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Uncertainty analysis of rocket motor thrust measurements with correlated biases
ISA TRANSACTIONS'" ELSEVIER
ISA Transactions 34 (1995) 253 259
Uncertainty analysis of rocket motor thrust measurements with correlated biases R o b e r t P . T a y l o r "'*, W . G l e n n S t e e l e ", F r e e d i e D o u g l a s , I I I b " Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA b NASA John C. Stennis Space Center, Stennis Space Center, MS 39529, USA
Abstract T h e u n c e r t a i n t y analysis of a rocket m o t o r thrust m e a s u r e m e n t and calibration system is p r e s e n t e d . T h e thrust m e a s u r e m e n t system has 10 load cells with 5 in tension and 5 in compression, and the calibration system has 6 load cells. T h e e l e m e n t a l sources of load-cell r a n d o m u n c e r t a i n t y c o n s i d e r e d are excitation voltage, amplifier stability and noise, and analog-to-digital conversion resolution. T h e e l e m e n t a l sources of load-cell systematic u n c e r t a i n t y (bias) considered arc calibration, excitation voltage, nonlinearity, hysteresis, system alignment, a n d o t h e r mechanical sources such as friction. T h e calibration hierarchy and excitation systematic uncertainties arc correlated since they have a c o m m o n source. T h e s e c o r r e l a t i o n effects d o m i n a t e the resulting thrust m e a s u r e m e n t u n c e r t a i n t i e s and form the focus of this study. T h e trust m e a s u r e m e n t u n c e r t a i n t y is up to t h r e e and a half times larger w h e n correlation effects are c o n s i d e r e d relative to the case where c o r r e l a t i o n is neglected. Keywords: U n c e r t a i n t y ; Systematic; Correlation; Rocket
1. Introduction
The uncertainty analysis of the thrust measurement system and calibration system design for the Advanced Solid Rocket Motor (ASRM) test facility at the NASA John C. Stennis Space Center, Bay St. Louis, Mississippi is presented in this study. The axial force is transmitted through 10 fully flexurized load cells with 5 load cells in tension and 5 in compression. An independent calibration system with 6 load cells is available for
T h i s paper was also presented at the 41st International Instrumentation Symposium in Aurora, CO, 7-11 May 1995. * Corresponding author. Tel.: 601 325 7319. Fax: 6()1 325 7332. E-mail: taylor(~me.msstate.edu.
calibration and balance checking of the main thrust measurement system. The specified allowable uncertainty for the thrust measurement is 0.5% of full load. The load cells are calibrated in a calibration laboratory by applying known weights and developing a table of force versus load-cell bridge output and excitation voltage. Since the same weights are used on all load cells, the elementary systematic uncertainty (bias) from the calibration hierarchy is common to all load cells. This common bias source causes the systematic uncertainties of the load cells to be correlated. Another common source of load cell uncertainty is the excitation voltage source. Since the load cell readings are summed, these correlated uncertain-
111119-11578/95/$119.511 © 1995 Elsevier Science B.V. All rights reserved SSDI 00 1q-0578(95)0D() 1 6-X
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R.P. Taylor et al. / ISA lg'ansactions 34 (1995) 25.?-259
ties compound and grow to dominate the uncertainty in the thrust measurement. The goal of this paper is to discuss the propagation of uncertainties when correlated systematic uncertainties are present and to present this example of a case where the correlation effects dominate the overall uncertainty. In the following, the uncertainty analysis methodology is briefly reviewed, the load cell calibration procedure is discussed, the uncertainty analyses of the thrust measurement and calibration systems are presented, and conclusions are discussed.
2. Uncertainty analysis methodology The two primary standards fox- the computation of m e a s u r e m e n t uncertainty are the A N S I / A S M E standard Measurement Uncertainty [1] and the ISO Guide to the Expression of Uncertainty in Measurement [2]. Steele et al. [3] have presented a detailed comparison of the two methodologies and concluded that for cases where the degrees of freedom are nine or more the standard methods can be replaced with a simplified uncertainty model. Since this is a design-level calculation, this simplified model is the most appropriate for our work. The experimental result is determined from a set of measured variables X~,
r = r(Xl,X
(1)
2 .... , X y ) .
The expanded uncertainty at the 95%-confidence level is computed in terms of the standard deviations of the random and systematic uncertainties using
As discussed above, the simplified model is used here and t,)5 = 2 is taken inside the sums ill Eq. (2) to produce U,-
i=1
J-I
J
+2 ~
~
i~l k=i+l
O~Okgk
i:-:1 J
1
+2 E
J
E
O,OkBik-
(3)
i-1 k-i+l
The random and systematic uncertainty estimators are determined by root-sum squaring the
elemental sources
U=(<)~
+ (<)~ + "'',
(4)
8? :
+
(5)
+ .-..
Brown et al. [4] have demonstrated that the covariance estimator can be determined by sun> ming the products of the elemental systematic uncertainties that arise from a common source. For example, assume that elemental sources 1 and 3 in measurements i and j arise from common SOtlrces, then
8.
:
+ (
((,)
3. Load cell operation The load cells are instrumented with two strain-gage bridges which are calibrated separately and have separate signal conditioning systems. Each bridge circuit is, therefore, an independent load cell as shown schematically in Fig. 1. Since the load cells are flexurized, they carry no moments and are either in pure compression or pure tension. The indicated axial force is determined by taking the axial component of the average forces indicated by the two bridges.
1
cos ox,(& + F,2).
(7)
Each load cell force is proportional to the change in strain-gage resistance which is proportional to the bridge voltage ratio
Oais] + ~_, 02b[ i
Y ~ ) ~i- + ~-. Oi2B,:~ 0.t
i=1
F, =
Ur2 = t
d E
=
(2)
F.... ~J
Voij Veil
(8)
R.P. Taylor et a l . / ISA Trmlsactiot~s .74 (1995) 253-259 Table I Load cell uncertainties
The calibration of the load-cell/bridge systems is performed in a calibration laboratory by applying standard weights and developing a table of force versus bridge voltage ratio. This information is transferred to the measurement system through an " R c a l " shunt resistor. The shunt resistor transfers the calibration lab results directly onto the entire data acquisition system as a unit. This procedure works in the following fashion. First, a reading is taken with an unloaded cell, and the analog-to-digital converter output, count~,~ t" is recorded. Next the shunt resistor is placed in parallel with a bridge m e m b e r as shown in Fig. l. This causes the bridge to become unbalanced and the signal to change to countl~3 ~"° which corresponds to a known force level determined in the calibration laboratory. The two relations, t ) . ( N I i~ 0 - %,tc(unti) - I +/7,,,
F,jl.?.c a I
- c % ( c o u n t (uRc'il'
)
)
255
Source
Systematic
Cal. hierarchy Excitation Load cell
• (correlated) • (correlated)
noll l inea rily
hysteresis Amplifier stability noise ADC resolution Alignment Other mechanical (i.e. friction)
• i
• •
4. Calibrator and thrust measurement system uncertainW analysis
(9) (111)
+/3,V ,
The calibrator thrust is determined bv summing the 6 load-cell readings
are solved for %i and /3~j which are used in the data reduction software to compute the force for any count 0 reading. This process replaces the amplifier and analog-to-digital converter systematic uncertainties with the calibration hierarchy systematic uncertainty contained in the shunt resistor. The elemental sources of systematic and random uncertainty in the load-cell measurements are indicated in Table 1.
L
¢'
1
~ ~cos O,,( F,, +F,:)+C,
where F o represents the other mechanical forces which are nominally zero and 0,~ is the misalignment angle which is nominally zero. F , is the load cell force and is nominally 600,000 lb r. The
esistor
Load Cell
Force
Y
,
~troln-GQg./ i
~
---'-''w
(11)
i=l
Shunt
~
Random
Bndge
Count i Fig. I. Load cell schernatic diagram.
R.P. Tayloret al./ISA Transactions34 (1995) 253-259
256
uncertainty in T~ is determined from Eq. (3) which is written here as U 2 =B 2
re
r~
+ P 2 c.
(12)
The random uncertainty limit for T~ is obtained by propagation through the uncertainty equation. 6
2 p2= E [½sin0x,(&+52)] 2 P4~ i=l 6 q- E [½COS0xi] 2 PF, 2 i=l
6
-1- E
[+COS
2
2
Source
Systematic (Ib r) Random (Ibf)
Cal hierarchy, Ben, Pcn Excitation, BEX , PEX Load cell nonlinearity hysteresis Amplifier stability noise ADC RSS BF, PF Correlation [Bcu + B2x] 1/2
120 300
170
240 120 30'0 190 290 490
t 420 320
i=1
+ P 2 o.
(13)
The systematic uncertainty limit for the thrust is also determined from the uncertainty equation 6
B'2"~= E [½sin 0x,( F,,
2
the precision and bias uncertainties are the same in all load cells, Eqs. (13) and (14) become
p2Tc = 3p12 + p~.2,
2
+ Fi2)] Bo, i B 7' 2~ =
6 6 q._ E [½CO S Oxi]2 BFiI-t2 Z [½COS i=1 i~l 5
+82,,+ 2 g
Oxl]2 BF.2 2
6
E [½coso,½cOSOx,],+,,
i=l j=i+l 6
6
+ E E [½cosO*,½cosO*,]BF.G2 i=l
j=l
6
6
+ E g [+cosox,lcoso+]8,,+ i=1 j = l 5
6
2 }-'. E i=1
(15) 1
i=1
+
Table 2 Elemental uncertainties for the calibrator system
[½c°sOx,½c°sOx,]BrJja •
]=i+l (14)
The correlated parts of the systematic uncertainty are those that result from a common source. As stated before, the calibration hierarchy is common for all load cells. Also, the excitation is correlated if all excitation sources have identical temperature sensitivity. Assuming that the calibration hierarchy and excitation are the only correlated portions of the systematic uncertainty, we replace the term Bii with B2n + B2EX. Also, using the nominal values of 0,. = 0 and assuming that
3B~. + B & 2 + ~-(13 2 ) (BET:,,+ B2x).
(16)
Note that because of the cosine function in Eq. (11), there are no first order uncertainties caused by the alignment since sin(0) = 0. The values of the elemental sources of uncertainty for the calibrator load cells are given in Table 2. The full scale load for each calibrator load cell is taken as 600,000 lb e. Substituting these B e and PF values into Eqs. (15) and (16) and neglecting the uncertainties in the other mechanical forces (PFo and B&) gives Pro = 8501be (0.024%), Brc= 19801b e (0.055%) and Ur~ = 2,1601bf (0.060%). Fig. 2 shows the relative contribution to the squared uncertainty, U2, of each elemental source listed in Table 2. The figure very clearly shows the dominant influence of the correlated terms. If the correlated terms had been neglected, the overall uncertainty would have been estimated to be 1,120 lbf which greatly underestimates the uncertainty in Tc. The measurement system thrust is determined by summing the 10 load cells 10
rm = E lcOSOxi(Fil
+/';2)
+Fo,
(17)
i=1
where F,, is the other mechanical effects and is again assumed to be nominally zero. The system-
257
R.P. Taylor et al. / I S A 7)ansactions 34 (1995) 253-259
Calibrator Uncertainty
Measurement System Uncertainty Correlation
CorrelationL ADC~
ADC
Amplifier Z
Amplifier
Load Cell
Load Cell
Excitation J
Excitation Col Hierarchy
Cal Hierarchy
o
ols
i
1£s
fi
£s
~
SquaredUncertaintyTerm
3.s
tb
0
1
Systematic Uric. ~
RandomUnc.
]
I1
Fig. 2. Relative contributions of the elemental uncertainty sources to the overall uncertainty in the calibrator thrust.
atic and random uncertainties in F o are also neglected. Following the same logic as before,
P2 = 51',~
(18)
1 B2r,,, = _5BF+. ~-(38{))( BcH + B ~ x ) .
(19)
Using the estimates of the elemental uncertainties given in Table 3, the uncertainty of the measured thrust is then calculated as Prm = 7801bf (0.021%), Brm = 5,6601bf (0.15%) and Urm = 5,7001bf (0.16%). Fig. 3 shows the relative contributions to the squared uncertainty, Ur2, of each elemental uncertainty source in Table 3. Table 3 Elemental uncertainties for the measurement system Source
Systematic (Ib r) Random (lb t)
Cal hierarchy, (0.02%) Excitation Load cell nonlinearity hysteresis Amplifier stability noise ADC resolution RSS B F, Pv Correlation [B~. u + B~:x] I/2
110 550
100
310 150 180 120
660 561
(s
zb
Squared Uncertainty Term (Millions)
(Millions)
260 350
Systematic Unc. ~
2s
30
RandomUnc.
Fig. 3. Relative contributions of the elemental uncertainty sources to the overall uncertainty in the measurement system thrust.
Once again the correlated systematic uncertainties are by far the controlling component of the uncertainty. If the correlated component had been neglected, the overall uncertainty would have been estimated to be 1660 lbf compared to the correct value of 5700 lbf.
5. Certification of the test stand
The calibrator is used to apply a force on the forward load train in the test stand. If both the calibrator and test stand measurement systems are properly calibrated and functioning correctly, then the two measured thrusts should agree within the uncertainty band established for the two readings. A balance check on the comparison can be defined as 6 Z = T c - T m = ~ , ½ c o s O , i(F~, n + Fc,i? ) i=1 10 -- E ½COS0xk(rm,kl + F r o , k 2 ) , k=l
(20) where the other mechanical forces Fo have been neglected again. Z should have a nominal value
258
R.P. Taylor el aL /ISA 7)zmsactions 34 (1995) 253-259
of zero with an uncertainty of Uz. As long as the absolute value of Z is less than or equal to Uz, the two thrust measurements agree.
121 gz,
(21)
U2 = B~ +PZz,
(22)
where PJ = p2. + p2T l l I • The systematic uncertainty term is more complicated. The proper treatment is to apply Eq. (3) to Eq. (20) with the strain-gage bridge readings Fi~, Fi2, F~l and Fk2 and the misalignment angle as the measured variables. Collecting common terms gives
B Z = B 27cq-BTm 6 10 [ c3Z
i)Z
+ 2 i=lE k=lE [ aFc,il aFm,kl B/%'.,tF,,,.kl
+
az
az
in the test stand measurement system (2,160 lb~ versus 5,700 lbf) and about 8 times less than the required measurement uncertainty of 0.5%. The most significant sources of uncertainty in each system are those which result from the assumption that the excitation and the calibration hierarchy systematic uncertainties are correlated for the load cells in each system. The Rcal shunt resistor calibration replaces the data acquisition systematic uncertainty leaving only the systematic uncertainties for calibration hierarchy, excitation, nonlinearity and hysteresis. Of these the excitation systematic uncertainty is the most significant• Any reduction in systematic uncertainty would require a significant reduction in BEX. Also, it was shown that if the alignment uncertainties are small, their effect on the overall uncertainty will be negligible.
OFc,il OE,I k~2 BI" iF I, •
ca
m.
"2
OZ
OZ + OFc,i2 aFro,k1BFc.'2F".k' OZ __OZ
7. Nomenclature Bi Bik
)
+ affc,i2 aFro,k2 BF~'el"'~',ke .
bi (23)
Assuming that the calibration hierarchy is the only common source of systematic uncertainty between the calibrator and the thrust measurement system (the excitation systems are separate and have separate temperature environments) and that the misalignment angle is nominally zero, Eq. (23) becomes 1
Bz2 = BT2 +
B 2Tm 2(240) -~BcH(T~)BcH(Tm). -
-
(24)
bik
countij
F,. Fo r Si
L
Substituting the numerical values from the previous sections, Uz = 5,970 lbf. If the comparison between the calibrator and test stand thrust measurement system is such that
t95
[Z] ~ 5,9701bf,
Voij Veij
Ur
then the balance check is satisfied.
6. Conclusion
The expected uncertainty in the calibration system is about 2.5 times less than the uncertainty
Z ot ij
Oi
0,.,
95% systematic uncertainty 95% covariance estimator systematic uncertainty-standard deviation level covariance estimator-standard deviation level analog-to-digital convertor output axial thrust on one load cell load-cell bridge reading other mechanical forces 95% random uncertainty result random uncertainty-standard deviation level total calibrator thrust total measurement system thrust student's t value for 95% confidence 95% uncertainty limit for result load-cell bridge output voltage load-cell bridge excitation voltage measurement balance check term in load-cell calibration term in load-cell calibration sensitivity coefficient = ar/aX~ load-cell axial misalignment angle
R.P. Taylor et a l . / ISA Transactions 34 (1995) 253-259
Acknowledgements This work was s u p p o r t e d in p a r t by the N A S A J o h n C. Stennis S p a c e C e n t e r . Mr. M i k e Y o c k e y of S v e r d r u p T e c h n o l o g y , Inc. p r o v i d e d the estim a t e s of the i n s t r u m e n t u n c e r t a i n t i e s in T a b l e s 2 and 3.
References [1] Measurement Uncertainty, American Society of Mechanical Engineers, ANSI/ASME PTCI9.1 -1985 Part 1, New York (1986) 68 pp.
259
[2] Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland, 1993, 101 pp. [3] W.G. Steele, R.A. Ferguson, R.P. Taylor and H.W. Coleman, "Comparison of ANSI/ASME and ISO models for calculation of uncertainty", ISA Transactions 33(4) (December 1994) 339-352. [4] K.K. Brown, H.W. Coleman, W.G. Steele and R.P. Taylor, "Evaluation of correlated bias approximations in experimental uncertainty analysis", American Institute of Aeronautics and Astronautics, paper AIAA-94-1)772, Washington, 1994.
ISA Transactions 34 (1995) 253 259
Uncertainty analysis of rocket motor thrust measurements with correlated biases R o b e r t P . T a y l o r "'*, W . G l e n n S t e e l e ", F r e e d i e D o u g l a s , I I I b " Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA b NASA John C. Stennis Space Center, Stennis Space Center, MS 39529, USA
Abstract T h e u n c e r t a i n t y analysis of a rocket m o t o r thrust m e a s u r e m e n t and calibration system is p r e s e n t e d . T h e thrust m e a s u r e m e n t system has 10 load cells with 5 in tension and 5 in compression, and the calibration system has 6 load cells. T h e e l e m e n t a l sources of load-cell r a n d o m u n c e r t a i n t y c o n s i d e r e d are excitation voltage, amplifier stability and noise, and analog-to-digital conversion resolution. T h e e l e m e n t a l sources of load-cell systematic u n c e r t a i n t y (bias) considered arc calibration, excitation voltage, nonlinearity, hysteresis, system alignment, a n d o t h e r mechanical sources such as friction. T h e calibration hierarchy and excitation systematic uncertainties arc correlated since they have a c o m m o n source. T h e s e c o r r e l a t i o n effects d o m i n a t e the resulting thrust m e a s u r e m e n t u n c e r t a i n t i e s and form the focus of this study. T h e trust m e a s u r e m e n t u n c e r t a i n t y is up to t h r e e and a half times larger w h e n correlation effects are c o n s i d e r e d relative to the case where c o r r e l a t i o n is neglected. Keywords: U n c e r t a i n t y ; Systematic; Correlation; Rocket
1. Introduction
The uncertainty analysis of the thrust measurement system and calibration system design for the Advanced Solid Rocket Motor (ASRM) test facility at the NASA John C. Stennis Space Center, Bay St. Louis, Mississippi is presented in this study. The axial force is transmitted through 10 fully flexurized load cells with 5 load cells in tension and 5 in compression. An independent calibration system with 6 load cells is available for
T h i s paper was also presented at the 41st International Instrumentation Symposium in Aurora, CO, 7-11 May 1995. * Corresponding author. Tel.: 601 325 7319. Fax: 6()1 325 7332. E-mail: taylor(~me.msstate.edu.
calibration and balance checking of the main thrust measurement system. The specified allowable uncertainty for the thrust measurement is 0.5% of full load. The load cells are calibrated in a calibration laboratory by applying known weights and developing a table of force versus load-cell bridge output and excitation voltage. Since the same weights are used on all load cells, the elementary systematic uncertainty (bias) from the calibration hierarchy is common to all load cells. This common bias source causes the systematic uncertainties of the load cells to be correlated. Another common source of load cell uncertainty is the excitation voltage source. Since the load cell readings are summed, these correlated uncertain-
111119-11578/95/$119.511 © 1995 Elsevier Science B.V. All rights reserved SSDI 00 1q-0578(95)0D() 1 6-X
254
R.P. Taylor et al. / ISA lg'ansactions 34 (1995) 25.?-259
ties compound and grow to dominate the uncertainty in the thrust measurement. The goal of this paper is to discuss the propagation of uncertainties when correlated systematic uncertainties are present and to present this example of a case where the correlation effects dominate the overall uncertainty. In the following, the uncertainty analysis methodology is briefly reviewed, the load cell calibration procedure is discussed, the uncertainty analyses of the thrust measurement and calibration systems are presented, and conclusions are discussed.
2. Uncertainty analysis methodology The two primary standards fox- the computation of m e a s u r e m e n t uncertainty are the A N S I / A S M E standard Measurement Uncertainty [1] and the ISO Guide to the Expression of Uncertainty in Measurement [2]. Steele et al. [3] have presented a detailed comparison of the two methodologies and concluded that for cases where the degrees of freedom are nine or more the standard methods can be replaced with a simplified uncertainty model. Since this is a design-level calculation, this simplified model is the most appropriate for our work. The experimental result is determined from a set of measured variables X~,
r = r(Xl,X
(1)
2 .... , X y ) .
The expanded uncertainty at the 95%-confidence level is computed in terms of the standard deviations of the random and systematic uncertainties using
As discussed above, the simplified model is used here and t,)5 = 2 is taken inside the sums ill Eq. (2) to produce U,-
i=1
J-I
J
+2 ~
~
i~l k=i+l
O~Okgk
i:-:1 J
1
+2 E
J
E
O,OkBik-
(3)
i-1 k-i+l
The random and systematic uncertainty estimators are determined by root-sum squaring the
elemental sources
U=(<)~
+ (<)~ + "'',
(4)
8? :
+
(5)
+ .-..
Brown et al. [4] have demonstrated that the covariance estimator can be determined by sun> ming the products of the elemental systematic uncertainties that arise from a common source. For example, assume that elemental sources 1 and 3 in measurements i and j arise from common SOtlrces, then
8.
:
+ (
((,)
3. Load cell operation The load cells are instrumented with two strain-gage bridges which are calibrated separately and have separate signal conditioning systems. Each bridge circuit is, therefore, an independent load cell as shown schematically in Fig. 1. Since the load cells are flexurized, they carry no moments and are either in pure compression or pure tension. The indicated axial force is determined by taking the axial component of the average forces indicated by the two bridges.
1
cos ox,(& + F,2).
(7)
Each load cell force is proportional to the change in strain-gage resistance which is proportional to the bridge voltage ratio
Oais] + ~_, 02b[ i
Y ~ ) ~i- + ~-. Oi2B,:~ 0.t
i=1
F, =
Ur2 = t
d E
=
(2)
F.... ~J
Voij Veil
(8)
R.P. Taylor et a l . / ISA Trmlsactiot~s .74 (1995) 253-259 Table I Load cell uncertainties
The calibration of the load-cell/bridge systems is performed in a calibration laboratory by applying standard weights and developing a table of force versus bridge voltage ratio. This information is transferred to the measurement system through an " R c a l " shunt resistor. The shunt resistor transfers the calibration lab results directly onto the entire data acquisition system as a unit. This procedure works in the following fashion. First, a reading is taken with an unloaded cell, and the analog-to-digital converter output, count~,~ t" is recorded. Next the shunt resistor is placed in parallel with a bridge m e m b e r as shown in Fig. l. This causes the bridge to become unbalanced and the signal to change to countl~3 ~"° which corresponds to a known force level determined in the calibration laboratory. The two relations, t ) . ( N I i~ 0 - %,tc(unti) - I +/7,,,
F,jl.?.c a I
- c % ( c o u n t (uRc'il'
)
)
255
Source
Systematic
Cal. hierarchy Excitation Load cell
• (correlated) • (correlated)
noll l inea rily
hysteresis Amplifier stability noise ADC resolution Alignment Other mechanical (i.e. friction)
• i
• •
4. Calibrator and thrust measurement system uncertainW analysis
(9) (111)
+/3,V ,
The calibrator thrust is determined bv summing the 6 load-cell readings
are solved for %i and /3~j which are used in the data reduction software to compute the force for any count 0 reading. This process replaces the amplifier and analog-to-digital converter systematic uncertainties with the calibration hierarchy systematic uncertainty contained in the shunt resistor. The elemental sources of systematic and random uncertainty in the load-cell measurements are indicated in Table 1.
L
¢'
1
~ ~cos O,,( F,, +F,:)+C,
where F o represents the other mechanical forces which are nominally zero and 0,~ is the misalignment angle which is nominally zero. F , is the load cell force and is nominally 600,000 lb r. The
esistor
Load Cell
Force
Y
,
~troln-GQg./ i
~
---'-''w
(11)
i=l
Shunt
~
Random
Bndge
Count i Fig. I. Load cell schernatic diagram.
R.P. Tayloret al./ISA Transactions34 (1995) 253-259
256
uncertainty in T~ is determined from Eq. (3) which is written here as U 2 =B 2
re
r~
+ P 2 c.
(12)
The random uncertainty limit for T~ is obtained by propagation through the uncertainty equation. 6
2 p2= E [½sin0x,(&+52)] 2 P4~ i=l 6 q- E [½COS0xi] 2 PF, 2 i=l
6
-1- E
[+COS
2
2
Source
Systematic (Ib r) Random (Ibf)
Cal hierarchy, Ben, Pcn Excitation, BEX , PEX Load cell nonlinearity hysteresis Amplifier stability noise ADC RSS BF, PF Correlation [Bcu + B2x] 1/2
120 300
170
240 120 30'0 190 290 490
t 420 320
i=1
+ P 2 o.
(13)
The systematic uncertainty limit for the thrust is also determined from the uncertainty equation 6
B'2"~= E [½sin 0x,( F,,
2
the precision and bias uncertainties are the same in all load cells, Eqs. (13) and (14) become
p2Tc = 3p12 + p~.2,
2
+ Fi2)] Bo, i B 7' 2~ =
6 6 q._ E [½CO S Oxi]2 BFiI-t2 Z [½COS i=1 i~l 5
+82,,+ 2 g
Oxl]2 BF.2 2
6
E [½coso,½cOSOx,],+,,
i=l j=i+l 6
6
+ E E [½cosO*,½cosO*,]BF.G2 i=l
j=l
6
6
+ E g [+cosox,lcoso+]8,,+ i=1 j = l 5
6
2 }-'. E i=1
(15) 1
i=1
+
Table 2 Elemental uncertainties for the calibrator system
[½c°sOx,½c°sOx,]BrJja •
]=i+l (14)
The correlated parts of the systematic uncertainty are those that result from a common source. As stated before, the calibration hierarchy is common for all load cells. Also, the excitation is correlated if all excitation sources have identical temperature sensitivity. Assuming that the calibration hierarchy and excitation are the only correlated portions of the systematic uncertainty, we replace the term Bii with B2n + B2EX. Also, using the nominal values of 0,. = 0 and assuming that
3B~. + B & 2 + ~-(13 2 ) (BET:,,+ B2x).
(16)
Note that because of the cosine function in Eq. (11), there are no first order uncertainties caused by the alignment since sin(0) = 0. The values of the elemental sources of uncertainty for the calibrator load cells are given in Table 2. The full scale load for each calibrator load cell is taken as 600,000 lb e. Substituting these B e and PF values into Eqs. (15) and (16) and neglecting the uncertainties in the other mechanical forces (PFo and B&) gives Pro = 8501be (0.024%), Brc= 19801b e (0.055%) and Ur~ = 2,1601bf (0.060%). Fig. 2 shows the relative contribution to the squared uncertainty, U2, of each elemental source listed in Table 2. The figure very clearly shows the dominant influence of the correlated terms. If the correlated terms had been neglected, the overall uncertainty would have been estimated to be 1,120 lbf which greatly underestimates the uncertainty in Tc. The measurement system thrust is determined by summing the 10 load cells 10
rm = E lcOSOxi(Fil
+/';2)
+Fo,
(17)
i=1
where F,, is the other mechanical effects and is again assumed to be nominally zero. The system-
257
R.P. Taylor et al. / I S A 7)ansactions 34 (1995) 253-259
Calibrator Uncertainty
Measurement System Uncertainty Correlation
CorrelationL ADC~
ADC
Amplifier Z
Amplifier
Load Cell
Load Cell
Excitation J
Excitation Col Hierarchy
Cal Hierarchy
o
ols
i
1£s
fi
£s
~
SquaredUncertaintyTerm
3.s
tb
0
1
Systematic Uric. ~
RandomUnc.
]
I1
Fig. 2. Relative contributions of the elemental uncertainty sources to the overall uncertainty in the calibrator thrust.
atic and random uncertainties in F o are also neglected. Following the same logic as before,
P2 = 51',~
(18)
1 B2r,,, = _5BF+. ~-(38{))( BcH + B ~ x ) .
(19)
Using the estimates of the elemental uncertainties given in Table 3, the uncertainty of the measured thrust is then calculated as Prm = 7801bf (0.021%), Brm = 5,6601bf (0.15%) and Urm = 5,7001bf (0.16%). Fig. 3 shows the relative contributions to the squared uncertainty, Ur2, of each elemental uncertainty source in Table 3. Table 3 Elemental uncertainties for the measurement system Source
Systematic (Ib r) Random (lb t)
Cal hierarchy, (0.02%) Excitation Load cell nonlinearity hysteresis Amplifier stability noise ADC resolution RSS B F, Pv Correlation [B~. u + B~:x] I/2
110 550
100
310 150 180 120
660 561
(s
zb
Squared Uncertainty Term (Millions)
(Millions)
260 350
Systematic Unc. ~
2s
30
RandomUnc.
Fig. 3. Relative contributions of the elemental uncertainty sources to the overall uncertainty in the measurement system thrust.
Once again the correlated systematic uncertainties are by far the controlling component of the uncertainty. If the correlated component had been neglected, the overall uncertainty would have been estimated to be 1660 lbf compared to the correct value of 5700 lbf.
5. Certification of the test stand
The calibrator is used to apply a force on the forward load train in the test stand. If both the calibrator and test stand measurement systems are properly calibrated and functioning correctly, then the two measured thrusts should agree within the uncertainty band established for the two readings. A balance check on the comparison can be defined as 6 Z = T c - T m = ~ , ½ c o s O , i(F~, n + Fc,i? ) i=1 10 -- E ½COS0xk(rm,kl + F r o , k 2 ) , k=l
(20) where the other mechanical forces Fo have been neglected again. Z should have a nominal value
258
R.P. Taylor el aL /ISA 7)zmsactions 34 (1995) 253-259
of zero with an uncertainty of Uz. As long as the absolute value of Z is less than or equal to Uz, the two thrust measurements agree.
121 gz,
(21)
U2 = B~ +PZz,
(22)
where PJ = p2. + p2T l l I • The systematic uncertainty term is more complicated. The proper treatment is to apply Eq. (3) to Eq. (20) with the strain-gage bridge readings Fi~, Fi2, F~l and Fk2 and the misalignment angle as the measured variables. Collecting common terms gives
B Z = B 27cq-BTm 6 10 [ c3Z
i)Z
+ 2 i=lE k=lE [ aFc,il aFm,kl B/%'.,tF,,,.kl
+
az
az
in the test stand measurement system (2,160 lb~ versus 5,700 lbf) and about 8 times less than the required measurement uncertainty of 0.5%. The most significant sources of uncertainty in each system are those which result from the assumption that the excitation and the calibration hierarchy systematic uncertainties are correlated for the load cells in each system. The Rcal shunt resistor calibration replaces the data acquisition systematic uncertainty leaving only the systematic uncertainties for calibration hierarchy, excitation, nonlinearity and hysteresis. Of these the excitation systematic uncertainty is the most significant• Any reduction in systematic uncertainty would require a significant reduction in BEX. Also, it was shown that if the alignment uncertainties are small, their effect on the overall uncertainty will be negligible.
OFc,il OE,I k~2 BI" iF I, •
ca
m.
"2
OZ
OZ + OFc,i2 aFro,k1BFc.'2F".k' OZ __OZ
7. Nomenclature Bi Bik
)
+ affc,i2 aFro,k2 BF~'el"'~',ke .
bi (23)
Assuming that the calibration hierarchy is the only common source of systematic uncertainty between the calibrator and the thrust measurement system (the excitation systems are separate and have separate temperature environments) and that the misalignment angle is nominally zero, Eq. (23) becomes 1
Bz2 = BT2 +
B 2Tm 2(240) -~BcH(T~)BcH(Tm). -
-
(24)
bik
countij
F,. Fo r Si
L
Substituting the numerical values from the previous sections, Uz = 5,970 lbf. If the comparison between the calibrator and test stand thrust measurement system is such that
t95
[Z] ~ 5,9701bf,
Voij Veij
Ur
then the balance check is satisfied.
6. Conclusion
The expected uncertainty in the calibration system is about 2.5 times less than the uncertainty
Z ot ij
Oi
0,.,
95% systematic uncertainty 95% covariance estimator systematic uncertainty-standard deviation level covariance estimator-standard deviation level analog-to-digital convertor output axial thrust on one load cell load-cell bridge reading other mechanical forces 95% random uncertainty result random uncertainty-standard deviation level total calibrator thrust total measurement system thrust student's t value for 95% confidence 95% uncertainty limit for result load-cell bridge output voltage load-cell bridge excitation voltage measurement balance check term in load-cell calibration term in load-cell calibration sensitivity coefficient = ar/aX~ load-cell axial misalignment angle
R.P. Taylor et a l . / ISA Transactions 34 (1995) 253-259
Acknowledgements This work was s u p p o r t e d in p a r t by the N A S A J o h n C. Stennis S p a c e C e n t e r . Mr. M i k e Y o c k e y of S v e r d r u p T e c h n o l o g y , Inc. p r o v i d e d the estim a t e s of the i n s t r u m e n t u n c e r t a i n t i e s in T a b l e s 2 and 3.
References [1] Measurement Uncertainty, American Society of Mechanical Engineers, ANSI/ASME PTCI9.1 -1985 Part 1, New York (1986) 68 pp.
259
[2] Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland, 1993, 101 pp. [3] W.G. Steele, R.A. Ferguson, R.P. Taylor and H.W. Coleman, "Comparison of ANSI/ASME and ISO models for calculation of uncertainty", ISA Transactions 33(4) (December 1994) 339-352. [4] K.K. Brown, H.W. Coleman, W.G. Steele and R.P. Taylor, "Evaluation of correlated bias approximations in experimental uncertainty analysis", American Institute of Aeronautics and Astronautics, paper AIAA-94-1)772, Washington, 1994.