Applied Mathematics and Mechanics >
Dirac method for nonlinear and non-homogenous boundary value problems of plates
Received date: 2023-07-03
Online published: 2023-12-26
Supported by
the National Natural Science Foundation of China(12002195);the National Science Fund for Distinguished Young Scholars(12025204);the Program of Shanghai Municipal Education Commission(2019-01-07-00-09-E00018);Project supported by the National Natural Science Foundation of China (No. 12002195), the National Science Fund for Distinguished Young Scholars (No. 12025204), and the Program of Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018)
Copyright
The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method (HBM) is utilized to solve coupled ordinary differential equations (ODEs) of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method (DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations; nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.
Xiaoye MAO, Jiabin WU, Junning ZHANG, Hu DING, Liqun CHEN . Dirac method for nonlinear and non-homogenous boundary value problems of plates[J]. Applied Mathematics and Mechanics, 2024 , 45(1) : 15 -38 . DOI: 10.1007/s10483-024-3066-7
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