Engineering Instrumentation and Measurement Experiments

MAE 300 - Engineering
Instrumentation and Measurement
Calibration of an Electronic Load Cell
and a Bourdon Tube Pressure Gage
Experiment No. 2
Kai Gemba, 003678517
California State University
Department of Mechanical and Aerospace Engineering
MAE 300 - Engineering Instrumentation and Measurement
Instructor: Rahai, Hamid R.
October 14, 2007

Abstract
Calibration procedures were performed by students on an electronic load-cell scale and mechanical
Bourdon tube-type pressure-gage scale. Both apparatuses were incrementally loaded to
full scale and data was recorded for both the load and unload direction. The data was plotted,
statistically evaluated and the method of least squares was applied to determine a best-fit analytical
expression for the calibration function. The Bourdon Gage output was closely correlated
to a linear calibration function throughout the test range, R 2 vales ranging from 0.996 to 0.9993.
Hysteresis in the unload direction was less than the output gage precision. The strain gage output
was less linear but a fist order polynomial was used, too. R 2 values ranging from 0.981 to 0.978.

Experiment No. 2
Contents
1 Objective of experiment 1
2 Background and Theory 1
2.1 Intercept Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Standard error of the linear relationship . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Experimental procedure 3
4 Experimental data 4
5 Calculations and Results 5
5.1 Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.2 Strain Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.3 Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6 Discussion of results 10
6.1 Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6.2 Strain Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7 Conclusions and recommendations 10
Bibliography 11
i

Experiment No. 2
List of Figures
1 Bourdon Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Bourdon Calibration Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Strain Gage Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Strain Gage Calibration Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
List of Tables
1 Stain Gage Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Bourdon Gage Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Curve Fit Calculation for Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . 9
ii

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
1 Objective of experiment
The stated objective of the experiment is to familiarize students with the calibration procedure and
method of least squares
2 Background and Theory
The following descriptions of the equipment and the analytic methods for evaluating the data were
obtained from the web resources noted in the bibliography and are consistent with the presentation
in the course textbook.
In a Bourdon tube gauge, a C shaped, hollow spring tube is closed and sealed at one end. The
opposite end is securely sealed and bonded to the socket. As the gauge pressure increases the tube
will tend to uncoil, while a reduced gauge pressure will cause the tube to coil more tightly. This
motion is transferred through a linkage to a gear train connected to an indicating needle. The needle
is presented in front of a card face inscribed with the pressure indications associated with particular
needle deflections.
Two points of the wheatstone bridge are connected to an exciter voltage (from the battery or AC
adapter) and an output analog voltage feed to a A/D Converter. The output voltage being fed in
the A/D varies in proportion to the load applied to the platform of the scale. This occurs since the
weighing platform of a scale is connected to the end of the load cell via a post. The applied force
is transferred from the platform, through the post and onto the aluminum beam. When the beam
bends the strain gauges bend resulting in their resistance value to change and the voltage changes
in proportion to the load applied to the platform.
Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when
given a series of measured data, attempts to find a function which closely approximates the data
(best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals)
between points generated by the function and corresponding points in the data. Specifically, it is
called least mean squares (LMS) when the number of measured data is 1 and the gradient descent
method is used to minimize the squared residual. LMS is known to minimize the expectation of
the squared residual, with the smallest operations (per iteration). But it requires a large number of
iterations to converge.
An implicit requirement for the least squares method to work is that errors in each measurement be
randomly distributed. The Gauss-Markov theorem proves that least square estimators are unbiased
and that the sample data do not have to comply with, for instance, a normal distribution. It is also
important that the collected data be well chosen, so as to allow visibility into the variables to be
solved for (for giving more weight to particular data, refer to weighted least squares).
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Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
2.1 Intercept Calculations
We sum the observations, the squares of the Ys and Xs and the products X × Y to obtain the
following quantities.
and Sy similarly.
and S + Y Y similarly
SX = x1 + x2 + ... + xn
SXX = x 2 1 + x 2 2 + ... + x 2 n
SXY = x1y1 + x2y2 + ... + xnyn
We use the summary statistics above to estimate the slope β.
β ≈ mSXY − SXSY
nSXX − SXSX
We use the estimate of β to estimate the intercept, α:
α ≈ SY − βSX
(2)
n
A consequence of this estimate is that the regression line will always pass through the center of the
data.
2.2 Standard error of the linear relationship
The following is excerpted from Applications, Basics, and Computing of Exploratory Data Analysis,
Velleman and Hoaglin, Wadsworth, Inc. 1981.
A fundamental step in most data analysis and in all exploratory analysis is the computation and
examination of residuals. [...] Most analysts propose a simple structure or model to begin describing
the patterns in the data. Such models differ widely in structure and purpose, but all attempt to
fit the data closely. We therefore refer to any such description of the data as a fit. The residuals
are, then, the differences at each point between the observed data value and the fitted value, or the
actual value and the predicted value: Residual = Actual value − Predicted value.
The median-median line provides one way to find a simple fit, and its residuals, r, are found for
each data value, (xi, yi) as
ri = yi − (axi + b)
A pessimist might view residuals as the failure of a fit to describe the data accurately. He might even
speak of them as errors, although a perfect fit, which leaves all residuals equal to zero, would arouse
suspicion. An optimist sees in residuals details of the data’s behavior previously hidden beneath
the dominant patterns of the fit. Both points of view are correct. The best fits leave small residual,
and systematically large residuals may indicate a poorly chosen model. Nevertheless, even a good
fit may do nothing more than describe the obvious.
Any method of fitting models must determine how much each point can be allowed to influence
the fit. Many statistical procedures try to keep the fit close to every data point. If the data include
an outlier, these procedures may permit it to have an undue influence on the fit. As always in
exploratory data analysis, we try to prevent outliers from distorting the analysis. Using medians in
fitting lines to data provide resistance to outliers, and thus the line fitting technique [...] is called
the resistant line [2].
2
(1)

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
What we will do is to look at a scatter plot of our residuals. This scatter plot may be able to tell us
what we might try if we do not have a good linear fit. In addition, the residuals measure a signed
distance of the predicted value from the actual value. We get the standard error by
�
�
� n�
�
� r
� i=1
std.err =
2 i
(3)
n − 2
This a measure of type of average distance from the data to the predicted values.
2.3 Coefficient of Determination
Excel reports the coefficient of determination R 2 . which is the proportion of variability in a data
set that is accounted for by a statistical model. In this definition, the term variability stands for
variance or, equivalently, sum of squares. There are several common and equivalent expressions for
R2. The version most common in statistics texts is based on an analysis of variance decomposition
as follows:
In the above definition,
R 2 = SSR
SST
= 1 − SSE
SST
SST = �
(yi − y) 2
i=1
SSR = �
( ˆyi − y) 2
i=1
SSR = �
( ˆyi − ˆyi) 2
i=1
That is, SST is the total sum of squares, SSR is the regression sum of squares, and SSE is the sum
of squared errors. In some texts, the abbreviations SSE and SSR have the opposite meaning: SSE
stands for the explained sum of squares (which is another name for the regression sum of squares)
and SSR stands for the residual sum of squares (another name for the sum of squared errors).
3 Experimental procedure
The Bourdon gage was accompanied by a set of weights labeled in nominal increments of 100 psi
and 500 psi. The weights were incrementally loaded on the Bourdon platform and the pressure
valve associated with the piston supporting the platform was opened until the piston raised to
mid-stroke. The platform was rotated to resolve any static friction and the output pressure values
were recorded. This procedure was repeated until the limit of the scale was reached and then the
procedure was reversed until the scale was unloaded. The pressure gage increments were 100 psi.
Output values were interpolated, nominal precision of the instrument should be considered +/-
50 psi. The strain gage load cell supported an aluminum platform approximately 8 inches square.
The tare was adjusted to zero on the digital display system before loading the platform. A series
of weights in nominal increments of 1 pounds were added to the platform. At maximum load, the
procedure was reversed until the scale was unloaded. The output was recorded to a precision of
0.001 mv for each load state however the nominal precision of the output was +/- .005 mv.
3
(4)

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
4 Experimental data
Table 1 and table 2 display the experimental data.
Table 1: Stain Gage Calibration Data
Input [lbs] Load [mv] Unload [mv]
0 0.000 0.000
1.5 0.075 0.100
2.5 0.150 0.170
3.5 0.210 0.230
4.5 0.270 0.290
5.5 0.330 0.350
6.5 0.390 0.400
7.5 0.450 0.440
8.5 0.490 0.470
9.5 0.520 0.510
11 0.570 0.560
13 0.640 0.640
Table 2: Bourdon Gage Calibration Data
Input [psi] Load [psi] Unload [psi]
0 0 0
100 125 120
300 555 550
500 990 1000
700 1400 1420
1200 2400 2450
1700 3410 3425
4

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
5 Calculations and Results
5.1 Bourdon Gage
The Bourdon Calibration Figure 1 shows the raw data and a least-squares linear curve fit to the
data. The Bourdon Calibration first order polynomial shows a first order least squares fit to the
data. The Bourdon Calibration Residuals illustrates the error of the data to the associated analytical
expression. The following analytical calibration were calculated from the data: Loading direction f(x)
= 2.0299x - 36.3082 +/- 50psi Calculated standard error of data 27 psi with a standard deviation
of errors 25 psi. Unloading direction, f(x) = 2.0509x - 37.7478 +/- 50psi. Calculated standard error
of data was found to be 36 psi and standard deviation of errors 35 psi.
Output (psi)
4000
3500
3000
2500
2000
1500
1000
500
0
-500
Bourdon Calibration
0 500 1000 1500 2000
Input (psi)
Figure 1: Bourdon Calibration
5
Load Linear Fit
y = 2.0299x - 36.3802
R² = 0.9996
Unload Linear Fit
y = 2.0509x - 37.7478
R² = 0.9993
Unload (psi)
Load (psi)

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
Output (psi)
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
Bourdon Calibration Residuals
0 500 1000 1500 2000
Input (psi)
Figure 2: Bourdon Calibration Residuals
6
Unload
(psi)
Load (psi)

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
5.2 Strain Gage
The Strain Gage Calibration Figure 3 shows the raw data. A cursory examination of this graph
reveals a linear character. The Strain Gage Polynomial Curve Fit shows a fist order polynomial
least squares fit to the data to the data points of the set. The Strain Gage Calibration Residuals
illustrates the error of the data to the associated analytical expression. The following analytical
calibration were calculated from the data. Calculated standard error of loading data; 0.029 mv, its
standard deviation 0.028 mv
f(x) = 0.0510 × x + 0.028 ± .029
Calculated standard error of unloading data: 0.030 mv, its standard deviation .0284 mv.
Output (mv)
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.000
f(x) = 0.048 × x + 0.049 ± 0.030
Strain Gage Curve Fit
0 5 10 15
Input (lbs)
Figure 3: Strain Gage Calibration
7
Load Curve Fit
y = 0.051x + 0.028
R² = 0.981
Unload Curve Fit
y = 0.048x + 0.049
R² = 0.978
Load (mv)
Unload (mv)
Linear (Load (mv))
Linear (Unload (mv))

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
Output (mv)
0.050
0.040
0.030
0.020
0.010
0.000
-0.010
-0.020
-0.030
-0.040
-0.050
-0.060
Strain Gage Calibration Residuals
0 2 4 6 8 10 12 14
Input (lbs)
Figure 4: Strain Gage Calibration Residuals
8
Unload (mv)
Load (mv)

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
5.3 Sample Calculation
The sample calculations are implicitly expressed in the calculated tabular data above and background
theory discussion. The calculations for Bourdon Gage linear coefficient are show below
Table 3: Curve Fit Calculation for Bourdon Gage
Xi Yi Error load Error 2 Xi × Y i X 2 i
0 0 36 1324 0 0
100 125 -42 1731 12500 10000
300 555 -18 309 166500 90000
500 990 11 131 495000 250000
700 1400 15 239 980000 490000
1200 2400 1 0 2880000 1440000
1700
� �
3410
�
-4
�
20
�
5797000
�
2890000
4500 8880 0 3753.7 10331000 5170000
The intercept can be calculated as follows, using summation values from table 3
a0 = XiXi × Yi − Yi × X 2 i
XiXi − 7X 2 i
= 4500 × 10331000 − 8880 × 5170000
4500 2 − 7 × 5170000
The slope can be calculated as follows, using summation values from table 3
a1 = XiYi − 7Xi × Yi
XiXi − 7X 2 i
= 4500 × 8880 − 7 × 10331000
4500 2 − 7 × 5170000
9
= −36.3802
= 2.0299

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
6 Discussion of results
6.1 Bourdon Gage
The Bourdon data was reported to a precision of +/- 10 psi, however the output gage increment was
100 psi so the nominal precision should be reported at +/- 50 psi. The Bourdon gage calibration
showed high correlation with a linear curve fit to the data throughout the range. The hysteresis in
the unload cycle was in all cases less then the nominal precision of the output gage which means
that one calibration curve for both loading directions would be sufficient for most applications.
Comparatively, a second order polynomial curve fit to the data produced a comparable correlation
coefficient. The plot of the errors associated with the calibration curves show increasing correlation
with higher loading as might be expected from losses associated with friction and other mechanical
factors. If the Bourdon gage was going to be used exclusively in the lowers section of the range,
errors are a lot smaller. The calculated standard error from the linear curve fit was less than the
nominal precision of the gage which reaffirms the suitability of a simple first order polynomial.
6.2 Strain Gage
The strain gage data was reported to a precision of .001 mv, however the fluctuation in the last
digit dictated a nominal precision of +/- .005 mv. The strain gage calibration data was less linear
throughout the range, but not irregular. Visual inspection of the raw data in Figure 3 shows that a
nonlinear fit would have better correlation with even the data points. The standard error associated
with the linear curve fit was about .027 mv. The lower number of data points in the curve-fit range
was a factor in the relatively high standard error. The strain gage scale also showed high hysteresis
or bias error in the unload direction. With a device like this, it is important to a specific procedure
to obtain accurate and repeatable results. A plot of the errors of the data to the analytical curve
show decreasing correlation with increasing load, which is self-evident in the raw data plot.
7 Conclusions and recommendations
A mechanical and an electronic scale were evaluated by students to familiarize them with calibration
procedures and the analytical methods of least squares curve fitting. The scales were incrementally
loaded with nominal weights and the gage outputs were recorded. The procedure was repeated
in the unload direction. Statistical evaluation comparing the data to the analytical results with
consideration for the nominal gage precision was used to determine the most suitable reporting
precision. The Bourdon mechanical scale calibration data showed good correlation with a linear
curve. The electronic strain gage calibration data was more nonlinear but a first order polynomial
was used for a good fit over the entire set of the range. The standard error was calculated for
both data sets. For both experiments, the standard error exceeded the output precision (the load
precision being close). Since Bourdon errors were smaller the Strain Gage errors, the first experiment
would be more reliable. This exercise was effective for illustrating the procedures and limitations of
calibration methodology as might be applied to a quality control process. The different apparatuses
also contrasted the different character of calibration data from linear throughout the range for the
Bourdon gage, to a more non-linear with for the strain gage. The concepts of bias error and precision
error were illustrated by hysteresis between the load and unload calibration direction.
10

Experiment No. 2 Bibliography
References
[1] Rahai, H.R et al, 2007, ”MAE 300 Instrumentation and Measurement”, CSULB MAE Department.
[2] Velleman, Paul F., 1984, ”Applications, Basics, and Computing of Exploratory Data Analysis ”,
Wadsworth Pub Co.
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