山东自考论文范文：变截面梁自由振动分析的精细积分法

摘要

传统方法分析梁的横向振动时，是求解同一种变量(挠度)表示的高阶微分方程。本文以材料力学中载荷集度、剪力、弯矩以及挠度之间的微分关系，将挠度、转角、弯矩和剪力作为混合变量称为状态变量，它代表了梁一个截面的变形和内力的状态，建立了以状态变量表示的、仅为一阶齐次线性常微分方程，形式简单，极易求解。用精细积分法求解此微分方程，得到一个分析梁横向振动的状态方程，用传递矩阵表示了梁段两端状态变量的传递关系。利用边界条件建立频率方程，用二分法求解出各阶频率。把各阶频率分别回代到频率方程中，得出一个二元一次方程组，由于此方程组是一个不互相独立方程组，所以只能求出边界上状态向量的相对值，把此状态向量代入状态方程就可以求出梁各点的相对状态向量，也就是振型。本方法适用于求解各种边界条件下的阶梯梁及带集中质量和弹性支承梁的横向振动固有频率和振型，具有普遍性，相应的Matlab程序也很容易编出。应用本文方法分析自由振动特性时，省略了求解微分方程的繁琐数学过程，计算方便、迅捷、且精度高。

关键词: 精细积分,传递矩阵，指数矩阵,自由振动,状态方程。

Abstract

Traditional methods for analyzing horizontal vibration of the beam, is to solve the high-order differential equations with deflection as variable. In this paper, the band-linear differential equations expressive with state vector was established, which form is simple and solution is easy to solve, this state vector is a mixed variables ,which is represented with the each other relations of the Load Collection Degree, shear, bending moment and deflection in mechanics of materials, it represents the state of deformation and internal forces in the cross-section of the beam. We solve this equations With precise integration of the method, as a result ,there is a beam of horizontal vibration variable equation of state ,which use of analysing free vibration of horizontal beam and indicating the transfer relations of the state at both ends of beam with the transfer matrix. The frequency equation is established whit Boundary conditions, to solve this equation with the second order method for the orders frequency. With the frequency back substituting into frequency equation, the simple equation group with two unknown quantity is obtained,because of this equation is an independent equations, we can only obtain on the relative state vector on border, and then with this vector substitute into the state equation can be obtained the relative status vector of every point on the beam, which is mode vector.This method can be applied to solving the natural frequency and mode voctor of horizontal vibration of beam with ladder,concentrated quality and flexibility on various boundary conditions a horizontal beam, which is universal, and the corresponding Matlab procedures can easily prepared.With this method to analyzing horizontal vibration, to omitted the tedious process of mathematics to solve differential equations,it is convenient, fast accuracy,to calculate.

Key wors: Precise Integration, Transition Matrix, exponential Matrix, Free Vibrations, Equation Of State.

目录

1 绪论 ..............................................................................1

1.1 引言 ........................................................................1

1.2 本文的目的和意义 ...............................................................1

1.3 本课题研究现状 .................................................................1

1.3.1 变截面自由振动的研究现状................................................1

1.3.2 精细积分法研究现状......................................................2

1.4 本文所要研究的内容..............................................................4

2 精细积分方法简介 ..................................................................5

2.1 引言 ...........................................................................5

2.2 现代结构动力学及其研究方法 .....................................................5

2.3 结构时域瞬态动力响应分析方法概述 ...............................................6

2.3.1坐标变换法 ...................................................................7

2.3.2直接积分法(逐步积分法) ........................................................8

2.4 结构动力方程的精细时程积分法....................................................14

2.4.1 方程的变换与时程积分.........................................................14

2.4.2 指数矩阵的精细计算...........................................................15

2.4.3 非齐次方程...................................................................16

2.4.4 精细积分的精度分析...........................................................17

2.4.5 指数矩阵的性质...............................................................17

3 变截面梁自由振动分析...............................................................19

3.1 引言............................................................................19

3.2 等截面梁普遍方程................................................................19

3.3 变截面梁自由振动的精细积分法....................................................21

4 数值算例...........................................................................23

结束语 ...............................................................................26

致谢 .................................................................................27

参考文献 .............................................................................28

附录 Matlab程序 .....................................................................31