Exact solutions for forced vibration of completely free orthotropic rectangular nanoplates resting on viscoelastic foundation

Fan, Junhai; Rong, Dalun; Zhou, Zhenhuan; Xu, Chi; Xü, Xinsheng

doi:10.1016/j.euromechsol.2018.06.007

Cited by 16 publications

(6 citation statements)

References 52 publications

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“…The Ritz method [13] can be used to get approximate solutions for any different boundary conditions, as long as this method can find a suitable approximation functions for the problem. The analytical solution is one of the most popular methods to get an exact solution for such that nanostructure [14,15].…”

Section: Solution Methods and Analysismentioning

confidence: 99%

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Analysis and vibration of rectangular nanoplates - An overview

Eraky

Salama

Abdelhameed

2021

The Egyptian International Journal of Engineering Sciences and

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This work presents a review on solution methods and analysis of nanoplates structures with different boundary conditions and load cases, under some effects such as magnetic field and the effect of other parameters on the vibration and analysis. Moreover, it represents a review about the theories that are used to study these nanoplates structures such; the nonlocal elasticity theory by Eringen which is introduced to take into consideration the small scale effect of such nano-structure. The equation of motion of the nanoplate is derived then it used to study this type of nano-structures. Nanoplates are used in a lot of branches of life and in different applications because of the high and excellent mechanical, thermal and electrical properties of these nanoplates. Applications of nanoplate structure are also introduced to show the importance of studying and getting a solution of such nanoplates. Also, some parametric studies are discussed to show the effect of the studied parameters on the dynamic behaviour and analysis of these nanoplates.

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Section: Solution Methods and Analysismentioning

confidence: 99%

“…Junhai et al [14] studied the forced vibration of a rectangular viscoelastic orthotropic nanoplate which was embedded in viscoelastic surrounding medium with a completely free ends. The viscoelastic foundation was modelled using Kelvin-Voigt foundation.…”

Section: Forced Vibrationmentioning

confidence: 99%

Analysis and vibration of rectangular nanoplates - An overview

Eraky

Salama

Abdelhameed

2021

The Egyptian International Journal of Engineering Sciences and

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“…Any trial function does not need to be assumed in symplectic method in advance, which makes the solution process more intuitive and rational. For example, Fan et al . (2019) obtain the exact solution of forced vibration of orthotropic rectangular nanoplates with four free edges on viscoelastic foundation based on symplectic method.…”

Section: Introductionmentioning

confidence: 99%

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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“…In view of the above shortcomings of symplectic elasticity method, in 2010, Liu and Li [19] proposed symplectic superposition method based on symplectic elasticity method and superposition method, which extends the range of analytical solutions of mechanical problems obtained by symplectic elasticity method. At present, the symplectic superposition method has been used to solve many mechanical problems, such as the bending [20] and vibration [21] of isotropic plates, the bending [22] and vibration [23] of orthotropic plates, the bending [24] of cylindrical shells, the vibration [25] of nanoplates, and so on.…”

Section: Introductionmentioning

confidence: 99%

Analytical solution of the bending problem of free orthotropic rectangular thin plate on two‐parameter elastic foundation

et al. 2021

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The bending problem of free orthotropic rectangular thin plate (RTP) on two‐parameter elastic foundation under a concentrated load is studied by the symplectic superposition method. Firstly, the original bending problem is decomposed into three subproblems by analyzing load effects and boundary conditions, each of which is the bending problem of the plate with two opposite edges slidingly supported. In order to solve the three sub‐problems based on the separation of variables method in Hamiltonian system, the Hamiltonian system for the orthotropic RTP with two opposite edges slidingly supported is studied, and then the eigenvalues and eigenfunctions of the Hamiltonian operator are obtained by combining the separation of variables and symbolic computation. Secondly, according to the symplectic orthogonality and the completeness of the eigenfunctions, the general solution of the Hamiltonian system with the two opposite edges slidingly supported is obtained. Furthermore, the solutions in the form of series of the three subproblems are obtained respectively. Finally, the symplectic superposition solution of the original bending problem is obtained by superposing the solutions of the three subproblems and the correctness of the symplectic superposition solution is verified by two numerical examples.

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Exact solutions for forced vibration of completely free orthotropic rectangular nanoplates resting on viscoelastic foundation

Cited by 16 publications

References 52 publications

Analysis and vibration of rectangular nanoplates - An overview

Analysis and vibration of rectangular nanoplates - An overview

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

Analytical solution of the bending problem of free orthotropic rectangular thin plate on two‐parameter elastic foundation

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