摘   要：推导Timoshenko梁振动微分方程的初参数解，结合边界条件，建立简支梁的频率方程. 当固有频率小于临界频率时，频率方程有双曲正弦函数与三角正弦函数之积的因式，当固有频率大于临界频率时，此因式变成为双三角正弦函数之积，此即Timoshenko梁产生第二频谱的理论原因. 推导出等截面等跨径的2～3跨连续Timoshenko梁的频率方程，并从理论上预测存在第二频谱现象的其他结构. 建立了简支Timoshenko梁第一、二频谱的频率计算公式. 通过实例验证第二频谱的存在. 通过微分方程求解，论证了临界频率是结构固有频率的有效组成部分，其对应的竖向位移模态无振幅、转角位移模态的振幅为常数;指出数值分析时，由于计算机截断误差的影响，所预测的临界频率有误差、所对应的竖向位移模态为不规则模态等特点.

关键词：频率分析;临界频率;第二频谱;Timoshenko梁

中图分类号：TU375                            文献标志码：A

Analysis on the Second Frequency Spectrum of Timoshenko Beam

XIA Guiyun

（School of Civil Engineering，Changsha University of Science and Technology，Changsha 410114，China）

Abstract：Incorporating the boundary conditions， initial parameter solutions of vibration differential equations for Timoshenko beam are used to derive the frequency equation of a simply-supported beam. When the natural frequency is less than the critical frequency， the frequency equation can be factorized into the hyperbolic sine function and the trigonometric sine function， while， when the natural frequency is greater than the critical frequency， the frequency equation can be factorized into double trigonometric sine functions， which is the crucial reason for the existence of the second frequency spectrum. Frequency equations for two-span and three-span continuous Timoshenko beams with uniform cross sections and equal spans are derived. Other structures with the second frequency spectrum are forecasted theoretically. The formulas for the first and second frequencies are deduced for simply-supported Timoshenko beam. The existence of the second frequency spectrum is confirmed through the examples. Through solving the differential equation of motion， the critical frequency is proven to be an efficient part of the natural frequencies for the framed structures. The corresponding mode shape of the critical frequency contributes to the displacement mode shape with zero amplitude and rotation mode shape with constant amplitude. Due to the truncation error of the computer， the critical frequency predicted by the finite element method shows error， and the mode shape of the displacement is very irregular.

Key words：frequency analysis;cutoff frequency;second frequency spectrum;Timoshenko beam

梁的弯曲振动是土木[1]、机械、石油、化工、航空航天等领域的重要问题，已有多种理论. 最早、最经典的理论是Euler梁模型，该模型考虑了梁的弯曲和截面惯性力，适应于细长杆系结构的分析计算，但对高跨比较大的深梁結构，存在静力问题计算挠度偏小[2]、动力问题高估振动频率和有无限阶次频率[3]等不足. Elishakoff等[4]评述Rayleigh梁考虑了截面转动惯量的影响，对Euler梁进行改进. Shear梁模型考虑结构弯曲、剪切变形的影响和截面惯性力，但没有考虑截面转动惯量影响（此模型不同于一般的不考虑结构弯曲……