Note on the dynamics of a slightly deformable body

f TECHNISCHE HOGESCHOOT VUEGTUIGBOUV, KUNDE

12 Juli 1950

i

Note on the Dynamics of a Slightly DeformaTjle Body By

-W. S. HEMP, M.A.,

of the Department of Aircraft DeslaTi.

^

j SUWIARY.

THE PURPOSE OP THIS NOTE 1 3 TO DEVELOP THE

EQUATIONS OP MOTION OP A SLIGHTLY DEPORMABLE BODY. APPEAL TO GENERAL PRINCIPLES SIOWS THE INDEPENDENCE OP THE TRAW3LATI0N

( P a r a . 2 ) . MOVING AXES ARE DEFINED IN PARA,3 -HICH CA:N BE TAKEN TO DEFINE THE ROTATION. MOTION RELATIVE TO THESE'AXES I S Dii'SCRIBED BY NORML CO-ORDINATES ( P a r a . 4 ) AND THE; KINETIC ENERGY OP THE MOTION REL.ATIVE TO THE CENTRE OP MASS I S S ^ L I T

INTO T-vO PARTS; THE ENERGY OP ROTATION AND THE ENERGY OP VIBRATION ( P a r a . 5 ) . EQUATIONS FOR THE .VIBRATION ARE THEN FORMULATED ( P a r a . 6 ) . ATTENTION 1 3 DRAWN TO THE COUPLING BETWEEN ROTATION kim VIBRATION, WHICH ONLY VANISHES WH^M THE ANGULAR VELOCITIES ARE SMALL ( P a r a . 7 ) .

1

2

-1. STATEMENT OF THE PROBLEM.

This note is concerned with the mathematical

description of the motion of a slightly deformable body and Fith the formulation of the equations which govern this m.otion< The general problem, of the motion of a deformable body I R

treated by Lamb in Art.72 of his "Higher Mechanics". The follo?;lng treatment is based upon that of Lamb and develops his foundations for the special case where the deformation is

small,

2 . MOTION OP THE CENTRE OF r/IASS.

The centre of mass m.oves, according to Vnovn d:;7ri.amical principles, as if it vre-^e a particle of mass equal to that of the body, subject to a force equal to the resultant of all the forces applied to the body. Again if ra.oving axes are used, ,which pass through the centre of mass, the equations governing

the variation of angular momentum can be written donrn without any refor-ence to the translation of the origin. Finally the Kinetic energy can be expressed as the sum. of two terms; tho

first is the energy of the ficticious particle i--hioh follows the path of the centre of mass, ;7hile the second is .the energy of the relative motion. It is clear then, that no di.i^ficulty arises in calculating the motion of the centre of mass nnd further, it can be disregarded in the studv of the relative motion. Only the relative motion will be consid.e-^ed in what follows.

^- CHOICE OF THE MOVING AXES.

Consider a system of moving a.xes pas=;lnfr throu.gh the centre of m.ass. The angular velocity vector of these a'x-es is denoted hj CO . If h is the angular momxentura vector and M the vector moment of the applied forces about the origin the equation of motion defining the variation of h

is:-^ 4.(is:-^x h a M . . . . . (1)

dt

Now if the body •^•ere rigid and were moving '"ith the axes it would possess angular momentum h-p given

by:-h;p = (Ap-Hq-Gr, -Hp-t-Bq-Fr,-Gp-Pq 4.Cr) . (2) il'here A,...P, ... are moments and products of ine"^tia and

(p>q.>ï') - €^ . The quantities A,P are not constants, but vary -'•ith the motion relative to our moving axes.

I

/ We

'.'e no^" choose our axes in such a --ay that the follo'^^ing equality holds

:-h = :-hj, (3)

The equations (1), (2) and (3) then define the motion of our axes, when the quantities A, P are known. The initial

orientation is as yet undefined.

MOTION RELATIVE TO THE A.KES.

Lamb in his treatment of this problem, chooses CO

by fitting the moving axes to t.ho actual motion, by means of a principle of least squares and obtains equation (3) above, ••Ve deduce then, for our case of a slightly deform.able body, that the motion relative to our moving axes is small. '-'e rem.ark further that this motion is without linear oi^ angular momentum (equation (3)). It follo'-'S • that the term, "vibration" m^ay -"ith justice be applied to this m-ïtion, but the term, must be used "'ith caution since the motion is referred to moving and not to fixed axes.

We describe the relative motion by means of

generalised co-ordinates Ö^, J':'-2 > ....,öj_, ... These may be defined as follo'-s:- Consider the case whnre the body is

at rest and referred to axes through its centre of mass. The body possesses an infinity of normal mod'-s of vibration which

are free from resultant mom^f^ntum. No"' sny kind of sm.all motion which is free from momentum can be expanded in an

infinite series of these normal modes. It follor's that the é*i can be taken as normal co-ordinates. The initial

orlp^ntation of our axes must be chosen to suit. For exam.ple In the case of a transient vibration, they must coincide, before the vibration begins, "-ith the axes used above in describing normal modes of vibration.

^^ The q u a n t l t l p s A, P can now be expressed in ter^ms of

t h e ' - ' j _ . S i n c e t h e Ö j_ a r e small ^^-e can

-^'rito:-A = -^'rito:-A^*C.I>-^'rito:-A\ -^'rito:-A . . . (4)

KINETIC ENERGY.

Ve now show that the Kinetic energy of the motion relative to the centre cjf mass can be split into t'"o parts, The first depends upon the (p,q,r) while the second dononis upon è j_.

If m is an elementary mass, i-ho-^e velocit3r vector is V, the Kinetic energy T = -|^mv^. If P is the position vector of m "e ean split the motion v into t"'o

parts:-V • (v -/^X F)+ 6JX f>

Substituting in the formula for T we obtain a mi-^^ed

term:-^ m

iv-'(^»fl{oj^f}-c^^mpx-7-'^m(o~iKf) -o.h -

2Tt^

••'•'here T R - Kinetic energy of the masses assumed rotatin.cr with our moving axes. Using (2), (3^ an-^ tho form.ula

2TR = ;.p2 4. -2Pqr - (5) we see that our mixed term is zero and so '"e can

'"'rite:-T = '"'rite:-Tp +.'"'rite:-Tv . ' . ' . . . (6) '.".'here T^ a Kinetic energy of the motion rclativ.e to the moving

axes.

EQ,U;.TIONS FOR THE 0^,

The equations cf motion for theöi can be formulated in the Lagranglan m.anner. V/e

find:-•-•here U is the strain energy of the deformation and Ssj_ is the generalised force corresponding t o ^ i .

We notice that ^ = 3>Tv and M" r ^Tr< , T R

depends upon ö ^ in virtue of the variation of A, P (equation (4)). Since Q^ are normal co-ordinates Tv and U h"ve the forms Tv = ll'Ji&i^' and U = i l c i ö i ^ , whore ai and ci are constants. Equation (11 thus transforms

tot-ale ^ +• c^ö ^ = d^i +• è ) / M \ p2 _ ^ -S/aP V qr.. ) (8)

(^^öijo \5öiio )

The terms in the curly brackets in (8) represent the effect of centrifugal forces on the vibration.

CASE OP SMALL ANGULAR VELOCITIES.

The equations developed above are complex. The equations (1),(2),(3) which define tho rotation Involve the vibration through ectnatiof} (4) . Likewise the vibrational equations (3) Involve the'angular velocity (p,q,r). These

complications disappear when the rotrtion is small. Neglecting term-s of second, order in (p,q,r) and. Öi we find using (iT

(2), (3), (4) and (8):-.

J_ (A p-HoO-Gor, -HoP + Boq-Por,-GoP-Foq4-Cor) =: M . (o) at

aj_Ö^-hC3_6^ =*^1 . . . (10) / ''••-' a « • a

^ T^

We see in this case that the rotation and vibration are independent,

8. CONCLUSION

We have separated, the motion of a slightly deform^able body into three parts:- translation, rotation and. x?-lbration. V/hlle the translation is independent of the other two, the rotation and vibration are coupled except in the case when both are sm.all.