Page 97 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

84 Harmonically Excited Vibration Chap. 3

Figure 3.11-7.
(x)/(o^^ < 1. As shown in Fig. 3.11-2, when = 0.70, the phase for co/co^^ < 1 ean be
expressed by the equation
77 to
^ = 2 ^ ,
Thus, for = 0, or 0.70, the phase distortion is practieally eliminated.
Example 3.11
Investigate the output of an aeeelerometer with damping ^ = 0.70 when used to
measure a periodie motion with the displaeement given by the equation
> = Tj sin iOj/ + 72 sin CO2 1
Solution: For ^ = 0.70, (j) = tt/ 2 X co/co,^ so that c/)| = tt/ 2 X coj/co,, and 02 = 7r/2 X
CO/C0 ,,. The output of the aeeelerometer is then
2
Z = Z, sin ( CO r - 0 , ) + Z 2 sin (60 2^ - 02)
,
By substituting for Z, and Z 2 from Eq. (3.12-6), the output of the instrument is
1 r 2 ^r • I \ • I ^
z = —z cofy. sin CO r - ---- + co^y^ sm co^ / - —
,
c o l l ^ ’ T 2co,J - - 2co,,
Beeause the time funetions in both terms are equal {t - tt/ 2 co,0, the shift of both
eomponents along the time axis is equal. Thus, the instrument faithfully reproduces
the acceleration y without distortion. It is obvious that if 0 , and 02 are both zero, we
again obtain zero phase distortion.

P R OB L E MS

3-1 A machine part of mass 1.95 kg vibrates in a viscous medium. Determine the damping
coefficient when a harmonic exciting force of 24.46 N results in a resonant amplitude
of 1.27 cm with a period of 0.20 s.
3-2 If the system of Prob. 3-1 is excited by a harmonic force of frequency 4 cps, what will
be the percentage increase in the amplitude of forced vibration when the dashpot is
removed?
3-3 A weight attached to a spring of stiffness 525 N /m has a viscous damping device.
When the weight is displaced and released, the period of vibration is 1.80 s, and the