Paraphrase lab report

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REPORT SUMMARY

Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.

Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.

As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).

In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.

We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differences in the results, including the finding that the period for all of the systems across all of the displacements are nearly identical (in some cases, up to two decimal places). A potential trend observed was the increase of damping as the magnetic damper increased its proximity to the track. For derivations, data visualization, and further results, please see the Results and Discussion section.

In this lab we took our theoretical vibration knowledge and applied it to data gathered in the real world. As our measurements and calculations show, the theoretical values of many components of vibrating systems very closely match their empirically measured corollaries. For detailed results, derivations, etc., please see the Results and Discussion section.

results and discussion

The following figures display a summary of our results. For further detail and the raw data, please see the Appendix.

Figure 1: Plot of all free vibration data

Figure 2: Plot of all underdamped vibration data

Figure 3: Plot of all overdamped data

In the experiment, we gathered three sets of displacement vs. time data. The first was from the "free" vibration system (plotted in Figure 1), the second was from the "underdamped" system (plotted in Figure 2), and the third was from the "overdamped" system (plotted in Figure 3). The "free" system clearing shows damping, but definitely oscillated for a greater amount of time than the "underdamped" or "overdamped" systems, which is at least partially in line with expectations. The "underdamped" and "overdamped" systems acted as expected, with "underdamped" oscillating much longer than "overdamped", which only oscillated for about 1.5 periods.

The first section of the results and discussion portion of the lab required a derivation of the scalar equation of motion of the "free" system. This was essentially a spring-mass system on an inclined plane of approximately 10 degrees. For a detailed derivation, see the work in the appendix, a brief overview follows. After creating the free body diagram of this system, Newton's second law was applied (see Equation 1).

Equation 1: Newton's second law

It is known from the free body diagram that the total force on the mass was the sum of the spring force and the gravity component along the mass' axis of motion. Substituting the spring force's constituents into the equation (spring displacement and spring constant) and then converting the resulting linear ordinary differential equation (ODE) into a scalar resulted in Equation 2.

Equation 2: "Free" system scalar equation

Then, in reference to the "free" system, the period (Τ), circular natural frequency (ωn), cyclical natural frequency (f), and phase angle (ɸ) were required. These quantities were determined from the graph. The period was found by taking the time difference of two sequential "peaks" of each curve, resulting in Τ-20 = 3 seconds, Τ-10 = 3 seconds, Τ+10 = 3.2 seconds, and Τ+20 = 3 seconds.

Equation 3: Natural circular frequency/period relation

Now that we had the period for each of the curves, we used Equation 3 to determine each of their natural frequencies: ωn -20 =2.09 rad/s, ωn -10 =2.09 rad/s, ωn +10 =1.96 rad/s, and ωn +20 =2.09 rad/s.

Equation 4: Cyclical frequency/period relation

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Figure 4: Phase angle definition from class notes

Equation 5: Phase angle equation

With the relation described in Equation 4, we were able to determine the following natural cyclical frequencies: f-20 = 0.33 hz, f-10 = 0.33 hz, f+10 = 0.3125 hz, f+20 = 0.33 hz. From the description of the phase angle in class (Figure 4), we determined that the difference referred to between the two curves would always be consistent with the definition at y(0). This allowed us to construct Equation 5 and generate the following results: ɸ-20 = 1.78 (lag), ɸ-10 = 1.25 (lag), ɸ+10 = 1.176 (lead), and ɸ+20 = 1.78 (lead).

Equation 6: Theoretical natural frequency

Next, the lab required a comparison of the theoretical and experimentally measured natural frequency. As the theoretical only depends on the spring constant (previously determined experimentally with the spring force relation) and the mass of the mass in the system, it will be a constant ωn = 2.09 for all initial displacements. Impressively, this identically matches 3 or 4 experimentally measured results (to two decimal places), with the third likely anomalous, but still very close. This is a good indicator that the theoretical equation very closely (again, in this case, identically) models the actual natural circular frequency. From the data gathered, it's clear that the circular natural frequency and period do not depend on the initial displacement (given all 4 values in these two cases are nearly identical). However, it's also clear that the phase angle data is significantly different, divided into two groups based on the magnitude of the initial displacement, therefore the phase angle is clearly affected by the initial displacement. Sadly, the graph of the "free" system shows that each curve has a decreasing amplitude as time increases. While it's disappointing that our "free" system is therefore clearly not perfectly free, it's perhaps worse that the lab then requires more work. To determine the damping coefficient (c), the logarithmic decrement method was used, as described in Equation 7, to determine c-20 = 0.01, c-20 = 0.009, c+10 = 0.008, c+20 = 0.01. All the values for the damping coefficient are approximately 0.01, showing that there is essentially no dependence on initial displacement (and therefore no variation in c due to variation in the initial displacement of the spring).

Equation 7: Damping coefficient

For the second portion of the results and discussion, the underdamped and overdamped data was analyzed. First, Newton's second law was again used to derive the scalar equation of motion. This was identical to the free system with the addition of a damping force.

Equation 8: Scalar equation of motion for damped system

Specifically requested were the period (Τd), circular natural frequency (ωd), cyclical natural frequency (f), and phase angle (ɸ) were required. These quantities were determined from the graph. The period was found by taking the time difference of two sequential "peaks" of each curve, resulting in Τd-20 = 3.3030 seconds, Τd-10 = 3.3030 seconds, Τd+10 = 3.0031seconds, and Τd+20 = 3.3031 seconds for the overdamped system and Τd-20 = 3.1999 seconds, Τd-10 = 3.0999 seconds, Τd+10 = 2.8999seconds, and Τd+20 = 3.0999 seconds for the underdamped system.

With the period for each of the curves, applying Equation 8 yielded the damped circular natural frequencies; ωd -20 =1.9022 rad/s, ωd -10 =2.092302 rad/s, ωd +10 =2.0922 rad/s, and ωd +20 =1.9023 rad/s for overdamped and ωd -20 =1.9636 rad/s, ωd -10 =2.0269 rad/s, ωd +10 =2.1667 rad/s, and ωd +20 =2.0269 rad/s for underdamped.

Equation 9: Damped circular natural frequency/period relation

With the relation described in Equation 9, it was possible to determine the following damped natural cyclical frequencies: f-20 = 0.5257 hz, f-10 = 0.47795 hz, f+10 = 0.47799 hz, f+20 = 0.526 hz for the overdamped system and f-20 = 0.3125 hz, f-10 = 0.3226 hz, f+10 = 0.3448 hz, f+20 = 0.3226 hz for the underdamped system.

Equation 10: Damped cyclical natural frequency/period relation

From the description of the phase angle in class (Figure 4), it was determined that the difference referred to between the two curves would always be consistent with the definition at y(0). This leads to Equation 5 and generates the following results: ɸ-20 = 0.201021(lag), ɸ-10 = 0.1809612 (lag), ɸ+10 = 0.21102 (lead), and ɸ+20 = 0.201002 (lead) for the overdamped system and ɸ-20 = 0.1709(lag), ɸ-10 = 0.1609 (lag), ɸ+10 = 0.2010 (lead), and ɸ+20 = 0.1609 (lead) for the underdamped system.

$\zeta = \frac{1}{\sqrt{1 + (\frac{2\pi}{\ln (x_0/x_1)})^2}},$

Equation 11: Damping ratio

Equation 10: damping ratio

Given Equation 10 (where xo and x1 are adjacent amplitudes), the damping ratios are -10 = 0.016465,-20 = 0.0295,10 = 0.01368,20 = 1.9029 for the overdamped system and -10 = 0.1686,-20 = 0.1779,10 = 0.1656,20 = 0.1858 for the underdamped system.

$\zeta = { c \over 2 \sqrt{m k} }.$

Equation 12: theoretical critical damping coefficient equation

To find the damping ratios using the decaying curve method, it was necessary to plot the magnitudes of several consecutive amplitudes. The overdamped graphs were far too short and inconsistent to use this decay method, but the underdamped curves supplied ample data. Fitting an exponential line to the curve we were able to obtain an equation for the exponential decay of our points, as shown in Figure 5.

Figure 5: Underdamped decay method

Using the equation shown in Figure 5, it was possible to use the obvious boundary condition and find that the constant, c = 1.1336. Equation 13 yielded β as β-10 = .0462, β-20 = .0297 β+10 = .0568 β+20 = .0338 for the underdamped system. These damping ratios are significantly different from those derived with the logarithmic method. For detailed work, see the Appendix.

Equation 13: Decaying curve

The theoretical damping coefficient equation was used as a comparison, yielding c-10 = 0.05518, c-20 = 0.98869 , c10 = 0.04585 , c20 =0.826815 for the overdamped system and c-10 = 0.565 , c-20 = 0.596 , c10 = 0.555 , c20 =0.623 for the underdamped system. For detailed work, see the Appendix. It's very likely that the damping coefficient is due to more than the magnet alone, evidenced by the damping observed in the "free" system, which occurred before the use of the magnetic damper. The damping should increase as the distance between the magnet and the track decreases. This wasn't found to be true at all times in the data, but this is likely an error in the calculations. The period is nearly the same across displacements and damping.

It's clear that extraneous variables are present, including the friction of the track. As shown by the damped behavior of the "free" system, there is an unknown (or unaccounted for) force acting on the system. It's difficult to identify these forces, but they may or may not include friction with the track, friction between the wheels and axels, damping forces due to shifting of cart internal components, and aerodynamic drag.

conclusions

In conclusion, we took our theoretical knowledge out of the classroom and gained valuable, practical, hands-on experience applying it to real, tangible systems. Even though theoretical models were incomplete, most of the time it was found that they were able to describe the real world behavior with nearly zero error. It was also discovered that some components of vibration are the same for a system, regardless of its damping or initial spring displacement (however, several vibration components are indeed effected by these factors).

Weaknesses in the experiment were minimal, as suggested by the strong, expected correlation visible in the results. The free body diagram for the systems were overly simplified and clearly didn't account for all forces involved (again, citing the very visible problem with the "free" state -- it damps out). It was also not possible to calibrate the equipment (range finder), although that may introduce negligible error as we were primarily concerned with distance displacement which should be independent of a simple sensor bias.

Appendix

Choo, Vincent (2013) Class Notes: Vibration notes cleaned up. ME 345 Class Notes Fall 2013. Las Cruces: New Mexico State University.

Underdamped system work

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Free

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Underdamped

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