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. In several recent papers [1-5] a method for treating discrete nonlinear vibration problems has been developed. By linearizing the nonlinear forces using a set of ultraspherical polynomials over the interval in which the motion takes place, approximate nonlinear amplitude-frequency relations with fair accuracy are obtained from the solutions of the resulting linear systems. Some more general linearizations have also been reported [6, 7]. The method is restricted essentially to systems which have one degree of freedom or may be characterized by a single amplitude [8]. In this note, the method of ultraspherical polynomials is extended to systems governed by nonlinear partial differential equations to obtain approximate nonlinear amplitude-frequency relations in the neighborhood of the linear eigenvibrations. It is assumed that the systems depend on one space variable and time, and the nonlinear terms in the equations depend on the displacement and its spatial derivatives but not on time explicitly. An obvious difficulty immediately arises when one attempts to follow the method developed for discrete systems because the amplitude of motion is a function of the space variable and not known in advance. To overcome this difficulty some appropriate "mode of deflection" must be assumed. In cases where a linear mode is known, it is taken to be the mode of deflection; otherwise some suitable approximation to the linear mode must be made. Next, an amplitude parameter is introduced so that the amplitude of motion is given by the product of the amplitude parameter and the normalized linear mode. Using ultraspherical polynomials, the nonlinear terms are replaced by terms that are linear in the displacement and linear partial differential equations result. With the given homogeneous boundary conditions one is led to solve linear eigenvalue problems. The determination of the eigenvalues leads to the desired amplitudefrequency relations.

Ultraspherical polynomials applied to nonlinear vibrations of continuous media