# Exact solution of vibrations of beams with arbitrary translational supports using shape function method

## Abstract

In this paper, the exact and explicit solution of free vibrations of Euler–Bernoulli beams supported by an arbitrary number of translational springs at arbitrary positions under variable boundary conditions is obtained by the proposed shape function method. The differential equation governing the vibration is formulated by incorporating the Dirac’s delta function and the frequency equation and mode shape function are solved by the variational iteration method and the Laplace transform. The highest order of the frequency equation is four, which is independent of the number of springs, making this method more efficient than solutions in the literature. A total of 49 possible cases of boundary conditions (pinned, clamped, free, sliding, translational spring, rotational spring, and combined translational-rotational) are accommodated by a system of four homogeneous equations, which can be conveniently programmed into a computer to solve engineering problems. The validity of the proposed solutions is justified by comparing with results from the literature. A parametric study is carried out to investigate the influences of weakened stiffness on the vibration frequencies. The solutions provided in this work is mathematically complete and may be used as benchmarks for future research. The derived frequency equation and mode shape function are potentially useful for solving the inverse problem, i.e., identifying weakened elastic supports, in a future study.

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## Acknowledgements

The authors are grateful to the referees for valuable comments and suggestions.

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Correspondence to Xingzhuang Zhao.

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