Page 149 - Intro Predictive Maintenance

140 An Introduction to Predictive Maintenance

In undamped forced vibration, the only difference in the equation for undamped free
vibration is that instead of the equation being equal to zero, it is equal to F 0 sin(wt):
2
M dX
()
+ KX = F 0 sin w t
g dt 2
c
Because the spring is not initially displaced and is “driven” by the function F 0 sin(wt),
a particular solution, X = X 0 sin(wt), is logical. Substituting this solution into the above
equation and performing mathematical manipulations yields the following equation
for X:

X st
X = C 1 sin (w n t) + C 2 cos (w n t) + 2 sin ()
t w
1 -(ww n )
where:

X = Spring displacement at time, t

X st = Static spring deflection under constant load, F 0
w = Forced frequency
w n = Natural frequency of the oscillation
t = Time
C 1 and C 2 = Integration constants determined from specific boundary
conditions

In the above equation, the first two terms are the undamped free vibration, whereas
the third term is the undamped forced vibration. The solution, containing the sum of
two sine waves of different frequencies, is not a harmonic motion.

Forced Vibration—Damped. In a damped forced vibration system such as the one
shown in Figure 7–14, the motion of the mass “M” has two parts: (1) the damped free
vibration at the damped natural frequency and (2) the steady-state harmonic motions
at the forcing frequency. The damped natural frequency component decays quickly,
but the steady-state harmonic associated with the external force remains as long as
the energy force is present.

With damped forced vibration, the only difference in its equation and the equation for
damped free vibration is that it is equal to F 0 sin(wt) as shown below instead of being
equal to zero.

2
M dX dX
()
+ c + KX = F 0 sin w t
g dt 2 dt
c
With damped vibration, the damping constant, “c,” is not equal to zero and the solu-
tion of the equation becomes complex assuming the function, X = X 0 sin(wt - f). In