Fundamental considerations

Many non-linear systems of engineering interest possess more than one stationary solution stable against small disturbances, that is, more than one attractor. Such solutions are termed locally stable. In practical applications usually only one of these is desirable; the others, for the most part, are unwelcome because they signal danger to safe and reliable operation of the device whose model is being analyzed. The problem is to establish the conditions which lead to a particular steady state, or alternatively, to examine the disturbances which are apt to cause one stationary state to change to another (for example, small-amplitude stationary vibration to large-amplitude motion, a non-oscillatory state to an oscillatory condition, etc.).

The solution to the first aspect of the problem, i.e. establishing the domains of the initial conditions which lead to different stationary solutions, is well known. These domains are called the domains of attraction of a particular solution. For systems which are directly described by a set of two first-order differential equations of the type (4.1, 1) or whose set of such equations is identical with the original second-order differential equation the problem is solved by means of phase portrait analysis outlined in Chapter 4. The differential equations of motion of one-degree-of-freedom systems excited harmonically by an external force or parametrically can be converted, by application of known procedures (van der Pol or Krylov and Bogoljubov methods) to a system of two first-order differential equations of the type (4.1, 1) which, however, are not identical with the original equation of motion.

Non-linear vibration, non-linear characteristics and basic definitions

As mentioned in the Introduction, vibration of linear systems is only a special case of vibration of non-linear systems. Two important principles, which are valid for linear systems, do not apply to non-linear systems:

1. (I) The principle of superposition,

2. (II) The principle of proportionality.

In linear systems described by linear differential equations (all terms are functions of dependent variables and their derivatives) the response to the various components of excitation can be added up; if the amplitude of harmonic excitation is increased n times, the amplitudes of steady vibration increase n times. It has been found advantageous to divide linear vibration into two large groups:

1. (a) Vibration of systems with constant parameters described by linear differential equations with constant coefficients. The term linear vibration is usually understood to mean vibration of systems of just this type. The theory of this vibration is essentially complete and will not be discussed in the present book.

2. (b) Vibration of systems with non-constant parameters described by linear differential equations whose coefficients are generally functions of a dependent variable, chiefly time. An important separate group is formed by systems whose parameters are periodic functions of a dependent variable. Such systems are called rheo-linear and their theory is usually studied together with that of non-linear vibration. One of the reasons for this is the fact that an investigation of stability for small disturbances of steady periodic solutions of non-linear differential equations generally leads to an investigation of stability of the trivial solution of systems of differential equations with periodically variable coefficients.

It has not been an easy decision to choose the general title of Non-linear vibrations. Recently there has been an overwhelming development of the theory and many applications of non-linear vibrations, development reflected in numerous books, many specialist journals and the Equa-Diff conferences and ten International Conferences on Non-linear Oscillations.

In view of these development, we would like this book to be considered not as an attempt to survey the state of the art and extent of current knowledge of non-linear vibrations, but rather as an account of some methods and results in the field of non-linear vibrations we have obtained or encountered in our personal experience. We do not believe in a single, comprehensive theory of non-linear vibrations, not even in the sense of linear vibration theory. On the contrary, we see the specific difficulty – and the attraction – of non-linear vibrations in the non-existence, more than in the existence, of certain rules of order. The nature of our endeavour may perhaps best be conveyed in Rilke's words: Uns überfüllts. Wir ordnens. Es zerfällt. Wir ordnens wieder…'.

We believe that the material presented here can be used for many kinds of vibration problems, although translating it into their terms is no simple task. Various methods and examples are traditionally associated with different notations, and we have not tried to make the notation uniform throughout this book.

The first author is responsible in particular for chapters 5, 6, 10, 11 and 12, the second for chapters 3, 4, 7, 8 and 9.

Finding examples for vibrations of systems which are linear is not a trivial matter; there is something in the remark by R. M. Rosenberg that dividing vibrations into linear and non-linear is like dividing the world into bananas and non-bananas. Nevertheless many important vibration problems must be and can be treated in the same way as linear ones, and the book Lineare Schwingungen by Müller and Schiehlen, for instance, is very helpful for all those grappling with vibration problems.

However, numerous vibration phenomena which are theoretically surprising as well as practically important can only be understood on the basis of non-linear vibration. For instance, the wide field of self-excited, parametric and autoparametric vibration, to which we give special consideration in this book, demands non-linear treatment from the very beginning.

The mathematical theory of non-linear vibrations, the numerical and experimental methods for their evaluation and their many applications in mechanics, in mechanical and civil engineering, in physics, electrotechnology, biology and other sciences, have all developed so rapidly of late that it would require many volumes and many specialists to give a comprehensive picture of non-linear vibration.

The difficulties faced by those engaged in the study of the subject are concerned, on the one hand, with the broad scope of problems and the diversity of systems involved, and on the other hand with the fact that some general laws, such as the principle of superposition and the principle of proportionality, which can be used to advantage in solutions of linear systems, do not apply to non-linear ones.

Basic considerations and methods of solution

Since self-excited vibration impairs reliable operation and endangers the safety of miscellaneous machinery and structures, its suppression constitutes one of the major tasks of vibration engineering. An ideal and very expedient means to this end is the removal of the source of self-excitation. However, this so-called active method is not applicable in all cases. The systems in which it fails include those where self-excitation is an inherent characteristic of the technological process (for example, the cutting forces in machine tools) or is inherent in the function of the device (for example, the hydrodynamic forces in journal bearings). Sometimes, as in the case of self-excited oscillations – galloping – of high-voltage transmission lines, application of the active method is not feasible because of economic or operational reasons. In cases of these sorts resort must be made to passive methods, that is, to paralyzing the destabilizing effect of negative damping, which is obtained in the equations of motion when expressing the action of forces producing self-excitation, by an increase in the level of positive damping. This chapter deals exclusively with the passive methods of quenching selfexcited vibration.

The practicability and efficacy of the various means used in connection with these methods will be examined using systems which belong to the class represented in its simplest form by the van der Pol oscillator. Only systems with a finite number of degrees of freedom will be considered.

Characteristic features of auxiliary curves, particularly the backbone curves and the limit envelopes

In harmonically excited systems, especially in systems with one degree of freedom, the specific characteristics of the auxiliary curves have been found to be very useful for preliminary qualitative analyses of stationary vibration as well as for identification of the various elements (e.g. damping) of the system being examined on the basis of experimental results. In the former case they enable the analyst to make a prompt estimate of the basic properties of the system and of the effect of various parameters on its behaviour, in the latter, to identify the specific properties of the system and in turn to formulate a suitable analytic expression of the forces acting in a particular element for the purpose of a mathematical model.

Let us first consider the characteristics of the so-called backbone (or skeleton) curve, and of the curves connecting the points at which sin ϕ = const (ϕ is the phase angle between response and excitation). The limiting case of the latter curves (sin ϕ = ± 1; the minus sign has no meaning except in special cases stated farther on) is the so-called limit envelope (this term was proposed by Tondl (1973d)). The backbone curves have been used in routine analyses for a long time; the application of the second type of curves, the limit envelope and the sin ϕ = const curves, is less common.