逆算符法及其在机械非线性动力分析中的应用
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摘要
本文对逆算符方法及其在机械系统非线性动力分析中的应用进行了研究。主要研
究工作如下：
利用Adomian的分解方法的思想，把机械系统中最一般的动力学模型转化为一阶
标准型微分方程组，以形式上的精确解的表达式为基础构造了求解机械系统非线性模
型近似解析解的逆算符方法（IOM）；针对机械系统非线性模型的特点，提出了直接处
理高阶方程（组）的不降阶逆算符方法；证明了该方法的收敛性。
在所建立的IOM的基础上，首次提出了基于IOM的符号-数值方法（S-N方法），
建立了适于计算的IOM-1方法及改进的IOM-1法。而精细积分法则成为IOM-1法的
一种特殊情况。
应用IOM-1法研究了齿轮系统的间隙非线性振动，表明所研究的系统随着某一参
数的变化，可通过倍周期分岔最终形成混沌响应；研究了非线性凸轮-从动件系统对不
同的输入运动的动态响应，表明系统中的弹簧非线性对系统的输出运动特性无大的影
响；给出了求解柔体系统动力学方程的S-N方法，算例表明，IOM-1法是求解非线性
刚性方程的高精度数值方法。
在对方程进行预处理的基础上，研究了具有非线性阻尼的自振系统的周期解，无
阻尼Duffing方程的周期解及van der Pol方程的周期解，应用逆算符方法求出了其近
似解析解的表达式。
ABSTRACT
The inverse operator method and its application in nonlinear dynamic analysis of mechanical systems are studied in this paper.
Firstly, by adopting the thought of Adomian’s decomposition method, the generic dynamic model in mechanical systems are transformed into a standard first-orderdifferential-equations, and then the inverse operator method (TOM) for the approximate analytic solution of nonlinear mechanical system is developed based on the exact solution in form. In accordance with the characteristics of the nonlinear mechanical models, the inverse operator method which can be used directly for the higher-order-differential-equations is presented hereby. The convergence of IOM is proved, which established the theoretical foundation of the method.
Secondly, the symbolic-numeric (S-N) method on the bases of the IOM is proposed for the first time in this paper. The one-step-inverse-operator-method (IOM- 1) and the improved IOM- 1 which are simple and practicable are therefore derived. In such a way that the prices time integration algorithm (P11) becomes a special case in the S-N method.
Thirdly, the three typical nonlinear mechanical systems are investigated by using TOM-i method. The investigation of the geared rotor-bearing system with clearance non-linearities shows the period-doubling route to chaos. The investigation regarding the dynamic responses of a nonlinear cam-follower systems to several input functions shows that the non-linearities of spring has little influence to the kinematics factors of the dynamic responses. The S-N method for the solution of flexible multibody dynamic equations is demonstrated, and the numerical results on two experimental models shows that TOM- 1 method is of high accuracy and high efficiency for solving nonlinear stiff equations.
Finally, the periodic solutions of a self-excited vibration systems with nonlinear damping, the van der Pol equation and the undamped Duffing equation are discussed. At the preliminary stage for analyzing the periodic responses, the saw-tooth time transformation and triangular time transformation are employed. And thereafter, by using IOM, the analytical approximate solution of the transformed systems are developed.
究工作如下：
利用Adomian的分解方法的思想，把机械系统中最一般的动力学模型转化为一阶
标准型微分方程组，以形式上的精确解的表达式为基础构造了求解机械系统非线性模
型近似解析解的逆算符方法（IOM）；针对机械系统非线性模型的特点，提出了直接处
理高阶方程（组）的不降阶逆算符方法；证明了该方法的收敛性。
在所建立的IOM的基础上，首次提出了基于IOM的符号-数值方法（S-N方法），
建立了适于计算的IOM-1方法及改进的IOM-1法。而精细积分法则成为IOM-1法的
一种特殊情况。
应用IOM-1法研究了齿轮系统的间隙非线性振动，表明所研究的系统随着某一参
数的变化，可通过倍周期分岔最终形成混沌响应；研究了非线性凸轮-从动件系统对不
同的输入运动的动态响应，表明系统中的弹簧非线性对系统的输出运动特性无大的影
响；给出了求解柔体系统动力学方程的S-N方法，算例表明，IOM-1法是求解非线性
刚性方程的高精度数值方法。
在对方程进行预处理的基础上，研究了具有非线性阻尼的自振系统的周期解，无
阻尼Duffing方程的周期解及van der Pol方程的周期解，应用逆算符方法求出了其近
似解析解的表达式。
ABSTRACT
The inverse operator method and its application in nonlinear dynamic analysis of mechanical systems are studied in this paper.
Firstly, by adopting the thought of Adomian’s decomposition method, the generic dynamic model in mechanical systems are transformed into a standard first-orderdifferential-equations, and then the inverse operator method (TOM) for the approximate analytic solution of nonlinear mechanical system is developed based on the exact solution in form. In accordance with the characteristics of the nonlinear mechanical models, the inverse operator method which can be used directly for the higher-order-differential-equations is presented hereby. The convergence of IOM is proved, which established the theoretical foundation of the method.
Secondly, the symbolic-numeric (S-N) method on the bases of the IOM is proposed for the first time in this paper. The one-step-inverse-operator-method (IOM- 1) and the improved IOM- 1 which are simple and practicable are therefore derived. In such a way that the prices time integration algorithm (P11) becomes a special case in the S-N method.
Thirdly, the three typical nonlinear mechanical systems are investigated by using TOM-i method. The investigation of the geared rotor-bearing system with clearance non-linearities shows the period-doubling route to chaos. The investigation regarding the dynamic responses of a nonlinear cam-follower systems to several input functions shows that the non-linearities of spring has little influence to the kinematics factors of the dynamic responses. The S-N method for the solution of flexible multibody dynamic equations is demonstrated, and the numerical results on two experimental models shows that TOM- 1 method is of high accuracy and high efficiency for solving nonlinear stiff equations.
Finally, the periodic solutions of a self-excited vibration systems with nonlinear damping, the van der Pol equation and the undamped Duffing equation are discussed. At the preliminary stage for analyzing the periodic responses, the saw-tooth time transformation and triangular time transformation are employed. And thereafter, by using IOM, the analytical approximate solution of the transformed systems are developed.
引文
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