摘要
将非线性动力系统化为连续变化的线性系统,并导出任意自治或非自治非线性动力系统的瞬时线性化方程,该线性方程的连续变化描述了系统的全部复杂动力行为.进一步采用Taylor变换法求解系统的线性化方程,得到一种非线性动力系统数值计算的新方法,避免了指数矩阵展开的乘积运算.计算实例表明该方法在不增加计算机时的前提下,精度高于传统的Houbolt,Wilson-θ及Newmark-β等方法.计算了Duffing方程和van Pol方程的混沌及周期特性.
Since the middle of last century, the nonlinear dynamics has been developed rapidly. The response analysis of nonlinear dynamic systems has become more and more important. The contributions from mathematics and physics are to develop new methods, and try to find naturallaws and to understand physical phenomena. Researches in this field have concentrated on lower dimension systems to find out the phenomena in nonlinear dynamic systems, such as bifurcation, chaos and strange attraction, etc. And the work based on engineering interests intends to avoid unwanted nonlinear vibrations and to estimate the effect of nonlinearity, etc. A lot of direct time integration methods, the step by step integration methods, have been developed by many authors for linear and nonlinear dynamic systems to have time history. Euler first proposed the trapezium method, and then the central difference method and Runge-Kutta method were presented. In the middle of last century, Houbolt method 1950, Newmark-/β method 1959, and Wilson-θ method 1973, etc. were published. They are the unconditional stable methods that can be used for high degrees of freedom (DOF) systems without under limited length of time steps, the details were demonstrated by Zheng 1986 and Wood 1990. In addition, some improved methods were developed and applied for practical nonlinear systems. In recent years, Chious et al. 1996 provided the Taylor transform method to transfer a nonlinear differential equation into a set of algebra equations, in which the algebra equations can be solved by the numerical iteration. However, all of these methods are based on the finite difference method. In the time steps, system equations are only satisfied at the end of time steps and are assumed to be satisfied in some pointed function. The step by step transfer form is an algebra form only, and not an integration form. There are many schemes, such as, single step and multiple steps, implicit and explicit form, conditional and unconditional stability. The single step central difference method, trapezoidal rule, was come from Euler forward (explicit) formula and backward formula. Houbolt presented a three-step unconditional stability scheme. Wilson, Newmark took off more terms in the expansion of state vector from Taylor series. This paper presents the continuous linearization model for the non-linear dynamic systems, and the time-depend linearization equation for an arbitrary autonomous or non-autonomous non-linear dynamic system is deduced, which can fully describe the complex system dynamic properties. Furthermore, the Taylor transform method is used to solve the linearization equation, therefore the complex matrixes product operation to obtain the matrix exponent is avoided. Then a new numerical calculation scheme for the non-linear system is obtained. Compare with the traditional method, such as Houbolt method, Wilson-θ method and Newmark-β method, etc, the calculation precision of this method is much higher, and the calculation time is also not increased. At last, this numerical method is applied to the chaos and periodic properties of Duffing and van der Pol equations.
出处
《力学学报》
EI
CSCD
北大核心
2002年第4期586-593,共8页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家重点基础研究发展规划项目(G1998020316)
国家自然科学基金项目(19972029)
中国博士后科学基金项目(1999
17号)资助
