Page 150 - Intro Predictive Maintenance

Vibration Monitoring and Analysis 141

dX
–KX –C
Spring dt

Mass Mass

F 0 Sin (wt )

Figure 7–14 Damped forced vibration system.

this equation, f is the phase angle, or the number of degrees that the external force,
F 0 sin(wt), is ahead of the displacement, X 0 sin(wt - f). Using vector concepts, the fol-
lowing equations apply, which can be solved because there are two equations and two
unknowns:
M
Vertical vector component: KX 0 - w 2 X 0 - F 0 cos f = 0
g c
Horizontal vector component: cX 0 - F 0 sin f = 0
w
Solving these two equations for the unknowns X 0 and f:

F 0
F 0 K
X o = 2 = 2 2
Ê M ˆ Ê w 2 ˆ Ê c w ˆ
2
c ( w ) + Á K - w 2 ˜ 1 - + 2 ¥
Ë g c ¯ Ë w 2 n ¯ Ë c c w n ¯
c w
2 ¥
cw c c w
tan f = M = 2 n 2
K - w 2 1 -(ww n )
g c
Where:
c = Damping constant

M
c c = Critical damping 2 w n
=
g c
c/c c = Damping ratio
F 0 = External force
F 0/K = Deflection of the spring under load, F 0 (also called static deflection, X st)
w = Forced frequency

w n = Natural frequency of the oscillation
w/w n = Frequency ratio