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Galerkin's method revisited and corrected in the problem of Jaworski and Dowell

机译：Galerkin的方法在Jaworski和Dowell的问题中重新审视并纠正

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This paper revisits the problem first studied by Jaworski and Dowell, namely, the free vibration of multi-step beams. Previous authors utilized approximate method of Ritz as well as the finite element method with attendant comparison with the experimental results. This study provides the exact solution for the Jaworski and Dowell problem in terms of Krylov-Duncan functions. Additionally, the Galerkin method is applied and contrasted with the exact solution. It is shown that the straightforward implementation of the Galerkin method, as it is usually performed in the literature, does not lead to results obtained by Jaworski and Dowell using the Ritz method. Moreover, the straightforward application of the Galerkin method does not tend to the results obtained by either exact solution or experiments. A modification of the Galerkin method is proposed by introducing generalized functions to describe both mass and stiffness of the stepped beam. Specifically, the unit step function, Dirac's delta function and the doublet function, are utilized for this purpose. With this modification, the Galerkin method yields results coinciding with those derived by the Ritz method, and turn out to be in close vicinity with those produced by the exact solution as well as experiments.
机译：本文重新审视了Jaworski和Dowell首次研究的问题，即多步梁的自由振动。以前的作者利用RITZ的近似方法以及与实验结果的伴随比较的有限元方法。本研究在Krylov-Duncan函数方面为Jaworski和Dowell问题提供了精确的解决方案。另外，将Galerkin方法应用并与精确的解决方案形成对比。结果表明，通常在文献中进行的Galerkin方法的直接实现不会导致Jaworski和Dowell使用Ritz方法获得的结果。此外，Galerkin方法的直接施加不倾向于通过精确溶液或实验获得的结果。通过引入阶梯束的全质量和刚度来提出Galerkin方法的修改。具体地，为此目的，使用单位步进功能，DIRAC的DELTA函数和双重功能。通过这种修改，Galerkin方法产生与RITZ方法得出的结果一致，并且结果与精确解决方案和实验一起产生的那些。

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