Mechanical Vibrations
Mechanical Vibrations
(MEE 321)
(MEE 321)
SDOF Free Vibration of Cantilever Beam
SDOF Free Vibration of Cantilever Beam
Date: 05 March 2014 Date: 05 March 2014 PERFORMED BY PERFORMED BY NORA GUZMAN NORA GUZMAN SOLAMAN MASON SOLAMAN MASON ZAK WHITMAN ZAK WHITMAN AND AND SUBMITTED SUBMITTED BY BY NORA GUZMAN NORA GUZMAN
Objective
Analyze the free vibration of a SDOF free vibration cantilever beam. To obtain the natural frequency of the cantilever beam and the damped frequency, and accelerometer is used. The accelerometer will help obtain the experimental results of the cantilever vibration. The natural and damped frequency obtained experimentally will be compared to the frequencies obtained theoretically.
Theory
A cantilever beam can be analyzed as a spring mass system with a single degree of freedom. It is considered a single degree of freedom due to it only moving in one coordinate. The equation for natural frequency is the following, where k eq is the stiffness of the beam, and meq is the equivalent mass of the beam.
In order to find the natural frequency and damped frequency of the beam, a measure of stiffness is needed. The equation for the stiffness of a cantilever be am is obtained from basic strength of materials concepts. Where E is the modulus of elasticity, I is the moment of inertia, and L is the effective length of the cantilever beam.
Usually mass is ignored and taken as being equal to zero. By ignoring mass, only approximate solutions can be obtained. To obtain more exact results ordinary differential equations that govern the motion of the cantilever beam can be used. This method is called continuous system dynamics. The total mass of the beam and block where m blockis the mass of the block and m beam is the mass of the beam is:
No damping will be applied to the cantilever beam during free vibration, but in real world scenarios, damping from the surroundings will be taken into considerations. The damped
frequency can be obtained from (ωn), the natural frequency, and the damping ratio (ζ).
By analyzing displacement over time of a SDOF damped system, δ can be obtained to be used in the equation for ζ. In the following equation x1is the measure of amplitude near time t1 and xn+1 is the measure of amplitude near time to tn+1and τdis the period of a damped frequency.
By algebraically manipulating the equation for δ we can then obtain a value for ζ.
√
Now that the value for the ζ is known it can be used to find the theoretical damping frequency of the cantilever beam system. ζ is expected to be less than one, which is true of
vicious damping. The damping and natural frequencies are expected to be about the same because ζ is less than one.
Equipment
Qty Device Serial #
1 Laptop 292348
1 Power Cord N/A
1 NI 9234 292390
1 NI USB-9162 292393
1 USB-B Cable N/A
1 Unidirectional Accelerometer 107440 1 Microdot to BNC Cable N/A
1 Amplifier N/A
1 BNC Cable N/A
1 Clamp N/A
1 Aluminum Beam N/A
1 Aluminum Block N/A
2 Screws N/A 1 Ruler N/A 1 Micrometer N/A 1 Scale BJ101060 1 Beeswax N/A 1 Tape N/A
1 Hex Driver N/A
1 Flash Drive N/A
Set-Up Procedure
1. Take mass measurements using the scale.
a. Measure the mass of the block with the screws. b. Measure the mass of the beam.
c. Measure the mass of the accelerometer. 2. Find dimensions of the beam
a. Using the ruler measure the width of the beam.
b. Using the ruler measure the total length of the beam. c. Using the micrometer measure the thickness of the beam. 3. Assemble the cantilever beam with the block
4. Clamp down the cantilever beam to the edge of the table. 5. Using the ruler measure the effective length of the beam.
6. Attach the accelerometer using beeswax, making sure that the arrow is pointing the vertical direction.
7. Wire the microdot cable to the BNC cable. Wire the BNC cable to the amplifier. Set the gain to one on the amplifier.
8. Connect the BNC cable to the NI 9234 DAQ.
9. Plug the NI USB-9162 to the computer with the USB-B cable. 10. Open Lab view on computer.
Experimental Procedure
1. In LabView Open a Blank VI and save as Lab 2.
2. Create a time domain graph, and a frequency domain graph.
3. Connect output from the DAQ assistant into the time do main graph.
4. Select a spectral measurement and connect that output to the frequency domain graph. 5. Select the appropriate channel in the DAQ assistant properties.
6. In the DAQ assistant properties set enough samples for 10 seconds.
7. Pull the beam, wait a second or two and press run in labview to begin recording samples. 8. Analyze the spectral graph to obtain values for the natural frequency.
9. Analyze the time domain graph to obtain values for x1, t1 near the beginning of the sample time, and x2, t2, near the end of the sample time.
Data
Table 1 Equipment Measurements
Description Value Unit Mass of Masses Plus Screws 76.30 g Mass of Beam 37.40 g Mass of Accelerometer 04.10 g Width of Beam 03.3 cm Thickness of Beam .1 cm Total Length of Beam 33.1 cm Effective Length of Beam 33.0 cm
Table 2 Experimental Data
Trial 1 2 3 4 5 f d Hz 3.600 3.600 3.600 3.600 3.600 +/- Hz 0.010 0.010 0.010 0.010 0.010 x1 V 0.047 0.550 0.225 0.210 0.540 +/- V 0.005 0.005 0.005 0.005 0.005 t1 sec 1.050 0.490 1.260 1.160 1.510 +/- sec 0.010 0.010 0.010 0.010 0.010 xn+1 V 0.032 0.358 0.122 0.103 0.251 +/- V 0.005 0.005 0.005 0.005 0.005 tn+1 sec 9.750 9.200 9.400 9.310 9.380 +/- sec 0.005 0.005 0.005 0.005 0.005 τd sec 0.280 0.290 0.280 0.280 0.280
Fig. 1 Block diagram
Fig 2. Front panel displaying time domain graph (left) and frequency domain graph (right).
Fig 3. Time domain and frequency domain graph modified to obtain experimental values Calculations
Example calculations from Trial 1
=
=31.07
=
=.012
√
=
√
=.0019
=
)
))
)))
=14.24 rad/s
=√
=22.61 %error= exp-theory/theory=22.61-14.23/14.23=59% ResultsThe natural frequency and damped natural frequency obtained approximately the same. The experimental damped frequency is much higher as can be seen in Table 3. This yields a high percent error.
Table 3 Experimental and Theoretical Results
Trial ζ Experiment ωn (rad/s) Theory ωd (rad/s) Theory ωd (rad/s) Experiment ωd % error 1 0.0019 14.24 14.23 22.62 59% 2 0.0022 14.24 14.24 22.62 58.8% 3 0.0034 14.24 14.24 22.62 58.8% 4 0.0038 14.24 14.24 22.62 58.8% 5 0.0038 14.24 14.24 22.62 58.8 Avg 0.0030 14.24 14.24 22.62 58.84
Questions
1. See Data fig.1 and fig. 2 2. See Data
3. It was unnecessary to use the micrometer to measure the width or length of the beam. This was because the micrometer is mean to measure very small, closer to one-hundredth of a millimeter.
4. Calibration of the accelerometer was unnecessary because it directly measures displacement of the tip of the beam.
5. The oscillation of the beam can be used to calibrate the accelerometer using the displacement.
6. The accelerometer is able to output a wave function based on the displacement caused by pulling on the beam. This wave output is proportional to the amplitude.
7. The beam behaves with viscous damping. The decay is gradual and ζ<<1.
8. This accelerometer was sensitive enough to obtain data without have to apply a large force on the beam.
9. The beeswax material worked out very well in this experiment. It remained stable enough to not move around as testing was done, yet it was easy to adjust. We mounted the
accelerometer at the tip of the block to obtain the best displacement of the tip of the beam.
Discussion
After comparing the theoretical values and experiments values for damped natural
frequency across all 5 trials there is a high percentage of error of about 59%. This may be due to error in measuring dimensions of the beam causing the theoretical data to be incorrect. The use of a ruler makes it more likely for there to be human error. There’s error in the time and
frequency graphs when observing data through each of the oscillations. The graphs have to be estimated significantly because the placement of the wave form graphs aren’t exactly on the lines indicating the time and amplitude. This experiment will need to be done again with much more attention paid to the accuracy of measurements. Due to the nature of this experiment human error is unavoidable but 59% is just too high.
Conclusion
The natural frequency and damped natural frequency were approximately the same. This was and expected result of viscous damping. There was consistency throughout all 5 trials of the
experiment in the values natural frequency and damped frequency. There was a large amount of error when compare to the experimental damped frequency. This is a cause for concern as
something was either measured incorrectly or there is too great of limitations in the measuring tools. This experiment will need to be done again to provide a better comparison on theoretical and experimental damped frequencies.
References
1. Class Notes provided by Professor Ryu 2. https://webcourses.niu.edu/lab2handout
