On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

Huang, Ming-Qiu; Zheng, Xinran; Zhou, Chao; An, Dongqi; Li, Rui

doi:10.1007/s00707-020-02829-x

Cited by 13 publications

(3 citation statements)

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“…Take the first 11 term (n 5 10) of symplectic superposition solution (62), at this time, it is necessary to calculate the determinant of order 44. Compare the numerical results with those in reference (Huang et al, 2021), as shown in Table 3 (the calculation results retain 5 significant digits).…”

Section: Symplectic Superposition Methodsmentioning

confidence: 76%

“…Taking the first 11 term (n 5 10) of series expansion solution (62), we calculated the first five frequencies of the free vibration of nanoplates with different values of the ratio of b/a, and compared it with the results in literature (Huang et al, 2021;Chakraverty and Behera, 2014), see Table 4 (The accuracy of calculation is 10 −4 ).…”

Section: Symplectic Superposition Methodsmentioning

confidence: 99%

“…The numerical results of series expansion solution (62) with n 5 10 are compared with those in reference Huang et al (2021), as shown in Table 1 (5 significant digits are reserved in the calculation results).…”

Section: Examplesmentioning

confidence: 99%

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Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Su,

Bai,

Hai

2023

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PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

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Section: Symplectic Superposition Methodsmentioning

confidence: 76%

Section: Symplectic Superposition Methodsmentioning

confidence: 99%