2010
Vibration Isolation
Experiment 2
Daniel Hammoud
309 185 106
Daniel Hammoud
309 185 106
AMME 2500 Engineering Dynamics LABORATORY
Experiment 2: Vibration Isolation
Objectives:
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
Aim:
1. To demonstrate the fundamental concepts of free and forced vibrations
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:
Part i: Spring Stiffness: Remove the spring from the engine-motor system and measure the
deflection when a static load is applied. Repeat for different known weights.
Part ii: Free vibration: Replace the spring. Using lab VIEW software set a time delay, number of
samples and samples per second. Running the software, lift up the engine component and
release at the same time. Repeat until an appropriate graph is attained
Part iii: 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 amplitude vs. frequency graph using lab
VIEW. From the graph determine at what frequency resonance occurs.
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Daniel Hammoud
309 185 106
Results:
Free Vibrations (using weights)
Unloaded Spring: x=0.0825m
Mass (Kg)
9.28
13.15
22.43
26.35
35.63
ΔX(m)
0.081
0.080
0.079
0.079
0.075
Force(N)
91.03
129.00
220.03
258.49
349.53
Spring Stiffness (K)
400
350
Force (N)
300
250
200
150
100
50
0
0.0000
0.0010
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
∆x (m)
[x-axis: deflection x, due to the load F]
[y-axis: weights of the loads]
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Daniel Hammoud
309 185 106
To find the stiffness of the spring then we use the formula below, which tells us that the stiffness of the
spring is equal to the constant load (F) over the deflection of the spring (x), (which is caused by the load
(F)) .
𝐾=
𝐹
𝑥
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,

inefficiencies of equipment,
 Placement of the spring on a non-level surface.
Therefore if we are to find the average value of the stiffness of the spring this could potentially give us
the closest value to the constant K. The average value of K can be found from the above graph, as it is
the gradient of the line of best fit.
Kaverage = 42551N/m
Free Vibrations (using Lab VIEW)
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Daniel Hammoud
309 185 106
For this report I chose graph 1 as my graph for the following clculations.
Calculations:
1. Logarithmic decrement, 
The logarithmic decrement was determined from equation 6 of the lab report handout, where x0
is the largest amplitude, xn is the smallest amplitude and n is the no. of oscillations between x0
and xn. The reason that decrement is calculated is simply to find the damping factor.
 = 0.1552
2. Damping factor, 
The damping factor was obtained through the use of equation 5, from the lab report handout,
where all values are constant and the logarithmic decrement is found to be 0.1552.
 = 0.0247
3. Damping constant, c
The damping constant c, can be obtained from equation 2, from the lab report handout.
c = 35.2147
4. Spring constant, K
The spring factor was already calculated earlier and obtained from the graph, as the average
value for K, which was given by the gradient of the line of best fit.
K = 42551 N/m
5. Natural system frequency,  n
The natural frequency can be calculated from equation 2 of the lab report handout, and can also
be determined by multiplying the frequency by 2
 n = 59.6903 rad/sec
6. Mass, M
The mass can be calculated from equation 2 of the lab report handout and is simply the spring
constant (42551N/m) over the natural system frequency (59.6903rad/sec) squared.
M = 11.9427 kg
7. Forced system frequency,
d
The forced system frequency, is obtained from equation 4, from the lab report handout, where
we need to have calculated the damping factor (0.0247) and the natural system frequency
(59.6903 rad/sec).
d = 59.6721 rad/sec
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Daniel Hammoud
309 185 106
Using excel the following graph was plotted:
Displacement vs Time
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Displacement(mm)
4
2
0
0
0.5
1
1.5
2
2.5
-2
-4
-6
Time (t)
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Daniel Hammoud
309 185 106
Discussion:
Spring Constant
The spring constant was found to be K = 42551 N/m. There are several errors in this, firstly the
measurements of the displacement of the spring was done with a standard home ruler and read with
the naked eye. Parallax errors and the ability to accurately measure displacements by the participants of
the experiment were not suffice to properly indicate the small increments of change in the spring. The
weights used were obviously worn and chipped, which would result in change of the true weight of the
loads to the weights specified on them. This shows the inaccuracies of the method used to take these
results. The measurements were only taken once, had they been taken more and averaged; more
reliable results would be achieved.
Free Vibration
It was interesting to note the need for us to take several attempts at this experiment in order to find a
graph that ‘was suitable for analysis’. Different time delays and heights that the motor was dropped
resulted in different values for the graph. Perhaps a standard height and time delay were necessary in
order to achieve more standard results, increasing the accuracy and reliability of the experiment. I also
found that the ‘dropping’ of the motor caused it bounce and vibrate, if only slightly, in a non vertical
direction, this will affect the one degree of freedom necessary to determine specific unknown values we
required. Our inexperience of what to expect and what was determined as a ‘good reading’ led to blind
acceptance of the results that were to be used for analysis. A greater understanding of the experiment
would greatly improve our ability to properly determine the unknowns required for this experiment, as
we would know what to calibrate and to an extent, expect. The y axis scale (amplitude) presented on the
lab view software jumped in increments of 2, requiring eyesight and judgment to determine the x0 and
xn values. Had more specific gridlines been available, more accurate results would be observed.
Forced Vibration
The graph of the forced vibration showed large peaks within a consistent wave form, allowing us tto
recognise its natural frequency to be approximately 22Hz. The frequency and amplitudes of the forced
vibration where 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
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Daniel Hammoud
309 185 106
seconds, corresponding the increase in the revs of the motor. This possibly indicats 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.
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