Dynamic response optimization of structures with viscoelastic material using the equivalent static loads method

Abstract

Viscoelastic material is widely used in automotive structures due to its outstanding vibration-damping characteristics with appropriate stiffness. Viscoelastic material, which has viscosity and elasticity, shows energy absorption and dissipation. The material properties of viscoelastic material are dependent upon time, temperature, and loading path. Hence, these characteristics have to be considered when performing structural optimization. Studies on the constitutive equations of viscoelastic material are widely carried out, and structural optimization using harmonic excitation in the frequency-domain is often reported. However, structural optimization in the time-domain is rarely performed. One of the reasons is that the cost of sensitivity analysis is quite expensive. The Equivalent Static Loads Method (ESLM) is a linear/nonlinear dynamic response structural optimization method. In this research, a practical structural optimization method to consider the characteristics of viscoelastic material is proposed using ESLM. Equivalent static loads (ESLs) are defined as the static loads that generate the same displacement field as that from dynamic analysis. In ESLM, dynamic analysis and linear static response optimization are alternatively repeated until convergence is achieved. Viscoelastic material reduces the vibration amplitude and the stored energy in a structural system. Thus, excellent damping performance is required for a part with viscoelastic material, while the proper stiffness is maintained. An appropriate design formulation is made for the design of viscoelastic material. In this research, the sum of damping ratios, the sum of weighted damping ratios, and the sum of squared displacements are considered as the objective functions. These three objective functions deal with the peak displacements of damped vibration. Three case studies are defined by optimizations of some typical automotive parts with viscoelastic material. They are a sandwich panel, a rubber bushing, and a seat cushion. The damping performances of the objective functions are compared and discussed.

Keywords

Viscoelastic material automotive parts structural optimization damping performance time-domain equivalent static loads

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References

Christensen

. Theory of viscoelasticity. Mineola, NY: Dover Publications, 2003.

Brinson

. Polymer engineering science and viscoelasticity. New York: Springer, 2008.

Rao

. Automotive applications of viscoelastic damping treatment. In: Fifth international congress on sound and vibration, Adelaide, South Australia, 15–18 December 1997. Gliwice, Poland: IIAV.

Rao

. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J Sound Vib 2003; 262(3): 457–474.

Kim

D-Y

Kim

, et al. Numerical prediction of the viscoelastic deformation of seat foam in response to long-term driving. Proc Inst Mech Eng D J Automob Eng 2015; 229(2): 214–225.

Maxwell

. Fundamental paper on relaxation and flow. Phil Mag 1868; 12(35): 134.

Kelvin

. On the theory of viscoelastic fluids. Math Phys Pap 1875; 3: 27–84.

Boltzmann

. On the theory of the elastic aftereffect. Poggendorff’s Ann Phys Chem 1876; 7: 624–645.

Simo

. On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Methods Appl Mech Eng 1987; 60(2): 153–173.

10.

Bergström

Boyce

. Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 1998; 46(5): 931–954.

11.

Kim

B-K

Youn

S-K

. A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation. Arch Appl Mech 2001; 71(11): 748–763.

12.

Abaqus Users’ Manual. ABAQUS Version 6.10. ABAQUS Inc. USA, 2010.

13.

LS-DYNA User’s Manual. LS-DYNA Version 971. Livermore Software Technology Corporation. USA, 2007, p. 354.

14.

Schmit

. Structural design by systematic synthesis. In: Proceedings of the second ASCE conference on electronic computation, Pittsburgh, PA, USA, 8–9 September 1960, pp.105–122. New York: ASCE.

15.

Dimitrovová

Rodrigues

. Optimization of passive vibration isolators mechanical characteristics. Struct Multidiscipl Optim 2010; 42(3): 325–340.

16.

Lumsdaine

Scott

. Shape optimization of unconstrained viscoelastic layers using continuum finite elements. J sound vib 1998; 216(1): 29–52.

17.

Lee

W-S

Youn

S-K

. Topology optimization of rubber isolators considering static and dynamic behaviours. Struct Multidiscipl Optim 2004; 27(4): 284–294.

18.

Subramanian

Surampudi

Thomson

, et al. Optimization of damping treatments for structure borne noise reductions. Sound vib 2004; 38(9): 14–19.

19.

Araújo

Martins

Soares

, et al. Damping optimization of viscoelastic laminated sandwich composite structures. Struct Multidiscipl Optim 2009; 39(6): 569–579.

20.

Chen

Liu

. Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus. Struct Multidiscipl Optim 2014; 50(2): 287–296.

21.

Fang

Zheng

. Topology optimization for minimizing the resonant response of plates with constrained layer damping treatment. Shock Vib 2015; 2015: 1–11.

22.

Chen

Liu

. Microstructural topology optimization of viscoelastic materials for maximum modal loss factor of macrostructures. Struct Multidiscipl Optim 2016; 53(1): 1–14.

23.

James

Waisman

. Topology optimization of viscoelastic structures using a time-dependent adjoint method. Comput Methods Appl Mech Eng 2015; 285: 166–187.

24.

James

Waisman

. On the importance of viscoelastic response consideration in structural design optimization. Optim Eng 2016; 17(4): 631–650.

25.

Yun

K-S

Youn

S-K

. Multi-material topology optimization of viscoelastically damped structures under time-dependent loading. Finite Elem Anal Des 2017; 123: 9–18.

26.

Choi

Park

. Structural optimization using equivalent static loads at all time intervals. Comput Methods Appl Mech Eng 2002; 191(19–20): 2105–2122.

27.

Shin

M-K

Park

K-J

Park

G-J

. Optimization of structures with nonlinear behavior using equivalent loads. Comput Methods Appl Mech Eng 2007; 196(4–6): 1154–1167.

28.

Kim

Y-I

Park

G-J

. Nonlinear dynamic response structural optimization using equivalent static loads. Comput Methods Appl Mech Eng 2010; 199(9–12): 660–676.

29.

Lee

Park

. Shape optimization of the initial blank in the sheet metal forming process using equivalent static loads. Int J Numer Methods Eng 2011; 85(2): 247–268.

30.

Lee

J-J

Jung

U-J

Park

G-J

. Shape optimization of the workpiece in the forging process using equivalent static loads. Finite Elem Anal Des 2013; 69: 1–18.

31.

Lee

J-J

Park

G-J

. Optimization of the structural and process parameters in the sheet metal forming process. J Mech Sci Technol 2014; 28(2): 605–619.

32.

Jang

H-H

Lee

Park

G-J

. Optimization of the loading path for the tube-hydroforming process. Proc Inst Mech Eng D J Automob Eng 2016; 230: 1605–1623.

33.

OptiStruct

. Altair OptiStruct user’s manual version 13.0. Altair Engineering Inc, Troy, MI, USA. 2014.

34.

Lee

H-A

Park

G-J

. A software development framework for structural optimization considering nonlinear static responses. Struct Multidiscipl Optim 2015; 52(1): 197–216.

35.

Lopez

La Sala

. A finite element approach to statical and dynamical analysis of geometrically nonlinear structures. Finite Elem Anal Des 2010; 46(12): 1093–1105.

36.

Thai

H-T

Kim

S-E

. Nonlinear static and dynamic analysis of cable structures. Finite Elem Anal Des 2011; 47(3): 237–246.

37.

Baroutaji

Morris

Olabi

A-G

. Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading. Thin-Walled Struct 2014; 82: 262–277.

38.

Park

Kang

. Validation of a structural optimization algorithm transforming dynamic loads into equivalent static loads. J Optim Theory Appl 2003; 118(1): 191–200.

39.

Lee

Kim

Park

, et al. Structural optimization of a joined wing using equivalent static loads. J Aircr 2007; 44(4): 1302–1308.

40.

Jeong

Kan

, et al. Structural optimization of an automobile roof structure using equivalent static loads. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering. 2008; 222: 1985-95.

41.

Lee

Park

. Crashworthiness design optimization using equivalent static loads. Proc Inst Mech Eng D J Automob Eng 2012; 226(1): 23–38.

42.

Yang

Chen

Kelly

. A topological optimization approach for structural design of a high-speed low-load mechanism using the equivalent static loads method. Int J Numer Methods Eng 2012; 89: 584–598.

43.

Leiva

. Topometry optimization: a new capability to perform element by element sizing optimization of structures. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference, Albany, NY, USA, 30 August–1 September 2004, p. 4. Reston, VA: AIAA.

44.

Ambrósio

Verissimo

. Improved bushing models for general multibody systems and vehicle dynamics. Multibody Syst Dyn 2009; 22(4): 341–365.

45.

Shintani

Azegami

. Shape optimization of a rubber bushing. JSIAM Letters 2014; 6: 89–92.

46.

Kaya

. Shape optimization of rubber bushing using differential evolution algorithm. Sci World J 2014; 2014: 379196.

47.

Kinkelaar

Neal

Crocco

. The influence of polyurethane foam dynamics on the vibration isolation character of full foam seats. SAE Technical Paper, 1998.

48.

Van Niekerk

Pielemeier

Greenberg

. The use of seat effective amplitude transmissibility (SEAT) values to predict dynamic seat comfort. J sound vib 2003; 260(5): 867–888.

49.

SAE International standards. Motor vehicle seat comfort performance measures. SAE: USA, 2012.

50.

Marzougui

Brown

Park

, et al. Development & Validation of a Finite Element Model for a Mid-Sized Passenger Sedan. In: Proceedings of the 13th international LS-DYNA users conference, Dearborn, MI, USA, 8–10 June 2014, pp.8–10. Livermore, CA: Livermore Software Technology Corporation.

51.

Reichert

Mohan

Marzougui

, et al. Validation of a Toyota Camry finite element model for multiple impact configurations. Warrendale, PA: SAE International, 2016.