利用Lie对称性法研究Duffing自由振动的精确解析动态响应

摘要

针对一类单自由度含阻尼的强非线性Duffing方程,在没有任何假设和近似的前提下,根据微分方程群不变性理论,经过Lie点变换,证明在参数满足一定关系时,系统的特殊无穷小对称可导出系统线性无关的不同形式积分因子,进而给出该类动力学方程的两个首次积分,结合消元法为非线性振动系统的精确解析解提供一个有效途径.%For a class of nonlinear Duffing equation with damping single degree of freedom, without any assumptions and approximations, according to differential equation group invariance theory, through Lie transform, it is proved that the special infinitesimal symmetry derived in different forms of integral factor system are linearly independent when the parameters satisfy certain relations, and that the kinetic equation of two first integral with elimination method can provide an effective way to exact analytical solutions of the system.