Page 84 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.7 Energy Dissipated by Damping 71

the energy lost per cycle. The energy lost per cycle due to a damping force is
computed from the general equation

fV, = dx (3.7-1)
In general, depends on many factors, such as temperature, frequency, or
amplitude.
We consider in this section the simplest case of energy dissipation, that of a
spring-mass system with viscous damping. The damping force in this case is
F^ = cx. With the steady-state displacement and velocity
X = X s\n{(x)t - (¡))

X = (oX cos {(Ot —(f))
the energy dissipated per cycle, from Eq. (3.7-1), becomes

= (^xdx = (j)cx^ dt

= CO) [^^'"‘“cos^i cot - cf>)dt = TTCwX^ (3.7-2)
•'n
Of particular interest is the energy dissipated in forced vibration at resonance. By
substituting (x)^ = yjk/m and c = , the preceding equation at resonance
becomes
W. = 2^7rkX^ (3.7-3)

The energy dissipated per cycle by the damping force can be represented
graphically as follows. Writing the velocity in the form

ù)Xcos{ù)t - (f)) = ±(x)X\l —sin^(o)t - (jy)
= ± 0) ^ x ^ - ,
the damping force becomes
F^ = cx = ±co)^X^ - , (3.7-4)
By rearranging the foregoing equation to
, 2
(i)
+ 1 ^ ! 1 (3.7-5)
UwA' /
we recognize it as that of an ellipse with F^ and x plotted along the vertical and
horizontal axes, respectively, as shown in Fig. 3.7-l(a). The energy dissipated per
cycle is then given by the area enclosed by the ellipse. If we add to F^ the force kx
of the lossless spring, the hysteresis loop is rotated as shown in Fig. 3.7-l(b). This
representation then conforms to the Voigt model, which consists of a dashpot in
parallel with a spring.