SAN D76-0266

Unlimited Release

Equations of Motion for Free-Flight Systems of Rotating-Translating Bodies

Albert E. Hodapp, Jr.

Prepared by Sandla Laboraqor~es, Albuquerque New .Mexico 87415 and L~vermore, California 94550 for the United Staw Energy R Q e a r d and Development Adminrstrat~on under Contract AT- t29-I )-786 .

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SAND76-0266 Unlimited Release

Printed September 1976

EQUATIONS OF MOTION FOR FREE-FLIGHT SYSTEMS O F ROTATING-TRANSLATING BODIES

Albert E. Hodapp, Jr. Aeroballistics Division 1331

Sandia Laboratories Albuquerque, New Mexico 87115

ABSTRACT

General vector differential equations of motion a re developed for a system of rotating-translating, unbalanced, constant mass bodies. Complete flexibility i s provided in placement of the refer- ence coordinates within the system of bodies and in placement of body fixed axes within each body. Example cases are presented to demonstrate the ease in reduction of these equations to the equa- tions of motion for systems of interest.

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3

Summary

A set of general vector differential equations is developed, relative to arbi t rwily placed ref-

' erence axes, for a system of unbalanced, constant mass bodies. The bodies can rotate and trans-

late relative to each other as well as relative to the reference coordinates.

flexibility in placement of both the body fixed coordinates and the reference coordinates, these gen-

eral equations a re more useful than those developed previously for multiple body free-flight systems.

Utilizing restraints and exploiting the flexibility in placement of coordinate systems, these general

equations are easily reduced to the equations of motion for many dynamic systems of interest.

example cases a re presented to demonstrate the ease of application and general utility of these equa-

tions.

Because of the complete

Two

4

CONTENTS

Summary

Nomenclature

Introduction

Theoretical Analysis

Particle Dynamics

jth - Body Dynamics

System Dynamics

Example Cases

Spinning Asymmetrical Body

Internal Vibrating Member

Conclusion

References

FIGURES

Figure

1 Coordinate Systems

2 Spinning Asymmetrical Body

3 Internal Vibrating Member

Page 4

7

9

9

10

11

13

13

14

16

19

23

Page

20

21

22

5-6

Nomenclature

cg Center of gravity

Derivative of ( ) with respect to time

F Total force (external + gravitational) acting on syetem of bodies [ cjFj], lb o r N

F . J

Total force (external + gravitational + interaction) acting on jth body

[ F. 1 1 b o r N P JP t

F Total force (external + gravitational + interaction) acting on particle "p"

of the jth - body [Eq. (111, lb o r N

Components of total external force a)ong the X, P, Z axes, respectively.

lb o r N

Components of gravitational acceleration along the X, Y, Z axes. respec-

tively, f t lsec o r m l s e c

Moment of momentum of jgbody about the origin of X Y Z [Eq. ( lo) ] ,

ft-lb-sec o r N-m-sec

Moments of inertia of jt& body about the X

j p

FXv F y P FZ

gx' gy' gz 2 2

j j j H. J

Y Z. axes, respectively, IX.' 5 . 9 IZ 2 2 j' j' J

J J J slug-ft o r kg-m 2

Products of inertia Qf the jgbody relative to the X.Y.Z. axes, Slug-ft o r J J J JXjYj' JYjZj' JXjZj kg-m 2

m

m

m

Mo,Mj . M

Total system mass[Eq. (17 ) l . slug o r kg

Mass of jth - body 1.9. (6)], slug o r kg

Mass of particle "p" in j g body, slug or kg

Moments of the forces F, F

axes), ft-lb o r N-m

Components of the moment of external farces about "0" (origin of XYZ

axes) relative to the X, Y, 2 axes, respectively, ft-Ib or N-m

Radius vector from "0" ( a r i g b d X Y Z axes) to the cg sf the system of

bodies, Figure 1, ft or rn

Radius vector from "0" lorigin of m Z axed to Ohe origin of the X.Y .z axes, Figure 1, f t or m

j

jp F , respectively, about "o" (origin of XYZ

j' jp jp

MXs My* Mz

r cg

J J j r . J

7

R jp

RO

t

V

XYZ

w. J

sz

Nomenclature (cont)

Radius vector from origin of X.Y.Z. axes to particle "p" in the jth body,

Figure 1, ft o r m 1 1 1 -

Radius vector from origin of X Y . Z . axes to "0" (origin of XYZ axes),

Figure 1, f t o r m

Time, sec

Velocity of "0" (origin of XYZ axe& relative to inertial space, f t /sec or m l sec

Reference coordinates for system of bodies (arbifrary location), Figure 1

Inertially fixed coordinates, Figure 1

Coordinates fixed in j s body (arbitrary location in body), Fig;ure 1

Radius vector from origin of X.Y .Z . axes to the cg of the j4_h body, 3 3 3

Figure 1, f t o r m

Radius vector from origin of X . Y . Z . axes to the particle "p" in the jth

body, Figure 1

i l l

- 5 3 3

Angular velocity of the j s body, radlsec

Angular velocity of the reference XYZ axes, radlsec

Denotes vectors resolved relative to the X Y Z , XYZ, and X Y 2 axes,

respectively

First and second derivatives with respect to time

Summations for all particles "p" of the jth body, and for all j bodies of

the system, respectively.

i i i j j j

A I

8

EQUATIONS OF MOTION FOR FREE-FLIGHT SYSTEMS OF ROTATING-TRANSLATING BODIES

Introduction

Often when dealing with the flight dynamics and control problems associated with aircraft ,

missiles, reentry vehicles, spacecraft, bombs, and shells the need a r i ses for describing the mo-

tion of systems of bodies rather than the motion of single bodies f ~ r which the equations a r e well

defined.'" 2, Examples of dynamic multiple body systems a r e aircraft, missiles, etc., with mov-

ing control surfaces and/or with moving internal components. Vector differential equations a r e

developed herein which describe the m o t i o ~ of a system of rotating-translating, constant mass

bodies. The angular velocities of the individual bodies and the angular velocity of the reference

coordinate system a re all assumed to differ. These general equations of motion differ f rom pre-

vious developments, notably Reference 3, in that the axis system in each body is not fixed at the

cg, and the reference coordinate frame for the system i s not fixed in any body. This flexibility in

placement of the reference coordinate system and each of the body fixed coordinate systems can a

be of great advantage in simplifying the equations for specific applications. For example, the re -

quirement for nonrolling reference axes is easily implemented, a s is the simulation of unbalanced

bodies.

By introducing restraints (e. g. , limiting the number of bodies, limiting their degrees of

freedom, defining the dynamic coupling of the bodies, etc. ), the general equations presented here-

in can be reduced in form to describe the equations of motion for many multiple body o r single body

dynamic systems of interest. To demonstrate the flexibility and usefulness of these equations, two

simplified example cases a r e considered. For the first, equations of motion for a spinning body

with mass asymmetries a r e developed relative to nonrolling coordinates. The second example in-

volves the development of equations of motion for a spinning body with an internal flexible-vibrating

member. The equations resulting from these example cases a r e ugeful for describing the free-

flight motions of both aerodynamically stabilized and gyroscopically stabilized spinning vehicles.

Theoretical Analysis

In Figure 1 the positions of the j rigid bodies (j = 1, 2, . . . , n), whiqh make up the system of

interest, a r e described relative to both inertial space (XiYiZi) and the reference coordinate system

(XYZ). The center of mass or center of gravity (cg) for the system of bodies is not required to be

coincident with the origin "of' of the reference coordinates. These bodies translate and rotate rela-

tive to one another a s well a s relative to the rotating reference frame (XYZ). The body fixed

coordinate system X.Y.Z. can be located arbitrarily in the j g body. As shown in Figure 1, the j J J J

bodies can have cg offsets.

For the following development, the symbol appearing a b ~ v e a vector quantity denotes the co-

ordinate frame that the vector is resolved relative to and also the coordinate frame that derivatives -, of the vector a re taken with respect to. Examples a r e r . and F r . and ?., or 9. and 4.. These

J j' J J J J , respective groups indicate vectors resolved relative to, and derivatives taken with respect to, the

X.Y.Z. inertial coordinates, the XYZ reference coordinates, and the X.Y.Z. body fizred coordinates. 1 1 1 J J J

Particle Dynamics

The force acting on the constant mass particle "p" (part of the jthbody, Figure 1) and the - moment of that force about the origin "o" of the reference XYZ coordinate frame a re given a s

The radius vector from the origin of the inertial reference system (X.Y .Z.) to the particle "p" of 1 1 1

the jth body (Figure 1) can be written a s -

Differentiating Eq. (3) and substituting it into Eq. (I),

Since the j4_h body is rigid (no relative motion of particles) the magnitude of is a constant; jp

however, the derivatives of 5 are nonzero because its d i~ec t ion is variable. jp

Substituting Eq. (4) into Eq. (2),

Recalling that

the previous equation can be written a s

jth Body Dynamics

Summing for all of the particles "p" of the j4_h body, the mass of the body ie obtained a s

Since the j 2 body i s rigid and its center of gravity Icg) is displaced from the origin of the X.Y Z J J ~

system, summing mass moments about the origin yields

F mjp'jp = mjPlcg * (7)

The force acting on the 49 body and the moment of &at foa*ce a b o ~ t "0" are obtained by

summing Eqs. (4) and f5) for a11 particles "p" vf the j s bady.

Substituting Eq. (6) along with Eq. (7) arrd its derivatives W o the above ,eqw.&bns, the force and

moment reduce to

where the moment of momentum of the jt& body aboqt the orisin of the X.Y.Z. coordinate system - H. is defined as 3 3 3

3

- 22

H. = p. X; m 3 p ( JP JP jp) = P j p X p j p d m

jz body

r

I-.

m

P

s (D 1

(D

El

u. a-

U.

0

11 +

U. Tl

I1

CI

Up

I: 0

Os

+ +

1 I:

1 1:

U. I1

U. +

U. E

;?b Os +

+ U

. i$

- 0

09 +

9 U.

0 I1

Cl.

+

x * N

U.

U.

U.

+

3

U. 4 0 Os

+ B

8 U

?k

U.

+

6 g +

E

U. % 31

- + 9

UP1

0

m -

6 2 2 N

P, 3 a

r-

9

0

m -

1

(D

m

(D

3

0

(D

0

0 0 1 2 E 5 W

M

9

W

h

m

e,

v k

a

I.

h

Y

W

System Dynamics

The total mass of the system and the cg location for the system relative to the origin "0" of

the reference coordinates (Figure 1) a r e obtained by summing the masses of the j bodies and their

mass moments about "0" a s follows,

E m j = m J

mj(T + F ~ ~ ~ ) = mit cg .

The total force acting on the system and the moment of that force about "0" are obtained by

summing Eqs. (15) and (16) for all j bodies and using Eq. (17) together with Eq. (18) and its deri-

vative s.

L

(17)

(18)

Motions of the j individual bodies are described by Eqs. (15 and (16), while Eqs. (19) and (20)

describe the motion of the system of bodies. The forces and moments acting on the j individual

bodies include those resulting from interaction of bodies, external forces, and body forces.

and moments resulting from body interactions occur in equal but opposite pairs; therefore, in the

summation to obtain Eqs. (19) and (20) these pairs cancel, leaving only the summations of the exter-

nal forces and moments and body forces and moments which a re represented by 3 and Go.

Forcers

Example Cases

In order to demonstrate the usefulness af these general equations of motion, two examples wil l

be given. For the f i r s t example, equations of motion wil l be developed relative to nonrol€ing coordi-

nates (aeroballistic axes) for a single spinning body having mass asymmetries (principal axis misalign-

ment and center of gravity offset). The equations of motion for a spinning body containing an internal

vibrating member (multiple body problem) will be developed a s the second example. Ae mentioned

13

earlier, the equations developed for these example cases are useful for describing the free-flight

motions of both aerodynamically stabilized and gyroscopically stabilized spinning vehicles.

Spinning Asymmetrical Body

The single rigid body considered here (Figure 2) is not required to have an axis of mass o r

For this example, the X Y Z body fixed axis system (Figure 2) has its

= 0). The XYZ nonrolling reference coordinate system

' aerodynamic symmetry.

origin fixed at the cg of the body (6 (Figure 2) is placed in the body such that its X axis is directed out the nose and its origin is offset

laterally from the cg 9 = (0. y z ). The X and X axes remain parallel while the positions of

Y relative to Y and Z1 relative to Z are described by the roll angle 6. Therefore, the reference

system pitches and yaws with the vehicle but does not roll with it. For a single body with the ar-

rangement of axis systems shown in Figure 2, Eqs. (19) and (20) reduce to

1 1 1

1cg

1 cg' cg 1

1

+ T1 + &F~ + 2 5 x 5 + i $ ~

Of the available options, the arrangement of coordinate systems shown in Figure 2 was

chosen because the resulting vector equations require fewer algebraic manipulations to reduce

them to scalar differential equation form. This approach does require that the moments and

products of inertia be obtained relative to the offset X Y Z axis system; however, because of

the simplicity of the transfer equations, this is not a disadvantage. 1 1 1

4 With the exception of H1, the vector quantities contained in Eqs. (21) and (22) a re defined

in terms of their components relative to the X Y Z nonrolling reference system as

where p, q, r a re the components of angular velocity relative to the body fixed X Y Z axis sys-

tern, 1 1 1

14

h

0.l

M

Y

h

0

M

Y

h

d

M

v

8

m 0

u

m c 0 ...I Y

2 % & 0

U

Q,

3

P)

2)

5 A

w

cu P

c rd Y

h

U

cu Y

i

@

w 0 c Y

.rl

h

d

M

v

I3

w a 9 fi z 9 n

E- w

Y

3

A

W

N

Y

i

@

w M

E

.rl u

h

n

N

0

M

m

Y

Y

n

* m

Y

E E

I1 It

+ +

+ *

N

h x

R

R

A

(D

M

N x

Y

9 8 8

.El &

H N

$ m

+ -- F a

C m .1

@d

&

a I

8

0

0

+ 8

m 0

8

m 3 m

un

N

"

*& N

T

&

e

H

(6 + qr) sin 6 - Jxlyl (p" - q2) cos Q, - JxlZk - qr) COS @ - Jxlyl

(p2 - r2) sin - + JXIZl

(4 + pr) cos - JYlzl(i. - pq) sin o - F X Y Jylzl

Possibly the most classic of the many applications of equations of this type (relative to non-

rolling coordinates) is in the description of artillery shell motion. The presence of products of

inertia, cg offsets, and unequal lateral moments of inertia in Eqs. (32) through (37) make them

different from the equations of motion relative to nonrolling coordinates often found in the litera-

ture.

flight vehicles.

These additional te rms can have important effects on the flight dynamic behavior of all

(37)

Internal Vibrating Member

The two body system considered herein consists of an outer rigid body (Body 1, Figure 3)

that has a small rigid body (Body 2, Figure 3) suspended within it by massless springs. Motion of

Body 2 is constrained to translation in a plane parallel to the Y Z -plane; i, e., i t can only move ' 1 1 laterally. The motion of Body 1 is unrestrained. The X Y Z body fixed axes (Body 1, Figure 3)

and the XYZ reference system a re coincident, with their origins fixed at the cg of Body 1. These

systems and the X Y Z system remain mutually parallel; therefore, both bodies have identical

angular velocity.

1 1 1

2 2 2 The X2Y2Z2 system has i ts origin placed at the cg of Body 2 (Figure 3). A s a

result of the constraints imposed and the placement of coordinate systems, a = + + Icg p2cg = I-1 = O*

The vector equations which describe the motion of this system, Eqs. (15), (18), (19), and

(20), then reduce to

m i ? = m F 2 2 cg

F - F ~ = ~ v+i ix i j ' l+ J + + 1

(39)

where from Eq. (17), m = m + m 1 2'

16

Like the previous example, the mass properties that define the moment of momentum of the

bodies wil l be given relative to the X.Y .Z. (j = 1, 2) body fixed coordinate systems. Therefore,

H. (j = 1, 2) can be obtained.

mutually parallel, H. = H - i. e., for this case, no angular transformation is required to obtain the

A J J J Because the X.Y.Z. systems and the X Y Z reference system a re

3 -b A ' J J J

J j' moment of momentum relative to the reference coordinates. Using the development of Eq. (30) a s

a model, the moments of momentum for the bodies relative to the reference coordinate system

' XYZ a re given a s

The remaining vector quantities contained in Eqs. (38) through (41) a r e defined relative to

the XYZ reference system a s

+ mgX, F + mgy, F Y

7 = (., v, w)

8 = (p, qy r)

where x2 , y2 , z

spring stiffness.

(Figure 3) describe the rest (unaecelerated) position of Body 2, and K is the 0 0 2o

By combining and expanding Eqs. (38) through (48), the scalar differential equations of

motion for the two-body system (Figure 3) can be written a s

.. Y 2 - (Y20 + Y2) y + r 2 ) + (1 + 2): Y2 - 2 P i 2

+ (z20 + z 2 ) (clr - 6) + x2h + E) + 5 = 0 1

(43)

(44)

(49)

17

I N

n

01

0 + N

Y*

1 +

&

N 3

ON

N

N + -t

N

W

N

xr

I $ tr T

1 + +

n

'D I

1 @

a

+

'2 '

N + 31 w

G-

N

0

+ N

W

N

T

N

0

+ Y - n P g

1

L\3

h

01

cn

Y

N

I

h

cn

w

Y

+ T

N

0

+ Y

W -

N e I

1. -

I

X

ON

T

+ 1

W

+ N N

e. N

I

N

1 9

R)

+ 9

9

1 +

1 I

1

4

-4

+ + N

W

N

I Nr T

I1

,

Mz = 0

m m m m - Ix - + (y20 + y2y]pq + + x2 0 (Y20 + Y2) (2 + r")

m

Equations (49) through (56) are useful for problems in which motion induced resonances of

internal vibrating members a re of concern, They are also useful for studying the effects on sys-

tem motion which result from the motion of the internal components.

Conclusion

A set of general vector differential equations has been developed that describes the motion

of a system of rotating-translating bodies relative to a rotating reference coordinate system.

These equations allow for: (1) complete flexibility in placement of the reference frame: (2) rela-

tive rotation and translation between the reference frame and each body; (3) relative rotation and

translation between the respective bodies of the system; and ( 4 ) the existence of mass asymmetries

in each of the bodies.

reduced to the governing equations for many dynamic systems of interest. Two examples are given

to illustrate this flexibility.

Through the use of simple restraints these general equations are easily

19

Po 0

Figure 2. Spinning Aeymmetrfcal Body (Nonrolling Coordinates)

2 1

i

Origin "0" and cg

z, z ,

Figure 3. Internal Vibrating Member (Body Fixed Coordinates)

I 22

References

i

1. W. T. Thomson, Introduction to Space Dynamics, New York, London: John Wiley and Sons Incorporated, 1961.

2. B. Etkin, Dynamics of Flight Stability and Control, New Yorh, London: John Wiley and Sons Incorporated, 1959.

3. D. T. Greenwood, Principles of Dynamics, Prentice-Hd 1 Incorporated, Engtewood Cliffs, New Jersey, 1965.

4. A. E. Hodapp, Jr., Equations of Motion for Constant Mass Entry Vehicles with Time Varying Center of Mass Position, SC-RR-70-691, Sandia Laboratories, Albuquerque, New Mexico, November 1970.

23

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