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Research Papers

Optimal Designs for Vibrating Structures Using a Moving Isosurface Threshold Method With Experimental Study

[+] Author and Article Information
Quantian Luo

School of Aerospace,
Mechanical and Mechatronic Engineering,
The University of Sydney,
Sydney, NSW 2006, Australia

Liyong Tong

School of Aerospace, Mechanical
and Mechatronic Engineering,
The University of Sydney,
Sydney, NSW 2006, Australia
e-mail: liyong.tong@sydney.edu.au

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 29, 2014; final manuscript received May 26, 2015; published online July 14, 2015. Assoc. Editor: Corina Sandu.

J. Vib. Acoust 137(6), 061005 (Dec 01, 2015) (13 pages) Paper No: VIB-14-1273; doi: 10.1115/1.4030771 History: Received July 29, 2014; Revised May 26, 2015; Online July 14, 2015

This paper studies optimal designs for vibrating structures using a moving isosurface threshold method (MIST). In the present study, a combination of strain and kinetic energy densities is selected as a response function of natural frequency and then formulations to maximize a specific frequency, frequency separation, and average-mean are derived. An efficient algorithm is developed to find a moving isosurface threshold level for evolving the design boundary and updating the weighting factor. The present algorithm coupled with commercial finite element analysis (FEA) software is used to study optimal designs for vibrating structures. The obtained optimal designs are fabricated and the experimental tests are conducted to validate the optimal topologies.

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References

Bendsoe, M. P., and Sigmund, O., 2003, Topology Optimization: Theory, Methods and Applications, Springer, New York.
Bendsøe, M. P., 1989, “Optimal Shape Design as a Material Distribution Problem,” Struct. Optim., 1(4), pp. 193–202. [CrossRef]
Zhou, M., and Rozvany, G. I. N., 1991, “The COC Algorithm—Part II: Topological, Geometrical and Generalized Shape Optimization,” Comput. Methods Appl. Mech. Eng., 89(1–3), pp. 309–336. [CrossRef]
Rozvany, G. I. N., Zhou, M., and Birker, T., 1992, “Generalized Shape Optimization Without Homogenization,” Struct. Optim., 4(3–4), pp. 250–252. [CrossRef]
Xie, Y. M., and Steven, G. P., 1993, “A Simple Evolutionary Procedure for Structural Optimization,” Comput. Struct., 49(5), pp. 885–896. [CrossRef]
Huang, X., Zuo, Z. H., and Xie, Y. M., 2010, “Evolutionary Topological Optimization of Vibrating Continuum Structures for Natural Frequencies,” Comput. Struct., 88(5–6), pp. 357–64. [CrossRef]
Allaire, G., and Jouve, F., 2005, “A Level-Set Method for Vibration and Multiple Loads Structural Optimization,” Comput. Methods Appl. Mech. Eng., 194(30–33), pp. 3269–3290. [CrossRef]
Sethian, J. A., 1999, “Fast Marching Methods,” SIAM Rev., 41(2), pp. 199–235. [CrossRef]
Yamada, T., Izui, K., Nishiwaki, S., and Takezawa, A., 2010, “A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy,” Comput. Methods Appl. Mech. Eng., 199(45–48), pp. 2876–2891. [CrossRef]
Xia, Q., Shi, T., and Wang, M., 2011, “A Level Set Based Shape and Topology Optimization Method for Maximizing the Simple or Repeated First Eigenvalue of Structure Vibration,” Struct. Multidiscip. Optim., 43(4), pp. 473–485. [CrossRef]
Wang, M. Y., Wang, X. M., and Guo, D. M., 2003, “A Level Set Method for Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 192(1–2), pp. 227–246. [CrossRef]
Tong, L. Y., and Lin, J. Z., 2011, “Structural Topology Optimization With Implicit Design Variable-Optimality and Algorithm,” Finite Elem. Anal. Des., 47(8), pp. 922–932. [CrossRef]
Vasista, S., and Tong, L. Y., 2012, “Design and Testing of Pressurized Cellular Planar Morphing Structures,” AIAA J., 50(6), pp. 1328–1338. [CrossRef]
Tong, L., and Luo, Q., 2014, “Selection of Integral Functions for Normal Mode Analysis in Topology Optimization,” Appl. Mech. Mater., 553, pp. 795–800. [CrossRef]
Vasista, S., and Tong, L. Y., 2014, “Isosurface, Stiffness Design, Three Dimensional Topology Optimisation,” Adv. Comput. Mech., 553, pp. 801–806. [CrossRef]
Lai, F. K., Mou, J. Q., See, I. B. L., and Lin, W. Z., 2013, “Modeling and Analysis of Notebook Computer Chassis Structure for Optimization of Component Mounting,” Int. J. Mech. Sci., 76, pp. 60–69. [CrossRef]
Vannucci, P., 2009, “Influence of Invariant Material Parameters on the Flexural Optimal Design of Thin Anisotropic Laminates,” Int. J. Mech. Sci., 51(3), pp. 192–203. [CrossRef]
Ma, Z.-D., Cheng, H.-C., and Kikuchi, N., 1994, “Structural Design for Obtaining Desired Eigenfrequencies by Using the Topology and Shape Optimization Method,” Comput. Syst. Eng., 5(1), pp. 77–89. [CrossRef]
Andkjaer, J., and Sigmund, O., 2013, “Topology Optimized Cloak for Airborne Sound,” ASME J. Vib. Acoust., 135(4), p. 041011. [CrossRef]
Du, J., and Olhoff, N., 2007, “Topological Design of Freely Vibrating Continuum Structures for Maximum Values of Simple and Multiple Eigenfrequencies and Frequency Gaps,” Struct. Multidiscip. Optim., 34(2), pp. 91–110. [CrossRef]
Ma, Z.-D., Kikuchi, N., and Cheng, H.-C., 1995, “Topological Design for Vibrating Structures,” Comput. Methods Appl. Mech. Eng., 121(1–4), pp. 259–280. [CrossRef]
Niu, B., Yan, J., and Cheng, G., 2009, “Optimum Structure With Homogeneous Optimum Cellular Material for Maximum Fundamental Frequency,” Struct. Multidiscip. Optim., 39(2), pp. 115–132. [CrossRef]
MSC Software, 2011, “MD Nastran 2011 & MSC Nastran 2011 Dynamic Analysis User's Guide,” MSC Software Corp., Santa Ana, CA.
Bathe, K.-J., 1996, Finite Element Procedures, Prentice Hall, Upper Saddle River, NJ.
Deaton, J., and Grandhi, R., 2013, “A Survey of Structural and Multidisciplinary Continuum Topology Optimization: Post 2000,” Struct. Multidiscip. Optim., 49(1), pp. 1–38. [CrossRef]
Sigmund, O., and Petersson, J., 1998, “Numerical Instabilities in Topology Optimization: A Survey on Procedures Dealing With Checkerboards, Mesh-Dependencies and Local Minima,” Struct. Optim., 16(1), pp. 68–75. [CrossRef]
Bendsoe, M. P., and Kikuchi, N., 1988, “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224. [CrossRef]
Huang, X., and Xie, Y.-M., 2010, “A Further Review of ESO Type Methods for Topology Optimization,” Struct. Multidiscip. Optim., 41(5), pp. 671–683. [CrossRef]
Bendsøe, M. P., and Triantafyllidis, N., 1990, “Scale Effects in the Optimal Design of a Microstructured Medium Against Buckling,” Int. J. Solids Struct., 26(7), pp. 725–741. [CrossRef]

Figures

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Fig. 1

Beamlike structure: (a) model A—fixed at two ends and (b) model B—simply supported

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Fig. 2

Frequency versus iteration and optimal designs to maximize frequencies ω1 and ω2 of model A: (a) frequency versus iteration, (b) optimal design to maximize ω1, (c) frequency versus iteration, and (d) optimal design to maximize ω2

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Fig. 3

Frequency versus iteration and optimal designs to maximize ω1 and ω2 of model B (a = 1 and b = 8): (a) frequency versus iteration, (b) optimal design to maximize ω1, (c) frequency versus iteration, and (d) optimal design to maximize ω2

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Fig. 4

Frequency variations for maximizing (a) ω2 and (b) ω2-ω1 of model B (a = 1 and b = 10)

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Fig. 5

Optimal designs for maximizing (a) ω2 and (b) ω2-ω1 of model B (a = 1 and b = 10)

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Fig. 6

Topology optimization to maximize Λ in Eq. (19a) for model A: (a) frequency versus iteration, (b) optimal design to maximize Λ (i = 1, j = 2, and w2=0.1), (c) frequency versus iteration, and (d) optimal design to maximize Λ (i = 2, j = 3, and w3=0.1)

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Fig. 7

Optimization to maximize Λ in Eq. (19a) (i = 2, j = 3, and w3=0.1) for model B (a = 1 and b = 10): (a) frequency versus iteration and (b) optimal design to maximize Λ

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Fig. 8

The Φ function surfaces for model A at (a) iteration 48, (b) iteration 49, (c) iteration 50, and (d) iteration 51

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Fig. 9

Frequencies ω1–ω5 versus iteration for different values of αK1: (a) αK1 = 0.1; ω1 = 60.6 Hz, (b) αK1 = 0.2; ω1 = 62.5 Hz, (c) αK1 = 0.3; ω1 = 65.7 Hz, (d) αK1 = 0.4; ω1 = 67.8 Hz, (e) αK1 = 0.5; ω1 = 72.1 Hz, and (f) αK1 = 0.54; ω1 = 76.9 Hz

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Fig. 10

Optimal topologies for different values of αK1: (a) αK1 = 0.1, (b) αK1 = 0.2, (c) αK1 = 0.3, (d) αK1 = 0.4, (e) αK1 = 0.5, and (f) αK1 = 0.54

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Fig. 11

The first-order frequency versus coefficient αK1 for Vf= 0.3, 0.4, and 0.5

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Fig. 12

Optimal topologies by using the Φ function with αK1= 0 and αK1= 0.3 for Vf=0.3 and Vf=0.4 : (a) Vf = 0.3; αK1 = 0; F1 = 49.2 Hz, (b) Vf = 0.3; αK1 = 0.3; F1 = 49.9 Hz, (c) Vf = 0.4; αK1 = 0; F1 = 51.5 Hz, and (d) Vf = 0.4; αK1 = 0.3; F1 = 55.6 Hz

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Fig. 13

Optimal designs for the aluminum beamlike structure: (a) Vf = 0.5; αK1 = 0; F1 = 1204 Hz and (b) Vf = 0.5; αK1 = 0.8; F1 = 1347 Hz

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Fig. 14

Frequency variations and optimal design to maximize ω1 of an aluminum plate: (a) frequency versus iteration and (b) optimal design of a plate F1 = 125.5 Hz

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Fig. 15

Photos of specimens 1 and 2 where an arrow indicates a hitting position and light is emitted by the laser vibrometer: (a) specimen 1 for the optimal design of a beamlike structure (αK1 = 0) and (b) specimen 2 for the optimal design of a beamlike structure (αK1 = 0.8)

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Fig. 16

Experimental setup and specimens 3 and 4: (1) antivibration table (KS2210); (2) DC power supply (BK PECISION 1761); (3) data logger (NI cDAQ 9172/9221); (4) polytec CLV-1000 compact laser vibrometer; (5) specimen 3; and (6) specimen 4

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Fig. 17

The FFT input data-signals measured by using laser vibrometer: (a) and (b) sampling frequency: 8000 Hz and (c) and (d) sampling frequency: 400 Hz

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Fig. 18

Frequency spectrums computed by FFT: (a) fundamental frequency = 1168 Hz; (b) fundamental frequency = 1279 Hz; (c) fundamental frequency = 88.75 Hz; and (d) fundamental frequency = 70.22 Hz

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Fig. 19

Contours of eigenvector and translation for mode 1 of specimens 1–4: (a) eigenvector contour and translation of mode 1 for specimen 1 (F1 = 1296 Hz), (b) eigenvector contour and translation of mode 1 for specimen 2 (F1 = 1426 Hz), (c) specimen 3 (F1 = 89.21 Hz), and (d) specimen 4 (F1 = 72.19 Hz)

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