中文摘要：

从Hamilton原理出发,建立了考虑剪切变形和转动惯量的Timoshenko梁振动微分方程,采用分离变量法求解了不同长细比（跨高比）的简支Timoshenko梁的特征值和特征函数。在此基础上,得到以Euler梁为基础考虑转动惯量的Rayleigh梁,以及在Euler梁基础上只考虑剪切变形的梁的精确解。注意到微分方程关于转动惯量项和剪切变形项的对称性,可以直观地得到简支Timoshenko梁的近似解。通过近似的数学运算和力学分析,证明了这个近似解构成精确解的下限解。针对实际工程中的工字形截面钢梁和矩形截面混凝土梁,给出了数值运算结果供实际工程应用参考。

英文摘要：

Based on the Hamilton＇ Law, the vibration governing equations of the Timoshenko beam were derived, then the eigen - value and eigen - function of a simply supported Timoshenko beam were obtained by solving the governing equations with the separating variable method. From governing equations and the eigen - function, the accurate natural frequencies were obtained for the simply supported Rayleigh beam that only the rotary inertia was added and the simply supported shear beam that only shear deformation was added, respectively, on the basis of the Euler - Bernoulli beam. Utilizing the symmetry of the governing equations about the rotary inertia and shear deformation, an approximate formula of the natural frequencies of the simply supported Timoshenko beam could be directly presented. The results show that the approximate solutions are the lower bound of the accurate ones, For the I- section steel beam and rectangular section RC beam, the calculation results of the approximate natural frequencies and the accurate natural frequencies are compared.