mech 391 instrumentation lab 9 vibration analysis of an aluminum cantilever beam performed: 03/15/04...
TRANSCRIPT
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] [mm] [mm] [kg] [Gpa] [m4]4.80 33.24 583.0 0.264 70 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 Length,L Weight,M Equivalent Mass,meq
No [mm] [kg] [kg]I 304 0.906 0.996II 304 0.226 0.316III 231 0.226 0.308IV 231 0.906 0.988V 353 0.678 0.773VI 353 1.132 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 [N/m] [Hertz] [Hertz] [%] [1] [kg/sec]I 2290 7.63 7.05 7.61 0.0031 0.2753II 2290 13.55 12.68 6.42 0.0060 0.3015III 5219 20.71 18.66 9.89 0.0216 1.5623IV 5219 11.56 10.73 7.22 0.0203 2.7054V 1463 6.92 6.49 6.24 0.0028 0.1742VI 1463 5.49 5.17 5.90 0.0023 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
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Figure 2 Acceleration vs. Time Plot of Exp#2
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Figure 3 Acceleration vs. Time Plot of Exp#3
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Figure 4 Acceleration vs. Time Plot of Exp#4
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Figure 5 Acceleration vs. Time Plot of Exp#5
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Figure 6 Acceleration vs. Time Plot of Exp#6
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Figure 7 Front Panel of the Vi Program
Figure 8 Block Diagram of the Vi Program