强非线性振动系统近似解研究与精度分析

There are many methods for solving the approximate solution of nonlinear oscillation. The traditional approximate solution method is only applicable to the weakly nonlinear system with a small parameter, not suitable for strongly nonlinear oscillations system. Some improved traditional analytic methods are only applicable to some particular single degree of freedom strongly nonlinear vibration system, not suitable for the general single degree of freedom and many degrees of freedom strongly nonlinear oscillation system.. This project applies the energy conservation method ,the homotopy perturbation method and combined with numerical methods to study on the general single degree and many degrees of freedom strongly nonlinear oscillation systems theoretically: On one hand, based on the mechanical characteristic that periodic solutions in a period the average energy should be conserved, we establish a quite new energy coordinate system and apply the energy conservation method to obtain sufficient and necessary conditions under which the periodic solutions of strongly nonlinear vibration system are existence and stability. The approximate expressions for the periodic solutions are given as well. On the other hand, by constructing the homotopy mapping, we convert the strongly nonlinear oscillation system into the weak nonlinear oscillation system, using the traditional analytical method, we construct the analytical expression of the approximate solution. Comparing the approximate solution with the numerical solution, we get the precision of the approximate solution and prove the efficiency of the expression of the approximate solution. The project provides some theoretical basis to limit harmful vibration, make use of profitable vibration and find out the vibration mechanism and reveal vibration inner laws and exterior factors.

非线性振动问题的近似求解方法有多种。传统的求近似解的方法只适用于含小参数的弱非线性系统，不适合强非线性振动系统；而一些改进的传统解析方法只适用于一些比较特殊的单自由度强非线性振动系统，对一般的单自由度或多自由度强非线性振动系统并不适用。.本项目利用能量守恒法和同伦摄动法并结合数值方法对一般单自由度和多自由度强非线性振动系统进行理论研究：一方面基于周期解在一个周期内其平均能量应该守恒这一力学性质，建立全新的能量坐标系，应用能量守恒法给出强非线性振动系统周期解存在与稳定的充分与必要条件并得出近似解的解析表达式；另一方面通过构造同伦映射化强非线性振动系统为弱非线性振动系统，再运用传统的解析方法构造近似解的解析表达式并与数值方法求得的解进行比较，得出近似解的精度，验证所给近似解表达式有效性。为限制有害振动和利用有用振动、弄清振动机理、揭示振动内在规律和外部影响因素提供理论依据。

振动存在于自然界各个领域，按其特性可分为线性振动和非线性振动两类。 严格地说，绝大多数振动系统都是非线性的, 非线性振动系统很难求得精确解析解。非线性振动问题的近似求解方法有多种, 传统的求近似解的方法只适用于含小参数的弱非线性系统，不适合强非线性振动系统；一些改进的传统解析方法大都只适用于一些比较特殊的单自由度强非线性振动系统, 而对一般的单自由度或多自由度强非线性振动系统的求解较少涉及。. 本项目利用同伦摄动法、变分迭代法和能量守恒法并结合微分不等式、不动点原理和Mawhin 重合度理论围绕一些可积的具有实际应用背景的一般单自由度和多自由度强非线性系统进行理论研究：一方面通过构造同伦映射化强非线性振动系统为弱非线性振动系统，运用改进的解析方法构造非线性系统近似解的解析表达式或利用变分原理构造广义变分迭代, 其次决定方程的初始近似, 再通过迭代表示式得到了对应方程的任意次近似解。所得的近似解结合微分不等式和不动点原理以及与数值模拟解进行比较，得出近似解的精度，验证所给近似解表达式有效性；另一方面基于周期解在一个周期内其平均能量应该守恒这一力学性质，建立新的能量坐标并结合Mawhin 重合度理论研究了一类具有一般非线性弹性力、广义阻尼力和强迫周期力项的相对转动强非线性动力系统周期解问题。. 由上述方法求得的强非线性振动近似解，不同于单纯模拟得到的数值近似解，它保留了解析表达式的特点，可以继续对它进行微分和积分等解析运算， 因此可以利用它对强非线性振动系统的定性性质做进一步的探讨和研究，为解决工程技术和实际生活中的非线性振动问题、最大限度限制有害振动和有效利用有用振动、弄清振动机理、揭示振动内在规律和外部影响因素提供理论依据和参考。