Hamiltonian System-Based Analytical Solutions to Free Vibration Problems of Functionally Graded Rectangular Plates

ZHANG Jichao; ZHONG Xinyu; CHEN Yiming; SHI Yueqing; GUO Chengjie; LI Rui

doi:10.21656/1000-0887.440279

Volume 45 Issue 9

Sep. 2024

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Article Navigation > Applied Mathematics and Mechanics > 2024 > 45(9): 1157-1171

ZHANG Jichao, ZHONG Xinyu, CHEN Yiming, SHI Yueqing, GUO Chengjie, LI Rui. Hamiltonian System-Based Analytical Solutions to Free Vibration Problems of Functionally Graded Rectangular Plates[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1157-1171. doi: 10.21656/1000-0887.440279

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Hamiltonian System-Based Analytical Solutions to Free Vibration Problems of Functionally Graded Rectangular Plates

doi: 10.21656/1000-0887.440279

ZHANG Jichao^1
,,
ZHONG Xinyu^2
,,
CHEN Yiming^1
,,
SHI Yueqing^1
,,
GUO Chengjie^1
,,
LI Rui^{1
,
,}

1.
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China
2.
Laboratory for Multiscale Mechanics and Medical Science, School of Aerospace Engineering, Xi'an Jiaotong University, Xi'an 710049, P.R.China

Received Date: 2023-09-20
Rev Recd Date: 2023-11-05
Publish Date: 2024-09-01

Abstract

Abstract

Finding vibration mode functions satisfying both high-order partial differential governing equations and various non-Lévy-type boundary conditions is extremely challenging for the free vibration problems of functionally graded rectangular plates, making it difficult to analytically solve the problem with traditional methods. Herein, the newly developed Hamiltonian system-based symplectic superposition method was extended and successfully applied to analytical solutions to the free vibration problems of functionally graded rectangular plates. For the solution methodology, the original vibration problem was divided into sub-problems and the physical neutral plane was introduced to eliminate the stretching-bending coupling effect caused by the transversely non-uniform materials. The sub-problems were analytically solved with some mathematical techniques, i.e., the variable separation and the symplectic eigenvector expansion, which are not applicable in the traditional Lagrangian system. The final solution to an original vibration problem was obtained through the superposition of sub-problems. The symplectic superposition method has the advantage of not requiring the pre-defined solution forms, which overcomes the limitations of traditional semi-inverse methods and allows for obtaining analytical solutions to more complex problems. Comparison of the obtained solutions with the numerical solutions proves the accuracy of the presented method. On this basis, quantitative parameter analyses on the natural frequencies were conducted to reveal the effects of boundary conditions, material distributions and aspect ratios.
- functionally graded plate,
- free vibration,
- Hamiltonian system,
- analytical solution

(Contributed by LI Rui, M.AMM Editorial Board)

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References(38)

References

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