Page 145 - Intro Predictive Maintenance

136 An Introduction to Predictive Maintenance

()
X = X 0 cos w t
Where:

X = Displacement at time t
X 0 = Initial displacement of the mass
w = Frequency of the oscillation (natural or resonant frequency)
t = Time

If this equation is differentiated and the result inserted into the equation that defines
motion, the natural frequency of the mass can be calculated. The first derivative of
the equation for motion yields the equation for velocity. The second derivative of the
equation yields acceleration.

dX
˙
()
Velocity = = X = -w X sin w t
0
dt
2
dX
˙˙
Acceleration = = X = -w 2 X cos w t
()
0
dt 2
2
dX
Inserting the expression for acceleration, or , into the equation for F s yields the
dt 2
following:
2
M dX
+ KX = 0
g dt 2
c
M
w
- w 2 X 0 cos () KX = 0
t +
g c
M M
+
- w 2 XKX = - w 2 + K = 0
g c g c
Solving this expression for w yields the equation:
Kg c
w =
M
Where:
w = Natural frequency of mass
K = Spring constant
M = Mass

Note that, theoretically, undamped free vibration persists forever; however, this never
occurs in nature, and all free vibrations die down after time because of damping, which
is discussed in the next section.