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.
Apparatus:
The vibrating machine consists of a lawn mower engine driven by an electric motor, which is hinged
on one side allowing the other side to vibrate only in the vertical direction. The engine and the
motor are firmly attached to the frame. This apparatus creates a one degree of freedom (vibrating in
the vertical direction) thus leaving it with only one displacement variable. On the rigid frame, a
accelerometer that measures the number of oscillations of vibration.
Method:
Part 1: Spring Stiffness
1. Remove the spring from the vibration machine.
2. Apply a load onto the spring and measure deflection.
3. Repeat for various weights.
4. Record the deflection.
Part 2: Free Vibration
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.
Part 3: Forced vibration
1. Set minimum frequency on the lab computer,
2. Set maximum frequency;
3. Measure in increments of 5 Hz every 5 seconds,
1
4. Obtain amplitude vs. frequency graph using lab VIEW,
5. From the graph determine at what frequency resonance occurs.
Results: Part 1: Spring stiffness
To calculate the force applied on the spring by the applied mass is given by 𝑤 = 𝑚 × 𝑔 and 𝑘 =
𝐹/𝑥 is the equation used to find the stiffness.
Mass (kg)
Force or Weight
Deflection(cm)
k
0.333
3.26673
0(very little
0
unreadable)
9.63
94.4703
0.2
47235.15
13.533
132.7587
0.3
44252.91
22.813
223.7955
0.4
55948.88
26.683
261.7602
0.5
52352.05
35.963
352.797
0.55
64144.91
Average
52786.78
Table 1: Spring Stiffness measurements
Free Vibration Spring Constant
400
352.79703
350
300
261.76023
Force (N)
250
223.79553
200
150
132.75873
100
94.4703
50
0
3.26673
0
0.2
0.4
0.3
Displacement (cm)
Graph 1 : Free Vibration spring constant
2
0.5
0.55
Part 2: Free vibration
Graph 2: Lab View of free vibration

The stiffness of k is 52786.78

The number of oscillations (N) is 19,

Total time (t) is 2 seconds,

Maximum amplitude is 5 mm (𝑥0 )

Minimum amplitude is 0.4 mm (𝑥𝑛 )

Natural frequency is given by the equation 𝜔𝑛 =
𝑁
.
2𝜋𝑡
The Natural frequency is
59.69026rad/s
1
𝑥

Logarithmic decrement is calculated using 𝛿 = 𝑁 𝑙𝑛 𝑥 0 which equals 0.132933

Damping Factor is calculated using the formula 𝜁 =

𝑀=

Damping constant 𝐶 = 2 ζ

Damping frequency is calculated using ωd = 𝜔𝑛 √1 − 𝛿 2 = 59.67691 rad/s
𝑛
𝑘
𝜔𝑛 2
𝛿
√4𝜋2 +𝛿 2
= 0.021152
= 14.81557
M = 37.40336
3
Using excel the following graph was plotted:
Displacement vs Time
6
Displacement(mm)
4
2
0
0
0.5
1
1.5
-2
-4
-6
Time (t)
Graph 3: Displacement Vs Time
Graph 4: Lab View of Forced Vibration
4
2
2.5
Discussion
Part 1: Spring Stiffness
When calculated, the sprig stiffness was found to be 52786.78. However, there were many errors
during this experiment and are as follows
1. The measurements of the displacement of the spring was done with a standard home ruler
and read with the naked eye. This was very difficult to get the correct deflection as it was
such a small change. This caused the parallax error 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.
2. The weights were chipped and this could have changed the true mass of the loads.
3. If the experiemnt was repeated several times and the weights measured on a scale, the
results would be much more reliable.
4. When the load of 22.813 kg was applied it gave a value for k of 55948.88 which was higher
than that of the next load of higher weight. This would result as an outlier and in turn skew the
results, making the less reliable.
Part 2: Free Vibration
The fact that we had to repeat the experiment several times until we found a graph, in which values
can be drawn, was in itself a problem. If there was a fixed height at which the motor was dropped
and time delay set which would have resulted in different values and thus more standard results
could have been achieved. Other errors are as follow:
1. 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.
2. 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.
Part 3: Forced vibration
From the graph attained from the lab view, revealed that its natural frequency is approxiamtely 2223 Hz, and this was revealed by the large spike. Its was also interesting to note that there was a spike
every 5 second. This was caused by the increase of the motors revs. 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.
Again as mentioned in the free vibration discussion, the graph acquired in the force vibration was
also very inaccurate in terms of y-axis increments and again made it hard to read values. However it
is more accurate tha our free vibration as the whole experiment for forced vibration was conducted
by the computer, thus removing all human errors. This in turn will produce in more accurate results.
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