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Frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Timoshenko-Ehrenfest beam with applications
Computers & Structures ( IF 4.4 ) Pub Date : 2025-01-08 , DOI: 10.1016/j.compstruc.2024.107599
J.R. Banerjee
Computers & Structures ( IF 4.4 ) Pub Date : 2025-01-08 , DOI: 10.1016/j.compstruc.2024.107599
J.R. Banerjee
Earlier research on the development of explicit algebraic expressions for the elements of the frequency-dependent mass, elastic and geometric stiffness matrices for free vibration analysis was carried out on Bernoulli-Euler, Timoshenko-Ehrenfest and axially loaded Bernoulli-Euler beams. Seeking solution for the correspondingly more difficult problem for an axially loaded Timoshenko-Ehrenfest beam seemed too difficult at the time when these earlier developments took place. Now, with the experience and knowledge gained, the difficulty is overcome in part by enhanced application of symbolic computing. Thus, the explicit algebraic expressions for the elements of the frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Timoshenko-Ehrenfest beam are derived from first principles. The equivalency of these matrices when added altogether, with the dynamic stiffness matrix is ensured. The derived matrices are then applied using the Wittrick-Williams algorithm as a solution technique to investigate the free vibration characteristics of some illustrative examples. The results are discussed, and significant conclusions are drawn. The proposed method preserves the exactness of results in the same way as the dynamic stiffness method, but importantly, it opens the possibility of including damping in free vibration and response analysis when using exact methods such as the dynamic stiffness method.
中文翻译:
轴向加载的 Timoshenko-Ehrenfest 梁的频率相关质量、弹性和几何刚度矩阵及其应用
早期对用于自由振动分析的频率相关质量、弹性和几何刚度矩阵单元的显式代数表达式的研究是在 Bernoulli-Euler、Timoshenko-Ehrenfest 和轴向加载的 Bernoulli-Euler 梁上进行的。在进行这些早期发展时,为轴向加载的 Timoshenko-Ehrenfest 梁寻找相应更困难的解决方案似乎太困难了。现在,随着获得的经验和知识,符号计算的增强应用在一定程度上克服了困难。因此,轴向加载的 Timoshenko-Ehrenfest 梁的频率相关质量、弹性和几何刚度矩阵的单元的显式代数表达式是根据第一性原理推导出来的。这些矩阵在完全相加时与动态刚度矩阵等效。然后使用 Wittrick-Williams 算法作为求解技术应用推导的矩阵,以研究一些说明性示例的自由振动特性。讨论了结果,并得出了重要的结论。所提出的方法与动态刚度法一样保留了结果的精确性,但重要的是,当使用动态刚度法等精确方法时,它为在自由振动和响应分析中包含阻尼提供了可能性。
更新日期:2025-01-08
中文翻译:
轴向加载的 Timoshenko-Ehrenfest 梁的频率相关质量、弹性和几何刚度矩阵及其应用
早期对用于自由振动分析的频率相关质量、弹性和几何刚度矩阵单元的显式代数表达式的研究是在 Bernoulli-Euler、Timoshenko-Ehrenfest 和轴向加载的 Bernoulli-Euler 梁上进行的。在进行这些早期发展时,为轴向加载的 Timoshenko-Ehrenfest 梁寻找相应更困难的解决方案似乎太困难了。现在,随着获得的经验和知识,符号计算的增强应用在一定程度上克服了困难。因此,轴向加载的 Timoshenko-Ehrenfest 梁的频率相关质量、弹性和几何刚度矩阵的单元的显式代数表达式是根据第一性原理推导出来的。这些矩阵在完全相加时与动态刚度矩阵等效。然后使用 Wittrick-Williams 算法作为求解技术应用推导的矩阵,以研究一些说明性示例的自由振动特性。讨论了结果,并得出了重要的结论。所提出的方法与动态刚度法一样保留了结果的精确性,但重要的是,当使用动态刚度法等精确方法时,它为在自由振动和响应分析中包含阻尼提供了可能性。