ME3112 E5 Lab Report
ME3112 E5 Lab Report
ME3112 E5 Lab Report
SEMESTER 5
SESSION 2013/2014
TAN ZI HAO
A0086885H
GROUP 3O1
4 September 2013
To introduce students to the use of certain measuring equipment commonly found in the
applied mechanics laboratories.
To use a cantilever beam as an object to illustrate the use of these equipments which include
strainmeter, accelerometer, shaker and real time analyzer.
Mode
Node Position
Theoretical (m)
Node Position
Experimental (m)
% Experimental
Error
Stroboscope
Frequency
(Hz)
4.53
0.398
0.371
6.78
26.87
26.85
28.41
0.240
0.235
2.08
75.97
75.40
79.55
0.430
0.413
3.95
0.170
0.170
0.306
0.305
0.33
151.63
151.50
155.89
0.440
0.431
2.05
257.66
Theoretical
Frequency (Hz)
Sample Calculations
Using mode 2 as an illustration,
E = 220GPa
L = 0.475m
b = 0.03m
t = 1.2 10-3 m
= 7903 kg/m3
I=
(
= 4.32 10-12 m4
m = bt
(
= 0.284508 kg/m
= 0.2845 kg/m (4 sig. fig.)
2 = 4.694
(
)( )
(
) (
)(
= 28.4069 Hz
= 28.41 Hz (4 sig. fig.)
Where a1 = 0.734
ai
1.00
for i 1
))
)
(
(
(
)
)
))
= 0.0518 m
Table 2:
0.00
Mode 1
0.000
Mode 2
0.000
Y
Mode 3
0.000
0.05
0.009
0.052
0.137
0.252
0.389
0.048
0.10
0.034
0.190
0.464
0.782
1.088
0.071
0.15
0.073
0.377
0.845
1.263
1.494
0.095
0.20
0.128
0.607
1.208
1.508
1.319
0.119
0.25
0.195
0.847
1.449
1.367
0.561
0.143
0.30
0.275
1.074
1.508
0.854
-0.443
0.166
0.35
0.361
1.261
1.373
0.140
-1.193
0.19
0.40
0.460
1.408
1.043
-0.630
-1.393
0.214
0.45
0.567
1.494
0.564
-1.203
-0.909
0.238
0.50
0.681
1.509
0.007
-1.410
0.022
0.261
0.55
0.797
1.455
-0.523
-1.208
0.911
0.285
0.60
0.922
1.327
-0.992
-0.640
1.397
0.309
0.65
1.052
1.131
-1.303
0.120
1.201
0.333
0.70
1.185
0.878
-1.410
0.845
0.417
0.356
0.75
1.314
0.592
-1.306
1.304
-0.531
0.38
0.80
1.451
0.265
-0.997
1.397
-1.260
0.404
0.85
1.589
-0.074
-0.533
1.070
-1.373
0.428
0.90
1.728
-0.407
0.014
0.421
-0.814
0.451
0.95
1.861
-0.703
0.538
-0.323
0.100
0.475
2.000
-0.972
1.001
-1.000
1.000
x (m)
x/L
0
0.024
Mode 4
0.000
Mode 5
0.000
0.1
0.2
0.3
0.4
0.5
-1.0000
-1.5000
0.1
0.2
0.3
0.4
-1.0000
-1.5000
-2.0000
0.5
0.1
0.2
0.3
0.4
0.5
-1.0000
-1.5000
-2.0000
From the graph, position of nodes are at x = 0.170m, x = 0.306m and x = 0.440m
Y against x/L
2.5
2
1.5
1
Mode 1
0.5
Mode 2
Mode 3
-0.5
0.1
0.2
0.3
0.4
0.5
Mode 4
Mode 5
-1
-1.5
-2
Discussion
a) What is resonance?
Resonance is the tendency of the system to oscillate at larger amplitude at some frequencies than at
others. The response amplitude is a relative maximum at the systems resonant frequencies. Since the
system stores vibrational energy, small periodic driving forces at these resonant frequencies are able
to produce large amplitude oscillations. Damping results in losses of energy from one cycle to the
next. With small damping effect, the resonant frequency is approximately equal to the natural
frequency of the system. Resonance phenomena occur in all types of vibrations and waves. For any
materials, there is infinite number of natural frequencies but in our experiment, we only measured 3
natural frequencies.
frequency of vibrations of atoms of fats, water and sugar, these atoms will vibrate with maximum
amplitude and maximum amount of energy is transferred to food to cook it.
Resonance also occurs in the basilar membrane in the cochlea of the ear. This enables people to
distinguish different frequencies or tones in the sounds they hear.
d) What is mode shape?
Mode shape describes the expected curvature or displacement of a surface vibrating at a particular mode.
Mode shape is usually sinusoidal and is dependent on the shape as well as the boundary conditions of the
surface. From the mode shape, we can find position of nodes. A node is a point where the amplitude of
vibration is zero. An anti-node is the opposite of a node, where the amplitude of vibration is a maximum. As
the number of nodes increase, the node positions become more compact since the distance between nodes are
shorter and the mode shape will include more sinusoidal waves within the fixed boundaries, where half a
sinusoidal wave is added for each increase in the number of nodes. A mode of vibration is characterized by a
modal frequency and a mode shape, and is numbered according to the number of half waves in the vibration.
Each mode is independent of all other modes. Thus all modes have different frequencies and different mode
shapes. Lower modes will have lower frequencies and greater amplitude. Since the lower modes vibrate with
greater amplitude, they cause the most displacement and stress in a structure. Thus they are called fundamental
modes.
e) Significance of mode shape in real life.
The study of mode shape is significant in real life because it allows us to determine the locations of nodes and
antinodes. This will assist engineers to make important design decision. For example, the locations of the
antinodes can be identified and shock absorbers can be installed at these locations to damp any oscillation.
This will reduce the amplitude of vibration and hence minimize of chances of any structure failure. Another
example where the knowledge of mode shape can be applied is to determine where to put vibration-sensitive
machineries. Any vibration will affect the operation of these machineries (e.g. wear and tear of gears) and
hence they should be placed at the locations of the nodes. Mode shape allows us to determine the positions of
the nodes.
f) Discussion of experimental results.
From the graph, we can observe that there is only one node for mode 2, two nodes for mode 3, three nodes for
mode 4 and four nodes for mode 5. It is also observed that at higher mode, the pitch of the sound becomes
higher.
In the experiment, mode 1 and mode 5 are not carried out. This is because at mode 1, the system will vibrate
with large amplitude and will damage the beam. Therefore, it is dangerous to carry out mode 1 in the
experiment. Mode 5 is not carried out in the experiment because it will be very difficult for us to see the
vibrations. As the mode gets higher, greater amount of energy is required to cause the vibration and more
energy will also be lost. Hence, the system will experience greater extent of damping and the amplitude of
vibration will decrease.
It can be seen that the CRO and stroboscope frequencies are always lower than the theoretical
frequency. This is because the transducer which is placed at the end of the ruler increases the mass of
the system. This will increase the damping of the system. From the formula
) ( ) , this
increase in mass will cause the experimental natural frequency to be lower than the theoretical natural
frequency.