If we pluck a string hard and look closely, we typically see the string whirling around in an elliptical pattern with a diminishing amplitude. Some understanding of these motions is obtained by considering the free planar oscillations of a string modeled by the two-dimensional equation
which is equation (3.7) with no forcing term. We noted in section 3.3 that the linear approximation to equation (3.7) results in elliptical motion (eq. (3.13)). We shall use this observation to calculate an approximate solution to equation (3.49) using a procedure put forth by Gough [2]; similar results were obtained by Elliot [9].
Transform the problem of nonlinear free vibrations to a reference frame
rotating with an angular frequency 
, with to be determined. In
this rotating frame, equation (3.49) becomes
where 
is the new radial displacement vector, subjected to the addition
of Coriolis and centrifugal accelerations.
Let us now look for a solution of the form
where 
and 
are small compared to 
and 
.
Looking at the x coordinate only, when we substitute equation
(3.51) into equation (3.50) and discard appropriate higher-order
terms, we get
a similar relation holds for the y coordinate. On equating sinusoidal terms of the same frequency we--after considerable algebra--discover
and
If there is no damping ( 
), then this approximate solution is
periodic in the rotating reference frame and is slightly distorted from an
elliptical orbit. The angular frequency 
is detuned from
by an amount proportional to the mean-square radius
.
In the original stationary reference frame, equation (3.53) shows us that
the orbit precesses at a rate proportional to the orbital
area 
. The angular frequency in equations
(3.52) to (3.54) is measured in the
rotating reference frame. It is
related to the angular frequency in the stationary reference frame 
by
Thus, in the stationary reference frame, the undamped motion is quasiperiodic
unless and are accidentally commensurate,
in which case the orbit is periodic.
The damped oscillations are also elliptical in character and precess at a rate
. In both cases, the detuning given by equation (3.55) is due to
two sources: the nonlinear planar motion detuning plus a detuning resulting from
the precessional frequency.
