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Fall 2018
Lab 7 – Position, Velocity, and Acceleration
Lab 7-1
Lab 7 – Position, Velocity, and Acceleration
Format
This lab will be conducted during your regularly scheduled lab time in a group format. You may
ask the lab instructors for assistance if needed, but successful completion of the lab is your
responsibility.
Report
An individual, informal report is due from each student at the beginning of the next lab (Lab 8).
Background
Position, velocity, and acceleration measurements are of obvious importance to mechanical
engineers. To learn more about these measurements, you will do the following during this
laboratory:
I.
Calibration of a linear potentiometer and measurement of displacement.
You will be provided a linear potentiometer; a potentiometer where the wiper position
on the potentiometer changes with the linear (translational) movement of a shaft. You
will calibrate the potentiometer by determining the change in output voltage as a
function of shaft displacement. You will then use the calibrated device to measure the
length of objects in the 360 laboratory.
II.
Measurement of rotational speed using a proximity sensor and an incremental encoder.
You will be provided a laboratory test-stand with a DC motor and the associated motor
driver. You will then measure the motor shaft speed using a proximity sensor and a
toothed-gear and then with an incremental encoder. In your laboratory report, you will
compare the two methods for measuring shaft speed.
III.
Measurement of beam vibration using a piezoelectric accelerometer.
You will be provided an accelerometer and accelerometer signal conditioner. You will
then use the combination to measure the vibration of a mass-ended aluminum beam
clamped to the lab station. The acceleration data will be used to determine the natural
frequency of the mass-ended beam with two different sets of masses on the beam. In
turn, you are to use the natural frequency of the beam to estimate the beam stiffness. In
your report, you must compare the measured and theoretical beam stiffnesses.
Because there are three parts to the lab, you will be rotating among different laboratory
workstations. You will have approximately 45 minutes to make all measurements at a given test
stand before rotating to the next workstation. As such, please be sure to stay focused and on-task
during the laboratory.
Fall 2018
I.
Lab 7 – Position, Velocity, and Acceleration
Lab 7-2
Position Measurement
You will be provided a linear potentiometer similar to the device shown in Figure 1 below.
As typical with potentiometers, there will be three wires coming out of one end of the “pot:”
an orange, a gray, and a yellow wire. The orange and gray wires are the “ends” of the
potentiometer. The resistance between those two points should not change with the pot
stroke. The yellow wire is attached to the potentiometer wiper. The resistance between the
yellow wire and the orange wire should change as the shaft of the pot is moved relative to the
pot body. If that resistance increases, then the resistance between the yellow wire and the gray
wire should decrease accordingly.
You will also find a set of digital calipers at your lab workstation. A similar set of calipers is
shown in Figure 2 below.
Figure 1. Linear potentiometer (Omega).
Figure 2. Digital calipers (Omega)
1. Use the DMM and the calipers at your lab station to make the following measurements:
a. Total potentiometer resistance as measured betweeen orange and gray wires:
_________________
b. Resistance between yellow and gray wires with shaft fully retracted:
_________________
c. Resistance between yellow and orange wires with shaft fully retracted:
_________________
d. Resistance between yellow and gray wires with shaft fully extended:
_________________
e. Resistance between yellow and orange wires with shaft fully extended:
_________________
f. Full extension length of shaft:
_________________
2. Plot the resistances between the yellow and orange wires and between the yellow and
gray wires as functions of shaft stroke using two of the measurements made in (1) above.
Determine the two equations that relate the yellow-orange resistance and the yellow-gray
resistance as functions of potentiometer stroke.
Fall 2018
Lab 7 – Position, Velocity, and Acceleration
Lab 7-3
3. Using the digital calipers as a guide, move the potentiometer shaft to the half-extension
position. In that position, measure the yellow-orange and yellow-gray potentiometer
resistances.
Yellow-orange:___________________
Yellow-gray:_____________________
Question: do the measurements made in (3) correlate with the graphs and equations
determined in (2)? If not, can you explain why not? Consider the possibility of saturation;
that is, the change in resistance stopping beyond a certain point, even if the shaft of the pot
keeps moving.
4. Connect the orange wire to +5V and the gray wire to GND on your breadboard. Then,
connect your breadboard to the power supply at your lab station.
5. Connect the Channel 1 input of your Analog Discovery Kit 2 (ADK2) to measure the
voltage of the yellow wire on the potentiometer. Do not forget to connect the Ch1 black
lead to GND on the breadboard. Observe the potentiometer output voltage (as indicated
on the yellow wire). See if there is any saturation that you can see on the potentiometer
(the voltage maxes/mins out before the shaft reaches max/min extension).
6. Using a DMM and the calipers, measure the voltages and shaft positions for the linear
range of operation of the pot. The “start” and “end” positions are the points where the
voltage starts to change with shaft movement and then stops changing with shaft
movement.
Start position:_____________
Start voltage:____________
End position: _____________
End voltage: ____________
7. Now make a plot of the voltage vs. shaft position and predict the voltage halfway between
the start and end positions.
Vhalf,pred:__________________
8. Move the pot shaft to the halfway point indicated in (7) and measure the voltage. How
does it compare to the predicted voltage from (7)?
Vhalf,act:___________________
Fall 2018
II.
Lab 7 – Position, Velocity, and Acceleration
Lab 7-4
Angular Velocity Measurement
This section of the laboratory will examine three different methods for measuring the rotational
speed of a motor’s shaft. The different sensors that you will use are:
1. Optical Tachometer
2. Incremental Encoder
3. Magnetic pickup
These are all shown in Figure 3 below.
In the first example, a handheld tachometer is pointed at an optical reflective tape on the shaft
coupling. The shaft speed in rpm is read directly off of the display on the front of the tachometer.
In the second example, a commercial incremental encoder is used. It has a resolution of 360 slots
per revolution and is attached to the end of the DC motor/tachometer. The output of this device
can be read with the Waveforms oscilloscope. To convert that reading into the motor speed in
rpm, use the following formula:
Motor' s
60 sec
1 rev
rev slots
.
*
*
=
min
sec 360 slots min
Finally, a circular plate with ferrous metal “gear-teeth” has been installed on the shaft of the
motor and is used to generate pulses as the teeth pass an external, fixed magnetic pickup coil.
There are 30 teeth on the plate and thirty “spaces.” As such, 30 pulses are generated for each
revolution of the plate. Those pulses are translated into the speed in rpm, which can be read on
the readout at the front of the experimental setup.
The process for testing these rotary velocity transducers is as follows:
1. Set the speed controller to a desired (low) setting.
2.
•⇒Measure the rotational speed of the shaft using each of the transducers listed above:
•
•
•
Handheld tachometer (direct reading in RPM)
Incremental encoder frequency in Hz from Waveforms oscilloscope. Make sure there are
between 5 and 20 complete cycles on the screen.
Magnetic gear pickup readout (direct reading in RPM)
3. Change the speed setting and repeat step #2.
4.
•⇒Continue step #3 until at least 10 different motor speeds (including the maximum speed)
have been tested and the data has been recorded for the report.
•⇒Outside Lab:
5. Treat the angular velocity from the magnetic gear pickup readout as ideal (ωideal) and plot the
other 2 directly measured speeds (ωhandheld, ωencoder) on the vertical axis of one plot.
6. Plot the deviations (ωhandheld – ωideal) vs. ωideal and (ωencoder – ωideal) vs. ωideal on a second plot.
Fall 2018
Lab 7 – Position, Velocity, and Acceleration
Lab 7-5
Handheld tachometer
(point here)
Incremental
Encoder
Magnetic gear pickup
Connections
for Encoder
Motor speed
controller
Magnetic gear
pickup readout
Figure 3. Velocity sensors and DC motor controller.
III.
Vibration Measurement with Accelerometers
As discussed in class, accelerometers are perhaps the most widely used sensors in the field of
vibration measurement. As you will see in the lab, they are quite easy to use, provided you have
the appropriate signal conditioning equipment.
For this portion of the lab, you will be measuring the acceleration response of a mass-ended,
cantilevered, aluminum beam. If the end-mass is significantly larger than the mass of the beam,
then as a first approximation, the system can be treated as a second-order mass-spring-damper
system. The natural frequency of such a system is given by the equation
K
3EI
,
K= 3 ,
ωn =
M end + 0.24 M beam
L
where E is the elastic modulus of aluminum, I is the cross-sectional inertia of the beam, and L is
the length from the base of the beam to the center of the mass at the end.
If the beam is pulled to a non-equilibrium position and released, then the beam displacement
(
)
response will be x (t ) = x0e −ζωnt sin ωn 1 − ζ 2 t . If you take the second derivative of that
expression, you will see (some algebra later) that the acceleration response of the beam will also
be a decaying sine with the same frequency. As such, if you measure the frequency of the
acceleration signal, you will be measuring the damped natural frequency which, for lightly
damped systems, is almost the same as the natural frequency. You can also use the log-decrement
method on the acceleration response to get an approximation of the system damping ratio.
You will find a mass-ended, cantilevered beam clamped to the lab bench. An accelerometer will
be attached to the end-mass; it is mounted using a synthetic wax (as opposed to bee’s wax, which
was one of the original accelerometer mounting adhesives). A thin layer of wax is all that is
Fall 2018
Lab 7 – Position, Velocity, and Acceleration
Lab 7-6
necessary; thicker layers allow for compliance between the device under test and the
accelerometer itself (which is a fancy way of saying that a thicker layer may act as an additional
spring-damper between the accelerometer and the mass to which it is attached). The
accelerometer will be wired to an XDCR (“transducer”) input on an ICP (integrated circuit
piezoelectric) signal conditioner (some lab stations will have multi-channel ICP signal
conditioners). The signal conditioner will have three gain settings: x1, x10, and x100. If a
10mV/g accelerometer is plugged into the ICP signal conditioner and that channel’s gain is set to
x10, then the output voltage on that signal conditioner will be 100mV/g, where one “g” is one
“gravity,” also known as an acceleration of 9.81m/s2 or 32.2ft/s2.
1. Measure the distance from the edge of the lab bench (where the beam is clamped) to the
center of the mass clamped to the end of the beam (a mass is clamped to the end of the
beam and the beam, in turn, is clamped to the table, so two clamps are in use).
L=______________
2. Use a BNC-BNC cable to connect the appropriate signal conditioner output to Channel 1
on the Analog Discovery Kit 2. Run the Waveforms software on your lab PC and set the
oscilloscope to run continuously. Gently pluck the end of the beam and observe the
resulting acceleration signal on the oscilloscope. Select the appropriate V/div vertical
setting so that you will be able to see all of the acceleration signal (you will be able to see
all of the peaks and valleys in the response). Select the appropriate s/div time setting so
that you can see at least eight periods of the signal.
3. Pluck the beam again and, at the same time, record a single trace of the oscilloscope data.
Save that data in a .csv file. Open that file in Excel and verify that you can observe at
least eight periods of the signal with sufficient time and voltage resolution so as to get a
good idea of each period’s peak value and the time that it occurs.
4. Remove the mass from the end of the beam. Use the scale at your lab station to measure
the mass of the end-mass (include the clamp used to hold the end-mass on the end of the
beam).
Mend-mass=_____________
5. Apply a different mass (say twice or half the original amount) to the end of the beam.
Repeat steps 3 and 4.
Outside of Lab:
6. Estimate the two different natural frequencies measured in the lab (in steps 3 and 5).
7. Use the two different natural frequencies and the two different known masses to obtain
the approximate stiffness of the beam. In your calculations, you should consider the mass
of the beam; you can estimate that value if you know the cantilevered length of the beam,
its cross-sectional area, and the density of aluminum.
8. Compare the measured stiffness of the beam to the theoretical stiffness listed in the
equation on the previous page. How do the two compare? You do NOT need to include
uncertainty in your analysis.
9. For each of the two tests, estimate the damping ratio of the beam using the log-decrement
method outlined in the class notes. Use the damping ratio and the theoretical natural
frequency as listed on the previous page to estimate the theoretical damped natural
frequency. Compare that value to the measured natural frequencies as estimated in (6)
above.