Engineering Instrumentation and Measurement Experiments
Engineering Instrumentation and Measurement Experiments
Engineering Instrumentation and Measurement Experiments
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
MAE 300 - <strong>Engineering</strong><br />
<strong>Instrumentation</strong> <strong>and</strong> <strong>Measurement</strong><br />
Calibration of an Electronic Load Cell<br />
<strong>and</strong> a Bourdon Tube Pressure Gage<br />
Experiment No. 2<br />
Kai Gemba, 003678517<br />
California State University<br />
Department of Mechanical <strong>and</strong> Aerospace <strong>Engineering</strong><br />
MAE 300 - <strong>Engineering</strong> <strong>Instrumentation</strong> <strong>and</strong> <strong>Measurement</strong><br />
Instructor: Rahai, Hamid R.<br />
October 14, 2007
Abstract<br />
Calibration procedures were performed by students on an electronic load-cell scale <strong>and</strong> mechanical<br />
Bourdon tube-type pressure-gage scale. Both apparatuses were incrementally loaded to<br />
full scale <strong>and</strong> data was recorded for both the load <strong>and</strong> unload direction. The data was plotted,<br />
statistically evaluated <strong>and</strong> the method of least squares was applied to determine a best-fit analytical<br />
expression for the calibration function. The Bourdon Gage output was closely correlated<br />
to a linear calibration function throughout the test range, R 2 vales ranging from 0.996 to 0.9993.<br />
Hysteresis in the unload direction was less than the output gage precision. The strain gage output<br />
was less linear but a fist order polynomial was used, too. R 2 values ranging from 0.981 to 0.978.
Experiment No. 2<br />
Contents<br />
1 Objective of experiment 1<br />
2 Background <strong>and</strong> Theory 1<br />
2.1 Intercept Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
2.2 St<strong>and</strong>ard error of the linear relationship . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
2.3 Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
3 Experimental procedure 3<br />
4 Experimental data 4<br />
5 Calculations <strong>and</strong> Results 5<br />
5.1 Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
5.2 Strain Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
5.3 Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
6 Discussion of results 10<br />
6.1 Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
6.2 Strain Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
7 Conclusions <strong>and</strong> recommendations 10<br />
Bibliography 11<br />
i
Experiment No. 2<br />
List of Figures<br />
1 Bourdon Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2 Bourdon Calibration Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />
3 Strain Gage Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
4 Strain Gage Calibration Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
List of Tables<br />
1 Stain Gage Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2 Bourdon Gage Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
3 Curve Fit Calculation for Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
ii
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
1 Objective of experiment<br />
The stated objective of the experiment is to familiarize students with the calibration procedure <strong>and</strong><br />
method of least squares<br />
2 Background <strong>and</strong> Theory<br />
The following descriptions of the equipment <strong>and</strong> the analytic methods for evaluating the data were<br />
obtained from the web resources noted in the bibliography <strong>and</strong> are consistent with the presentation<br />
in the course textbook.<br />
In a Bourdon tube gauge, a C shaped, hollow spring tube is closed <strong>and</strong> sealed at one end. The<br />
opposite end is securely sealed <strong>and</strong> bonded to the socket. As the gauge pressure increases the tube<br />
will tend to uncoil, while a reduced gauge pressure will cause the tube to coil more tightly. This<br />
motion is transferred through a linkage to a gear train connected to an indicating needle. The needle<br />
is presented in front of a card face inscribed with the pressure indications associated with particular<br />
needle deflections.<br />
Two points of the wheatstone bridge are connected to an exciter voltage (from the battery or AC<br />
adapter) <strong>and</strong> an output analog voltage feed to a A/D Converter. The output voltage being fed in<br />
the A/D varies in proportion to the load applied to the platform of the scale. This occurs since the<br />
weighing platform of a scale is connected to the end of the load cell via a post. The applied force<br />
is transferred from the platform, through the post <strong>and</strong> onto the aluminum beam. When the beam<br />
bends the strain gauges bend resulting in their resistance value to change <strong>and</strong> the voltage changes<br />
in proportion to the load applied to the platform.<br />
Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when<br />
given a series of measured data, attempts to find a function which closely approximates the data<br />
(best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals)<br />
between points generated by the function <strong>and</strong> corresponding points in the data. Specifically, it is<br />
called least mean squares (LMS) when the number of measured data is 1 <strong>and</strong> the gradient descent<br />
method is used to minimize the squared residual. LMS is known to minimize the expectation of<br />
the squared residual, with the smallest operations (per iteration). But it requires a large number of<br />
iterations to converge.<br />
An implicit requirement for the least squares method to work is that errors in each measurement be<br />
r<strong>and</strong>omly distributed. The Gauss-Markov theorem proves that least square estimators are unbiased<br />
<strong>and</strong> that the sample data do not have to comply with, for instance, a normal distribution. It is also<br />
important that the collected data be well chosen, so as to allow visibility into the variables to be<br />
solved for (for giving more weight to particular data, refer to weighted least squares).<br />
1
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
2.1 Intercept Calculations<br />
We sum the observations, the squares of the Ys <strong>and</strong> Xs <strong>and</strong> the products X × Y to obtain the<br />
following quantities.<br />
<strong>and</strong> Sy similarly.<br />
<strong>and</strong> S + Y Y similarly<br />
SX = x1 + x2 + ... + xn<br />
SXX = x 2 1 + x 2 2 + ... + x 2 n<br />
SXY = x1y1 + x2y2 + ... + xnyn<br />
We use the summary statistics above to estimate the slope β.<br />
β ≈ mSXY − SXSY<br />
nSXX − SXSX<br />
We use the estimate of β to estimate the intercept, α:<br />
α ≈ SY − βSX<br />
(2)<br />
n<br />
A consequence of this estimate is that the regression line will always pass through the center of the<br />
data.<br />
2.2 St<strong>and</strong>ard error of the linear relationship<br />
The following is excerpted from Applications, Basics, <strong>and</strong> Computing of Exploratory Data Analysis,<br />
Velleman <strong>and</strong> Hoaglin, Wadsworth, Inc. 1981.<br />
A fundamental step in most data analysis <strong>and</strong> in all exploratory analysis is the computation <strong>and</strong><br />
examination of residuals. [...] Most analysts propose a simple structure or model to begin describing<br />
the patterns in the data. Such models differ widely in structure <strong>and</strong> purpose, but all attempt to<br />
fit the data closely. We therefore refer to any such description of the data as a fit. The residuals<br />
are, then, the differences at each point between the observed data value <strong>and</strong> the fitted value, or the<br />
actual value <strong>and</strong> the predicted value: Residual = Actual value − Predicted value.<br />
The median-median line provides one way to find a simple fit, <strong>and</strong> its residuals, r, are found for<br />
each data value, (xi, yi) as<br />
ri = yi − (axi + b)<br />
A pessimist might view residuals as the failure of a fit to describe the data accurately. He might even<br />
speak of them as errors, although a perfect fit, which leaves all residuals equal to zero, would arouse<br />
suspicion. An optimist sees in residuals details of the data’s behavior previously hidden beneath<br />
the dominant patterns of the fit. Both points of view are correct. The best fits leave small residual,<br />
<strong>and</strong> systematically large residuals may indicate a poorly chosen model. Nevertheless, even a good<br />
fit may do nothing more than describe the obvious.<br />
Any method of fitting models must determine how much each point can be allowed to influence<br />
the fit. Many statistical procedures try to keep the fit close to every data point. If the data include<br />
an outlier, these procedures may permit it to have an undue influence on the fit. As always in<br />
exploratory data analysis, we try to prevent outliers from distorting the analysis. Using medians in<br />
fitting lines to data provide resistance to outliers, <strong>and</strong> thus the line fitting technique [...] is called<br />
the resistant line [2].<br />
2<br />
(1)
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
What we will do is to look at a scatter plot of our residuals. This scatter plot may be able to tell us<br />
what we might try if we do not have a good linear fit. In addition, the residuals measure a signed<br />
distance of the predicted value from the actual value. We get the st<strong>and</strong>ard error by<br />
�<br />
�<br />
� n�<br />
�<br />
� r<br />
� i=1<br />
std.err =<br />
2 i<br />
(3)<br />
n − 2<br />
This a measure of type of average distance from the data to the predicted values.<br />
2.3 Coefficient of Determination<br />
Excel reports the coefficient of determination R 2 . which is the proportion of variability in a data<br />
set that is accounted for by a statistical model. In this definition, the term variability st<strong>and</strong>s for<br />
variance or, equivalently, sum of squares. There are several common <strong>and</strong> equivalent expressions for<br />
R2. The version most common in statistics texts is based on an analysis of variance decomposition<br />
as follows:<br />
In the above definition,<br />
R 2 = SSR<br />
SST<br />
= 1 − SSE<br />
SST<br />
SST = �<br />
(yi − y) 2<br />
i=1<br />
SSR = �<br />
( ˆyi − y) 2<br />
i=1<br />
SSR = �<br />
( ˆyi − ˆyi) 2<br />
i=1<br />
That is, SST is the total sum of squares, SSR is the regression sum of squares, <strong>and</strong> SSE is the sum<br />
of squared errors. In some texts, the abbreviations SSE <strong>and</strong> SSR have the opposite meaning: SSE<br />
st<strong>and</strong>s for the explained sum of squares (which is another name for the regression sum of squares)<br />
<strong>and</strong> SSR st<strong>and</strong>s for the residual sum of squares (another name for the sum of squared errors).<br />
3 Experimental procedure<br />
The Bourdon gage was accompanied by a set of weights labeled in nominal increments of 100 psi<br />
<strong>and</strong> 500 psi. The weights were incrementally loaded on the Bourdon platform <strong>and</strong> the pressure<br />
valve associated with the piston supporting the platform was opened until the piston raised to<br />
mid-stroke. The platform was rotated to resolve any static friction <strong>and</strong> the output pressure values<br />
were recorded. This procedure was repeated until the limit of the scale was reached <strong>and</strong> then the<br />
procedure was reversed until the scale was unloaded. The pressure gage increments were 100 psi.<br />
Output values were interpolated, nominal precision of the instrument should be considered +/-<br />
50 psi. The strain gage load cell supported an aluminum platform approximately 8 inches square.<br />
The tare was adjusted to zero on the digital display system before loading the platform. A series<br />
of weights in nominal increments of 1 pounds were added to the platform. At maximum load, the<br />
procedure was reversed until the scale was unloaded. The output was recorded to a precision of<br />
0.001 mv for each load state however the nominal precision of the output was +/- .005 mv.<br />
3<br />
(4)
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
4 Experimental data<br />
Table 1 <strong>and</strong> table 2 display the experimental data.<br />
Table 1: Stain Gage Calibration Data<br />
Input [lbs] Load [mv] Unload [mv]<br />
0 0.000 0.000<br />
1.5 0.075 0.100<br />
2.5 0.150 0.170<br />
3.5 0.210 0.230<br />
4.5 0.270 0.290<br />
5.5 0.330 0.350<br />
6.5 0.390 0.400<br />
7.5 0.450 0.440<br />
8.5 0.490 0.470<br />
9.5 0.520 0.510<br />
11 0.570 0.560<br />
13 0.640 0.640<br />
Table 2: Bourdon Gage Calibration Data<br />
Input [psi] Load [psi] Unload [psi]<br />
0 0 0<br />
100 125 120<br />
300 555 550<br />
500 990 1000<br />
700 1400 1420<br />
1200 2400 2450<br />
1700 3410 3425<br />
4
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
5 Calculations <strong>and</strong> Results<br />
5.1 Bourdon Gage<br />
The Bourdon Calibration Figure 1 shows the raw data <strong>and</strong> a least-squares linear curve fit to the<br />
data. The Bourdon Calibration first order polynomial shows a first order least squares fit to the<br />
data. The Bourdon Calibration Residuals illustrates the error of the data to the associated analytical<br />
expression. The following analytical calibration were calculated from the data: Loading direction f(x)<br />
= 2.0299x - 36.3082 +/- 50psi Calculated st<strong>and</strong>ard error of data 27 psi with a st<strong>and</strong>ard deviation<br />
of errors 25 psi. Unloading direction, f(x) = 2.0509x - 37.7478 +/- 50psi. Calculated st<strong>and</strong>ard error<br />
of data was found to be 36 psi <strong>and</strong> st<strong>and</strong>ard deviation of errors 35 psi.<br />
Output (psi)<br />
4000<br />
3500<br />
3000<br />
2500<br />
2000<br />
1500<br />
1000<br />
500<br />
0<br />
-500<br />
Bourdon Calibration<br />
0 500 1000 1500 2000<br />
Input (psi)<br />
Figure 1: Bourdon Calibration<br />
5<br />
Load Linear Fit<br />
y = 2.0299x - 36.3802<br />
R² = 0.9996<br />
Unload Linear Fit<br />
y = 2.0509x - 37.7478<br />
R² = 0.9993<br />
Unload (psi)<br />
Load (psi)
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
Output (psi)<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
-10<br />
-20<br />
-30<br />
-40<br />
-50<br />
-60<br />
Bourdon Calibration Residuals<br />
0 500 1000 1500 2000<br />
Input (psi)<br />
Figure 2: Bourdon Calibration Residuals<br />
6<br />
Unload<br />
(psi)<br />
Load (psi)
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
5.2 Strain Gage<br />
The Strain Gage Calibration Figure 3 shows the raw data. A cursory examination of this graph<br />
reveals a linear character. The Strain Gage Polynomial Curve Fit shows a fist order polynomial<br />
least squares fit to the data to the data points of the set. The Strain Gage Calibration Residuals<br />
illustrates the error of the data to the associated analytical expression. The following analytical<br />
calibration were calculated from the data. Calculated st<strong>and</strong>ard error of loading data; 0.029 mv, its<br />
st<strong>and</strong>ard deviation 0.028 mv<br />
f(x) = 0.0510 × x + 0.028 ± .029<br />
Calculated st<strong>and</strong>ard error of unloading data: 0.030 mv, its st<strong>and</strong>ard deviation .0284 mv.<br />
Output (mv)<br />
0.800<br />
0.700<br />
0.600<br />
0.500<br />
0.400<br />
0.300<br />
0.200<br />
0.100<br />
0.000<br />
f(x) = 0.048 × x + 0.049 ± 0.030<br />
Strain Gage Curve Fit<br />
0 5 10 15<br />
Input (lbs)<br />
Figure 3: Strain Gage Calibration<br />
7<br />
Load Curve Fit<br />
y = 0.051x + 0.028<br />
R² = 0.981<br />
Unload Curve Fit<br />
y = 0.048x + 0.049<br />
R² = 0.978<br />
Load (mv)<br />
Unload (mv)<br />
Linear (Load (mv))<br />
Linear (Unload (mv))
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
Output (mv)<br />
0.050<br />
0.040<br />
0.030<br />
0.020<br />
0.010<br />
0.000<br />
-0.010<br />
-0.020<br />
-0.030<br />
-0.040<br />
-0.050<br />
-0.060<br />
Strain Gage Calibration Residuals<br />
0 2 4 6 8 10 12 14<br />
Input (lbs)<br />
Figure 4: Strain Gage Calibration Residuals<br />
8<br />
Unload (mv)<br />
Load (mv)
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
5.3 Sample Calculation<br />
The sample calculations are implicitly expressed in the calculated tabular data above <strong>and</strong> background<br />
theory discussion. The calculations for Bourdon Gage linear coefficient are show below<br />
Table 3: Curve Fit Calculation for Bourdon Gage<br />
Xi Yi Error load Error 2 Xi × Y i X 2 i<br />
0 0 36 1324 0 0<br />
100 125 -42 1731 12500 10000<br />
300 555 -18 309 166500 90000<br />
500 990 11 131 495000 250000<br />
700 1400 15 239 980000 490000<br />
1200 2400 1 0 2880000 1440000<br />
1700<br />
� �<br />
3410<br />
�<br />
-4<br />
�<br />
20<br />
�<br />
5797000<br />
�<br />
2890000<br />
4500 8880 0 3753.7 10331000 5170000<br />
The intercept can be calculated as follows, using summation values from table 3<br />
a0 = XiXi × Yi − Yi × X 2 i<br />
XiXi − 7X 2 i<br />
= 4500 × 10331000 − 8880 × 5170000<br />
4500 2 − 7 × 5170000<br />
The slope can be calculated as follows, using summation values from table 3<br />
a1 = XiYi − 7Xi × Yi<br />
XiXi − 7X 2 i<br />
= 4500 × 8880 − 7 × 10331000<br />
4500 2 − 7 × 5170000<br />
9<br />
= −36.3802<br />
= 2.0299
Experiment No. 2 Calibration of a Electronic Load Cell <strong>and</strong> a Bourdon Tube-Type Pressure Gage<br />
6 Discussion of results<br />
6.1 Bourdon Gage<br />
The Bourdon data was reported to a precision of +/- 10 psi, however the output gage increment was<br />
100 psi so the nominal precision should be reported at +/- 50 psi. The Bourdon gage calibration<br />
showed high correlation with a linear curve fit to the data throughout the range. The hysteresis in<br />
the unload cycle was in all cases less then the nominal precision of the output gage which means<br />
that one calibration curve for both loading directions would be sufficient for most applications.<br />
Comparatively, a second order polynomial curve fit to the data produced a comparable correlation<br />
coefficient. The plot of the errors associated with the calibration curves show increasing correlation<br />
with higher loading as might be expected from losses associated with friction <strong>and</strong> other mechanical<br />
factors. If the Bourdon gage was going to be used exclusively in the lowers section of the range,<br />
errors are a lot smaller. The calculated st<strong>and</strong>ard error from the linear curve fit was less than the<br />
nominal precision of the gage which reaffirms the suitability of a simple first order polynomial.<br />
6.2 Strain Gage<br />
The strain gage data was reported to a precision of .001 mv, however the fluctuation in the last<br />
digit dictated a nominal precision of +/- .005 mv. The strain gage calibration data was less linear<br />
throughout the range, but not irregular. Visual inspection of the raw data in Figure 3 shows that a<br />
nonlinear fit would have better correlation with even the data points. The st<strong>and</strong>ard error associated<br />
with the linear curve fit was about .027 mv. The lower number of data points in the curve-fit range<br />
was a factor in the relatively high st<strong>and</strong>ard error. The strain gage scale also showed high hysteresis<br />
or bias error in the unload direction. With a device like this, it is important to a specific procedure<br />
to obtain accurate <strong>and</strong> repeatable results. A plot of the errors of the data to the analytical curve<br />
show decreasing correlation with increasing load, which is self-evident in the raw data plot.<br />
7 Conclusions <strong>and</strong> recommendations<br />
A mechanical <strong>and</strong> an electronic scale were evaluated by students to familiarize them with calibration<br />
procedures <strong>and</strong> the analytical methods of least squares curve fitting. The scales were incrementally<br />
loaded with nominal weights <strong>and</strong> the gage outputs were recorded. The procedure was repeated<br />
in the unload direction. Statistical evaluation comparing the data to the analytical results with<br />
consideration for the nominal gage precision was used to determine the most suitable reporting<br />
precision. The Bourdon mechanical scale calibration data showed good correlation with a linear<br />
curve. The electronic strain gage calibration data was more nonlinear but a first order polynomial<br />
was used for a good fit over the entire set of the range. The st<strong>and</strong>ard error was calculated for<br />
both data sets. For both experiments, the st<strong>and</strong>ard error exceeded the output precision (the load<br />
precision being close). Since Bourdon errors were smaller the Strain Gage errors, the first experiment<br />
would be more reliable. This exercise was effective for illustrating the procedures <strong>and</strong> limitations of<br />
calibration methodology as might be applied to a quality control process. The different apparatuses<br />
also contrasted the different character of calibration data from linear throughout the range for the<br />
Bourdon gage, to a more non-linear with for the strain gage. The concepts of bias error <strong>and</strong> precision<br />
error were illustrated by hysteresis between the load <strong>and</strong> unload calibration direction.<br />
10
Experiment No. 2 Bibliography<br />
References<br />
[1] Rahai, H.R et al, 2007, ”MAE 300 <strong>Instrumentation</strong> <strong>and</strong> <strong>Measurement</strong>”, CSULB MAE Department.<br />
[2] Velleman, Paul F., 1984, ”Applications, Basics, <strong>and</strong> Computing of Exploratory Data Analysis ”,<br />
Wadsworth Pub Co.<br />
11