Analytical studies of nonliner non autonomous systems

Abstract

This thesis presents the results of some analytical studies in the field of non linear analysis. The main aim of this research has been to develop analytical methods leading to closed form solutions of the governing differential equations of large classes of non linear, non autonomous systems of engineering interest. The emphasis here is more on the analytical aspects of the methods of non linear analysis rather than on the computational aspects of any particular problem. With the advent of high speed digital computers, theoretically most non linear problems can be tackled by numerical methods on a one off basis. Most of the recent literature in the field of non linear analysis concerns numerical methods of solving individual non linear problems. This investigation was therefore undertaken with a view to revive interest in the field of analytical methods of non linear analysis. In the first chapter, a review of the extensive literature in the field of non linear analysis has been presented, and the various analytical techniques available therein have been classified to bring out their relative merits. Some analytical methods, based on the idea of transformation techniques, are detailed in the second chapter. The object of these transformation techniques is to study the given non linear, non autonomous systems (with deterministic or random parameters) by converting them into equivalent systems amenable to existing methods in the fields of stochastic processes and linear/non linear analysis. Such transformations often lead to equations which admit exact or closed form solutions. These solutions of the transformed equations can then be mapped back through the transformation laws to obtain the response of the given systems. The scope of existing analytical tools (which can lead to closed form solutions of particular equations) is thereby considerably widened by these transformation techniques to cover large classes of non linear problems. In the next chapter, the studies on non linear, non autonomous systems have been carried out using the approach of synthesis. Here, various classes of systems are synthesised from an a priori knowledge of their behaviour in the time domain. The advantage of this approach is that once a certain class of systems is synthesised, the study of any member of this class will be merely a formal mathematical exercise. Some interesting studies, which involve the use of the techniques developed earlier (as well as techniques available elsewhere in the literature), are presented in the fourth chapter. The analysis of gyro systems subjected to random inputs, the study of systems subjected to combined deterministic and stochastic loading, and the response of systems with deterministic or random parameters are some of the studies detailed in this chapter. The significance of the results of this investigation in the study of non linear, non autonomous systems has been highlighted in the concluding chapter.