| dc.description.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. | |
