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| 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 under Arbitrary Boundary Conditions |
| 投稿时间：2022-10-01 修订日期：2022-11-21 |
| DOI： |
| 中文关键词: Timoshenko梁 自振频率 修正Timoshenko梁 Rayleigh-Ritz法 IFSM法 APP设计 |
| 英文关键词:Timoshenko beam Modified Timoshenko beam natural vibration characteristics Rayleigh-Ritz method IFSM method App design |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
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| 中文摘要: |
| 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，本文所采用的方法具有较好的收敛性以及较高的计算精度；通过数值计算得出：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的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，对于不同情性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。 |
| 英文摘要: |
| A new method for solving the natural frequencies and mode shapes of 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 functional of the Timoshenko beam is derived by the Rayleigh-Ritz method. According to the Hamilton principle, the original problem is transformed into a matrix generalized 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 literature solution 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 calculation results. Finally, the MATLAB programming software is used to program the App. For different beams, only the parameters need to be modified, which can provide efficient and convenient calculation scheme and reliable theoretical basis for practical engineering. |
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