Some approximate methods of analysis of Non-Linear Dynamical systems

Abstract

The present investigation concerns some approximate methods of analysis in the area of non-linear oscillations. The analysis includes the steady-state as well as the transient response of non-linear systems. In both cases, the study has been restricted to single degree of freedom systems. Emphasis is placed on the proposition and applications of the approximate methods rather than on their mathematical justifications.

The problems of free-undamped oscillation of non-linear systems have been investigated by an equivalent linearization technique called the Weighted Mean Square Method of Linearization. In this technique, the original non-linear differential equation is replaced by an equivalent linear differential equation such that the error between the two is least in the weighted mean square sense. The accuracies in approximate periods resulting from the choice of various weighting functions have been examined in detail for different types of non-linearities. It is shown that all the linearization techniques available so far in the literature lead to this kind of linearization by a suitable choice of the weighting function, and the method proposed is of a higher order of generalization than the existing ones.

The proposed method has been extended to the case of steady-state response of forced oscillations under sinusoidal excitation and to the problems governed by Liénard's equations. The approximate results have been compared with the results obtained by the well-established methods.

A method is also proposed for the study of transient response of non-linear systems. It is based on an averaging technique through the application of ultraspherical polynomials. The method is illustrated with its application to the transient response of non-linear autonomous systems. It is shown that the proposed method provides a generalized concept of the averaging technique, and the Krylov–Bogoliubov results are given as a special case of the method.

The above technique has also been extended to study the non-linear, non-conservative systems subjected to the step function excitation. The step response of a cubic spring-mass system in the presence of different forms of damping has been investigated in detail, and the approximate results are compared with the digital and analog computer solutions.

The extension of the above methods to non-linear systems of higher degrees of freedom is indicated as a future program.