Page 153 - Intro Predictive Maintenance

144 An Introduction to Predictive Maintenance

k 1

M 1
X 1
k 3
M 2
X 2
k 2

Figure 7–16 Undamped two-degrees-
of-freedom system with a spring
couple.

forced to act together. In this example, the movement of the two masses is limited to
the vertical plane and, therefore, horizontal movement can be ignored. As in the single-
degree-of-freedom examples, the absolute position of each mass is defined by its ver-
tical position above or below the neutral, or reference, point. Because there are two
coupled masses, two locations (i.e., one for M 1 and one for M 2 ) are required to locate
the absolute position of the system.

To calculate the free or natural modes of vibration, note that two distinct forces are
acting on mass, M 1 : the force of the main spring, K 1 , and that of the coupling spring,
K 3 . The main force acts upward and is defined as -K 1 X 1 . The shortening of the cou-
pling spring is equal to the difference in the vertical position of the two masses, X 1 -
X 2 . Therefore, the compressive force of the coupling spring is K 3 (X 1 - X 2 ). The com-
pressed coupling spring pushes the top mass, M 1 , upward so that the force is
negative.

Because these are the only tangible forces acting on M 1 , the equation of motion for
the top mass can be written as:

M 1 ˙˙
(
X 1 =- K X 1 - K X 1 - X 2 )
1
3
g c
or
M 1 ˙˙
)
X 1 + ( K 1 + K X 1 - K X 2 = 0
3
3
g c
The equation of motion for the second mass, M 2 , is derived in the same manner. To
make it easier to understand, turn the figure upside down and reverse the direction of
X 1 and X 2. The equation then becomes: