非线性Zener模型周期解的稳定性与鞍结分岔集  

Stability of periodic solutions and saddle-node bifurcation set of nonlinear Zener models

作　　者：俞力洋 吴少培 李国芳[1] 丁旺才[1] YU Liyang;WU Shaopei;LI Guofang;DING Wangcai(School of Mechanical Engineering,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学机电工程学院,甘肃兰州730070

出　　处：《华中科技大学学报（自然科学版）》2023年第4期75-81,共7页Journal of Huazhong University of Science and Technology(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11962013,11732014,12162020,12262017);甘肃省优秀研究生创新之星资助项目(2021CXZX-543);甘肃省教育厅高校大学生就业创业能力提升工程资助项目(2021);甘肃省青年科技基金资助项目(21JR7RA328,21JR7RA335)。

摘　　要：采用非线性Zener模型表征橡胶等黏弹隔振系统的动力学特性,利用谐波平衡法求解了系统的无量纲运动微分方程,计算了系统周期解稳定的边界条件,给出了一种基于系统稳定性判别条件获取系统双参平面内鞍结分岔集与多不变集共存区的方法.结果表明:随着激励幅值的不断增大,系统骨架线逐渐向右弯曲,质量块幅频响应曲线从近似线性转变为出现多不变集共存的共振滞后区;在多不变集共存区内,系统对外界条件比较敏感,其不稳定周期解将依初始状态的选取而稳定到不同的稳定周期解上;系统刚度比的选取不仅直接决定了骨架曲线的起点,还从一定程度上决定了系统上下稳定边界的交点及系统上跳鞍结分岔集与下跃鞍结分岔集的交点.The nonlinear Zener model was used to characterize the dynamic characteristics of viscoelastic vibration isolation system such as rubber,and the dimensionless differential equations of motion were solved by harmonic balance method.The boundary conditions for the stability of the periodic solution of the system were calculated,and a method to obtain the coexistence region of saddle-node bifurcation set and multiple invariant set in the two-parameter plane based on the stability criterion of the system was given.Results show that with the increase of excitation amplitude,the system skeleton curve gradually bends to the right,and the mass amplitude-frequency response curve changes from approximately linear to the resonance lag region with the coexistence of multiple invariant set.In the coexistence region of multiple invariant set,the system is sensitive to external conditions,and its unstable periodic solution will be stabilized to different stable periodic solutions according to the selection of initial state.The selection of system stiffness ratio not only directly determines the starting point of skeleton curve,but also to a certain extent determines the intersection point of upper stable boundary and lower stable boundary and the intersection point of“jump up”saddle-node bifurcation set and“jump down”saddle-node bifurcation set of the system.

关 键 词：非线性Zener模型 橡胶隔振系统 稳定边界 鞍结分岔集 多不变集共存 

分 类 号：O322[理学—一般力学与力学基础]