Zhou Ding. Vibration of Arbitrarily Shaped Membranes with Elastical Supports at Points[J]. Applied Mathematics and Mechanics, 1989, 10(12): 1107-1113.
 Citation: Zhou Ding. Vibration of Arbitrarily Shaped Membranes with Elastical Supports at Points[J]. Applied Mathematics and Mechanics, 1989, 10(12): 1107-1113.

# Vibration of Arbitrarily Shaped Membranes with Elastical Supports at Points

• Publish Date: 1989-12-15
• This paper presents a new method for solving the vibration of arbitrarily shaped membranes with elastical supports at points. The reaction forces of elastical supports at points are regarded as unknown external forces acting on the membranes. The exact solution of the equation of motion is given which includes terms representing the unknown reaction forces. The frequency equation is derived by the use of the linear relationship of the displacements with the reaction forces of elastical supports at points. Finally the calculating formulae of the frequency equation of circular membranes are analytically performed as examples and the inherent frequencies of circular membranes with symmetric elastical supports at two points are numerically calculated.
•  [1] Steele,C.W.,Numerical computation of electric and magnetic field in a uniform waveguide of arbitrary cross-section,Journal of Computational Physics,3(1968). [2] Bates,R.H.T.,The theory of point-matching method for perfectly conducting waveguides and transmission lines,Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques,MTT-17(1969). [3] Bullcy,R.M.and J.B.Davies,Computation of approximate polynomial solutions to TE modes in an arbitrary shaped waveguide,Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques,MTT-17(1969). [4] Laura,P.A.A.,A simple method for the caleulation of cut-off frequencies of waveguide with arbitrary cross-section,Proceedings of the Institution of Electrical and Electronic Engineers(1966),54. [5] Arlett,P.L.,A.K.,Bahrani and O.C.Zienkiewicz,Application of finite elements to the solution of Helmholtz's equation,Proceeding of the Institution of Electrical Engineers(1968),115. [6] Laura,P.A.A.,W.,Romanelli and M.J.Maurizi,On the analysis of waveguides of doublyconnected cross-section by the method of conformal mapping,Journal of Sound and Vibration,20(1972). [7] Mazumdar,J.,Transverse vibration of membranes of arbitrary shape by the method of constant-deflection contours,Journal of Sound and Vibration,27(1973). [8] Nagaya,K.,Vibrations and dynamic response of membranes with arbitrary shape,American Society of Mechanical Engineers Journal of Applied Mechanics,45(1978). [9] Nagaya,K.,Vibration of an arbitrarily shaped membrane with point supports,Journal of Sound and Vibration,65(1979).

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