Page 87 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

74 Harmonically Excited Vibration Chap. 3

motion. From Eq. (3.7-2),
(3.8-2)
where must be evaluated from the particular type of damping force.
Example 3.8-1
Bodies moving with moderate speed (3 to 20 m/s) in fluids such as water or air are
resisted by a damping force that is proportional to the square of the speed. Determine
the equivalent damping for such forces acting on an oscillatory system, and find its
resonant amplitude.
Solution: Let the damping force be expressed by the equation

where the negative sign must be used when x is positive, and vice versa. Assuming
harmonic motion with the time measured from the position of extreme negative
displacement,
jc = —X cos cot
the energy dissipated per cycle is
= 2 f ax^ dx = laco^X^ f sin^cotd(cot)

= ^aco^X^
The equivalent viscous damping from Eq. (3.8-2) is then

The amplitude at resonance is found by substituting c = in Eq. (3.8-1) with
(O = CO'.
3^Fq
X =
Sacol
Example 3.8-2
Find the equivalent viscous damping for Coulomb damping.
Solution: We assume that under forced sinusoidal excitation, the displacement of the
system with Coulomb damping is sinusoidal and equal to x = X sin cot. The equiva­
lent viscous damping can then be found from Eq. (3.8-2) by noting that the work done
per cycle by the Coulomb force is equal to X 4X. Its substitution into Eq.
(3.8-2) gives
TTC^^coX^ = 4F^X

=
TTCOX