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Kin C. WongGlenn Research Center, Cleveland, Ohio
Derivation of the Data Reduction Equationsfor the Calibration of the Six-ComponentThrust Stand in the CE–22 AdvancedNozzle Test Facility
NASA/TM—2003-212326
April 2003
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Kin C. WongGlenn Research Center, Cleveland, Ohio
Derivation of the Data Reduction Equationsfor the Calibration of the Six-ComponentThrust Stand in the CE–22 AdvancedNozzle Test Facility
NASA/TM—2003-212326
April 2003
National Aeronautics andSpace Administration
Glenn Research Center
Available from
NASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076
National Technical Information Service5285 Port Royal RoadSpringfield, VA 22100
Available electronically at http://gltrs.grc.nasa.gov
The Propulsion and Power Program atNASA Glenn Research Center sponsored this work.
Derivation of the Data Reduction Equations for the Calibration of the Six-Component Thrust Stand in the CE�22 Advanced
Nozzle Test Facility
Kin C. Wong National Aeronautics and Space Administration
Glenn Research Center Cleveland, Ohio 44135
SUMMARY This paper documents the derivation of the data reduction equations for the calibration of the six-component thrust stand located in the CE�22 Advanced Nozzle Test Facility. The purpose of the calibration is to determine the first-order interactions between the axial, lateral, and vertical load cells (second-order interactions are assumed to be negligible). In an ideal system, the measurements made by the thrust stand along the three coordinate axes should be independent. For example, when a test article applies an axial force on the thrust stand, the axial load cells should measure the full magnitude of the force, while the off-axis load cells (lateral and vertical) should read zero. Likewise, if a lateral force is applied, the lateral load cells should measure the entire force, while the axial and vertical load cells should read zero. However, in real-world systems, there may be interactions between the load cells. Through proper design of the thrust stand, these interactions can be minimized, but are hard to eliminate entirely. Therefore, the purpose of the thrust stand calibration is to account for these interactions, so that necessary corrections can be made during testing. These corrections can be expressed in the form of an interaction matrix, and this paper shows the derivation of the equations used to obtain the coefficients in this matrix. SYMBOLS CX1 Aft-left axial calibration load cell CX2 Aft-right axial calibration load cell CY1 Forward-right lateral calibration load cell CY2 Aft-right lateral calibration load cell CZ1 Forward-left vertical calibration load cell CZ2 Forward-right vertical calibration load cell CZ3 Aft-left vertical calibration load cell CZ4 Aft-right vertical calibration load cell RX1 Aft-left axial reaction load cell RX2 Aft-right axial reaction load cell RY1 Forward-left lateral reaction load cell RY2 Aft-left lateral reaction load cell RZ1 Forward-left vertical reaction load cell
1NASA/TM—2003-212326
RZ2 Forward-right vertical reaction load cell RZ3 Aft-left vertical reaction load cell RZ4 Aft-right vertical reaction load cell FX Axial force, or force in the x-direction FY Lateral force, or force in the y-direction FZ Vertical force, or force in the z-direction MX Rolling moment, or moment about the x-axis MY Pitching moment, or moment about the y-axis MZ Yawing moment, or moment about the z-axis [FMX] The 5×1 force matrix that contains the rolling moment component [FMY] The 5×1 force matrix that contains the pitching moment component [R] The 5×1 reaction load cell matrix that contains the rolling moment
component [Q] The 5×1 reaction load cell matrix that contains the pitching moment
component [S] The 5×5 interaction matrix that contains the coefficients for the rolling
moment calculation [U] The 5×5 interaction matrix that contains the coefficients for the pitching
moment calculation sij The coefficients in the matrix [S] uij The coefficients in the matrix [U] σij The coefficients in the matrix [S]�1 µij The coefficients in the matrix [U]�1 LP1
Forward moment arm for the pitching moment (distance from centroid to the CZ1 and CZ2 load cells)
LP2 Aft moment arm for the pitching moment (distance from centroid to the
CZ3 and CZ4 load cells) LY Moment arm for the yawing moment (distance from centroid to the CY1
and CY2 load cells) LR Moment arm for the rolling moment (distance from centroid to the CZ1
and CZ3 load cells, and to the CZ2 and CZ4 load cells) THE CE�22 NOZZLE TEST FACILITY The CE�22 Advanced Nozzle Test Facility is located in the Engine Research Building (ERB) at the NASA Glenn Research Center. A schematic of the CE�22 Test Facility is shown in figure 1. The test section consists of a 23-ft long by 7-½-foot-inside-diameter vacuum tank. The upstream section is fixed while the downstream section slides open on rails, allowing for easy access to the test model. An inflatable rubber seal is used to prevent leakage when the two halves are closed for testing. The inlet air and exhaust is supplied by compressors and exhausters operated by Central Process Systems (CPS). The primary air system can supply inlet air at a pressure of 40 psig with a maximum flow rate of 40 lbm/sec. There are also provisions for two one-inch secondary supply lines that enter through the top of the test tank. The secondary air
2NASA/TM—2003-212326
can be supplied at 40, 150, or 450 psig with maximum flow rates of 21, 2, and 10 lbm/sec, respectively. The exhaust system can pull a vacuum down to approximately 1.9 psia to simulate altitude conditions at 48,000 feet. The CE�22 Test Facility also has a six-component thrust stand, which allows for simultaneous force and moment measurements along the three coordinate axes. The thrust stand and the altitude capability make the CE�22 Test Facility a unique asset to NASA. The facility is ideally suited for testing sub-scale advanced aircraft nozzle concepts which employ thrust vectoring. For further details about the CE�22 Test Facility, refer to references (1) and (2). THE SIX-COMPONENT THRUST STAND The installation of the thrust stand is shown in figure 2. The thrust stand is comprised of two major parts: the ground frame and the live frame. The ground frame is bolted to the floor of the test tank and the live frame is in turn attached to the ground frame through the 8 reaction load cells. The inlet piping and the experimental model are mounted to the live frame. Thus, forces produced by the experimental model cause a displacement of the live frame relative to the ground frame. This displacement, on the order of thousandths of an inch, is converted to a voltage by each of the 8 reaction load cells. This voltage is then converted to a force reading by the data acquisition system. The interaction matrix is then applied by the data acquisition program to correct the individual force readings to take into account the interactions between the load cells.
3NASA/TM—2003-212326
Figure 2.––The six component thrust stand.
LiveframeLiveframe
GroundframeGroundframe
ASMEcalibrationnozzle
ASMEcalibrationnozzle
Forward test tankForward test tankStepper motorStepper motor
SteppermotorSteppermotor Load
cellLoadcell
Load cellLoadcell
FlowdirectionFlowdirection
There are three other items concerning the operation of the thrust stand that need mentioning: (1) There is a metric break that is located in the inlet air supply line just upstream of the thrust stand. The metric break is a physical break in the piping that isolates the experimental model and the part of the piping attached to the live frame of the thrust stand from the rest of the upstream air supply line. This break in the air supply line is enclosed by a labyrinth seal to minimize leakage; (2) The live frame of the thrust stand is actually comprised of two parts joined together by an elastic hinge. The purpose of this elastic hinge is to minimize the interaction between the axial and the lateral components. Because of the elastic hinge, off-center axial forces will not produce a yawing moment. A yawing moment can only be produced by offset lateral forces; and (3) the weight of the live frame, the piping, and the experimental model are zeroed out before the start of each test run. Figure 3 shows a schematic of the thrust stand. The sign convention used is as follows. The axial direction, or x-axis, is positive in the upstream direction. The lateral direction, or y-axis, is positive on the right hand side, aft-looking-forward. And the vertical direction, or z-axis, is positive in the downward direction. The terms used for the forces and moments are summarized in Table I.
4NASA/TM—2003-212326
RY1
MYMZ
MXX
Y
Z
CX2
CY1
RX2
CZ4
CZ3
CZ1
RZ1
CZ2
RZ2
RZ3
RY2
RX1
RZ4
CY2
CX1
Live frame
Ground frameInlet piping(model interfacestation)CD-03-82285
Air flow
Figure 3.––Schematic of the thrust stand showing location of load cells.
Table I: Symbols and Terms Used for Forces and Moments. Symbol Force or Moment
FX Axial force, or force in the x-direction FY Lateral force, or force in the y-direction FZ Vertical force, or force in the z-direction MX Rolling moment, or moment about the x-axis MY Pitching moment, or moment about the y-axis MZ Yawing moment, or moment about the z-axis
LOAD CELL DESCRIPTION As previously mentioned, there are 8 reaction load cells that measure forces by detecting the displacement of the live frame relative to the ground frame. In addition, there are 8 calibration load cells mounted in-line with the 8 reaction load cells. These calibration load cells are used to measure the applied load on the thrust stand during calibration. The calibration load cells are designated by a C (e.g., CX1, CY1, etc.) and the reaction load cells with an R (e.g., RX1, RY1, etc). The positions of the 8 reaction load cells are summarized in Table II, and their exact locations relative to the centroid are shown in
5NASA/TM—2003-212326
Table II: Load Cell Locations. Load Cell Location
1RX Aft-Left 2RX Aft-Right 1RY Forward-Left 2RY Aft-Left 1RZ Forward-Left 2RZ Forward-Right 3RZ Aft-Left 4RZ Aft-Right
LP1=22.25
LY=17.75
LY=17.75LP2=22.0
LR=11.0
LR=11.0
14.0
14.0
X
Y
Z
Figure 4. ––Schematic of Load Cell locations.
RY1
CZ1
RZ1
RY2CX1
CZ3
RX1RZ3
RX2
RZ4
CY2
RZ2CY1
CX2
CZ2
CZ4
CD-03-82340
figure 4. Forward and aft is relative to the direction of the airflow, and left and right is referenced from an aft-looking-forward perspective. Before the thrust stand itself is calibrated, each of the 16 load cells (8 calibration and 8 reaction load cells) are sent out for calibration at the calibration laboratory. The load cells are calibrated against a standard that can be traced to a national standard. Each individual load cell works by converting a displacement into an electrical voltage. The calibration laboratory applies a known force on the load cell and plots a Force versus Voltage curve. This curve should be linear within the working range of the load cell. Once the slope and intercept from this plot are determined, these constants are entered into the data acquisition program. During an experiment, the displacement of the thrust stand is
6NASA/TM—2003-212326
measured as a voltage from the load cells, which in turn is converted to a force by the data acquisition system. A close-up of one of the vertical load cell pairs mounted on the thrust stand is shown in figure 5. The metal fixtures on each end of the load cell are flexures. Their job is to minimize the effect of off-axis forces on the load cell. Thus, ideally, an axial force produced by the experimental model is measured only through the axial load cells and should have little or no affect on the lateral and vertical load cells.
Figure 5.––Calibration/reaction load cell pair.
CZ4
RZ4
Flexure
LiveframeLiveframe
GroundframeGroundframe
Flexure
FlexureFlexure
FlexureFlexure
FlexureFlexure
CALIBRATION PROCEDURE The purpose of the thrust stand calibration is to determine the first-order interactions between the load cells. For example, what effect does applying an axial force have on the vertical load cells? When a 100 lb. force in the x-direction is applied, one would expect the sum of the axial load cells to read 100 lbs. But due to constraints from the other attached load cells, this force may not be exactly 100 lbs. In addition, a force may be induced on the lateral and or vertical load cells. If the thrust stand is designed properly, such interactions can be minimized, but are hard to eliminate completely. The purpose of the calibration is to obtain a correction factor to account for these interactions. This correction factor is expressed as a matrix and is referred to interchangeably as the interaction matrix or the calibration matrix.
7NASA/TM—2003-212326
The calibration of the thrust stand is performed by applying known forces and moments and recording its effects on the reaction load cells. Forces are applied through the use of the stepper motors mounted in-line with the calibration load cells. The calibration load cells are used to determine the amount of force applied by the stepper motor. Forces and moments are applied in the manner shown in Table III. The first column indicates the type of load applied to the thrust stand. The second column indicates the calibration load cells used to measure this applied load. For the applied moments, the load is applied on one side of the thrust stand, then on the other side, as indicated by the "/". The third column shows the range and increment of the applied loads. The range and increment may be adjusted depending on the range of interest for upcoming test programs. As an example, when applying the moment about the y-axis, or MY, the stepper motors in line with CZ1 and CZ2 are activated to apply 0, �25, �50, �75, �100, �125, �100, �75, �50, �25, and 0 lbs. each to the thrust stand. The load is then applied in the other direction from 0 to 100 lbs. and back down to 0. At each point the readings indicated by the 8 reaction load cells are recorded. Readings are taken on the way up and on the way back down to determine the repeatability of the thrust stand. The same procedure is then repeated for the stepper motors mounted in-line with CZ3 and CZ4.
Table III: Calibration Loadings. Applied Force or Moment
Calibration Load Cell(s) used to Measure Applied Load
Range and Increment of Applied Load
FZ CX1, CX2 0 to 3000, in. 250 lb increments
FY CY1, CY2 0 to �500 and 0 to 500, in 100 lb increments
FZ CZ1, CZ2, CZ3, CZ4 0 to �250 and 0 to 250, in 50 lb increments
MX CZ1, CZ3 / CZ2, CZ4 0 to �250 and 0 to 250, in 50 lb increments
MY CZ1, CZ2 / CZ3, CZ4 0 to �250 and 0 to 250, in 50 lb increments
MZ CY1 / CY2 0 to �500 and 0 to 500, in 100 lb increments
The applied calibration loads are split equally between the two load cells when the FX, FY, MY, and MZ loads are applied. However, when the CZ1 and CZ3 (or CZ2 and CZ4) load cells are loaded together, as in the case when applying the FZ and MX loads, care must be taken in order to prevent an unwanted pitching moment. Due to the fact that the forward vertical load cells (CX1 and CX2) have a different moment arm than the aft vertical load cells (CX3 and CX4), the forces must be applied such that: 1 2CZ CZ= , (1)
8NASA/TM—2003-212326
3 4CZ CZ= , (2) and
2
1
1 3p
p
LCZ CZ
L= ⋅ , (3)
where LP1
= 22.25 in. is the distance along the x-axis from the centroid to CZ1 and CX2, and LP2
= 22.0 in. is the distance along the x-axis from the centroid to CZ3 and CZ4. THE FORCE AND MOMENT EQUATIONS The basic equation used to calculate forces and moments is: [ ] [ ][ ]F S R= , (4) where [ ]F is a 6×1 matrix containing the 3 forces and 3 moments to be measured, [R] is the 8×1 matrix containing the 8 reaction load cell readings, and [S] is the interaction or calibration matrix. During the calibration, known forces and moments, [F], are applied to the thrust stand and the load cell readings, [R], are recorded from the data system. In order to obtain [S], its inverse, [S]�1, must be determined from the following form of equation (4): 1[ ] [ ] [ ]S F R− = (5) The problem with modeling the system using equations (4) and (5) is that [S] is a 6×8 matrix, and is therefore not invertible. In other words, because there are 8 reaction load cells and only 6 components that need to be resolved, the system is indeterminate. The way around this problem is to divide the system into two parts. (This technique was developed by Roger Werner, 1991, NASA Glenn Research Center, OH, personal communication.) The load cells are combined, as shown below, so that there are 5 load cell readings and 5 components in each system. The two systems are represented below as: [ ] [ ][ ]MXF S R= (6)
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
1 212
1 32 4
s s s s sFX RX RXs s s s sFY RYs s s s sFZ RYs s s s sMX RZ RZs s s s sMZ RZ RZ
+ = + +
(7)
9NASA/TM—2003-212326
[ ] [ ][ ]MYF U Q= (8)
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
1 212
1 23 4
u u u u uFX RX RXu u u u uFY RYu u u u uFZ RYu u u u uMY RZ RZu u u u uMZ RZ RZ
+ = + +
(9)
In equation (7), the vertical load cells are combined left (RZ1 + RZ3) and right (RZ2 + RZ4) such that they produce a rolling moment, while in equation (9), the vertical load cells are combined forward (RZ1 + RZ2) and aft (RZ3 + RZ4) such that they produce a pitching moment. See figure 3 for the load cell locations. In addition, the RX1 and RX2 terms are combined, while the RY1 and RY2 terms are kept separate. This is due to the elastic hinge. Because of this elastic hinge, an offset axial force would not produce a yawing moment; therefore, the terms can be combined. The RY1 and RY2 terms need to remain separate in order to resolve the yawing moment, MZ. Both [FMX] and [FMY] are usually calculated by the data acquisition program during an experiment. But because nozzles usually provide pitch and/or yaw vectoring, the components in [FMY] are more relevant since it contains the equation for the pitching moment. The calculation of [FMX] serves as a check on [FMY], and is also used to monitor the test article for unexpected rolling moments. THE INTERACTION MATRICES In order to better illustrate the function of the interaction matrix, the FX component as represented in equation (7) is written out as follows: 11 12 13 14 15( 1 2) 1 2 ( 1 3) ( 2 4)FX s RX RX s RY s RY s RZ RZ s RZ RZ= + + + + + + + (10) In this form, it is easier to see what each of the coefficients represents. For instance, the coefficient s11 represents the weighting factor for the effect of the sum of the two axial load cells, RX1 and RX2, on FX. The coefficient s12 represents the weighting factor for the effect of the RY1 load cell on FX, and so on. Thus, for small interactions, one would expect the coefficient s11 to be approximately one, and the other coefficients (s12, s13, s14, and s15) to be close to zero. To get a better feel for the expected values of the coefficients in the interaction matrices [S] and [U], assume for the moment that there are no interactions between the load cells. If this were the case, it is expected that the axial force would be the sum of the two axial load cells (RX1 + RX2), the lateral force would be the sum of the two lateral load cells (RY1 + RY2), and the vertical force would be the sum of the four vertical load cells
10NASA/TM—2003-212326
(RZ1 + RZ2 + RZ3 + RZ4). Similarly, the moments would be equal to the product of the applied force and its corresponding moment arm (see figure 4 for the lengths of the moment arms). For this ideal case, the following equations would be used to determine the three forces and three moments: 1 2FX RX RX= + (11) 1 2FY RY RY= + (12) 1 2 3 4FZ RZ RZ RZ RZ= + + + (13) ( 1 3) ( 2 4)R RMX L RZ RZ L RZ RZ= × + + × + (14)
1 2( 1 2) ( 3 4)P PMY L RZ RZ L RZ RZ= × + + × + (15)
1 2Y YMZ L RY L RY= × + × (16) These equations can be represented in the form of equations (7) and (9) as:
1 0 0 0 0 1 20 1 1 0 0 10 0 0 1 1 20 0 0 1 30 0 0 2 4
R R
Y Y
FX RX RXFY RYFZ RY
L LMX RZ RZL LMZ RZ RZ
+ = + +
(17)
1 2
1 0 0 0 0 1 20 1 1 0 0 10 0 0 1 1 20 0 0 1 2
3 40 0 0P P
Y Y
FX RX RXFY RYFZ RY
L LMY RZ RZMZ RZ RZL L
+ = + +
(18)
Again, equations (17) and (18) represent the ideal case where there are no interactions between the load cells. However, for small interactions, the coefficients for the [S] and [U] matrices obtained from the calibration should be fairly close to the above values. Thus the values in equations (17) and (18) can be used as a sanity check to determine if the calibration and data reduction were done correctly.
11NASA/TM—2003-212326
DERIVATION OF THE INTERACTION MATRIX COEFFICIENTS The previous sections explained how the interaction matrices, [S] and [U], are used to correct the forces and moments measured by the thrust stand. The following sections will explain the method by which the coefficient terms in the interaction matrices are derived. Equations (7) and (9), rewritten in the form of equation (5), are shown below: 1[ ] [ ] [ ]MXS F R− = (19)
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
1 212
1 32 4
FX RX RXFY RYFZ RYMX RZ RZMZ RZ RZ
σ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σ
+ = + +
(20)
1[ ] [ ] [ ]MYU F Q− = (21)
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
1 212
1 23 4
FX RX RXFY RYFZ RYMY RZ RZMZ RZ RZ
µ µ µ µ µµ µ µ µ µµ µ µ µ µµ µ µ µ µµ µ µ µ µ
+ = + +
(22)
Equations (20) and (22) are the basis for the calibration data reduction equations. The idea here is to apply a single known force or moment, for example, FX, record the reaction load cell readings, RX1 + RX2, RY1, etc., and determine the individual coefficients σ and µ. Once the individual coefficients are determined, the [S]�1 and [U]�1 matrices are inverted to obtain the interaction matrices [S] and [U]. The next three sections explain in detail how this is done. Examples are given for the axial force (FX), the pitching moment (MY), and the yawing moment (MZ) calibrations.1 The lateral force (FY) and vertical force (FZ) calibrations are similar to the axial force calibration, and the rolling moment (MX) calibration is similar to the pitching moment calibration.
1 The term axial force calibration is used to indicate the loading of the axial calibration force. The axial force is not �calibrated� until all the interaction coefficients are determined from all the different loadings. Likewise for the terms lateral force calibration, pitching moment calibration, etc.
12NASA/TM—2003-212326
THE EQUATIONS FOR THE FX CALIBRATION For the axial force calibration, the stepper motors in line with the CX1 and CX2 load cells are used to apply the loads shown previously in Table III. For this case, all the terms in [FMX] are zero except for FX. Thus, equation (20) becomes:
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
1 20 10 20 1 30 2 4
FX RX RXRYRY
RZ RZRZ RZ
σ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σ
+ = + +
(23)
From this, the first column of the inverse matrix can be determined. Writing out the individual equations in the system, discarding the zero terms, and solving for σi1, equation (23) becomes:
11 1 2FX RX RXσ = + ⇒ 111 2RX RXFX
σ += (24)
21 1FX RYσ = ⇒ 211RY
FXσ = (25)
31 2FX RYσ = ⇒ 312RY
FXσ = (26)
41 1 3FX RZ RZσ = + ⇒ 411 3RZ RZFX
σ += (27)
51 2 4FX RZ RZσ = + ⇒ 512 4RZ RZFX
σ += (28)
Each of the σi1 terms can be determined from the slope of the plot of the corresponding load cell readings versus FX. For example, the coefficient σ11 is the value of the slope of the plot of RX1 + RX2 versus FX. The coefficients µi1, for the matrix [U]�1 are found in a the same manner:
111 2RX RXFX
µ += (29)
211RY
FXµ = (30)
312RY
FXµ = (31)
13NASA/TM—2003-212326
411 2RZ RZFX
µ += (32)
513 4RZ RZFX
µ += (33)
Similarly, the FY and FZ calibrations will produce the coefficients σi2 and σi3 for [S]�1, and the coefficients µi2 and µi3 for [U]�1. THE EQUATIONS FOR THE MY CALIBRATION The coefficients for the moment calibrations are determined in a similar fashion as the pure force calibrations, except there is an extra term due to the fact that an offset vertical force is used to produce the moment. For the MY calibration, the pitching moments are applied through two separate loadings. Forces are first applied through the stepper motors in line with the forward vertical load cells, CZ1 and CZ2. The procedure is then repeated with the aft vertical load cells, CZ3 and CZ4. The coefficients are computed separately for each loading, and then averaged in the final result. For both cases, the non-zero components of [FMY] are FZ and MY. Therefore, equation (22) becomes:
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
0 1 20 1
21 2
0 3 4
RX RXRY
FZ RYMY RZ RZ
RZ RZ
µ µ µ µ µµ µ µ µ µµ µ µ µ µµ µ µ µ µµ µ µ µ µ
+ = + +
(34)
Writing out each of the individual equations, discarding the zero terms, and solving for µi4, equation (34) becomes:
13 14 1 2pFZ L FZ RX RXµ µ+ = + ⇒
14 131 1 2
P
RX RXL FZ
µ µ+ = −
(35)
23 24 1pFZ L FZ RYµ µ+ = ⇒
24 231 1
P
RYL FZ
µ µ = −
(36)
33 34 2pFZ L FZ RYµ µ+ = ⇒
34 331 2
P
RYL FZ
µ µ = −
(37)
14NASA/TM—2003-212326
43 44 1 2pFZ L FZ RZ RZµ µ+ = + ⇒
44 431 1 2
P
RZ RZL FZ
µ µ+ = −
(38)
53 54 3 4pFZ L FZ RZ RZµ µ+ = + ⇒
54 531 3 4
P
RZ RZL FZ
µ µ+ = −
(39)
The µi3 terms in the above equations are determined from the FZ calibration and the
ratios ( 1 2RX RXFZ+ , 1RY
FZ, etc.) are determined from the slope of the plot of the
corresponding load cell readings versus the applied force, FZ. The σi4 terms are determined in a similar fashion through the MX calibration. THE EQUATIONS FOR THE MZ CALIBRATION For the MZ calibration, the yawing moments are applied through two separate loadings. Forces are first applied through the stepper motor in line with the CY1 load cell. The procedure is then repeated for the CY2 load cell. The coefficients are computed separately for each loading, and then averaged in the final result. In this case, equation (20) becomes:
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
0 1 21
0 20 1 3
2 4
RX RXFY RY
RYRZ RZ
MZ RZ RZ
σ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σσ σ σ σ σ
+ = + +
(40)
Writing out each of the individual equations, discarding the zero terms, and solving for σi5, equation (40) becomes:
12 15 1 2YFY L FY RX RXσ σ+ = + ⇒
15 121 1 2
Y
RX RXL FY
σ σ+ = −
(41)
22 25 1YFY L FY RYσ σ+ = ⇒
25 221 1
Y
RYL FY
σ σ = −
(42)
32 35 2YFY L FY RYσ σ+ = ⇒
15NASA/TM—2003-212326
35 321 2
Y
RYL FY
σ σ = −
(43)
42 45 1 3YFY L FY RZ RZσ σ+ = + ⇒
45 421 1 3
Y
RZ RZL FY
σ σ+ = −
(44)
52 55 2 4YFY L FY RZ RZσ σ+ = + ⇒
55 521 2 4
Y
RZ RZL FY
σ σ+ = −
(45)
The σi2 terms in the above equations are determined from the FY calibration and the
ratios ( 1 2RX RXFY+ , 1RY
FY, etc.) are determined from the slope of the plot of the
corresponding load cell readings verus the applied force, FY. The µi5 terms for the [U]�1 matrix are found in a similar fashion. SUMMARY OF THE INTERACTION COEFFICIENTS A summary of all the equations used for determining the coefficients in the [S]�1 and [U]�1 matrices are shown in Tables IV and V. Once all the coefficients for the inverse matrices are found, the matrices can be inverted to obtain the interaction matrices [S] and [U]. A sample set of calibration data is listed in Appendix A.
Table IV: Coefficients ijσ for the 1[ ]S − Matrix
i j 1 2 3 4 5
1 ( )1 2RX RXFX
∆ +∆
( )1 2RX RXFY
∆ +∆
( )1 2RX RXFZ
∆ +∆
( )1 1 213
RX RXL FZR
σ ∆ +
− ∆
( )1 1 2
12RX RX
L FYYσ
∆ +−
∆
2 1RYFX
∆∆
1RYFY
∆∆
1RYFZ
∆∆
1 123
RYL FZR
σ ∆
− ∆
1 122
RYL FYY
σ ∆
− ∆
3 2RYFX
∆∆
2RYFY
∆∆
2RYFZ
∆∆
1 233
RYL FZR
σ ∆
− ∆
1 232
RYL FYY
σ ∆
− ∆
4 ( )1 3RZ RZFX
∆ +∆
( )1 3RZ RZFY
∆ +∆
( )1 3RZ RZFZ
∆ +∆
( )1 1 343
RZ RZL FZR
σ ∆ +
− ∆
( )1 1 3
42RZ RZ
L FYYσ
∆ +−
∆
5 ( )2 4RZ RZFX
∆ +∆
( )2 4RZ RZFY
∆ +∆
( )2 4RZ RZFZ
∆ +∆
( )1 2 453
RZ RZL FZR
σ ∆ +
− ∆
( )1 2 4
52RZ RZ
L FYYσ
∆ +−
∆
16NASA/TM—2003-212326
Table V: Coefficients ijµ for the 1[ ]U − Matrix
i j 1 2 3 4 5
1 ( )1 2RX RXFX
∆ +∆
( )1 2RX RXFY
∆ +∆
( )1 2RX RXFZ
∆ +∆
( )1 1 213
RX RXL FZP
µ ∆ +
− ∆
( )1 1 212
RX RXL FYY
µ ∆ +
− ∆
2 1RYFX
∆∆
1RYFY
∆∆
1RYFZ
∆∆
1 123
RYL FZP
µ ∆
− ∆
1 122
RYL FYY
µ ∆
− ∆
3 2RYFX
∆∆
2RYFY
∆∆
2RYFZ
∆∆
1 233
RYL FZP
µ ∆
− ∆
1 232
RYL FYY
µ ∆
− ∆
4 ( )1 2RZ RZFX
∆ +∆
( )1 2RZ RZFY
∆ +∆
( )1 2RZ RZFZ
∆ +∆
( )1 1 243
RZ RZL FZP
µ ∆ +
− ∆
( )1 1 242
RZ RZL FYY
µ ∆ +
− ∆
5 ( )3 4RZ RZFX
∆ +∆
( )3 4RZ RZFY
∆ +∆
( )3 4RZ RZFZ
∆ +∆
( )1 3 453
RZ RZL FZP
µ ∆ +
− ∆
( )1 3 452
RZ RZL FYY
µ ∆ +
− ∆
CONCLUSION This report documents the derivation of the interaction matrix for the six-component thrust stand in the NASA Glenn Research Center�s CE�22 Advanced Nozzle Test Facility. It is intended to give the reader a better understanding of the theory behind the calibration and also insight into how the interaction coefficients are used during testing. Although the equations derived in the paper are specific to the thrust stand in the CE�22 Test Facility, the theory and procedure can be applied to other similar thrust stands.
17NASA/TM—2003-212326
APPENDIX A�SAMPLE DATA Tables A1 to A9 show the raw data acquired from a typical calibration. The first column is the applied load, which is measured through the calibration load cells. The other columns are the reaction load cell readings for that particular applied load. All the load cell readings are in units of lbs-force. In order to reduce the data, each column of reaction load cell readings is plotted against the applied load. The slope of each plot is determined and is shown in bold below each column of reaction load cell readings. The slopes are then plugged into the appropriate equations in Tables IV and V to obtain the [S]�1 and [U]�1 matrices as shown in equations (A1) and (A2), respectively. Finally, the matrices are inverted to obtain the interaction matrices, [S] and [U], as shown in equations (A3) and (A4), respectively. NOTE: The slopes can be determined without plotting the data by using the SLOPE function from a spreadsheet application or by using some other curve fitting application. However, it is recommended that the plots be made to verify that the relationships are linear and that the load cell readings are repeatable. A non-linear relationship or non-repeating load cell readings may indicate possible impingement or other problems with the thrust stand. When plotting the reaction load cell readings that are in-line with the applied load (i.e., RX1 + RX2 versus FX), it may be easier to spot any non-linearity by plotting the
differences ( 12
FXRX − and 22
FYRX − versus FX). For the FY calibration data
(Table A2), plot 12
FYRY − and 22
FYRY − versus FY, etc.
19NASA/TM—2003-212326
Tab
le A
1.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Axi
al (FX
) Cal
ibra
tion
Loa
d FX
RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4 RZ
1+RZ
2RZ
3+RZ
4-0
.076
0.
0811
6 -0
.016
16
0.09
777
0.00
806
-0.0
2446
-0
.016
33-0
.000
07-2
48.2
73
-244
.184
23
-0.0
1621
0.
1847
5 -2
.119
85
1.62
589
0.13
440
-0.6
2837
-496
.850
-4
88.3
6049
0.
0055
6 0.
5539
8 -3
.683
00
1.41
625
-0.1
1567
-2.1
5108
-741
.769
-7
28.8
0176
-0
.134
96
1.31
452
-5.1
7018
0.
9682
0 -0
.430
34-3
.771
64-9
95.1
60
-977
.716
18
-0.1
2427
1.
9445
3 -6
.744
03
0.67
170
-0.7
6709
-5.3
0524
-119
1.34
0 -1
170.
8676
0 -0
.059
19
2.24
859
-7.9
7051
0.
4623
9 -0
.876
02-6
.632
10-1
506.
270
-147
9.90
848
-0.1
2404
3.
0307
4 -9
.805
19
-0.0
5211
-1
.288
64-8
.568
66-1
749.
960
-171
8.99
245
-0.1
9981
3.
5739
6 -1
1.28
090
-0.3
9213
-1
.592
84-1
0.08
020
-199
6.21
0 -1
960.
8703
8 -0
.178
16
4.31
263
-12.
7465
0 -0
.569
14
-1.7
6691
-11.
5487
0-2
247.
730
-220
8.06
632
-0.2
2164
4.
8558
0 -1
4.57
040
-0.9
0945
-2
.168
84-1
3.31
100
-249
3.76
0 -2
449.
8359
7 -0
.232
33
5.37
711
-16.
1337
0 -0
.956
18
-2.3
1041
-14.
7795
0-2
245.
690
-220
6.30
791
-0.1
8907
4.
3994
8 -1
4.41
840
-0.9
2054
-2
.049
62-1
3.28
930
-198
8.62
0 -1
953.
4723
9 -0
.124
16
4.11
726
-12.
9093
0 -0
.547
64
-1.7
6706
-11.
6899
0-1
716.
560
-168
6.63
940
-0.1
3479
3.
6174
1 -1
1.46
590
-0.2
0700
-1
.560
37-1
0.11
260
-150
2.47
0 -1
475.
9028
1 -0
.145
76
3.00
909
-9.7
7222
-0
.117
35
-1.3
8639
-8.5
0318
-122
7.92
0 -1
206.
2099
3 -0
.102
33
2.55
280
-8.0
7914
0.
2559
3 -0
.973
44-6
.849
78-1
082.
120
-106
2.68
568
-0.1
2427
1.
9008
7 -6
.971
62
0.31
164
-0.7
9985
-5.8
6013
-734
.171
-7
21.4
5830
-0
.059
19
1.20
591
-5.0
7227
0.
9575
2 -0
.430
40-3
.684
36-5
04.0
28
-495
.209
46
-0.0
2696
0.
6409
9 -3
.498
27
1.23
252
-0.1
4781
-2.1
1794
-256
.595
-2
52.1
8475
-0
.102
73
0.07
607
-2.1
0914
1.
5614
5 0.
0803
9-0
.628
08-2
.034
-2
.520
99
0.08
132
0.07
605
0.09
475
-0.0
8899
0.
1140
4-0
.108
27
Sl
ope
9.82
289E
-01
9.03
304E
-05
-2.2
1641
E-03
6.
3370
1E-0
3 8.
4546
6E-0
4 1.
0386
5E-0
36.
1438
3E-0
3
20NASA/TM—2003-212326
Tab
le A
2.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Lat
eral
(FY)
Cal
ibra
tion
Loa
d FY
RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4 RZ
1+RZ
2RZ
3+RZ
40.
173
1.72
279
0.13
895
-0.0
3260
0.
1665
3 -0
.567
56
0.46
330
-0.8
6432
-199
.017
1.
0217
3 -1
00.0
4375
-9
8.23
386
3.97
433
-1.7
7150
0.
8832
81.
3195
5-3
98.0
89
0.23
809
-200
.014
56
-196
.674
08
7.87
209
-3.3
3785
1.
5882
72.
9459
7-5
98.6
39
-0.5
6701
-3
00.6
6609
-2
95.6
5779
11
.780
50
-4.8
2801
2.
3802
84.
5722
4-7
98.1
02
-0.9
1570
-4
00.8
7485
-3
93.5
7694
15
.700
10
-6.3
4095
3.
1935
16.
1656
9-9
97.7
70
-1.6
7719
-5
01.9
6905
-4
90.9
7454
19
.543
80
-7.8
9665
3.
8442
77.
8028
9-8
02.9
78
-1.0
2454
-4
04.8
0565
-3
94.1
6332
15
.820
20
-6.3
0867
3.
2802
06.
2313
3-5
99.4
96
-0.0
4541
-3
01.6
1662
-2
94.5
9337
11
.857
40
-4.6
9863
2.
5535
74.
6052
2-4
00.3
15
0.08
629
-201
.558
94
-196
.283
06
7.91
596
-3.2
6284
1.
8376
22.
8155
0-2
01.2
23
0.80
389
-101
.048
39
-98.
5600
3 4.
0402
9 -1
.870
35
0.93
740
1.23
254
-1.5
54
0.03
669
-0.2
3489
-0
.054
32
0.01
238
-0.1
0756
0.
0589
7-0
.154
15-1
.586
-0
.113
80
-0.3
3209
-0
.162
95
-0.0
5281
-0
.085
41
-0.1
1456
-0.0
2366
101.
302
1.52
951
50.7
6034
50
.523
74
-1.9
3118
0.
6000
3 -0
.364
04-0
.967
1120
1.19
1 1.
5948
9 10
0.92
412
99.7
9834
-3
.972
34
1.10
063
-0.8
1932
-2.0
5239
400.
904
2.68
374
201.
0140
0 19
8.15
191
-8.1
3028
2.
6118
4 -1
.676
41-3
.842
0460
0.08
3 3.
3800
0 30
0.98
468
296.
4616
7 -1
2.29
910
3.97
063
-2.5
5511
-5.7
7337
800.
849
4.37
951
401.
7870
5 39
4.98
880
-16.
5110
0 5.
1887
3 -3
.758
93-7
.563
3010
00.1
40
5.53
105
500.
9918
8 49
4.05
984
-20.
7882
0 6.
5693
3 -4
.811
14-9
.407
7680
2.12
1 4.
1629
5 40
2.15
437
395.
9448
7 -1
6.51
130
5.37
297
-3.6
8327
-7.4
5502
601.
170
3.05
559
301.
2006
9 29
7.22
208
-12.
3752
0 3.
9921
4 -2
.739
62-5
.643
4040
1.73
9 2.
4251
1 20
1.78
045
198.
4557
6 -8
.195
59
2.57
888
-1.8
5003
-3.7
6667
201.
462
1.87
866
101.
1941
2 99
.754
90
-4.0
3738
1.
2953
2 -0
.884
77-1
.857
30-0
.978
-0
.156
74
-0.2
7809
0.
2933
0 0.
0237
5 0.
0648
8 -0
.017
370.
1060
0
Sl
ope
3.24
399E
-03
5.02
184E
-01
4.93
022E
-01
-2.0
1384
E-02
7.
2737
3E-0
3 -4
.322
04E-
03-8
.542
66E-
03
21NASA/TM—2003-212326
Tab
le A
3.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Ver
tical
(FZ)
Cal
ibra
tion
Loa
d FZ
RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4 RZ
1+RZ
2RZ
3+RZ
4-0
.182
0.
2109
6 0.
3027
0 0.
6035
2 0.
1302
3 -0
.181
84
0.03
653
-0.0
8813
988
-122
.114
-0
.056
53
-0.0
7552
0.
3208
3 -6
6.93
741
-55.
3681
8 -5
4.47
783
-67.
8277
7-1
95.4
05
0.23
145
-0.5
6161
1.
3420
1 -9
7.10
230
-98.
5318
2 -9
6.38
174
-99.
2523
8-3
96.8
76
0.22
783
-1.5
7722
2.
2327
4 -1
98.2
2435
-1
98.2
7911
-1
96.3
0354
-200
.199
9-5
95.7
04
0.50
627
-2.7
5482
3.
1669
6 -2
96.3
7602
-2
99.0
1475
-2
95.2
9661
-300
.094
13-7
97.5
05
1.12
958
-4.2
6704
4.
0573
3 -3
97.0
0668
-4
00.0
8529
-3
94.0
8145
-403
.010
53-9
97.2
80
1.23
316
-5.8
9845
4.
8611
1 -4
96.7
6134
-5
00.4
2699
-4
92.5
8238
-504
.605
94-7
91.4
01
0.95
530
-4.5
3703
3.
9705
4 -3
98.0
4806
-3
93.3
7946
-3
96.6
9803
-394
.729
492
-591
.878
0.
6783
2 -2
.992
41
3.03
643
-298
.458
71
-293
.262
27
-297
.697
42-2
94.0
2356
-394
.760
0.
0538
6 -1
.987
43
2.12
405
-198
.565
21
-196
.202
39
-198
.007
44-1
96.7
6015
7-1
95.4
68
-0.1
5899
-0
.820
68
1.42
891
-99.
6144
2 -9
5.92
333
-98.
4328
5-9
7.10
4903
-0.3
56
-0.2
0055
0.
1191
6 0.
7556
6 -0
.177
47
-0.2
9079
-0
.102
39-0
.365
874
-0.3
56
-0.1
9993
0.
3244
7 0.
9076
3 -0
.177
26
-0.1
3897
-0
.091
67-0
.224
5624
106.
225
-0.2
4085
0.
6915
9 0.
2124
3 46
.972
58
57.9
6355
49
.809
8655
.126
2720
2.89
7 -0
.195
20
1.19
928
-0.1
5672
99
.614
29
101.
5335
1 99
.139
9710
2.00
7826
403.
314
-0.4
7372
2.
3339
8 -0
.786
66
199.
5478
5 20
1.64
844
197.
6555
520
3.54
074
600.
397
-0.6
6589
3.
1877
7 -1
.634
00
297.
8901
8 30
0.25
246
294.
7775
130
3.36
5180
2.78
1 -0
.857
36
3.92
213
-2.2
2038
39
8.75
876
401.
3567
8 39
5.11
877
404.
9967
610
03.7
20
-1.2
0230
4.
4625
7 -3
.089
39
499.
7361
3 50
1.06
468
495.
2898
450
5.51
097
799.
046
-0.8
8040
4.
0084
7 -2
.220
38
398.
3629
0 39
7.90
517
395.
6134
840
0.65
459
601.
312
-0.9
6911
3.
2842
8 -1
.677
40
299.
7014
4 29
9.31
722
298.
1633
230
0.85
5371
403.
950
-0.3
0180
2.
4306
0 -0
.960
61
200.
6779
9 20
0.96
201
200.
0222
120
1.61
7792
203.
367
-0.0
2257
1.
5986
9 -0
.026
42
99.5
5675
10
1.63
349
101.
0039
010
0.18
635
-0.2
15
0.25
533
0.52
920
0.71
212
-0.6
1246
-0
.066
36
-0.5
2553
-0.1
5330
04
Sl
ope
-1.1
2824
E-03
5.
1857
7E-0
3 -3
.914
94E-
03
4.98
625E
-01
4.99
398E
-01
4.95
026E
-01
5.02
996E
-01
22NASA/TM—2003-212326
Tab
le A
4.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Rol
ling
(MX
) Cal
ibra
tion
Mom
ent (CZ2
+CZ4
) FZ
(CZ2
+CZ4
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4-0
.095
0.
4060
5 -0
.018
89
0.09
239
0.07
597
0.02
983
-98.
092
0.94
584
-0.8
4028
0.
3312
6 -0
.053
02
-97.
9853
0-1
98.7
16
1.39
963
-1.6
8273
0.
8745
4 -0
.398
18
-198
.121
15-2
85.0
66
1.83
268
-2.6
3401
1.
2004
1 -0
.558
56
-284
.145
77-3
95.6
03
3.25
951
-3.3
7940
1.
8304
2 -0
.601
34
-394
.516
21-4
96.6
71
4.27
402
-4.4
1651
2.
1995
6 -0
.695
48
-496
.057
51-3
93.7
18
3.17
302
-3.4
5507
1.
6998
8 -0
.427
69
-393
.193
46-2
95.6
35
1.96
152
-2.7
5224
1.
3090
6 -0
.227
50
-295
.249
44-2
03.2
42
1.42
101
-1.8
8824
0.
9831
3 -0
.078
28
-203
.024
38-9
8.30
4 0.
5561
7 -0
.861
67
0.37
476
-0.0
2587
-9
8.40
839
0.12
2 0.
2764
7 -0
.094
61
0.11
414
0.24
322
-0.1
0298
0.12
2 0.
5784
3 -0
.246
16
-0.0
1630
0.
2214
4 -0
.157
3247
.468
0.
1894
1 0.
5323
4 -0
.190
04
0.41
537
46.2
9615
101.
030
-0.5
4614
1.
0937
2 -0
.472
58
0.58
859
99.5
8819
201.
307
-1.1
3021
2.
2824
3 -1
.037
42
0.84
653
199.
4538
530
1.50
5 -1
.670
07
3.25
461
-1.4
0687
1.
0607
0 29
9.19
772
393.
713
-2.0
1518
4.
1515
1 -1
.645
60
1.25
342
391.
1504
750
2.79
8 -2
.685
65
5.19
957
-2.1
0197
1.
5117
1 49
9.99
444
400.
466
-2.3
4019
4.
1188
9 -1
.667
38
1.21
956
397.
9566
031
4.83
8 -2
.125
38
3.30
868
-1.4
7203
1.
0259
8 31
2.82
024
203.
245
-1.1
5181
2.
2499
7 -1
.015
81
0.76
717
201.
7178
910
2.64
5 -0
.545
78
1.13
709
-0.5
3788
0.
4231
4 10
1.65
790
0.06
8 0.
3191
7 0.
0243
2 0.
4401
4 -0
.040
47
0.99
723
Slop
e -6
.702
80E-
03
9.71
169E
-03
-4.4
0648
E-03
2.
2995
3E-0
3 9.
9576
5E-0
1
23NASA/TM—2003-212326
Tab
le A
5.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Rol
ling
(MX
) Cal
ibra
tion
Mom
ent (CZ1
+CZ3
) FZ
(CZ1
+CZ3
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4-0
.119
0.
0098
3 0.
0407
1 0.
1359
2 -0
.029
80
-0.0
2729
-96.
137
-0.5
3339
-0
.110
55
0.26
619
-95.
7391
4 -0
.360
48-1
96.8
37
-0.9
2484
-0
.240
37
0.76
593
-196
.047
80
-0.7
2147
-296
.575
-1
.620
91
-0.4
7803
1.
0267
1 -2
95.1
7781
-1
.102
89-3
98.1
64
-2.2
9420
-0
.964
60
1.50
476
-396
.794
52
-1.1
8055
-496
.597
-2
.750
75
-1.3
2089
1.
8088
9 -4
95.1
4043
-1
.420
26-4
00.7
60
-2.4
4669
-1
.147
92
1.63
503
-399
.374
56
-1.2
1455
-300
.914
-1
.772
05
-0.8
1278
1.
1787
8 -2
99.7
4948
-0
.862
74-2
01.1
42
-1.0
3284
-0
.553
52
0.72
244
-200
.407
14
-0.4
8717
-102
.089
-0
.381
29
-0.5
1004
0.
4183
9 -1
01.6
8782
-0
.184
570.
097
-0.1
6213
-0
.348
21
0.09
239
0.15
703
0.08
560
0.07
5 0.
2044
9 -0
.337
52
0.11
405
0.17
888
0.09
594
100.
807
0.40
289
-0.4
9924
-0
.298
74
99.7
6244
0.
3182
120
0.54
8 0.
5991
1 -0
.455
92
-0.6
6822
19
9.07
447
0.66
960
301.
766
1.25
080
-0.6
0673
-0
.711
50
299.
9975
1 0.
9379
339
9.61
2 1.
4037
8 -0
.833
25
-1.1
6765
39
7.36
804
1.28
183
501.
018
1.79
513
-1.1
1411
-1
.754
57
498.
1364
7 1.
6392
540
2.17
9 1.
7942
9 -0
.725
59
-1.3
8518
39
9.86
265
1.12
246
302.
363
1.27
253
-0.4
6667
-1
.059
34
300.
7155
4 0.
8633
220
3.72
4 0.
9892
9 -0
.185
70
-0.5
8125
20
2.27
007
0.61
982
104.
093
0.31
563
-0.1
8620
-0
.211
76
102.
9855
0 0.
2744
7-0
.185
0.
0096
6 -0
.056
88
0.04
890
-0.0
3888
-0
.152
13
Sl
ope
4.76
143E
-03
2.44
622E
-04
-3.4
8135
E-03
9.
9525
0E-0
1 3.
0529
6E-0
3
24NASA/TM—2003-212326
Tab
le A
6.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Pitc
hing
(MY)
Cal
ibra
tion
Mom
ent (CZ3
+CZ4
) FZ
(CZ3
+CZ4
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
2 RZ
3+RZ
40.
114
0.08
619
-0.0
2446
-0
.065
35
-0.1
7562
-0
.028
55-9
5.96
7 0.
1706
0 -0
.013
71
0.47
773
-0.0
5575
-9
5.91
599
-197
.705
0.
5136
4 -0
.024
35
0.89
045
-0.0
3389
-1
97.5
4135
-294
.995
0.
9650
1 -0
.046
12
1.52
049
0.05
348
-294
.690
16-3
98.3
20
1.02
797
-0.0
6789
2.
0635
4 -0
.098
16
-398
.026
02-4
93.0
10
1.19
765
-0.0
4634
2.
7152
8 -0
.130
32
-493
.128
50-4
20.8
68
1.39
415
-0.0
2468
2.
2157
3 -0
.130
51
-420
.708
52-3
05.2
38
0.98
726
-0.0
4606
1.
6507
8 -0
.195
56
-305
.136
99-2
02.4
74
0.59
973
-0.0
2462
1.
0642
2 -0
.012
32
-202
.403
78-1
05.3
53
0.19
133
0.02
972
0.56
467
-0.0
8825
-1
05.2
7046
-0.1
36
-0.0
0148
0.
0298
3 -0
.087
07
0.04
166
0.02
727
-0.1
04
-0.0
8750
0.
0297
8 -0
.021
92
-0.1
3168
-0
.016
2798
.888
-0
.367
10
-0.0
2440
-0
.652
03
0.01
987
98.1
0948
202.
037
-0.6
0105
-0
.089
27
-1.3
0360
0.
0305
2 20
1.26
211
298.
033
-0.7
5012
0.
0192
5 -1
.977
12
-0.0
2396
29
7.18
196
388.
805
-0.9
4450
0.
0626
8 -2
.454
99
0.03
009
388.
0286
050
3.68
2 -1
.221
89
-0.0
2390
-3
.041
42
0.16
009
502.
7125
241
4.06
8 -0
.855
63
-0.0
1332
-2
.368
03
0.17
226
413.
1690
432
4.28
5 -0
.620
45
-0.0
3493
-1
.846
60
-0.0
3363
32
3.33
389
211.
681
-0.5
3763
-0
.056
48
-1.1
2968
-0
.077
30
210.
6791
210
2.71
0 -0
.128
80
-0.0
5675
-0
.369
38
-0.0
4404
10
1.80
023
0.07
1 0.
1504
1 -0
.110
98
-0.1
0883
-0
.022
10
0.20
512
Slop
e -2
.581
32E-
03
3.43
725E
-05
-5.6
4588
E-03
2.
0736
2E-0
4 9.
9852
4E-0
1
25NASA/TM—2003-212326
Tab
le A
7.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Pitc
hing
(MY)
Cal
ibra
tion
Mom
ent (CZ1
+CZ2
) FZ
(CZ1
+CZ2
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
2 RZ
3+RZ
4-0
.057
-0
.086
72
0.07
834
0.13
033
-0.0
5664
-0
.029
31-9
8.23
9 0.
0630
4 -0
.905
29
0.19
539
-97.
8961
3 -0
.058
17-1
98.3
64
0.04
114
-2.0
3966
0.
3474
0 -1
97.7
2341
-0
.231
36-2
97.1
11
-0.3
2899
-3
.141
36
0.69
507
-296
.458
96
-0.2
2986
-394
.386
-0
.177
31
-4.5
8892
1.
1729
9 -3
93.5
3480
-0
.419
79-4
95.4
19
0.05
887
-6.1
3363
1.
3901
2 -4
94.3
2711
-0
.580
19-4
01.1
70
-0.4
1605
-4
.891
15
0.99
907
-400
.042
13
-0.4
6138
-301
.643
0.
0199
4 -3
.356
96
1.04
276
-300
.917
63
-0.3
3220
-200
.902
-0
.326
68
-2.3
4172
0.
4777
5 -2
00.2
8124
-0
.247
33-1
00.3
86
-0.2
1731
-1
.261
17
0.26
045
-99.
9433
7 -0
.277
220.
203
0.12
980
-0.3
6496
0.
1520
0 0.
3088
9 -0
.042
12-0
.057
0.
1517
8 -0
.213
25
0.15
208
0.29
809
0.02
324
99.4
36
0.17
428
0.67
216
-0.1
9566
98
.699
10
0.06
921
199.
062
-0.1
7067
1.
6880
2 -0
.477
97
198.
0760
8 0.
2558
529
9.20
4 0.
1544
4 2.
6276
6 -0
.543
03
298.
0078
3 0.
2333
839
6.69
9 0.
1122
2 3.
3626
1 -0
.890
77
395.
3613
8 0.
3785
049
8.64
8 0.
0243
2 4.
0325
9 -0
.912
42
497.
1060
7 0.
3373
642
6.11
8 -0
.278
32
3.65
409
-0.8
2546
42
4.58
068
0.39
550
307.
580
-0.1
4877
2.
9195
3 -0
.738
65
306.
5537
5 0.
3449
420
4.00
4 -0
.019
04
2.08
739
-0.3
6915
20
3.02
115
0.18
254
103.
048
0.28
332
1.06
090
-0.1
7367
10
2.25
487
0.10
756
-0.4
91
0.19
501
0.01
309
0.08
694
-0.3
0891
-0
.047
60
Sl
ope
1.87
379E
-04
1.01
435E
-02
-2.3
6003
E-03
9.
9694
0E-0
1 9.
9533
2E-0
4
26NASA/TM—2003-212326
Tab
le A
8.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Yaw
ing
(MZ)
Cal
ibra
tion
Mom
ent (CY1
) FY
(CY1
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4 RZ
1+RZ
2RZ
3+RZ
4-0
.060
0.
3463
6 0.
0054
0 0.
0924
4 -0
.146
67
0.07
093
-0.0
9759
0.02
185
-98.
736
-0.4
7761
-9
9.35
273
1.11
350
2.55
698
-1.7
3772
0.
7806
60.
0385
9-1
96.9
99
-0.0
6588
-1
98.6
5658
1.
8951
1 5.
1185
1 -3
.717
43
1.37
741
0.02
367
-296
.414
-0
.196
44
-298
.792
72
3.17
686
7.70
208
-5.6
4329
1.
9956
80.
0631
1-3
97.4
79
0.14
908
-400
.807
95
4.30
644
10.5
5750
-7
.536
78
2.73
333
0.28
740
-497
.490
-0
.175
15
-501
.861
22
5.39
250
12.9
2390
-9
.668
88
3.08
069
0.17
431
-401
.864
0.
0840
8 -4
05.3
9702
4.
1976
0 10
.590
10
-7.7
1076
2.
6900
50.
1893
1-3
01.3
10
0.23
670
-303
.825
11
3.17
683
7.85
460
-5.6
4357
2.
0065
80.
2044
5-2
02.3
30
-0.0
8751
-2
03.8
9467
1.
9821
3 5.
0862
2 -3
.674
57
1.43
141
-0.0
1976
-102
.221
-0
.043
92
-102
.711
36
1.00
481
2.46
999
-1.8
3554
0.
7263
5-0
.091
90-0
.733
0.
2384
6 -0
.221
40
-0.0
1628
-0
.190
06
0.02
653
-0.0
7603
-0.0
8751
-0.8
42
-0.1
9514
-0
.253
80
0.11
423
-0.0
1623
0.
0157
1 0.
0107
7-0
.011
2999
.735
0.
1309
5 99
.989
92
-1.0
3731
-2
.796
19
2.18
243
-0.4
1219
-0.2
0157
197.
792
0.54
200
198.
7219
8 -1
.732
04
-5.6
2926
4.
1625
6 -1
.269
18-0
.197
5229
7.86
8 0.
6516
3 29
9.10
552
-2.6
2260
-8
.266
98
5.88
162
-2.0
5029
-0.3
3507
397.
706
0.89
018
399.
2090
8 -3
.382
77
-10.
9372
0 7.
7743
4 -2
.755
37-0
.407
4749
7.54
4 1.
1937
0 49
9.29
104
-4.2
2984
-1
3.80
250
9.59
137
-3.7
0989
-0.5
0122
402.
266
1.30
217
403.
7667
3 -3
.339
31
-11.
0786
0 7.
9380
6 -2
.766
24-0
.374
3130
3.60
1 0.
6731
3 30
4.70
024
-2.7
9655
-8
.365
32
5.92
616
-2.0
3915
-0.4
0002
202.
754
0.32
711
203.
3553
0 -1
.992
96
-5.5
3228
4.
1315
5 -1
.312
32-0
.088
4110
2.64
4 0.
3052
9 10
3.07
865
-0.9
0669
-2
.775
03
2.07
552
-0.5
6359
-0.1
3593
-0.2
99
0.15
410
-0.2
9699
0.
0489
6 0.
0149
9 -0
.035
13
0.05
490
-0.0
7504
Slop
e 1.
2686
5E-0
3 1.
0060
7E+0
0 -9
.638
70E-
03 -2
.687
59E-
02
1.93
614E
-02
-6.7
9093
E-03
-7.2
3528
E-04
27NASA/TM—2003-212326
Tab
le A
9.�
Rea
ctio
n L
oad
Cel
l Dat
a fo
r an
App
lied
Yaw
ing
(MZ)
Cal
ibra
tion
Mom
ent (CY2
) FY
(CY2
) RX
1+RX
2 RY
1 RY
2 RZ
1+RZ
3 RZ
2+RZ
4 RZ
1+RZ
2RZ
3+RZ
40.
122
-0.2
7797
-0
.000
13
0.00
000
-0.1
9291
-0
.011
09
-0.1
4659
-0.0
5741
-98.
354
1.05
326
-0.0
2201
-9
7.73
551
1.31
651
0.00
043
0.03
793
1.27
901
-195
.700
-0
.096
98
-0.1
3074
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Sl
ope
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28NASA/TM—2003-212326
T
he In
vers
e M
atri
ces
[]1
9.82
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(A1)
[]1
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(A2)
29NASA/TM—2003-212326
T
he In
tera
ctio
n M
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[]
1.01
804
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(A4)
30NASA/TM—2003-212326
REFERENCES
1. Luis Beltran, Richard L. Del Roso, and Ruben Del Rosario, �Advanced Nozzle and Engine Components Test Facility,� NASA Technical Memorandum 103684, January 1992.
2. Joyel M. Kerl, Gwynn A. Severt, and Kurt Loos, �Advanced Nozzle Test Facility at NASA Glenn Research Center,� AIAA�2002�3245, June 2002.
31NASA/TM—2003-212326
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NASA TM—2003-212326
E–13921
WBS–22–708–90–66
36
Derivation of the Data Reduction Equations for the Calibration of the Six-Component Thrust Stand in the CE–22 Advanced Nozzle Test Facility
Kin C. Wong
Thrust measurement; Loads (forces); Nozzle thrust coefficients; Thrust loads; Test stands;Test facilities; Transducers; Pitching moments; Yawing moments; Rolling moments; Thrustvector control
Unclassified -UnlimitedSubject Category: 09 Distribution: Nonstandard
Responsible person, Kin C. Wong, organization code 7180, 216–433–8712.
This paper documents the derivation of the data reduction equations for the calibration of the six-component thruststand located in the CE–22 Advanced Nozzle Test Facility. The purpose of the calibration is to determine the first-orderinteractions between the axial, lateral, and vertical load cells (second-order interactions are assumed to be negligible).In an ideal system, the measurements made by the thrust stand along the three coordinate axes should be independent.For example, when a test article applies an axial force on the thrust stand, the axial load cells should measure the fullmagnitude of the force, while the off-axis load cells (lateral and vertical) should read zero. Likewise, if a lateral force isapplied, the lateral load cells should measure the entire force, while the axial and vertical load cells should read zero.However, in real-world systems, there may be interactions between the load cells. Through proper design of the thruststand, these interactions can be minimized, but are hard to eliminate entirely. Therefore, the purpose of the thrust standcalibration is to account for these interactions, so that necessary corrections can be made during testing. These correc-tions can be expressed in the form of an interaction matrix, and this paper shows the derivation of the equations used toobtain the coefficients in this matrix.
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