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Vibration Isolation
Free Vibration and Forced Vibration
Introduction
Vibration is the mechanical oscillations about an equilibrium point. Vibration occurs when a body
responds to applied forces with a restoring force. In most mechanical structures vibration is
undesirable as it wastes energy and creates noise. Vibration is generally caused by unbalanced
rotating fixtures or uneven friction. In this case the system being analysed is a lawnmower engine
that is driven by an electric motor. The vibration is caused by unbalanced rotating components in the
engine and friction from air being forced by the piston through the inlet and exhaust valves. The
system is hinged at the base on one side, allowing vibration in the vertical direction only. Having one
degree of freedom allows for the system to be reduced to a discrete model with one displacement
variable.
Background
Free vibration is when a spring mounted body is disturbed from its equilibrium position. The system
is controlled by 2 forces; gravity and a restoring force. Free vibration occurs at a natural frequency
which is the rate of transfer of energy between the body and the spring. The natural frequency is
unique for each system. In every free vibration case there is a retarding or damping force that
diminishes the motion.
Forced vibrations are caused by external forces or are generated internally. The frequency of the
vibration depends on the external forces.
Vertical acceleration of the system is measured using a piezo-resistive accelerometer. Lab VIEW
software is used for data analysis.
Aim
The aim of this experiment is to model a vibrating system so that an understanding of free vibration
and forced vibrations is gained. Also to reduce a vibrating system to a one degree of freedom model
so that the system’s parameters may be estimated. Under free vibration various system properties
can be calculated to give the amplitude of vibration as a function of shaft frequency.
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Method
1. Remove the spring from the engine-motor system and measure the deflection when a static
load is applied. Do this at least 3 times and repeat for various different loads. Plot the load F
against the deflection 𝑥 to get the spring constant k using 𝐹 = 𝑘𝑥.
2. Free vibration:
Replace the spring. Using the lab VIEW software set a time delay, number of samples and
samples per second. Run the software, lift up the engine component and release at the
same time. Repeat until a useable plot is obtained.
3. From the plot determine the total number of oscillations N, the total time t, the maximum
amplitude 𝑥0 and minimum amplitude 𝑥𝑛 .
4. Forced vibration:
Set the minimum frequency at 10 Hz and the maximum frequency at 50 Hz. Measure in
increments of 5 Hz every 5 seconds. Obtain an amplitude vs. frequency graph using lab
VIEW. From the graph determine at what frequency resonance occurs.
Results and Calculations
Spring Stiffness
Figure 1: Spring Stiffness Data
weight (kg)
average spring length (mm)
X deflection (m)
F weight.gravity (N)
0
83.0667
0.0000
0
9.28
80.7233
0.0023
91.0368
13.15
80.2033
0.0029
129.0015
22.78
79.5233
0.0035
223.4718
26.35
79.2200
0.0038
258.4935
35.63
78.2800
0.0048
349.5303
Figure 2: Spring Constant Graph
Free Vibration -Spring Constant
400
y = 73244x - 36.947
R² = 0.9163
Force F (N)
300
200
Series1
100
0
0.0000
-100
Linear (Series1)
0.0010
0.0020
0.0030
Deflection x (m)
0.0040
0.0050
0.0060
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Figure 3: Amplitude vs. Samples graph from lab VIEW
Figure 4: System Values
Property
Description
Formula
Calculated
Value
Spring constant k
Linear spring stiffness.
Total number of
Measured from peak to peak of the
oscillations N
Lab View Free vibration data graph.
Total time t
The total time of the experiment.
1.42 s
Maximum amplitude
The maximum amplitude observed
5.0m
𝑥0
from the Lab View Free vibration
𝐹
𝑥
73244
14
data graph.
Minimum amplitude
The minimum amplitude observed
𝑥𝑛
from the Lab View Free vibration
1.0
data graph.
Natural frequency
The unique frequency of the free
𝑁
2π 𝑡
61.95 rad/s
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𝜔𝑛
vibrating system.
Logarithmic
decrement
δ
1 𝑥0
ln
𝑁 𝑥𝑛
0.1150
𝛿
0.01829
Damping factor ζ
√4𝜋 2 + 𝛿 2
Damping constant
2 ζ𝜔𝑛 M
C
(𝑀 =
Damped frequency
𝑘
𝜔𝑛 2
𝜔𝑛 √1 − 𝛿 2
𝜔𝑑
Using Matlab, the following graph of equation x(t) was made.
𝑥(𝑡) = 𝑥0 𝑒 −𝜁𝜔𝑛 𝑡 [𝑐𝑜𝑠𝜔𝑑 𝑡 +
𝜁𝜔𝑛
𝑠𝑖𝑛𝜔𝑑 𝑡]
𝜔𝑑
Figure 5: Motion of Free Vibration Displacement vs. Time Graph
43.26
)
61.54 rad/s
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Figure 6: Amplitude vs. Frequency graph from lab VIEW
The 2 large peaks are caused by resonance in the system. Resonance occurs when the natural
frequency of the system is equal to the frequency caused by external forces.
Discussion
Spring Stiffness (k)
The spring stiffness was found to be 73244 N/m, however there were numerous errors encountered
while trying to measure the spring stiffness. These were manual errors due to the nature that the
measurements were taken. The accuracy in measuring the spring deflection is reduced when
alligning the calliper as we rely on the asccuracy of our eye sight. We reduce this margin of error by
taking multiple readings of each static load.
Free Vibration
Free Vibration is when a spring at equilibrium is disturbed without the effect of external forces. In
the experiment we were able to model a system of free vibration with one degree of freedom.
During the course of the experiment various errors were encountered including human error in the
actual lifting of the system. Each time we repeated the experiment we were not able to replicate
exactly the height the system was lifted. Also the time delay of the software means that
measurements are not taken at the axact same moment. However the time delay also allows us to
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gain a clearer reading of the amplitude as the piezo-resistive accelorometer has a maximum
amplitude that can be measured. The time delay means that excessive amplitudes are ruled out.
Another error occurs when we get the maximum and minimum amplitude values from the graph, we
rely on our eye sight once again.
Forced Vibration
Forced vibration is caused by an external force on a system. In the forced vibration experiment the
electric motor causes the motor to vibrate. The amplitude of this vibration was recorded in the lab
VIEW software and analysed. We can see from figure 6 that particular frequencies from the engine
are the same as the natural frequency of the system . This is wat causes the large peakes in the
amplitude and resonance occurs.
Looking at free vibration and forced vibration comparatively we can see that forced vibration gives
us a more accurate reading as there is less human error involved. The graph obtained (Figure 5) from
calculated data shows a clearer reading of displacement vs. time.
Conclusion
We were able to successfully model a system of vibration with one degree of freedom. Under free
vibration we were able to perform various calculations giving us the natural frequency and damping
frequency. We were also able to graph the displacement of the system vs. Time, giving us a plot that
decreases at a more uniform rate than the amplitude vs. Samples graph. From the forced vibration
section we were able to obtain a clearer and more accurate reading.
References
http://www.thermotron.com/resources/vibration_handbook.html, accessed 19/03/10
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)