中文摘要：

基于考虑剪切变形所引起转动惯量的Timoshenko梁,采用分离变量法和高阶线性微分方程组特征值问题求解方法,系统地给出了修正Timoshenko简支梁模态特性的分析方法,推导得到了修正Timoshenko简支梁自振频率计算公式和振型函数表达式;并给出了Euler梁模型相对于修正Timoshenko梁模型的误差计算公式.分析结果表明：影响Euler梁模型计算误差的因素包括四个方面：振型阶数、材料泊松比、梁剪应力不均匀系数和回转半径与梁高跨比;随着振型阶数和高跨比的增加,Euler梁模型计算误差值迅速增长;在建筑材料泊松比的分布范围内,Euler梁模型计算误差随泊松比大约呈线性增长趋势;典型截面对Euler梁模型计算误差影响的排序为：圆形＜矩形＜T字型＜圆管＜箱型＜工字型＜H型,采用Euler模型计算工字型和H型截面梁振型频率时,需要特别加以注意.

英文摘要：

Associated with the modified motion equation of Timoshenko, in which the rotational inertia induced by shear deformation is considered, a system approach to solving the eigenvalue problems of simply supported beam is presented. The natural frequency and mode function of simply supported beam based on modified motion equation of Timoshenko was deduced, according to the method of separation of variables and the solution method of eigenval- ue problem for higher order linear differential equations. Compared with modified Timoshenko beam model, errors existing in Euler beam model are also showed in the paper. The presented theoretical and numerical results show that the influencing factors of errors existing in Euler beam include four aspects ： vibration modes, Poisson＇ s ratio, sectional shear coefficients and radius of gyration and the depth-span ratio. With the increase of vibration modes and the depth-span ration, errors existing in Euler beam are also increasing fast. For ordinary building materials, errors are approximately linear growing with the rise of Poisson＇ s ratio. Error values existing in Euler beam are greatly influenced by the cross sections, they were followed the order of circular cross-section 〈 rectangular cross- section 〈“T” shaped cross-section 〈 tube cross-section 〈 “I” shaped cross-section 〈 “H” shaped cross-sec- tion. Hence, special attention should be paid when Euler beam is used as the model for the calculation of the natural frequency of “I” shaped cross-section or“H”shaped cross-section beam.