z-axis. The points used to represent the primary profile with its evaluation length and the sampling lengths, as parts of the primary profile, are considered to be the sample, since they are a part (sample) of the points that can be used to represent the entire surface topography. The points used to represent the entire surface topography are considered to be the population. Hence, the standard deviation for n data points is s, while the mean value for n data points z . For the graphical presentation of the deviations sizes in Figs. 2 and 3, we have adopted the normal (Gaussian) distribution, although it is possible to have an abnormal distribution of irregularities within the sampling lengths. In the first case, as a result of the scattering of the mean values of the sampling lengths around the mean line (mean value) of the primary profile, the standard deviation determined for the primary profile will be greater than the standard deviations of the individual sampling lengths. It can be observed that the lower the scatter of the mean values of the sampling lengths, the lower the standard deviation for the primary profile, which means that it will converge to the standard deviation values of the sampling lengths. In order to describe this interdependence mathematically, we introduce the coefficient Ks, which we propose to calculate as follows: 2

Ks =

s , (1) s 2p

where s is the standard deviation calculated as the mean value of the individual standard deviations within the sampling length, i.e.:

2

s =

s12 + s22 + s32 + s42 + s52 . (2) 5

Eq. (2) applies when the total profile contains five sampling lengths, while, if there are n sampling lengths, s will be calculated as follows:

2

s =

s12 + s22 + ... + sn2 , (3) n

where sp, in Еq. 1, is the standard deviation calculated for the primary profile. The standard deviations s1, s2, ..., sp for n data points will be calculated as [22] to [24]:

s1 , s2 ,..., s p =

2

1 n ∑ zi − z , (4) n − 1 i =1

(

)

where z is the mean value of the points zi which represent the profile, and if there are n points, the calculation is as follows: 1 n z = ∑ zi . (5) n i =1 As shown in Eq. (4), deviations are determined using a mathematical formulation that is adequate for a normal (Gaussian) distribution. This prerequisite is necessary because if we first check the shape of the distribution of the irregularities and then use an appropriate mathematical formulation to calculate the deviation, we will inevitably face the problem of comparing the values obtained using different mathematical formulations, which is unacceptable. Although the magnitude of s2 is mathematically equal to the variance in this study, we have chosen to use the standard deviation as it is a more elementary form than the distributions measures. The value of the coefficient Ks will converge to one if the mean values of the sampling length data points are equal or approximately equal to the mean value of the primary profile, provided that the sampling length data points have equal or approximately equal standard deviations. The value of the coefficient Ks will converge to zero if the mean values of the data points in the sampling lengths are significantly different from the mean value of the primary profile. For the second case, we propose to introduce a coefficient Ksm which will be calculated as follows:

K sm =

2 2 smax − smin , (6) 2 smax

where smax is the value of the maximal standard deviation calculated within the sampling lengths, while smin is the value of the minimal standard deviation calculated within the sampling lengths.

A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

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The value of the coefficient Ksm will converge to zero if there are no differences between the values of the standard deviations determined within the sampling lengths. If the value of Ksm converges to one, then that means that there are large differences among the values of the standard deviations determined within the sampling lengths. The fact that real primary profiles are a combination of the two previously described elementary cases suggests that the expression of the scattering of the irregularities around the mean line, represented using points in a reference system, requires the interdependence of the coefficients Ks and Ksm. Basically, the coefficient Ksm complements Ks on two grounds. Firstly, the introduction of Ksm removes the condition that the values of the standard deviations within a given sampling length are approximately equal to each other. Secondly, the value s is a mean value, which means that if the value of any standard deviation within a sampling length deviates significantly from the others, its real impact will be reduced due to the averaging. It should be emphasized that the values that the coefficients Ks and Ksm may have are inversely proportional, which means that for a primary profile where the irregularities are evenly accumulated around the mean value (mean line), Ks gets a value that converges to one, while Ksm gets a value that converges to zero, and vice versa. In practice, the real profiles obtained by measuring the real topography of surfaces usually exhibit irregularity deviations, which are a combination of the profiles from the first and the second case. As a consequence of this, we propose to introduce a new non-dimensional parameter called the parameter of statistic equality of sampling lengths, denoted by SE, which will be calculated as:

SE =

K sm . (7) Ks

In the opposite case: K sm → 1 → ∞. Ks → 0

The correlations described above suggest that the parameter of statistic equality of sampling lengths SE is an increasing parameter in the case where there is increasing scattering of the data points around the mean line. 4 VERIFICATION OF THE SE PARAMETER The research activities related to the verification of the SE parameter have considered 70 different primary profiles obtained by measuring the etalon surfaces representative of various processes and real surfaces. The paper will show only a part of the analyzed primary profiles. The surfaces were measured using a MarSurf XR20 stylus measuring system. The nominal form was removed from the total profile using the least squares method.

3 2 1

z

0

[µm] -1 -2 -3

Psk = -0.050; Pku = 1.754; Ks = 0.987; Ksm = 0.045; SE = 0.046;

-4 0

344

K sm → 0 → 0. Ks → 1

0.5

1

1.5

2

2.5

3

3.5

4

x [mm]

Fig. 4. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of turning with Ra = 1.6 μm (The specified values for the parameter Ra on the etalon surface is declared by the manufacturer) 8

Gaussian mean line

Primary profile

6

In the case of a primary profile where the irregularities are evenly accumulated around the mean value (mean line), or, in other words, the data points within the sampling lengths have approximately equal statistic characteristics with each other as well as with the primary profile, then the parameter SE will have the following value:

Gaussian mean line

Primary profile

4

4

z

2 0

[µm] -2 -4

Psk = 0.452; Pku = 2.216; Ks = 0.994; Ksm = 0.078; SE = 0.078;

-6 0

0.5

1

1.5

2

2.5

3

3.5

x [mm]

Fig. 5. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of planning with Ra = 1.6 μm Tomov, M. – Kuzinovski, M. – Cichosz, P.

4


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

Primary profile

Gaussian mean line

Primary profile

1

0.4

0.5

0.2

0

z

[µm]

z

[µm]

-0.5

0 -0.2 -0.4 -0.6

-1

-0.8

Psk = -0.526; Pku = 2.048; Ks = 0.994; Ksm = 0.117; SE = 0.118;

-1.5 0

0.5

1

1.5

2

2.5

3

3.5

0

4

1.5

2

2.5

3

3.5

4

Primary profile

3

Gaussian mean line

2 1

z

0

[µm] -1 -2

Psk = -0.265; Pku = 3.126; Ks = 0.884; Ksm = 0.224; SE = 0.253; 0

0.5

1

1.5

2

2.5

3

3.5

-3 0

0.5

1

1.5

2

2.5

3

3.5

4

x [mm]

Fig. 7. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of flat grinding with Ra = 0.4 μm

Fig. 10. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of radial grinding with Ra = 0.8 μm

Gaussian mean line

Primary profile

0.6

Psk = -0.139; Pku = 2.711; Ks = 0.968; Ksm = 0.648; SE = 0.670;

-4

4

x [mm]

0.4

Primary profile

8

0.2

[µm]

1

x [mm]

Gaussian mean line

Primary profile

0.5

Fig. 9. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of face grinding with Ra = 0.1 μm

Fig. 6. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of turning with Ra = 0.4 μm 2.5 2 1.5 1 0.5 0 z -0.5 [µm] -1 -1.5 -2 -2.5 -3

Psk = -0.437; Pku = 3.758; Ks = 0.815; Ksm = 0.410; SE = 0.503;

-1

x [mm]

z

Gaussian mean line

0.6

Gaussian mean line

6

0

4

-0.2

2

-0.4

z

-0.6 -0.8

[µm]

Psk = -0.411; Pku = 3.701; Ks = 0.881; Ksm = 0.364; SE = 0.413;

-1 0

0.5

1

1.5

2

2.5

3

3.5

0 -2 -4 -6

4

x [mm]

Fig. 8. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of cross-face grinding with Ra = 0.1 μm

The Figs. 4 to 15 also show the Gaussian filter mean lines determined for the relevant primary profiles using Matlab (R2009b). The mathematical formulation provided in ISO 11562:1996 [7] and ISO 16610-21:2011 [13] was used as the weight function for the Gaussian filter. In order to investigate whether there is a dependence between the new parameter SE and the Gaussian (normal) nature of the primary profiles, skewness (Psk) and kurtosis (Pku) were calculated for the primary (P) profiles. The values for (Psk) and (Pku) are shown in the same figures.

-8 -10

Psk = -0.658; Pku = 3.021; Ks = 0.842; Ksm = 0.671; SE = 0.797;

-12 0

0.5

1

1.5

2

2.5

3

3.5

4

x [mm] Fig. 11. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of circular grinding with Ra = 1.6 μm

The disadvantages of the Gaussian mean lines can be seen from the profiles shown in Figs. 13 to 15. Fig. 14 (Detail B) and Fig. 15 (Details C and D) show notable distortions at the ends of the filter mean lines, while Fig. 13 shows that the filter mean line withdraws from the middle of the primary profile due to the deep grooves, which is clearly presented in Detail A.

A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

345


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

Primary profile

1

Gaussian mean line

0.5 0

z

[µm]

-0.5 -1 -1.5

Psk = -1.137; Pku = 5.013; Ks = 0.783; Ksm = 0.723; SE = 0.923;

-2 0

0.5

1

1.5

2

2.5

3

3.5

4

For primary profiles where the value of SE is less than one, disadvantages of the mean line determined using the Gaussian filter should not be expected. The application of the filtering procedure prescribed in ISO 13565-1:1996 [12] is justified due to the deep grooves of the primary profile, shown in Fig. 13. To recognize such primary profiles, the authors of this paper propose the use of the value of the Ks coefficient as an additional criterion.

x [mm]

0.05 0 -0.05

0

-0.15 -0.2

0

0.5

1

0.2

0.4

0.6

0.8

1

1.2

x [mm]

Psk = -5.526; Pku = 42.280; Ks = 0.960; Ksm = 0.970; SE = 1.01;

-12

Psk = -1.253; Pku = 4.158; Ks = 0.589; Ksm = 0.934; SE = 1.59; 0

Gaussian mean line

-8 -10

Gaussian mean line

-0.25

Primary profile

[µm] -6

Detail C

Detail D

[µm] -0.1

-2 -4

z

0.1

z

Detail A

2

Primary profile

0.15

Fig. 12. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of lapping with Ra = 0.2 μm

1.5

2

2.5

3

3.5

4

Fig. 15. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of super finish with Ra = 0.025 μm

x [mm]

Fig. 13. Primary profile, Gaussian mean, line and value of the SE parameter for the real surface obtained with honing 1.5

Primary profile

1

Detail B

0.5 0

z

[µm]

-0.5 -1

Gaussian mean line

-1.5

It can be concluded that for primary profiles with a periodic structure (the ones that do not have random height distribution), the value of SE is very low. In these primary profiles, the Gaussian mean line has no disadvantages (the mean line passes through the center of the irregularities), so the low values for SE are not significant as a criterion for selection of the Gaussian filter.

-2 -2.5

Psk = -0.736; Pku = 7.918; Ks = 0.556; Ksm = 0.724; SE = 1.303;

-3 0

0.5

1

1.5

2

2.5

3

3.5

4

x [mm]

Fig. 14. Primary profile, Gaussian mean line, and value of the SE parameter for the etalon surface representative of circular grinding with Ra = 0.1 μm

For the other profiles, the form of the filter mean line has no disadvantages and passes through the middle of the irregularities. An analysis of the calculated values of the SE parameter will show that its value increases in the event of an increased scattering of the irregularities around the mean value of the primary profile. For the considered primary profiles where the mean lines have disadvantages, the value of SE is greater than one or close to one. Hence, the authors of this paper came to the conclusion that one is the key value of the SE parameter from the point of view of software filtration using a standard Gaussian filter. 346

5 IMPLEMENTATION OF SE IN THE PROCEDURES FOR MEASURING SURFACE ROUGHNESS Based on the value of the SE parameter, this paper proposes an expansion of the current procedure for obtaining the primary profile, the roughness profile, and the waviness profile shown in Fig. 1 using the algorithm shown in Fig. 16. The proposal is to replace Position 9 of Fig. 1 with the algorithm shown in Fig. 16. 6 CONCLUSIONS The new SE parameter has proven to be a successful attempt to qualitatively determine the status (shape and character) of primary profiles. An SE value of one was shown to be a significant value from the point of view of filtration using a Gaussian filter. For primary profiles where the value of SE is less than one, disadvantages of the mean line determined using

Tomov, M. – Kuzinovski, M. – Cichosz, P.


Strojni拧ki vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

Calculation of SE and Ks

SE

1

No

Application of standard Gaussian profile filter 位c

Yes

Application of the filtering procedure ISO 13565-1:1996

Yes Ks > 0.9 No Application of one of the recommendations given in ISO / TS 16610-28:2010

Whether mean line has disadvantages

No

Roughness profile, Position 10

Yes Application of other profile filter (not standard Gaussian) from ISO 16610-series

Roughness profile, Position 10

Fig. 16. Algorithm modifying Position 9 of the measuring procedure shown in Fig. 1

the standard Gaussian filter should not be expected, and vice versa. These investigations propose a new approach to the determination of surface roughness, where the choice of the most appropriate profile filter directly depends on the shape of the primary profile. This paper proposes that the shape of the primary profile should be determined quantitatively, instead of qualitatively (as is the current practice), using the new parameter of statistic equality of sampling lengths SE. The calculated values for Psk and Pku showed that there is no dependence between the Gaussian nature of the primary profiles and the new parameter SE, since there is no dependency between the Gaussian nature of the primary profiles and the disadvantages of using the filter mean lines. The SE parameter can also be used as a tool for providing information about the stability of the production process. If the surface topography is constituted within a stable production process, then the irregularities will be accumulated around the mean line (value). An increased value of the SE parameter may, in some cases, indicate the following:

- Fig. 14 indicates the use of an inappropriate method for removing the shape from the total profile; - Inappropriate selection of the sampling length: In spite of the recommendations provided in [25] for the selection of the sampling length value depending on the form (character) of the profile, the selection can be an inappropriate selection if the profile is of a combined character and there are difficulties with the classification of the profile, i.e. it is unclear whether it belongs to nonperiodic or periodic profiles. In the future, it would be significant to investigate the dependence of the parameter SE on: - The differences in the shape of the primary profiles and their parameters obtained from two different measuring instruments that have different mechanical references. - Since the parameter SE is calculated for the primary profile, i.e. after the levelling of the total profile, it can be used as a parameter for comparing the various methods of levelling.

A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, 339-348

7 REFERENCES [1] Jiang, X., Scott, P.J., Whitehouse, D.J., Blunt, L. (2007). Paradigm shifts in surface metrology. Part I. Historical philosophy. Proccedings of the Royal Society, vol. 463, p. 2049-2070, DOI:10.1098/rspa.2007.1874. [2] Jiang, X., Scott, P.J., Whitehouse, D.J., Blunt, L. (2007). Paradigm shifts in surface metrology. Part II. The current shift. Proccedings of the Royal Society, vol. 463, p. 2071-2099, DOI:10.1098/rspa. 2007.1873. [3] Stanisław, A., Edward, M., Čuš, F. (2009). A model of surface roughness constitution in the metal cutting process applying tools with defined stereometry. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 1, p. 45-54. [4] Edward, M., Łukasz, N. (2012). Analysis and verification of surface roughness constitution model after machining process. Procedia Engineering, vol. 39, p. 395-404, DOI:10.1016/j.proeng.2012.07.043. [5] ISO 4287:1997. Geometrical Product Specifications (GPS) - Surface texture: Profile method - Terms, definitions and surface texture parameters. International Organization for Standardization, Geneva. [6] ISO 3274:1996. Geometrical Product Specifications (GPS) - Surface texture: Profile method - Nominal characteristics of contact stylus instruments. International Organization for Standardization, Geneva. [7] ISO 11562:1996. Geometrical Product Specifications (GPS) - Surface texture: Profile method - Metrological characteristics of phase correct filters. International Organization for Standardization, Geneva. [8] ASME B46.1 (2009). Surface Texture (Surface Roughness, Waviness, and Lay). The American Society of Mechanical Engineers, New York. [9] Raja, J., Muralikrishnan, B., Fu, S. (2002). Recent advances in separation of roughness, waviness and form. Journal of the International Societies for Precision Engineering and Nanotechnology, vol. 26, no. 2, p. 222-235, DOI:10.1016/S0141-6359(02)001034. [10] Whitehouse, D.J. (2011). Handbook of Surface and Nanometrology, Second edition. CRC Press, Taylor & Francis Group, Boca Raton, USA. [11] Muralikrishnan, B., Raja, J. (2009). Computational Surface and Roughness Metrology. Springer, London. [12] ISO 13565-1:1996. Geometrical Product Specifications (GPS) - Surface texture: Profile method; Surfaces having stratified functional properties - Part 1: Filtering and general measurement conditions. International Organization for Standardization, Geneva.

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[13] ISO 16610-21:2011. Geometrical product specifications (GPS) - Filtration: Linear profile filters: Gaussian filters. International Organization for Standardization, Geneva. [14] ISO/TS 16610-28:2010. Geometrical product specifications (GPS) – Filtration - Part 28: Profile filters: End effects. International Organization for Standardization, Geneva. [15] ISO/TS 16610-31:2010. Geometrical product specifications (GPS) - Filtration: Robust profile filters: Gaussian regression filters. International Organization for Standardization, Geneva. [16] ISO/TS 16610-32:2009. Geometrical product specifications (GPS) - Filtration: Robust profile filters: Spline filters. International Organization for Standardization, Geneva. [17] ISO/TS 16610-22:2006. Geometrical product specifications (GPS) - Filtration: Linear profile filters: Spline filters. International Organization for Standardization, Geneva. [18] ISO/TS 16610-29:2006. Geometrical product specifications (GPS) - Filtration: Linear profile filters: Spline wavelets. International Organization for Standardization, Geneva. [19] ISO/TS 16610-41:2006. Geometrical product specifications (GPS) - Filtration: Morphological profile filters: Disk and horizontal line-segment filters. International Organization for Standardization, Geneva. [20] ISO/TS 16610-49:2006. Geometrical product specifications (GPS) - Filtration: Morphological profile filters: Scale space techniques. International Organization for Standardization, Geneva. [21] Wozencraft, J.M., Jacobs, I.M. (1965). Principles of communication engineering. John Wiley & Sons, New York, London, Sydney. [22] ISO 13565-3:1998. Geometrical Product Specifications (GPS) - Surface texture: Profile method; Surfaces having stratified functional properties - Part 3: Height characterization using the material probability curve. International Organization for Standardization, Geneva. [23] Bury, K. (1999). Statistical distributions in engineering. Cambridge University Press, Cambridge, DOI:10.1017/ CBO9781139175081. [24] Taylor, J.K., Cihon, C. (2004). Statistical techniques for data analysis (Second edition). Chapman & Hall/ CRC Press, Boca Raton, USA. [25] ISO 4288:1996. Geometrical product specifications (GPS) - Surface texture: Profile method - Rules and procedures for the assessment of surface texture. International Organization for Standardization, Geneva.

Tomov, M. – Kuzinovski, M. – Cichosz, P.


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5 Vsebina

Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 59, (2013), številka 5 Ljubljana, maj 2013 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki člankov Dunja Ravnikar, Primož Mrvar, Jožef Medved, Janez Grum: Mikrostrukturna analiza lasersko oplastene aluminijeve zlitine EN AW-6082-T651 s keramičnima sestavinama TiB2 in TiC Gang Cheng, Peng Xu, De-hua Yang, Hui Li, Hou-guang Liu: Analiza kinematike novega vzporednega manipulatorja 3CPS s pomočjo Rodriguesovih parametrov Hamid Reza Vosoughifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, Seyed Reza Hashemi Nezhad: Vrednotenje toka fluida čez stopničasti preliv z novim pristopom po metodi končnih volumnov Milosav Ognjanovic, Miroslav Milutinovic: Konstruiranje za zanesljivost z namenom zagotavljanja zmogljivosti avtomobilskih menjalnikov Jiehui Zou, Qungui Du: Model kocke funkcijskega utemeljevanja za konceptualno snovanje mehatronskih sistemov Aniruddha Ghosh, Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: Študija toplotnih razmer pri obločnem varjenju pod praškom Mite Tomov, Mikolaj Kuzinovski, Piotr Cichosz: Nov parameter statistične enakosti referenčnih dolžin pri merjenju površinske hrapavosti

SI 55 SI 56 SI 57 SI 58 SI 59 SI 60 SI 61


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 55 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-12-07 Prejeto popravljeno: 2013-02-26 Odobreno za objavo: 2013-03-13

Mikrostrukturna analiza lasersko oplastene aluminijeve zlitine EN AW-6082-T651 s keramičnima sestavinama TiB2 in TiC Dunja Ravnikar1 – Primož Mrvar2 – Jožef Medved2 – Janez Grum1,* 1 Univerza

2 Univerza

v Ljubljani, Fakulteta za strojništvo, Slovenija v Ljubljani, Naravoslovnotehniška fakulteta, Slovenija

Članek predstavlja lasersko oplastenje aluminijeve zlitine EN AW-6082-T651 s prednanosom keramičnih sestavin TiB2-TiC z dodatkom aluminija v prašnati obliki. Za lasersko oplastenje sta značilni dobra ponovljivost procesa z dobro kemično zvezo med oplastenim slojem in substratom ter manjša poroznost. Laserska tehnologija omogoča tudi naknadno toplotno obdelavo s ciljem izboljšanja končnih lastnosti izdelka. Oplastenje aluminijevih zlitin s keramičnimi sestavinami je zelo zahteven postopek zaradi združevanja dveh različnih vrst materialov z različnimi fizikalnimi lastnostmi. Keramične sestavine se od aluminijeve zlitine zelo razlikujejo po temperaturi tališča in toplotnem raztezanju, imajo pa tudi slabo omočljivost. Zato je zelo pomembna ustrezna izbira keramičnih sestavin z dodatki in optimalna izbira laserskih parametrov, da se doseže kakovosten oprijem obloge s substratom. Za dobro kemično zvezo med oplastenim slojem in substratom smo morali zagotoviti zadosten vnos energije, ki je delno natalil vnaprej naneseno mešanico praška, kakor tudi tanek površinski sloj substrata. Uporabljen je bil vlakenski laserski vir proizvajalca IPG Photonics, maksimalne moči 3 kW s kontinuiranim delovanjem v infrardečem območju pri valovni dolžini 1070 nm. Izbrali smo dve mešanici praška (40 mas.% TiB2 / 40 mas.% TiC / 20 mas.% Al in 60 mas.% TiB2 / 20 mas.% TiC / 20 mas.% Al) in ju nanesli na čisto površino substrata pred izvedbo laserskega oplastenja. Oplastenje je bilo izvedeno s tremi različnimi močmi laserskega snopa (800, 1000 in 1200 W), s hitrostjo pomika 60  mm/s in velikostjo pege na površini 1,0  mm. Postopek je bil izveden z dvema stopnjama prekrivanja laserskih sledi (30 in 50%). Nastalo oblogo smo preučili z mikrostrukturno in mikrokemično analizo ter izmerili mikrotrdoto v oblogi in v delu substrata pod oblogo. Delci TiC in TiB2 v nastali aluminijevi matrici se razlikujejo po obliki in velikosti. Obloga je dobro oprijeta s substratom, brez razpok in s povprečno poroznostjo nižjo od 2%. Termodinamična analiza sistema je pokazala možen obstoj aluminijevega karbida Al4C3 v oblogi, medtem ko je energijska disperzijska spektroskopija oz. analiza EDS pokazala možen nastanek aluminijevih oksikarbidov v oblogi poleg prisotnosti TiB2, TiC in Al. Dodatno izvedena termična analiza pa je nakazala obstoj dveh ločenih eksotermnih vrhov, iz katerih lahko sklepamo o domnevni postopni precipitaciji faze Mg2Si kot tipičnega precipitata v aluminijevi zlitini 6082. Mikrotrdota v oblogi je v povprečju za 40% višja kot pri substratu, medtem ko je mikrotrdota v lasersko pretaljeni in toplotno vplivani coni znatno nižja od neoplastene aluminijeve zlitine. Spremenjena mikrostruktura v pretaljenem področju pod oblogo se odraža v nižji trdoti, kar potrjuje spremembo iz izločevalno utrjenega stanja v gašeno stanje po oplastenju. Toplotni vplivi med procesom oplastenja in hitrega ohlajanja prispevajo k nastanku prisilne raztopine z zmanjšano mikrotrdoto. Z nanosom različnih keramičnih sestavin z različnimi dodatki lahko povečamo trdoto v oplastenem sloju ter zagotovimo izboljšano obrabno in kemično obstojnost. Oplastenje aluminijevih zlitin s keramičnimi sestavinami je vedno bolj zanimivo za procesno, kemično in avtomobilsko industrijo ter za uporabo pri gradnji delov v različnih termoenergetskih strojih in napravah. Lahki konstrukcijski materiali dobijo s tako oblogo izboljšane lastnosti površine, kar prinaša pomemben tehnološki napredek tudi pri gradnji lahkih motorjev in sestavnih delov strojev z izboljšano učinkovitostjo. Ključne besede: lasersko oplastenje, aluminijeva zlitina, keramični sestavini TiC-TiB2, mikrostrukturna analiza, termodinamična analiza, termična analiza, mikrotrdota

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

SI 55


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 56 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-08-06 Prejeto popravljeno: 2012-10-11 Odobreno za objavo: 2013-02-14

Analiza kinematike novega vzporednega manipulatorja 3CPS s pomočjo Rodriguesovih parametrov Gang Cheng1,* – Peng Xu1 – De-hua Yang2 – Hui Li2 – Hou-guang Liu1 1 Višja

šola za strojništvo in elektrotehniko, Kitajska rudarska in tehniška univerza, Kitajska astronomski observatoriji/Institut za astronomsko optiko in tehniko v Nanjingu, Kitajska akademija znanosti, Kitajska

2 Cacionalni

Astronomija se ukvarja z opazovanjem oddaljenih in šibkih nebesnih teles. Veliki optični sistemi so pomembni za astronomske raziskave, saj lahko zberejo več svetlobe in imajo boljšo kotno ločljivost. Z dimenzijami astronomskih teleskopov pa se povečuje tudi njihova občutljivost na zunanje motnje in večina velikih teleskopov bo zato v prihodnje razdeljena na segmente. Pri teh teleskopih so velika primarna zrcala razdeljena v več manjših in tankih segmentov. Glavna težava takšne izvedbe v primerjavi s teleskopi z enim samim zrcalom so napake pozicioniranja in orientiranja segmentnih zrcal. Zato je treba razviti mehanizem za aktivno nastavljanje večjega števila zrcal z več prostostnimi stopnjami. Članek podaja predlog prototipa platforme za aktivno nastavljanje segmentnih zrcal, katerega srce je nov paralelni manipulator 3CPS. Prototip ima kompaktno zgradbo in omogoča gibanja z ločevanjem prostostnih stopenj. Analizirana je gibljivost manipulatorja, ki ima tri translacijske in tri rotacijske prostostne stopnje. Kinematična analiza paralelnega manipulatorja je osnova za dinamično analizo in snovanje krmilnega sistema, zaradi zahtev krmiljenja v realnem času in visoke računske učinkovitosti opisovanja položaja pa so za popis položaja gibljive platforme idealni Rodriguesovi parametri. Po metodi Rodriguesovih parametrov so izpeljane formule za reševanje inverzne/direktne kinematike – premikov, hitrosti in pospeškov. Skladno s topološkimi značilnostmi konstrukcije manipulatorja in zasnovo trajektorij gibljive platforme so nato z numerično simulacijo in na osnovi inverzne kinematike rešene dolžine, hitrosti in pospeški šestih premičnih nog. Numerična simulacija inverzne kinematike je bila opravljena s pomočjo programa Matlab znotraj obdobja 1 sekunde z intervalom 0,01 s. Iz rezultatov je razvidno, da so krivulje hitrosti in pospeškov zvezne ter nimajo nenadnih skokov. Pri gibanju premične platforme torej ne bo prihajalo do udarcev, kar je pomembno za dinamične lastnosti premične platforme. Za preverjanje pravilnosti teoretičnega kinematičnega modela je bila uporabljena fotogrametrična merilna metoda z eno kamero, ki meri položaj in orientacijo premične platforme. Po predstavitvi preskusnega okolja za fotogrametrične meritve je opisan tudi proces preizkušanja kinematične zmogljivosti ter konkretni koraki za merjenje različnih položajev. Z ozirom na učinkovitost in stroške meritev ni treba preizkušati celotnega obravnavanega območja. Za intuitivno primerjavo rezultatov eksperimenta z zasnovanimi trajektorijami so prikazane samo napake položaja in orientacije med dvema skupinama od 0 do 0,1 s pri času vzorčenja 0,01 s. Rezultati eksperimentov odstopajo od zasnovane trajektorije: napaka položaja je približno 0,2 mm, napaka orientacije pa približno 0,3°. Ob upoštevanju napak merilnega orodja, razlik med fizikalnim prototipom in simulacijskim modelom ter vplivov okolja pa se računske vrednosti v splošnem dobro ujemajo z rezultati eksperimenta, s čimer je potrjena primernost kinematičnega modela. Ključne besede: astronomski teleskop, segmentno zrcalo, aktivna nastavitvena platforma, paralelni manipulator 3CPS, Rodriguesovi parametri, kinematika, fotogrametrija

SI 56

*Naslov avtorja za dopisovanje: Višja šola za strojništvo in elektrotehniko, Kitajska rudarska in tehniška univerza 221008, Xuzhou, Kitajska, chg@cumt.edu.cn


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 57 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-06-18 Prejeto popravljeno: 2012-11-23 Odobreno za objavo: 2013-01-17

Vrednotenje toka fluida čez stopničasti preliv z novim pristopom po metodi končnih volumnov Hamid Reza Vosoughifar1,* – Azam Dolatshah2 – Seyed Kazem Sadat Shokouhi1 – Seyed Reza Hashemi Nezhad1 1 Islamska

univerza Azad, oddelek v južnem Teheranu, Fakulteta za gradbeništvo, Teheran, Iran univerza Azad, oddelek v Dezfulu, Fakulteta za gradbeništvo, Dezful, Iran

2 Islamska

Stopničasti preliv je hidravlična konstrukcija, ki kot integralni del jezu omogoča varno prelivanje tokov. Pri gradnji stopničastih prelivov se v zadnjih letih zaradi nizke cene in razmeroma visoke hitrosti gradnje pogosto uporablja tehnika dela z valjanim cementnim betonom. Rešitev z intenzivnim vključevanjem zraka (aeracijski tok) in stopnicami nad prelivom omogoča učinkovito disipacijo energije. To pomeni, da se zmanjša tveganje kavitacije, manjše pa so tudi dimenzije struktur za disipacijo energije pod prelivom. Zato je treba poiskati robustno in hitro metodo za ugotavljanje lastnosti tokov na stopničastih prelivih. Predstavljena raziskava je korak naprej pri iskanju učinkovite metode za preučevanje tokov na stopničastih prelivih. Zaradi natančnosti in hitrosti numeričnih metod je bila po principih računalniške dinamike fluidov (CFD) razvita numerična koda V-Flow za dvodimenzionalno (2D) modeliranje nestacionarnih tokov na stopničastih prelivih. Za reševanje vodilnih enačb toka fluida je bila uporabljena metoda končnih volumnov (FVM) in mreženje po Voronoiu. V-Flow je mogoče povezati s programsko opremo GAMBIT za triangulacijo. V-Flow lahko tako modelira različne geometrije prelivov s pomočjo mrežnih elementov po Voronoiu. Pri diskretizaciji je bila uporabljena shema Power Law, implicitna časovna aproksimacija, Gauss-Seidlova metoda in algoritem SIMPLE. Privzet je bil laminaren tok brez turbulenc. V-Flow lahko enostavno vrednoti lastnosti tokov na stopničastih prelivih, zlasti vektorje hitrosti, tokovnice, statični, dinamični in skupni tlak. Modelira lahko pojave kot so nastajanje vrtincev in recirkulacija na vogalih stopnic, visok pozitiven tlak zaradi stika toka z vodoravnimi ploskvami stopnic, negativen tlak blizu vertikalnih ploskev stopnic, kakor tudi tveganje kavitacije. Rezultati so bili validirani s pomočjo rezultatov, dobljenih iz modela aplikacije FLUENT. Gonzalezov eksperimentalni model je bil simuliran z V-Flowom in s FLUENT-om. Primerjava med rezultati modelov obeh aplikacij kaže dobro ujemanje z ozirom na masno neravnotežje, ki je uporaben kazalnik konvergence postopka reševanja ter kriterij ujemanja rezultatov V-Flowa in FLUENT-a. V-Flow je bil za namene tega članka prilagojen za modeliranje toka brez proste površine, ker je bil v eksperimentalnem modelu uporabljen spremenjen profil s širokim grebenom. Kot grebena je bil 15,94°, kar ustreza naklonu kanala. V prihodnjih raziskavah bo V-Flow mogoče prilagoditi tudi za tokove s prosto površino. Vključiti bo mogoče tudi teorijo turbulentnega toka ter uporabiti modele turbulence za boljše ocenjevanje pojava turbulence pri toku prek stopnic ter za primerjanje laminarnih in turbulentnih tokovnih stanj v postopku reševanja. Raziskava podaja kodo CFD za reševanje vodilnih enačb toka fluida prek stopničastih prelivov, prvič pa so bila uporabljena tudi numerična sredstva in metode kot sta FVM in mreženje po Voronoiu. Heksagonalni elementi mreže po Voronoiu pokrijejo računsko območje ter omogočajo dober izračun lastnosti v središčni točki na osnovi podatkov iz šestih sosednjih točk. Rezultati so zato natančnejši v primerjavi z drugimi mrežnimi elementi kot so trikotniki ali pravokotniki. V-Flow je uporaben za vrednotenje toka čez stopničaste prelive, dragi in zamudni eksperimenti pa praktično niso več potrebni. Ključne besede: stopničasti prelivi, valjani cementni beton, numerično modeliranje, metoda končnih volumnov, diskretizacija, mreža po Voronoiu

*Naslov avtorja za dopisovanje: Islamska univerza Azad, oddelek v južnem Teheranu, Fakulteta za gradbeništvo, Teheran, Iran, vosoughifar@azad.ac.ir

SI 57


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Prejeto v recenzijo 2012-05-21 Prejeto popravljeno: 2012-10-16 Odobreno za objavo: 2012-11-29

Konstruiranje za zanesljivost z namenom zagotavljanja zmogljivosti avtomobilskih menjalnikov Milosav Ognjanovic1,* – Miroslav Milutinovic2 2 Univerza

1 Univerza v Beogradu, Fakulteta za strojništvo, Srbija v Vzhodnem Sarajevu, Fakulteta za strojništvo, Bosna in Hercegovina

Glavni namen raziskave, predstavljene v članku, je razvoj integrirane metodologije, ki vključuje konstruiranje na osnovi značilnosti, robustno in aksiomatično konstruiranje za identifikacijo funkcijskih zahtev in opredelitev konstrukcijskih parametrov za kompleksne izdelke. Raziskava opredeljuje in uporablja zanesljivost kot konstrukcijsko omejitev, ter učinkovito in robustno usklajuje konstrukcijske parametre avtomobilskega menjalnika z delovnimi pogoji. Novi V-model konstrukcijskega procesa je bil osnova za razvoj integrirane metodologije za opredeljevanje konstrukcijskih parametrov (KP) in identifikacijo funkcijskih zahtev (FZ). Model vključuje dekompozicijo strukture zasnove, lastnosti usklajevanja osnovnih komponent in končno integracijo strukture zasnove in lastnosti za pridobivanje kazalnikov vedenja strukture zasnove oz. kazalnikov kakovosti zasnove. Zanesljivost je v tej raziskavi uporabljena kot funkcijska zahteva (želena celotna zanesljivost menjalnika), kot lastnost zasnove osnovnih komponent menjalnika, kot konstrukcijska omejitev pri opredelitvi KP in identifikaciji FZ (elementarna zanesljivost), ter kot kazalnik kakovosti menjalnika (izračunana celotna zanesljivost). Elementarna zanesljivost je osnovni element razvite in uporabljene metodologije. Ta zanesljivost je opredeljena na poseben način, primeren za ta namen, ter povezuje verjetnost obratovalnih pogojev (obremenitveni režim oz. obremenitveni spekter) in verjetnost odpovedi komponent konstrukcije, oz. verjetnost odpovedi strojnih elementov za določene vrste odpovedi. Obremenitveni režimi so identificirani z meritvami izhodnega momenta menjalnika v izbranih delovnih pogojih ter z vrednotenjem deležev posameznih pogojev. Vrednotenje je bilo opravljeno na osnovi rezultatov uporabljene metodologije intervjuvanja uporabnikov in vzdrževalcev. Verjetnost odpovedi je opredeljena na osnovi rezultatov laboratorijskih preizkusov osnovnih komponent, kot so zobniki, ležaji, sklopke in tesnila. Razviti računski model je osnova za razvoj računalniškega programa DRAG, ki omogoča izračunavanje konstrukcijskih parametrov, npr. zobniških dvojic, obremenljivosti zobniških dvojic in zmogljivosti celotnega menjalnika. Določena je metodologija za opredeljevanje konstrukcijskih parametrov in identifikacijo zmogljivosti avtomobilskih menjalnikov. Vhodni podatki za ta namen so bili delovni režim menjalnika, delovna doba in želena stopnja zanesljivosti menjalnika ob koncu življenjske dobe. Razviti računalniški program DRAG vsebuje pet modulov ter omogoča interaktivno izračunavanje zmogljivosti menjalnika za opredeljeni delovni režim, življenjske dobe in zanesljivosti ob koncu življenjske dobe. Omogoča tudi izračunavanje konstrukcijskih parametrov osnovnih komponent in stopnje celotne zanesljivosti med delovno dobo. Preizkušanje zanesljivosti avtomobilskih menjalnikov zahteva izjemno zahtevne in dolgotrajne postopke. Za preverjanje rezultatov izračunov je treba preizkusiti zanesljivost menjalnika v izbranih pogojih, trenutno pa ni pogojev za izvedbo takšnih eksperimentov. Glavni prispevek članka je v razvoju integrirane metodologije za robustno in aksiomatično konstruiranje kompleksnih izdelkov. Rezultati niso občutljivi na spreminjajoče se delovne pogoje. Zahtevana zanesljivost je glavna omejitev ter elementarna zanesljivost je nov pojem, ki je bil opredeljen in prilagojen za ta namen. Za preračunavanje avtomobilskih menjalnikov oz. usklajevanje konstrukcijskih parametrov in zmogljivosti z delovnimi pogoji je bil razvit računalniški program DRAG. Ključne besede: avtomobilski menjalniki, zanesljivost, robustno konstruiranje, aksiomatično konstruiranje

SI 58

*Naslov avtorja za dopisovanje: Univerza v Vzhodnem Sarajevu, Fakulteta za strojništvo, Vuka Karadzica 30, 71123 Vzodno Sarajevo, Bosna in Hercegovina, m.milutinovic82@gmail.com


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 59 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-08-27 Prejeto popravljeno: 2013-02-14 Odobreno za objavo: 2013-02-18

Model kocke funkcijskega utemeljevanja za konceptualno snovanje mehatronskih sistemov Jiehui Zou* – Qungui Du Fakulteta za strojništvo in avtomobilsko tehniko, Tehniška univerza Južne Kitajske, Kitajska

Funkcijsko utemeljevanje je pristop, ki vse bolj pridobiva na pomenu v tehniki. Bistvo funkcijskega utemeljevanja je ustvarjanje podfunkcij naslednje ravni na osnovi dane funkcije. V zgodovini se je pojavljalo več vplivnih pristopov ali modelov funkcijskega utemeljevanja za konceptualno snovanje. Vsak podaja okvir za celoten sistem utemeljevanja, a ko se ti modeli uporabijo za dejansko funkcijsko razgradnjo, se vedno pojavi splošno vprašanje: kako nastanejo podfunkcije nižje ravni? Avtorji članka so prepričani, da je problem predvsem posledica nejasnih relacij med podfunkcijami in funkcijo. Garbacz Pawel je pokazal, da je še vedno neznana tudi semantika relacije “x je podfunkcija od y”. Funkcijsko utemeljevanje je zato bolj kot od znanja v veliki meri odvisno od navdiha in od izkušenj konstruktorjev. Sistematični model, ki sta ga predlagala Pahl in Beitz, se razlikuje od drugih pristopov funkcijskega utemeljevanja. V njunem modelu so relacije med podfunkcijami in funkcijo opisane s tokovi. Relacije tako postanejo razmeroma jasne, model pa kljub vsemu ni popoln. Prvič je težko izluščiti (opredeliti) in izčrpno popisati vse vhodne in izhodne tokove po zahtevah stranke. Drugič funkcijska razgradnja nima definitivnega cilja za tri vrste tokov, ki so vključeni v opis celotne funkcije in v opis podfunkcij. Zato se spet pojavlja staro vprašanje: kako nastanejo podfunkcije nižje ravni? Če se tri vrste tokov analizirajo ločeno, je treba pri funkcijski razgradnji slediti samo spremembam tokov ene vrste. Cilj je tako jasen in funkcijsko utemeljevanje postane enostavno. Na ta način se izognemo napakam sistematičnega modela. Pri tej strategiji najprej prečistimo predstavitve funkcij. Ker nekateri vhodni in izhodni tokovi tehničnega sistema niso neposredno povezani z njegovo funkcijo, prečistimo tokove in funkcije opredelimo kot spremembo samo ene vrste toka (tok energije, materiala ali informacij). V naši prečiščeni različici tokovi funkcije torej vedno pripadajo samo eni vrsti. Tokovi te vrste se imenujejo ciljni tokovi, ostali tokovi pa se imenujejo pogojni tokovi. Holonomen mehatronski sistem je sestavljen iz treh funkcijskih podsistemov: izvršni, pogonski in upravljalni podsistem. Njihova naloga je ravnanje z materialom, pretvarjanje energij in upravljanje v realnem času, ciljni tokovi treh podsistemov pa so tokovi materiala, energije in informacij. Ključni prispevek te prečiščene predstavitve funkcij je v ločitvi treh vrst tokov. Na osnovi prečiščene predstavitve funkcij lahko celotno funkcijsko utemeljevanje holonomnega mehatronskega sistema ločimo v tri funkcijske razgradnje. Razgradnje morajo slediti samo spremembam ene vrste toka (ciljnega toka). Most med dvema funkcijskima razgradnjama je pogojni tok, ki ga opredeljuje reševanje koncepta. Rezultate funkcijske razgradnje in funkcijskega reševanja je mogoče jasno predstaviti v obliki kocke. V članku je podrobno pojasnjen postopek gradnje modela kocke, kakor tudi področja uporabe modela kocke. Model kocke je mogoče obravnavati kot izboljšan sistematični model. Razen prednosti sistematičnega modela ima tudi naslednje značilnosti: jasno utemeljevanje na osnovi fizikalnega znanja, enostavne rezultate in široke možnosti uporabe pri snovanju različnih tehničnih sistemov. Metodi funkcijskega utemeljevanja pri današnjem konceptualnem snovanju manjkajo definitivna pravila in v veliki meri je odvisna od izkušenj in navdiha konstruktorjev. Pomanjkanje pravil utemeljevanja ovira uporabo računalniških orodij pri funkcijskem utemeljevanju. Model kocke, ki je predlagan v tem članku, pretvori funkcijsko utemeljevanje v sledenje spremembam toka ene vrste. Model kocke lahko tako celovito izkorišča fizikalna znanja pri funkcijskem utemeljevanju ter odpira možnosti za računalniško podprto konceptualno snovanje in samodejno računalniško konceptualno snovanje. Ključne besede: model kocke, prečiščena predstavitev funkcij, funkcijsko utemeljevanje, proces snovanja, konceptualno snovanje, inženirsko konstruiranje

*Naslov avtorja za dopisovanje: Fakulteta za strojništvo in avtomobilsko tehniko, Tehniška univerza Južne Kitajske, Kitajska, zoujiehui@yahoo.cn

SI 59


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 60 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-09-03 Prejeto popravljeno: 2013-02-18 Odobreno za objavo: 2013-03-29

Študija toplotnih razmer pri obločnem varjenju pod praškom Aniruddha Ghosh1,* – Nilkanta Barman2 – Himadri Chattopadhyay2 – Sergej Hloch3 Vladna višja šola za tehniko in tekstil, Oddelek za strojništvo, Indija 2 Univerza Jadavpur, Oddelek za strojništvo, Indija 3 Tehniška univerza v Košicah, Fakulteta za proizvodne tehnologije, Slovaška republika 1

Za vzdrževanje kakovosti obločnega varjenja plošč iz konstrukcijskega jekla pod praškom je pomembna kritična raziskava prehodne porazdelitve temperature. Pri ugotavljanju prehodne porazdelitve temperature na varjenih ploščah je treba upoštevati konvektivne toplotne izgube. Konvektivne toplotne izgube obravnava le malo raziskovalnih člankov, pričujoči članek pa je eden od njih. Predstavljena študija obravnava termične razmere pri obločnem varjenju pod praškom ob upoštevanju ovalnega vira toplote. Proces varjenja je predstavljen z enačbo o ohranitvi energije, pri čemer je premična elektroda opisana kot Gaussov vir toplote ovalne oblike. Vodilna enačba je rešena na osnovi rešitve Greenove funkcije, nato pa je upoštevan vpliv toplote v talilni kopeli in konvekcije po varjenju. Napoved se dobro ujema z rezultati eksperimentov in ugotovljeno je bilo, da je ovalna oblika boljši približek za toplotni vir. Članek obravnava modeliranje prenosa toplote pri postopku obločnega varjenja pod praškom. Študija prehodne porazdelitve temperature v toplotno vplivani coni je pomembna za postopke obločnega varjenja pod praškom in je tudi cilj tega članka. Prehodna porazdelitev temperature je bila ugotovljena po analitični poti. Članek razkriva primerno obliko vira toplote za proces obločnega varjenja pod praškom in podaja analitično analizo konvektivnih toplotnih izgub. Pri analitičnem določanju prehodne porazdelitve temperature je bil upoštevan vpliv toplote v talilni kopeli in konvekcije. Napoved se dobro ujema z rezultati eksperimentov in ugotovljeno je bilo, da je ovalna oblika boljši približek za vir toplote. Praktični podatki o prehodni porazdelitvi temperature so uporabni tako za varilne inženirje kot za raziskovalce. Ključne besede: obločno varjenje pod praškom, Gaussov vir toplote ovalne oblike, parabolična talilna kopel, konvektivne toplotne izgube, sklopljeni sistem

SI 60

*Naslov avtorja za dopisovanje: Vladna višja šola za tehniko in tekstil, Oddelek za strojništvo, Indija, agmech74@gmail.com


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 61 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-05-23 Prejeto popravljeno: 2013-02-11 Odobreno za objavo: 2013-03-29

Nov parameter statistične enakosti referenčnih dolžin pri merjenju površinske hrapavosti Mite Tomov1,* – Mikolaj Kuzinovski1 – Piotr Cichosz2 1 Univerza

2 Institut

sv. Cirila in Metoda v Skopju, Fakulteta za strojništvo, Makedonija za proizvodni inženiring in avtomatizacijo pri Tehniški univerzi v Wroclawu, Poljska

V članku je podan pregled veljavnih priporočil standardov ISO za določanje hrapavosti s kontaktnimi instrumenti z referenčno ploskvijo. Shematsko je prikazan postopek določanja primarnega profila, hrapavosti in valovitosti. Pri realnih merilnih procesih se operaterji soočajo s tem, da morajo izbrati primeren filter λc z merilnotehničnimi lastnostmi, ki so odvisne od značilnosti in oblike primarnega profila. Operater pa stanje (obliko in karakteristike) primarnega profila vrednoti kvalitativno (visoka ali nizka stopnja valovitosti, pomembna ali nepomembna končna popačenja itd.). Kvalitativno ovrednotenje primarnega profila je odvisno od izkušenj in prepričanj operaterja, ki pa lahko privedejo tudi do izbire filtra neustrezne vrste. Avtorji članka so zaključili, da bi bilo treba kvalitativno vrednotenje celotnega oz. primarnega profila zamenjati s parametrom, ki bi vključeval informacije o odstopanjih celotnega oz. primarnega profila od srednjice. Zato predlagajo uvedbo novega brezdimenzijskega parametra, ki se imenuje parameter statistične enakosti referenčnih dolžin SE. Za matematično izpeljavo novega parametra sta uvedena dva teoretična primera porazdelitve odstopanj od srednjice primarnega profila. V prvem primeru imajo točke, ki predstavljajo primarni profil znotraj referenčnih dolžin, različne srednje vrednosti in približno enako standardno deviacijo, matematično pa so opisane s koeficienti Ks. V drugem primeru imajo točke, ki predstavljajo primarni profil znotraj referenčnih dolžin, približno enake srednje vrednosti in različne standardne deviacije, matematično pa so opisane s koeficienti Ksm. Pri verifikaciji parametra SE je bilo upoštevanih več različnih primarnih profilov, dobljenih z merjenem etalonov, ki predstavljajo različne procese in realne površine. Srednjice pri obravnavanih primarnih profilih imajo nepravilnosti, vrednost SE pa je večja od 1 ali blizu 1. Avtorji zato sklepajo, da je 1 ključna vrednost parametra SE z ozirom na programsko filtriranje s standardnim Gaussovim filtrom. Pri primarnih profilih, kjer je vrednost SE manjša od 1, ni pričakovati nepravilnosti srednjice, določene z Gaussovim filtrom. Članek predlaga razširitev postopka določanja primarnega profila, hrapavosti in valovitosti z ustreznim algoritmom na osnovi vrednosti parametra SE. Predlagani algoritem omogoča nov pristop k ugotavljanju površinske hrapavosti, kjer je izbira najprimernejšega filtra neposredno odvisna od oblike primarnega profila. Novi pristop je še posebej pomemben za merilne instrumente, ki ne omogočajo grafičnega prikaza primarnega profila (kar velja za večino prenosljivih instrumentov). V prihodnje bo treba raziskati, ali je parameter SE mogoče uporabiti tudi za razlike v obliki in parametrih primarnih profilov, ki so pridobljeni z dvema različnima merilnima instrumentoma z različnimi mehanskimi referencami. Parameter SE je izračunan za primarni profil, torej po glajenju celotnega profila, zato je uporaben za primerjavo različnih načinov glajenja. Parameter SE je uporaben tudi kot orodje za spremljanje stabilnosti proizvodnega procesa. Če topografijo površine določa stabilen proizvodni proces, bodo odstopanja zbrana okrog srednjice. Ključne besede: parameter statistične enakosti referenčnih dolžin, površinska hrapavost, profil hrapavosti, profil valovitosti, primarni profil, srednja vrednost, standardna deviacija

*Naslov avtorja za dopisovanje: Univerza sv. Cirila in Metoda v Skopju, Fakulteta za strojništvo, Karpoš II. bb, 1000 Skopje, Makedonija, mitetomov@yahoo.com

SI 61


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 62-63 Osebne objave

Doktorske disertacije, diplomske naloge

DOKTORSKE DISERTACIJE Na Fakulteti za strojništvo Univerze v Ljubljani so obranili svojo doktorsko disertacijo: ●    dne 11. aprila 2013 Gašper CAFUTA z naslovom: »Napredno načrtovanje preoblikovalnih orodij z upoštevanjem elastične povračljivosti preoblikovane pločevine« (mentor: prof. dr. Boris Štok, somentor: doc. dr. Nikolaj Mole); V delu so predstavljeni različni pristopi k načrtovanju geometrije preoblikovalnega orodja ob upoštevanju elastične povračljivosti in stanjšanja preoblikovane pločevine, ki sta pogost pojav pri preoblikovalnih procesih. Prikazan je tudi postopek razvoja nove iteracijske metode, ki ob natančnemu napovedovanju geometrije orodja z uporabo računalniških simulacij, zasnovanih na metodi končnih elementov, omogoča optimizacijo geometrije preoblikovalnega orodja tako, da se pri tem kompenzira odstopanje geometrije izdelka, ki je posledica elastične povrnitve izdelka po odstranitvi iz orodja in spremembe debeline stene izdelka. Metoda poleg tega omogoča tudi načrtovanje oblike platine in upoštevanje obreza izdelka, ki je izveden po preoblikovanju, pri načrtovanju geometrije orodja. Delovanje metode je prikazano na več numeričnih primerih, pri čemer se izkaže, da je njeno delovanje izredno robustno in učinkovito. Možnost uporabe metode v praksi je bila dokazana z eksperimentalno potrditvijo. DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv univerzitetni diplomirani inženir strojništva: dne 2. aprila 2013: Jernej OMOVŠEK z naslovom: »Sledenje procesov na osnovi brezžičnih senzorskih mrež« (mentor: prof. dr. Alojz Sluga); Gašper RESMAN z naslovom: »Postopek konstruiranja spenjalnih spojev iz polimernih materialov« (mentor: izr. prof. dr. Jože Tavčar, somentor: prof. dr. Jožef Duhovnik); dne 23. aprila 2013: Živa KERNIČ z naslovom: »Karakterizacija odkovka gonilne lopatice Francisove turbine« (mentor: izr. prof. dr. Roman Šturm, somentor: doc. dr. Tomaž Pepelnjak); SI 62

Anej MATEŽIČ z naslovom: »Optimiranje procesa varjenja okrova motorja« (mentor: prof. dr. Janez Tušek); METLIKOVIČ Klemen z naslovom: »Optimizacija časa menjav orodja na montažni liniji filtrov« (mentor: izr. prof. dr. Niko Herakovič); dne 24. aprila 2013: Blaž CIMPERMAN z naslovom: »Primerjava stroškov ogrevanja z različnimi energenti na nivoju koristne energije« (mentor: prof. dr. Vincenc Butala); Boštjan MEZINEC z naslovom: »Razvoj torzijskega zaznavala za preskuševališče dinamične trdnosti« (mentor: izr. prof. dr. Jernej Klemenc); Matej ŠKRJANC z naslovom: »Model učne tovarne za napredno izobraževanje proizvodnih inženirjev in tehnikov« (mentor: prof. dr. Peter Butala); Matej ŠTUPAR z naslovom: »Načrtovanje naprave za razvod čistih medijev« (mentor: izr. prof. dr. Ivan Bajsić). Aleš MARKOČIČ z naslovom: »Skupna učinkovitost opreme« (mentor: prof. dr. Marko Starbek, somentor: izr. prof. dr. Janez Kušar); Matic PAKIŽ z naslovom: »Obdelava kamnitih materialov na CNC obdelovalnem stroju« (mentor: prof. dr. Janez Kopač); Simon POZNIČ z naslovom: »Vrtanje slepe luknje v kompozitno leterv« (mentor: doc. dr. Davorin Kramar, somentor: prof. dr. Janez Kopač); dne 25. aprila 2013: Domen ČEMAŽAR z naslovom: »Zasnova in razvoj puhala recirkulacijo vodika v gorivnih celicah« (mentor: izr. prof. dr. Roman Žavbi Somentor: izr. prof. dr. Mihael Sekavčnik); Primož PETAČ z naslovom: »Tolerančna analiza leče in ohišja avtomobilskega žarometa« (mentor: doc. dr. Samo Zupan); Žiga RANČOV z naslovom: »Določitev porabe goriva in izpustov onesnažil mestnih avtobusov z različnimi zasnovami pogonskih sklopov« (mentor: izr. prof. dr. Tomaž Katrašnik); Urh SREDENŠEK z naslovom: »Energetski sistemi z elektrolizerjem, shranjevalnikom vodika in gorivno celico« (mentor: izr. prof. dr. Mihael Sekavčnik).


Strojniški vestnik - Journal of Mechanical Engineering 59(2013)5, SI 62-63

* Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister inženir strojništva: dne 22. aprila 2013: Marjan BUKŠEK z naslovom: »Preračun vrtilnega stolpa za kontinuirano litje jekla« (mentor: prof. dr. Iztok Potrč, somentor: prof. dr. Tone Lerher). * Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv univerzitetni diplomirani inženir strojništva: dne 26. aprila 2013: Simon PODGRAJŠEK z naslovom: »Razvoj lopatice 50 kilovatne vetrne turbine« (mentor: izr. prof. dr. Stanislav Pehan); Tomaž TISAJ z naslovom: »Uporaba bionike pri razvoju kompozitnih sestavnih delov vetrne elektrarne EKO 50 kW« (mentor: izr. prof. Vojmir Pogačar).

Luka NOVAK z naslovom: »Spravilo lesa z vprežno živino« (mentor: prof. dr. Jožef Duhovnik, somentor: prof. dr. Rajko Bernik); dne 15. aprila 2013: Andraž BAJDE z naslovom: »Preizkuševališče tesnil« (mentor: izr. prof. dr. Andrej Senegačnik); Luka KOCBEK z naslovom: »Določitev izstrelitvene tirnice večstopenjske rakete« (mentor: doc. dr. Viktor Šajn, somentor: prof. dr. Franc Kosel); Črt SAMBOLEC z naslovom: »Koncept in preliminarni preračun lahkega športnega reakcijskega letala« (mentor: izr. prof. dr. Tadej Kosel). *

Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani inženir strojništva (UN): dne 26. aprila 2013: Janez BITENC z naslovom: »Numerična analiza toka okoli lopatičnega profila vetrne turbine« (mentor: doc. dr. Ignacijo Biluš, somentor: prof. dr. Brane Širok);

Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 18. aprila 2013: Gregor URBANČIČ z naslovom: »Vzdrževanje avtoplinskih sistemov« (mentor: izr. prof. dr. Igor Drstvenšek, somentor: dr. Primož Pogorevc); dne 25. aprila 2013: Jure ŽILIČ z naslovom: »Pomen CIP sistemov čiščenja naprav za delovanje procesnih naprav« (mentor: prof. dr. Matjaž Hriberšek); dne 26. aprila 2013: Silvo DREVENŠEK z naslovom: »Računalniško podprto načrtovanje in planiranje proizvodnje v podjetju Tehcenter Ptuj« (mentorica: doc. dr. Nataša Vujica Herzog, somentor: Matjaž Vuzem).

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Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 11. aprila 2013: Roman HAFNAR z naslovom: »Razvoj nihajne rotacijske žage za razrez kamene volne« (mentor: prof. dr. Marko Nagode); Rok JELOVČAN z naslovom: »Visokotlačni preizkus hidravličnega valja« (mentor: doc. dr. Franc Majdič); Marjan KOVAČ z naslovom: »Hidravlični pogon polžnega transporterja za končno praznjenje silosov« (mentor: doc. dr. Franc Majdič);

Na Fakulteti za strojništvo Univerze v Mariboru sta pridobila naziv diplomirani inženir strojništva (VS): dne 26. aprila 2013: Andrej GOLOB z naslovom: »Vzdrževanje helikopterja BELL 412 v informacijskem sistemu Maximo 7« (mentor: doc. dr. Marjan Leber, izr. prof. dr. Igor Drstvenšek); Borut PRICA z naslovom: »Merilna proga s frekvenčno reguliranim elektromotorjem za črpalne sisteme« (mentor: doc. dr. Mitja Kastrevc, somentor: doc. dr. Ignacijo Biluš).

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SI 63


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 Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu, http://www.sv-jme.eu Print DZS, printed in 450 copies Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia

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59 (2013) 5

Chamber of Commerce and Industry of Slovenia Metal Processing Industry Association

Since 1955

Strojniški vestnik Journal of Mechanical Engineering

s

mož Mrvar, Jožef Medved, Janez Grum: alysis of Laser Coating Ceramic Components TiB2 and TiC y EN AW-6082-T651

Cover: Laser coating process of aluminum alloy with ceramics components is shown on metallographic cross sections and corresponding X-ray elemental distribution of elements.

Xu, De-hua Yang, Hui Li, Hou-guang Liu: ics of a Novel 3CPS Parallel Manipulator Based on ters

, Miroslav Milutinovic: ity as a Key Component in Automotive Gearbox Load tion

Du: oning Cube Model for Conceptual Design of Mechatronic

Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: l Behaviour during Submerged Arc Welding

aj Kuzinovski, Piotr Cichosz: of Statistic Equality of Sampling Lengths in Surface rement

Journal of Mechanical Engineering - Strojniški vestnik

ghifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, mi Nezhad: Flow over Stepped Spillways Using the Finite Volume Approach

year

no. 5 2013 59

X-ray elemental map of Ti

Specimen

X-ray elemental map of Al

volume

Optical microscopy crosssection

Image Courtesy: Laboratory for Materials Testing and Heat Treatment, Faculty of Mechanical Engineering, University of Ljubljana

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 Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. 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 Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. 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 President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia 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. We would like to thank the reviewers who have taken part in the peerreview process.

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http://www.sv-jme.eu

59 (2013) 5

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

281

Dunja Ravnikar, Primož Mrvar, Jožef Medved, Janez Grum: Microstructural Analysis of Laser Coating Ceramic Components TiB2 and TiC on Aluminium Alloy EN AW-6082-T651

291

Gang Cheng, Peng Xu, De-hua Yang, Hui Li, Hou-guang Liu: Analysing Kinematics of a Novel 3CPS Parallel Manipulator Based on Rodrigues Parameters

Hamid Reza Vosoughifar, Azam Dolatshah, Seyed Kazem Sadat Shokouhi, Seyed Reza Hashemi Nezhad: Evaluation of Fluid Flow over Stepped Spillways Using the Finite Volume Method as a Novel Approach

Milosav Ognjanovic, Miroslav Milutinovic: Design for Reliability Based Methodology for Automotive Gearbox Load Capacity Identification

301 311

Jiehui Zou, Qungui Du: 323 A Functional Reasoning Cube Model for Conceptual Design of Mechatronic Systems 333

Aniruddha Ghosh, Nilkanta Barman, Himadri Chattopadhyay, Sergej Hloch: A Study of Thermal Behaviour during Submerged Arc Welding

Mite Tomov, Mikolaj Kuzinovski, Piotr Cichosz: A New Parameter of Statistic Equality of Sampling Lengths in Surface Roughness Measurement

339

Journal of Mechanical Engineering - Strojniški vestnik

Contents

5 year 2013 volume 59

X-ray elemental map of Ti

no.

Specimen

X-ray elemental map of Al

Optical microscopy crosssection

Journal of Mechanical Engineering 2013 5  

The Strojniški vestnik – Journal of Mechanical Engineering publishes theoretical and practice oriented papaers, dealing with problems of mod...