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.

Strong interactions, local instabilities and turbulence: a postscript

Short waves and long waves

Small-wavelength disturbances may ride on large-amplitude long gravity waves. The orbital velocities of fluid particles due to the long waves provide a variable surface current through which the short waves propagate. When this current is comparable with the propagation velocity of the short waves relative to the long ones, their interaction is no longer weak. Nevertheless, the characteristics of the short waves may still be described, at least in part, by Whitham's theory of slowly-varying wave-trains in an inhomogeneous medium (see §11.3). Phillips (1981b) deduced from wave action conservation that capillary waves are likely to be ‘blocked’ by steep gravity waves. In much the same way, Gargett & Hughes (1972) earlier showed that short gravity waves may be trapped by long internal waves, so leading to caustic formation and local wave breaking. Untrapped modes also undergo amplitude modulations by the straining of the dominant wave field.

Computations of Longuet-Higgins (1978a, b) and McLean et al. (1981), already described in §22.2, display generation of short waves by high-order instability of steep gravity waves. In addition, finite-amplitude wave-trains necessarily contain bound harmonics, which travel with the fundamental Fourier component. Weakly-nonlinear interaction of neighbouring frequency components may also give rise to phase-locking of modes.

For these reasons, and doubtless others besides, measurements of the phase speeds of Fourier components of wave fields often reveal significant departures from the linear dispersion relation, even after allowance is made for wave-induced mean currents.

Introduction to nonlinear theory

Introductory remarks

Nonlinear theories are of three more or less distinct kinds. In one, properties of arbitrarily-large disturbances are deduced directly from the full Navier–Stokes equations. Consideration of integral inequalities yields bounds on flow quantities, such as the energy of disturbances, which give stability criteria in the form of necessary or sufficient conditions for growth or decay with time. An admirable account of such theories is given by Joseph (1976). They have the advantage of supplying mathematically rigorous results while incorporating very few assumptions regarding the size or nature of the disturbances. Sometimes, these criteria correspond quite closely to observed stability boundaries. The bounds for onset of thermal (Bénard) instability and centrifugal (Rayleigh–Taylor) instability in concentric rotating cylinders are particularly notable successes. Often, however, the bounds are rather weak: this is especially so for shear-flow instabilities, where local details of the flow typically play an important rôle which cannot be (or, at least, has not been) incorporated into the global theory.

The second class of theories relies on the idea that linearized equations provide a satisfactory first approximation for those finite-amplitude disturbances which are, in some sense, sufficiently small. Successive approximations may then be developed by expansion in ascending powers of a characteristic dimensionless wave amplitude. These are known as weakly nonlinear theories, and they have proved successful in revealing many important physical processes.