402 Lab 5 writeup complete Fall 2016- as taught by Yahya Modarres-Sadeghi PDF

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Lab 5: DOF Vibrations Test
402 Lab 5 writeup complete Fall 2016- as taught by Yahya Modarres-Sadeghi...

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Lab 5: DOF Vibrations Test

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XXXXXXXXX November 2, 2016

MIE 402 Mechanical Engineering Lab II Fall 2016 Professor Yahya Modarres-Sadeghi

Abstract: In this lab, a one degree-of-freedom mass-spring system was tested. This was done using a cart of known mass and attaching two springs of known stiffness on either side of the cart to make the one degree-of-freedom system. The cart was then displaced a number of times at

various initial disturbances. Data was then collected from this and the natural frequency and damping ratio was calculated. A plot of amplitude of its response versus frequency of an external sinusoidal force was also created. Introduction: This lab is designed to determine the natural frequency and damping ratio of a 1 degreeof-freedom mass-spring system as well as plotting the amplitude of the response against an external sinusoidal force. When a sinusoidal force is applied to a first order system it affects the system’s response and changes the output signal. In this experiment, a cart of known mass will be placed on a track and attached to two anchored springs on either end of the track. The cart will be connected wirelessly to the lab computer and CAPSTONE software will be used to record the position of the cart and it’s time response under various frequency inputs. This data will later be analyzed to determine the natural frequency of the cart/mass as well as the damping ratio of the system.

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The general solution to first order differential equation for a system with a sinusoidal input gives the time response, y(t):

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The first term to the right of the equals sign is the trivial response, which is not the focus of this lab since it decays to 0 with time. The second term, also known as the steady response, is what we’ll be investigating during this lab. As can be seen from the equation, the amplitude depends on the value of the applied frequency and that’s what will be modified during the lab and analyzed.

Equipment: For Laboratory Station (6) in Gunness Lab, the equipment that was used to conduct the lab was: 1. PASCO Wireless Smart Cart 2. ACCULAB Electronic Digital Scale Model 5001 3. Two Springs from Dynamics Track Spring Set 4. PASCO Track for Cart 5. TENMA Laboratory DC Power Supply 72-7245 6. CAPSTONE Software 7. PASCO 250g Weights 8. PASCO Oscillator/Driver 9. Digital Tachometer DT-2234C+ 10. Stopwatch 11. Stanley Tape Measure Experimental Procedure: 1. The mass of the PASCO Wireless Smart Cart was obtained using the balance and value noted. 2. Two springs from the Dynamics Tracks Spring Set were selected and attached to a vertical scale to obtain the spring constant and the stiffness was also measured through a linear fit on a graph with various weights. 3. The cart was placed on the PASCO track and the two springs were attached to the cart on either side and to the ends of the track and the equilibrium position of the cart was noted. 4. The CAPSTONE software was setup and the cart was connected to it. The number of samples and the sampling frequency were set. 5. The cart was displaced from equilibrium a particular distance and the data of the cart motion time history were collected through the CAPSTONE software. 6. This displacement was repeated at other previously determined initial disturbances. 7. A known mass was added to the cart and the test was repeated again.

8. One of the springs were detached from the cart and that end of the cart was connected to the PASCO Oscillator. 9. The frequency of rotation for various voltages of the motor was measured using the digital tachometer and verified using the stopwatch. 10. The spring and cart system were driven using the oscillator starting at 0.4 Hz with increments of 0.1 Hz and that data was collected. 11. This was continued till the frequency of oscillation reached twice the measured natural frequency. Data Analysis: To begin data analysis the spring constants of our two spring was obtained by displacing them using several different weights. This data was used to construct a linear curve in Excel that gave the respective spring constants. Next, the cart was displaced 5 times with two different weights in order to calculate the natural frequency and damping ratio of the system under the two weight conditions. This data was recorded by the CAPSTONE software on the lab computer into an Excel format for later analysis. The values obtained from these two results were used in the following experimental calculations of the externally forced spring-mass system. Before applying the external force, the tachometer was also calibrated at various input voltages/ frequencies in order to develop its own plot. This data was also recorded an analyzed in Excel. Therefore, the resulting amplitude results could be associated with their appropriate frequencies as calculated from the voltage input to the oscillator. Again, using the CAPSTONE software on the lab computer data was recorded for various input frequencies of 0.4Hz to 2.4Hz in 0.1Hz increments. After obtaining all the data it was compiled into excel along with its associated mass and input frequency. A FFT analysis was performed on each set of data in order to calculate the natural frequency of the system under the various conditions. This was then used to calculate the amplitude as outlined in the introduction. This data was the compiled into tables along with the theoretical values in order to evaluate the characteristics of our system which are discussed below in the results section.

Results and Discussion: The lab’s results closely agree with what was expected. The damping ratio was determined to be 0.14, which is very low and considered an underdamped value since it is below the critical ratio of 1. This makes sense, since the only source of damping in the system is in the springs and friction in the cart’s wheel assemblies with their bearings and the track. The value for the spring constant of springs 1 and 2 is 6.5051 and 6.8591 respectively. As can be seen in figures 4 and 5 there were 3 displacement vs force measurements taken and fitted with a linear trendline and the slope gave the spring constants. Using these two spring constants the natural frequency, damping frequency, and damping ratio were determined and the results can be seen in tables 1 and 2. Table shows the results of the cart with no added mass and clearly shows that the values for values for the damping frequency and ratio change very little by changing the initial displacement conditions. The largest observed change is just over 4% and considered insignificant. These results are expected as the natural frequency, damping frequency, and damping ratio are calculated from the springs constants and mass which remain constant with changing displacement. When the test is repeated there is a similar pattern to the results. Table 2 again shows that the natural frequency, damping frequency, and damping ratio remain constant with changing initial displacement. However, the added mass does result in a decrease in these values compared to previous test which is as expected. The results of the test involving an applied sinusoidal external force are shown in table 3. The results show the experimental amplitude for each frequency from 0.4Hz to double the natural frequency (the natural frequency being 1.17Hz, the results go to 2.4Hz). As expected when the applied frequency approaches the natural frequency the system experiences resonance and the amplitude climbs to a maximum value of 22.3 cm of displacement at 1.2Hz. This value would have been greater but the external force was removed as the cart was going to collide with the end of the track. Comparing these experiment results to the theoretical results as is done in figure 3 it is clear to see that the experimental data very closely approximates the theoretical values. As was explained earlier there is a significant decrease between the theoretical and experimental amplitude at resonance but that is a results of the limited track space. It was not

possible to plot and compare the data using dimensionless values since the force data was not recorded during testing. It is important to note that all of these calculated values should have an associated error with each of them due to bias and precision. In this lab, the T.A. was unable to provide the bias error associated with the instruments so there is no error associated with bias displayed in our results. This is clearly wrong and will be corrected in future labs. Also, there is no precision error in our results as we were only able to run each test a single time due to time constraints. There are only 3 hours available for the lab and nearly all that time was taken to perform the nearly 25 separate runs. Without at least 3 runs of the same test it is not statistically possible to calculate the precision error and thus it is not included in our results. Conclusion: The stated objective of measuring the natural frequency, damping ratio, and plotting the amplitude vs. frequency of an external force was successfully achieved during this lab. By using the experimental data to plot this curve it was possible to compare these values to the theoretical ones which are shown in figure 3. As expected, it is shown that there is a large increase in amplitude around the natural frequency of 1.17Hz for this particular spring-mass system. For the theoretical curve this value should go to infinity. However, the closest frequency that could be achieved in lab without running out of track was 1.2Hz which was used for both curves. In the experimental case this resulted in a value of 22.3cm while the theoretical shows an expected value of 25.6cm. This discrepancy is explained by the fact that there is a small amount of damping in the real world experiment performed and is an important conclusion of this lab. When designing similar systems in future work it is important to include the real world factors such as damping when performing engineering calculations. For future labs, it is suggested that a larger number of samples are taken in order to calculate the statistical precision of the results. Also, greater care should be taken to provide/obtain the bias error associated with the instruments, particularly the software’s bias.

Appendix Table 1. Shows the natural frequency of the cart, by itself, at various displacements. M=246g (cart only) Displacement (cm)

Natural Frequency

Damping Frequency

Zeta

5

1.2

1.19

0.14

10

1.15

1.14

0.14

15

1.17

1.16

0.14

7.5

1.16

1.15

0.14

12.5

1.18

1.17

0.14

Table 2. Shows the natural frequency of the cart, and 2 of the 250g weights, at various displacements. M=751g (cart and (2) 250g weights) Displacement (cm)

Natural Frequency

Damping Frequency

Zeta

5

0.71

0.70

0.13

10

0.69

0.68

0.13

15

0.69

0.68

0.13

Table 3. Shows the experimental amplitude of the cart, by itself, at various frequencies. Note that the highest amplitude is found at the natural frequency of around 1.2Hz. This is resonance. Frequency (Hz)

Amplitude (cm)

0.405

0.764

0.51

0.78

0.615

0.69

0.7

0.82

0.801

1.07

0.911

1.77

0.998

4.13

1.095

12.57

1.2

22.3

1.307

11.83

1.495

2.07

1.81

0.88

2.11

0.38

2.34

0.17

Table 4. Shows the theoretical amplitude of the cart, by itself, at various frequencies. Note that the highest amplitude is found at the natural frequency of around 1.2Hz. This is resonance. Frequency (Hz)

Amplitude (cm)

0.405

0.15

0.51

0.86

0.615

0.95

0.7

0.93

0.801

1.22

0.911

1.84

0.998

3.66

1.095

9.63

1.2

25.56

1.307

17.89

1.495

13.03

1.81

2.09

2.11

0.51

2.34

0.14

Figure 1. Shows the amplitude vs. frequency plot for the experimental data.

Figure 2. Shows the amplitude vs. frequency plot for the theoretical curve.

Figure 3. Shows the amplitude vs. frequency plot for both the theoretical curve and experimental data overlaid.

Figure 4. Shows the force vs. extension plot along with spring constant for the first spring. Error = +/- 0.07N/m due to resolution.

Figure 5. Shows the force vs. extension plot along with spring constant for the second spring. Error=+/-0.06N/m due to resolution.

Figure 6. Tachometer calibration showing the linear trendline and formula....