变截面Euler-Bernoulli梁在轴力作用下固有振动的级数解

建立了截面特性、轴力和质量分布均连续变化的Euler-Bernoulli梁固有振动方程。采用Frobeniu方法求解方程,得到了方程的级数解析解;并计算了结构的振动频率与振型。算例证明了该方法的有效性,并表明轴向力的几何非线性效应对振动的低阶频率和自振振型有一定的影响,级数项数的取值对算法的收敛性有影响。

Series Solution of Natural Vibration of the Variable Cross-Section Euler-Bernoulli Beam Under Axial Force

The differential equation for the natural vibration of Euler-Bernoulli beam with continuously varying cross-section, in consideration of axially acted force and its mass distribution is derived, and the series analytical solution is obtained by the Frobenius method. Introducing boundary conditions, the natural frequencies and modal solutions are computed. Numerical results demonstrate the accuracy of the proposed method. It concluds that the geometrically nonlinear effect of axial force has influence on the lower order natural frequencies and mode shapes to some extent. The effect of the number of terms of the series on the computation results is analyzed.