内部含不连续的变截面梁自由振动分析

顾业清,常婷婷,鲍四元,沈峰

振动与冲击 ›› 2022, Vol. 41 ›› Issue (19) : 164-171.

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PDF(1049 KB)
振动与冲击 ›› 2022, Vol. 41 ›› Issue (19) : 164-171.
论文

内部含不连续的变截面梁自由振动分析

  • 顾业清,常婷婷,鲍四元,沈峰
作者信息 +

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

  • GU Yeqing, CHANG Tingting, BAO Siyuan, SHEN Feng
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文章历史 +

摘要

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也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,把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,本文的方法简便合理,具有较好的推广价值。
关键字:欧拉梁;内部不连续;改进傅里叶级数;自由振动

Abstract

Aiming at solving the free vibration of an Euler-Bernoulli beam with variable cross-section and internal spring subjected to axial compression load under arbitrary boundary conditions, a strategy based on the improved Fourier series and Garlerkin method is proposed to study the vibrational characteristics of beams. In order to simulate arbitrary boundaries, the transverse displacement spring and rotational spring are set at both ends of the beam. By assigning different values to the stiffness coefficient of the spring, the internal discontinuity condition is simulated. First, the displacement admissible function of the vibration displacement is established, in which the auxiliary polynomial is derived from the value of the end function or derivative. Then, the Garlerkin method is used to deal with the governing differential equation for free vibration of beams, thus the vibration problem is transformed into matrix eigenvalue problem. Finally, the numerical simulation is carried out by using the numerical calculation software. When the boundary conditions and mid-span discontinuity are changed, the mass matrix and stiffness matrix of the system can be determined by changing the spring stiffness coefficient, and the corresponding vibrational characteristics can be obtained. Numerical results show that the proposed method is simple, reasonable and of good practical significance.
Keywords: Euler-Bernoulli beam; internal discontinuity; the improved Fourier series; free vibration

关键词

欧拉梁 / 内部不连续 / 改进傅里叶级数 / 自由振动

Key words

Euler-Bernoulli beam / internal discontinuity / the improved Fourier series / free vibration

引用本文

导出引用
顾业清,常婷婷,鲍四元,沈峰. 内部含不连续的变截面梁自由振动分析[J]. 振动与冲击, 2022, 41(19): 164-171
GU Yeqing, CHANG Tingting, BAO Siyuan, SHEN Feng. Free vibration analysis of variable cross-section beam with internal discontinuities[J]. Journal of Vibration and Shock, 2022, 41(19): 164-171

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