Page 21 - Rotating Machinery Pratical Solutions to Unbalance and Misalignment

Rotating Machinery: Practical Solutions

For a simple mass spring system that vibrates within the elas-
tic limits of the spring, Hooke’s Law applies and is expressed as:

F = –kx (1.5)

Where:
k= the spring constant
x= the distance from the rest position

The minus sign in this equation indicates that the force of the
spring is acting opposite to the direction in which the spring is
stretched.
If the spring were stretched from x = 0 to x = x work is done
0
which is equal to the average force times the distance moved. The
average force is half the maximum force and the distance is x .
0
Therefore, the energy in the spring is:

PE = (1/2k x )(x ) = 1/2k x 2 (1.6)
0 0 0

The sum of the potential energy [PE] and the kinetic energy
[KE] must remain constant and equal to the maximum potential
2
energy, 1/2k x . Kinetic energy is the energy due to motion and
0
2
is expressed as 1/2 mv . Thus expressed in equation form:
PE + KE = 1/2k x 2 (1.7)
0
Substituting:

2
2
1/2k x + 1/2 mv = 1/2k x 2 (1.8)
0
Or:
2
2
mv = k(x 2 - x ) (1.9)
0
Now substituting from Equation (1.1) into Equation (1.5):
–kx = ma (1.10)
Or:
a = –(k/m)× (1.11)