非线性振动方程多重解求解方法  

A new solving method for multiple solutions to nonlinear vibration equations

作　　者：钱征文[1] 程礼[1] 陈卫[1] 李应红[1] 

机构地区：[1]空军工程大学工程学院,西安710038

出　　处：《振动与冲击》2011年第10期14-18,共5页Journal of Vibration and Shock

基　　金：国家自然科学基金(51175509)

摘　　要：非线性振动方程多重解的求解过程中,迭代初值难以有效确定,不稳定周期解收敛域很小,利用通常的微分方程解法无法直接求解。针对这个问题,引入同伦算法,使得初始值的选取无任何限制;同时利用预测-校正算法对外激励参数变化下的解曲线进行追踪,得到系统的多重解。该方法不但可以计算稳定的周期解,而且不稳定的周期解也可以求出。采用Duffing振子运动方程对该方法进行了计算验证,通过与理论近似解以及龙格-库塔法计算结果的对比,验证了该方法的有效性。The initial iterative value is difficult to choose when there are multiple periodic solutions in solving nonlinear vibration equations.Moreover,the simple iterative solving process of conventional approaches is inefficient to track periodic solutions under different external excitations.For these two problems,the homotopy method was employed so that the initial iterative value could be chosed easily.The solution curve varying with external excitation was tracked with the predict-correct method.Both the stable and unstable periodic solutions could be calculated by using this method.The feasibility of the method was verified through calculating a Duffing oscillator equation.It was shown that the simulation results using this method agree well with the theoretical approximate ones and the numerical ones using Rung-Kuta method.

关 键 词：非线性 振动 多重解 谐波平衡法 预测-校正算法 同伦算法 

分 类 号：O322[理学—一般力学与力学基础]