Page 86 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.8 Equivalent Viscous Damping 73

component of power, the average value of which is zero over any interval of time that
is a multiple of the period.

Example 3.7-2
A force F = lOsiuTr/ N aets on a displacement of x = Isiniirt - tt/6) m. D eter­
mine (a) the work done during the first 6 s; (b) the work done during the first \ s.
Solution: Rewriting Eq. (3.7-1) as IE = jFxdt and substituting F = sin ca/ and x =
X sin(ic>/ - (¡)) gives the work done per cycle of
W - 7tF()A sin (f)

For the force and displacement given in this problem, F„ = 10 N, = 2 m, 0 = 7t/ 6,
and the period r = 2 s. Thus, in the 6 s specified in (a), three complete cycles take
place, and the work done is

W = 3(^^F^^Xsm (/>) = Stt X 10 X 2 X sin 30° = 94.2 N • m
The work done in part (b) is determined by integrating the expression for work
between the limits 0 and \ s.

W cos s 30° i'^^sin 77/cos 77/(7/ + sin 30° i'^^sin^ 77/J/
A) A)
0.866 ^ ^ t sm 2 t7/ \
= TT X 10 X 2 — i------ c o s 2 t7 / + 0.50 ^ ----------- ^--------
4 t7 \ 2 4 t7 j
= 16.51 N • m

3.8 EQUIVALENT VISCOUS DAMPING
The primary influence of damping on oscillatory systems is that of limiting the
amplitude of response at resonance. As seen from the response curves of Fig.
3.1-3, damping has little influence on the response in the frequency regions away
from resonance.
In the case of viscous damping, the amplitude at resonance, Eq. (3.1-9), was
found to be

(3.8-1)

For other types of damping, no such simple expression exists. It is possible,
however, to approximate the resonant amplitude by substituting an equivalent
damping in the foregoing equation.
The equivalent damping is found by equating the energy dissipated by the
viscous damping to that of the nonviscous damping force with assumed harmonic