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0593_C13_fm Page 449 Monday, May 6, 2002 3:21 PM
Introduction to Vibrations 449
where µ is:
µ= km (13.5.8)
In this case, the block moves to its static equilibrium position without oscillation. This
phenomenon is called critical damping.
Finally, suppose the damping constant is larger than critical damping, that is, larger
than 2 km . Then, from Eq. (13.5.4), we see that ω is imaginary and that the solution for
the displacement x of B becomes:
+
x = Ae −(µν t ) + Be −(µ −ν t ) (13.5.9)
where µ and ν are:
/
c 2 k 12
µ = cm and ν = 2 − (13.5.10)
2
4 m m
Observe that µ is always larger than ν and we have a relatively rapidly decaying motion
of B. This phenomenon is called overdamping.
13.6 Forced Vibration of a Damped Linear Oscillator
Consider next the forced vibration of a damped linear oscillator as depicted in Figure
13.6.1. From the principles of dynamics, we readily find the governing equation of motion
to be:
mx cx kx = () (13.6.1)
+
+
˙
˙˙
F t
where as before m is the mass of the block B, c is the viscous damping coefficient, k is the
linear spring constant, and F(t) is the forcing function. Suppose that as in Section 13.4 F(t)
has the periodic form:
Ft () = F cos pt (13.6.2)
0
c
k B
m
F(t)
FIGURE 13.6.1
Forced motion of a damped mass–spring x
system.