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New exact solutions for free vibrations of rectangular thin plates

Bo Liu's picture

(1) The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported (S) and clamped (C) boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

(2) In this paper, a novel separation of variables is presented for solving the exact solutions for the free vibrations of thin orthotropic rectangular plates with all combinations of simply supported (S) and clamped (C) boundary conditions, and the correctness of the exact solutions are proved mathematically. The exact solutions for the three cases SSCC, SCCC, and CCCC are successfully obtained for the first time, although it was believed that they are unable to be obtained. The new exact solutions are further validated by extensive numerical comparisons with the solutions of FEM and those available in the literature.

Reply to Rui Huang and Ahmadpeik: Thank you for your comments. The new solutions are exact for free vibrations of thin plates. Levy, Midlin and others gave exact solutions for free vibrations of thin and moderate thick plates with a pair of opposite edges simply supported. The new method can solve free vibrations of thin and moderate thick plates with any combinations of simply supported and clamped boundary conditions. Thank you.


Rui Huang's picture

The governing equation of plate (Eq. 1 of the first paper) is valid only for low-frequency vibrations. Recall the classical example of flexural waves in a beam and the correction made by Timoshenko. Thus your solution is "exact" only under the approximate theory. Similar solutions have been worked out by Mindlin and others based on more accurate plate theories at high frequencies.


Ahmad Rafsanjani's picture


I did not find in your article what you mean "novel separation of variables " and also any result which has been obtained "for the first time" !

As Rui Huang said: Lots of "similar solutions have been worked out by Mindlin and others"


LG's picture

To my understanding, it will be much better for you to do some researches relate to free vibration of shell with big cutout rather than a thin plate without any faults.

Since there are no shear force inside across section of the plate, regarding to boundary condition of clamped(W=0, dW/dx=0), simply supported(W=0, d^2W/dx^2=0), or the free edge(no bending moment, no twisting moment, and no shear force; these will induce three restrictions, however, the fourth order governing equation only need two restrctions on each edge along x axis or y axis, so the equivalent shear force was introduced by a france scientist). The fourth order PDE (Bi-harmonic) of the deflection W(for your model of homogenous equation after the seperation of variables)  can be easily solved by Mathematic or Maple. Subsequently, the unknown coefficients can be determined according to the boundary restrictions. For the model of static, the Levy or Navier method(fourier series solution) for non-homogeneous fourth order equation can be found in many textbook.

 A good reference should be " Thin plates and shells: theory, analysis and spplications" by  Eduard Ventsel, Theodor Krauthammer.


Are you doing some research in thin shell? Thanks.

Bo Liu's picture

Thank you for your suggestions. Exact solutions based on the theory of elasticity are only limited to some simple cases, some simple combinations of simply supported, clamed and free boundary conditions. It may be regarded as a contribution to analytical methods to find some new exact solutions for the classical problem of free vibrations of thin plates. Numerical methods can solve problems of real structure and are my present research interest. Thank you.

These days I am study the exact solutions for vibration of trctangular plates and shell structire. Can you give me the program about the "New exact solutions for free vibrations of thin orthotropic rectangular plates" and "Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells"  Thanks

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