轴压粘弹性圆柱壳在横向扰动下的混沌行为

Chaotic behavior of viscoelastic cylindrical shell under axial pressure and transverse periodic excitation

基于大挠度薄壳的Donnell-Ká rmá n理论和Kelvin-Voigt粘弹性本构关系,对轴压粘弹性圆柱壳在横向扰动下的混沌行为进行了研究。导出了关于挠度和应力函数的控制方程,借助Galerkin原理将粘弹性圆柱壳的控制方程转化为二阶三次非线性微分动力系统,用Melnikov函数给出了系统发生Smale马蹄型混沌的临界条件。数值计算分析了轴压载荷和粘性阻尼系数对混沌运动的影响。通过分岔图、位移时程曲线、相平面图和Poincar啨映射描述了系统的运动行为。研究表明:当轴压载荷与圆柱壳的材料参数满足一定关系时,系统才有可能发生Smale马蹄型混沌;随着轴压载荷的增大,混沌运动区域逐渐减小;随着粘性系数与外阻尼系数比值的增大,混沌运动区域逐渐减小;轴压粘弹性圆柱壳在横向扰动下既会发生定常运动也会发生混沌运动。

Chaotic behavior of viscoelastic cylindrical shell under axial pressure and transverse periodic excitation was investigated on the basis of Donnell-Kármán theory of thin shell with large deflection and Kelvin-Voigt constitutive relation.The governing equations for deflection and stress function were derived,and the governing equations of viscoelastic cylindrical shell were transfered into a second order cubic nonlinear differential equation system by using Galerkin method.The critical conditions of horseshoe-type chaos were obtained by using Melnikov function.The influences of axial pressure and viscous damping coefficient on chaotic motion of system were analysed with numerical calculation.The motion behaviors of system were described through bifurcation diagrams,time-history curve,phase portrait and Poincar map.The results show when the axial pressure and material parameters of cylindrical shell satisfy a certain relation between themselves,the horseshoe-type chaotic motion will appear.The chaotic motion region will be reduced with the increase of axial pressure and the raise of the ratio of viscous damping coefficient to external damping coefficient.The system of viscoelastic cylindrical shell under axial pressure and transverse periodic excitation may present steady motion or chaotic motion.