简支形状记忆合金层合梁的混沌及安全盆侵蚀  

Chaos and Fractal Boundary of Safe Basin of Simpliy Supported Laminated Shape Memoray Aolly Beam

作　　者：葛根[1] 竺致文[2] 许佳[2] 

机构地区：[1]天津工业大学机械工程学院,天津300387 [2]天津大学机械学院,天津300072

出　　处：《振动．测试与诊断》2013年第4期602-608,723,共7页Journal of Vibration,Measurement & Diagnosis

基　　金：国家自然科学基金重点资助项目(10732020);天津市教委科技项目(20120902)

摘　　要：为研究简支形状记忆合金层合梁受轴向激励下的混沌阈值和安全盆的分形边界问题,利用vanderpol环模型模拟了形状记忆合金在加载和卸载过程中的应力应变迟滞环特性。根据弹性理论和Galerkin方法建立了形状记忆合金简支层合梁在受轴向激励时的振动模型,得出系统发生混沌的Melnikov判据。利用待定固有频率法研究了模型的非线性参数对系统固有频率的影响,根据待定固有频率法的计算结果和时间尺度变化提出了系统Melnikov函数的改进表达式,得到了模型在参数激励下发生混沌的较精确阈值。用数值方法得到初值对系统安全性的影响,及激励参数对系统安全盆边界的侵蚀现象。观察结果发现,随着激励幅值的增大,安全盆的边界出现分形特性。In order to study the chaos and safe basin with fractal boundaries of the simply supported shape memory alloy(SMA)composite beam,the Vanderpol cycle is applied to describe the hysteretic nonlinear characteristic of the strain-stress relation of a shape memory alloy.A dynamical model of a simply supported SMA beam subject to parametrical harmonic excitation is proposed based on Galerkin′s approach.At first,the undermined fundamental frequency and normal form method is utilized to study the influence of the disturbing parameters to the fundamental frequency.Secondly,the improved Melnikov expression for the oscillator is built based on the results of the undermined fundamental frequency method and time scale transformation to obtain the approximate threshold value of chaotic motion.To quantify the chaos,the boundary of the system′s safe basin is studied and it is inclusively fractal when chaos arises based on the Melnikov method.The numerical results show the efficiency of the theoretical analysis.

关 键 词：形状记忆合金层合梁 轴向激励 改进的Melnikov方法 安全盆 混沌 vanderpol滞后环 

分 类 号：O322[理学—一般力学与力学基础] O326[理学—力学]