Introduction

So far, we have seen the appearance of essentially linear behavior for relatively small-amplitude forced oscillations together with the occurrence of (local) instability phenomena and subharmonic oscillations. The main features were a growth of amplitude with forcing magnitude and proximity to resonance and a response frequency the same as the input (forcing) frequency - features not dissimilar to the purely linear oscillator. The appearance of hysteresis signaled the enhanced role played by nonlinearity and initial conditions. However, considered locally from a topological viewpoint, all these responses can still be classified as periodic attractors, and in many important ways they are not substantially different from their point attractor counterparts in dissipative, gradient systems. Predicting the future behavior in these cases is relatively easy. But for nonlinear dynamical systems (flows) that exist in a phase space of three or more dimensions thoroughly more complicated and less predictable behavior becomes possible (Lorenz, 1963).

Chaos

It is the fascinating (and universal) nature of chaos that will be the main focus of attention in this chapter. The discussion will be somewhat constrained to the types of behavior exhibited directly by the experimental system, with a focus on invariant measures. A number of excellent books on chaos are available. A sample includes those covering theoretical (numerical) approaches (Guckenheimer and Holmes, 1983; Thompson and Stewart, 1986; Wiggins, 1990; Marek and Schreider, 1991; Ott, 1993), experimental aspects (Moon, 1992; Tufillaro Abbot, andReilly, 1992), and general treatments (Jackson, 1989;Mullin, 1993). This subject has reached a sufficient level of maturity that there are even books using pedagogical approaches (Abraham and Shaw, 1982; Strogatz, 1994; Baker and Gollub, 1996) and more general expositions for the general public (Gleick, 1987; Stewart, 1989).

In keeping with the progression of the previous chapters we introduce further strengthening of the external driving, thus encouraging significant nonlinear effects. It will be shown later that a progression toward chaotic behavior follows some very generic routes, but since many of the subtle interactions are of a global nature, they will be left until later (Dowell and Pezeshki, 1986).

Introduction

We have seen that in order to design a close mechanical analogue of the equation of motion (developed in the previous chapter) a number of objectives have to be achieved. An initial approach might consist of allowing a small ball to roll on a grooved guide (Marion and Thornton, 1988). However, it is then difficult to monitor the motion of the ball (at least in an accurate manner) and significant slippage can occur. Suppose we wish to build a small cart or roller coaster. In this case it is relatively easy to measure the position of the cart based on the output of a rotational potentiometer attached to an axle. The rotary inertia of the cart can be minimized by keeping the cart small, and the a term can also be adjusted through the track geometry. To avoid slippage between the cart wheels and the track, a chain-sprocket system can be used. This does have the drawback of complicating the damping modeling but it will be seen that the overall damping in the model is quite small. Another advantage of the chain-sprocket guide is that it minimizes the possibility of the cart actually leaving the track during fast motions (Gottwald, Virgin, and Dowell, 1992).

We will also see that to replicate Duffmg's equation it is desirable to have a relatively shallow curve so as to minimize those nonlinear terms in the accurate equation of motion (5.38) that do not appear in the standard Duffing's equation (4.1). We do not wish to stray too far from the familiar form of Duffing's equation, although we realize that typical nonlinear features are by no means restricted to a narrow class of ordinary differential equations. The features to be described are actually quite generic and robust. The theoretical development in the previous chapter showed that these effects can be grouped together by consideration of a single nondimensional parameter a in the experiment. Equation (5.37) showed that a can be made smaller by reducing the vertical distance between the unstable equilibrium (hilltop) and the symmetrically positioned stable equilibria to reduce the vertical component of the acceleration.

Introduction

In this chapter we go back and reexamine the transition between the periodic behavior described in Chapter 8 and the thoroughly nonlinear chaotic behavior described in Chapter 9. This is an interesting aspect of the double-well oscillator (or indeed any system with a hilltop): Trajectories, initially contained within a single potential energy well, might escape over a nearby local maximum (hilltop). In the doublewell case, this means that the motion may traverse through to the adjacent well. Clearly, this is a situation completely alien to the confined, parabolic-well, linear oscillator. Dynamical systems characterized by this possibility of escape from a local potential energy well occur in many physical problems including a rigid-arm pendulum passing over its inverted equilibrium position (Baker and Gollub, 1996), snap-through buckling in arch and shell structures (Bazant and Cedolin, 1991), capsizing of ships (Virgin, 1987), and the toppling of rigid blocks (Virgin, Fielder, and Plaut, 1996).

The escape of trajectories from a local minimum of an underlying potential energy function is essentially a transient phenomenon. Given a single-degree-of-freedom system at rest in a position of stable equilibrium, it is often desirable to find the range of harmonic excitation that causes the subsequent motion to overcome an adjacent barrier defined by the limit of the catchment region (basin of attraction) surrounding the minimum. Escape occurs as the motion within the potential well grows “large enough.” This is clearly more likely to occur when the forcing magnitude is “large” in relation to some system characteristic. However, even in linear dynamics, the response of a sinusoidally forced system will be magnified close to resonance as we have seen. In nonlinear systems, the size of the basins of attraction surrounding an attractor depend crucially on certain system parameters. Nonstationary changes are incorporated in this chapter to simulate quasi-steady escape.

First, the unforced system is used as an introduction to escape based on initial conditions. Second, a slowly evolving harmonic excitation is applied to the system. The evolution is achieved by changing the forcing amplitude or frequency very slowly, either in an increasing or decreasing manner. In this way transients are minimized such that the evolving trajectory remains “close” to the underlying steady-state solution.

Introduction

All of the preceding chapters have dealt with single-degree-of-freedom oscillators, generally with a periodic excitation and, hence, a threedimensional phase space. Many real dynamical systems are continuous, modeled by partial differential equations with both space and time as independent variables. Despite the fact that the dynamics often take place on a relatively low-order subspace of the (infinite) phase space of the full system, there are still many situations in which an analysis in a high-order space is necessary. For continuous systems such as beams and plates (see Chapter 4), modal analysis has proved to be a powerful technique for extracting the dominant dynamic characteristics from complex systems, especially for linear systems. In a theoretical context, Galerkin's method can be used to reduce a partial differential equation into a set of coupled ordinary differential equations, which can then be analyzed using standard techniques. The success with which a reducedorder model captures the full range of behavior is a very complicated issue (especially in fluid mechanics (Lorenz, 1963; Ruelle and Takens, 1971)), but, for example, a continuous beam excited close to its fundamental natural frequency will display behavior dominated by the lowest mode, and hence a lumped parameter model will likely be good enough in an engineering context.

Some of the earliest studies in chaos were generated by the consideration of thin beams, which under certain circumstances could be very successfully modeled by Duffing's equation (see (Moon, 1992) and Chapter 4). The presence of multiple equilibria and periodic excitation provided conditions under which a wide range of nonlinear behavior could be observed and measured. In this appendix, we take a brief look at a continuous (i.e., high-order) experimental system that displays behavior that is qualitatively similar to the single-degree-of-freedom examples encountered earlier. The practical context for this example occurs in certain aerospace systems where thin metal panels are subject to intense acoustic excitation and are often in a postbuckled equilibrium configuration owing to thermal effects (Tauchert, 1991).

The theoretical treatment is rather involved, and the reader is referred to Refs. (Murphy, Virgin, and Rizzi, 1996a; Murphy, Virgin, and Rizzi, 1997; Murphy, 1994) for more details. Here, we shall concentrate on experimental results and try to contrast the similarities and differences with some of the (low-order) results presented earlier in this book.

Preface

Those who have presumed to make pronouncements about nature as if it were a closed subject, whether they were speaking from simple confidence or from motives of ambition and academical habits, have done very great damage to philosophy and the sciences. They have been successful in getting themselves believed and effective in terminating and extinguishing investigation. They have not done so much good by their own abilities as they have done harm by spoiling and wasting the abilities of others. Those who have gone the opposite way and claimed that nothing at all can be known, whether they have reached this opinion from dislike of the ancient sophists or through a habit of vacillation or from a kind of surfeit of learning, have certainly brought good arguments to support their position. Yet they have not drawn their view from true starting points, but have been carried away by a kind of enthusiasm and artificial passion, and have gone beyond all measure. The earlier Greeks however (whose writings have perished) took a more judicious stance between the ostentation of dogmatic pronouncements and the despair of lack of conviction (acatalepsia); and though they frequently complained and indignantly deplored the difficulty of investigation and the obscurity of things, like horses champing at the bit they kept on pursuing their design and engaging with nature; thinking it appropriate (it seems) not to argue the point (whether anything can be known), but to try it by experience.

Introduction

In earlier chapters, we saw the means by which motion, bounded initially within a potential energy well, might spill over or escape either to infinity or to an adjacent energy well. In this chapter we take a closer look at global issues. We will see how basin boundaries and unstable fixed points have a considerable influence on behavior in the large. Dependence on initial conditions has been encountered earlier in this book in terms of multiple (point and periodic) attractors and the extreme sensitivity of chaos. We shall see that extreme sensitivity to initial conditions may also appear when the boundaries separating domains of attraction become fractal, causing transients to have arbitrarily long lengths (Eschenazi, Solari, and Gilmore, 1989; Grebogi, Ott, and Yorke, 1987; Gwinn and Westervelt, 1986). This is often a precursor of steady-state chaos. One specific aspect of interest is the appearance of indeterminate bifurcations. For the purposes of illustrating this behavior, we will revert back to the double-well Duffing oscillator of earlier chapters of this book. A chronological note here is that the experimental results to follow were obtained a couple of years after those described in Chapters 8 and 9 and, hence, some small adjustments appear in the basic system coefficients (Todd and Virgin, 1997b).

Dependence on Initial Conditions

One of the fundamental differences between a linear and a nonlinear system is that nonlinear systems often possess multiple stable solutions, and, hence, the final solution depends to an extent on the starting conditions. The standard theory of linear vibrations, even for high-order systems, obviates the need to consider this, with unique solutions capturing all possible initial conditions. We have seen that nonlinear systems (even unforced problems) typically have a variety of long-term solutions for a fixed set of parameter values. Although it can be argued that persistent (stable) solutions perhaps have the most practical importance (certainly in relation to their local region of phase space), it is the unstable solutions that have a profound influence on global behavior (Grebogi, Ott, and Yorke, 1986b).

General Comments

• Aims and Scope

Nonlinear dynamics and chaos have been popular areas of research for more than twenty years. One of the most interesting aspects of nonlinear behavior is its ubiquity. Population biology, celestial mechanics, cardiac fibrillation, and aircraft wing flutter are just a few of the highly disparate areas in which nonlinear dynamics and chaos have shed new light. Interdisciplinary research and the cross fertilization of ideas between different areas has provided much of the impetus. There is hardly a branch of applied science that is not touched by nonlinear dynamics: Witness the range of interdisciplinary books and meetings spread within, and between, research boundaries. However, it is the sheer intrigue provided by nonlinear systems, and chaos in particular, that has caught the imagination of so many people in science and engineering and has led to continued growth in the field.

The aim of this book is to provide a relatively concise pedagogical approach to nonlinear dynamics based on the evolution of an experimental paradigm. In contrast to most other books in this general area, this book will use real data, generated from the laboratory, to introduce, rather than occasionally confirm, the fascinating array of behavior encountered even in a relatively restricted subset of the nonlinear world. In this way, the material will span the gap between abstract, often chaos-driven, ideas and practical engineering reality.

This book uses experimental mechanical vibration as the backbone, or theme, on which the material is developed. We focus attention on this specific branch of engineering to tell a story: one of expanding the horizon from a linear to a nonlinear view, without losing sight of the exigent world of experimentally verifiable behavior. An advantage of this approach is that mechanical vibration, as well as being an important area of research in its own right, has a well-founded history based on linear behavior, and using experimental data as a key component emphasizes the robustness of the described behavior. Thus the narrative of the present book evolves from a local, linear, largely analytical, foundation toward the rich and often unpredictable world of nonlinearity. This book does not attempt to cover the whole range of nonlinear phenomena and experimental techniques.

He became aware that the human intellect is the source of its own problems, and makes no sensible and appropriate use of the very real aids which are within man's power; the consequence is a deeply layered ignorance of nature, and as a result of this ignorance, innumerable deprivations. He therefore judged that he must make every effort to find a way by which the relation between the mind and nature could be wholly restored or at least considerably improved. But there was simply no hope that errors which have grown powerful with age and which are likely to remain powerful for ever would (if the mind were left to itself) correct themselves of their own accord one by one, either from the native force of the understanding or with the help and assistance of logic. The reason is that the first notions of things which the mind accepts, keeps and accumulates (and which are the source of everything else), are faulty and confused and abstracted from things without care; and in its secondary and other notions there is no less passion and inconsistency. The consequence is that the general human reason which we bring to bear on the inquiry into nature is not well founded and properly constructed; it is like a magnificent palace without a foundation.