【作者】 邓浩；

【导师】 方玺；

【作者基本信息】 武汉理工大学 ， 数学， 2020， 硕士

【摘要】 轴承转子系统作为核心部件被广泛地应用于发电机、燃气轮机和航空发电机等机械设备.针对其动力学特性,目前应用最广泛的数值分析方法是传递矩阵法和有限元法.然而传递矩阵法精度低,容易出现数值不稳定,有限元法耗时且占内存.因此,本文将上述两种算法相结合,构建出兼顾计算精度和运行速度的算法,不仅能丰富算法理论,也具有重要的现实意义.构建保辛的数值算法是确保长时间数值稳定性的关键所在.因此,本文在Hamilton系统背景下,构建传递辛矩阵并验证该方法在临界转速和振型求解上具有高精度.构建基于Magnus级数的精细积分格式以解决逐步积分法在求解动力学方程时的精度不足和稳定性限制等问题.本文的主要工作如下:1.推导了经典传递矩阵法的基本格式,给出了 Hamilton矩阵和辛几何的几个重要结论,介绍了求解线性结构动力方程常用的精细积分法,并给出了矩阵指数的精细计算步骤.2.将有限元法和传递矩阵法相结合,构建系统运动的数学模型,推导出Hamilton系统下转子的传递辛矩阵,给出了转子振型,临界转速,动力响应的计算公式.通过数值实验验证了在计算高阶临界速度时,传递辛矩阵法的数值精度和数值稳定性相较于传统的传递矩阵法有所提高.3.对精细积分法进行改进,构建了基于Magnus级数的精细积分格式.首先将动力学方程导向Hamilton系统,得到方程解的结构v(t)=eΩ(t)v0.然后通过详细的推导证明求出Magnus级数Ω(t)的具体表达形式,以Gauss-Legendre求积公式做Magnus级数的数值逼近得到迭代系数矩阵exp(Ωn),再应用矩阵指数的精细计算给出了迭代系数矩阵exp(Ωn)的精细积分格式.最后通过数值算例,从算法的计算精度,稳定性,计算效率等方面验证了基于Magnus级数的精细积分法优于Newmark-β法.本文将Hamilton系统下的传递辛矩阵方法应用于转子的动态特性分析,与传统的传递矩阵法相比,避免了求解转子梁的聚集模型,一定程度上简化了运算且物理意义明确.利用基于Magnus级数的精细积分法求解动力学方程的瞬态效应,为转子系统的实时工况提供了一种高效稳定的分析手段,为转子的工作参数和结构优化设计提供了理论依据.更多还原

【Abstract】 As the core component,bearing rotor system is widely used in generators,gas turbine,aviation generator and other mechanical equipment.In view of its dynamic characteristics,the most widely used numerical analysis methods are transfer matrix method and finite element method.However,the accuracy of transfer matrix method is low and numerical instability is easy to occur.Finite element method is time-consuming and takes up a lot of memory.Therefore,this dissertation combines the above two algorithms to build an algorithm with both calculation accuracy and running speed,which not only enriches the algorithm theory,but also has important practical significance.Constructing a symplectic numerical algorithm is the key to ensure long-term numerical stability.Therefore,under the background of Hamilton system,this dissertation constructs the symplectic transfer matrix and verifies that the method has high accuracy in the critical speed and mode.The precise integration scheme based on Magnus series is constructed to solve the problem of insufficient precision and limited stability of the stepwise integration method in solving dynamic equations.The main work of this dissertation is as follows:1.The basic format of the classical transfer matrix method is derived.Some important conclusions about Hamilton matrix and symplectic geometry are given.The precise integration method used to solve the dynamic equations of linear structures is introduced,and the precise calculation steps of the matrix exponential are given.2.The finite element method and transfer matrix method are combined to construct the mathematical model of system motion.The transfer symplectic matrix of the rotor under Hamilton system is derived and the calculation formulas of rotor vibration mode,critical speed and dynamic response are given.Numerical experiments verify that the numerical accuracy and numerical stability of the transfer symplectic matrix method are improved compared with the traditional transfer matrix method when calculating the high order critical speed.3.This dissertation improves the precise integration method and constructs the precise integration scheme based on Magnus series.Firstly,the dynamic equation is directed to the Hamilton system to obtain the structure of the equation solution v(t)=eΩ(t)v0.Then,the specific expression of the Magnus series Q(t)is obtained through detailed derivation and proof,and the iterative coefficient matrix exp(Qn)is obtained by using the Gauss-Legendre quadrature formula to approximate the numerical value of the Magnus series.The precise integral scheme of the iterative coefficient matrix exp(Qn)is given by using the precise calculation of matrix exponent.Finally,through a numerical example,the precise integration method based on Magnus series is verified superior to the Newmark-β method in the algorithm of calculation accuracy,stability,the computational efficiency and other aspects.In this paper,the transfer symplectic matrix method under Hamilton system is applied to the analysis of the dynamic characteristics of the rotor.Compared with the traditional transfer matrix method,the transfer symplectic matrix method proposed in this paper avoids solving the aggregation model of the rotor beam,and simplifies the calculation to some extent and its physical significance is clear.The precise integration method based on Magnus series is used to solve the transient effect of the dynamic equation,which provides an efficient and stable analysis method for the real-time working condition of the rotor system,and provides a theoretical basis for the optimal design of the rotor’s working parameters and structure.更多还原