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Vibration of Circular Plates
Thu, 2012-02-16 23:40 - hrbeu221
Dear All,
These days I am do some research on vibration of circular plates (out-of-plane and in-plane). In the reference, the displacement in radial direction can be expressed with Bessel polynomial. I want to use the Rayleigh-Ritz method to solve these problems. The strain energy and kinetic energy can obtain with integral. If there is the bessel function, how to calculate the integral. Also I want to consider some spring in the boundary to change the boundary conditions.
Or there are some other method to do? Express the displacement in the radial direction with trigonometric function?
Hope all can give me some advice on it.
Thanks.
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Hi, from what you said i
Hi, from what you said i realized that you want to get the analytical solution ratehr then numerical calculation for the displacements inside the vibrated circular plates. As you mentioned, the in plane and out plane displacements are taken into account, so the probles should be three dimensional (r, sita, z). the circular plate should be central symmetrical, then you can simplify the second/fourth order PDE into two dimensional. I am not sure the initial conditions of you problem, i just suppose the governing equation can be homogeneous. if you loading is point source, then the governing equation can be inhomogeneous. There are some classical solutions of displacements for the vibrated circular plates, the bessel function is well known for the solution of general linear wave propagation/vibration. For the boundary condition around the edge of the circular plate, you can assign the unknown coefficient of the series. For certain BC, you can analytically obtain the corresponding coefficient. Since you want to get the strain energy, you can use the recurrence of bessel function for the integration. there are two references may be useful to you :
Book : Special Functions
Journal paper: The Integration of Bessel Functions, R. A. WALDRON
Another way, you may want to use the laurent series for bessel function reversely to get the exponential function for the integration.