变截面修正Timoshenko梁自振频率的回传射线矩阵法分析

考虑剪切变形对转动惯量的影响，对经典Timoshenko梁理论进行修正，并基于回传射线矩阵法，将横截面线性变化的圆台均匀分为多段分段等截面等效梁，推导并求解三种经典边界(两端简支、两端固支、一端固支一端自由)条件下变截面修正Timoshenko梁的自振频率，进一步分析分段数目和梁长度的变化对变截面修正Timoshenko梁自振频率的影响；将计算结果与相同边界条件下经典Timoshenko梁的相应结果进行对比。研究表明：回传射线矩阵法用于分析变截面梁的自振频率时具有良好的计算精度和收敛性；相同边界条件下修正Timoshenko梁的自振频率小于经典Timoshenko梁，且梁越短粗，修正造成的影响(剪切变形对转动惯量的影响)就越大。

Natural frequency of a modified Timoshenko beam with variable cross-section with the method of reverberation ray matrix

Considering the influence of shear deformation on the moment of inertia, the classical Timoshenko beam theory was modified in this paper. Based on the method of reverberation ray matrix (MRRM), the bar with linearly varying circular cross-section was evenly divided into multiple constant section beams. The formulation was deduced for solving the natural vibration frequencies of the modified Timoshenko beam with variable cross-section under three classical boundaries (simply supported-simply supported, clamped-clamped, and clamped-free). The effects of the number of segments and the length of beam on the natural frequency of modified Timoshenko beam with variable cross-section were analyzed. The calculated results were then compared with those of the classical Timoshenko beam under the same boundary conditions. The results show that the MRRM has good accuracy and fast convergence. Under the same boundary conditions, the natural frequencies of modified Timoshenko beam are lower than those of classical Timoshenko beam, and the shorter the beam, the greater the influence of shear deformation on the moment of inertia.