Transient and steady-state response of second and higher order non-linear systems

Abstract

The present investigation deals with the transient flnfl steady-state response o£ second and higher order nonlinear systems. The method of attack consists of the ultraspherical polynomial approximation and the weighted mean square linearisation. The former method provides the concept of a generalised averaging technique xirhere the averaging method proposed by Bogoliubov and Mitropolsky comes out to be a special case given by a particular set of ultraspherical polynomials. A large class of approxiiiiate solutions can be generated for a given problem by this method, for different values of the parameter X, The weighted mean square linearisation is a general method, from which other linearisations can be derived, by choosing the proper weight function. This method with the value of the parameter m = 3, provides accurate results for small oscillations. The above methods are used in the analysis of transient and steady-state response of non-linear systems. The pulse response of a second-order system is obtained by transforming the displacement variable; The response of the linear system subjected to the pulse is used as the additional transformation function, after transformingj,the equation is in the form where Anderson's ultraspherical polynomial approximation SIRACI method can be applied, as an example, the response of a damped Duffins oscillator subjected to pulses like cosine, sine, exponentially decaying. Asymptotic exponential step and the step function, is obtained. The analytical results are compared with digital solutions. The transient response of a third-order non-linear system is obtained by ultraspherical polynomial approximation method based on Maksimov’s method and Popov’s method, A degenerate third-order system is also analysed. Examples are worked out and the analytical results agree v/ell with the digital solutions. The concept of the variation of parameters is extended to study the transient response of thirdorder system. The resulting first order equations are averaged on the basis of ultraspherical polynomial averaging. The resulting first order equations are averaged on the basis of ultraspherical polynomial averaging. The results compare well with the digital solutions. The steady-state forced response of a third-order nonlinear system subjected to harmonic forming is studied through ultraspherical polynomial linearisation. The relation between steady-state amplitude and forcing frequency is obtained by means of the linearised equations. These are compared v;ith digital and analog solutions. Analog solutions are obtained on Pace TR-48 computer, vrhereas digital solutions are obtained on IBM 360/44 digital system. A two degrees of freedom system is analysed bnrough the weighted mean square linearisation. Free, Forced and self-excited oscillations are considered. The results tally with the results obtained from established methods. In this thesis emphasis is placed on the proposition and application of approxiiaate methods. Even though rigorous mathematical justifications are not provided, the results are compared with numerical results obtained from digital and analog computers for justification.