Ch 3.9: Forced Vibrations. We continue the discussion of the last section, and now consider the presence of a periodic external force:. Forced Vibrations with Damping. Consider the equation below for damped motion and external forcing funcion F 0 cos t .
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where the general solution of the homogeneous equation is
and the particular solution of the nonhomogeneous equation is
while in the second case
is a steady oscillation with same frequency as forcing function.
the transient solution uC(t) enables us to satisfy whatever initial conditions might be imposed.
can be rewritten as
depends on the driving frequency . For low-frequency excitation we have
where we recall (0)2 = k /m. Note that F0 /k is the static displacement of the spring produced by force F0.
where (0)2 = k /m. Note max < 0, and max is close to 0 for small . The maximum value of R is
where the last expression is an approximation for small . If
2 /(mk) > 2, then max is imaginary. In this case, Rmax= F0 /k,which occurs at = 0, and R is a monotone decreasing function of . Recall from Section 3.8 that critical damping occurs when 2 /(mk) = 4.
we see that Rmax F0 /( 0)for small .
which occurs at = 0.
is characterized by the equations
it follows that
sin[(0 + )t/2] oscillates more rapidly than sin[(0 - )t/2].
nearly equal frequency.
frequency of 0.9, slightly less than natural frequency 0 = 1.