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

Transcription

1 SA D Unliited Release Equations of Motion for Free-Flight Systes of Rotating-Translating Bodies Albert E. Hodapp, Jr. Prepared by Sandla Laboraqor~es, Albuquerque ew.mexico 8745 and L~verore, California 9455 for the United Staw Energy RQeard and Developent Adinrstrat~on under Contract AT- t29-i )-786. Printed Septeber 9'6 When printing a copy of any digitized SAD Report, you are required to update the arkings to current standards.

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3 SAD Unliited Release Printed Septeber 976 EQUATIOS OF MOTIO FOR FREE-FLIGHT SSTEMS OF ROTATIG-TRASLATIG BODIES Albert E. Hodapp, Jr. Aeroballistics Division 33 Sandia Laboratories Albuquerque, ew Mexico 875 ABSTRACT General vector differential equations of otion are developed for a syste of rotating-translating, unbalanced, constant ass bodies. Coplete flexibility is provided in placeent of the reference coordinates within the syste of bodies and in placeent of body fixed axes within each body. Exaple cases are presented to deonstrate the ease in reduction of these equations to the equations of otion for systes of interest. Printed in the United State. of Aerics Available fro ational Technical Idoration Service U. S. Dsprrtent CdCoerce 5285 Port Royal Road Springflcld. VireM. 285 Price: Prhted Copy $4.5: Microfiche $2.25 3

4 Suary A set of general vector differential equations is developed, relative to arbitrwily placed ref- ' erence axes, for a syste of unbalanced, constant ass bodies. The bodies can rotate and translate relative to each other as well as relative to the reference coordinates. Because of the coplete flexibility in placeent of both the body fixed coordinates and the reference coordinates, these general equations are ore useful than those developed previously for ultiple body free-flight systes. Utilizing restraints and exploiting the flexibility in placeent of coordinate systes, these general equations are easily reduced to the equations of otion for any dynaic systes of interest. Two exaple cases are presented to deonstrate the ease of application and general utility of these equations. 4

5 COTETS Suary oenclature Introduction Theoretical Analysis Particle Dynaics jth - Body Dynaics Syste Dynaics Exaple Cases Spinning Asyetrical Body Internal Vibrating Meber Conclusion References Page FIGURES Figure Coordinate Systes 2 Spinning Asyetrical Body 3 Internal Vibrating Meber Page

6 oenclature cg Center of gravity Derivative of ( ) with respect to tie F Total force (external gravitational) acting on syete of bodies [ cjfj], lb or F. J F jp FXv F y P FZ Total force (external gravitational interaction) acting on jth body [ F. bor P JP t Total force (external gravitational interaction) acting on particle "p" of the jth - body [Eq. (, lb or Coponents of total external force a)ong the X, P, Z axes, respectively. lb or gx' gy' gz H. J Coponents of gravitational acceleration along the X,, Z axes. respec- 2 2 tively, ftlsec or lsec Moent of oentu of jgbody about the origin of X Z [Eq. (lo)], j j j ft-lb-sec or --sec Moents of inertia of jt& body about the X Z. axes, respectively, IX.' 5. 9 IZ 2 2 j' j' J J J J slug-ft or kg- 2 Products of inertia Qf the jgbody relative to the X..Z. axes, Slug-ft or JXjj' JjZj' JXjZj kg- 2 J J J j jp Mo,Mj.M jp MXs My* Mz r cg r. J Total syste ass[eq. (7)l. slug or kg Mass of jth - body.9. (6)], slug or kg Mass of particle "p" in jg body, slug or kg Moents of the forces F, F j' axes), ft-lb or - Coponents of the oent of external farces about "" (origin of XZ axes) relative to the X,, 2 axes, respectively, ft-ib or - Radius vector fro "" (arigb d XZ axes) to the cg sf the syste of bodies, Figure, ft or rn F, respectively, about "o" (origin of XZ jp Radius vector fro "" lorigin of Z axed to Ohe origin of the X..z J J j axes, Figure, ft or 7

7 oenclature (cont) R jp RO t V XZ Radius vector fro origin of X..Z. axes to particle "p" in the jth body, Figure, ft or - Radius vector fro origin of X.Z. axes to "" (origin of XZ axes), Figure, ft or Tie, sec i l l Velocity of "" (origin of XZ axe& relative to inertial space, ft/sec or l sec Reference coordinates for syste of bodies (arbifrary location), Figure Inertially fixed coordinates, Figure Coordinates fixed in j s body (arbitrary location in body), Fig;ure Radius vector fro origin of X..Z. axes to the cg of the j4_h body, Figure, ft or Radius vector fro origin of X..Z. axes to the particle "p" in the jth body, Figure - w. J sz Angular velocity of the j s body, radlsec Angular velocity of the reference XZ axes, radlsec Denotes vectors resolved relative to the X Z, XZ, and X 2 axes, iii j j j respectively First and second derivatives with respect to tie Suations for all particles "p" of the jth body, and for all j bodies of the syste, respectively. 8

8 EQUATIOS OF MOTIO FOR FREE-FLIGHT SSTEMS OF ROTATIG-TRASLATIG BODIES Introduction Often when dealing with the flight dynaics and control probles associated with aircraft, issiles, reentry vehicles, spacecraft, bobs, and shells the need arises for describing the o- tion of systes of bodies rather than the otion of single bodies f ~ which r the equations are well defined.'" 2, Exaples of dynaic ultiple body systes are aircraft, issiles, etc., with ov- ing control surfaces and/or with oving internal coponents. Vector differential equations are developed herein which describe the otio~ of a syste of rotating-translating, constant ass bodies. The angular velocities of the individual bodies and the angular velocity of the reference coordinate syste are all assued to differ. These general equations of otion differ fro pre- vious developents, notably Reference 3, in that the axis syste in each body is not fixed at the cg, and the reference coordinate frae for the syste is not fixed in any body. This flexibility in placeent of the reference coordinate syste and each of the body fixed coordinate systes can a be of great advantage in siplifying the equations for specific applications. For exaple, the re- quireent for nonrolling reference axes is easily ipleented, as is the siulation of unbalanced bodies. By introducing restraints (e. g., liiting the nuber of bodies, liiting their degrees of freedo, defining the dynaic coupling of the bodies, etc. ), the general equations presented herein can be reduced in for to describe the equations of otion for any ultiple body or single body dynaic systes of interest. To deonstrate the flexibility and usefulness of these equations, two siplified exaple cases are considered. For the first, equations of otion for a spinning body with ass asyetries are developed relative to nonrolling coordinates. The second exaple involves the developent of equations of otion for a spinning body with an internal flexible-vibrating eber. The equations resulting fro these exaple cases are ugeful for describing the freeflight otions of both aerodynaically stabilized and gyroscopically stabilized spinning vehicles. Theoretical Analysis In Figure the positions of the j rigid bodies (j =, 2,..., n), whiqh ake up the syste of interest, are described relative to both inertial space (XiiZi) and the reference coordinate syste (XZ). The center of ass or center of gravity (cg) for the syste of bodies is not required to be coincident with the origin "of' of the reference coordinates. These bodies translate and rotate relative to one another as well as relative to the rotating reference frae (XZ). The body fixed

9 coordinate syste X..Z. can be located arbitrarily in the jg body. J J J bodies can have cg offsets. As shown in Figure, the j For the following developent, the sybol appearing ab~ve a vector quantity denotes the co- ordinate frae that the vector is resolved relative to and also the coordinate frae that derivatives -, of the vector are taken with respect to. Exaples are 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..Z. inertial coordinates, the XZ reference coordinates, and the X..Z. body fizred coordinates. J J J Particle Dynaics The force acting on the constant ass particle "p" (part of the jthbody, - Figure ) and the oent of that force about the origin "o" of the reference XZ coordinate frae are given as The radius vector fro the origin of the inertial reference syste (X..Z.) to the particle "p" of the jth body (Figure ) can be written as - Differentiating Eq. (3) and substituting it into Eq. (I), Since the j4_h body is rigid (no relative otion of particles) the agnitude of jp however, the derivatives of 5 are nonzero because its di~ection is variable. jp is a constant; Substituting Eq. (4) into Eq. (2), Recalling that the previous equation can be written as

10 jth Body Dynaics Suing for all of the particles "p" of the j4_h body, the ass of the body ie obtained as Since the j2 body is rigid and its center of gravity Icg) is displaced fro the origin of the X. Z J J ~ syste, suing ass oents about the origin yields (7) F jp'jp = jplcg * The force acting on the 49 body and the oent of &at foa*ce abo~t "" are obtained by suing Eqs. (4) and f5) for a particles "p" vf the j s bady. Substituting Eq. (6) along with Eq. (7) arrd its derivatives Wo the above,eqw.&bns, the force and oent reduce to - where the oent of oentu of the jt& body aboqt the orisin of the X..Z. coordinate syste H. is defined as H. = p. X; 3 p ( JP JP jp) = PjpXpjpd jz body

11 6 2 2 (D (D 2 3 (D E 5 W x * U. U. U. P, 3 a s (D (D r I-. P U. El u. a- U. E U. Tl I CI ;?b 9 U. I Cl. Up I: Os Os U. i$- U. 9 Os E - r 9 M 9 W 6 g U UP - U. I: I : B U?k - 3 % U. 8 U. h v e, k h W a I.

12 Syste Dynaics The total ass of the syste and the cg location for the syste relative to the origin "" of the reference coordinates (Figure ) are obtained by suing the asses of the j bodies and their ass oents about "" as follows, Ej = J j(t F ~ = it ~ cg ~. ) (7) (8) The total force acting on the syste and the oent of that force about "" are obtained by suing Eqs. (5) and (6) for all j bodies and using Eq. (7) together with Eq. (8) and its deri- vative s. L Motions of the j individual bodies are described by Eqs. (5 and (6), while Eqs. (9) and (2) describe the otion of the syste of bodies. The forces and oents acting on the j individual bodies include those resulting fro interaction of bodies, external forces, and body forces. and oents resulting fro body interactions occur in equal but opposite pairs; therefore, in the Forcers suation to obtain Eqs. (9) and (2) these pairs cancel, leaving only the suations of the external forces and oents and body forces and oents which are represented by 3 and Go. Exaple Cases In order to deonstrate the usefulness af these general equations of otion, two exaples will be given. For the first exaple, equations of otion will be developed relative to nonrol ing coordinates (aeroballistic axes) for a single spinning body having ass asyetries (principal axis isalignent and center of gravity offset). The equations of otion for a spinning body containing an internal vibrating eber (ultiple body proble) will be developed as the second exaple. Ae entioned 3

13 earlier, the equations developed for these exaple cases are useful for describing the free-flight otions of both aerodynaically stabilized and gyroscopically stabilized spinning vehicles. Spinning Asyetrical Body The single rigid body considered here (Figure 2) is not required to have an axis of ass or ' aerodynaic syetry. For this exaple, the X Z body fixed axis syste (Figure 2) has its origin fixed at the cg of the body (6 = ). The XZ nonrolling reference coordinate syste cg (Figure 2) is placed in the body such that its X axis is directed out the nose and its origin is offset laterally fro the cg 9 = (. y z ). The X and X axes reain parallel while the positions of cg' cg relative to and Z relative to Z are described by the roll angle 6. Therefore, the reference syste pitches and yaws with the vehicle but does not roll with it. For a single body with the arrangeent of axis systes shown in Figure 2, Eqs. (9) and (2) reduce to T &F~ 25x5 i $~ Of the available options, the arrangeent of coordinate systes shown in Figure 2 was chosen because the resulting vector equations require fewer algebraic anipulations to reduce the to scalar differential equation for. This approach does require that the oents and products of inertia be obtained relative to the offset X Z axis syste; however, because of the siplicity of the transfer equations, this is not a disadvantage. 4 With the exception of H, the vector quantities contained in Eqs. (2) and (22) are defined in ters of their coponents relative to the XZ nonrolling reference syste as where p, q, r are the coponents of angular velocity relative to the body fixed X Z axis sys- tern, 4

14 h.l M h M h d M v 8 u c...i 2 % & U Q, 3 P) 2) 5 A w cu P c rd h U cu w c.rl h d M v I3 w a 9 fi z 9 n E- w 3 A W w M E.rl u h n M n * E E I It * h x R R A (D M x El & H $ -- F a & a I un " *& T & e H

15 - Jxlyl JXIZl (6 qr) sin 6 - Jxlyl (p2 - r2) sin - (p" - q2) cos Q, JxlZk - - qr) (4 pr) cos - Jlzl(i. - pq) sin o - FX Jylzl (37) Possibly the ost classic of the any applications of equations of this type (relative to nonrolling coordinates) is in the description of artillery shell otion. The presence of products of inertia, cg offsets, and unequal lateral oents of inertia in Eqs. (32) through (37) ake the different fro the equations of otion relative to nonrolling coordinates often found in the literature. These additional ters can have iportant effects on the flight dynaic behavior of all flight vehicles. Internal Vibrating Meber The two body syste considered herein consists of an outer rigid body (Body, Figure 3) that has a sall rigid body (Body 2, Figure 3) suspended within it by assless springs. Motion of Body 2 is constrained to translation in a plane parallel to the Z -plane; i, e., it can only ove ' laterally. The otion of Body is unrestrained. The X Z body fixed axes (Body, Figure 3) and the XZ reference syste are coincident, with their origins fixed at the cg of Body. These systes and the X Z syste reain utually parallel; therefore, both bodies have identical angular velocity. The X22Z2 syste has its origin placed at the cg of Body 2 (Figure 3). As a result of the constraints iposed and the placeent of coordinate systes, a = p2cg = I- = O* Icg The vector equations which describe the otion of this syste, Eqs. (5), (8), (9), and (2), then reduce to i? =F 2 2 cg viixij' l J F - F ~ = ~ (39) where fro Eq. (7), = 2' 6

16 Like the previous exaple, the ass properties that define the oent of oentu of the bodies will be given relative to the X..Z. (j =, 2) body fixed coordinate systes. Therefore, A J J J H. (j =, 2) can be obtained. Because the X..Z. systes and the XZ reference syste are 3 -b A ' J J J utually parallel, H. = H - i. e., for this case, no angular transforation is required to obtain the J j' oent of oentu relative to the reference coordinates. Using the developent of Eq. (3) as a odel, the oents of oentu for the bodies relative to the reference coordinate syste ' XZ are given as The reaining vector quantities contained in Eqs. (38) through (4) are defined relative to the XZ reference syste as gx, F gy, F (43) (44) 7 = (., v, w) 8 = (p, qy r) where x2, y2, z 2o spring stiffness. (Figure 3) describe the rest (unaecelerated) position of Body 2, and K is the By cobining and expanding Eqs. (38) through (48), the scalar differential equations of otion for the two-body syste (Figure 3) can be written as (2 2) y r2) ( 2): 2 (z2 z2) (clr - 6) x2h - 2Pi2 E) 5 = (49) 7

17 9 9 I 4-4 I r W T I I n & * 3 O I $ T tr n I '2 ' T - W e I.- I -t W xr 3 w G- W T - L\3 n P g I X O T W e. I 9 R) h cn h cn w

18 Mz = - Ix - y2y]pq (y2 x2 (2 2) (2 r") Equations (49) through (56) are useful for probles in which otion induced resonances of internal vibrating ebers are of concern, They are also useful for studying the effects on sys- te otion which result fro the otion of the internal coponents. Conclusion A set of general vector differential equations has been developed that describes the otion of a syste of rotating-translating bodies relative to a rotating reference coordinate syste. These equations allow for: () coplete flexibility in placeent of the reference frae: (2) relative rotation and translation between the reference frae and each body; (3) relative rotation and translation between the respective bodies of the syste; and (4) the existence of ass asyetries in each of the bodies. Through the use of siple restraints these general equations are easily reduced to the governing equations for any dynaic systes of interest. Two exaples are given to illustrate this flexibility. 9

19 Po

20 Figure 2. Spinning Aeyetrfcal Body (onrolling Coordinates) 2

21 i Origin "" and cg z, z, Figure 3. Internal Vibrating Meber (Body Fixed Coordinates) 22

22 i References. W. T. Thoson, Introduction to Space Dynaics, ew ork, London: John Wiley and Sons Incorporated, B. Etkin, Dynaics of Flight Stability and Control, ew orh, London: John Wiley and Sons Incorporated, D. T. Greenwood, Principles of Dynaics, Prentice-Hd Incorporated, Engtewood Cliffs, ew Jersey, A. E. Hodapp, Jr., Equations of Motion for Constant Mass Entry Vehicles with Tie Varying Center of Mass Position, SC-RR-7-69, Sandia Laboratories, Albuquerque, ew Mexico, oveber

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