Abstract
This paper studies the optimization design of sandwich structures with lattice core and viscoelastic layers for suppressing structural resonance response in the frequency domain. A concurrent optimization scheme is proposed to simultaneously optimize the damping material topology in the viscoelastic layers and the size distribution of the lattice core. The damping effect of the viscoelastic layers is simulated as hysteretic damping model, and the full method is used to accurately calculate the dynamic responses. Based on the adjoint method, the corresponding design sensitivities are analytically derived efficiently and the Globally Convergent Method of Moving Asymptotes algorithm is adopted. To ensure a smooth convergence in case of mode switching, the mode tracking technique based on the Modal Assurance Criteria is introduced to track the targeted resonant mode. Numerical examples demonstrate the effect of the concurrent optimization in suppressing structural resonance response.
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References
Alfouneh M, Tong L (2017) Maximizing modal damping in layered structures via multi-objective topology optimization. Eng Struct 132:637–647. https://doi.org/10.1016/j.engstruct.2016.11.058
Alvelid M (2008) Optimal position and shape of applied damping material. J Sound Vib 310:947–965. https://doi.org/10.1016/j.jsv.2007.08.024
Ansari M, Khajepour A, Esmailzadeh E (2013) Application of level set method to optimal vibration control of plate structures. J Sound Vib 332:687–700. https://doi.org/10.1016/j.jsv.2012.09.006
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Castanie B, Bouvet C, Ginot M (2020) Review of composite sandwich structure in aeronautic applications. Compos Part C Open Access 1:100004. https://doi.org/10.1016/j.jcomc.2020.100004
Chen W, Liu S (2016) Microstructural topology optimization of viscoelastic materials for maximum modal loss factor of macrostructures. Struct Multidiscip Optim 53:1–14. https://doi.org/10.1007/s00158-015-1305-1
Delgado G, Hamdaoui M (2019) Topology optimization of frequency dependent viscoelastic structures via a level-set method. Appl Math Comput 347:522–541. https://doi.org/10.1016/j.amc.2018.11.014
Delissen A, van Keulen F, Langelaar M (2020) Efficient limitation of resonant peaks by topology optimization including modal truncation augmentation. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-019-02471-9
Eldred MS, Venkayya VB, Anderson WJ (1995) Mode tracking issues in structural optimization. AIAA J 33:1926–1933. https://doi.org/10.2514/3.12747
Fang Z, Zheng L (2015) Topology optimization for minimizing the resonant response of plates with constrained layer damping treatment. Shock Vib. https://doi.org/10.1155/2015/376854
Fang Z, Hou J, Zhai H (2018) Topology optimization of constrained layer damping structures subjected to stationary random excitation. Shock Vib. https://doi.org/10.1155/2018/7849153
Guo X, Cheng GD (2010) Recent development in structural design and optimization. Acta Mech Sinica 26:807–823. https://doi.org/10.1007/s10409-010-0395-7
Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88:357–364. https://doi.org/10.1016/j.compstruc.2009.11.011
Johnson CD (1995) Design of passive damping systems. J Mech Des Trans ASME 117:171–176. https://doi.org/10.1115/1.2836451
Johnson CD, Kienholz DA (1982) Finite element prediction of damping in structures with constrained viscoelastic layers. AIAA J 20:1284–1290. https://doi.org/10.2514/3.51190
Kang Z, Zhang X, Jiang S, Cheng G (2012) On topology optimization of damping layer in shell structures under harmonic excitations. Struct Multidiscip Optim 46:51–67. https://doi.org/10.1007/s00158-011-0746-4
Kim TS, Kim YY (2000) MAC-based mode-tracking in structural topology optimization. Comput Struct 74:375–383. https://doi.org/10.1016/S0045-7949(99)00056-5
Kim SY, Mechefske CK, Kim IY (2013) Optimal damping layout in a shell structure using topology optimization. J Sound Vib 332:2873–2883. https://doi.org/10.1016/j.jsv.2013.01.029
Li H, Luo Z, Gao L, Wu J (2018) An improved parametric level set method for structural frequency response optimization problems. Adv Eng Softw 126:75–89. https://doi.org/10.1016/j.advengsoft.2018.10.001
Lim JW, Lee DG (2011) Development of the hybrid insert for composite sandwich satellite structures. Compos Part A Appl Sci Manuf 42:1040–1048. https://doi.org/10.1016/j.compositesa.2011.04.008
Ling Z, Ronglu X, Yi W, El-Sabbagh A (2011) Topology optimization of constrained layer damping on plates using Method of Moving Asymptote (MMA) approach. Shock Vib 18:221–244. https://doi.org/10.3233/SAV-2010-0583
Liu T, Zhu JH, Zhang WH, Zhao H, Kong J, Gao T (2019) Integrated layout and topology optimization design of multi-component systems under harmonic base acceleration excitations. Struct Multidiscip Optim 59:1053–1073. https://doi.org/10.1007/s00158-019-02200-2
Przemieniecki JS (1968) Theory of matrix structural analysis, New York
Radovcic Y, Remouchamps A (2002) BOSS QUATTRO: an open system for parametric design. Struct Multidiscip Optim 23(2):140–152
Rao MD (2003) Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J Sound Vib 262:457–474. https://doi.org/10.1016/S0022-460X(03)00106-8
Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8:207–227. https://doi.org/10.1007/BF01742705
Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48:1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Takezawa A, Daifuku M, Nakano Y, Nakagawa K, Yamamoto T, Kitamura M (2016) Topology optimization of damping material for reducing resonance response based on complex dynamic compliance. J Sound Vib 365:230–243. https://doi.org/10.1016/j.jsv.2015.11.045
Wang Y, Luo Z, Zhang X, Kang Z (2014) Topological design of compliant smart structures with embedded movable actuators Smart Mater Struct 23:. https://doi.org/10.1088/0964-1726/23/4/045024
Wang R, Shang J, Li X, Luo Z, Wu W (2018) Vibration and damping characteristics of 3D printed Kagome lattice with viscoelastic material filling. Sci Rep 8:1–13. https://doi.org/10.1038/s41598-018-27963-4
Xu Y, Gao W, Yu Y, Zhang D, Zhao X, Tian Y, Cun H (2017) Dynamic optimization of constrained layer damping structure for the headstock of machine tools with modal strain energy method Shock Vib 2017. https://doi.org/10.1155/2017/2736545
Xu YJ, Zhu JH, Wu Z, Cao Y, Zhao Y, Zhang W (2018) A review on the design of laminated composite structures: constant and variable stiffness design and topology optimization. Adv Compos Hybrid Mater 1:460–477. https://doi.org/10.1007/s42114-018-0032-7
Yan K, Wang BP (2020) Two new indices for structural optimization of free vibration suppression. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-019-02451-z
Yang X, Li Y (2014) Structural topology optimization on dynamic compliance at resonance frequency in thermal environments. Struct Multidiscip Optim 49:81–91. https://doi.org/10.1007/s00158-013-0961-2
Yang XW, Li Y, Li YM (2015) Structural topology optimization on dynamic compliance at resonance frequency in thermal environments. Sci China Phys Mech Astron 58:1–12. https://doi.org/10.1007/s11433-014-5539-5
Yang JS, Ma L, Schmidt R, Qi G, Schröder KU, Xiong J, Wu LZ (2016) Hybrid lightweight composite pyramidal truss sandwich panels with high damping and stiffness efficiency. Compos Struct 148:85–96. https://doi.org/10.1016/j.compstruct.2016.03.056
Yun KS, Youn SK (2018) Topology optimization of viscoelastic damping layers for attenuating transient response of shell structures. Finite Elem Anal Des 141:154–165. https://doi.org/10.1016/j.finel.2017.12.003
Zargham S, Ward TA, Ramli R, Badruddin IA (2016) Topology optimization: a review for structural designs under vibration problems. Struct Multidiscip Optim 53:1157–1177. https://doi.org/10.1007/s00158-015-1370-5
Zhang X, Kang Z (2016) Vibration suppression using integrated topology optimization of host structures and damping layers. JVC/Journal Vib Control 22:60–76. https://doi.org/10.1177/1077546314528368
Zhang H, Ding X, Li H, Xiong M (2019) Multi-scale structural topology optimization of free-layer damping structures with damping composite materials. Compos Struct 212:609–624. https://doi.org/10.1016/j.compstruct.2019.01.059
Zhao J, Yoon H, Youn BD (2019) An efficient concurrent topology optimization approach for frequency response problems. Comput Methods Appl Mech Eng 347:700–734. https://doi.org/10.1016/j.cma.2019.01.004
Zheng H, Cai C, Pau GSHH, Liu GR (2005) Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments. J Sound Vib 279:739–756. https://doi.org/10.1016/j.jsv.2003.11.020
Zheng W, Lei Y, Li S, Huang Q (2013) Topology optimization of passive constrained layer damping with partial coverage on plate. Shock Vib 20:199–211. https://doi.org/10.3233/SAV-2012-00738
Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23:595–622. https://doi.org/10.1007/s11831-015-9151-2
Zhu JH, Zhou H, Wang C, Zhou L, Yuan S, Zhang W (2021) A review of topology optimization for additive manufacturing: status and challenges. Chinese J Aeronaut 34:91–110. https://doi.org/10.1016/j.cja.2020.09.020
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This work is supported by Key Project of NSFC (51790171, 51761145111, 51735005).
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Zhu, JH., Liu, T., Zhang, WH. et al. Concurrent optimization of sandwich structures lattice core and viscoelastic layers for suppressing resonance response. Struct Multidisc Optim 64, 1801–1824 (2021). https://doi.org/10.1007/s00158-021-02943-x
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DOI: https://doi.org/10.1007/s00158-021-02943-x