ME 375 – Vibrations Laboratory

Lab Report One

October 7, 2010Section 001

Group 6

Submitted by:Josh M Jacoboski

Submitted to:Thomas Smith

Lab Partners:Mason Thompson

Peter Ginsberg

I. Introduction

The purpose of this lab is to reinforce the knowledge gained after taking the Mechanical Vibrations introduction course and provide an introduction to data acquisition. Data was collected for a translational and a torsional test stand using a linear potentiometer and a rotary potentiometer respectively. The collected data will be fed into SimuLink where the responses will be captured and analyzed.

The experimental data was compared to an analytical response using MatLab and their graphs were produced. The graphs show how well the experimental collection of data matches with the expected results, showing the minor flaws of the test stands. The discussion explains what parameters needed tweaking and how the systems should be interpreted.

II. Results

Potentiometers are never perfect because they have friction acting on them through the test stand and internally. Calibration factors needed to be found for the two potentiometers so this friction could be accounted for in the experiment. The calibration factors were implemented into SignalCalc to better the results. The difference in calibrating the torsional stand and the translational stand was that the torsional stand involved a disk with a connecting rod and the risk was turned to a certain displacement and the voltage output was recorded. For the translational stand, the block was connected to four thin pieces of metal connected to a mounted bottom plate and the top plate was shifted to a known displacement and the output voltage was recorded. Figure 1 shows the calibration plot produced for the translational system. The block was moved to a known displacement, the voltage was recorded using SignalCalc, then the data was placed into excel and a plot was produced.

-8 -6 -4 -2 0 2 4 6 8-2

-1

0

1

2

3

4

5

f(x) = 0.394821428571429 x + 1.40285714285714

Translational Calibration

Disp. (mm)

Volta

ge (V

)

Figure 1: Plot displaying the recorded points from the calibration test. A trend line was added, with its equation, to get the calibration factor, which is the slope of the line.

Figure 2 shows the calibration plot for the torsional system. The data was collected in the same fashion, rotating the disk to a know displacement, recording the voltage and using excel to produce the plot.

-15 -10 -5 0 5 10 15

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

f(x) = − 0.0238888888888889 x − 0.966666666666667

Torsional Calibration

Displacement (Deg)

Volta

ge (V

)

Figure 2: Plot displaying the calibration plot for the torsional stand. A trend line was added, with its equation, to get the calibration factor, which is the slope of the line.

Both sets of data collected for the calibration of the stands can be found in the attached data sheets. Only the slope of the line is of importance to the experiment because the offset (intercept of the line) is meaningless to the calibration.

After the calibration of the potentiometers, the test stands were ready for testing. To test the free vibration response of the translational system, the block was displaced to -6 mm and then let go to vibrate until it settled. The data was retrieved using SignalCalc by the user stopping the test when the full signal was visible. This data is inserted into MatLab using a predefined code. The code is tweaked to better to gain a better fit for the graph. Figure 3 is a MatLab plot showing the free response of the translational system.

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6

-2000

-1000

0

1000

2000

3000

4000

5000

6000

Translational Test Stand Free Vibration Response

Time (sec)

Dis

plac

emen

t (m

m)

Figure 3: Plot showing the free vibration response of the translational test stand. Data was collected using SignalCalc and the plot was generated using MatLab.

Figure 4 shows the free vibration response of the torsional system. The data was collected by rotating the disk to -4 degrees and let go and allowed to vibrate freely until the disk settled. The data was collected trough SignalCalc and the data was inserted into MatLab.

1 1.5 2 2.5 3 3.5 4

35

40

45

50

55

60Torsional Test Stand Free Vibration Response

Time [ sec ]

Ang

ular

Dis

plac

emen

t [

deg

]

Figure 4: Plot of the free vibration response of the Torsional test stand. Data was collected using SignalCalc and the plot was generated using MatLab.

The model for the translational was set up to compute an analytical comparison for the experimental data gathered. The system was modeled as a second order linear differential equation because the system consists of a mass, springs and dampers. The mass of the system was the top plate. This is what was given the initial displacement when gathering the experimental data. The springs were the supports that attached the top plate to the bottom plate. The dampers were the pieces of tape placed on the inside of the supports. All these aspects of the system governed the decision to model the system as a second order linear system. In lab 4, the experimental data was compared to the analytical equation using MatLab. Offsets were used to make the graphs fit better to each other. Figure 5 shows the plot comparing the experimental and analytical data. The stiffness and the damping coefficient had to be changed in the analytical code in MatLab to make the graph a better fit to the experimental data.

0 0.5 1 1.5 2 2.5-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000Translational Test Stand Response Comparison

Hor

izon

tal D

ispl

acem

ent

[ m

m ]

Time [ sec ])

Analytical

Experimental

Figure 5: Plot showing the experimental data for the translational test stand compared to the analytical equation. The end of the experimental graph doesn’t match up perfectly because of the friction in the potentiometer.

The model for the torsional system was set up to compute an analytical comparison for the experimental data gathered. The mass of the system was the disk. The disk was displaced and allowed to vibrate till it settled. The spring of the system was the connecting rod that connected the disk to the test stand. The damping of the system came from the bearing attached to the disk. A second order non-linear differential equation was used to model this system because there is damping from the bearing, stiffness in the connecting rod and a mass moment of inertia from the disk. The system is non-linear because we assume coulomb friction. The second order non-linear equation was used to produce an analytical response of the system with the experimentally determined parameters. Figure 6 shows the experimental data for the torsional test plotted against the analytical solution.

0 0.5 1 1.5 2 2.5 3-15

-10

-5

0

5

10

15Torsional Test Stand Free Vibration Comparison

Ang

ular

Dis

plac

emen

t [d

eg]

Time [sec]

Simulated

Experimental

Figure 6: Plot showing the relation of the experimental data gathered for the torsional system against the analytical solution. The graphs match well, without tweaking the parameters for the analytical solution, showing that the experiment was done without flaws.

III. Discussion

Many assumptions were made between these two test stands to make sense of the experimental data gathered. For the translational test stand, it was assumed that no energy was lost through any other part of the test stand other than the plate and supports. Also it was assumed that the support acted like linear, mass less spring (ideal springs) and that the base was rigid. For the torsional system, it was assumed that the disk didn’t shift as it was oscillating. Also it was assumed that the base was rigid, the connecting rod acted like a mass less spring, and that the deformation on the rod was elastic. These assumptions allowed the systems to be modeled easily.

For the translational system, the damping ratio effected how long it took for the response to settle and the stiffness of the supports affected the amplitude of the peaks. The assumption of linear viscous damping is not appropriate for the entire length of the

signal for the translational test because of the friction from the potentiometer. For the torsional system, the frictional moment acted like a damping ratio because it effected the time it took for the signal to settle. The spring constant affected the amplitude of the peaks of the signal. The assumption of a constant frictional moment over the entire range of the signal is not valid again because of the friction.

To avoid the avoid the friction in the potentiometer when running these kinds of tests, one could run tests to figure out the friction of the potentiometer, then account for the friction in future experiments. For the translational system, the stiffness and damping coefficient needed to be changed in order for the analytical data to better fit the experimental graph. This could be for a number of reasons, namely that the assumptions made about the system were not completely accurate. For the torsional system, none of the parameters needed to be changed in order for the two plots to match up. This shows that the torsional test stand stood valid to the assumptions made.