Note on rigid body motion

Jnl. MechanismsVolume 6, pp. 259-266/Pergamon Press1971/Printed in Great Britain

Note on Rigid Body Motion Professor K. E. Bisshopp*

Received 22 July 1970 Abstract The equations for the velocity and acceleration fields of a rigid body are obtained solely from the definition of an n dimensional projection field. Z u s a m m e n f a s s u n g - B e m e r k u n g zur Bewegung des starren Kbrpers: K. E. Bisshopp. Die Gleichungen for das Geschwindigkeits- und Beschleunigungsfeld eines starren Kbrpers werden allein aus der Definition eines n-dimensionalen Projektionsfeldes abgeleitet. P e l l o M e - - O ~BH)KeHHH TaepJ1oro Te~a. K. B a t l i o n n . BblBe~eHb[ ypaBaeHH~t rtoJlefl CKOpOCTH H yCKOpeHHR TBcp~oFo Te~a H3 Orlpe.~eIIeHH.q n-pa3MepHol'o rlOng npoerultR.

Introduction

THE OBJECTIVE of this pedagogical note is to show how the basic equations of rigid body motion can be obtained from a single principle, namely the invariance of distance between any two distinct points of a rigid body during motion. On this basis the existence of angular velocity and acceleration is established as derived concepts which are associated with free vectors of a velocity and acceleration field. It will be advantageous to formulate this problem in n dimensions by using matrices, since the structure of the rigid body constraints on its particles has the same mathematical representation. Then, the classical equations of rigid body motion are obtained for the special cases where n = 2 or 3 by reverting to vectorial notation. It is axiomatic that the distance between any two distinct points P~ and P> fixed in a rigid body, is constant for all time irrespective of the motion of the body. This single fundamental property, when expressed vectorially, implies that the corresponding velocities V~ and Vj respectively, have projections, equal in magnitude and direction, on the line through P; and P~ as shown in the figure. Let 0' be an arbitrary origin, fixed in space, a n d 0 likewise fixed in the moving body. Either 0' or 0 can be chosen as a reference point for vectors to points P~ and Pj; then P~Pi = R~ - R~ = R i' - R j

since

R[--Rj = 0'0+R~--0'0--Rj. During motion of the body the vectors R[, R~, and 0'0 vary in direction and *Professorof MechanicalEngineering.RensselaerPolytechnicInstitute.Troy, New York. 259

260

R'

O'

Figure 1. magnitude while Ri, Rj, and R~--Rj are translated and rotated without change in length: hence in order for the distance l&&l to be constant for all time it is r e q u i r e d [ l , 2]* that (R~--Rj)- (Ri--Rj)--' (R{--Rf).

(Ri'--Rf

-constant.

The Velocity Field Let V , be the velocity vector at P , then differentiation of equation to time yields (R,-

Ri) • ( R I -

Rj) = ( V , -

I l)

t with respect

V0) • ( R , - - Rj) = 0.

But d

dt

(PiPi)

----

R i - R j = V,-- Vj,

therefore [1,2] V i . ( R i - - R j ) = Vj" (R/--R))

(2)

which is called the projection theorem. This equation shows that the origin can be any point fixed in the moving body, or fixed in space. A vector field {Vi} in which any pair of distinct vectors satisfies equation (2) is called a projection field, whether or not the V i represent velocities of points in a rigid body. Since scalar products are valid in n dimensions, equation (2) still applies and the theorems of rigid body dynamics can be deduced as special cases for n = 2, or 3. Equation (2) determines the vector field e v e r y w h e r e when compatible field vectors are prescribed at n points PI, Pe . . . . . P,, and R,, Re . . . . . R:, are linearly independent. Consider, for example, the case where n = 4 and Vo is unknown, then the scalar equations g o - R i = Vi. R1. ( i = 1 , 2 , 3 , 4 ) (2a) have a unique solution for the c o m p o n e n t s of Vo in terms of a representative basis and :~Denotes

references

at the end

of

note.

261

the projection field is unique. Since the base point 0 is arbitrary, the entire field is determined. The quantities of equation (2) are related such that an alternative representation of a projection field can be constructed from products of matrices U~ with the position vectors to field points Ps and the vector V0 at P0, so that V~ = V0 when Ri = 0, thus V, - Vo = U s ~ .

(3)

The restrictions imposed on the U~ by equation (2) in order for {V~} to be a projection field are (V,-Vo)

-~

= 1~. • U i ~

(4)

= 0

for all values of i. It can be shown by expansion that ~ • U i ~ = 0 has the non-trivial solution Us----Us r, and therefore U~ is an anti-symmetric matrix, which is a direct consequence of the projection theorem. Equation (3) for any two field points yields

( v , - v j ) . (1~-a~)= ( ~ . - R j ) . (usa,.-ujaj) (4a) = - aj. usa,-

~.. uj~.

The left side vanishes by equation (2) and reciprocation o f - R j • U ~ gives

~.. ( u , - uj)aj = o. For arbitrary ~ and Rj this result shows that Ui = Uj = U is a free matrix, independent of the field point, so that in general V = V0+ UR.

(U = - u r ) .

(5)

A unique matrix U for a given field can be constructed when vectors which satisfy equation (2) at n suitable field points are prescribed. The procedure is as follows: For n field vectors, by introducing a basis, equation (5) yields n ( n - 1) linear relations among the ½n(n- I) unknown elements uij of the matrix U. These n ( n - 1) relations include equivalent sets of½n(n-- l) linearly independent equations resulting from the following constraints imposed by equation (2), where V~- Vo = Gs, and i = 1,2 . . . . . n-l: (1) ~ • G~ = 0 n - 1 equations (2) ~ . G j + R j - G i = 0 ½ n ( n - 3) + 1 equations. Thus there are ½n(n-- 1) constraints leaving ½n(n--1) linearly independent equations. These constraints involve the uu linearly, therefore all systems of ½n(n--1) linearly independent equations included among the original n(n--1) equations are equivalent and have the same solution.* The numerical example in the Appendix illustrates this conclusion. *This and the following uniqueness proof show that an n-dimensional projection field can always be represented by equation (5).

262

T h e matrix U is unique, that is, no other matrix Z generates the same projection field derived from the vectors which determine U. Obtain by the m e t h o d of e q u a t i o n (51 a n o t h e r set of n v e c t o r s which belong to the field. T h e s e vectors determine a free matrix Z, then !6)

V = V,)--ZR.

Set U = Z + u. Let Ve be any one of the vectors from which Z is obtained. N o w ifVt is any field v e c t o r R.~.-(V~-V,)) = R k, ( Z + u ) R ~ [ F r o m equation (5)] =R~.ZR~+R~.uR t = -- Rt ' ZRk + R~. ' aRt.

7)

N o w Z was obtained from a set of v e c t o r s which include V k and Rk, therefore ZR~ = V~ -- V o and e x p a n s i o n of equation I'7) with the aid o f equation (2) gives V/. (R~-Rl)--Vk-(R~.--Rt)= R~-uRl.

81

T h e left side vanishes, while R~. and R~ are arbitrary: therefore n = [0] and Z = U is unique. So far the properties of a general projection field have been developed. T h e matrix U can be identified with the angular velocity matrix [3],

~o.

0

L °-" --oJ~

--

z

----=~)<

(.o c

w h e n V is the velocity of a point P in a rigid b o d y then, [2, 3] V = V o + o J x R.

(9)

S o m e special properties of two and three dimensional v e c t o r fields are more easily d e d u c e d by o p e r a t i o n s with the v e c t o r p r o d u c t which c a n n o t be used in general. In three dimensions, the equation ( V i - Vo) " l~ = 0

10)

has the identical solution* Vi = V0-r- ( W ~ + ~.,R~) × R, where .Wi _ V 0 -- V~ x R,. Ri-' * T h i s r e p r e s e n t a t i o n is the m o t i v a t i o n for e q u a t i o n ~3 I.

11)

263

At this point the vector W/cannot be identified with to without invoking equation (2). For another point Pj in the field i is replaced withj. If R~ × ~ ,= 0. the scalars hl and ;~ can be chosen such that the angular velocity t o - W / + h , ~ = W~ + X~R~

12)

in order to satisfy equation (2). which makes to a free vector over the field. Scalar multiplication on either side of this equation with ~ and Rj yields two linear equations which can be solved for h / a n d ~.j so that the velocity at any point P in a rigid body is equation (9). The uniqueness proof at equation (8) applies to to. The vector to can be obtained in any particular case by substituting appropriate values into equation (9) for example, when V0, V~ and Vj satisfy equation (2), V~ - V0 = to x

v ~ - V o = to x R~. Routine methods of vector algebra [2.3] show that. if the vector products of both sides of these equations are equated and expanded, the solution is [4].

to=

(v,-Vo)X(v~-Vo)_(Vo-V,)×(vj-Vo) (V,--V o ) . R j -(V~--Vo).~

( V , - V o ) . R j ~ O (13) ( V j - V o) R, # 0

otherwise R/× ~ • to = 0, then to = R,(R~ × Rj) • ( V : - Vo) -- a/(l~. × R~) • (V/--Vo) (1~. x R~)'-'

(14)

For two dimensional fields, the vector to corresponding to equation (13) can be obtained by observing that V _ V ° = (Vj--Vo) • R, R~'-' 1~' +/3(V°-- Vi)'

(15)

where evaluation of the factor/3 is not required. Substitution of equation (15) into equation (13) gives [2.3], the unique vector to=

Vo ~.,Vi

R/-

X~.

(16)

The relation to-R~ = 0 is trivially true. On the other hand let R = a R ~ + 7 ( V 0 - V ~ ) be any vector in the plane ofVo, V~, and R , then \

R~-

~

"[aRl+y(Vo--Vi)]--0

(17)

which shows that every vector R in a two dimensional field has the property to • R = 0 without invoking the definition of angular velocity. Incidentally, equation (16) shows that At = Aj = 0 [equation (12)]. But At --- ,~j = 0 does not imply plane motion since for any three dimensional field R~ and Rj can lie in a plane perpendicular to to so that

264 o.~ • R~ = oJ - Rj = 0 w h i c h s h o w s that ~.~ = a., = O. S i n c e V0 c a n h a v e a n y d i r e c t i o n the v e c t o r field is still t h r e e d i m e n s i o n a l . F o r the s p e c i a l c a s e w h e r e V,, V , a n d V~ are d i s t i n c t field v e c t o r s s u c h that R s - / V ~ - V o ) × ( V j - V o ) = 0 a n d I V y - V0) × ( V , - V o ) O. X, = O, t h e n oJ c a n be o b t a i n e d f r o m the field v e c t o r s at the two p o i n t s , P o a n d P~, b u t ,kj d o e s n o t n e c e s s a r i l y v a n i s h , i f the m a t r i x r e p r e s e n t a t i o n o f the v e l o c i t y field w e r e u s e d . t h e s e s p e c i a l p r o p e r t i e s w o u l d n o t be q u i t e so o b v i o u s . T h e r e m a i n i n g p r o p e r t i e s o f the v e l o c i t y field c a n be o b t a i n e d b y v e c t o r i a l o p e r a t i o n s o n the r e l a t i o n s d e r i v e d from the projection theorem.

The Acceleration Field* T i m e d i f f e r e n t i a t i o n of the v e l o c i t y field, e q u a t i o n (2), y i e l d s u s e f u l p r o p e r t i o s o f the a c c e l e r a t i o n field, a l t h o u g h it is n o t , p e r s o , a p r o j e c t i o n field. T h u s A i • ( R i --

R2 ) - At • ( R i -- R~ ) = Vj • ( V i - V2 ) -- V i • ( V i - V2 )

therefore, A2-(Ri--Rj)--Ai'(RI--Ri

) = ( V i - - V ; ) e.

(18)

In particular ( A o - - A i ) ' Ri = ( V i - - V o ) ~ =

UR i • UR i

=

R i • UrURi

:

_

(19)

R i • U"-Ri"

T h e m a t r i x U ~ is s y m m e t r i c s i n c e it is the s q u a r e o f a n a n t i - s y m m e t r i c m a t r i x . T h e g e n e r a l s o l u t i o n o f e q u a t i o n (19), b y i n s p e c t i o n , is (20)

A i - - A~) =- K i R i q - U " - R i

w h e r e K~ is a n a r b i t r a r y a n t i - s y m m e t r i c m a t r i x . It f o l l o w s f r o m e q u a t i o n s (18) a n d ( 19) t h a t Ki -----Kj = K is a free m a t r i x o v e r t h e a c c e l e r a t i o n field b y w r i t i n g ( A i - - Aj) • Ri = Ri" K i R i - - R i " K j R / + Ri • U : ~ R i - - R i • U"-R2 ( A , - As) "R~ = R 2 • K , R g - - K - K j K + Rj • U2R~-- Rj " U'-'Rj A~- (R,--R;)-A,. (R,--R;) = R~ . K ~ R j - - R , .K,R; - - R/• U'-'Ri + 2Ri • U2R~ - Rj • U~R~ ( v , - v~)-' = K ( K ~ - - , q ) R ~ + ( V , - - V~)-'.

(2t)

T h e last t e r m o n the right side is o b t a i n e d b y o b s e r v i n g that (v,-v~)"

= [v,-Vo-('%-v0)]" = u(K-K) • u(K-K)

= [u(K-l%)]~

=- (K-K) - u~(K-K) = - K U"R~ + 2Rq. U"Rj -- R~. U~Rj. :"The acceleration field could be obtained by differentiating equation 151 which would not emphasize the role of the projection theorem and intrinsic differentiation.

265

The vectors ~ and Rj are arbitrary, therefore the only non-trivial solution of 1~. (Kj--•i)Rj = 0 is K~= Ks = K.* From equation (20) it is seen that K has dimensions sec-", therefore it can be none other than angular acceleration. For three dimensions U:R = oJ × (ca × R), then equation (20) yields, ca x R i = Ai--A0-- ca X (ca× Ri) (22) cax R , = A~--Ao--ca X (ca X Rj) which are the usual textbook equations [2, 3] for the acceleration field of a rigid body. The vector d~ now can be obtained by routine methods of vector algebra.? The result is too unwieldly to record here. The uniqueness of dJ is essentially proved at equation (2 1). Again only velocity and acceleration vectors were used to derive equation (22). Obviously. this method can be extended to derivatives of higher orders. It is to be observed that equations (19) and (22) were obtained intrinsically without evaluating derivatives in a moving coordinate system.

Conclusions The well-known properties of the velocity and acceleration fields[I-3] of a rigid body have been deduced by matrix and vector algebra from the sole hypothesis, equation (2), which is a consequence of the mathematical definition of a rigid body from which the existence of the angular velocity and acceleration vectors also has been proved. Thus the projection theorem is of fundamental importance in rigid body kinematics.

References [I] [2] [3] [4]

BANACH STEFAN, M e c h a n i c s , translated by E. J. Scott. Hafner, New York (1951). COE C. J., T h e o r e t i c a l M e c h a n i c s , Macmillan, New York (1938). FOX E. A.. M e c h a n i c s . Harper and Row, New York (1967). BEGGS J. S , , A d v a n c e d M e c h a n i s m . Macmillan, New York (1966).

Appendix Construction of 4-dimensional matrix U. Let

R0 = ( 0 . 0 , 0 . 0 ) RI= (1,1,0,0)

V0= (1, 1, 1, 1) VI= (2,0,1,1)

R , = (1.0, 1,0) Ra= (1,0,0.1)

Ve = (0, 1 , 2 , - - 1 ) V3= ( 2 , - - 1 , 3 , 0 )

Equation (5) yields ul,, = 1 - - t t z 2 -~" - - 1

- - 1 1 1 3 - ll23 = 0 --1114

--

t124 = 0

ula --ttt..,+

=

--

l

U23 =

lll4 = 0

1

- - t r y , + It., 4 = - - 2

--1113 = 1

--1113 "~ lt34 = 2

--/~14--/J34 = --2

- - t i t 4 = - - 1.

*Uniqueness of the matrix ~¢can be proved by a method similar to that employed for U at equations (6), (7). and (8). *See equation (13).

266 Here. for e x a m p l e , t w o equivalent sets of six linearly i n d e p e n d e n t equations are 1112 -= I

lit: ) =- lt.,3 .= 0

ll12 ~

H13 =

lll4 ~- 1124 ~- 0

It1: l -= - - 1

Ill. 2 - - l t 2 4

= 2

Ill4-'-//34

1114 ~

lti: ~ - l t : ¢ )

~

-- I

lti4 =

|

=

2

T h e r e f o r e , the solution is

U=

io,,, ,1 -

0 11 -i o l', 1 1--1 o j.

I

]

Ill. -, - - H , 2 : 3 ~

0

--2