黏弹性屈曲梁非线性内共振稳态周期响应  

Steady-State Periodic Responses of a Viscoelastic Buckled Beam in Nonlinear Internal Resonance

作　　者：熊柳杨 张国策[1] 丁虎[1] 陈立群[1,2,3] 

机构地区：[1]上海大学上海市应用数学和力学研究所,上海200072 [2]上海大学力学系,上海200444 [3]上海市力学在能源工程中的应用重点实验室

出　　处：《应用数学和力学》2014年第11期1188-1196,共9页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(重点项目)(11232009);国家自然科学基金(11372171;11422214);上海市教委科研创新项目(12YZ028)~~

摘　　要：研究了内共振下简支边界屈曲黏弹性梁受迫振动稳态周期幅频响应.考虑Kelvin黏弹性本构关系,并通过对非平凡平衡位形做坐标变换,建立屈曲梁横向振动的非线性偏微分-积分模型.基于对控制方程的Galerkin截断,得到多维非线性常微分方程组.在前两阶模态内共振存在的条件下,运用多尺度法分析截断后的控制方程,利用可解性条件消除长期项,获得一阶主共振下的幅值与相角方程.通过数值算例以展示系统稳态幅频响应关系以及失稳区域,从而聚焦系统共振中存在的非线性现象,如跳跃现象、滞后现象,并讨论了双跳跃现象随轴向荷载的演化.通过直接数值方法处理截断方程,数值验证近似解析解,计算结果表明多尺度法具有较高精度.Nonlinear vibration of a hinged-hinged viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance was investigated.The governing integro-partial differential equation was derived via introduction of coordinate transform for the non-trivial equilibrium configuration,with the viscoelastic constitutive relation taken into account.Based on the Galerkin method,the governing equation was truncated to a set of infinite ordinary differential equations and the condition for internal resonance was obtained.The multiple scales method was applied to derive the modulation-phase equations.Steady-state periodic solutions to the system as well as their stability were determined.The numerical examples were focused on the nonlinear phenomena,such as double-jump and hysteresis.The generation and vanishing of a double-jumping phenomenon on the amplitude-frequency curves were discussed in detail.The R unge-Kutta method was developed to verify the accuracy of results from the multiple scales method.

关 键 词：屈曲梁 黏弹性 内共振 多尺度方法 稳定性 

分 类 号：O322[理学—一般力学与力学基础] O345[理学—力学]