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A direct method for analyzing the forced vibrations of continuous systems having damping

Journal of Sound and Vibration
Publication Date
DOI: 10.1016/s0022-460x(78)80150-3


The classical procedure for analyzing the forced vibrations of a continuous system having damping is well known—one expresses the forcing function in terms of the eigenfunctions of free vibration, and determines the amplitude and time response of each mode. However, exact eigensolutions are seldom known and, if they are available, the procedure is usually a long and laborious method. The present paper extends the Rayleigh-Ritz-Galerkin approximate methods used widely on free vibration problems to forced vibration problems having damping. The procedure developed is simple and straightforward and the accuracy of the results depends, in a manner similar to the free vibration case, upon the closeness of the assumed mode shapes to the actual excited mode shapes. The procedure is developed for a general class of forced vibration problems of one-, two- and three-dimensional continuous bodies and is demonstrated by two classical examples. The first example, a vibrating string, has a simple forced vibration solution by the classical normal mode expansion procedure. The present approximate method is seen to give reasonable accuracy for amplitude response with a single-function assumed deflection mode. An extremely accurate solution is obtained by taking two trial functions. The second example, a clamped circular plate, has an exact, but intricate and laborious solution by the classical procedure. The present method is used with a single deflection function to obtain useful results quite easily. Further extensions of the method are also discussed.

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