Spring ‘11
Lab 6 - System Response
Lab 6 - 1
Lab 6 - System Response
Format
This lab will be conducted during your regularly scheduled lab time in a group format. I
strongly recommend that you rotate roles during the lab. You may ask the lab instructor for
assistance if needed, but successful completion of the lab is your responsibility – not theirs!
Reports
An individual, informal report covering the first lab exercise (First Order System Response)
is due from each student by 8:00 AM on Monday, 3/7/11. This required report will count as
two homework exercises.
An individual, formal report covering the second lab exercise (Second Order System
Response) is due from each student by 8:00 AM on Friday, 3/11/11. Do not include any
material from the first lab exercise in the formal report. Use Microsoft Word and Excel to
create the report. Two “hard” copies (stapled in the upper left-hand corner) must be turned in
– one complete copy and one copy without an appendix (marked “English”). In addition, an
electronic copy of the report without figures or appendix must be emailed to the course
instructor on the due date.
Use your complete name as part of the file name for the electronic copy.
Procedures
I. First Order System Response
Your lab group will build the circuit shown in Figure 1, which will be connected to both the
function generator and the National Instruments data acquisition system. A first voltage
follower buffers the output from the function generator to the 1st order RC system. The
function generator output is connected to the data acquisition system with the orange (Or)
and black (Blk) twisted wire pair. A second voltage follower buffers the output from the 1st
order RC system to the data acquisition system. The second voltage follower output is
connected to the data acquisition system with the yellow (Yel) and black (Blk) twisted wire
pair.
Or Ch0 Hi
Ch0 Lo
Blk
Function Generator
TTL
50
Or
Ch1 Hi
Ch1 Lo
+
+
Yel
R
Blk
Blk
C
Blk
Figure 1. 1st order RC system time constant set-up.
Spring ‘11
Lab 6 - System Response
Lab 6 - 2
1. Build the two voltage follower op-amp circuits shown above. The resistance R is
nominally 240 k(220 k kin series) and the capacitance C is nominally
0.022 F. Measure and record the actual series resistance and capacitance of the initially
installed components for this 1st order RC system. Connect the function generator and
data acquisition system as shown.
2. Adjust the function generator to produce an approximately 4 to 6 Hz square wave with an
amplitude of about 2 volts peak-to-peak.
3. Set Signal Express to read 1000 samples at 5 kHz (5,000 samples/sec). This will be data
set #1. The vertical position setting for both traces should be approximately centered
vertically in the screen.
4. You should obtain a pair of traces that are “similar to” those in Figure L5-2. The square
wave ("step input" to RC system) comes from the function generator. The exponential
(1st order) response curve comes from the output of the 1st order RC system.
It is important that the RC system output reach a constant steady-state value, i.e., the
output response should “flatten out” after the 1st order exponential rise or decay. If this
does not happen with your RC system, check your resistor and capacitor values and your
setting for Signal Express carefully.
6
4
Voltage
2
0
IMPORTANT!
-2
-4
-6
0
0.02
0.04
0.06
0.08
0.1
Time, sec
Figure 2. Input and output waveforms for 1st order RC system.
5. Have the lab monitor check the results from your RC circuit at this point!
6. When the lab monitor “OK’s” your results, save the data shown on the Signal Express
screen in Excel. If you cannot open the file in Excel and see the data for plotting (time,
Spring ‘11
Lab 6 - System Response
Lab 6 - 3
input voltage, output voltage), then you have not stored the data correctly from Signal
Express.
7. Estimate the experimental time constants from your plotted data.
8. Compute the theoretical time constant for the 1st order RC system from the measurements
of the resistance, R and the capacitance, C. Compare to the experimental time constant
estimated in Step #7. These values should agree to within ~10%.
Note - it is a "good idea" to perform this step in the lab. If your experimental
and theoretical time constants are not "close," you need to find out why
before you leave the lab!
9. Measure a 0.0047F capacitor, and add it in parallel with the existing 0.022 F capacitor
(keep the existing 240 k resistor combination). This will be your second 1st order RC
system.
10. Collect a set of data (1000 samples at 5 kHz ) from the second 1st order RC system and
save to Excel. This will be data set #2.
Outside of lab:
11. The experimental time constant of each 1st order RC systems can be accurately
determined from the input and output vs. time plots. Determine the time constants for
each of the 1st order RC systems.
12. Clearly and completely show both experimental time constants on separate plots of the
experimental data. Show where the step input initially occurred and when the output
reaches the t= value.
13. The theoretical time constant of each 1st order RC system can be determined from the
measured resistance and capacitance values. Since the values used in the theoretical
computation are measured, there will be uncertainty in the theoretical value for time
constant. Use the instrument data from your course notes to determine this computed
uncertainty.
14. There will be also be uncertainty in the experimental values for time constant due to the
inexact knowledge of when the step input occurred and when the RC output value equals
the theoretical prediction.
15. Determine whether the theoretical (±uncertainty) time constants and the experimental
time constants (±uncertainty) for both 1st order RC systems overlap. If they don’t overlap,
give a reasoned engineering explanation.
Spring ‘11
Lab 6 - System Response
Lab 6 -
1
II. Second Order System Response
A load cell / differential amplifier circuit will be used to measure the dynamic response of a
2nd order, spring – mass combination. Four different masses will be used to generate four different
natural frequencies for the system. The natural frequencies will be used to estimate the spring rate
(or spring constant) of the spring. These spring rates will be compared to a theoretical spring rate
determined from material properties and the physical dimensions of the spring.
1. Build the differential amplifier circuit for the load cell shown in the Load Cell Wiring
Diagrams section of the ME 360 Lab Notes folder provided at each lab station. Note that the
color code for the load cell does not follow the ME 360 conventions. A circuit gain in the
range of -50 to -75 (G = -Rf /Ri) will give a reasonable sensitivity for the 2nd order system.
2. Connect the output from the differential amplifier (Eout) to one of the two input channels of the
data acquisition system / Signal Express. Verify that Signal Express is reading the load cell by
pulling on the spring with 5 or so pounds of force.
3. Weigh your four loads (one at a time!) on your digital scale and record the weights.
4. Each station will have an extension spring that attaches to the load cell via a hook eye. Weigh
the extension spring on your digital scale and record the weight.
5. Attach the largest weight (W1) to the free end of the extension spring. See the lab monitor for
the correct hardware and mounting procedure.
6. Deflect the spring a small amount and release. You should see a damped sine wave response
on the Signal Express screen.
Note – the coils of the spring should never completely “collapse” and touch each other at any
point in the oscillation. If they do you will get an incorrect trace that does not always look like
a damped sine wave. You will need to use a larger initial weight or smaller initial deflection of
the spring if this happens.
7. Set the Samples to Read and Rate in Signal Express to obtain 4 to 7 oscillation periods from
the spring-mass system. Collect enough data at a sufficiently high rate to have at least 10 to 20
samples per cycle. Save the Signal Express data for subsequent analysis (natural frequency)
and plotting.
8. Add one of the smaller weights to the first weight to obtain weight W2. Repeat the steps above
for a second set of experimental damped sinusoidal data.
9. Add another smaller weight to the first two to obtain weight W3. Repeat the steps above for a
third set of experimental damped sinusoidal data.
10. Add the last small weight to the first three to obtain weight W4. Repeat the steps above for a
fourth set of experimental damped sinusoidal data.
11. You will need to obtain three other measurements in the lab in order to determine the
theoretical spring rate, Ktheo:
Measure these two diameters very carefully and several
 wire diameter, d
times each with different students – the calculation for
 spring outer diameter, DO
Ktheo is very sensitive to these values, particularly d.
 number of total coils, NT.
Spring ‘11
Lab 6 - System Response
Lab 6 -
2
The easiest way to determine the number of coils for a tightly wound spring is to measure the
coiled length of the spring and divide by the wire diameter, see Figure 3.
Lcoil
NT 
Lcoil
d
d
DO
Figure 3. Determining Total Number of Coils, NT.
Outside Lab:
12. For each of the four sets of data collected in lab use the experimental damped sine wave
oscillations to determine the experimental period, Texp and the experimental natural frequency,
exp. Also determine the uncertainty in the experimental natural frequency, Uexp.
13. Clearly and completely show the experimental period Texp on plots of the experimental data.
14. Calculate four experimental spring constants, Kexp for the spring from the four measured total
weights (W1 to W4) and corresponding experimental natural frequencies, exp1 to exp4. See the
Unit Conversion Examples section in your ME 360 course manual (or on the website) for an
example. Unit conversions are critical in these calculations!
15. Calculate four additional experimental spring constants, Kexp for the spring by adding one third
of the spring weight to each total weight, W1* = W1 + Wspring/3, etc. to account for the mass of
the spring. Use the corresponding experimental natural frequencies, exp1 to exp4.
16. Calculate the uncertainty in each experimental spring constant due to the uncertainty in the
weights (UWi) and the uncertainty in the corresponding experimental natural frequency, Uexp.
17. Calculate a single theoretical spring constant based on physical dimensions. One commonly
used formula for determining the theoretical spring constant, Ktheo is from Mechanical
Engineering Design, 4th Ed. by Shigley and Mitchell (1983),
d 4G
K theo 
8D 3 N
where
d is the wire diameter (in),
G is the shear modulus of elasticity (Gsteel ~ 11.5·106 lb/in2, UGsteel=±0.5·106 lb/in2)
D is the mean spring diameter, D=DO - d (in), and
N is the number of active coils in the spring (will equal the total number of coils NT for an
extension spring like the ones used in this lab).
18. Compare your eight experimentally determined spring constants and uncertainties, Kexp±UKexp
to the single theoretical spring constant, Ktheo. Does the theoretical formula accurately predict
the experimental spring constant? If not, give a reasonable explanation based on engineering
judgment.