Non linear vibrations of stretched strings

Abstract

The results obtained in this chapter can be summarized as follows. It is not possible to induce purely transverse oscillations since longitudinal and transverse oscillations are coupled together. However, if the order of transverse modes is small compared to ?, the equations for transverse and longitudinal oscillations can be decoupled. It is found that nonlinearity causes inter?modal coupling, but does not generate modes that are not present at the initial instant. For sinusoidal initial conditions, the partial differential equations can be reduced to a pair of ordinary differential equations. These equations can be solved by the method of variable amplitude and phase. The amplitudes and frequencies of the mutually perpendicular components of transverse displacement possess an oscillatory character. The frequencies, which depend upon the respective amplitudes, decay with time and approach the ‘linear’ value as t??t \to \inftyt??. But the amplitude and frequency of the total transverse displacement Y2+Z2\sqrt{Y^2 + Z^2}Y2+Z2? do not exhibit this oscillatory behaviour. The oscillation of amplitude and frequency causes an oscillation of energy between the Y and Z components. For an undamped string, the energy oscillation is sinusoidal and the frequency of energy oscillation is proportional to the nonlinearity parameter ?. The time?averages of the energies of the components are the same, whatever be the initial distribution of energy. This phenomenon also suggests that planar vibrations of an undamped string are unstable. If terms of order ?² are taken into account, it is found that the potential energies of the components depend on the path also, even though the total potential energy does not. The part of the energy associated with the path is termed the energy of interaction. The acceleration of a point on the string is purely radial just as in the case of the linear theory. The curve traced by a particle on the string is an ellipse with slowly rotating and shrinking axes. Nonlinear oscillations of continuous systems is a field that is still practically unexplored. Even the vibrations of a stretched string, which forms the simplest example of a continuous system, present many interesting features which have no counterparts in the behaviour of discrete systems. The contribution of this thesis lies in formulating the equations of motion and developing methods of analysis which give a certain physical insight into the large?amplitude oscillatory behaviour of the string. The approach adopted in the foregoing analysis treats the string as a discrete system with infinitely many degrees of freedom. Actually, it is a system with doubly infinite degrees of freedom since the string has two transverse components of vibration. The degrees of freedom, each of which represents a particular mode of vibration, are mutually coupled. Hence, theoretically, all the modes of vibration exist, whatever be the initial conditions or mode of excitation. In practice, however, only a finite number of modes are dominant in any particular case, so that the system can be considered to possess a finite number of degrees of freedom. The problem is thus brought down to a more familiar level. Though the theory of nonlinear vibrations of coupled discrete systems is by no means well developed, it is at least possible to extend some of the techniques of analysis of single?degree?of?freedom systems. The analysis is further simplified if it is restricted to cases where only a single mode is predominant. Both the cases considered in this thesis, viz., resonant harmonic oscillations and free oscillations under sinusoidal initial conditions, fall under this category. The motion of the string can therefore be described by a pair of coupled nonlinear ordinary differential equations. The solution for the case of forced vibrations has been obtained by a straightforward application of the principle of harmonic balance, while extensions of standard procedures have been used to solve the initial value problem and the problem of stability. The analysis in this thesis shows that damping plays a significant role in determining the behaviour of the string. In most investigations of nonlinear oscillatory systems, damping of the motion is completely ignored presumably on the supposition that its effect can be understood by analogy with the linear case. This, however, is not true. Consideration of damping leads to several results which cannot be anticipated otherwise. As shown in Chapter 3, it is impossible even to define the resonance frequency or determine the magnitude of the amplitude jumps without considering damping. Further, in the presence of damping, the character of the amplitude response depends in a qualitative way on the magnitude of the driving force. While this result is not of much practical importance for the case of light damping considered here, it will assume great significance when the damping is heavy. Damping, however small it may be, is also responsible for rendering planar free vibrations stable. In general, damping plays an important role in determining the stability of different types of motion. It should be noted that all these results concerning the effect of damping are of fundamental importance and are applicable to many other nonlinear oscillatory systems also. Though the analysis in this thesis is confined to some special cases which are chosen for their relative simplicity, the cases considered reveal some of the basic features of nonlinear behaviour and pave the way for a more detailed analysis. Such an analysis, which should consider driving forces and initial conditions of a more general type, can be expected to reveal a complex coupling between modes and exchange of energy between modes of different order. The phenomenon of generation of ultraharmonics and subharmonics, which has not been studied in this thesis, also needs to be investigated. The information gathered from the study of nonlinear oscillations of strings can serve as a useful guide to the analysis of more complex nonlinear systems.