非线性能量阱的曲梁设计研究

PDF(1715 KB)

振动与冲击 ›› 2024, Vol. 43 ›› Issue (22) : 53-61.

论文

非线性能量阱的曲梁设计研究

郑智伟1，黄修长*1,2，华宏星1，袁志豪3，杨咏2

作者信息 +

Design method of Euler curved beams in nonlinear energy sinks

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

Author information +

文章历史 +

摘要

非线性能量阱（Nonlinear energy sink, NES）在减振和能量采集领域具有重要价值。尽管立方刚度NES及含立方刚度的双稳态NES已受广泛研究，但精确实现指定立方刚度的方法鲜有讨论。为此针对基于欧拉曲梁实现的NES开展研究，通过减小曲梁回复力与一个特定的理想非线性回复力之间的相对偏差，来实现NES中精确的立方刚度。基于欧拉梁理论得到圆弧梁和折线梁的初始刚度公式，用于设计曲梁长度。基于有限元方法求解了不同曲梁形状的非线性回复力，确定了能够实现立方刚度的圆弧梁和折线梁形状，并得到了满足相对偏差要求的临界位移拟合公式。基于以上两个公式总结出一套快速设计曲梁的方法，通过合理调节形状和截面尺寸使曲梁的回复力在需要的变形区间内逼近于理想非线性回复力。与有限元仿真进行对比，推导的解析公式可以对大初始挠度曲梁的初始刚度进行精确计算，设计出的NES回复力与目标之间的相对偏差绝对值小于1%。提出的设计方法有助于更精准、高效地设计NES，为曲梁实现非线性弹簧提供了新的设计方法。

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.

导出引用

郑智伟1, 黄修长1, 2, 华宏星1, 袁志豪3, 杨咏2. 非线性能量阱的曲梁设计研究[J]. 振动与冲击, 2024, 43(22): 53-61

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

参考文献

[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.