Design method of Euler curved beams in nonlinear energy sinks

PDF(1715 KB)

Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (22) : 53-61.

ZHENG Zhiwei1，HUANG Xiuchang*1,2，HUA Hongxing1，YUAN Zhihao3，YANG Yong2

Author information +

History +

Abstract

Nonlinear Energy Sinks (NES) are important in the realms of vibration mitigation and energy harvesting due to their target energy transfer phenomenon. While extensive research has been conducted on cubic stiffness NES and bistable NES incorporating cubic stiffness, there exists a noticeable gap in discussions regarding the precise realization of cubic stiffness, thereby constraining the practical applications of NES. This study investigates the methodology designing curved beams to approximate its restoring force to a specific ideal nonlinear restoring force, ultimately achieving precise cubic stiffness in the NES. Initial stiffness formulas for circular and folded beams are derived based on Euler beam theory and used to design the beam lengths. Nonlinear restoring forces for various beam shapes are solved using finite element methods, identifying circular and folded beam shapes capable of achieving cubic stiffness. Critical displacement fitting formulas meeting relative deviation requirements are obtained. A rapid beam design method is summarized based on these formulas, allowing adjustment of shape and cross-sectional dimensions to make the restoring force of the beam approach the ideal nonlinear restoring force within the required deformation range. A comparison with finite element simulations shows that the derived analytical formulas can accurately calculate the initial stiffness of beams with large initial deflections, and the relative deviation absolute value between the designed NES restoring force and the target is less than 1%. The proposed design method contributes to a more precise and efficient NES design, offering a new approach for implementing nonlinear springs in curved beams.

Key words

Euler curved beam / nonlinear energy sink / cubic stiffness / nonlinear restoring force

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

ZHENG Zhiwei1, HUANG Xiuchang1, 2, HUA Hongxing1, YUAN Zhihao3, YANG Yong2. Design method of Euler curved beams in nonlinear energy sinks[J]. Journal of Vibration and Shock, 2024, 43(22): 53-61

References

[1] VAKAKIS A F, GENDELMAN O V., BERGMAN L A, et al. Nonlinear targeted energy transfer in mechanical and structural systems [M]. Dordrecht: Springer, 2008.
[2] TRIPATHI A, GROVER P, KALMÁR-NAGY T. On optimal performance of nonlinear energy sinks in multiple-degree-of-freedom systems[J]. Journal of Sound and Vibration, 2017, 388: 272-297.
[3] ZHENG Z, HUANG X, SU Z, et al. H∞ optimization of cubic stiffness nonlinear energy sink attached to a linear system[J]. Nonlinear Dynamics, 2023, 111(17): 15653-15673.
[4] QIU D, LI T, SEGUY S, et al. Efficient targeted energy transfer of bistable nonlinear energy sink: application to optimal design[J]. Nonlinear Dynamics, 2018, 92(2): 443-461.
[5] WANG G X, DING H. Mass design of nonlinear energy sinks[J]. Engineering Structures, 2022, 250(10): 113438.
[6] ZENG Y cheng, DING H, DU R H, et al. Micro-amplitude vibration suppression of a bistable nonlinear energy sink constructed by a buckling beam[J]. Nonlinear Dynamics, 2022, 108(4): 3185-3207.
[7] 姚红良, 刘帅, 王钰玮, 等. 可调永磁双稳态非线性能量阱及应用研究[J]. 振动与冲击, 2020, 39(3): 8.
YAO Hong Liang, LIU Shuai, WANG Yuwei, et al. Adjustable permanent magnet bi-stable nonlinear energy sink and its application[J]. Journal of Vibration and Shock, 2020, 39(3): 8
[8] GOURDON E, ALEXANDER N A, TAYLOR C A等. Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[J]. Journal of Sound and Vibration, 2007, 300(3-5): 522-551.
[9] 刘海平, 杨建中, 罗文波, 等. 新型欧拉屈曲梁非线性动力吸振器的实现及抑振特性研究[J]. 振动与冲击, 2016, 35(11): 7.
LIU Haiping, YANG Jianzhong, LUO Wenbo, et al. Realization and vibration supperession ability of a new novel Euler buckled beam nonlinear vibration absorber[J]. Journal of Sound and Vibration, 2016, 35(11): 7
[10] 王菁菁, 李浩博, 刘志彬. 高层结构附加轨道非线性能量阱减振性能研究[J]. 振动与冲击, 2020, 39(9): 8.
WANG Jingjing, LI Haobo, LIU Zhibin. Track nonlinear energy sink attached to a high-rise building for response mitigation[J]. Journal of Sound and Vibration, 2020, 39(9): 8
[11] 赵跃宇, 康厚军, 冯锐, 等. 曲线梁研究进展[J]. 力学进展, 2006, 36(2): 170-186.
ZHAO Yueyu, KANG Houjun, FENG Rui, et al. Advances of research on curved beams[J]. Advances in Mechanics, 2006, 36(2): 170-186
[12] LIN K C, LIN C W. Finite deformation of 2-D laminated curved beams with variable curvatures[J]. International Journal of Non-Linear Mechanics, 2011, 46(10): 1293-1304.
[13] BATISTA M. Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions[J]. International Journal of Solids and Structures, 2014, 51(13): 2308-2326.
[14] CEDOLIN L, BAZANT Z. Stability of structures[M]. Oxford University Press: New York, 1991.
[15] 刘兴天, 黄修长, 张志谊, 等. 激励幅值及载荷对准零刚度隔振器特性的影响[J]. 机械工程学报, 2013, 49(6): 6.
LIU Xingtian, HUANG Xiuchang, ZHANG Zhiyi, et al. Influence of excitation amplitude and load on the characteristics of quasi-zero stiffness isolator[J]. Journal of Mechanical Engineering, 2013, 49(6): 6
[16] AWTAR S, SEN S. A generalized constraint model for two-dimensional beam flexures: Nonlinear load-displacement formulation[J]. Journal of Mechanical Design, 2010, 132(8): 0810081-08100811.
[17] 叶康生, 姚葛亮. 平面曲梁有限元静力分析的p型超收敛算法[J]. 工程力学, 2017, 34(11): 9.
YE Kangsheng, YAO Geliang. A p-type superconvergent recovery method for FE static analysis of planar curved beams[J]. Engineering Mechanics, 2017, 34(11): 9
[18] 曾森, 陈少峰, 曲婷, 等. 大位移小转角空间曲梁的弹性力学方程[J]. 工程力学, 2010, 27(12): 14-20.
ZENG Sen, CHEN Shaofeng, QU Ting, et al. Elasticity equations for spatial curved beams with large displacement and small rotation[J]. Engineering Mechanics, 2010, 27(12): 14-20
[19] 曾森, 陈少峰, 王焕定, 等. 大变形空间曲梁有限元分析[J]. 东南大学学报(英文版), 2010, 26(004): 591-596.
ZENG Sen, CHEN Shaofeng, WANG Huanding, et al. Finite element analysis of spatial curved beam in large deformation[J]. Journal of Southeast University(English Edition), 2010, 26(004): 591-596
[20] JUTTE C V, KOTA S. Design of nonlinear springs for prescribed load-displacement functions[J]. Journal of Mechanical Design, 2008, 130(8): 0814031-08140310.