求解非线性振动周期解的参数延续打靶法

Parameter continuation-shooting method for the calculation of periodic solutions of nonlinear vibration

高维非线性振动系统复杂周期解的计算是非线性的研究难点之一,传统解析方法求解过程复杂,局部延拓打靶法会在转折点处失效,限制了非线性理论在工程中的应用。本文将同伦延拓的思想和打靶法结合,形成参数延续打靶法,利用切向预估和垂直于切向的超平面的牛顿校正,使得打靶法在求解过程中顺利通过转折点。最后利用含外激励和阻尼的Duffing方程、转子系统碰摩两个算例,通过与谐波平衡法、龙格-库塔法结果的对比,说明本方法合理有效,且能够用于分析参数大范围变化时系统周期解的全局行为。

The calculation of complex periodic solutions of high dimensional nonlinear vibration system is one of the difficulties in nonlinear dynamics.The traditional analytical method has a complicated process which is too difficult for the users,and the local continuation shooting method will fail at the turning point.All of those limit the engineering applications of the nonlinear vibration theory.In this paper,homotopy continuation and the traditional shooting method are combined to a new method:parameter continuation-shooting method.Estimating the initial values at the tangent direction,and then correcting with Newton method at the hyperplane which is perpendicular to the tangent direction,the shooting method can get through the turning point in the solving process.Finally,examples of Duffing equation with external excitation and damping,and the rotor rubbing system are given.The comparison with the results of harmonic balance method and Runge-Kutta method shows that this method is accurate and effective,and it can be used for the investigation of global behavior of the periodic solutions of the nonlinear system.