Page 147 - Intro Predictive Maintenance

138 An Introduction to Predictive Maintenance

to the displacement. Using Newton’s equation of motion, where SF = Ma, the sum of
the forces acting on the mass can be represented by the following equation, remem-
bering that X is positive in the downward direction:

2
M dX Mg dX
= - c - KX h)
+
(
g dt 2 g c dt
c
2
M dX dX
= Kh c - KX - Kh
-
g dt 2 dt
c
2
M dX dX
=- c - KX
g dt 2 dt
c
M
Dividing by :
g c
2
dX cg dX Kg X
c
c
=- -
dt 2 M dt M
In order to look up the solution to the above equation in a differential equations table
(such as in CRC Handbook of Chemistry and Physics), it is necessary to change the
form of this equation. This can be accomplished by defining the relationships, cg c /M
2
= 2m and Kg c /M = w , which converts the equation to the following form:
2
dX dX 2
=- 2m -w X
dt 2 dt
Note that for undamped free vibration, the damping constant, c, is zero and, therefore,
m is zero.
2
dX 2
=-w X
dt 2
2
dX 2
=+w X = 0
dt 2
The solution of this equation describes simple harmonic motion, which is given as
follows:

X = A cos w t + B sin w t
()
()
dX
Substituting at t = 0, then X = X 0 and = 0, then
dt

X = X 0 cos(wt)

This shows that free vibration is periodic and is the solution for X. For damped free
vibration, however, the damping constant, c, is not zero.