任意边界条件下Timoshenko梁及其修正理论的自振特性分析

Analysis on natural vibration characteristics of Timoshenko beam and its modified theory with arbitrary boundary conditions

提出一种求解任意边界条件下经典Timoshenko梁以及修正Timoshenko梁自振频率和振型的新方法。利用改进的傅立叶级数消除传统傅立叶级数的边界不收敛问题,然后通过Rayleigh-Ritz法导出Timoshenko梁的拉格朗日泛函,根据Hamilton原理将原问题转化为求解矩阵广义特征值问题。通过与解析解对比,本文采用的方法具有较好的收敛性以及较高的计算精度;通过数值计算发现,经典Timoshenko梁的自振频率略高于修正的Timoshenko梁,随着振型阶数的提高,经典Timoshenko梁的计算结果逐渐偏离文献解和有限元结果,而修正的Timoshenko梁能够保持较好的一致性;对于不同边界条件下修正Timoshenko梁的计算结果均能与有限元的计算结果吻合得很好。最后运用MATLAB编程软件将程序设计为App,对于不同情形的梁只需要修改参数即可,可为实际工程提供高效便捷的计算方案和可靠理论依据。

A new method for determining the natural frequencies and mode shapes of a classical Timoshenko beam and modified Timoshenko beam with arbitrary boundary conditions is presented.The improved Fourier series is used to eliminate the boundary non-convergence problem of the traditional Fourier series,and then the Lagrange functionals of the Timoshenko beams are derived by the Rayleigh-Ritz method.According to the Hamilton principle,the original problem is transformed into a generalized matrix eigenvalue problem.Compared with the analytical solution,the method used in this paper has better convergence and higher calculation accuracy.The numerical results show that the natural frequency of the classical Timoshenko beam is slightly higher than that of the modified Timoshenko beam.With the increase of the vibration mode order,the calculation results of the classical Timoshenko beam gradually deviate from the solution from the published literature and the finite element results,while the modified Timoshenko beam can maintain good consistency.The calculation results of the modified Timoshenko beam under different boundary conditions are in good agreement with the finite element results.Finally,the MATLAB software is used to produce an App program.Using this App,for different beams,only the parameters need to be modified,which can provide an efficient and convenient calculation scheme and reliable theoretical basis for practical engineering.