Transient vibrations of one and two degree of freedon non-linear systems

Abstract

This thesis is devoted to the study of transient vibrations of non-linear systems, with emphasis on two degree?of?freedom systems. After a résumé of the free and forced response of non-linear two degree?of?freedom systems, a method of analysis using ultraspherical polynomials is presented. It is shown that this method can be construed as a particular case of the more general method due to Galerkin. This approach is used to study the transient response of non-linear non?conservative systems enjoying two degrees of freedom. The approximate solutions obtained by using the method of averaging of Krylov and Bogoliubov are shown to be a particular set of solutions generated by this general method of averaging proposed in this thesis. The method of averaging of Krylov and Bogoliubov is extended to obtain the step?function response of non?linear, non?conservative systems enjoying two degrees of freedom. A discussion of the effects of force amplitude and non?linearity on the response features is then taken up. The study of an important class of problems concerning non?linear two degree?of?freedom systems subjected to transient disturbances and pulses of finite duration is undertaken. It is shown that the method of averaging can also be extended to non?linear, non?conservative systems. Three types of excitations namely, blast pulse, half?sinusoidal pulse and triangular pulse are considered and in each case the complete response of a typical two degree?of?freedom system is determined. Two approximate methods to treat the step?function excitation problem of non?linear single degree?of?freedom systems are considered. The period of oscillation in terms of amplitude is determined first by the application of shifted Chebyshev polynomials and secondly by a method based on weighted mean square approximation of the restoring force characteristic. The methods proposed are illustrated with the help of examples and the adequacy of the methods is demonstrated by comparing the results with those obtained on a digital computer using numerical techniques.