Download MEMS 1041 Rocket Thrust Test Report

MEMS 1041
Rocket Thrust Test Report
Date:
December 3, 2013
Lab Instructor:
Emre Biyikli
Submitted by:
Stanley Schurdak
Introduction
The goal of this project was to design a force transducer to measure the thrust of
an Estes C-65 model rocket engine. The transducer used in this project consisted of a
cantilever beam with a hole at the free end in which the rocket engine could be placed.
Strain gauges were mounted at a specified offset from the fixed end of the beam, and
were used to measure the strain in the beam as it deflected due to the force of the rocket.
Once the relationship between the measured strain and the applied force was established,
the transducer was used to measure the force of the rocket engine. A total of three
engines were tested using this transducer, and the resulting thrust data were analyzed and
compared to those provided by the manufacturer.
Theory
The ultimate goal of this project was to measure the thrust from a model rocket
engine. However, since the force could not be measured directly, a transducer was
necessary to convert the analog force signal into a digital voltage signal that could be
recorded and stored for analysis. Resistance strain gauges mounted on the cantilever
beam served to convert the input force to a measurable change in voltage.
Resistance strain gauges are based on the principle that the electrical resistance of
a material depends on the material’s length and cross-sectional area according to the
following formula:
R=
"e L
Ac
(1)
Where !e is the electrical resistivity of the material, L is the overall length, and Ac is the
cross-sectional area (normal to the direction of electrical current). If any of these
!
dimensions are changed the electrical
resistance of the material will change accordingly.
However, the electrical resistivity of a material is generally assumed to remain constant,
except when the material exhibits piezoelectric characteristics. The infinitesimal change
in electrical resistance of a material may be expressed according to the formula:
dR =
1
(" A dL + Ac Ld"e # LdAc )
Ac 2 e c
(2)
Dividing (2) by (1) gives:
!
dR dL d" e dAc
=
+
#
R
L
"e
Ac
(3)
The cross-sectional area Ac may be expressed in terms of a characteristic length as:
!
!
Ac = bD2
"!
(4)
Where b is a constant and D is the characteristic length. Taking the derivate of (4) yields:
dAc = 2bDdD
(5)
dAc 2dD
=
Ac
D
(6)
Dividing (5) by (4) gives:
Substituting (6) into (3) and noting that the term dL/L in equation (3) is equal to axial
strain (!a) and dD/D in equation (6) is equal to transverse strain (!t) yields the following
!
formula:
dR
d#
= " a + e $ 2" t
R
#e
(7)
Dividing equation (7) by the axial strain !a and noting that – !t/!a is equal to Poisson’s
Ratio, ", the gauge factor may be defined as:
!
GF =
dR
d#e
= 1+
+ 2$
"a R
" a #e
(8)
Rearranging equation (8) to solve for the change in resistance yields:
!
dR = GF"aR
(9)
If the gauge factor for a strain gauge is known, the change in resistance due to a sensed
strain can be calculated. Since the function of transducer was to convert an input force to
a sensible strain, the theoretical axial strain as a function of applied force was derived
from beam theory equations.
The normal stress in a beam due to a bending moment M is:
"=
My
I
(10)
Where # is the normal stress at the point of consideration, M is the moment at that
location, y is the perpendicular distance from the neutral axis (located at middle of a
rectangular beam) to the point !
of consideration, and I is the area moment of inertia. The
strain at this location is related to the stress by Hooke’s law:
# = E"
Where " is the strain at that point. Substituting equation (10) into (11), rearranging to
solve for the strain, and noting that the moment is a function of the offset x gives:
!
#!
(11)
" (x) =
M(x)y
EI
(12)
The bending moment M(x) at any point in the beam is defined as the product of EI and
the second derivative of the deflection of the beam. From beam theory, the deflection at
! the beam is given by2:
any point along the length of
Fx 2 (x " 3l)
d=
6EI
(13)
Where x is the distance from the fixed end of the beam, F is the applied load, l is the
length of the cantilever beam, and E is the elastic modulus of the beam material.
! EI, and taking the second derivative with respect to x gives
Multiplying equation (13) by
the moment as function of offset (x):
M(x) = F(x – l)
(14)
For a beam with a rectangular cross-section the area moment of inertia is defined as:
bh 3
I=
12
(15)
Where b is the width of the beam and h is the thickness. Substituting equations (14) and
(15) into (12), and recognizing that y = h / 2 gives the strain as a function of offset:
!
6F(x # l)
(16)
" (x) =
Ebh 2
Using equations (9) and (16), the change in resistance due to an applied load on the
cantilever beam can be calculated. This change in resistance can then be translated into a
change in voltage and sent !
to a data acquisition unit to store the voltage data and relate
the voltage change to input force.
In designing a device to measure periodic input signals, it is necessary to design
the device with a resonant frequency well above the sensor bandwidth. Therefore, it is
necessary to have knowledge of the anticipated signal input frequencies. If the frequency
content of the input signal is not known, but sampled data of the input is available, a
Fourier analysis of the data can be used to determine the dominant frequencies present in
the signal.
The main concept behind the Fourier analysis is that any complex signal can be
modeled as consisting of sines and cosines of different periods and amplitudes, which are
added together in an infinite trigonometric series1. Mathematically, this means that any
signal y(t) can be described as:
! !
!
$!
!
!!!!!(17)
Where An and Bn are the coefficients of the nth cosine and sine terms. A0, An, and Bn can
be determined from:
(18)
However, equations (18) are useful only if the continuous time signal is known. If only
discrete sampled data are available a discrete Fourier transform must be used to analyze
the frequency content of the signal. The relationship between a continuous signal and its
discrete approximation may be thought of as a set of subsequent impulses:
{y(r$t)} = y(t)$(t – r$t)
r = 0, 1, 2,…, N – 1
(19)
Where $(t – r$t) is the delayed unit impulse function. For a discrete data set of N data
points sampled at $t intervals, the approximation to the integral for the continuous signal
is given by:
(20)
fk = k$f
"f =
k = 0, 1, 2, …,(0.5N – 1)
1
N"t
Computer algorithms have been developed that use the Discrete Fourier Transform
(DFT) of equation (20) for discrete signal analysis. An improvement on the
! in equation (20), known as the Fast Fourier Transform (FFT),
approximation presented
has been developed and is widely used in signal analysis.
If the cantilever sensor design is modeled as an equivalent spring-mass system
with negligible damping, the natural frequency can be determined from:
"n =
keq
meq
(21)
Where %n is the natural frequency, meq is the equivalent mass, and keq is the equivalent
spring constant. Because a cantilever beam is not a system consisting of a simple spring
with a known spring constant!and an attached mass, the beam-equivalent spring constant
!
%!
and equivalent mass must be calculated. For a spring, Hooke’s law states that the applied
force is directly proportional to the length of extension or compression of the spring:
F = kd
(22)
Where F is the applied force, d is the distance of extension or compression, and k is the
spring constant. Rearranging equation (22) to solve for the spring constant gives:
k=
F
d
(23)
From beam theory2, the tip deflection of a cantilever beam due to an applied tip load is
given by:
!
d=
Fl 3
3EI
(24)
Where F is the applied load, l is the length of the cantilever beam, E is the elastic
modulus of the beam material, and I is the area moment of inertia for the beam cross! in the form of (23) gives the equivalent spring
section. Rearranging equation (24)
constant of the beam:
F 3EI
= 3
d
l
(25)
Where b is the width of the beam and h is the thickness. Substituting equation (15) for the
area moment of inertia into (25) gives the equivalent spring constant of the beam in terms
!
of the beam dimensions:
keq =
Ebh 3
4l 3
(26)
The resonant frequency in equation (21) is based on the assumption that the mass of the
system is equal to the attached mass, and that the mass of the spring is negligible. The
mass of the beam and that of!the rocket used in this project were comparable, so an
equivalent mass that accounted for the mass of the spring component (i.e., the beam) had
to be computed. The equivalent mass of the system was assumed to be:
meq = 0.25mbeam + mrocket
(27)
Where mbeam is the mass of the entire beam and mrocket is the mass of the rocket. Since the
mass of the beam depended on the dimensions of the beam, it was necessary to define the
mass of the beam in terms its dimensions:
mbeam = !blh
!
&!
(28)
Where ! is the density of the beam material, and b, l, and h are the dimensions as defined
before. Combining (26), (27), and (28) gives the natural frequency of the beam solely in
terms of beam dimensions and material constants:
Ebh 3 4l 3
"n =
0.25 #blh + mrocket
(29)
Equations (16) and (29) can be used to determine the appropriate dimensions of the beam
that satisfy the resonant frequency and minimum strain requirements.
!
Since the change in resistance from a strain gauge is generally small, and
resistance is not directly measureable, strain gauges are often arranged in a Wheatstone
bridge circuit to convert the change in resistance to a change in voltage. Figure 1 depicts
a general schematic of a Wheatstone bridge.
Figure 1: Schematic of a Wheatstone bridge for strain measurement. Any one or all of the
resistances may be replaced with strain gauges.
The output voltage of the bridge (Eo) may be represented in terms of the input
voltage (Ei) and the branch resistances (R1, R2, R3, R4) according to the formula:
# R1
R3 &
EO = E i%
"
(
$ R1 + R2 R3 + R4 '
(26)
If all of the resistances are replaced with strain gauges, the change in output voltage due
to some applied strain is:
!
# R dR " R1dR2 R3 dR4 " R4 dR3 &
dE O = E i % 2 1
+
(27)
(
2
(R3 + R4 ) 2 '
$ (R1 + R2 )
Substituting equation (9) into equation (12) for all four resistances gives:
!
!
'!
$ R R (GF1" a1R1 # GF2" a 2 R2 ) R3 R4 (GF4" a 4 # GF3" a 3 R3 ) '
dE O = & 1 2
+
)
(R1 + R2 ) 2
(R3 + R4 ) 2
%
(
(28)
If all of the resistances are equal and all of the gauge factors are the same, equation (28)
reduces to:
!
E GF
(29)
dE O = i
("1 # " 2 + " 4 # " 3 )
4
Where GF is the common gauge factor and "i is the strain in each gauge i. Dividing
equation (29) by the input voltage Ei gives the ratio of the change in output voltage to the
!
input voltage as a function
of the four strains.
dE O GF
=
(" # " + " # " 3 )
Ei
4 1 2 4
(30)
If more than one strain gauge is used in a bridge circuit, the arrangement can be
characterized by a bridge constant, #, which is defined as the ratio of the actual bridge
!
output to the bridge output
if there were only one gauge sensing the maximum strain. The
use of multiple gauges has the effect of multiplying the output of the bridge with multiple
gauges over that of a single gauge by a factor of #:
& = dEo,actual / dEo,single
(31)
The transducer design for this project incorporated a full bride circuit arrangement, with
pairs of gauges sensing equal and opposite strains. The corresponding bridge constant for
this arrangement is given by Figliola1 to be: # = 4.
The Wheatstone bridge alone is not capable of producing a substantial voltage
change, so active amplifiers are used to amplify the output voltage to a measurable level.
Figure 2 depicts the circuit schematic for the amplifier used in this project.
Figure 2: Amplifier circuit used for signal conditioning
!
(!
The amplifier increases the input signal by a factor of Ei(R2 / R1), where Ei is the input
voltage signal. Different combinations of resistors can be used to vary the amplification
or gain of the circuit to the required level.
In order to minimize noise in the data obtained from the device, an active highpass filter is used to attenuate useless signals of high frequencies. Figure 3 depicts the
circuit schematic for the high-pass filter used in this project.
Figure 3: Schematic of the filter used for signal conditioning
The main function of the filter is to attenuate signals with frequencies above the cutoff
frequency of: fc = 1 / R2C. Because the filter has an input voltage (i.e., because it is an
active filter), it also functions to amplify the input signal by a factor of R4 / R3. When
amplifying components are used in series, the total gain is equal to the product of the
individual gains. The signal conditioning circuitry for this project used an amplifier in
series with a low-pass active filter, giving a total gain of: R2R4 / R1R3.
Procedure
In order to design an adequate transducer, the expected magnitude and frequency
of the input forces (time-dependent thrust from the rocket) needed to be quantified. A
frequency analysis of the force versus time data provided by the manufacturer3 revealed
that the frequency content of the expected input signal consisted of only a few significant
low-level frequencies (i.e., less than 5 Hz). From this analysis the bandwidth of the
sensor was recommended to be no less than 100 Hz. In order to avoid exciting the
cantilever design at its resonant frequency, the beam dimensions were determined using
equations (16) and (29) so that the resonant frequency was greater than the expected input
frequency range.
Once the appropriate beam dimensions were determined, the beam was machined
according to the schematic in Figure 4. First, the overall length of the beam (5.75 in) was
cut from 1.00x0.125 in aluminum bar and the cut edge was ground. The 0.7 in hole was
then cut with a drill press and the burrs inside the cut were removed. The middle section
of the beam was then machined in a CNC press using a 3/16 in bit. Finally, the 0.0625 in
setscrew hole was drilled and tapped.
!
)!
After the machining was completed, four strain gauges were attached to the beam
at the specified offset (0.25 in). Each gauge had a nominal resistance of 120 ohms and a
gauge factor of 2.11. Two gauges were mounted on top of the beam and two were
mounted on the opposite side. This full bridge configuration was chosen in order to
maximize the accuracy of the sensor.
Figure 4: Schematic of the cantilever beam design
After mounting the strain gauges, the signal amplifier and filter were built as
shown in Figures 2 and 3, respectively. The amplifier was designed with resistor R1 equal
to 120 ohms, and R2 equal to 3300 ohms. The designed gain for this circuit component
was: R1/R2 = 27.5. The resistors in the low-pass filter were designed with R3 equal to
1500 ohms and R4 equal to 10000 ohms, giving a gain of: R3/R4 = 6.67. The capacitor
element had a value of 0.33 µF, and the resulting cutoff frequency of the filter with the
designed component values was calculated to be: fc = (2'R4C)-1 = 48.22 Hz. The total
gain of these two circuit components with the values as designed was: (27.5)(6.67) =
183.3.
Once the signal conditioning circuitry was complete the sensor was calibrated.
Seven different masses ranging from 100 to 1900 grams were hung from the free end of
the beam for two seconds, and the resulting voltage was recorded using a MatLab data
acquisition session. Five trials were run for each mass, giving a total of thirty-five data
points. The data were then plotted and the equation of the best-fit line was computed.
The MatLab data acquisition script used to record the data obtained from the
rocket test was the same script used to gather the calibration data. The script was set to
run for a specified amount of time particular to the session: two seconds for the
calibration session and forth-five seconds for the rocket testing session. The script for
these sessions may be found in Appendix I.
After the sensor was calibrated, the rocket engines were tested. Because the actual
testing took place in a much cooler outdoor environment, the device needed to be recalibrated to account for the ambient conditions and to ensure that the sensor was
!
*!
operating properly. The calibration readings taken on site indicated that the sensor was
working correctly. However, immediately after calibration one of the wires broke off
from its solder joint and rendered the sensor useless. An attempt to re-solder the wire
failed and two other solder connections were broken off in the process. A stock sensor
and its corresponding conditioning circuitry, instead of the designed sensor and circuitry,
were then used to gather the rocket thrust data. Additionally, the MatLab script ran into
runtime errors, so another script was used instead.
Uncertainty
The factors considered as contributing to the overall uncertainty of the sensor
included the tolerances in the machining of the beam, variations in the circuit element
values, data acquisition unit resolution, and calibration uncertainty. Using the root-sumsquare method of calculating overall uncertainty due to various contributing factors, a
total device uncertainty of 8.211% was estimated. Additionally, from the measured data it
was determined that a total of 20 tests would have to be made to keep the total
uncertainty in the results below 10%. The calculations and methods used may be found in
Appendix II.
Results
Figure 5 shows the results of the frequency analysis from the thrust-versus-time
data provided by the manufacturer3. The magnitude of the frequencies began to fall off
above 5 Hz, so the dominant frequencies in the input signal were expected to fall below 5
Hz.
!
"+!
Figure 5: Frequency analysis of the manufacturer-provided force-time data. Since the
magnitudes of the frequencies greater than 5 Hz were comparatively negligible, the input
signal was expected to consist of dominant frequencies below 5 Hz.
Table 2 summarizes the beam dimensions, beam material properties, equivalent
mass, resonant frequency of the beam, and strain at expected peak force. The equivalent
mass, resonant frequency, and the strain at the expected peak force were calculated using
equations (23), (16), and (25), respectively.
Table 2: Final beam design dimensions and material parameters.
Beam Material
Aluminum
Modulus of Elasticity (E, psi)
1040000
Density ($, lbf/in3)
0.098
Beam Length (in)
2.75
Beam Width (in)
0.375
Beam Thickness (in)
0.125
Strain Gauge Offset (x, in)
0.25
Equivalent Mass (lbm)
0.069
Resonant Frequency (Hz)
111.4
Peak Strain at 3.37 lbf (15 N)
871.6
The expected gains of the amplifier and filter using the nominal circuit element
values, as well as the predicted cutoff frequency of the filter are listed in Table 3.
Table 3: Circuit element values for the amplifier and filter components
Circuit Element
Active Low-Pass Filter
Amplifier
R1 (ohm)
1500
120
R2 (ohm)
10000
3300
C (µF)
0.33
Gain (R2/R1)
6.67
27.5
fc (Hz)
48.22
Total Circuit Gain
183.3
183.3
Table 4 lists the calibration data from both the designed beam and the stock beam
used in the rocket testing session. Figures 6 and 7 show the calibration data and the bestfit line for the designed beam and the beam actually used, respectively.
Table 4: Calibration data for the designed beam and the actual beam used in testing
Mass (g) Force (N) Force (lbf) Designed Vout (V) Theoretical (V) Used Vout (V)
100
0.981
0.221
-0.151
0.315
9.397
200
1.962
0.441
-0.055
0.630
9.323
400
3.924
0.882
0.117
1.260
9.184
500
4.905
1.103
0.212
1.575
9.082
1000
9.810
2.205
0.628
3.150
8.668
1500
14.715
3.308
1.046
4.725
8.314
!
""!
1900
18.639
4.190
1.40
5.985
Figure 6: Calibration data and best-fit line for the designed cantilever beam
!
"#!
-
Figure 7: Calibration data and best-fit line for the beam used in the rocket test
The thrust versus time curves from the rocket testing session are shown in Figure
8. The voltage read into the data acquisition unit was converted to force using the
equation in Figure 6.
Figure 8: Comparison of the rocket test results to the manufacturer data. Test 3 showed a significant
offset, so the data was normalized by the minimum recorded force.
Thrust data gathered from the three rocket tests indicated that there was some constant
offset for each run, most noticeable in test three. In order to compare the test data with
the manufacturer data, each run was normalized by its minimum thrust value in the time
interval of interest. Figure 8 shows the comparison of the normalized tests to the
manufacturer data.
!
"$!
Figure 9: Comparison of the test data normalized by the minimum recorded thrust and the
manufacturer data. The test data indicate significantly lower peak thrusts, but slightly longer burn
times.
Figure 10 shows the thrust versus time data for the three tests and compares the time
delay to ejection charge.
Figure 10: Comparison of the time delay for the three rocket tests. Test 1 showed a delay of
approximately 6 seconds, Test 2 showed a delay of approximately 6.8 seconds, and Test 3 showed a
delay of about 7.3 seconds.
!
"%!
Table 5 lists the peak thrust, time delay for ejection charge, total impulse, and
total burn time for both the rocket tests and the manufacturer data.
Table 5: Comparison of manufacturer data to pertinent data obtained from tests
Manufacturer
Test 1
Test 2
Test 3
Peak Thrust (lbf)
3.167
1.879
2.306
2.192
Time Delay (sec)
6.240
7.336
5.888
6.640
Impulse (lbf-sec)
1.983
1.644
1.641
1.802
Burn Time (sec)
1.860
2.048
1.960
2.152
Table 6: Comparison of average test data and standard deviation to manufacturer
data
%
Manufacturer
Test Average
sx
Peak Thrust (lbf)
3.167
0.661
2.13
0.221
Impulse (lbf-sec)
1.983
0.041
1.70
0.092
Burn Time (sec)
1.860
0.310
2.053
0.096
Discussion
The calibration data obtained from measurements did not corresponded with the
theoretically predicted data. The theoretically predicted output voltage was approximately
four times greater than the experimentally determined voltage, which could have been
caused by variations in the circuit elements such as the resistors, which generally showed
significant deviations from their nominal values (e.g., ±200 ohms for a 1.5 kohm
resistor). Likewise, many of the operational amplifiers used in the circuit did not function
properly, often distorting input signals. The resistors used in the circuitry were obtained
from a generally sorted supply of resistors and it is possible that a resistor with a
significantly different value than the designed value was used in the circuit, which could
explain the large discrepancies.
From the thrust data obtained from the three tests, the largest peak thrust
measured was 2.31 lbf (from Test 2), approximately 0.8 lbf less than that given by the
manufacturer. The tests indicate that on average the measured peak thrust is nearly 1 lbf
less than that predicted by the manufacturer. There was very little variation in peak thrust
measured across all three runs, which had a standard deviation (after normalization) of
only 0.221 lbf.
The ejection charge time delays were, in general, not very consistent. The three
test showed a maximum time delay of 7.3 seconds, and a minimum delay of 5.9 seconds,
which fall within a range of approximately ±1 second of the manufacturers predicted
value. Considering that the total burn time for the engine was approximately two seconds,
these variations in time delay could be significant.
All three tests indicated total impulses that were less than the manufacturer
predicted value. Test 3 showed the value closest to that predicted by the manufacturer,
but it did not fall within the stated range (±0.041 lbf-sec). On average, the impulse
!
"&!
indicated by the tests was significantly lower than the predicted value, with a difference
of approximately 0.28 lbf-sec (1.25 N-sec).
The use of the sample beam and circuit, instead of the designed components,
could be responsible for some of the differences between the observed and manufacturer
predicted measurements. The calibration data obtained for the sample beam used in the
test came from only one trial, and, in the interest of time, no check was made to make
sure that sequential measurements produced similar results. However, both Test 1 and
Test 2 showed fairly similar results, indicating that the device was at least consistent. The
data from test three indicate that the overall load on the beam, assuming that the device
remained consistent, was much greater than the previous two tests. However, when the
data from Test 3 were normalized by the approximate bias (approximately 3.82 lbf (17
N)), the results appeared to be consistent with the other tests. This apparent offset could
have been caused by mishandling the sensor between tests, or by a circuit element
malfunctioning. Many of the operational amplifiers available for circuit construction did
not work properly and had to be discarded, and could this have been responsible for the
significant offset. Additionally, observations of the input signal between Tests 2 and 3
indicated that the zero force signal was increasing steadily, but no steps were taken to
assess the cause of the phenomenon. This could have also contributed to the differences
in the third data set. Finally, it was observed that the flames from the ejection charge
consistently contacted the beam, possibly heating it as well as the strain gauges. It was
not known how sensitive the gauges were to extreme thermal conditions, but it was
assumed that the gauges were not designed for such rapid temperature changes, so it is
possible that the ejection charges could have affected the performance of the gauges.
Conclusion
The results of the tests indicate that the specimens tested did not meet the
manufacturer specifications. All peak thrusts measured were significantly less than that
predicted by the manufacturer, as were the calculated impulses. However, the tests
indicated that overall, the total burn time was actually longer than that predicted by the
manufacturer. Finally, it must be noted that the test data was compared to manufacturer
data that was obtained nearly thirteen years prior to the current tests. It is not known
whether this somewhat antiquated data is truly representative of the rockets used in the
test, and this could possibly account for the observed discrepancies.
References
1.
2.
3.
!
Figliola, Richard S., and Donald E. Beasley. Theory and Design for Mechanical Measurements.
Hoboken, NJ: Wiley, 2011. Print.
Budynas, Richard G., and J. Keith Nisbett. Shigley's Mechanical Engineering Design. New Delhi:
McGraw Hill, 2011. Print.
NAR Published Data: Estes C6. Rep. N.p.: NAR Standards and Testing, 1998. Web. 1 Dec. 2013.
<https://courseweb.pitt.edu/bbcswebdav/pid-15591779-dt-content-rid9207211_2/courses/2141_UPITT_MEMS_1041_SEC1050/Project/Thrust_Curve_C6.pdf>.
"'!
Appendix I: MatLab script written for data acquisition
clear all
%Stephanie Lee
%Mechanical Measurements 1
ansyn = 'y';
runnum = 0;
while ansyn == 'y'
runnum = runnum + 1;
trial = input('trial: ','s');
%%grams = input('grams: ','s');
%Connect to Device
%daq.getDevices
s = daq.createSession('ni');
s.addAnalogInputChannel('dev3','ai0','Voltage');
%Establish Data Acquisition Parameters
s.Rate = 500;
s.DurationInSeconds = 20;
daq.Range(-5,5,'Volts');
%Event Listener
listen = s.addlistener('DataAvailable',@(s,event)plot(event.TimeStamps,event.Data));
%Collect Data
[data, time] = s.startForeground();
switch runnnum
case 1
data1 =
time1 =
case 2
data2 =
time2 =
case 3
data3 =
time3 =
case 4
data4 =
time4 =
case 5
data5 =
time5 =
case 6
data6 =
time6 =
case 7
data7 =
time7 =
end
data;
time;
data;
time;
data;
time;
data;
time;
data;
time;
data;
time;
data;
time;
%End Session
if s.IsRunning == 0
save(strcat('RocketTest_',trial','.mat'),'data', 'time');
s.stop
delete(s)
clear('s')
end
ansyn = input('Run again? (y/n) :','s');
end
plot(time,data);
xlabel('Time');
ylabel('Voltage');
title('Voltage vs Time');
grid on
Appendix II: Uncertainty Calculations
The total uncertainty in the measurement device consisted of two main sources:
uncertainty in the components of the device and uncertainties in calibration. The
following calculations enumerate the sources of uncertainty taken into consideration.
In order to determine the overall uncertainty, the uncertainties each component
was taken into account using the root-sum-square method indicated in equation A2.1:
" ua1 % 2 " ua 2 % 2
" uan % 2
ux
= $ ' + $ ' + ...+ $ '
x
# a1 & # a2 &
# an &
(A2.1)
This gives the percent uncertainty in x due to the percent uncertainties from the
contributing factors a1 through an. If a contributing factor ai is related to other variables
! ai = f(y1, y2, … yk), then the uncertainty uai is calculated using
y1, y2, … , yk, such that
equation A2.2:
# "ai & 2 # "ai
&2
# "ai & 2
uai = %
uy1 ( + %
uy 2 ( +… + %
uyk (
$ "y1 ' $ "y 2 '
$ "y k '
(A2.2)
If the relationship between ai and y1, y2, … , yk is of the form ai = y1c1y2c2…ykcn, then, in
terms of percent uncertainty, equation A2.2 becomes:
!
" uy1 % 2 " uy 2 % 2
" uyk % 2
uai
= $c1 ' + $ c 2
' +… + $c n
'
ai
# y1 & # y 2 &
# yk &
(A2.3)
The uncertainty of each element in the device (e.g., that of the resistors) was
calculated using equation (A2.3), and that arising from the components of the device was
! equation (A2.1). The overall uncertainty was also calculated using
calculated using
equation (A2.1), considering the uncertainty of the components and that of the calibration
as the only two contributing factors. The elements considered and results of the
calculations are listed in Table A2.1.
The theoretical strain sensed by the strain gauge is given by equation A2.4:
" (x) =
6Fl
Ebh 2
(A2.4)
Where l is the distance from the tip to the point of measurement, E is Poisson’s ratio for
the beam material, and b and h are the width and thickness of the beam, respectively. The
total percent uncertainty in !
the strain using equation (A2.3) is:
!
"(!
# ul & 2 # ub & 2 # uh & 2
u"
= % ( + % ) ( + % )2 (
$l' $ b' $
"
h'
(A2.5)
In general, the uncertainty introduced due to machining tolerances is typically
within ±0.005 in, so this range was used for the uncertainty of b. Placement of the strain
! to human error, so an uncertainty of ±0.0625 in was assumed for
gauges were susceptible
l. Likewise, an uncertainty of ±0.002 in was assumed for variations in the thickness, h, of
the bar.
The change in resistance of a strain gauge is determined from:
dR = GFR"
(A2.6)
Therefore, the uncertainty in the change in resistance is:
" uGF % 2 " uR % 2 " u( % 2
udR
= $
' +$ ' +$ '
dR
# GF & # R & # ( &
(A2.7)
The uncertainty in the gauge factor (GF) was not explicitly given, so a minimal
uncertainty of 1% was assumed. The nominal resistance, R, of the gauges was stated to be
!
20 ohms, but measurements
of the resistance revealed a consistent value of 123 ohms, so
a variation of ±4 ohms was assumed.
The output voltage from the Wheatstone bridge is given by:
Eo = Ei
" dR
4 R
(A2.8)
Where Ei is the bridge input voltage, and & is the bridge constant. The uncertainty in the
output voltage is therefore:
!
" uEi % 2 " uR % 2 " u( % 2
uEo
(A2.9)
= $ ' +$ ' +$ '
Eo
# Ei & # R & # ( &
The bridge constant was assumed to have negligible variation, so the uncertainty was
assumed to be 0. The uncertainty in the resistance was assumed to be ±4 ohms as before,
! the input voltage was assumed to be ±0.5 V.
and the uncertainty in
The resolution of the data acquisition unit used to collect the voltage data was
stated to be 12 bits. Converting this value into a resolution in voltage for a ±5 V range
(10 V full-scale range) gives:
Q=
!
!
10V
= 0.00244V
212
")!
(A2.10)
Assuming the zero-order uncertainty to be half of the data acquisition unit resolution (see
reference 1), the uncertainty of the unit was found to be 0.00122 V (1.22 mV), giving a
percent uncertainty of: u0 / EFSR = 0.00122 / 10 = 0.000122 (0.0122%).
The overall design stage uncertainty of the device was estimated using equation
(A2.1) and using the percent uncertainties obtained from equation (A2.9) and the zeroorder uncertainty as arguments. This gives the design stage uncertainty of the readout
voltage as:
" uEo % 2 " u0 % 2
uVo
= $ ' +$
'
Vo
# E o & # E FSR &
(A2.11)
A least-squares fit of the average calibration data was made using the cftool in
MatLab. The fit yielded the voltage as a linear function of input force as:
!
V = 0.3867F – 0.2263
(A2.12)
The uncertainty between the data and the fit was estimated from (see reference 1):
sVF =
# (V
i
" Vci ) 2
$
( = N – (m + 1), N = 7, m = 1
!
ucal =
(A2.13)
t" ,95 sVF
N
Equations (A2.13) yielded an uncertainty in the fit of ±0.0075 V, at 95% confidence.
Dividing this result by the full-scale voltage range gives the percent uncertainty in
!
calibration as:
ucal
0.0075V
=
= 0.00075(0.075%)
(A2.14)
E FSR
10V
Table A2.1 summarizes the sources of uncertainty considered and their
corresponding uncertainties, component uncertainties, and overall uncertainty of the
!
device.
!
"*!
Table A2.1: Elements considered in total measurement-device uncertainty
Component
Value
Uncertainty
Uncertainty
Component
(%)
Uncertainty (%)
Strain Gauge
2.75
0.0625
2.273
Placement (in)
Thickness (in)
0.125
0.002
1.600
Width (in)
0.375
0.005
1.333
Strain Uncertainty
4.145
Gauge Factor
2.11
0.021
1.000
Gauge Resistance (&)
120
4.000
3.333
Strain
4.145
dR Uncertainty
5.412
Input Voltage (V)
15
0.500
3.333
Nominal Resistance
120
4.000
3.333
(&)
dR (&)
5.412
Vout Uncertainty
7.247
R1 (&)
1500
20.000
1.333
R2 (&)
10000
20.000
0.200
Vout (V)
7.247
Vout, amplifier
7.737
Uncertainty
R3 (&)
120
4.000
3.333
R4 (&)
3300
20.000
0.606
C (F)
3.3E-8
0.000
0.000
Vout, amplifier (V)
7.737
Vout, filter Uncertainty
8.112
Calibration
10
0.00075
0.075
0.075
Uncertainty (V)
DAQ Uncertainty
10
0.00122
0.0122
0.012
Total Uncertainty
8.112
In order to determine the total number of runs necessary to maintain an overall
uncertainty below 10%, the following formula was used (see reference 1):
(A2.15)
Where NT is the total number of runs required, s1 is the sample standard deviation of the
mean, tN-1, 95 is the t-statistic computed at N – 1 degrees of freedom and 95% confidence,
and d is half the size of the interval required to be below the specified uncertainty. From
the three tests performed, the mean peak thrust was computed to be 2.13 lbf with a
standard deviation of 0.221. In order to be within ±10% uncertainty of the mean value,
the standard deviation should have a range of ±0.213 lbf. This gives d = 0.213. Since
!
#+!
there are only three tests (N = 3), the t-statistic is evaluated at two degrees of freedom
and 95% confidence. This gives t2, 95 = 4.303. Substituting these values in to equation
(A2.15) gives the total number of runs required:
Therefore, a total of at least twenty tests are required to maintain a total uncertainty
below 10%.
!
#"!