非线性包装系统自由振动特性的对称性解法

目的基于Lie积分法精确解析求解包装系统的非线性自由振动响应。方法考虑到弹簧恢复力三次多项式形式的非线性关系,依据分析力学准则建立系统单自由度含阻尼动力学模型;首次运用微分方程Lie群变换理论求解系统的对称性和2个首次积分,证明在结构设计参数满足一定关系时,包装系统自由振动的精确解是一类椭圆积分函数。结果实际算例仿真计算表明,系统的自共振频率随着初始振幅条件的增大而增大,非线性系数项使得位移响应振幅的衰减变快。结论从推演过程可看出,将Lie对称性理论应用到包装系统非线性动力学特性研究中,系统的非线性系数以及阻尼系数无需满足小参数假设,因此适用范围更广。

A Symmetric Solution for the Free Vibration Characteristics of a Nonlinear Packaging System

The work aims to accurately solve the nonlinear free vibration response of a packaging system based on Lie integral method. Considering the nonlinear relation of the cubic polynomial form in the spring restoring force, the system model with single degree of freedom and damping dynamics was established based on the analytical mechanics principle. The differential equation Lie group transformation was used for the first time to solve the symmetry and two first integrals of the system, for the purpose of proving that, the exact solution of the free vibration of the packaging system was a class of elliptic integral functions when structural design parameters satisfied certain relations. The actual numerical simulation calculations showed that, the self-resonance frequency of the system increased with the increase of the initial amplitude conditions, and the nonlinear coefficient made the attenuation of the displacement amplitude faster. From the perspective of deduction process, the Lie symmetry theory is applied to the study on the nonlinear dynamic characteristics of packaging system, and the nonlinear coefficient and damping coefficient of the system do not need to satisfy the hypothesis of small parameters; therefore, the application scope is wider.