00

00c0 NAWCWPNS T71 8134

Equations of Motion of a HingedBody Over a Spherical Earth

byMegaera C. Halter

Intercept Weapons Deparment

AUGUST 1993

NAVAL AIR WARFARE CENTER WEAPONS DIVISIONCHINA LAKE, CA 93555-6001

(UG2 6 1993A\ E

Approved for public release; distribution isunlimited.

93-19906

51 8 25 011l

Naval Air Warfare Center Weapons Division

FOREWORD

This report describes the derivation of the equations of motion of a hinged vehicle.This work was accomplished at the Naval Air Warfare Center Weapons Division, ChinaLake, during the period from May 1990 to May 1992 and was funded by the AirLaunched Weaponry Block, NW1A, sponsored by the Office of Naval Technology.

Approved by Under authority ofJ. A. WUNDERLICH, Head W. E. NEWMANIntercept Weapons Department RAdm., U.S. Navy27 July 1993 Commander

Released for publication byS. HAALANDDeputy Commander for Research and Development

NAWCWPNS Technical Publication 8134

Published by .............................................. Technical Information DepartmentCollation ........................................................................ Cover, 18 leavesFirst printing ........................................................................... 80 copies

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4. TITLE AND SUBTITLE S. FUNDING NUSER

Equations of Motion of a Hinged Body Over a Spherical Earth

6. AUTNDMS) I

Megaera C. Halter, Code C2915, 939- 8241

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) S. PERFORMING ORGANIZATIONREPORT NUMBE

Naval Air Warfare Center Weapons Division NAWCWPNS TP 8134China Lake, Calif.

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Office of Naval Technology

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Approved for public release. Distribution is unlimited.

I. ABSTRACT (MSNMrBnE AV PrA

The six-degree-of-freedom (6-DOF) equations of motion are derived in vector form for a hinged vehicleflying over a spherical Earth. Initially, the most general form of the equations is developed assumring that thecombustor may roll with respect to the body, as well as deflect in pitch and yaw. Next, the primary case isdeveloped in which the combustor cannot roll with respect to the body. Finally, these equations arespecialized to three simpler cases. The motivation for deriving this set of equations is to support feasibilitystudies of the low drag ramjet (LDR), which is streamlined by deleting all aerodynamic surfaces. Alldirectional control comes from thrust vectoring, which is accomplished by deflecting the entire motor ratherthan just the exhaust nozzle, as in more conventional designs. Since the motor is a large fraction of the totalvehicle mass, its inertia has a significant effect on the dynamics of the overall vehicle. Sufficient detail isincluded to serve as a tutorial.

m4 SUSAJECT TERMS IS. NUMBER OF PAGES

equations of motion 1 71S. PRICE CODE

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NAWCWPNS TP 8134

CONTENTS

Introduction ...................................................................................... 2

General Six-Degree-of-Freedom Motion Over Spherical Earth With Rolling Hinge ...... 2Newton's Second Law Applied to Whole Vehicle ........................................ 3Conservation of Angular Momentum for Main Body ..................................... 8Conservation of Angular Momentum for Combustor .................................. 9Kinematics ................................................................................. 12

General Six-Degree-of-Freedom Motion Over Spherical Earth With No-Roll Hinge .... 17

Special Case: Single Rigid Body ............................................................ 27

Special Case: Flat Earth ...................................................................... 28

Special Case: Flat Earth and Pitch Plane ................................................... 29

Conclusions ..................................................................................... 31

Nomenclature ................................................................................... 32General Notation .......................................................................... 32Symbols ................................................................................... 34

References ..................................................................................... 37

Figures:1. Vehicle Over Spherical Earth ............................................................ 42. Forces and Moments in the Hinge ................................................... 53. Planar Vehicle Over Flat Earth .............................................. . ..... 30

ACKNOWLEDGEMENT

The author appreciates the help of Allen Robins, who traced through this Eldevelopment and exercised the resulting equations in his 6-degree-of-freedom (DOF)simulation of the low drag ramjet (LDR) missile. Thanks also to Richard Boss for hisindependent derivation of the planar form of these equations, which served as helpfulcheck of this work. Diýt, ibutior I

DrIC QUALITY Z11 3ECT72D 3 Availability Codes

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NAWCWPNS TP 8134

INTRODUCTION

The six-degree-of-freedom (6-DOF) equations of motion are derived in vector form

for a hinged vehicle flying over a spherical Earth. Initially, the most general form of the

equations is developed assuming the combustor may roll with respect to the body, as well

as deflect in pitch and yaw. Next, the primary case is developed in which the combustor

cannot roll with respect to the body. Finally, these equations are specialized to three

simpler cases. The motivation for deriving this set of equations is to support feasibility

studies of the low drag ramjet (LDR), which is streamlined by deleting all aerodynamic

surfaces. All directional control comes from thrust vectoring, which is accomplished by

deflecting the entire motor rather than just the exhaust nozzle, as in more conventional

designs. Since the motor is a large fraction of the total vehicle mass, its inertia has a

significant effect on the dynamics of the overall vehicle.

GENERAL SIX-DEGREE-OF-FREEDOM MOTION OVER SPHERICAL

EARTH WITH ROLLING HINGE

To fully describe the trajectories of two rigid bodies, 24 first-order scalar differential

equations are needed. For each body there are 12 equations, three for each of the

following: position, attitude, velocity, and angular rate. In this case, however, the task

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of developing the angular and translational acceleration equations is simplified by the fact

that the combustor is attached to the body. The hinge makes the velocity and acceleration

of the combustor's center of gravity (CG) a function of the combustor's attitude, angular

velocity, and the body states. In this section, the combustor is allowed to roll, pitch, and

yaw with respect to the body.

An additional simplification results from the assumption that the mass properties of

each body change slowly, PhB=0, Mc=0, IBj0,. II B and

'rIa 0. The missile geometry is illustrated in Figure 1, and the notation is explained in

detail in the notation section at the end of this report.

NEWTON'S SECOND LAW APPLIED TO WHOLE VEHICLE

Apply Newton's second law to the whole vehicle, where the forces and moments in

the hinge are as illustrated in Figure 2. (Reference 1 has a good statement of Newton's

second law.) External forces and moments are assumed to act at the combustor and body

CGs.

d(mv~v) ()=t MVfv = TB + PC + MVU (1)

In Equation 1, the subscript "V" refers to a quantity that applies to the overall vehicle. A

subscript "C" refers to the combustor or motor section, and the "B" subscript refers to the

main body.

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ZB BODYCOMBUSTOR

"x" points North

y- points EastzH points "down"

FIGURE 1. Vehicle Over Spherical Earth.

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Mh(BODY

COMBUSTOR

FIGURE 2. Forces and Moments in the Hinge.

Since the location of the vehicle CG relative to both bodies varies with combustor

deflection angle, and the inertial measurement unit will be located in the main body, the

body CG is the reference point for these equations. The body acceleration is derived

from the vehicle acceleration as follows:

mvav = mBGB + P1c2Zf

mva'v = mtBa + inca8 + mcic- mc'fB

mvi =(MB +M0)4 + MC(UC-aB)

Mivav = Mvas + "caC/ B

MVnB = mvav -mncc,8 (2)

The " Ac/B" is the acceleration of the combustor with respect to the body. This is a

noninertial acceleration. Since the combustor is attached to the main body, its position

relative to the main body can be described algebraically.

XC = X+ + XCII

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XC = XB + X*IB + XC/h

XC =XB+47B+4C

Differentiate for combustor velocity:

VC = • VB + VcIBdt

"VC =VB + (A +& +O)B XPC)

Remember, PB - 0 and PC 0

VC= VB + c0B X PB + &0C X PC

Differentiate a second time for combustor acceleration:

AC= VC = B+ ac,,dt

ZfC=4 a+[MB X13 + WB XOB+ O)BX(&BX #B)

+[- XC & + C X -C +iuWC X(1PC X c)]

Since (i8 and Pc are assumed to be near zero,

AC=ZB + )BXAB +OB X (OBX #B) +WC X #C + X(O)C X #C)

Isolating the acceleration of th,2_ combustor with respect to the body,

_C:_BXB )B(0Xf5)ACXJ+ONC X(WC XI3C) (3)

Inserting Equations I and 3 into Equation 2 yields an expression for the inertial

acceleration of the main body CG

mvaB = B + Fc + mvG

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Because the body and combustor angular rates are relative to inertial space, they

include the rates attributable to the spherical, rotating Earth.

a = WC/E + OE/H + WyHIG + a

S= cE/H + 6THIG + +G

The inertial acceleration of the body CG is now known. But it is more useful toknow the body acceleration relative to the body frame, VB/GL (see the notation section),

since this is what a strapdown accelerometer located at the body CG would measure.

Expanding the body CG acceleration in terms of body angular velocity yields

Sd2 . 8 d - XXB)

tf3 d i2

aB = BIG + 2NG X VBIG + aG XB (&G X(YG Y xD)

where

VB/GIG = VB/GIB + WB/G X V/G

and wG is assumed to be zero.

4B= V'B/GIB + (OBIIH + WHIG) X VB/G + 2 0G X VBIG + 6YG X PdG X x8B)

Reazrange UB to isolate the different types of acceleration.

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a8i = V'BIG + &BIH X VB/ Flat Earth

+UHIG xV, ,- Spherical Earth

+20C "' BIG Coriolis Rotating Earth

+&'G X (NG x x8 ) Centrifugal relief I

"Taserting this into Equation 4 and placing all derivatives on the left side yields the

first of the three vector differential equations for translational and angular velocities.

nhVVB/GIB +MC PiC X #C + WB X AB)

= TB+ F X MVU -mc[N.c X (&C X #C) + &B X (6 X #B)] (5)

-mv[NIJRG X VBIG + 20G X 7BIG + &G X( PG X B)]

CONSERVATION OF ANGULAR MOMENTUM FOR MAIN BODY

Conservation of angular momentum for the main body about its own CG is the

starting point for the next derivation.

At this point, an assumption is made about the forces internal to the hinge. The servo is

assumed to exert a pure couple (see Figure 2). Thus, the force internal to the hinge

between the body and the combustor is defined by Newton's third law. (Reference I has

a good statement of Newton's third law.) This force is not generally measured, so it is

eliminated by applying Newton's second law to the combustor, solving for the force in

the hinge and substituting for it in the angular momentum equation for the body.

"IU = -h+ Tc + C

mcc-, , +cmmn cGn nnn NNM lm

NAWCWPNS TP 8134

Substitute for P,

B xT + "CUh + CjC)

Expar " out the derivative on the left side.

IB1B WB + 1BCB + a BiB = -WB + Mjh + ý X (TC + mC - mCC)

Remember, I = 0, expand out and place all derivatives on the left hand side:

iBfWB +J 6B I~C (VBIGIB +WC X #JC +OB X #B)

= M-B + -h + #B X(FC +mcG')- &B XJB&B

-#B XlC[•,B/G X V-,, + 20G X VB-G + •XG X (,G X XB)]

IAB X nC([iCX (&Ci X #3C) + UB X (&B X AB)] (6)

CONSERVATION OF ANGULAR MOMENTUM FOR COMBUSTOR

The development of the angular momentum equation of the combustor about its own

CG is very similar to that for the body. At this point, the combustor is assumed to be

able to pitch, yaw, and roll with respect to the body.

d (-

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Expand out, assuming ILj - 0 and rearrange.

Wc +f#C xmrc (VBGIB +uWC X# ~+ Iu5BX AD)

= I - Rh+ #CX + "ICU) - &C X Cifc

-#c x mc[[iBIG X V'BIG + 2IG X V•/B + G X (UG Xx IB)]

Because the two bodies are connected, Equations 5, 6, and 7 are coupled and must be

solved simultaneously. This can be done by rearranging so that the whole set of

equations can be treated as a single nine-element vector differential equation, which is

then inverted to solve for the state derivatives. To do this, a common coordinate system

must be chosen for all three equations. The simplest choice is the body frame. Second,

all vector cross products to the left of the equal sign must be replaced by matrix products.

In general, a vector cross product can be replaced by a matrix product as follows:

where

0 -a2 ayd a. 0 -a,

[-ay a, 0

Also in general,

a X 6 = -9 X< a

These two rules must be applied to the left side of the equal sign, but for aesthetics, they

are also applied to the right side. The result after rearranging is

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P3fClB irVBlGI

MVI nCJ -nC+ A IB

BI- B -B-B -B -B--B -) -B -BB-B

-CB -B--- -B-B--B WBIG VBIG -GB- -B-(8)

-B - B ~B-B-- +)GB ~B-B -B-B-

The matrix premultiplying the derivative vector serves as an expanded inertia tensor.

Written in partitioned form above, it appears to be nearly skew symmetric. However,

when (tediously) expanded, it is symmetric.

To make use of this equation, each vector in Equation 8 must be written explicitly in

the body frame. Selected vectors are listed below. Figures 1 and 2 and the nomenclature

section prvide further information.

Body CG location with respect to the hinge and hinge location with respect to the

combustor CG:

Gravitational acceleration:

11

i' IC IIC&C #C +&I &II #B I / ;RI G +2 BI G + I G B

NAWCWPNS TP 8134R 20

=[TH2B190o(j-) {E }Body position: The position of the body CG, XB, is relative to the inertial reference point

at the Earth's center. Thus, the z-component of XBH (body CG position expressed in

the local horizontal frame) is (RE + h).

Angular velocity of the ground frame:

[WE1W

WG [TG2I 0=[TH2B][TG2HI{O}

Angular velocity of local horizontal frame with respect to the ground frame:

(OAIG = [T124{ -i}-+ R1 Y~}

Combustor inertia tensor. I: = [TR2 c WC

KINEMATICS

In the preceding section, nine of the 18 differential equations describing the motion of

the two connected bodies were developed. Additional differential equations that describe

the position and attitude of the vehicle are now needed. Start with the attitude of the main

body with respect to local horizontal. Remember, on the spherical Earth the local

horizontal frame rotates with respect to the inertial frame. The body frame is related to

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local horizontal by the Euler rotation sequence V' about z, 0 about y, and 0 about x.

The inertial angular velocity of the body is

&B= WE/IH +OJHIG + G

OE/H =WB-EiHIG -&G

Again, use the body frame.

1 o 0 0' -1 0 0 cosO0 -sin 01'BBIH 0 + 0 cos, sine + 0 coso sin| 0 1 0 0

0 0 -sin# cosoJ[0J [0 -sino cos4.sin6 0 cos6 J,

[1 0 -sin6WEIH [0 coso sincosO

-0 -sino coscosOVf,

Solve for the Euler angle rates using Cramer's rule:

os* -sin¢ ]DIH (9)sin__. cos• 9

coso coso j

An important drawback exists to using the differential equations for Euler rotation

sequence to describe the body angular motion. There is a singularity in the

transformation matrix at 0 = 90 deg, i.e., when the body is vertical. An alternate way to

relate the inertial and the horizontal frames is by using quaternions. This method yields a

transformation matrix that has no singularities and that can be calculated more rapidly by

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computer. However, quaternions are not physically intuitive, and this vehicle should

never reach a large pitch angle, so Euler angles were used in this development.

Next, determine the position on the surface of the spherical Earth of the body CG

with respect to a reference point, usually the intersection of the prime meridian and the

equator.

XBIG XHIG + XBIH

XBIG =(XH -XG)+XBIH

Note that the origins of the body and local horizontal frames coincide, so XBIH = 0.

Differentiate with respect to the ground frame:

VBIG =XE/IGL

VBIG XHIG-XGIG

Assume the Earth rotates at a constant rate about the axis through the poles, XGIG =0.

VEG XHI +OJHIG XXH

To solve for the latitude and longitude rates, this equation must be expressed in scalar

form. Choose the local horizontal frame:

-H H -H -HVBIG =5tB IH+_HGX

H OMIG XX11

-H r -BVAIG [TB2HJVBIG

-HNote that for a spherical earth, Vi/G is a non-inertial quantity. In a later section, when

the equations are reduced to flat Earth, it will be inertial. To avoid confusion with the

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NAWCWPNS TP 8134

inertial velocity, we will add the superscript "H" to each element of the vector, keeping in

mind that the two become identical for flat Earth.

u HVBI G -- V

wH

jvH}{} I .-k }isi X{0}+ )

Solving for the spherical coordinate rates:

RE + h (10)

V H (11)(RE + h)cos(I

h=-wH (12)

The geographic position is

x =RE4Y=REA.

The final equations needed are for the combustor deflection relative to the body.

Write out in the body frame.

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8 cos 6y cos8, - S sin5 8'p

f., cos8y sin82, +8S, cosz} 6 NcB-Z,0=q (13)

-S i y + .2r I

By convention, p, q, and r are assumed to be angular rates about the axes of the body

frame. B and &IB are available from integrating Equations 8. Solving for the

combustor deflections yields

S.,' - I - cosby tan 6, sin Sy ]p'

{Sy}-[tan3, cosby sin 8ytan 6{ } (14)S..J I o o 1 rL. CIB

Remember, the combustor deflection angles are modeled as an Euler sequence in this

development. This choice was made for convenience in the formulation and would

represent a set of nested gimbals. Depending on how the actual servo is designed, some

other model might be more appropriate.

The combustor CG velocity is available algebraically. Though it is not needed for

simulation of the vehicle's motion, it is provided here for reference. Remember, the

subscript " h" refers to the hinge.

VCIG = lcIB +VBIG

= V•ch + VhIB + VBIG

= -CIG X AC +&BIG X #B + RBIG

For interest, the surface distance flown along a great circle since launch is also included

(Reference 2):

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R =RECOS- {cos(1 - 4)o(- I) + sinj I-2 E2sr( - 41 )COS(1 2 -A)

The motion of the missile and combustor have been fully described. Equations 8

through 12 and Equation 14 describe the dynamics of the vehicle's 18 independent states,

subject to the following assumptions:

1. Combustor and body are rigid

2. Control servo in the hinge only exerts pure couple

3. Hinge can be modeled as a gimbal set

4. Earth is perfectly spherical, rotating at a constant rate

5. Mass properties of the two vehicle sections change "slowly"

GENERAL SIX-DEGREE-OF-FREEDOM MOTION OVER SPHERICAL

EARTH WITH NO-ROLL HINGE

Adding the constraint that the combustor may not roll with respect to the body

removes a degree of freedom; the combustor roll rate, 4,, = Pc, is no longer

independent and is deleted from the state vector. Further, ML is just a reaction moment

and is eliminated from the equations. Doing this will reduce the size of the state vector to

be integrated, but will also destroy the nice, compact form of Equation 8.

Define Pc using Equation 13, remembering that 58, = 0.

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{SYcios,:1= q:

The y and z elements yield differential equations for 8y and 3,

SY qCIB (15)eos=

r-B=r - (16)

Use the x element to relate Pc to the other states.

Pc/B = PC -B -8, sin8.

PC PB -8y, sin6Z

PC =P B-(qc-qB)tan~z (17)

Differentiate and use the expression for S.-

/ PB -(qc -qB)tanZ (qc -qBXrC -rB)PC = - 4S- Cs 2 8z (18)

Set up the following definitions to make subsequent work more compact.

'n C -B-CB--B+-B~B--Bl+ -B• -BI -B-B-B)

o~& [MV (&BBVB +& + 2@iGiT, G + &G -IB~B-BB& B -m -- C -m-C-B I -B @ -BB -B-B-B)

- -B "B -B B -BIGVBIG + 2BVBIG +CB-B + -)

{k OCICBZYLCJ "kng(A&CBA'# + &B&#B+ &BIGVBBG + 2GVB + C')B&XB~B)j

(19a)

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tWCB + AB B+ MCUB)J

[ill j12 j13"[ my! -~"

J22 i2=n ýcI (IBB- mcjB 2) -MCI3 (19c)

jl 32 j33 nCAC -mCACB B (IC nCCB2

Equation 8 then becomes

jJ2l j22 j23 dý fB b .(+OjW

h31 132 j3 it ý b(0

Remember,

#BBIGIB T(OR UBIGIB VB/GJ 8 WB/IG1 5 B 4B iB pC P C

{MA.}{0 0 0Mh Mx My-~-~-~

Solve the seventh equation in Equation 20 for Bhi

[ B11/t + 13112OBBIGIB + J31313 4B1GI,] +[J32j IPB +J+3213B

+[J331iPc + J33124C + J3313k] =fCx -Ml~ + kC,

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ML = -[J3 l11U/GI + J3112A,/GIB + J3113 hia/GIB]-[J3211 ps + J32 12 qB + J3213 *B]

-[J3311PC + J33 u12ic + J3313rl +fC, + kC,

Substitute into the fourth differential equation in Equation 20

[J21114BGIB + J2112 ,BIG + J2113i'B/G IBI+[J2211 p + J2212 q4 + J2213 i1

+[lJ23 1Pc + J2312 + J2313i']= f' A + ML +kB,

[J2111,iB/GIB + J2112 BIGIB + J2113 + GIB [J2211PB J221 2 qB +J2213*B]

+[J23iipc + J 23124c + J2313 lCI

= 1J3111(iB/G1e+ J3112iBIGf8 + J3l13w4/GII

-[J321l.B + J3212 4B + J3213i ]-[J33I Pc + J3312 qC + J3313 'c]

+fBx + fc, + kBz + kc,

Place all the derivatives on the left side and write out the full matrix equation. The

result is shown in Equation 22 at the end of this section. To eliminate the extra state Pc,

insert Equation 18 into Equation 22 and rearrange. The result is shown in Equation 23 at

the end of this section.

Now, for the specific vehicle being modeled, this equation can be simplified. First,

the deflection angles are fairly small, and second, the deflection rates are large in only one

direction at a time, i.e., S8 and S, will not be large at the same time. Referring to

Equation 18,

PC = PB -(4c - 4B)tans- (qC -qB)(rc -r 8)

Cos2

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Recall that 6 -(qc - q,) / cosob and Z(rc - rB).

PC-PB -(We - 4B) sin ~ 686yzcos6z

Apply small angle assumption:

PC-PB -(WC - 4B)6- iy

Remember that 5y and 6z will not be large at the same time:

PC "PB (21)

Applying this yields Equation 24.

Equation 22

0 0 0

0 0 0

0 0 0ill J12 J13 ---

----- ------------------------------------ J311 J3112 J3113 J'J21 J22 J23 +

J3121 J312 J312 J3221 J3222 J32 -2 J332 .337 0 0 0

J3131 J313 2 J3133 J3231 J3232 J3233 J3331 J3332 J3333 0 0 0------------------ 4----

0 0 0

0 0 0:

21

(21)

0 0 0 0 0 0 0 0 0

0 0 0o0 0 0 0 0 0

0 0 0 0 0 00 0 0

J3111 J3112 J3113 : J3211 J3212 J3213 J331 , J3312 J3133 B Iv fcx+k Lv

0 0 0:0 0 0:0 0 0 Afcy-k

0 00 o 0 o 0 0

o~~~~~ o z

0 0 o o 0 0 0 0 0

NAWCWPNS TP 8134

Equation 23

J1312 J1313 0 0 0 J1311

Jll j12 J1322 J132 0 0 0 J1321

J1331 J1333 0 0 0 J1331-------------------------. 13 ----------- 0----0---0-

J3111

,I21 j22 J23z J232 + +3131 + J3311:o -00 J2321

J233 2 J233313121 J312 J31,• J322 J3222 J32•3 J3332 J33. 0 0 0 J2332

J313, J3132 J3133 J3231 J323 2 J3233 -J3333 0 0 0 J3321

0 0 0 J3331

J1311

0 J1321

Iv fc, + kc v 11331

MBAB Mh k, (qc-qB Xrc rB) J231 +J3311

fc y MhB + : y + Cos 26 z J232

fc~ z M y kc~ J2331

J3331

•,7 '23

J1311 J131, tan6z 0 -J1311 tan 6. 0J1321 11321 tan 6Z 0 -J1321 tan 61 0 #/BIGI

13 J133 tan6 -J13.. . 1133 - - - ---11 - - - - -- - - - --- 3--- -- ta - -- -- 0 -Pr J3211 1 J3212 J33

J2321 + 1J2321 tan5 6z0 -(J23 21 + J33)6 0 PB

J2331 J2331 tan 6z 0 -J2331 tan 6z 0 qc

J332, J332, tan 6z 0 -J3321 tan 6z 0i

J3331 J3331 tan 6z 0 -J3331 tan 6z 0

NAWCWPNS TP 8134

Equation 24

J13 2 J1313 0 0 0 J1311

i1 .12 J132 J1323 0 "0 0 J1321

I I22J2l~J13 0 0 0 1

J3111 J3112 J3113 J3211 +J2311 +J331,121 j22 1 J232 J2323 +

0 0 0 J2321

332J33 0 0 0 J2331

J317a J3122 J313 ]J3221 J3222 J322:--------------------------------3 ---

J3131 J3132 J313 3 :J3231 J3232 J3233 J3332 J3333 0 0 0 J33 21

0 0 0 J3331

25

( ,,, , jI

J131 , 0 0 0 0

J1321 0 0 0 0 tB/ GIBJ133 1 0 0 0 0 PB 0J21 3ý J321 2 J3213 :J3312 JA13 AB f~ z kv

210 0:0 0 A +B

J2332 1 0 0 0 0 fcz -- ii I zJ-- - - ----- ------c-- --- fI IkIgn B

J3331 0 0 0 0

NAWCWPNS TP 8134

SPECIAL CASE: SINGLE RIGID BODY

When the second body is removed, Equations 8 decouple and can be treated as three

separate vector equations, rather than as one 9-element equation. Further, the differential

equations for •c, 6, Sy, and 8 , are typically replaced by three second-order equations

to adequate fidelity. This is the case for most missiles, both thrust vector controlled and

aerodynamically controlled, because the mass of the deflecting nozzle or control surface

is negligible relative to the total mass of the rest of the vehicle, and the distance from the

hinge to the CG is small. The effect of the second body on the dynamics is restricted to

its contribution to the forces and moments on the body. From Equation 8, the equations

for the body velocity and angular rate are

vIY B + C+ B m( -BG + WZ)+XB+

C B _ &B I-44= +- 0B )

The remaining Equations 9 though 12 are unchanged.

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SPECIAL CASE: FLAT EARTH

For flat Earth, the radius of the Earth becomes infinite and its angular rate zero. The

gravitational field is usually assumed to be uniform, and the local horizontal frame

becomes parallel to the ground frame, which is now inertial. Assuming the combustor

can roll with respect to the main body, most of these changes can be applied to the initial

set of equations by zeroing out the appropriate terms.

mV7 B - 8 ]VB-IGIB [ TB B

M~fiCB _JCIýBB (CfB n4 I B I~A?%+ABc(TC'+MecUB)}

MC- ("--B --B -+ -B-B) --B-B0~~~' U ~ cJJ oBJIB~v.BVBIG

-. OB a yB-B "B ( -B -CB -- B - B -B -B - B -B1 UBiBB {B&B&BB +DO~j~+BIG;VBIG)1 (25)B-B B B66J +OBIGVBIG)J

The exception is the set of equations describing the missile CG position. In this case it

is necessary to shift from motion on the surface of a sphere to motion over a flat surface.

The missile position rate in the inertial frame is simply the velocity in the body frame,

transformed, or

{ [ W. { (26)

where h=-wB

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NAWCWPNS TP 8134

The surface distance traveled is simply

R=-X+ Y (27)

Equations 9 and 14 are unaffected.

In this section, the combustor can roll with respect to the body. For the LDR,

however, this roll freedom is gone, so equations 9, 15, 16, 19, 24 (with UG =0) and 26,

along with assumption 21, were implemented in the simulation.

SPECIAL CASE: FLAT EARTH AND PITCH PLANE

The assumption that motion is restricted to the pitch plane results in an enormous

simplification: the three vector equations in Equation 8 are reduced to three scalar

equations, which can be solved separately. The notation for the scalar forces and

moments is illustrated in Figure 3.

An additional assumption here is that both sections are symmetric homogeneous

cylinders, so the inertia tensors of the body and combustor are diagonal with the y and z

elements equal, i.e.,1&i 0 00 [,z 0 0]= 0 1' 0 ,and iC= 0 C

S01B2 0 Ic

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NB

COMBUSTOR mB

~M mv =MB + rnc

MCg

T

FIGURE 3. Planar Vehicle Over Flat Earth.

Since this is the planar case, neither the body nor the combustor roll.

mvu + mc t xc sin8 = AB + (Ac + T)cos8- Nc sin8

-mvg 0 sinG - ntt - XB) + t 2xC cosS]- _mvOw

mvlw+mc1xB) +m MXC x costS = -NB - Nc cos - (AC + T) sinl8

+mvg 0 cos8 + mc 2 xc sin S + mvOu

IBd+n;C(I-XB)(Wý+ d(l- XB) +fXc COS8]-= MB + Mh

+(I- XB-Nc cosS - (Ac + T)sin 8 + mTgo cosO + mc(C2Xc sin -6ul

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Ic + mcxc[wcos8+ usin6+ 6(1- xB)cos• + Cxc]= Mc -Mh

+Xc{-Nc + nmcg0 cosý Pmc[62(1 - xB)srn6 + 6(ucos3 - wsin8)]}

Equation 9 becomes 6= qB. The differential equations for position reduce to

XB = uBcosO+wBsinO

!B = -u 8 sin 0 + wB cos0

CONCLUSIONS

Equations of motion for a missile consisting of two rigid bodies connected by a hinge

have been developed for flight over a spherical rotating Earth. These equations have been

specialized to three simpler cases: flight over a spherical rotating Earth where the second

body is much smaller than the first, flight over a stationary flat Earth, and pitch plane

flight over a flat Earth.

For modeling the dynamics of more complex vehicles, i.e., with more sections or

with more complex connections, the development by Abzug in Reference 3 would be

useful. The notation, while intricate, is well laid out for handling the complexity of that

type of vehicle. For the LDR vehicle, we used both approaches for developing the first

set of equations in this report, and concluded that the more intuitive approach presented

here was workable for two bodies and would be clearer to present in this report. As the

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NAWCWPNS TP 8134

number of bodies increases and the connections become more complex, the more intuitive

approach taken here becomes impractical.

NOMENCLATURE

GENERAL NOTATION

Because this is a two-body vehicle, reference is made to the two portions, the front

body and the aft combustor. These two parts make up the vehicle.

]matrix

() - (tilde overbar) - matrix

-) - (overbar) - vector

Transformations and Coordinate systems

Several coordinate systems are used in this development. First, of course, is the

inertial frame, which is assumed to be fixed in inertial space. Next, the ground frame is

attached to a fixed point on the surface of the Earth. For the flat Earth, the ground and

inertial frames are identical. The body frame travels and rotates with the main body. The

combustor frame travels and rotates with the combustor. The local horizontal frame

travels with the body, but remains aligned with the local horizon. For flat Earth, the local

horizontal and ground frames remain aligned.

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NAWCWPNS TP 8134

The coordinate frames are related to each other by a sequence of Euler rotations. In

this discussion, all Euler rotations are in the sequence x,y,z. To illustrate with the

rotation from local horizontal to body frames:

&B = [TH2B Ii

where

[TH2B]=[111[1

1 0 0 o cosG 0 -sino oosV siny, 0]

[TH21=[0 cosO sino 0 1 0 -siny, cosy,0 -sin 0 cos sin0 0 cos, 0 0 0 1

[ cosOcosly cosOsin V -sin 1[TH2B]= sinosin0cosyf-cososinyr sinosin~siny +cosocosyt sinocosG

Lcos~sin~cosV,+sin~sinW cososin~sinyf-sinocosyf cosocosO]

Other transformations

[TB2c] = [456x[8}8, - body to combustor

[TG2H]= [-Iy[l-x "ground to local horizontal

[TH2B] = [¢x[0]y[VI] - local horizontal to body

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Derivatives

A final relevant subject is that of noninertial derivatives. Since there are several

noninertial frames in this development, the notation becornts a little more intricate than

usual.

- - inertial time derivative of vector i

dt

NBIG - angular velocity of body frame "B" with respect to the ground frame "G"

a-G - time derivative of vector f with respect to (usually) noninertial frame "G"

Note that a-BG11 = da = IGIG + Gx a& . Since the angular velocity

of "G" is with respect to inertial space, the "I" in the subscript usually is

dropped, i.e., WGII = &G.

UB - vector i expressed in components along the body frame

Note that a vector may be expressed in any coordinate system, regardless of its physical

meaning. For example, the angular velocity of the body with respect to the ground

(&BIG) may be expressed in components in the body frame (-BIG), in components

along the ground frame (1W/G), or any other frame, such as the combustor frame

(-'BIG). References 1 and 6 both contain good discussions of transformations.

SYMBOLS

Superscripts and subscripts

B -body

C - combustor (motor)

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G - ground

h - hinge

H - local horizontal

I - inertial

V - vehicle (body + combustor)

Scalar variables

RE - radius of Earth

R - distance travelled by main body

h - altitude

X - position from a reference point along constant meridian of the spherical Earth

Y - position from a reference along constant parallel of the spherical Earth

1 - latitude position

- longitude position

m - mass

go - magnitude of gravitational acceleration at sea level

u,v,w - components of velocity in body frame

p, q, r - components of angular velocity in body frame

x, y, z - position components

WE - magnitude of Earth's angular velocity

Vector Variables

S- a general vector

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NAWCWPNS TP 8134

b" - a general vector

- external forces except due to gravity and due to the other body via the hinge

G" - gravitational acceleration

M - external moments except due to the other body via the hinge

S- position

S- velocity

S- acceleration

S- angular velocity

- location of component CG within that component

I- without subscript, identity matrix

I - with subscript, inertia tensor

Abbreviations

CG - center of gravity

LDR - low drag ramjet

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REFERENCES

1. D. T. Greenwood. Principles of Dynamics. Englewood Cliffs, N.J., Prentice-Hall,Inc., Sections 8-1, 1965.

2. C. I. Palmer and C. W. Leigh. Plane and Spherical Trigonometry, 3rd ed. NewYork, McGraw-Hill Book Co., Ch 7, 1925.

3. B. Etkin. Dynamics of Atmospheric Flight. New York, John Wiley and Sons, Inc.,Ch. 5, Sections 1-9, 1972.

4. M. Abzug. "Active Satellite Attitude Control," in Guidance and Control of AerospaceVehicles, ed. by C. T. Leondes. New York, McGraw-Hill Book Co., 1963.

5. E. Cliff. Class Notes for Dynamics of Aerospace and Ocean Vehicles (AOE 5210).Blacksburg, Va., VPI&SU, 1982.

6. F. Lutze. Class notes for Vehicle Performance II (AOE 4270). Blacksburg, Va.,VPI&SU, 1981.

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