For a very small forcing amplitude the string responds with a linear resonance , such as that illustrated in Figure 3.9.
Figure 3.9: Response curve for a harmonic oscillator.
According to linear theory, the response of
the string is maximum when 
.
In other words, it is maximum
when the forcing frequency 
exactly equals the natural frequency
.
A primary
(or main) resonance
exists when the natural frequency and the
excitation frequency are close.
The resonance diagram (Fig. 3.9) is called a linear response
because it can be obtained by solving the periodically forced, linearly
damped harmonic oscillator ,
which has a general solution of the form
The constants 
and 
are initial conditions.
Equation (3.32) is a solution to equation (3.31) if we discard
higher-order terms in 
. The maximum amplitude of x, as a function of
the driving frequency 
, is found from the asymptotic solution of
equation (3.32),
which produces the linear response diagram shown in Figure 3.9, since
After the
transient solution dies out, the steady-state response has the same frequency
as the forcing term, but it is phase shifted by an amount 
that
depends on , , and F. As with all damped linear systems, the
steady-state response is independent of the initial conditions so that we can
speak of the solution.
In the linear solution, motions of significant amplitude
occur when F
is large or when 
.
Under these circumstances the nonlinear term in
equation (3.28) cannot be neglected. Thus, even for planar motion, a
nonlinear model of string vibrations may be required when a resonance occurs
or where the excitation
amplitude is large.
