基于双势阱系统的混沌振动研究

Chaotic vibration study based on two-well potential theory

应用多尺度法求出了双势阱系统自由振动响应的近似解析解,分析了系统响应中慢变分量和快变分量对响应的影响。采用平均法得到了系统的幅频关系表达式,给出了振动幅值和频率调节因子之间的舌状结构曲线,揭示了系统的非线性本质特征。为了证实理论分析的正确性,利用双势阱理论设计了单端磁吸式混沌振动试验装置。研究了不同激励幅值、不同激励频率下系统的响应。观察到了系统中出现的次谐波现象、超谐波现象以及系统中周期1运动的不同模式。通过参数改变,该系统在一定频率和激励幅值条件下可以产生持续稳定的混沌振动。应用相空间重构技术重构了实测信号的奇怪吸引子图,计算了混沌信号的最大Lyapunov指数和关联维数,研究结果为混沌工程应用提供了有益参考。

The multiscale method was applied to solve the approximate analytical solution to a two-well potential system's free vibration response. The effects of fast varying components and slow varying ones in the system response on the response were analyzed. The amplitude-frequency relationship of the system was obtained with the averaging method.The tongue shape structural curve of the vibration amplitude versus the frequency regulation factor was gained,it revealed the system's nonlinear essential characteristics. To verify the theoretical results,a test rig of magnetic absorbing leaf spring based on the two-well potential theory was designed here. The response characteristics of this system subjected to different excitation amplitudes and frequencies were studied. Subharmonic phenomena,superharmonic phenomena and different modes of period-1 motion of the system were observed. Meanwhile,through changing parameters of the system,continuous and stable chaotic vibration was excited under the condition of a certain frequency and excitation amplitude.The strange attractors of the measured signals were reconstructed with the phase space reconstruction technique. The maximum Lyapunov exponent and correlation dimension number of chaotic signals were calculated. Results were beneficial to the application of chaos phenomena in engineering.