essay 335
Ayoub-94
123.docxAbstract
The dynamic analysis of structural members can be an intensive process heavily dependent on the analysis of first and second-order systems. The primary objectives of frequency analysis for structural members is determining the resonant and damping frequencies, as well as the damping coefficients of the system under different conditions. In this experiment a cantilever beam was excited in such a way that the resultant vibration could be analyzed using signal analyses including as FFT. The damping coefficients, damping frequencies, and natural frequencies of two systems with unique weights acting as dampeners were determined through experimentation and subsequent calculation.
Introduction
Differential equations can be used to solve many natural problems that happen in the world. One of the real life problems that it helps with is with a cantilever beam. There will be a beam that will be set up with a Wheatstone bridge to try to find the natural frequency of the beam. This is important in terms of vibration analysis. From here there will be different masses placed upon it to see the effects of damping on the system. With the Wheatstone bridge, the damping and natural frequencies, as well as the damping coefficients, can be determined through experimentation and subsequent calculations. This damping coefficient will change depending on the different type mass that was added to the cantilever beam.
Theory
Using a Wheatstone bridge connected to strain gauges and a data acquisition system one can find out the different natural and damping frequencies of a cantilever beam with different dampeners. The Wheatstone bridge circuit is utilized by using the circuit to record the changes in material that is being tested. Since they are very sensitive it works by measuring the local strain in the beam using a strain gauge. This will then be measured and recorded into a frequency produced by the beam once the mass and beam are “excited.” From here one can use the recorded data to find the frequency and damping coefficient of the harmonic reaction.
The function can be used to find the log decrement. The ln(y) is from the first peak as the 2nd lny is from a certain number n of peaks away. This then is used to find the damping coefficient through the equation . Once our damping coefficient is found we can therefore find the natural frequency of the system through . The damping frequency can be found by creating an Amplitude-Response plot. Solving for provides the natural frequency of the system.
Refer to instructions titled “ME 335 Lab Experiment #3 – Static and Dynamic Response of a Cantilever Beam Scale” in the lab manual, “ME-335 Handout,” pgs 48-50.
Results
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Table 3.1 - Summary of results for 105 g weight
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105 g Weight Results |
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δ |
Damping Coefficient, C |
Ringing frequency, ωR |
Natural frequency, ωn |
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0.0528 |
0.00841 |
14.677 Hz |
14.677 Hz |
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Table 3.2 - Summary of results for 457 g weight
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457 g Weight Results |
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δ |
Damping Coefficient, C |
Ringing frequency, ωR |
Natural frequency, ωn |
ωn |
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0.0914 |
0.0146 |
6.75 Hz |
6.75 Hz |
13.50 Hz |
The calibration plot for the beam is shown in Fig. 3.1. The measured weights were compared to the calculated weights and the results are shown in Fig 3.2. The standard deviation for voltage Fig 3.1 was found to be 5.07 volts. The standard deviation for a calculated weight in Fig 3.2 is 8.80. The confidence interval was calculated to 10.9 V.
The data analysis was applied to two weight samples, the 105g and 457g. In order to find the damping frequency, an FFT analysis was performed. For the 105g sample, Figure 3.4 shows a damped frequency value of 14.6771 Hz. The damped frequency is equal to the ringing frequency. For the 457g sample, from Figure 3.6 the ringing frequency is equal to 6.751468 Hz. As we can notice from the results, the ringing frequency decreases with the increase of mass. Table 3.1 and Table 3.2 show the values obtained through calculations based on the results found from the FFT analysis.
The second step was determining the damping coefficient of our system. The system we have is an underdamped system, which oscillates and decays until it reaches the equilibrium. The damping coefficient can be found through the equation stated in the theory. The damping coefficient for the 105g is 0.008408. While the Table 3.2 for the 457g shows a damping coefficient 0.01455. The two damping coefficients are unique for each case. The damping coefficient increased with the increase of mass.
The proportional prediction did match for the case. Multiplying the natural frequency of the 1 lb by yielded a value of 13.501 Hz. The natural frequency of the 0.231 lb was 14.675 Hz. The values fall within a reasonable range. The variation could be due to errors performed during the experiment. These errors include a systematic error in the calibration process, indicated by the fact that the calculated weights did not perfectly match the true value of the weights. Other important sources of error include instrument error and resolution error.
Conclusions
Seeing as the FFT gives the ringing frequency it’s natural that one would then find the natural frequency of the system to compare the two and see how much of a difference there are between the two. The FFT function is used to bring in multiple wavelengths and run them through a theoretical prism to split the waves and then find the proper ones to add them together to get the proper frequency of the system. This is why it was important in finding the damping coefficient among other things. Through this experiment it was possible to successfully determine the damping and natural frequencies, as well as the damping coefficients of a system consisting of a cantilever beam and a weight acting as a dampener. Our collected data and corresponding calculations appear to align with the theory behind the experiment, indicating that the procedure was carried out successfully. The information obtained through this experiment could be used to carry out further structural analysis of the system.
