UK Nonlinear News, November 2000

Autoparametric resonance in mechanical systems

By Ales Tondl, Thijs Ruijgrok, Ferdinand Verhulst and Radoslav Nabergoj

Reviewed by Alan Champneys

Cambridge University Press

196 Pages ISBN: 0-521-65079-8 (hardback)

The book introduces itself

``When a mechanical system consists of two or more coupled vibrating components, the vibration of one of the component subsystems may destabilise the motion of the other components. This destabilising effect is called autoparametric resonance... This book is the first completely devoted to the subject of autoparametric resonance in an engineering context.''
The authors are careful to tell us in the preface that the book represents a collaboration between an engineer, a physicists and two mathematicians. (In my view, in the field of nonlinear dynamics, it is becoming increasingly unimportant which of the above labels we give ourselves.) The emphasis here is more on the engineering side, with a traditional nonlinear vibrational feel to it, with occasional glimpses of the modern geometric theory of nonlinear dynamics, bifurcation chaos etc.

The book's structure is to present the key phenomenon of parametric resonance through a series of archetypical examples, typically taking the form of second-order-oscillator systems (either forced, parametrically forced or self-exciting) being coupled through various nonlinear terms. Each different nonlinearity and forcing is motivated by a physical example of a simple mechanical system, and each is analysed separately. One key step of analysis is reinforced over and over again. Namely ``the determination of sets in parameter space in which the semi-trivial solution is unstable'' since these are defined to be the region of autoparametric resonance. Here the semi-trivial solution is defined to be where the primary oscillator is in motion, but the secondary one has zero response. So autoparametric resonance is when the secondary system moves, being parametrically excited by the motion of the first. The techniques used to perform this first step are in general approximate, relying on averaging, harmonic balance and the Poincaré-Lindstedt method. In keeping with the example-driven approach, an explanation of these techniques, along with methods from bifurcation theory are relegated to the final chapter `Mathematical Methods and Ideas'.

Chapter 1, Introduction is highly readable and introduces the key concept of autoparametric resonance (although it does not tell the reader what parametric resonance is - more of that later). It also illustrates the mechanics behind some of the example systems that come up later in the book, and gives a brief literature survey. More references are given in the course of the analysis presented in each chapter - although a quarter of the 84 cited works are by the first author (which is perhaps excusable given that this is a monograph rather than a text book). Finally, the chapter ends by giving the scope of the book; ``partly a literature survey and partly a workbook''.

Chapter 2, Basic Properties, introduces three introductory examples which differ in whether the primary system is periodically forced, either parametrically or not, or is self-excited. Let us briefly look at the style of analysis applied to the first of these (section 2.2). The reader is given an ansatz to find the semi-trivial solution (this is easy because the decoupled primary system is linear). Then a linear approximation is taken and we are told that the second equation is of `Mathieu type' (the reader is invited to go to Chapter 9 if he needs more background on Mathieu equations) and its main instability domain is found at a certain ratio between two parameters. Then a parameter perturbation is introduced (without justification other than assuming the reader knows how and why to do this for Mathieu-type equations) and another linear stability condition is derived. Then we are told that the boundary of the main instability domain can be found by averaging (how? go to Section 9.5) leading a simple condition on the parameters. The implication of this condition is then discussed using two different paradigms (excitation-oriented or response-oriented). The nontrivial solution is analysed via a different scaling of parameters leading to a solution via the Poincaré Lindstedt method (see Chapter 9). We are later told in the interpretation of the results that one of the instabilities is a period-doubling bifurcation (bifurcations are introduced in Chapter 9). The unfortunate thing is that if the non-expert reader is trying to follow all this, Chapter 9 is not entirely helpful. It is a good self-contained introduction to some of these concepts (in fact period-doubling is not one of the bifurcations treated there) but it does not refer back enough to the actual examples used in the main body, so the reader is not really given the necessary intuition that would have been required to have produced these calculations for themselves.

Much of the rest of Chapters 2-8 continues in the same vain. I do not wish to be unduly critical, I learned a lot from this book that I did not know. I just think that course tutors need to be wary that, despite its claims, this is not really a `workbook' suitable for students; at least not unless they already have a good grasp of asymptotic methods, parametric resonance etc. In my view the book would be greatly improved in a second edition if the authors could try a little harder to link the material on methods and the modern theory of dynamical systems to the main body of examples. It would be most useful to include a new chapter, before chapter 2, that defines parametric resonance, introduces the Mathieu equation via a physical example, and shows how to perform asymptotic calculations on it. Then the analysis in the current chapter 2 would not come as such a bolt from the blue. Finally, much of the book would benefit from presenting more phase portraits of solutions; but perhaps the intention is for readers to perform numerical calculations for themselves in order to visualise solutions.

Having got my main criticism off my chest, let me try to draw out the highlights of this book while going through the rest of the contents. Chapter 2 ends with a `concluding remarks' section (as do most of the other chapters) that summarises, in a nontechnical way, what we have seen qualitatively may occur in parametrically excited systems. The phenomenon of `saturation' may be useful as an amplitude limiting device. Synchronisation is an interesting issue, as are the possibility of autoparametric resonance at other than the dominant Mathieu instability, and the effect of varying the nonlinear coupling.

Chapter 3, an Elementary Discussion of Single-Mass Systems, considers three more examples in a similar spirit. Concepts such as domains of attraction are introduced (perhaps not in the most modern context, but highly readably). Chapter 4 analyses a Mass-Spring-Pendulum System in some detail and introduced the new phenomenon of `quenching', where the pendulum system effectively quenches the effects of external vibration on the spring system. The latter half of the chapter takes on a different style. Hopf bifurcations are found in the nontrivial solution and a hint of chaotic behaviour via \u{S}ilnikov (the correct spelling should be Shil'nikov) bifurcations. Normal form calculations are presented near a certain `Quasi-degenerate' Hopf bifurcation in order to prove the existence of a `strongly quenched solution'. The analysis is backed up with numerical simulations.

The brief Chapter 5 extended some of the analysis of the previous chapter to Models with More Pendulums, taking a three-degree of freedom system as an illustrative example. Chapter 6, Ship Models, then analyses two and three degree-of freedom models inspired by the rigid body motion of a ship. The results are able to predict the onset of parametrically induced roll oscillations through coupling to either heave or heave and roll, and an interesting parallel is drawn with an observation by Froude in 1861.

Chapter 7 analyses simple mechanical models inspired by Flow-Induced Vibrations; a two-degree-of-freedom `critical velocity model' and three-degree-of-freedom `vortex-shedding model'. In the former, the effects of adding a dry friction term are also considered, and it is shown by direct numerical simulation that there are chaotic dynamics (any nonstandard effects due to the piecewise-linear nature of the dry friction term are masked in the analysis by the heavy use of harmonic balance methods). The chapter ends with a section (7.7) that uses the theory of perturbed normal forms and \u{S}ilnikov (sic) dynamics to give a flavour of the proof of chaos for a generalisation of the critical velocity system. This is a quick `show and tell' tour through the normal form of a pitchfork-Hopf bifurcation, Melnikov methods, and Shil'nikov's theorem. Advanced stuff. But appropriate references are given to Wiggins and Guckenheimer & Holmes (surprisingly not Kuznetsov, which I would have thought was more up to date).

Chapter 8 looks at a model arising in Rotor Dynamics that takes the form of two coupled Mathieu-like equations. Linear and nonlinear damping are added, with the latter leading to hysteresis and phase locking. The former leads to a counter-intuitive result that linear damping can lead to a discontinuity in the shape of the bifurcation diagram. This property is motivated analytically by performing normal form analysis on a general class of coupled two-degree of freedom rotational systems. The book closes with the aforementioned Chapter 9 which on its own provides a readable introduction to some of the Mathematical Methods and Ideas encountered in the preceding exposition.

Despite the above-mentioned shortcomings as a teaching aid, this book clearly has merits as a pot boiler of old and new ideas around the important, yet previously much overlooked, topic of autoparametric resonance. In the closing section of the preface, it is stated that the book is a monograph that:

``constitutes a first inventory of problems important from both engineering and mathematical points of view. We are convinced that it can be an inspiration for a considerable amount of research in both fields''.
It certainly certainly fulfills these objectives. It will be a useful as a source of example problems in nonlinear dynamics and as an inspiration to engineers to understand autoparametric resonance in practice via analogy with the examples treated.

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks Cambridge University Press for providing a review copy of this book.

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Last Updated: 31st October 2000.