Page 144 - Intro Predictive Maintenance

Vibration Monitoring and Analysis 135

Spring Static Deflection (X ) F s

Mass Mass

Weight (W )
Figure 7–12 Undamped spring-mass system.

vertically, which is illustrated in Figure 7–12. In this figure, the mass “M” is sup-
ported by a spring that has a stiffness “K” (also referred to as the spring constant),
which is defined as the number of pounds tension necessary to extend the spring
one inch.

The force created by the static deflection, X i, of the spring supports the weight, W, of
the mass. Also included in Figure 7–12 is the free-body diagram that illustrates the
two forces acting on the mass. These forces are the weight (also referred to as the
inertia force) and an equal, yet opposite force that results from the spring (referred to
as the spring force, F s).

The relationship between the weight of mass, M, and the static deflection of the spring
can be calculated using the following equation:

W = KX i

If the spring is displaced downward some distance, X 0, from X i and released, it will
oscillate up and down. The force from the spring, F s, can be written as follows, where
“a” is the acceleration of the mass:

Ma
F s =- KX =
g c
2
dX
It is common practice to replace acceleration, a, with , the second derivative of
dt 2
the displacement, X, of the mass with respect to time, t. Making this substitution, the
equation that defines the motion of the mass can be expressed as:

2
2
M dX M dX
=- KX or + KX = 0
g dt 2 g dt 2
c
c
Motion of the mass is known to be periodic. Therefore, the displacement can be
described by the expression: