Kin C. Wong

Glenn Research Center, Cleveland, Ohio

Derivation of the Data Reduction Equations

for the Calibration of the Six-Component

Thrust Stand in the CE–22 Advanced

Nozzle Test Facility

NASA/TM—2003-212326

April 2003

https://ntrs.nasa.gov/search.jsp?R=20030054464 2018-05-15T16:38:38+00:00Z
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Kin C. Wong

Glenn Research Center, Cleveland, Ohio

Derivation of the Data Reduction Equations

for the Calibration of the Six-Component

Thrust Stand in the CE–22 Advanced

Nozzle Test Facility

NASA/TM—2003-212326

April 2003

National Aeronautics and

Space Administration

Glenn Research Center
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.
Derivation of the Data Reduction Equations for the Calibration 

of the Six-Component Thrust Stand in the CEñ22 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 CEñ22 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 

NASA/TM—2003-212326 1
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]

ñ1

μij The coefficients in the matrix [U]

ñ1

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 CEñ22 NOZZLE TEST FACILITY 

The CEñ22 Advanced Nozzle Test Facility is located in the Engine Research Building 

(ERB) at the NASA Glenn Research Center. A schematic of the CEñ22 Test Facility is 

shown in figure 1. The test section consists of a 23-ft long by 7-1⁄2-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 oneinch secondary supply lines that enter through the top of the test tank. The secondary air 

NASA/TM—2003-212326 2
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 CEñ22 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 CEñ22 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 CEñ22 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. 

NASA/TM—2003-212326 3
Figure 2.––The six component thrust stand.

Live

frame

Ground

frame

ASME

calibration

nozzle

Forward test tank

Stepper motor Stepper motor

Stepper

motor

Stepper

motor Load 

cell

Load

cell

Load 

cell

Load

cell

Flow

direction

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. 

NASA/TM—2003-212326 4
RY1

MY MZ

MX

X

Y

Z

CX2

CY1

RX2

CZ4

CZ3

CZ1

RZ1

CZ2

RZ2

RZ3

RY2

RX1

RZ4

CY2

CX1

Live frame

Ground frame

Inlet piping

(model interface CD-03-82285 station)

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 

NASA/TM—2003-212326 5
Table II: Load Cell Locations. 

Load Cell Location 

RX1 Aft-Left 

RX 2 Aft-Right 

RY1 Forward-Left 

RY 2 Aft-Left 

RZ1 Forward-Left 

RZ2 Forward-Right 

RZ3 Aft-Left 

RZ 4 Aft-Right 

LP1=22.25

LY=17.75

LY=17.75

LP2=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

RY2

CX1

CZ3

RX1

RZ3

RX2

RZ4

CY2

RZ2 CY1

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 

NASA/TM—2003-212326 6
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

Live

frame

Ground

frame

Flexure

Flexure

Flexure

Flexure

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. 

NASA/TM—2003-212326 7
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, ñ25, ñ50, ñ75, ñ100, ñ125, ñ100, ñ75, 

ñ50, ñ25, 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 ñ500 and 0 to 500, 

in 100 lb increments 

FZ CZ1, CZ2, CZ3, CZ4 0 to ñ250 and 0 to 250, 

in 50 lb increments 

MX CZ1, CZ3 / CZ2, CZ4 0 to ñ250 and 0 to 250, 

in 50 lb increments 

MY CZ1, CZ2 / CZ3, CZ4 0 to ñ250 and 0 to 250, 

in 50 lb increments 

MZ CY1 / CY2 0 to ñ500 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: 

 12 CZ CZ = , (1) 

NASA/TM—2003-212326 8
 34 CZ CZ = , (2) 

and 

2

1

1 3 p

p

L

CZ 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]

ñ1, must be determined from the following form of 

equation (4): 

1 [][ ] [] SF 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: 

 [ ] [ ][ ] FMX = 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 2

1

2

1 3

2 4

FX RX RX sssss

FY RY sssss

FZ RY sssss

MX RZ RZ sssss

MZ RZ RZ sssss

     +

    

=

+

     +

 (7) 

NASA/TM—2003-212326 9
[ ] [ ][ ] FMY = 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 2

1

2

1 2

3 4

FX RX RX uuuuu

FY RY uuuuu

FZ RY uuuuu

MY RZ RZ uuuuu

MZ RZ RZ uuuuu

     +

    

=

+

     +

 (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 FX s RX RX s RY s RY s RZ RZ s RZ RZ = + + + + ++ + ( 1 2) 1 2 ( 1 3) ( 2 4) (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 

NASA/TM—2003-212326 10
(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: 

 12 FX RX RX = + (11) 

 12 FY RY RY = + (12) 

 1234 FZ RZ RZ RZ RZ =+ ++ (13) 

 ( 1 3) ( 2 4) MX L RZ RZ L RZ RZ =× + +× + R R (14) 

1 2 ( 1 2) ( 3 4) MY L RZ RZ L RZ RZ =× + +× + P P (15) 

 12 MZ L RY L RY =× +× Y Y (16) 

These equations can be represented in the form of equations (7) and (9) as: 

10 0 0 0 1 2

01 1 0 0 1

00 0 1 1 2

00 0 1 3

0 00 2 4

R R

Y Y

FX RX RX

FY RY

FZ RY

MX RZ RZ L L

MZ RZ RZ L L

     +

    

=

+

     +

 (17) 

1 2

10 0 0 0 1 2

01 1 0 0 1

00 0 1 1 2

00 0 1 2

0 00 3 4

P P

Y Y

FX RX RX

FY RY

FZ RY

MY RZ RZ L L

MZ RZ RZ L 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. 

NASA/TM—2003-212326 11
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 [][ ] [] MX SF 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 2

1

2

1 3

2 4

FX RX RX

FY RY

FZ RY

MX RZ RZ

MZ RZ RZ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

    +

   

=

+

    +

 (20) 

1 [ ][ ] [ ] UF Q MY− = (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 2

1

2

1 2

3 4

FX RX RX

FY RY

FZ RY

MY RZ RZ

MZ RZ RZ

μμμμμ

μμμμμ

μμμμμ

μμμμμ

μμμμμ

    +

          =    +   

 

     +

 (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]

ñ1 and [U]ñ1

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 ìcalibratedî until all the interaction coefficients are determined from all the different loadings. 

Likewise for the terms lateral force calibration, pitching moment calibration, etc. 

NASA/TM—2003-212326 12
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 2

0 1

0 2

0 13

0 24

FX RX RX

RY

RY

RZ RZ

RZ RZ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

    +

   

=

+

    +

 (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 σ FX RX RX = +1 2 ⇒ 11

RX RX 1 2

FX

σ

+ = (24)

21 σ FX RY = 1 ⇒ 21

RY1

FX

σ = (25)

31 σ FX RY = 2 ⇒ 31

RY 2

FX

σ = (26)

41 σ FX RZ RZ = +1 3 ⇒ 41

RZ RZ 1 3

FX

σ

+ = (27)

51 σ FX RZ RZ = +2 4 ⇒ 51

RZ RZ 2 4

FX

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]

ñ1 are found in a the same manner: 

11

RX RX 1 2

FX μ

+ = (29) 

21

RY1

FX μ = (30) 

31

RY 2

FX μ = (31) 

NASA/TM—2003-212326 13
41

RZ RZ 1 2

FX μ

+ = (32) 

51

RZ RZ 3 4

FX μ

+ = (33) 

Similarly, the FY and FZ calibrations will produce the coefficients σi2 and σi3 for [S]

ñ1, 

and the coefficients μi2 and μi3 for [U]

ñ1. 

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 12

0 1

2

1 2

0 34

RX RX

RY

FZ RY

MY RZ RZ

RZ RZ

μμμμμ

μμμμμ

μμμμμ

μμμμμ

μμμμμ

    +

          =    +   

 

     +

 (34) 

Writing out each of the individual equations, discarding the zero terms, and solving for 

μi4, equation (34) becomes: 

13 14 1 2 μ μ FZ L FZ RX RX + =+p ⇒

14 13

1 12

P

RX RX

L FZ μ μ

  + = −     (35) 

23 24 1 μ μ FZ L FZ RY + =p ⇒

24 23

1 1

P

RY

L FZ μ μ

  = −     (36) 

33 34 2 μ μ FZ L FZ RY + =p ⇒

34 33

1 2

P

RY

L FZ μ μ

  = −     (37) 

NASA/TM—2003-212326 14
43 44 1 2 μ μ FZ L FZ RZ RZ + =+

p ⇒

44 43

1 12

P

RZ RZ

L FZ μ μ

  + = −     (38) 

53 54 3 4 μ μ FZ L FZ RZ RZ + =+p ⇒

54 53

1 34

P

RZ RZ

L FZ μ μ

  + = −     (39) 

The μi3 terms in the above equations are determined from the FZ calibration and the 

ratios ( RX RX 1 2

FZ

+ , RY1

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 12

1

0 2

0 13

2 4

RX RX

FY RY

RY

RZ RZ

MZ RZ RZ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

σσσσσ

    +

          =    +   

 

     +

 (40) 

Writing out each of the individual equations, discarding the zero terms, and solving for 

σi5, equation (40) becomes: 

12 15 1 2 σ σ FY L FY RX RX + =+ Y ⇒

15 12

1 12

Y

RX RX

L FY

σ σ

  + = −     (41) 

22 25 1 σ σ FY L FY RY + = Y ⇒

25 22

1 1

Y

RY

L FY

σ σ

  = −     (42) 

32 35 2 σ σ FY L FY RY + = Y ⇒

NASA/TM—2003-212326 15
35 32

1 2

Y

RY

L FY

σ σ

  = −     (43) 

42 45 1 3 σ σ FY L FY RZ RZ + =+ Y ⇒

45 42

1 13

Y

RZ RZ

L FY

σ σ

  + = −     (44) 

52 55 2 4 σ σ FY L FY RZ RZ + =+ Y ⇒

55 52

1 24

Y

RZ RZ

L FY

σ σ

  + = −     (45) 

The σi2 terms in the above equations are determined from the FY calibration and the 

ratios ( RX RX 1 2

FY

+ , RY1

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]

ñ1 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]

ñ1 and 

[U]

ñ1 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 σ ij for the 1 [ ] S − Matrix 

i j 1 2 3 4 5 

1 (RX RX 1 2)

FX

∆ +

∆

(RX RX 1 2)

FY

∆ +

∆

(RX RX 1 2)

FZ

∆ +

∆

1 ( ) 1 2

13

RX RX

LR FZ σ   ∆ +   −   ∆

1 ( ) 1 2

12

RX RX

LY FY σ   ∆ +   −   ∆

2 RY1

FX

∆

∆

RY1

FY

∆

∆

RY1

FZ

∆

∆

1 1

23

RY

LR FZ σ   ∆

  −   ∆

1 1

22

RY

LY FY σ   ∆

  −   ∆

3 RY 2

FX

∆

∆

RY 2

FY

∆

∆

RY 2

FZ

∆

∆

1 2

33

RY

LR FZ σ   ∆

  −   ∆

1 2

32

RY

LY FY σ   ∆

  −   ∆

4 (RZ RZ 1 3)

FX

∆ +

∆

(RZ RZ 1 3)

FY

∆ +

∆

(RZ RZ 1 3)

FZ

∆ +

∆

1 ( ) 1 3

43

RZ RZ

LR FZ σ   ∆ +   −   ∆

1 ( ) 1 3

42

RZ RZ

LY FY σ   ∆ +   −   ∆

5 (RZ RZ 2 4)

FX

∆ +

∆

(RZ RZ 2 4)

FY

∆ +

∆

(RZ RZ 2 4)

FZ

∆ +

∆

1 ( ) 2 4

53

RZ RZ

LR FZ σ   ∆ +   −   ∆

1 ( ) 2 4

52

RZ RZ

LY FY σ   ∆ +   −   ∆

NASA/TM—2003-212326 16
Table V: Coefficients μij for the 1 [ ] U − Matrix 

i j 1 2 3 4 5 

1 (RX RX 1 2)

FX

∆ +

∆

(RX RX 1 2)

FY

∆ +

∆

(RX RX 1 2)

FZ

∆ +

∆

1 ( ) 1 2

13

RX RX

LP FZ μ

  ∆ +   −   ∆

1 ( ) 1 2

12

RX RX

LY FY μ

  ∆ +   −   ∆

2 RY1

FX

∆

∆

RY1

FY

∆

∆

RY1

FZ

∆

∆

1 1

23

RY

LP FZ μ

  ∆

  −   ∆

1 1

22

RY

LY FY μ

  ∆

  −   ∆

3 RY 2

FX

∆

∆

RY 2

FY

∆

∆

RY 2

FZ

∆

∆

1 2

33

RY

LP FZ μ

  ∆

  −   ∆

1 2

32

RY

LY FY μ

  ∆

  −   ∆

4 (RZ RZ 1 2)

FX

∆ +

∆

(RZ RZ 1 2)

FY

∆ +

∆

(RZ RZ 1 2)

FZ

∆ +

∆

1 ( ) 1 2

43

RZ RZ

LP FZ μ

  ∆ +   −   ∆

1 ( ) 1 2

42

RZ RZ

LY FY μ

  ∆ +   −   ∆

5 (RZ RZ 3 4)

FX

∆ +

∆

(RZ RZ 3 4)

FY

∆ +

∆

(RZ RZ 3 4)

FZ

∆ +

∆

1 ( ) 3 4

53

RZ RZ

LP FZ μ

  ∆ +   −   ∆

1 ( ) 3 4

52

RZ RZ

LY FY μ

  ∆ +   −   ∆

CONCLUSION 

This report documents the derivation of the interaction matrix for the six-component 

thrust stand in the NASA Glenn Research Centerís CEñ22 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 CEñ22 

Test Facility, the theory and procedure can be applied to other similar thrust stands. 

NASA/TM—2003-212326 17

APPENDIX AóSAMPLE 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]

ñ1 and [U]ñ1 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 nonrepeating 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 ( 1

2

FX RX − and 2

2

FY RX − versus FX). For the FY calibration data 

(Table A2), plot 1

2

FY RY − and 2

2

FY RY − versus FY, etc. 

NASA/TM—2003-212326 19
Table A1.óReaction Load Cell Data for an Applied Axial (FX) Calibration Load 

FX RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4 RZ1+RZ2 RZ3+RZ4

-0.076 0.08116 -0.01616 0.09777 0.00806 -0.02446 -0.01633 -0.00007

-248.273 -244.18423 -0.01621 0.18475 -2.11985 1.62589 0.13440 -0.62837-496.850 -488.36049 0.00556 0.55398 -3.68300 1.41625 -0.11567 -2.15108-741.769 -728.80176 -0.13496 1.31452 -5.17018 0.96820 -0.43034 -3.77164-995.160 -977.71618 -0.12427 1.94453 -6.74403 0.67170 -0.76709 -5.30524

-1191.340 -1170.86760 -0.05919 2.24859 -7.97051 0.46239 -0.87602 -6.63210-1506.270 -1479.90848 -0.12404 3.03074 -9.80519 -0.05211 -1.28864 -8.56866-1749.960 -1718.99245 -0.19981 3.57396 -11.28090 -0.39213 -1.59284 -10.08020-1996.210 -1960.87038 -0.17816 4.31263 -12.74650 -0.56914 -1.76691 -11.54870-2247.730 -2208.06632 -0.22164 4.85580 -14.57040 -0.90945 -2.16884 -13.31100-2493.760 -2449.83597 -0.23233 5.37711 -16.13370 -0.95618 -2.31041 -14.77950-2245.690 -2206.30791 -0.18907 4.39948 -14.41840 -0.92054 -2.04962 -13.28930-1988.620 -1953.47239 -0.12416 4.11726 -12.90930 -0.54764 -1.76706 -11.68990-1716.560 -1686.63940 -0.13479 3.61741 -11.46590 -0.20700 -1.56037 -10.11260-1502.470 -1475.90281 -0.14576 3.00909 -9.77222 -0.11735 -1.38639 -8.50318-1227.920 -1206.20993 -0.10233 2.55280 -8.07914 0.25593 -0.97344 -6.84978-1082.120 -1062.68568 -0.12427 1.90087 -6.97162 0.31164 -0.79985 -5.86013

-734.171 -721.45830 -0.05919 1.20591 -5.07227 0.95752 -0.43040 -3.68436-504.028 -495.20946 -0.02696 0.64099 -3.49827 1.23252 -0.14781 -2.11794-256.595 -252.18475 -0.10273 0.07607 -2.10914 1.56145 0.08039 -0.62808

-2.034 -2.52099 0.08132 0.07605 0.09475 -0.08899 0.11404 -0.10827

Slope 9.82289E-01 9.03304E-05 -2.21641E-03 6.33701E-03 8.45466E-04 1.03865E-03 6.14383E-03

NASA/TM—2003-212326 20
Table A2.óReaction Load Cell Data for an Applied Lateral (FY) Calibration Load 

FY RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4 RZ1+RZ2 RZ3+RZ4

0.173 1.72279 0.13895 -0.03260 0.16653 -0.56756 0.46330 -0.86432

-199.017 1.02173 -100.04375 -98.23386 3.97433 -1.77150 0.88328 1.31955-398.089 0.23809 -200.01456 -196.67408 7.87209 -3.33785 1.58827 2.94597-598.639 -0.56701 -300.66609 -295.65779 11.78050 -4.82801 2.38028 4.57224-798.102 -0.91570 -400.87485 -393.57694 15.70010 -6.34095 3.19351 6.16569-997.770 -1.67719 -501.96905 -490.97454 19.54380 -7.89665 3.84427 7.80289-802.978 -1.02454 -404.80565 -394.16332 15.82020 -6.30867 3.28020 6.23133-599.496 -0.04541 -301.61662 -294.59337 11.85740 -4.69863 2.55357 4.60522-400.315 0.08629 -201.55894 -196.28306 7.91596 -3.26284 1.83762 2.81550-201.223 0.80389 -101.04839 -98.56003 4.04029 -1.87035 0.93740 1.23254

-1.554 0.03669 -0.23489 -0.05432 0.01238 -0.10756 0.05897 -0.15415-1.586 -0.11380 -0.33209 -0.16295 -0.05281 -0.08541 -0.11456 -0.02366

101.302 1.52951 50.76034 50.52374 -1.93118 0.60003 -0.36404 -0.96711201.191 1.59489 100.92412 99.79834 -3.97234 1.10063 -0.81932 -2.05239400.904 2.68374 201.01400 198.15191 -8.13028 2.61184 -1.67641 -3.84204600.083 3.38000 300.98468 296.46167 -12.29910 3.97063 -2.55511 -5.77337800.849 4.37951 401.78705 394.98880 -16.51100 5.18873 -3.75893 -7.56330

1000.140 5.53105 500.99188 494.05984 -20.78820 6.56933 -4.81114 -9.40776

802.121 4.16295 402.15437 395.94487 -16.51130 5.37297 -3.68327 -7.45502601.170 3.05559 301.20069 297.22208 -12.37520 3.99214 -2.73962 -5.64340401.739 2.42511 201.78045 198.45576 -8.19559 2.57888 -1.85003 -3.76667201.462 1.87866 101.19412 99.75490 -4.03738 1.29532 -0.88477 -1.85730

-0.978 -0.15674 -0.27809 0.29330 0.02375 0.06488 -0.01737 0.10600

Slope 3.24399E-03 5.02184E-01 4.93022E-01 -2.01384E-02 7.27373E-03 -4.32204E-03 -8.54266E-03

NASA/TM—2003-212326 21
Table A3.óReaction Load Cell Data for an Applied Vertical (FZ) Calibration Load 

FZ RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4 RZ1+RZ2 RZ3+RZ4

-0.182 0.21096 0.30270 0.60352 0.13023 -0.18184 0.03653 -0.08813988

-122.114 -0.05653 -0.07552 0.32083 -66.93741 -55.36818 -54.47783 -67.82777-195.405 0.23145 -0.56161 1.34201 -97.10230 -98.53182 -96.38174 -99.25238-396.876 0.22783 -1.57722 2.23274 -198.22435 -198.27911 -196.30354 -200.1999-595.704 0.50627 -2.75482 3.16696 -296.37602 -299.01475 -295.29661 -300.09413-797.505 1.12958 -4.26704 4.05733 -397.00668 -400.08529 -394.08145 -403.01053-997.280 1.23316 -5.89845 4.86111 -496.76134 -500.42699 -492.58238 -504.60594-791.401 0.95530 -4.53703 3.97054 -398.04806 -393.37946 -396.69803 -394.729492-591.878 0.67832 -2.99241 3.03643 -298.45871 -293.26227 -297.69742 -294.02356-394.760 0.05386 -1.98743 2.12405 -198.56521 -196.20239 -198.00744 -196.760157-195.468 -0.15899 -0.82068 1.42891 -99.61442 -95.92333 -98.43285 -97.104903

-0.356 -0.20055 0.11916 0.75566 -0.17747 -0.29079 -0.10239 -0.365874-0.356 -0.19993 0.32447 0.90763 -0.17726 -0.13897 -0.09167 -0.2245624

106.225 -0.24085 0.69159 0.21243 46.97258 57.96355 49.80986 55.12627202.897 -0.19520 1.19928 -0.15672 99.61429 101.53351 99.13997 102.007826403.314 -0.47372 2.33398 -0.78666 199.54785 201.64844 197.65555 203.54074600.397 -0.66589 3.18777 -1.63400 297.89018 300.25246 294.77751 303.3651802.781 -0.85736 3.92213 -2.22038 398.75876 401.35678 395.11877 404.99676

1003.720 -1.20230 4.46257 -3.08939 499.73613 501.06468 495.28984 505.51097

799.046 -0.88040 4.00847 -2.22038 398.36290 397.90517 395.61348 400.65459601.312 -0.96911 3.28428 -1.67740 299.70144 299.31722 298.16332 300.855371403.950 -0.30180 2.43060 -0.96061 200.67799 200.96201 200.02221 201.617792203.367 -0.02257 1.59869 -0.02642 99.55675 101.63349 101.00390 100.18635

-0.215 0.25533 0.52920 0.71212 -0.61246 -0.06636 -0.52553 -0.1533004

Slope -1.12824E-03 5.18577E-03 -3.91494E-03 4.98625E-01 4.99398E-01 4.95026E-01 5.02996E-01

NASA/TM—2003-212326 22
Table A4.óReaction Load Cell Data for an Applied Rolling (MX) Calibration Moment (CZ2+CZ4) 

FZ(CZ2+CZ4) RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4

-0.095 0.40605 -0.01889 0.09239 0.07597 0.02983

-98.092 0.94584 -0.84028 0.33126 -0.05302 -97.98530

-198.716 1.39963 -1.68273 0.87454 -0.39818 -198.12115-285.066 1.83268 -2.63401 1.20041 -0.55856 -284.14577-395.603 3.25951 -3.37940 1.83042 -0.60134 -394.51621-496.671 4.27402 -4.41651 2.19956 -0.69548 -496.05751-393.718 3.17302 -3.45507 1.69988 -0.42769 -393.19346-295.635 1.96152 -2.75224 1.30906 -0.22750 -295.24944-203.242 1.42101 -1.88824 0.98313 -0.07828 -203.02438

-98.304 0.55617 -0.86167 0.37476 -0.02587 -98.40839

0.122 0.27647 -0.09461 0.11414 0.24322 -0.102980.122 0.57843 -0.24616 -0.01630 0.22144 -0.15732

47.468 0.18941 0.53234 -0.19004 0.41537 46.29615

101.030 -0.54614 1.09372 -0.47258 0.58859 99.58819201.307 -1.13021 2.28243 -1.03742 0.84653 199.45385301.505 -1.67007 3.25461 -1.40687 1.06070 299.19772393.713 -2.01518 4.15151 -1.64560 1.25342 391.15047502.798 -2.68565 5.19957 -2.10197 1.51171 499.99444400.466 -2.34019 4.11889 -1.66738 1.21956 397.95660314.838 -2.12538 3.30868 -1.47203 1.02598 312.82024203.245 -1.15181 2.24997 -1.01581 0.76717 201.71789102.645 -0.54578 1.13709 -0.53788 0.42314 101.65790

0.068 0.31917 0.02432 0.44014 -0.04047 0.99723

Slope -6.70280E-03 9.71169E-03 -4.40648E-03 2.29953E-03 9.95765E-01

NASA/TM—2003-212326 23
Table A5.óReaction Load Cell Data for an Applied Rolling (MX) Calibration Moment (CZ1+CZ3) 

FZ(CZ1+CZ3) RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4

-0.119 0.00983 0.04071 0.13592 -0.02980 -0.02729

-96.137 -0.53339 -0.11055 0.26619 -95.73914 -0.36048

-196.837 -0.92484 -0.24037 0.76593 -196.04780 -0.72147-296.575 -1.62091 -0.47803 1.02671 -295.17781 -1.10289-398.164 -2.29420 -0.96460 1.50476 -396.79452 -1.18055-496.597 -2.75075 -1.32089 1.80889 -495.14043 -1.42026-400.760 -2.44669 -1.14792 1.63503 -399.37456 -1.21455-300.914 -1.77205 -0.81278 1.17878 -299.74948 -0.86274-201.142 -1.03284 -0.55352 0.72244 -200.40714 -0.48717-102.089 -0.38129 -0.51004 0.41839 -101.68782 -0.18457

0.097 -0.16213 -0.34821 0.09239 0.15703 0.085600.075 0.20449 -0.33752 0.11405 0.17888 0.09594

100.807 0.40289 -0.49924 -0.29874 99.76244 0.31821200.548 0.59911 -0.45592 -0.66822 199.07447 0.66960301.766 1.25080 -0.60673 -0.71150 299.99751 0.93793399.612 1.40378 -0.83325 -1.16765 397.36804 1.28183501.018 1.79513 -1.11411 -1.75457 498.13647 1.63925402.179 1.79429 -0.72559 -1.38518 399.86265 1.12246302.363 1.27253 -0.46667 -1.05934 300.71554 0.86332203.724 0.98929 -0.18570 -0.58125 202.27007 0.61982104.093 0.31563 -0.18620 -0.21176 102.98550 0.27447

-0.185 0.00966 -0.05688 0.04890 -0.03888 -0.15213

Slope 4.76143E-03 2.44622E-04 -3.48135E-03 9.95250E-01 3.05296E-03

NASA/TM—2003-212326 24
Table A6.óReaction Load Cell Data for an Applied Pitching (MY) Calibration Moment (CZ3+CZ4) 

FZ(CZ3+CZ4) RX1+RX2 RY1 RY2 RZ1+RZ2 RZ3+RZ4

0.114 0.08619 -0.02446 -0.06535 -0.17562 -0.02855

-95.967 0.17060 -0.01371 0.47773 -0.05575 -95.91599

-197.705 0.51364 -0.02435 0.89045 -0.03389 -197.54135-294.995 0.96501 -0.04612 1.52049 0.05348 -294.69016-398.320 1.02797 -0.06789 2.06354 -0.09816 -398.02602-493.010 1.19765 -0.04634 2.71528 -0.13032 -493.12850-420.868 1.39415 -0.02468 2.21573 -0.13051 -420.70852-305.238 0.98726 -0.04606 1.65078 -0.19556 -305.13699-202.474 0.59973 -0.02462 1.06422 -0.01232 -202.40378-105.353 0.19133 0.02972 0.56467 -0.08825 -105.27046

-0.136 -0.00148 0.02983 -0.08707 0.04166 0.02727-0.104 -0.08750 0.02978 -0.02192 -0.13168 -0.0162798.888 -0.36710 -0.02440 -0.65203 0.01987 98.10948

202.037 -0.60105 -0.08927 -1.30360 0.03052 201.26211298.033 -0.75012 0.01925 -1.97712 -0.02396 297.18196388.805 -0.94450 0.06268 -2.45499 0.03009 388.02860503.682 -1.22189 -0.02390 -3.04142 0.16009 502.71252414.068 -0.85563 -0.01332 -2.36803 0.17226 413.16904324.285 -0.62045 -0.03493 -1.84660 -0.03363 323.33389211.681 -0.53763 -0.05648 -1.12968 -0.07730 210.67912102.710 -0.12880 -0.05675 -0.36938 -0.04404 101.80023

0.071 0.15041 -0.11098 -0.10883 -0.02210 0.20512

Slope -2.58132E-03 3.43725E-05 -5.64588E-03 2.07362E-04 9.98524E-01

NASA/TM—2003-212326 25
Table A7.óReaction Load Cell Data for an Applied Pitching (MY) Calibration Moment (CZ1+CZ2) 

FZ(CZ1+CZ2) RX1+RX2 RY1 RY2 RZ1+RZ2 RZ3+RZ4

-0.057 -0.08672 0.07834 0.13033 -0.05664 -0.02931

-98.239 0.06304 -0.90529 0.19539 -97.89613 -0.05817

-198.364 0.04114 -2.03966 0.34740 -197.72341 -0.23136-297.111 -0.32899 -3.14136 0.69507 -296.45896 -0.22986-394.386 -0.17731 -4.58892 1.17299 -393.53480 -0.41979-495.419 0.05887 -6.13363 1.39012 -494.32711 -0.58019-401.170 -0.41605 -4.89115 0.99907 -400.04213 -0.46138-301.643 0.01994 -3.35696 1.04276 -300.91763 -0.33220-200.902 -0.32668 -2.34172 0.47775 -200.28124 -0.24733-100.386 -0.21731 -1.26117 0.26045 -99.94337 -0.27722

0.203 0.12980 -0.36496 0.15200 0.30889 -0.04212

-0.057 0.15178 -0.21325 0.15208 0.29809 0.0232499.436 0.17428 0.67216 -0.19566 98.69910 0.06921

199.062 -0.17067 1.68802 -0.47797 198.07608 0.25585299.204 0.15444 2.62766 -0.54303 298.00783 0.23338396.699 0.11222 3.36261 -0.89077 395.36138 0.37850498.648 0.02432 4.03259 -0.91242 497.10607 0.33736426.118 -0.27832 3.65409 -0.82546 424.58068 0.39550307.580 -0.14877 2.91953 -0.73865 306.55375 0.34494204.004 -0.01904 2.08739 -0.36915 203.02115 0.18254103.048 0.28332 1.06090 -0.17367 102.25487 0.10756

-0.491 0.19501 0.01309 0.08694 -0.30891 -0.04760

Slope 1.87379E-04 1.01435E-02 -2.36003E-03 9.96940E-01 9.95332E-04

NASA/TM—2003-212326 26
Table A8.ó Reaction Load Cell Data for an Applied Yawing (MZ) Calibration Moment (CY1) 

FY(CY1) RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4 RZ1+RZ2 RZ3+RZ4

-0.060 0.34636 0.00540 0.09244 -0.14667 0.07093 -0.09759 0.02185

-98.736 -0.47761 -99.35273 1.11350 2.55698 -1.73772 0.78066 0.03859

-196.999 -0.06588 -198.65658 1.89511 5.11851 -3.71743 1.37741 0.02367-296.414 -0.19644 -298.79272 3.17686 7.70208 -5.64329 1.99568 0.06311-397.479 0.14908 -400.80795 4.30644 10.55750 -7.53678 2.73333 0.28740-497.490 -0.17515 -501.86122 5.39250 12.92390 -9.66888 3.08069 0.17431-401.864 0.08408 -405.39702 4.19760 10.59010 -7.71076 2.69005 0.18931-301.310 0.23670 -303.82511 3.17683 7.85460 -5.64357 2.00658 0.20445-202.330 -0.08751 -203.89467 1.98213 5.08622 -3.67457 1.43141 -0.01976-102.221 -0.04392 -102.71136 1.00481 2.46999 -1.83554 0.72635 -0.09190

-0.733 0.23846 -0.22140 -0.01628 -0.19006 0.02653 -0.07603 -0.08751-0.842 -0.19514 -0.25380 0.11423 -0.01623 0.01571 0.01077 -0.0112999.735 0.13095 99.98992 -1.03731 -2.79619 2.18243 -0.41219 -0.20157

197.792 0.54200 198.72198 -1.73204 -5.62926 4.16256 -1.26918 -0.19752297.868 0.65163 299.10552 -2.62260 -8.26698 5.88162 -2.05029 -0.33507397.706 0.89018 399.20908 -3.38277 -10.93720 7.77434 -2.75537 -0.40747497.544 1.19370 499.29104 -4.22984 -13.80250 9.59137 -3.70989 -0.50122402.266 1.30217 403.76673 -3.33931 -11.07860 7.93806 -2.76624 -0.37431303.601 0.67313 304.70024 -2.79655 -8.36532 5.92616 -2.03915 -0.40002202.754 0.32711 203.35530 -1.99296 -5.53228 4.13155 -1.31232 -0.08841102.644 0.30529 103.07865 -0.90669 -2.77503 2.07552 -0.56359 -0.13593

-0.299 0.15410 -0.29699 0.04896 0.01499 -0.03513 0.05490 -0.07504

Slope 1.26865E-03 1.00607E+00 -9.63870E-03 -2.68759E-02 1.93614E-02 -6.79093E-03 -7.23528E-04

NASA/TM—2003-212326 27
Table A9.óReaction Load Cell Data for an Applied Yawing (MZ) Calibration Moment (CY2) 

FY(CY2) RX1+RX2 RY1 RY2 RZ1+RZ3 RZ2+RZ4 RZ1+RZ2 RZ3+RZ4

0.122 -0.27797 -0.00013 0.00000 -0.19291 -0.01109 -0.14659 -0.05741

-98.354 1.05326 -0.02201 -97.73551 1.31651 0.00043 0.03793 1.27901

-195.700 -0.09698 -0.13074 -194.94930 2.47839 0.50088 0.18960 2.78967-297.907 -0.35991 -0.12028 -297.05226 3.80308 0.90376 0.40640 4.30044-392.284 -0.99036 -0.19633 -390.57267 5.11724 1.45849 0.60145 5.97428-486.835 -1.72973 -0.14288 -484.96217 6.36609 1.81731 0.79642 7.38697-404.638 -1.09835 -0.16381 -402.97931 5.13934 1.50159 0.55786 6.08307-306.325 -0.51151 -0.17439 -305.00536 3.96669 1.09907 0.48232 4.58344-210.687 0.07625 -0.22795 -209.72551 2.76120 0.52253 0.26554 3.01819-105.826 0.94605 -0.16236 -105.31903 1.51264 0.08697 0.17889 1.42072

-0.084 -0.10406 -0.09727 -0.13037 -0.16004 -0.13053 -0.11388 -0.17669-0.095 -0.19144 -0.04288 -0.08692 -0.15998 -0.04380 -0.08137 -0.1224197.849 1.75407 -0.12940 97.62693 -1.48436 -0.92545 -0.34154 -2.06827

195.956 2.25764 0.00116 195.53638 -2.85236 -1.35007 -0.59093 -3.61149294.585 2.45579 -0.03074 293.66294 -4.11190 -1.77491 -0.68830 -5.19851394.236 2.99926 0.05651 392.96253 -5.63228 -2.21091 -0.82937 -7.01382493.790 3.71653 0.12176 492.08922 -7.00022 -2.65797 -0.91597 -8.74222411.604 3.45377 0.14275 410.28030 -5.92587 -2.36298 -0.93769 -7.35116327.352 3.03964 0.10939 326.14689 -4.55752 -1.92726 -0.76420 -5.72058201.709 1.82409 0.13060 201.27272 -2.96133 -1.16533 -0.49331 -3.63335105.679 1.69122 0.00010 105.31893 -1.54982 -0.76203 -0.41737 -1.89448

-0.019 0.32908 -0.03247 0.06518 -0.13802 0.02135 -0.13558 0.01890

Slope 5.30436E-03 3.39443E-04 9.96382E-01 -1.36043E-02 -4.61535E-03 -1.84459E-03 -1.63751E-02

NASA/TM—2003-212326 28
The Inverse Matrices 

[ ] 1

9.82289E-01 3.24399E-03 -1.12824E-03 -5.21101E-04 -1.13682E-04

9.03304E-05 5.02184E-01 5.18577E-03 4.30321E-04 2.83304E-02

-2.21641E-03 4.93022E-01 -3.91494E-03 -4.20514E-05 -2.83386E-02

6.33701E-03 -2.01384E-02 4

S −

=

.98625E-01 -4.51341E-02 -3.73846E-04

8.45466E-04 7.27373E-03 4.99398E-01 4.51233E-02 6.75402E-04

                (A1) 

[ ] 1

9.82289E-01 3.24399E-03 -1.12824E-03 -6.25890E-05 -1.13682E-04

9.03304E-05 5.02184E-01 5.18577E-03 -2.28486E-04 2.83304E-02

-2.21641E-03 4.93022E-01 -3.91494E-03 -7.42812E-05 -2.83386E-02

1.03865E-03 -4.32204E-03

U −

=

4.95026E-01 -2.25248E-02 -1.39334E-04

6.14383E-03 -8.54266E-03 5.02996E-01 2.25429E-02 4.40889E-04

                (A2) 

NASA/TM—2003-212326 29
The Interaction Matrices 

[ ]

1.01804 -1.55676E-03 -5.41036E-03 -4.74953E-03 7.01590E-03

2.15971E-03 1.00515 1.00467 3.03588E-03 -5.58783E-03

-7.28563E-03 7.67051E-03 1.83743E-02 1.00195 1.00206

6.18255E-02 -5.08750E-01 -9.85532E-02 -11.0884 11.07

S

=

59

-4.11345E-02 17.4869 -17.8107 -6.87765E-02 -2.52631E-01

                (A3) 

[ ]

1.01800 -1.29503E-03 -5.33780E-03 -2.72514E-04 2.52342E-03

2.14928E-03 1.00491 1.00473 -8.08604E-03 5.42246E-03

-7.28416E-03 7.67704E-03 1.83836E-02 1.00235 1.00166

-1.13301E-01 -1.32250E-01 3.21076E-01 -22.3640 22.01

U

=

32

-4.09244E-02 17.4823 -17.8106 -2.20509E-01 -1.01940E-01

                (A4) 

NASA/TM—2003-212326 30
REFERENCES 

1. Luis Beltran, Richard L. Del Roso, and Ruben Del Rosario, ìAdvanced Nozzle 

and Engine Components Test Facility,î NASA Technical Memorandum 103684, 

January 1992. 

2. Joyel M. Kerl, Gwynn A. Severt, and Kurt Loos, ìAdvanced Nozzle Test Facility 

at NASA Glenn Research Center,î AIAAñ2002ñ3245, June 2002. 

NASA/TM—2003-212326 31
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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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 SixComponent 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; Thrust

vector control

Unclassified -Unlimited

Subject 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 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.