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Research Papers

Damping Design of Flexible Structures With Graded Materials Under Harmonic Loading

[+] Author and Article Information
Mahmoud Alfouneh

School of Aerospace,
Mechanical and Mechatronic Engineering,
The University of Sydney,
Camperdown, NSW 2006, Australia
e-mail: alfoone@uoz.ac.ir

Liyong Tong

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

1Present address: Mechanical Engineering Department, Zabol University, Zabol 98613-35856, Iran.

2Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 18, 2017; final manuscript received March 6, 2018; published online April 26, 2018. Assoc. Editor: A. Srikantha Phani.

J. Vib. Acoust 140(5), 051011 (Apr 26, 2018) (12 pages) Paper No: VIB-17-1162; doi: 10.1115/1.4039571 History: Received April 18, 2017; Revised March 06, 2018

This article presents a novel moving isosurface threshold (MIST) method for designing flexible structures using graded materials with multivolume fractions and constraints and viscous or hysteretic damping under harmonic loadings. By employing a unit dynamic load with the same frequency of an applied load, the displacement amplitude at chosen degrees-of-freedom (DOFs) can be expressed in an integral form in terms of mutual modal strain and kinetic energy densities over the entire design domain. Such integrals enable the introduction of novel physical response functions for solving a range of topology optimization problems, including single and multiple objectives with single and multiple volume fractions and/or constraints, e.g., single-input and single-output (SISO) and multi-input and multi-output (MIMO). Numerical examples are presented to validate the efficiency and capability of the present extended MIST method. Experiments are also conducted on rectangular plates with and without damping layer, fully and optimally covered, to demonstrate the benefits of the optimal damping layer design.

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References

Min, S. , Kikuchi, N. , Park, Y. C. , Kim, S. , and Chang, S. , 1999, “ Optimal Topology Design of Structures Under Dynamic Loads,” Struct. Optim., 17(2–3), pp. 208–218.
Olhoff, N. , and Du, J. , 2005, “ Topological Design of Continuum Structures Subjected to Forced Vibration,” Sixth World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May 30–June 3, pp. 1–8. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.6006&rep=rep1&type=pdf
Takezawa, A. , Daifuku, M. , Nakano, Y. , Nakagawa, K. , Yamamoto, T. , and Kitamura, M. , 2016, “ Topology Optimization of Damping Material for Reducing Resonance Response Based on Complex Dynamic Compliance,” J. Sound Vib., 365, pp. 230–243. [CrossRef]
Du, J. , and Olhoff, N. , 2007, “ Minimization of Sound Radiation From Vibrating Bi-Material Structures Using Topology Optimization,” Struct. Multidiscip. Optim., 33(4–5), pp. 305–321. [CrossRef]
Jog, C. , 2002, “ Topology Design of Structures Subjected to Periodic Loading,” J. Sound Vib., 253(3), pp. 687–709. [CrossRef]
Zheng, H. , Cai, C. , Pau, G. S. H. , and Liu, G. R. , 2005, “ Minimizing Vibration Response of Cylindrical Shells Through Layout Optimization of Passive Constrained Layer Damping Treatments,” J. Sound Vib., 279(3–5), pp. 739–756. [CrossRef]
Kang, Z. , Zhang, X. , Jiang, S. , and Cheng, G. D. , 2012, “ On Topology Optimization of Damping Layer in Shell Structures Under Harmonic Excitations,” Struct. Multidiscip. Optim., 46(1), pp. 51–67. [CrossRef]
Phani, A. S. , and Woodhouse, J. , 2007, “ Viscous Damping Identification in Linear Vibration,” J. Sound Vib., 303(3), pp. 475–500. [CrossRef]
Adhikari, S. , and Phani, A. S. , 2009, “ Experimental Identification of Generalized Proportional Viscous Damping Matrix,” ASME J. Vib. Acoust., 131(1), p. 011008. [CrossRef]
Phani, A. S. , and Adhikari, S. , 2008, “ Rayleigh Quotient and Dissipative Systems,” ASME J. Appl. Mech., 75(6), p. 061005. [CrossRef]
Pydimarry, K. , Mozumder, C. , Patel, N. , and Renaud, J. , 2009, “ Synthesis of a Dynamically Loaded Structure With Topology Optimization,” SAE Int. J. Passenger Cars—Mech. Syst., 2(1), pp. 1143–1150. [CrossRef]
Gaynor, A. T. , Meisel, N. A. , Williams, C. B. , and Guest, J. , 2014, “ Multiple-Material Topology Optimization of Compliant Mechanisms Created Via PolyJet Three-Dimensional Printing,” ASME J. Manuf. Sci. Eng., 136(6), p. 061015. [CrossRef]
Tong, L. , and Lin, J. , 2011, “ Structural Topology Optimization With Implicit Design Variable—Optimality and Algorithm,” Finite Elem. Anal. Des., 47(8), pp. 922–932. [CrossRef]
Luo, Q. , and Tong, L. , 2015, “ Design and Testing for Shape Control of Piezoelectric Structures Using Topology Optimization,” Eng. Struct., 97, pp. 90–104. [CrossRef]
Luo, Q. , and Tong, L. , 2015, “ Optimal Designs for Vibrating Structures Using a Moving Isosurface Threshold Method With Experimental Study,” ASME J. Vib. Acoust., 137(6), p. 061005. [CrossRef]
Vasista, S. , and Tong, L. , 2012, “ Design and Testing of Pressurized Cellular Planar Morphing Structures,” AIAA J., 50(6), pp. 1328–1338. [CrossRef]
Alfouneh, M. , and Tong, L. , 2015, “ MIST Topology Optimization for Bending Plates Under Static Loading,” 11th World Congress on Structural and Multidisciplinary Optimization, Sydney, Australia, June 7–12, pp. 906–911. http://web.aeromech.usyd.edu.au/WCSMO2015/papers/1416_paper.pdf
Luo, Q. , and Tong, L. , 2016, “ An Algorithm for Eradicating the Effects of Void Elements on Structural Topology Optimization for Nonlinear Compliance,” Struct. Multidiscip. Optim., 53(4), pp. 695–714. [CrossRef]
Tong, L. , and Luo, Q. , 2016, “ Design of Cellular Structures With Multi-Volume Fractions Using Topology Optimization,” Chin. J. Comput. Mech., 33(4), pp. 516–521.
Bendsøe, M. P. , and Sigmund, O. , 2003, Topology Optimization: Theory, Methods and Applications, Springer, Berlin.
Paultre, P. , 2013, Dynamics of Structures, Wiley, New York. [CrossRef]
Alfouneh, M. , and Tong, L. , 2017, “ Maximizing Modal Damping in Layered Structures Via Multi-Objective Topology Optimization,” Eng. Struct., 132, pp. 637–647. [CrossRef]
Ozgen, G. O. , Erol, F. , and Batihan, A. C. , 2012, “ Dynamic Stiffness-Based Test Systems for Viscoelastic Material Characterization: Design Considerations,” Topics in Modal Analysis I, Vol. 5, Springer, New York, pp. 287–297. [CrossRef]
Bhowmick, A. K. , and Stephens, H. , 2000, Handbook of Elastomers, CRC Press, Boca Raton, FL. [PubMed] [PubMed]
Gallimore, C. A. , 2008, “ Passive Viscoelastic Constrained Layer Damping Application for a Small Aircraft Landing Gear System,” Master thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. https://vtechworks.lib.vt.edu/handle/10919/35350

Figures

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Fig. 1

A cellular design domain includes of: a solid part, Ar1; a porous part Ar2 and a void part Ar3

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Fig. 2

(a) A structure with domain Ω subject to a true load fleiωτ and a virtual load {e¯} ei ω τ (dashed line indicates deformed domain) and (b) schematic of subdomains Ω⌢ξ (ξ = 0, 1, 2,…, n) and isosurface thresholds tξ (ξ = 1, 2,…, n)

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Fig. 3

Flowchart of MIST method

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Fig. 4

A rectangular cantilever two-layer plate with a base layer and a VEM layer subject to a true load f1(τ) and a virtual load f¯1(τ)={e¯1} eiωτ

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Fig. 5

Objective function evolution versus iteration number for loading frequency fp = 100 Hz, of SISO damped structure with n = 1 and n = 2

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Fig. 6

The displacement frequency response before and after optimization of SISO optimization problem for material with n = 2

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Fig. 7

Optimal shapes of VEM layer of two layer structure excited by a harmonic load with frequency fp = 100 Hz: (a) n = 1 (p0 = ε, p1 = 1), (b) n = 2 (Vf1 = 0.2, Vf2 = 0.1, p0 = ε, p1 = 0.5, p2 = 1), (c) 3D Ф surface of the case of n = 1, and (d) 3D Ф surface of the case of n = 2 in which lower isosurface is for p1 = 0.5 and upper one is for p2 = 1

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Fig. 8

(a) The objective function evolution versus iteration number for load frequency fp = 100 Hz of the SISO problem of maximizing output DOF for the structure with n = 1 and (b) the corresponding optimal layout of the VEM layer at final iteration

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Fig. 9

A clamped plate with thickness 0.0021 m, true load f1(τ) and dynamic virtual loads f¯j(τ)={e¯j} eiωτ (j = 1, 2,…, 4) for the problem of multiple output DOFs minimization (SIMO)

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Fig. 10

Convergence curves for the amplitudes of four output DOFs at the four points in Fig. 9 versus the iteration number for the optimization problem of multiple output DOFs (SIMO) with n = 2

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Fig. 11

The displacement frequency response before and after optimization of SIMO optimization problem for material with n = 2. (BO: before optimization and AO: after optimization).

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Fig. 12

Optimal shapes of the structure excited by a harmonic load with a frequency of fp = 100 Hz (SIMO problem): (a) one cellular structure (n = 1, p0 = ε, p1 = 1), (b) cellular structure with two volume fractions (n = 2, Vf1 = 0.3, Vf2 = 0.2, p0 = ε, p1 = 0.75, p2 = 1), (c) 3D Ф surface and one isosurface for n = 1, and (d) 3D Ф surface with two isosurfaces (lower and upper isosurfaces represent p1 = 0.75 and p2 = 1) for n = 2

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Fig. 13

(a) Cantilever square plate with four applied true forces fl(τ), (l = 1, 2,…,4) and a virtual force f¯1(τ)={e¯1} eiωτ and (b) amplitude of output DOF versus iteration number histories of multi-objective optimization problem case 2 with n = 1 and n = 2

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Fig. 14

The displacement frequency response before and after optimization of multiple input single output optimization problem for material with n = 2. (BO: before optimization and AO: after optimization).

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Fig. 15

Optimal shapes for viscous damping of case 2 of multi-objective optimization: (a) optimal shape for two cellular structure (n = 2, Vf1 = 0.3, Vf2 = 0.2, p0 = ε, p1 = 0.75, p2 = 1) and (b) optimal shape for one cellular structure (n = 1, p0 = ε, p1 = 1)

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Fig. 16

Optimal shapes for hysteretic damping of case 2 of multi-objective optimization: (a) optimal shape for n = 2 (Vf1 = 0.3, Vf2 = 0.2, p0 = ε, p1 = 0.75, p2 = 1) and (b) optimal shape for n = 1 (p0 = ε, p1 = 1)

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Fig. 17

Optimal layout of the VEM layer in two-layer cantilever plate (left side is clamped)

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Fig. 18

Experimental setup for the optimally covered plate with the following system used: (1) data logger NI cDAQ-9172; (2) DC power supply BK PECISION 1761; (3) signal controller NIPCI 6723 CB-68 LPR; (4) bar plate with optimized cover attached; (5) displacement sensor (Indikon 100-400 series); (6) TMS, the modal shop, INC k2004e01mini smart shaker; and (7) antivibration table KS2210

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Fig. 19

Curves of measured amplitude versus time for the plates with no rubber layer (bare), with fully covered rubber layer (full cover), with both top and bottom optimum patches (optimum damping), with only bottom optimum patch (optimum damping-bottom) and with only top optimum patch (optimum damping-top) under the selected harmonic loading for a 1.2 s time interval

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