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Description
The purpose of this thesis is to conduct a stability analysis of a forced, damped two degree-of-freedom spring mass system with a cubic nonlinearity in the main spring. The analysis has attempted to answer the following questions: (1) What analytical techniques are available for performing stability analyses of nonlinear systems? (2) Is the dynamical system under investigation unstable in the operating area known as the "jump" region of the amplitude-frequency response curve? (3) Can constraints on the parameters of the physical system be established for purposes of predicting the stability of motion? (4) What type of stability applies near -the "jump" region, i.e., asymptotic, uniform, neutral, stability in the large, or stability in the small? In order to answer these questions, a survey of nonlinear phenomena and the characteristics of stability analyses for nonlinear systems is presented. This includes background information on the historical development of nonlinear theory to the present time. The problems which resulted in the breakdown of linear theory and a description of the four significant nonlinear physical phenomena are discussed. Of these, the jump characteristic is explained in greater detail since this is associated with the basic problem under investigation in this study. Analytical and physical definitions of the various types of stability which can be encountered in nonlinear problems are presented. Comparisons are given to representative linear systems to illustrate that the complexities of a nonlinear system cannot be accurately described using classical methods. An explanation of each of the seven analytical stability analysis techniques is given. Complete descriptions of these methods has been presented by the author in a previous paper, TRW Systems Report No. 7221.6-192, Redondo Beach, Calif., 1967. In order to select which of these techniques is the most desirable for the investigation of the nonlinear problem under study, a discussion describing the limitations and advantages of each is presented. A derivation of the equations of motion for the physical problem being investigated is reviewed. Transforming to a set of dimensionless coordinates enables the retention of only significant system parameters for the subsequent analyses. A selection of the analytical techniques to be applied in this study is made based upon their ability to provide sufficient information in order to answer the questions posed above. From the results of rigorous and detailed investigations using four analytical techniques, it is shown that the Caughey-Miles method is capable of establishing if the system is stable in and near the vicinity of the jump region. This method also provides for the evaluation of the constraints on the parameters in the system which are required to insure stability. Two other techniques, amplitude variations in orbital stability, and the Klotter-Pinney scheme, only provided a minimal amount of information concerning the constraints on the system. The Second Method of Lyapunov was also used but the results were inconclusive. This is because no well defined analytical procedure exists for implementing this method. The task of applying this scheme reduces to trial and error. The results of the stability analyses performed here have shown that the physical system under investigation is unstable within the jump region of operation and is asymptotically stable in other regions. The Caughey-Miles technique is a very valuable analytical tool for calculating the behavior of nonlinear systems. The application of this method should be expanded to include other classes of problems in nonlinear theory. Particular emphasis should be given to the areas of nonlinear damping, non-sinusoidal periodic forcing functions, and impulsive type forcing functions.
