变截面Euler-Bernoulli梁稳态谐振动的微分求积法研究

Differential Quadrature Method for Steady-state Harmonic Vibration of Beams with Variable Cross-section

研究运用微分求积法分析了均质变截面梁的稳态谐振动问题。首先,基于Euler-Bernoulli梁的基本理论,将均质变截面梁的横向稳态谐振动响应问题转化为一个变系数常微分方程的两点边值问题。再根据微分求积法理论,将该常微分方程的两点边值问题转化为高斯主元消去法求解线性代数方程组,从而获得均质变截面梁稳态谐振动的位移及内力正确解。通过等直梁和变截面梁两个数值算例,验证了微分求积法研究变截面梁稳态谐振动的可行性和精确性;同时,对数值计算结果进行数据分析,由梁的位移和内力等物理参量的急剧变化这一现象,可以定性判定梁发生共振的频率范围。

The differential quadrature method is used to analyze the steady-state harmonic vibration of a beam with variable cross-section.Firstly,based on the basic theory of Euler Bernoulli beam,the transverse steady-state harmonic vibration response of homogeneous beam with variable cross-section is transformed into a two-point boundary value problem of ordinary differential equation with variable coefficients.Then,according to the theory of differential quadrature method,the two-point boundary value problem of the ordinary differential equation is transformed into the Gauss principal component elimination method to solve the linear algebraic equations,so as to obtain the correct solutions of the amplitude and internal force of the steady-state harmonic vibration of the homogeneous variable cross-section beam.The feasibility and accuracy of differential quadrature method to study the steady-state harmonic vibration of variable cross-section beam are verified by two numerical examples of equal straight beam and variable cross-section beam.Meanwhile,through the data analysis of the numerical results,the frequency range of resonance can be determined qualitatively by the rapid changes of physical parameters such as displacement and internal force of the beam.