MECH 391 Instrumentation
Lab 9 Vibration Analysis of an
Aluminum Cantilever Beam
Performed: 03/15/04
Sinan Ozcan : I believe I performed 100% of this lab
ABSTRACT
• In this experiment, the Cantilever Beam configuration was modeled
as a simple spring-mass system. Then an equivalent system was
built.
• The theoretical natural frequencies were compared with the natural
frequencies obtained from the Data Acquisition System. The
percentage difference between those two values ranges between 5
to 10 %.
• For constant length, the natural frequency of the system increases
as the load decreases. For constant load, the natural frequency
decreases as length increases.
• The damping coefficient seems to have a strong relation with the
length of the beam under vibration. As length decreases, the
damping coefficient of the system increases.
• Since all the damping ratios are less than one, the system we
modeled was an under-damped system.
Table 1 Characteristics of an Aluminum
Cantilever Beam
Thickness,T
Width,W
Length , Lbeam
Mass of Beam,mbeam
Young's Modulus,E
Area Moment of Inertia,I
[mm]
4.80
[mm]
33.24
[mm]
583.0
[kg]
0.264
[Gpa]
70
[m4]
3.06E-10
•This table presents the measured values (T,W,L ,m beam) and calculated
values (E,I) of an Aluminum Cantilever Beam.
•Area Moment of Inertia of a beam which has a rectangular cross section
can be found from the equation: I = 1/12*W*T3.
Table 2 Experimental Data of the System
Experiment
No
I
II
III
IV
V
VI
Length,L
[mm]
304
304
231
231
353
353
Weight,M
[kg]
0.906
0.226
0.226
0.906
0.678
1.132
Equivalent Mass,meq
[kg]
0.996
0.316
0.308
0.988
0.773
1.227
•This Table presents the values of the variables in each configuration.
•Equivalent mass of the system was calculated from the relation
M+0.23m where m is the mass of the beam that was subjected to
vibration also M value includes the masses of the screw, nut and
accelerometer used during the experiment.
Table 3 Characteristics of an Aluminum
Beam under Vibration
Experiment Spring Constant,k Theor. Natural Frequency,fN Lab View Natural Frequency,fN Error in f N Damping Ratio, ζ Damping Coefficient, c
No
I
II
III
IV
V
VI
[N/m]
2290
2290
5219
5219
1463
1463
[Hertz]
7.63
13.55
20.71
11.56
6.92
5.49
[Hertz]
7.05
12.68
18.66
10.73
6.49
5.17
[%]
7.61
6.42
9.89
7.22
6.24
5.90
[1]
0.0031
0.0060
0.0216
0.0203
0.0028
0.0023
[kg/sec]
0.2753
0.3015
1.5623
2.7054
0.1742
0.1828
•Spring constant of the system increases dramatically as the length of the
aluminum beam that is under vibration decreases.
•Natural Frequency of the system is also sensitive to the changes in Length
and mass. The theoretical natural frequencies of the system are very close to
the actual natural frequencies obtained from the Data Acquisition system.
The percentage error is between 5 to 10 %. The error can be decreased by
improving the experimental set-up.
•Since all the Damping Ratios are less than one, the System we modeled is
an under-damped system.
Figure 1 Acceleration vs. Time Plot of Exp#1
0.3
Acceleration,a [mV]
0.2
0.1
0
0
0.5
1
1.5
2
-0.1
-0.2
-0.3
Time,t [sec]
2.5
3
3.5
4
Figure 2 Acceleration vs. Time Plot of Exp#2
1.5
Acceleration,a [mV]
1
0.5
0
0
0.5
1
1.5
2
-0.5
-1
-1.5
Time,t [sec]
2.5
3
3.5
4
Figure 3 Acceleration vs. Time Plot of Exp#3
1.5
Acceleration,a [mV]
1
0.5
0
0
0.2
0.4
0.6
0.8
1
-0.5
-1
-1.5
Time,t [sec]
1.2
1.4
1.6
1.8
2
Figure 4 Acceleration vs. Time Plot of Exp#4
0.8
0.6
Acceleration,a [mV]
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
-0.2
-0.4
-0.6
-0.8
Time,t [sec]
1.2
1.4
1.6
1.8
2
Figure 5 Acceleration vs. Time Plot of Exp#5
0.4
0.3
Acceleration,a [mV]
0.2
0.1
0
0
0.5
1
1.5
2
2.5
-0.1
-0.2
-0.3
-0.4
Time,t [sec]
3
3.5
4
4.5
5
Figure 6 Acceleration vs. Time Plot of Exp#6
0.3
Acceleration,a [mV]
0.2
0.1
0
0
0.5
1
1.5
2
2.5
-0.1
-0.2
-0.3
Time,t [sec]
3
3.5
4
4.5
5
Figure 7 Front Panel of the Vi Program
Figure 8 Block Diagram of the Vi Program