A survey on non-linear oscillations
TL;DR: A comprehensive review of work in the field of non-linear oscillations can be found in this paper, where a brief discussion of second-order systems is followed by a presentation of exact criteria, approximate analytical methods and computational techniques for limit cycles in single variable systems.
Abstract: This survey paper presents a comprehensive review of work in the field of non-linear oscillations. A brief discussion of second-order systems is followed by a presentation of exact criteria, approximate analytical methods and computational techniques for limit cycles in single variable systems. Multivariate systems are then covered from an analogous viewpoint which allows the reader to clearly identify both how single variable methods have been extended and the possibilities for further research. Particular emphasis is placed on describing function methods since it is believed that, where exact solutions are not possible, the approach may not only give approximate solutions but provides good insight for further computational or simulation studies. The coverage is essentially restricted to continuous lumped parameter systems and includes both free and forced oscillations. Several applications of the theories in various fields of engineering and science are discussed and indicate the broad interest in non-l...
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TL;DR: The harmonic balance method is applied to the analysis of period-doubling bifurcations in a general class of nonlinear feedback systems and compact conditions for the prediction and stability analysis are obtained.
138 citations
Cites background or methods from "A survey on non-linear oscillations..."
...The main HB tool for stability analysis of limit cycles is the Loeb criterion (see e.g. Popov, 1962; Gelb and Vander Velde, 1968; Atherton, 1975; Atherton and Dorrah, 1980; 1260 A. Tesi et al. Krenz and Miller, 1986; Mohler, 1991; Slotine and Li, 1991)....
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...For a complete discussion of this approach, see e.g. Goldfarb (1947) Gelb and Vander Velde (1968) Siljak (1969), Atherton (1975), Atherton and Dorrah (1980), Mees (1981) Khalil (1992) and Vidyasagar (1992)....
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DOI•
01 Oct 1986TL;DR: In this paper, a new approach for the analysis of nonlinear microwave circuits, the named frequency-domain continuation method, is presented, which allows the unified, but nevertheless efficient, computer analysis of autonomous and forced nonlinear circuits by utilizing the globalization properties of the continuation technique, and provides stability information in parallel.
Abstract: A new approach for the analysis of nonlinear microwave circuits, the named frequency-domain continuation method, is presented. It allows the unified, but nevertheless efficient, computer analysis of autonomous and forced nonlinear circuits by utilizing the globalization properties of the continuation technique, and provides stability information in parallel. Its theoretical background and relation to bifurcation theory is outlined. Its high potential is demonstrated by application to problems with single- and multiple-frequency input, strong nonlinearities, hysteresis phenomena and subharmonics.
96 citations
TL;DR: In this paper, a first-order harmonic balance approach based on describing function representations for the flame response is proposed to investigate the dynamics of thermoacoustic systems with multiple linearly unstable modes, and conditions for the existence and stability of single or multi-mode steady-state oscillations.
Abstract: We investigate the dynamics of thermoacoustic systems with multiple linearly unstable modes. If a linear analysis reveals more than one mode with positive growth rate, nonlinear methods have to be used to determine the existence and stability of steady-state oscillations. One possible way to engage this problem is a first-order harmonic balance approach based on describing function representations for the flame response. In contrast to the case of a single unstable mode, the nonlinearity output to multiple sinusoidal components with different frequencies and amplitudes has to be known. Based on this approach, we present conditions for the existence and stability of single- or multi-mode steady-state oscillations. We apply this method to a thermoacoustic model system having two linearly unstable modes. By varying one of the system parameters, we find stable and unstable single-mode steady-states as well as unstable simultaneous oscillations. Associated with the stability of the single-mode limit cycles, we...
40 citations
15 Dec 1993
TL;DR: In this paper, the use of the harmonic balance method for the control of period doubling bifurcations is considered and the objectives of the control design can include delay of a given period doubling Bifurcation as well as stabilization.
Abstract: The use of the harmonic balance method for the control of period doubling bifurcations is considered. The objectives of the control design can include delay of a given period doubling bifurcation as well as stabilization. The results are inherently approximate in nature. >
31 citations
TL;DR: In this article, the authors developed and discussed the generation of new limit cycles in chemical systems by external periodic perturbations, and derived a deterministic analysis based on prior work by Loud.
Abstract: We develop and discuss the generation of new limit cycles in chemical systems by external periodic perturbations. The deterministic analysis is based on prior work by Loud. We choose a model derived from experiments, the conversion of 2,3 epoxy‐1‐propanol to glycerine. The autonomous system of equations which govern the dynamics of this system has a limit cycle arising from a normal Hopf bifurcation, and a stable focus, and a limit cycle associated with an inverted Hopf bifuraction. For specific choices of periodic perturbations a prescribed number of periodic attractors can be generated from a single periodic attractor (normal Hopf bifurcation); and two periodic attractors, a periodic and biperiodic, or two biperiodic attractors can be generated from a stable focus and periodic attractor (inverted Hopf bifurcation). Noise is then imposed on the attractors of the autonomous system and the multiple attractors of the externally forced system. Noise imposed on the autonomous system with a stable focus and limit cycle causes random transitions, analogous to first order phase transitions. This behavior has some resemblance to intermittency noted in some biological systems. From the results of noise imposed on systems with different numbers of isoperiodic limit cycles we suggest, after a limited number of observations, the trend that as the number of attractors increases, the lines in the power spectrum narrow and the area in the phase plane within contours of equal probability increases.
25 citations
References
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Book•
01 Jan 1955
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.
7,071 citations
Book•
01 Jan 1961
TL;DR: In this article, a wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering.
Abstract: : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.
2,259 citations
Book•
01 Jan 1965
TL;DR: In this paper, a new chapter on computational methods that present the basic principles on which most modern computer programs are developed is presented, which introduces an example on rotor balancing and expands on the section on shock spectrum and isolation.
Abstract: This edition features a new chapter on computational methods that presents the basic principles on which most modern computer programs are developed It introduces an example on rotor balancing and expands on the section on shock spectrum and isolation
2,196 citations
Book•
01 Jan 1975
TL;DR: In this article, the authors provide contemporary coverage of the primary concepts and techniques in vibration analysis, and more elementary material has been added to the first four chapters of this second edition for an updated and expanded introduction to vibration analysis.
Abstract: This book provides contemporary coverage of the primary concepts and techniques in vibration analysis. More elementary material has been added to the first four chapters of this second edition-making for an updated and expanded introduction to vibration analysis. The remaining eight chapters present material of increasing complexity, and problems are found at the end/of each chapter.
1,545 citations