ME3112E Lab 1
Vibration Measurement LIN SHAODUN A0066078X by Group Date
1A 11-Sept-2012
TABLE OF CONTENTS
O BJECTIVE
1
E XPERIMENT D ATA
1
S AMPLE C ALCULATION
3
D ISCUSSION
4
C ONCLUSION
6
O BJECTIVES 1. To familiarize with the techniques in measuring dynamic quantities as well as using the related equipment such as Accelerometer, Shaker, Function generator and Stroboscope. 2. To determine the resonance frequencies and the corresponding mode-shapes of a vibrating beam with several different techniques. E XPERIMENT D ATA Table 1a Vibration Measurement
Mode
Theoretical Natural Frequency (Hz)
Position of Nodes (m)
Experimental Natural Frequency (Hz)
Stroboscope
CRO (Hz)
Cycles per min
Hz
Experimental
Theoretical
1
4.53
-
-
-
-
-
2
28.41
27.17
1638
27.30
0.375
0.394
3
79.55
76.34
4569
76.15
0.245,0.411
0.238,0.428
4
155.89
152.7
9138
152.3
0.175,0.315,0.435
0.166,0.308,0.442
Frequency (Hz)
200 150
Theoretical Natural Frequency Oscilloscope Stroboscope
100 50
155.89 152.7 152.3
79.55 76.34 76.15 28.41 27.17
27.3
0 Mode 2
Mode 3
Mode 4
Table 1b Experimental error comparison Theoretical Mode Natural Frequency (Hz)
2 3 4
28.41 79.55 155.89
Experimental Natural Frequency (Hz) CRO 27.17 76.34 152.7
Err (%) Stroboscope Err (%) 4.36% 4.04% 2.05%
27.30 76.15 152.3
3.91% 4.27% 2.30%
Position of Nodes (m) Experimental
Theoretical
Err (%)
0.375 0.245,0.411 0.175,0.315,0.435
0.399 0.238,0.427 0.170,0.305,0.441
-6.34% -1.48% 0.91%
From above table one can see that the experimental result is quite close to theoretical value.
1
Table 2 Mode shape calculation Modes x
x/L
0.0000
Mode1
Mode2
Mode3
Mode4
0.0
0.00000
0.00000
0.00000
0.00000
0.0475
0.1
0.00749
0.18589
0.45604
0.77008
0.0950
0.2
0.02931
0.60719
1.20804
1.50771
0.1425
0.3
0.06448
1.06951
1.50896
0.86804
0.1900
0.4
0.11201
1.40827
1.04279
-0.62996
0.2375
0.5
0.17094
1.50981
0.01897
-1.41012
0.2850
0.6
0.24028
1.32690
-0.99163 -0.64035
0.3325
0.7
0.29870
0.88332
-1.41012 0.83205
0.3800
0.8
0.37596
0.26515
-0.99732 1.39688
0.4275
0.9
0.45795
-0.40007
0.00215
0.43655
0.4750
1.0
0.54341
-0.97229
1.00141
-1.00041
0.6
2.0 Amplitude
Amplitude
Mode 1 0.4 0.2
x/L 0.0
1.0
x/L
-2.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2.0
2.0
Mode 3
1.0
Amplitude
Amplitude
Experimental Node Position
-1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
-1.0
Mode 2
0.0
Experimental Node Position
0.0
-1.0
x/L
-2.0
Mode 4
1.0 Experimental Node Position
x/L
-2.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2
S AMPLE C ALCULATION 1. Theoretical Natural Frequency, for Mode 4: √
√
2. Amplitude of Vibration, for Mode 4 and x/L=0.9: ( (
)
) (
)
[
(
)
(
)]
3. Node position, for Mode 4: (
)
3
D ISCUSSION 1. Discuss the significance of the resonant frequencies, modes shapes and the effect of accelerometer's mass on these quantities. Significance of the resonant frequencies: Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, Engineers must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster. (Source: http://en.wikipedia.org/wiki/Resonance) For industrial application, engineers usually need to design the system structure in higher resonance frequencies to reduce vibration level. For example, by changing the linear bearing of an assembly from cross roller to needle bearing, the resonance frequencies of the assembly significantly improved so does the vibration level.
4
Significance of the mode shape: For mechanical engineers, mode shapes are useful because they represent the shape that the structure will vibrate in free motion. By study the mode shape, it is possible to find out what is the weakest link in the structure so that engineer knows how to improve the structure to reduce vibration level. Mode shape predict by FEA software also tells how the structure behaves during vibration. Below is an example of FEA study of mode shape.
Effect of accelerometer's mass In the experiment, the accelerometer's mass affects the resonant frequencies of a one-end fixed beam by 2~4% ( See Page 1). The resonant frequency of a system is described as
√ ,
from this equation one can see that the accelerometer only contribute mass to the system without improve the system stiffness, so the frequency will be reduced. Hence the resonant frequency measured in experiment is smaller than the theoretical calculation. As for mode shape, the accelerometer’s mass will reduce the amplitude of vibration as it acting as a mass damper to dissipate the stored system energy. It may change the position of node and antinode as well as the mass distribution of the beam is different from initial assumption. 2. Discuss what is Node and Anti-Node, and the significance of Node and AntiNode in industrial application. A node is a point along a standing wave where the wave has minimal amplitude. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is a maximum. These occur midway between the nodes. (Source: http://en.wikipedia.org/wiki/Node_(physics) ).
5
The illustration of node and anti-node is as follow:
The significance of node and anti-node in industrial application : When install precision equipment which is sensitive to vibration in a building, it is advisable to study the mode shape of the floor and identify the location of nodes, the equipment should be installed on the nodes to minimize the effect from vibration. While in the case the vibration needs to be amplified, for example a musical instrument, the structure can be modified so that the sound generation component is located near to anti-node. C ONCLUSION
After completed this experiment, I have a better understanding about mechanical system vibration , resonance frequency and the mode shape. I also learned how to use accelerometer and stroboscope to measure the resonance frequency of the beam at different modes, and gained hands on experience on these techniques in measuring dynamic quantities.
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