Vibration Isolation
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Introduction
The mechanical oscillations about a point of equilibrium are called vibrations. It occurs due to the response of a
restoring force by a body undergoing an applied force.
Free vibration experiences two forces; restoring and gravitational. The rate of transfer of energy between the body and
the restoring force is called the natural frequency, and is unique for each system. There is a damping force in the system
that decreases its motion over time.
Forced vibrations are caused by external forces that are external or internal, with its frequency dependent upon the
frequency of the external forces.
Aim
1. To demonstrate the fundamental concepts of free and forced vibration
2. To model a vibrating machine as a one degree-of-freedom vibrating system and to estimate the system
parameters from the way the amplitude of vibration varies as a function of shaft frequency.
Method:
a) Spring Constant
1.
2.
3.
4.
Remove the spring from the vibration machine.
Apply a load onto the spring and measure deflection.
Repeat for various weights.
Record the deflection.
b) Free Vibration Test
1. Replace the spring onto the vibrating machine.
2. Using the Lab software on the computer, set the time delay, samples per second and the number of sample you
want to read.
3. Pressing the start button on the computer you lift the rigid frame and release to obtain a graph on the
computer.
4. Repeat several times until an appropriate graph is obtained.
c) Forced Vibration
1.
2.
3.
4.
5.
Set minimum frequency on the lab computer,
Set maximum frequency;
Measure in increments of 5 Hz every 5 seconds,
Obtain amplitude vs. frequency graph using lab VIEW,
From the graph determine at what frequency resonance occurs.
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Results:
a) Spring Constant:
Mass (Kg)
Force(N)
Avg ΔX(m)
k
0
0
0
na
9.28
91.0368
0.002
45518.4
13.15
129.0015
0.00117
110258
13.2
129.492
0.00167
77540.1
Table 1: Data of Spring Stiffness measurements
𝑘𝑎𝑣𝑔 = -77772.1
Free Vibration Spring Constant
160
140
120
100
80
60
40
20
0
Free Vibration
Spring Constant
Linear (Free
Vibration
Spring
Constant)
0
0.002 0.00117 0.00167
Figure 1: Free Vibration Spring Constant
b) Free Vibration Test:
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Figure 2: Lab view of free vibration
Formula
Property
Calculated
Value
Mass
35.63 kg
Spring constant k
F/x
-77772.1 (N.m)
Total number of oscillations n
Obtained from figure 2
4
Total time t
Obtained from figure 2
0.35 sec
Obtained from figure 2
1.75
Obtained from figure 2
1
Maximum amplitude
Equation 2
Natural frequency
Logarithmic decrement
Damping factor ζ
Damping constant C
Damped frequency
δ
46.720 rad/s
Equation 6
0.1399
Equation 5
0.0222
Equation 2
Equation 4
73.9097
46.708
Table 2: Calculations
Using excel, the following function was plotted:
Displacement Vs Time
2
1
0.5
0
-0.5
-1
0
0.105
0.21
0.315
0.42
0.525
0.63
0.735
0.84
0.945
1.05
1.155
1.26
1.365
1.47
1.575
1.68
1.785
1.89
1.995
Displacement (mm)
1.5
-1.5
-2
Time (sec)
Figure 3: Excel generated image of free
vibration
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c) Forced Vibration Test:
Figure 4: Lab view of forced Vibration
From graph, resonance occurs at 23.24 Hz.
Resonance frequency (natural angular frequency) = max amp x gear ratio
= 23.24 x 2/5
= 9.296 rad/s
Discussion:
a) Spring constant:
Using the formula 𝐾 =
−𝐹
𝑥
we are able to determine the spring stiffness. However, the stiffness (K) for any given spring
should be a constant, however as we apply different loads to the spring, we witness different deflections of the spring,
this as a result changes the value of (K) and so the stiffness does not give us a constant. The reason that this occurs is
mainly due to experimental errors. These errors may include:



Incorrect measurements of the spring. Displacement taken with naked eye, making it difficult to get the correct
deflection as the change was minimal. From table 1, it could be seen that a mass of 9.28 kg gave a deflection of
0.002cm, whereas a heavier mass of 13.16 kg gave a less deflection. Theoretically a heavier mass should give a
larger deflection.
Inefficiencies of equipment. Weights were chipped, which could change the true mass of the weights. Masses
would have been weighed before addition onto spring.
Use of only 3 weights, making extrapolation less accurate. More weights could have been used to increase the
accuracy of deflection.
Therefore the average of the constant was used to make any such errors minimal in calculation.
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b) Free Vibration:

The graph obtained from the lab view were not very accurate as the y –axis jumped at increments of 2
rather than a more precise increment of 0.5 which would have made it easier to determine the values of
𝑥0 and 𝑥𝑛 more accurately. This would also have made the results more reliable.

The fact that we had very minimal knowledge about the topic may have caused us to blindly accepted
the graphs that were obtained. If we were more familiar with this topic, we would know what a good
graph would be and would continue to repeat the experiment until an appropriate graph was obtained.
c) Forced Vibration:
The graph of the forced vibration showed large peaks within a consistent wave form, allowing us to recognise its
natural frequency to be approximately 23.24Hz. The frequency and amplitudes of the forced vibration were
significantly smaller than the free vibration, as they responded to the vibrations of the revolving motor and its
parts. Once again the scale of the lab VIEW software was too large to be able to accuralty take readings from it.
It was interesting to note the small spikes in the graph every five seconds, corresponding the increase in the revs
of the motor. This possibly indicates the linear impulse of the piston that governs the motion of the motor, or
more precisly, the frequency matched the frequency of the external force of the motor, as you would expect in a
forced vibration. The results of the forced vibration should be significantly more accurate than the free
vibrations experiment as it was computer controlled, removing many human related errors.
References
J. L. Meriam, L. G. Kraige. Engineering Dynamics. Wiley (6th edition 2008)
Thomson, W.T. Theory of Vibration with Applications. Allen & Unwin (3rd edition 1998)
Rao, S.S. Mechanical Vibration. Addison-Wesley (3rd edition 1995)
C.E. Shock and Vibration concepts in Engineering Design. Prentice-Hall (1965)