非线性能量阱系统的强调制响应研究

Investigation on Strongly Modulated Response of Nonlinear Energy Sink System

为了得到谐波激励下本质非线性能量阱系统出现强调制响应的充要条件,开展非线性能量阱系统的强调制响应研究.利用复变量平均法推导主共振下系统的慢变动力流方程;结合多尺度法实现系统快、慢变流形的分离,得到不同参数下系统的慢不变流形以及全局分岔特性;通过构建快变流形的一维映射函数,揭示非线性能量阱系统呈现强调制响应状态的充要条件;搭建非线性能量阱系统的模拟电路,进行强调制响应检测电路试验.仿真和试验结果表明,强调制响应是由耦合系统慢变动力流中极限环的鞍结分岔引起的且真实存在,系统出现稳定的强调制响应必须满足两个条件:一是非线性能量阱系统响应超越慢不变流形上的极值点,又不吸引至慢不变流形某一分支;二是形成连续跳跃环路而不陷入局部循环时.

In order to obtain the necessary and sufficient conditions for the strongly modulated response(SMR)of a nonlinear energy sink(NES)system under harmonic excitation,the SMR study of the NES system is carried out.Firstly,the equation of systematic slow variation flow is derived by using the complex-averaging method.Secondly,the multi-scale method is used to realize the separation of the rapid variable manifold and the slow variable manifold.And the systematic slow invariant manifold under the different parameters and the global bifurcation property are ob⁃tained.Then,by building the one-dimensional mapping function of the rapid variable manifold,the sufficient and necessary conditions for the NES system in the SMR state are revealed.Finally,the simulation circuit of the NES sys⁃tem is constructed,and the response detection circuit test is carried out.The simulation and experimental results show that the SMR is caused by the saddle-node bifurcation of limit cycles in the slowly varying power flow of the coupling system,and it is an actual phenomenon in engineering.The system in which SMR state can appear must sat⁃isfy the following two conditions:the response of the NES system exceeds the amplitude of the extremum point on the slow invariant manifold,but it does not attract to a branch of the slow invariant manifold,and a continuous jump loop without falling into a local cycle is formed.