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Transients
We saw, in Section 3.7, that when a onedimensional dynamical system,
close to a stable equilibrium point, is subject to a sinusoidal external
force of the form (104) then the equation of motion of the
system is written

(132) 
We also found that the solution to this equation which oscillates in sympathy
with the applied force takes the form

(133) 
where and are specified in Equations (110) and (111), respectively.
However, (133) is not the most general solution to Equation (132).
It should be clear that we can take the above solution and add to it
any solution of Equation (132) calculated with the righthand side set to zero,
and the result will also be a solution of Equation (132).
Now, we investigated the solutions to (132) with the righthand
set to zero in Section 3.5. In the underdamped regime (),
we found that the most general such solution takes the
form

(134) 
where and are two arbitrary constants [they are in fact the integration constants
of the secondorder ordinary differential equation (132)],
and
. Thus,
the most general solution to Equation (132) is written

(135) 
The first two terms on the righthand side of the above equation
are called transients, since they decay in time. The transients
are determined by the initial conditions. However, if we wait
long enough after setting the system into motion then the transients will
always decay away, leaving the timeasymptotic solution (133), which is independent of the
initial conditions.
Figure 8:
Transients.

As an example, suppose that we set the system into motion at time
with the initial conditions
.
Setting in Equation (135), we obtain

(136) 
Moreover, setting in Equation (135), we get

(137) 
Thus, we have now determined the constants and , and, hence,
fully specified the solution for . Figure 8
shows this solution (solid curve) calculated for
and
. Here,
. The associated timeasymptotic solution (133)
is also shown for the sake of comparison (dashed curve). It can
be seen that the full solution quickly converges to the timeasymptotic
solution.
Figure 9:
A simple pendulum.

Next: Simple Pendulum
Up: OneDimensional Motion
Previous: Periodic Driving Forces
Richard Fitzpatrick
20110331