A method is described of investigation of nonstationary oscillatory processes in nonlinear systems with slowly varying parameters. The problem of passing through resonance in nonlinear oscillatory systems with one and more degrees of freedom is considered in detail as well as the problem of passing through parametric resonance and some other typical problems of physics and technics connected with nonstationary oscillations (resonance phenomena in accelerators, passing through resonance in gyroscopic systems, etc.). (Author).
Results of investigations of transient processes of linear systems with any number of degrees of freedom (rods, shafts, plates and disks) are presented. Nonstationary oscillation systems described by equations with periodically changing coefficients and the nonstationary oscillations of nonlinear systems were studied with the aid of the Yu. A. Mitropol'sky method and with direct integration of equations. Considerable attention was paid to nonstationary flexural vibration in shafts and to nonstationary vibrations of systems with restricted excitation. As examples, numerous problems were solved with the aid of a digital computer. (Author).
The study of complex dynamic processes governed by nonlinear and nonstationary characteristics is a problem of great importance in the analysis and control of power system oscillatory behavior. Power system dynamic processes are highly random, nonlinear to some extent, and intrinsically nonstationary even over short time intervals as in the case of severe transient oscillations in which switching events and control actions interact in a complex manner. Phenomena observed in power system oscillatory dynamics are diverse and complex. Measured ambient data are known to exhibit noisy, nonstationary fluctuations resulting primarily from small magnitude, random changes in load, driven by low-scale motions or nonlinear trends originating from slow control actions or changes in operating conditions. Forced oscillations resulting from major cascading events, on the other hand, may contain motions with a broad range of scales and can be highly nonlinear and time-varying. Prediction of temporal dynamics, with the ultimate application to real-time system monitoring, protection and control, remains a major research challenge due to the complexity of the driving dynamic and control processes operating on various temporal scales that can become dynamically involved. An understanding of system dynamics is critical for reliable inference of the underlying mechanisms in the observed oscillations and is needed for the development of effective wide-area measurement and control systems, and for improved operational reliability.
This book suggests a new common approach to the study of resonance energy transport based on the recently developed concept of Limiting Phase Trajectories (LPTs), presenting applications of the approach to significant nonlinear problems from different fields of physics and mechanics. In order to highlight the novelty and perspectives of the developed approach, it places the LPT concept in the context of dynamical phenomena related to the energy transfer problems and applies the theory to numerous problems of practical importance. This approach leads to the conclusion that strongly nonstationary resonance processes in nonlinear oscillator arrays and nanostructures are characterized either by maximum possible energy exchange between the clusters of oscillators (coherence domains) or by maximum energy transfer from an external source of energy to the chain. The trajectories corresponding to these processes are referred to as LPTs. The development and the use of the LPTs concept a re motivated by the fact that non-stationary processes in a broad variety of finite-dimensional physical models are beyond the well-known paradigm of nonlinear normal modes (NNMs), which is fully justified either for stationary processes or for nonstationary non-resonance processes described exactly or approximately by the combinations of the non-resonant normal modes. Thus, the role of LPTs in understanding and analyzing of intense resonance energy transfer is similar to the role of NNMs for the stationary processes. The book is a valuable resource for engineers needing to deal effectively with the problems arising in the fields of mechanical and physical applications, when the natural physical model is quite complicated. At the same time, the mathematical analysis means that it is of interest to researchers working on the theory and numerical investigation of nonlinear oscillations.
Nonlinear System Analysis focuses on the study of systems whose behavior is governed by nonlinear differential equations. This book is composed of nine chapters that cover some problems that play a major role in engineering and physics. The opening chapter briefly introduces the difference between linear and nonlinear systems. Considerable chapters are devoted to engineering and physics related problems and their applications to particle accelerators, frequency measurements, and masers. Included in these chapters are important practical problems, such as synchronization, stability of systems with periodic coefficients, and effect of random disturbances. The remaining chapters examine random fluctuations of the motion and self-oscillators. This book is intended primarily for engineers and physicists.
Nonlinear systems with stationary sets are important because they cover a lot of practical systems in engineering. Previous analysis has been based on the frequency-domain for this class of systems. However, few results on robustness analysis and controller design for these systems are easily available. This book presents the analysis as well as methods based on the global properties of systems with stationary sets in a unified time-domain and frequency-domain framework. The focus is on multi-input and multi-output systems, compared to previous publications which considered only single-input and single-output systems. The control methods presented in this book will be valuable for research on nonlinear systems with stationary sets.
