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 differ ...
REPORT SUMMARYVibration refers to a mechanical.docx
1. 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
2. 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
3. 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,
4. 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
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
5. 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).
6. 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
7. 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.
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.
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 =
8. 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
9. 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