A model of a string oscillating in its fundamental mode is presented in Figure 3.3 and consists of a single mass fastened to the central axis by a pair of linearly elastic springs [7]. Although the springs provide a linear restoring force, the resulting force toward the origin is nonlinear because of the geometric configuration.
The ends of the massless
springs are fixed a distance l apart where the relaxed length of the spring is
and the spring constant is k. In the center a mass
is attached that is free to make oscillations in the x-y
plane centered at the
origin.
The
motion in the two transverse directions, x and y, is coupled directly,
and also indirectly, via the longitudinal motion of the spring. Both of these
coupling mechanisms are nonlinear. The multimode extension of this
single-mode model would consist of n masses hooked together by n+1 springs.
The restoring force on the mass shown in Figure 3.3 is
where the position of the mass is given by polar coordinates 
of
the transverse plane (see Prob. 3.8).
Expanding the right-hand side of equation
(3.2) in a Taylor series
(2r ;SPMlt; l), we find that
The force can be written as
so
Note the cubic restoring force.
Also note that nonlinearity dominates when 
.
That is, the nonlinear effects are accentuated
when the string's tension is low.
Define
and
Then from equation (3.3) we get, because of symmetry in the angular
coordinate, the vector equation for 
,
which is
the equation of motion for a two-dimensional conservative cubic
oscillator.
The behavior of equation (3.6) depends critically upon the ratio 
. If 
, the coefficient of the nonlinear term, K, is
positive,
the
equilibrium point at r = 0 is stable, and we have a model for a string
vibrating primarily in its fundamental mode. On the other hand,
if 
,
then K is negative, the
origin is an unstable equilibrium point, and two stable equilibrium points exist
at approximately 
. This latter case
models the motions of a single-mode elastic beam [8]. For our purpose
we will mostly be concerned with the case , or K ;SPMgt; 0.
In general, we will want to consider damping and forcing, so equation (3.6) is modified to read
where 
is a periodic forcing term and 
is the
damping coefficient. Usually, the forcing term is just
a sinusoidal function applied in one radial direction, so that
it takes the form
. For simplicity, we have assumed that the
energy losses are linearly proportional to the radial velocity of the string,
. We also assumed that
the ends of the string are symmetrically fixed, so that
is a scalar. In general, the damping rate depends on the radial direction,
so the damping term is a vector function. This is the case,
for instance, when a string is strung over a bridge that breaks
the symmetry of the
damping term.
Equation (3.7) was also derived by Gough [2] and Elliot [9], both of whom
related 
and K to actual string parameters that arise in experiments.
For instance, Gough showed that the natural frequency is given by
and the strength of the nonlinearity is
where
is the longitudinal extension of a string of equilibrium length l,
is the low-amplitude angular frequency of free vibration, and
c is the transverse wave velocity.
Again, we see that the nonlinearity parameter, K, increases as the
longitudinal extension, , approaches zero.
That is, the nonlinearity is enhanced when the longitudinal extension--and
hence the tension--is small. Nonlinear effects are also amplified when the
overall string length is shortened, and they are easily observable in common
musical instruments.
For a viola D-string with a
vibration amplitude of 1 mm, typical values of the string parameters showing
nonlinear effects are: l = 27.5 cm, 
Hz, 
mm,
K = 0.128 
[2].
Equation (3.7) constitutes our single-mode model for nonlinear string vibrations and is the central result of this section. For some calculations it will be advantageous to write equation (3.7) in a dimensionless form. To this end consider the transformation
which gives
where the prime denotes differentiation with respect to 
and
Before we begin a systematic investigation of the single-mode model it is
useful to consider the unforced linear problem, 
.
If the nonlinearity parameter K is zero, then equation (3.7) is simply a
two-degree of freedom linear harmonic oscillator with
damping that admits solutions of the form
where 
and 
are the initial amplitudes in the x and y directions.
Equation (3.13) is a solution of (3.7) if we discard second-order
terms in .
In the conservative limit ( 
), the orbits are ellipses centered
about the z-axis. As we show in section 3.7, one effect of the
nonlinearity is to cause these elliptical orbits to precess. The trajectories of
these precessing orbits resemble Lissajous figures, and these precessing
orbits will be one of our first examples of quasiperiodic motion on a torus
attractor.
