When I began the chapter entitled the Conclusion, in the former part of The Rights o/Man, published last year, it was my intention to have extended it to a greater length; but in casting the whole matter in my mind which I wished to add, I found that I must either make the work too bulky or contract my plan too much. I therefore brought it to a close as soon as the subject would admit, and reserved what I had further to say to another opportunity.

Several other reasons contributed to produce this determination. I wished to know the manner in which a work, written in a style of thinking and expression at variance with what had been customary in England, would be received before I proceeded further. A great field was opening to the view of mankind by means of the French revolution. Mr. Burke's outrageous Opposition thereto brought the controversy into England. He attacked principles which he knew (from information) I would contest with him, because they are principles I believe to be good and which I have contributed to establish and conceive myself bound to defend. Had he not urged the controversy, I had most probably been a silent man.

Another reason for deferring the remainder of the work was that Mr. Burke promised in his first publication to renew the subject at another opportunity and to make a comparison of what he called the English and French constitutions. I therefore held myself in reserve for him. He has published two works since without doing this; which he certainly would not have omitted had the comparison been in his favor.

In his last work, his “Appeal from the New to the Old Whigs,” he has quoted about ten pages from The Rights ofMan, and having given himself the trouble of doing this, says “he shall not attempt in the smallest degree to refute them,” meaning the principles therein contained. I am enough acquainted with Mr. Burke to know that he would if he could. But instead of contesting them, he immediately after consoles himself with saying that “he has done his part.”

This chapter will set out the motivation and define the context for the material contained in this book. The recurrent themes to be described should help to establish what this book is about (and what it isn't).

The Physical Context

• Mechanical Vibration

This book uses an aspect of traditional mechanics to provide a focus and solid context for the material development. The field of mechanical vibration provides a rich source of material at the interface between applied mathematics and practical design. Classical vibration is very well developed, especially for linear, low-order systems. Progress is being made in stochastics, control, and nonlinear effects. It is this latter aspect of vibration theory (and its experimental verification) that is pursued in this book.

• Modeling

In this book we are primarily interested in low-order, time-varying, deterministic, nonlinear, experimental, mechanical systems! During the modeling process we make a number of trade-offs between the desire for mathematical simplicity and practical usefulness (and phenomenological interest). Although the experimental paradigm developed is a discrete mechanical system, the extension to continuous systems becomes clear. Classical methods (e.g., Lagrange's equation) are used to derive governing equations, with the major modeling challenge presented by energy dissipation.

• Duffing's Equation

Throughout we will use Duffing's equation. Why focus so much on one type of system? The development of nonlinear dynamics is framed by intense scrutiny of certain archetypal systems, typically named after the researchers who first studied them: van der Pol, Duffing, Lorenz, Hénon, and Rossler. The nonlinear ordinary differential equation that has come to be known as Duffing's equation has particular importance in engineering, and indeed it was first considered by the German experimentalist Georg Duffing to study the hardening spring effect observed in many mechanical systems.

The specific double-well form of Duffing's equation, which provides the backbone of this book, offers compelling pedagogical insights. This is partly based on the fact that it is symmetric (about the origin) in a global sense, a feature nominally present in a variety of practical situations. But since the underlying equilibria are offset, it also subsumes a variety of types of asymmetric behavior and is globally bounded.

Francis Bacon was born in 1561, the fifth and last surviving son of Sir Nicholas Bacon, Lord Keeper to Queen Elizabeth I, the second surviving child of his second wife. Left a widower in 1552, with six children under twelve to bring up, Nicholas had rapidly married Anne Cooke, one of five highly educated daughters of Edward VI's tutor, Sir Anthony Cooke, celebrated, like their father, for their learning and piety. All made extremely advantageous marriages: Margaret to a prominent goldsmith; Elizabeth to Sir Thomas Hoby and then to the son of the earl of Bedford; Katherine to Sir Henry Killigrew. Most significantly, Mildred became the second wife of William Cecil, later Queen Elizabeth's Principal Secretary of State. Thus Francis was kin to some of the most powerful and influential figures of his time.

This was just as well, since he had to contend, throughout his life, with the fact that his father left him inadequately provided for financially. Sir Nicholas was in the process of making suitable long-term purchases of land for Francis and his elder brother Anthony when he died unexpectedly in 1579. Had that settlement been complete, Bacon later claimed, he would have been able to devote his entire life to study, and his grand plan for an entirely new system of learning might have reached completion in his lifetime.

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).