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Generalized multi-symplectic method for vibration of cracked simply supported beam

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Abstract

The effects of cracks on the dynamic behavior of engineering structures affect the structure safety. In this paper, a generalized multi-symplectic approach is developed to investigate the effects of crack on a dynamic behavior of a simply supported beam in detail. First, the effect of the crack on the bending stiffness of the simply supported beam is expressed by an equivalent bending stiffness and the dynamic model with variable coefficients is established to describe the transverse vibration of the cracked beam. Then, the approximate symmetric form of the dynamic model is constructed and its dynamic symmetry breaking factors are discussed. The Preissmann scheme of the approximate symmetric form is constructed based on the generalized multi-symplectic theory. In numerical simulations, the structure-preserving properties of the Preissmann scheme are illustrated and the validity of the dynamic model is verified. Then, the effects of the crack on the vibration of the beam are investigated by using the generalized multi-symplectic method in detail. The numerical results presented prove that the proposed generalized multi-symplectic method is an effective approach to investigate the dynamic behavior of cracked structures.

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Acknowledgements

The authors wish to thank Professor Thomas J Bridges of Surrey University and Yinghui Li of Southwest Jiaotong University for giving us several good suggestions. The research is supported by the National Natural Science Foundation of China (12172281, 11972284 and 12102339), Fund for Distinguished Young Scholars of Shaanxi Province (2019JC-29), Foundation Strengthening Programme Technical Area Fund (2021-JCJQ-JJ-0565), the Fund of the Science and Technology Innovation Team of Shaanxi (2022TD-61), the Fund of the Youth Innovation Team of Shaanxi Universities (22JP055), Natural Science Basic Research Program of Shaanxi (2022JM-029) and the Doctoral Dissertation Innovation Fund of Xi'an University of Technology (252072016, 252072115). The authors declare that they have no conflict of interest with the present manuscript.

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  1. School of Civil Engineering and Architecture, Xi’an University of Technology, Xi’an, 710048, Shaanxi, People’s Republic of China

    Weipeng Hu, Peijun Zhang & Jianmei Wang

  2. State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, 710048, Shaanxi, People’s Republic of China

    Weipeng Hu

  3. China Academy of Ordnance Science, Beijing, 100089, People’s Republic of China

    Zhe Zhai

  4. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Zichen Deng

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Hu, W., Zhang, P., Wang, J. et al. Generalized multi-symplectic method for vibration of cracked simply supported beam. Acta Mech 233, 4805–4816 (2022). https://doi.org/10.1007/s00707-022-03350-z

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