非线性弹性直杆纵振时的混沌行为

Chaotic behavior of nonlinear elastic straight bar under lonitudinal vibration

研究了S形本构关系的弹性直杆纵振时的混沌行为。用Galerkin原理将杆纵振时的动力控制方程转化为二阶三次非线性微分动力系统;给出了其产生同宿轨道和异宿轨道的条件,得到了同宿轨道的参数方程;借助Melnikov函数给出了系统发生混沌的临界条件;数值计算给出了混沌运动区域随β和γ的变化规律,用分岔图、位移时程曲线、相平面图和Poincaré映射判断了系统的运动行为即定常还是混沌。进一步的研究还表明本构关系中的二次非线性项对系统的动力响应具有很大的影响。

The chaotic motion of nonlinear elastic straight bar with S-shape constitutive relation under longitudin vibration was investigated.The dynamic equation of bar during lonitudinal vibration was transfered into differential dynamic system by Galerkin principle.The conditions that homoclinic orbit or heteroclinic orbit exists were given out,and the parameter equations of homoclinic orbit were solved.Using the Melnikov function,the critical conditions that the system enters into chaotic states were provided.The regularity of chaotic motive region changing with parameters β and γ was obtained by numerical analysis,and the bifurcation diagrams,displacement-time history diagram,phase-plane diagram and Poincar map indicate that the system exibits steady motion or chaotic motion.The further research shows that the quadric nonlinear item in constitutive relation has great effect on the dynamic behavior.