内部含不连续的变截面梁自由振动分析

Free vibration analysis of variable cross-section beam with internal discontinuities

针对带有内弹簧的变截面欧拉梁受轴压载荷的自由振动问题,提出一种根据改进傅里叶级数和伽辽金法研究其振动特性的策略。在梁两端分别设置弹簧以模拟任意边界,包括横向位移弹簧和旋转约束弹簧,梁内部的不连续变形条件通过设置横向及旋转弹簧的刚度系数来实现;具体计算过程中,首先建立振动位移的容许函数,其中的辅助多项式由端部函数值及导数值得到具体的形式;然后采用伽辽金法处理梁的自由振动控制微分方程,将振动问题转化为矩阵特征值问题;最后进行数值仿真验证,仿真中当边界条件和跨中弹性弹簧改变时需通过改变弹簧刚度系数,以改变系统的质量矩阵和刚度矩阵,可得到其振动特性。数值结果表明,该方法简便合理,具有推广价值。

Here,to solve the problem of free vibration of a variable cross-section Euler beam with internal spring under axial compression load,a strategy based on improved Fourier series and Galerkin method was proposed to study the beam’s free vibration characteristics.Springs were set at both ends of the beam to simulate any boundary including lateral displacement springs and rotating constraint springs.The discontinuous deformation condition inside the beam was also realized by setting stiffness coefficients of transverse and rotating springs.In the specific calculation process,allowable functions of vibration displacements were established firstly,and forms of auxiliary polynomials were obtained with end function values and derivative values.Then,the beam’s free vibration control differential equation was treated using Galerkin method,and the free vibration problem was converted into matrix eigenvalue problem.Finally,the numerical simulation verification was performed,changing boundary conditions and midspan elastic spring were realized by changing spring stiffness coefficients,and the system’s mass matrix and stiffness matrix also were changed to obtain its free vibration characteristics.The numerical results showed that the proposed method is simple,reasonableand worth popularizing.