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Motion of a symmetric rigid body subjected to a specialized body-fixed force
hr.
J. Sofkfs .Structwes,
1972, Vol. 8, pp. 415 to 438.
MOTION SUBJECTED
Pergamon Press. Printed in Great Britain
OF A SYMMETRIC
TO A SPECIALIZED T. R.
KANE-~
RIGID
BODY
BODY-FIXED
FORCE
and B. S. CHLQ
Department of Applied Mechanics, Stanford University, Stanford, California
Abstract-Exact dyflamical and kinematical equations governing the attitude and translational motions of a symmetric rigid body under the action of a specialized body-fixed force are formulated. From these equations, an analytical, but approximate, description of the behavior of the system is obtained and the validity of the solution is then tested by comparing its predictions with those of digital computer solutions of the exact equations of motion.
INTRODUCTION THE problem of determining the motion of a rigid body under the action of a constant, body-fixed force whose line of action fails to pass through the center of mass is so simple from a physical point of view that it may be surprising that no complete, analytical solution appears to be possible. In fact, not even the Euler dynamical equations for this problem have been solved in closed form for the general case, that is, for a body with unequal centroidal principal moments of inertia and for an arbitrary o~en~tion of the line of action of the force. (Leimanis and Lee [l, 23 developed expressions for the angular velocity for the special case of a force placed so that its moment about the center of mass is parallel to a centroidal principal axis, but these expressions involve integrals which, in general, cannot be evaluated by quadrature ; and Grammel [3] constructed approximate solutions of Euler’s dynamical equations.) Hence it is natural to seek solutions of related, but somewhat simpler problems. A major step in the direction of decreased analytical complexity can be taken by confining attention to a body whose inertia ellipsoid for the mass center is axisymmetric. One of the first to deal with this problem was Biidewadt [4], who showed that the solution of Euler’s dynamical equations can be expressed in terms of integrals of the Fresnel type.5 Unfortunateiy,the next step of his solution, that of integrating a set ofkinematica1 equations, was based on a false premise, because, as may be verified by substitution from equation (49) into (42) of [4], the method proposed by BGdewadt is valid only when the matrices U and W’ commute, which is not the case for the problem at hand. Consequently, the orientation of the body remains to be determined as a function of time. The same problem was also attacked by Auelmann [Sj, who attempted to find a closed-form solution to the kinematical equations. While falling short of this goal, he succeeded in obtaining an expression for the angle between the symmetry axis and the initial direction of the symmetry axis for the special case when the initial angular velocity is normal to the symmetry axis. t Professor of Applied Mechanics. $ Research Assistant. 9 An English account of Biidewadt’s work appears in chapter 10 of Leimanis’ book [l]. 415
416
T. R. KANE and
B. S. CHIA
A further simplification results from restricting oneself to consideration of body-fixed forces whose lines of action lie in the plane containing the mass center and normal to the symmetry axis. A very elegant treatment of one facet of this problem was presented by Valeev [6], who used continued fractions to obtain a description of the motion of a spacefixed unit vector relative to three body-fixed unit vectors parallel to principal axes of inertia for the mass center. It is the purpose of the present paper to present a complete, albeit approximate, solution of the same problem, for both the rotational and translational motion. The paper begins with a detailed description of the system to be studied and the mathematical quantities to be used in the analysis are defined. Next, exact dynamical and kinematical equations governing attitude motions are formulated ; the dynamical equations are solved exactly ; and an approximate solution to the kinematical equations is obtained. The validity ofthe approximate solution is then tested by comparison with digital computer integrations of the exact equations. Finally, the motion of the mass center is studied in terms of an approximate analytical solution and results are once again tested by means of comparisons with solutions obtained by numerical integrations.
SYSTEM
DESCRIPTION
The system to be analyzed is shown in Fig. 1, where B represents an axially symmetric rigid body of mass m. The center of mass of B, designated B*, is located by a position vector X relative to a point 0 that is fixed in an inertial reference frame A. Mutually perpendicular unit vectors a,, a2 and a3 = a, x a2 are fixed in A and mutually perpendicular unit vectors bl , b, and b3 = b, x b, are fixed in B parallel to principal axes of inertia of B for B*, with b3 parallel to the symmetry axis of B. The moments of inertia of B with respect to lines passing through B* and parallel to bi , b2 and b3 have the values I, I and J, respectively. f
A
h
FIG. 1. Model of an axially symmetric
rigid body.
Motion of a symmetric rigid body subjected to a specialized body-fixed force
417
It is presumed that B is subjected to the action of a force F of constant magnitude and fixed direction in B. This force is applied to B at a fixed point Q which is located relative to B* by a vector q. Furthermore, it is assumed that both F and q are perpendicular to b,. Hence, if the nine scalar quantities xi, Fi and qi(i = 1,2, 3) are defined as xi P X .ai
(1)
Fi 4 F . bi
(i = 1,2,3)
(2)
qi ’ Q. bi
(3)
then F3 = q3 = 0. When the force F exerted on B at Q is replaced with a force applied at B* together with a couple of torque T, then T=qxF=
7h,
(4)
where T is defined as T s qlF2-q2F1.
ATTITUDE
MOTION
The work that follows is facilitated by introducing a neither in the body B nor in the inertial reference frame such a way that a unit vector c3, fixed in C, remains at all which is fixed in B. The angular velocity of B relative to necessarily parallel to c3 and can be expressed as
reference frame C which is fixed A, but is constrained to move in times equal to the unit vector b,, C, to be denoted by %I?, is then
ceP = SC3
(6)
where s is a function of time t. As will be seen presently, it is the choice of s which furnishes the key to the solution of the problem at hand. However, no matter how s is chosen, the angular velocity of C relative to A, AceC,can be expressed as A
”
C-
-
Plcl
(7)
+P2cZ+P3c3
where pi(i = 1,2,3) are functions oft, and c 1 and c2 are unit vectors fixed in C, perpendicular to each other and such that c1 x c2 = c3. It then follows that the angular velocity of B relative to A is given by AeP =
plc,+p2c2+(p3+s)c3.
(8)
As b3 = c3, it follows from equation (4) that the moment about B* of the force F can be expressed as T = Tc,
(9)
and, in accordance with the angular momentum principle, the quantities pi, p2, p3 and s are thus related to each other as follows : rp1 +p,[Js-(I-J)p,]
zp2 -
p1 [Js
-
(I
-
= 0
(10)
J)p,] = 0
(11)
Jo3+S)
= T.
(12)
418
T. R. KANE and B. S. CHIA
Equations (10)-(12) contain four unknown quantities, p, , p2, p3 and s. However, s was introduced solely for the purpose of facilitating the analysis and may, therefore, be chosen at will; and a choice which, indeed, simplifies the subsequent analysis is s = Lp,
(13)
I-J J
(14)
where L is defined as L
p
-.
Now, if pi0 denotes the initial value of pi(i = 1,2,3) and A is defined as A-‘;
(15)
thensubstitutionfromequation(l3)into(l0)-(12),followed yields Pl = PlO,
Furthermore,
by integration oftheseequations, P3 = At+P3o.
P2 = P20,
(16)
from equations (13) and (16), s = L@t + p30)
(17)
and the angular velocity of B in A can now be found by substitution into equation (8). However, as it is advantageous to express ‘aB in terms of b, , b2 and b, , rather than cl, q, and cg , a brief digression will prove useful. If c, is taken to be equal to bl at t = 0 and 8 is defined as 84
* s dt = L(@t2 fp30t) s0
(18)
then 8 is the radian measure of the angle between bl and c1 ; and if Oi(i = 1,2,3) is now defined as Wi “oB.bi
(i =
1,2,3)
(19)
it follows from equation (8) that w1 = p1 cos 8+p, w2 = -pl
sin 8
(20)
sin@+p,cos@
(21)
w3 = p3+s.
(22)
Hence, if oiO denotes the initial value of Oi(i = 1,2,3), then Pl = WlO,
s=L
and
i
P2 = w209
At+% l+L
I
pa = At+%
l+L
(23) (24)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
419
so that, substituting into equations (20H22), one finds o1 = or0 cos @+w,e sin = J(o:, w2
=
+ O&J cos e + tan [ +(-:)I
--~,,sin8+~,,cos8
=
= J(f$, 03
e
+ &)
sin e + tan [ -‘( -~)I
(27)
(1 +L)&+o,,.
(28)
Equations (25j(28) constitute a complete solution of the dynamical equations of rotational motion for the problem at hand. In order to describe the instantaneous orientation of B in A, one must integrate yet one more set of differential equations, the so-called kinematical equations. Conceptually, the simplest way to obtain a complete description of the attitude motion of B in A would be to solve the nine first-order differential equations governing the elements of the direction cosine matrix relating the body-fixed unit vectors b, , b,, b, to the spacefixed unit vectors a,, a*, a3. However, since these equations are strongly coupled and involve the time-dependent functions Wi(i = 1,2,3) [see equations (26)-(28)], there appears to be little hope of carrying such a solution to a successful conclusion. Alternatively, one may take advantage of the relative simplicity of the differential equations governing the elements of the direction cosine matrix relating c, , q, c3 to a 1, a2, a, and of the fact that, once the orientation of C in A is known, the orientation of B in A can be found easily, because B performs a simple rotational motion in C and the time-dependence of the angle 0 associated with this rotational motion is already known [see equation (251-J. Accordingly, one may begin by defining aij(i, j = 1,2,3) as aij A ai. cj
(i, j = 1,2,3)
(29)
and then take advantage of the fact that the first time-derivative of ai in A can be expressed as Adai = Cdai -~+ROCXtIi dt
(i =
1,2,3).
Since ai is fixed in A, this derivative is equal to zero. Using equation (7), one is thus led to ci,lCl+ (ii*C* +cii3C3 =
(P3%2
-p2@3)ct
+(plai3-_p3ail)c2+(p,ai,
-plai&3 (i
=
1,2,3).
(31)
The quantities pi(i = 1,2,3) appearing in equation (31) are known functions of t [see equations (2311; furthermore, pZ may be set equal to zero, for this implies only that @20
=
0
(32)
that is, that AeP is initially perpendicular to bz, which can always be arranged by choosing the orientations of b, and b2 in B suitably. Consequently, substitution from equations (16)
420
T. R. KANE and B. S. CHIA
into (31), and subsequent rearrangement,
results in
Liii-P~eUiz = ItUi, lif2_Pi@i3+P3oUil
=
-AlUi$_
(33) (i =
1,2,3)
(34)
cii3+Pioaiz = 0.
(35)
The initial value of ajj, to be denoted by Uij(O),depends on the initial orientation
of
ci , c2 and cg relative to a,, a2 and a3. If ci = ai at t = 0, which can always be arranged by
choosing aj(i = 1,2,3) properly, then [see equations (29)J aij(O) = 6ij
(i, j = I, 2,3)
(36)
where I&,p 1
(i = 8,
dij k 0
(i # j).
(37)
These initial conditions also imply that bl , b, and b3 are respectively paralfel to a,, a2 and a3 at t = 0, since cr , c2 and c3 initially coincide with bit b2 and b,, respectively. Despite their relative simplicity, equations (33)-(35) do not appear to admit of an exact solution. Hence we shall resort to the following method of successive approximations: integrate equations (33)-(35) with I = 0, and require the solution to satisfy equation (36). Next, substitute the resulting expressions for a,, and ui2 into the right-hand members of equations (33H35), integrate these equations and again require the solution to satisfy equation (36). Finally, repeat this process once more. Since the expressions for aij(i, j = 1,2,3) obtained in this way are quite lengthy, it is advantageous to cast the associated approximate description of the attitude motion of C in A in a simpler, equivalent form. This can be accomplished by defining pot Pi and P3 as PO 9 J(P:o + P30),
pi c ~lO~P0,
(38)
p3 4% P30/~0
and then introducing a dextrai set of orthogonal unit vectors, e, , e2 and e3, fixed in reference frame A in such a way that e, has the same directions as the initial angular velocity of C in A, e, = a2 and e3 = e, xe,. Keeping in mind that ci = ai at t = 0, one can then express the relation between ei and ai(i = 1,2,3) as
(39)
from which it follows by reference to equations (29) that
(40)
where ~1@11+~3a3, a21
i -P3a,,+Pla3,
~P%2+~3a32 a,, -P3a,2+Pla32
PA3
+P3a33
(41)
a23 -P3a13+Pla33
I
Motion of a symmetric rigid body subjected to a specialized body-fixed force
421
and
(43)
smpot+P, +P~P3(~)(~) . 2(2i)2(P~+P~SP,-3P~P~)(1-cospot)
e21
M P3 sin pot+PT
+zpzp3
-Pi
1 3 0 _& P$
f
$
sin pot - 2P:F,
z (1 0& At
-’
3
cosp,t-~P:
At 2
PO sin pot + f Pi -“, ( I 0PO 0 -;i Iz PO
1
(I-cosp,t)+~P:P,
i
I
i
-2
0 PO It
4
sin
e22
x
cos --
I
_z PO
j$
(
g
Ii
-I
a* ~0
sin pot I
0PO
cos
pot
2 It -z cospot i POI( PO1
*sinpOt-_P:P, I
(45)
sin pot-
1 A -= At 4 0 8pot p2“20- 5P3 00po cospot+P,
- ;P:
Rt =
pot
Pi Ii ;It PO1 +p*p1 3 ( A_
1
-I
W)
1
:Pf
1 -l
T
0
it3
I
0PO
2~~,~-~~~)~cos~~pot g
(44)
422
T. R. KANE and B. S. CHIA
(47)
e31 x
-P,cosp,t-Pi
(48) e32 x sinpot+-P,
1 2
1 -I
-i 0PO
1 At2 cospot+gPf 0PO
;1 -I At 3 cos pot 7 0PO 0PO
(49)
sin pot
(50) It is worth noting that, if pSo = 0, so that P, = 1 and P3 = 0 [see equations (38)], then eij is equal to aij(i, j = 1,2,3) because ei is then equal to a,(i = 1,2,3).
Motion of a symmetric rigid body subjected to a specialized body-fixed force
423
Equations (42)-(50) have been written in such a way that t is always multiplied by Aand divided by p. when it appears outside of the argument of a trigonometric function and Ais divided by pg when it is not multiplied by t. This representation of e&i, j = 1,2,3) permits one to carry out a systematic simp~i~cation of the equations and thus to arrive at approximate results which, as will be seen in the sequel, can furnish an entirely satisfactory description of the motion under consideration. Specifically, dropping from equations (42)-(50) all terms involving (A/&” with n > 1 and all terms involving (It/pJ” with m 2 3, replacing At/p0 and (&‘po)z in equations (59H67) with sin(~~/~~) and 2[1 -cos(&,/po)], respectively, and introducing new dimensionless p~ameters x and z as (51) one arrives at e,, % k~-~~~~(l-cosz~)(~-cosx)
(52)
cl2 R -PIP&--coszx)sinx
(53)
e13 3 k, +P:P,(l
(54)
-cos zx)(l -cos x)
ezf w k,sinx-t-k,cosx
@22 --
= ~~k~+k~)cos
-k5 sinx+kscosx
= ,/(kffkz
ez3 RJ -k,sinx-k,cosx
e31
x
[
xi-tan
-‘( -~)I
cos
= ~(k~~k~~cos
(56)
[
x-!-tan
-‘(-~)]
= q’(kg-bki) sin[x+tan-‘(
-k,cosx$k,+sinx
-~)I
e32 x k, cos x + k, sin x = J(k: c kz) sin
(57) (58) (59)
e33 z k, cos x - kz sin x = ,/(kf -t-k$ sin x + tan
[
where kl,...,
(55)
-‘( -:)]
k6 are defined as k, 4 PI --PIP, sin 2x--P,(l-4F$)(l k2 g F&z-‘(1
-cos
-cos zx)
2x)
(61) (62)
k, k P3 + Pi sin 2x - 4PfP3( 1 - cos 2x)
(63)
k4 A Pp-“(1 -cos 2x)
(64)
k5 A+P3z-l(1 -cos
(65)
k, k 1.
zx)
(66)
Now that approximate expressions for the direction cosines eij (i,j = I, 2, 3) are at hand, approximate expressions for angles describing the orientation of the unit vectors
424
T. R. KANE
and B. S. CHIA
and c3 relative to e, , e, and e3 can be constructed. One set of such angles can be generated by aligning ci with ei (i = 1,2,3) and then performing successive right-handed rotations of C of amount @I about an axis parallel to c3, O2 about an axis parallel to cz , and e3 about an axis parallel to ca. These angles are then related to eij (i,j = 1,2,3) as follows :
cl, c2
sin81=
J(e{l;e;3).COS@l = J(e;:;;3)
(67)
-1
sin8, =
cos0,
(e13e3d-e23e32e33),
=
e33
(68)
e22-J(ei3+43)
To determine the motion of the body-fixed unit vectors bl, b, and b3 relative to e, , e2 and e3, it is only necessary to observe that equations (18), (38) and (51) permit one to express 0, the angle between b, and c, , as 0 =
(70)
L($zX2fP3X)
and that three angles, say ‘pl, (p2 and (p3, analogous to f?,, e2 and 8,, but governing the orientation of bl, b, and b3 relative to el, e2 and e3, are given by 91
=
4,
402 =
825
93
=%,-I-&
(71)
The parameters P, and P3, introduced in equations (38) for purposes of analytical convenience, can be expressed in a physically more meaningful form after introducing R, and Sz, as *l p &;L:,)’
n3 A J(o;L:O).
(72)
Referring to equations (16), (23) and (72), one then obtains (73) Evidently, ifeither Q, = 1 (so that Q3 = 0) or 52r = 0 (so that S23 = I), then either P1 = Q, and P3 = 0 or PI = 0 and P3 = &, regardless of the value of L. This means that the initial angular velocity of C is equal to that of B if B initially has a pure tumbling or pure spinning motion. In summary, given the dimensionless parameters Q, , S& , L, z and x, one proceeds as follows to evaluate (pl, (p2 and (p3 : 1. 2. 3. 4. 5. 6.
compute P1 and P3 from equations (73); use equations (61H66) to evaluate k, , . . . , k6 ; find eij (tj = 1,2,3) from equations (52x60); evaluate Bi, 8, and e3 by reference to equations (67)(69); find 8 from equation (70) ; determine ql, p2 and pp3by using equations (71).
Motion of a symmetric rigid body subjected to a specialized body-fixed force
425
A measure of the utility of the procedure described above may be obtained by comparing values of 01, Oz and 8, found by means of this procedure with values resulting from a numerical integration of equations (33)-(35), subsequent evaluation of eij (i,j = 1,2,3) by reference to equation (41), and determination of 01, 13~and e3 by inversion of equations (67H69). Figures 2-5 each show 01, f12 and 0, as functions of x, the values corresponding to the digital computer solution of the exact equations being represented by solid curves, whereas the approximate solution is represented by crosses ; and each figure applies to a different combination of values of Q, , Q3, L and z. The accuracy of the approximate solutions is seen to be relatively insensitive to the numerical values of R, , R, (see Figs. 2-4) and L (see Figs. 3 and 4), but to decrease both
*
Exact solution I
x
Approximote
solution
-100
c
0
2.5
5-o
7.5
10-O x
12.5
15.0
17.5
20.0
FIG.2. Comparison of solutions for orientation angles. R, = 1.0, R3 = 0, z = 0.01.
x
x
Exact
a
solution
Approximate
solution
-100 0
2.5
50
7.5
10-O
12.5
15.0
17.5
20.0
x
FIG. 3. Comparison of solutions for orientation angles. Q, = 0.70711, f& = 0.70711, z = 0.01, L = 9.0.
426
T. R. KANE and B. S. CHIA
0
2.5
50
75
10.0 Y
12.5
15.0
17.5
20.0
FIG. 4. Comparison of solutions for orientation angles. R, = O-70711, Q3 = 0.070711, z = 0.01, L = 1.0.
when z increases (cf. Fig. 3 with Fig. 5) and when x increases future reference that for z = 0.01 (see Figs. 24) the exact and (i = I, 2,3) are nearly indistinguishable from each other so than 15.00 and that similarly good agreement obtains for z =
TRANSLATIONAL
(see Figs. 2-5). We note for the approximate value of oi long as x remains smaller 0.05 (see Fig. 5) for x < 2.5.
MOTION
To study the translational motion of B, that is, the motion of the mass center B* of B, we recall the definitions of m,X and F; and, after defining Xi (i = 1,2,3) andA (i = 1,2) as Xi 4 X.ei
(74)
A B Fi/m
(75)
express Newton’s Second Law of Motion in the form 8, = [err(fr cos O-fi sin O)+e,,(fI
sin O+fz cos 0)]
(76)
X, = [ezI(fi cos 0-f*
sin f3)+e,,(fi
sin
e+f2 cos e)j
(77)
ifs = [e3,(fi cos S-f2
sin 8)+e,,(fl
sin e+f,
cos @I.
(78)
Equations (76)-(78) are coupled to the rotational motion of B by the quantities 8 and eij (i, j = 1,2,3). If these are eliminated by using equations (70) and (52x60) and if new dimensionless quantities Di, 6, i and pi are defined as (i = 1,2,3)
f;:
d
J(2Jlm)
(i = 1,2,3)
(79)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
0
7.5
-
Exact soluticm
x
X * App~ximote solution
KI.0
12.5
15.0
17.5
427
20.0
X
FIG. 5. Comparison of solutions for orientation angles. a, = O-70711, as = 0.70711, z = 0S05,L = 9.0.
4
4% +Lz
p1 A &LP,
& p (LP,+z)/2 83 k (LP,--2)/Z j.Q A (LP,- q/2 /?s h (LP,+1)/2 & A (LPs-z- 1112 87 A (LP,-z-U)/:! & k (LP,+z--1)/2 &I e (LP,+z+1)/3 then, when primes are used to denote differentiation found to be governed by
with respect to x, Di (i = 1,2,3) are
p3
0’; w FIPI
(4P$ -P$ cos(W + 2/31x)--Z_[sin(&2 -I-2/$x)-
sin@* 4 2/&x)
i + cos(W
+ 2&x)
+ cos(~x2 +2&x)]
f P3cos(rx2 + 2#&x)
+ $(l - 4Pi?,-f-PJ [cos(Tx2 + 2&x) -t cos(l;x2 + 2&x>] I -&PI
- P3) sin@* + 2B,x) +[cos(~x2
+ 2/&x) - cos(ex2 f Z&X)
428
T. R. KANE and B. S. CHIA + sWx2 + $(I -
+2&x)
+
sin(ix’ f 2&c)] + P3 sin([x2
-t-2p5x)
4P5+ P3)[sin(ix* + 2B2x) + sin([x* + 2p3x)]
(91)
i D’i X FI +(P, - 4PfP,)[sin([x* i
- cos(jx2 +
+ 2&x) - sin(@ + 2/3,x) + sin(cx’ +
-sWx2+2&x)lj
- 6
I
+2/&x)
28,x) - cos(jx2 + 2&x) + cos(
+ PfPJsin(ix’
-i[cos(ix2
+2/3,x) - sin([x2 +2/3*x)] +:[cos(cx’
2&x) i
+s;P:,--‘rt[c0s(lx2+2~~~)+coe(jx’+2~~x~~ + 2&x)
+ costixz +2&x)
+ cos(~x2 +2/53x) + cos(gx2 + 2/.&x)]
%P, - 4P_fP3)[cos(ix2 + 2/34x)- cos(~xZ + 2&x)]
+ 2 [sin(Cx2 + 2&x) - sin(5x2 + 2&x) - sin(ix2 + 2/&x) +
sin(ix* +2&x) + P~P3[cos(~x2 + 2/3,x) -
+ cos([x2
+
+
COS(~$~ + 2p,~)
2&$) - cos({x2 + 2&x)] 1 - 9 2 P2z 3 -
sin(ix* + 2p5x)] -$[sin(ix2
‘{~bin(~x2+2~4x)
+2&c) + sin(
+ sinKx2 + 2&x) + sin(ix2 + 2&x)] + .FI +Jsin(cx2 + 2j14x) I i fsin(ix*
+~~,x)]-~z-~[cos(~x~
+Zg,x)-COS([X*
+2p5x)3
+~z-1~~os(~x~+2~~x)-cos(~x*+2~~x)+cos{~x*+2~~x) -cos(~x2 +2&x)]
1
+ 35 f[cos(yx2 + 2B4x) +cosgx* +2/35x)] i
-$z-‘[sin(ixZ+2/l,x)-sin([x’
+2~~x)J+~~-‘[sin(~x*
-sin(ix* + 2&x) + sin(jx* + 2p,x) - sin([x2 +2&x)] D”3 * &
1
$-
I
1 +4P:)[cos(~x*+2p4X)+COS(5X*+2&X)]
-~[sin(jx2+2&x)+sin(bc2
-I-2&x)-sin(&x2 +2/3,x)
- sWx* f 2/&x) - PiP3[cos(~x2 + 2/3,x) + cos(~x2
+
2&x)
+2&x) z#O
(92)
Motion of a symmetric rigid body subjected to a specialized body-fixed force + cos(@ -‘
sin(ilx2
+ +
2/3,x) + cos(5x2 -I-2&x)] 1 + 9 I P2z 3 -
’ {%sWx’
429
+2&x)
2f14x)]- $[sin(cx’ +2&x) - sin(&? +2/&x) + sin(cx2 + 2/&x) - sz
- sWx2 +2&x)] I
i
?( - 1 -t-4Pf) [sin(W +2&x) + sin([x* +2&x)]
-~[c0s(5x~+2j!I&+c0s(~x~+2~,x)-c0s(~x~+2~.& -c&ix2
+2/&x)] - P:Ps[sin@Y + Z&x) + sin(cx’ + Z&x) - SzP$z- l {$[cos([x2 + 2f14x)
+ sin(5x’ + 2&x) +sin(5x2 +2&x)] I -cos(gx2 + 2/I&x)]-$[cos(ix2
+ 2/$3x)- cos([x2 + 2p,x)
-I-C0S(jX2 + 2fi6x) -cos(@
$.z- ‘[sin([x2 f 2fi4x)
+ sin(ix2 + Z/Q)] +~[cos(~x2 +2&x) -cos([x2 - ?z-
+2/Q)]
‘[sin([x2 f 2&x) + sin([x’ +2&x) + sin@? +2&x)
+ sin([x2 f 2/&x)] + Fz FzI
+f[sin(5x2 +2&x)-sir@?
l [cos(Cx*
f
2&x) f c0s(l;x2 + 2pSx)]
i
+2p4x)] -?z-
‘[cos(
+2&x)
+ cos(Yx2 +2&x) + cos([x2 f 2&x) + cos(gx2 + 2/3,x)] 1 Furthermore,
2 # 0.
(93)
Di(X) (i = 1,2,3) must satisfy the initial conditions Di(0)= 0,
D;(o) = f$,
(i = 1,2,3).
(94)
[The reason for taking Oi(O) (i = 1,2,3) equal to zero is that no loss of generality results from regarding B* as initially coincident with point 0 (see Fig. l), since the choice of point 0 is arbitrary.] The solution of equations (91)--(93) can be expressed in terms of certain functions related to Fresnel integrals. The functions to be employed are defined as follows :
(95)
(96)
T. R. KANE and B. S. CHM
430
9
C(a,b,x)
[~~(~)(uX+b)]{coS(~)C*[~(ux+b)i]
+sin(~)S,[$ax+b)‘]}--ksin(ax’+?bx)
a > 0
(97)
a > 0
(98)
S(a,b,x) A [;J(;)‘aX+b)]{cos(5JS2[$2X+b,2] -sin(~)C2[~(ax+b)‘]}+&cos(ax2+2hn) C,(a, b, x)
e J(%)
S,(a, b, x) p J(E) x C*(a, b, x)
A
”
ss0 0
{cos(~)C2[&zx+b)2]+sin(~)S2[$ax+b)2]~
a > 0
(99)
{cos(~)S2[~(ax+b)2]
a > 0
(100)
-sin(‘)
cos(au2 + 2bu) du du
= C(a, b, x) - C,(a, b, 0)x - C(u, b, 0)
a>0
(101)
a > 0.
(102)
ss x
S*(a, b, x) p
C2[i(ax+b)2]j
”
sin(au2 + 2bu) du do
0 0
= S(a, b, x) - S,(u, b, 0)x - S(a, b, 0)
Two successive integrations of equations (91H93) and use of equations (94) now lead to the following expression for D, , D, and D, : D, = &f’~
+
(4P:--p,)C*K,P,,x)-
c*(L
+ s*ti,
89
7
x)1+p&*(I, B53x) - 3u-4p: + P3) [C*(L 83,x)
P9, xl + pJ*(I,
+v1:ox
~[~~(~,B~.x)--S*(I,PJ,X)~C*(~,L,,X)
P5 7 xl ++u
-4p:
+ P3) P*(i,
+ c*K-,
P2 9 x) + s*tr,
82,
83 9
xl
41
[>O
D 2 = 9% t(P3-4P:P3)[S*(i,PJ,X)--*(5,84,41
(103)
+~[C*(i,B*,x)-C*(~,Bg, 4
- c*c, P6,XI+ c*K, 87,x)1+ p~p,[s*(i, 8794 - s*a 86,X)+ s*Ki B934
+ c*ti, 87, XI- c*(r, Ps7x)+ c*K B9941
w3 - 4p:p,)[C*(L P4, XI
Motion of a symmetric rigid body subjected to a specialized body-fixed force
431
432
T. R. KANE and B. S. CHIA
which permits one to express the solution to equationi (91~(93) satisfying equations (94) as follows :
--Y~P”,z- l
-~[s*(i,84tX)+S*(%,85,X)l+ars*cr,P,,X)+S*(r,Bs,X)
+s*(l,87tx)+s*(r,86,x)1
I
+4
Ii
-~~s*ce,8~,X~+S*~i,B5rX11-~z-*~C*tbr84rX~
-c*~~,P,.~~l+~~-‘~c*~~,8.,~~-c*~i’,~~,x~+c*~i.P~,~,-c*ciB,,x~lj +% f~r~~,84,X)fC*(ir85,Xll+~z-1rS*~~,B5,X~-S*(lr84rX~l i -~z-1[s*~~,~,,~~-s*i;.~~,~~~s*r~,B~,x)-s*~e.B,,x~l~ +v*,x
r-co
(1W
D3 = eFl ~~-1+4P:~rc*~i,8,,x~+c*~~,P,,x~l+~~s*~~,8,,x~+s*~i^,B,,x~l i - s*c, 87 9XI- s*G 86 >$I- mw*(L
86 3XIf c*G 87, XI+ c*cb, A37XI
Motion of a symmetric rigid body subjected to a specialized body-fixed force
+
mw*(B, &?f x) + s*K,Bs, xl + s*
-szpsz -“kw*(%,
84,-4 -
+c*(5,D6,x)--*(%,87,~)1
433
I
c*tc, 85,x)1-arc*cr, B*,x1- c*(L 89,x) I
+&
i
-~z-‘[s*~b,Bl,x)+s*(T,B~,n)l
+f[c*(i,84,X)--C*(I,85,X)1+~z-1rs*c~,P,,n,+s*(i,Bs.x)+s*(T,87,X) +~*uL~)l
+v,,x
I
+e
~z-‘[c*(i,BJ,x)+c*(!,85,x)l-t[s*(i,B5,1)
1
5 < 0.
W@t
Finally, for c = 0, the solution to equations (91)-(93) that satisfies the initial conditions in equation (94) is
- c*G%3x) + W.&f XI+ s*w9 3a + p$*&,
XI+ $0 - 4G + PA @*(A 3-4 (110)
D2 - c*vb >4 +” c*w7 3$I+ f9w*(B7
3XI- s*(AT,XI+ s*@g 2XI- s*u&, x)1
!dc*vL x) + c*w5 7xl1- $[c*&i, x) + c*uh , XI+ C*u&l,x)
- s*& 9XI- s*uL XI+ s*u&3t XII+
mw*aj ,XI-
c*fB,, XI+ c*tAj, XI
T. R. KANE and
434
B. S. CHIA
--c*(ir,~x)!) -WW’ ~~iSI(~~,x~+S*~Pr.x~l-~~s~~~
I 1 +6
+S*(B7,x)+S*(&,x)l
:[s*(~,,r)+s*(Bs,x)l-~z-1[C*(a4,X)
i
-c*(P,,*)1+~z-'~c*(P*,x)-c*(II9,x)-cc*(~~,~~-~*~~~,x~1
+@? it~C*(Bd,X)fc*(PS,x)]-~z-1[s*(~,,X)-S*(~~rX)] I
-~*(P,,x)+~*(P,,x)--*(~s,~,l
+&ox
z#O
(fll)
D, = .Fl ~(-l+4P:)ic*~~~,x)+c*~~5,x)]-~ts*(~~,x)+s*(~~,x)-s*(~~,x) i -
s*uL
3
PlPp,Z 2 -i
x)1-
Q~s*cBs ?4 - s*(84,x)l- tD*(p,, xl - s*ub 4 + s*(& -4 3
i
-s*~Ps,x)l
+
I
mw*(B6 >4 + CYB,, 4 + c*(P,, 4 + c*@-J,41
7
I-e i
~(-l+4~f)~S*(B,,xf+s*(P,,x)l-~~C*(B,,x)
c*@7 4 - c*(P*,x) - c*uL XII- ww*uL 7
+s*u?6,x)l} -*F2P:z-l { w*uL,
-4 + s*&. ,x1+ s*(fb, 4
xf- c*w,,41 -w*(P*? xl-
c*uh 4
+95 ~~-l~c*(alrx,+c*~~,,X)!+i~S*~~S,X)-S*(P~,X)i i -~z-1~c*(B~,x)+c*~B7,x)+c*(P8,x)+c*(po,x)1
i
+vj,x
z # 0.
(112)
where the functions C*(pi, x) and S*(8,, x) are defined as C*(p,, X) ’ j: l: COS(2piU) du do
= ~[l z
-cOS(2PiX)]
(i =
1,. . . ,9)
(113)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
435
and S*(8i, X)
r J”
+&
sin(2fiiu) du dv
0 0
= --J-sin(2/?ix)+z
4Piz
(i
=
1,. . . ) 9).
(114)
2Bi
In summary, given the dimensionless parameters SZ!r, R,, I,, z, x, PI, S2 and I&, (i = 1,2,3), one may proceed as follows to evaluate D,, D, and D, : 1. determine P1 and P3 by reference to equations (73) ; 2. use equations (81~90~ to evaluate I and /Ii (i = 1,. . . ,9); 3. if 5 > 0, evaluate D, , D, and D, by reference to equations (103~105) together with the definitions in equations (101) and (102); 4. if [ > 0, form [ and pi (i = I,. . . , 9) by using equations (106), then find D,, D, and D3 from equations (107~(109), using the definitions in equations (101) and (102); 5. if i = 0, determine D,, D, and D, by using equations (110)-(112) together with the definition in equations (113) and (114). Once again we turn to the computer to obtain a measure of the accuracy of the solution just obtained. In choosing parameter values for this purpose it is well to keep in mind that the approximate solution was generated by using approximate expressions for eij (i,j = 1,2,3), which suggests that the magnitudes of z and x bear directly on the accuracy of I), , D, and D, ;that is, one must expect the discrepancies between the approximate and the exact solution to grow as z and x increase. Figures 6-10 each show D, , I),and D3 plotted as a function of x. Numerical solutions of the exact equations of motion are represented by solid curves, while values obtained by using the approximate solution are represented by crosses. In all cases, B* is presumed to be at rest initially. Cursory examination of Figs. 6-10 leads to the immediate conclusion that the approximate solution furnishes qualitatively correct results in all cases and that, as expected, quantitative discrepancies increase as x increases. The effect of increasing z can be assessed by comparing Fig. 7 with Fig. 9. As for the effect of other parameters on accuracy, this appears to be negligible; that is, for a given value of z, the accuracy of the approximate solution is independent of the values of St 1, Qt,, c&, F2 and L. Figures 6-10 also permit one to explore the effect of parameter values on the nature of the translational motion. Consider, for example, the parameter L,which characterizes the inertia ellipsoid of the body. Figures 6 and 10 suggest that the mass centers of bodies possessing differently shaped inertia ellipsoids must be expected to follow markedly different trajectories. The force level, on the other hand, plays a relatively minor role as regards the shape of the trajectory, as may be seen by reference to Figs. 7 and 8, which show that, although the values of Pi and FS used to generate Fig. 8 are twice as large as those for Fig. 7, the trajectories followed by the mass center are almost identical. This is not to say, however, that the force level is totally irrelevant to the motion. On the contrary, as shown by equations (103)--(105), (107H109) and (110)-(112), D,,D2 and D, are directly proportional to the magnitude of the applied force F, since P1 and & are proportional to this quantity ; and this proportionality is, in fact, the reason underlying the similarity of Figs. 7 and 8.
T. R. KANE and
436
IO
x
FIG. 6. Comparison
FIG. 7. Comparison
Exact
-
x
x
B. S. CHIA
solution
Approximate
solution
of solutions for position coordinates of mass center. RI = 1.0, R, = 0, z = 0.01, L = 9.0, FI = 0.2, 4 = 0.2, & = 0 (i = 1,2,3).
of solutions for position coordinates of mass center. R, = 0.70711, a3 = 0.70711, z = 0.01, L = 9.0, 9, = 0.2, Fz = 0.2, &:., = 0 (i = 1,2,3).
Motion
of a symmetric
rigid body subjected
to a specialized
body-fixed
force
241 22..
-
20..
I
Exact solution * x Approximate
solution
16.. 16.. 14.. 12,. 10.. 8.. 6.. 4.. 2..
-6 .. -6, 0
FIG. 8. Comparison
.
.
.
.
. 5
.
.
.
. IO x
.
.
.
. 15
...+ 20
of solutions for position coordinates of mass center. R, = 0.70711, R3 = 0.70711, z = 0.01, L = 9.0, .FI = 0.4, Sz = 0.4, I& = 0 (i = 1,2,3).
6 Exact
-
solution
5 x
x
): Approxtmote
solution
4
FIG. 9. Comparison
of solutions for position coordinates of mass center. R, = 0.70711, RS = 0.70711, z = 0.05, L = 9.0, SI = 0.2, .Fz = 0.2, I& = 0 (i = 1,2,3).
431
438
T. R. KANE
and B. S.
CHIA
40
35
-
30
x
Exact salutiofl x
x Approximate
solutiin
25..
FIO. 10.
Comparison
of
solutions for position coordinates of mass center. Q, = l*O,Q, = 0, z = 0.01, L = -0.25, FJ f 0.2, .Ez = 0.2, 6, = 0 (i = I, 2,3).
In conclusion, it can be said that the approximate expressions for Di (i = 1,2,3) describe all essential features of the motion of the mass center and that the agreement between the approximate and exact solution is such as to justify considerabte confidence in the former. Furthermore, it may be worth pointjng out that this paper applies to a somewhat broader class of problems than that covered by the title. ~pe~~~a~~y, the approximate solutions derived here produce good results for certain gyrostats, as will be shown in a subsequent paper.
REFERENCES LEIMANIS, The General Problem of the Motion of Coupled Rigid Bodies About a Fixed Point. SpringerVerlag (196sj. [Z] R. LIZ The motion of a self-excited asymmetric rigid body. Arc&.+ration. Mech. Analysis 22, 129-142 (1966). [33 R. GRAMMEL, Der Selbsterregte Unsymrnetrische Kreisel. J. appf. Mu&. Me& 18, 73-97 (1954). [4] U. I’. I%DEWADT~ Der Symmet~~he Kreisel bei Zeitfester Drehkraft. Mark 2. 55, 310-320 (1952).
[I] E.
/5] R. R, A~EL~~~, On the Motion of a Symm~t~~I Rigid Body with Constant Body-Fixed Torques, Private Commun~cafion. [6J K. G. VALEEY, OR the probIem of the motion of an axially symmetrical body under the action of a constant moment. J. appl. M&h. Mech. 29, 162-165 (1965). [7’j B. S. CHIA and T. R. KANE, Motion of a Symmetric Rigid Body Under the Action of a Body-Fixed Force, Technical Report No. 203, Department of Applied Mechanics, Stanford University (1971). (Received 8 September 1971)
A~c.T~wT---~IT~~J~TM~Tc~
QlOpbiynbJ ToYHbJx AutJaMwYecxnx li XmJeMaTfiYecxMx ypaBHeHuR. yYnTbJBalOllIUX ABtrxeHJGJ J-JOJJO’HteHUJJ N TpaHCJJSJJJfJf,CWMMtTTfWECKO~O XeCTKOrO YWJa, JJ0.U BRHUHReM CtJeUHaNbJJO8, CBK3aWJofi C TWO+.%,CHIlbI. &%33TBX ypaBKeHJi8, JJOJlyYaJoTcXaHaWWJeCK&, HO npr16mmceHHb& 3armcb rJo=aeHnsJ CMCTeMbJ. Cpasnweae-rcs, 3a~eM, BamHocTb peiiIeHnff nyreh4 cpasuefm era npeacrra3a~& c T~K~MW wre, a~T~~a~~~M~ 3~3 peuJeH%n Ha q3M TOSHOTO ypaBJJeHBR ABW~JWJ.
J. Sofkfs .Structwes,
1972, Vol. 8, pp. 415 to 438.
MOTION SUBJECTED
Pergamon Press. Printed in Great Britain
OF A SYMMETRIC
TO A SPECIALIZED T. R.
KANE-~
RIGID
BODY
BODY-FIXED
FORCE
and B. S. CHLQ
Department of Applied Mechanics, Stanford University, Stanford, California
Abstract-Exact dyflamical and kinematical equations governing the attitude and translational motions of a symmetric rigid body under the action of a specialized body-fixed force are formulated. From these equations, an analytical, but approximate, description of the behavior of the system is obtained and the validity of the solution is then tested by comparing its predictions with those of digital computer solutions of the exact equations of motion.
INTRODUCTION THE problem of determining the motion of a rigid body under the action of a constant, body-fixed force whose line of action fails to pass through the center of mass is so simple from a physical point of view that it may be surprising that no complete, analytical solution appears to be possible. In fact, not even the Euler dynamical equations for this problem have been solved in closed form for the general case, that is, for a body with unequal centroidal principal moments of inertia and for an arbitrary o~en~tion of the line of action of the force. (Leimanis and Lee [l, 23 developed expressions for the angular velocity for the special case of a force placed so that its moment about the center of mass is parallel to a centroidal principal axis, but these expressions involve integrals which, in general, cannot be evaluated by quadrature ; and Grammel [3] constructed approximate solutions of Euler’s dynamical equations.) Hence it is natural to seek solutions of related, but somewhat simpler problems. A major step in the direction of decreased analytical complexity can be taken by confining attention to a body whose inertia ellipsoid for the mass center is axisymmetric. One of the first to deal with this problem was Biidewadt [4], who showed that the solution of Euler’s dynamical equations can be expressed in terms of integrals of the Fresnel type.5 Unfortunateiy,the next step of his solution, that of integrating a set ofkinematica1 equations, was based on a false premise, because, as may be verified by substitution from equation (49) into (42) of [4], the method proposed by BGdewadt is valid only when the matrices U and W’ commute, which is not the case for the problem at hand. Consequently, the orientation of the body remains to be determined as a function of time. The same problem was also attacked by Auelmann [Sj, who attempted to find a closed-form solution to the kinematical equations. While falling short of this goal, he succeeded in obtaining an expression for the angle between the symmetry axis and the initial direction of the symmetry axis for the special case when the initial angular velocity is normal to the symmetry axis. t Professor of Applied Mechanics. $ Research Assistant. 9 An English account of Biidewadt’s work appears in chapter 10 of Leimanis’ book [l]. 415
416
T. R. KANE and
B. S. CHIA
A further simplification results from restricting oneself to consideration of body-fixed forces whose lines of action lie in the plane containing the mass center and normal to the symmetry axis. A very elegant treatment of one facet of this problem was presented by Valeev [6], who used continued fractions to obtain a description of the motion of a spacefixed unit vector relative to three body-fixed unit vectors parallel to principal axes of inertia for the mass center. It is the purpose of the present paper to present a complete, albeit approximate, solution of the same problem, for both the rotational and translational motion. The paper begins with a detailed description of the system to be studied and the mathematical quantities to be used in the analysis are defined. Next, exact dynamical and kinematical equations governing attitude motions are formulated ; the dynamical equations are solved exactly ; and an approximate solution to the kinematical equations is obtained. The validity ofthe approximate solution is then tested by comparison with digital computer integrations of the exact equations. Finally, the motion of the mass center is studied in terms of an approximate analytical solution and results are once again tested by means of comparisons with solutions obtained by numerical integrations.
SYSTEM
DESCRIPTION
The system to be analyzed is shown in Fig. 1, where B represents an axially symmetric rigid body of mass m. The center of mass of B, designated B*, is located by a position vector X relative to a point 0 that is fixed in an inertial reference frame A. Mutually perpendicular unit vectors a,, a2 and a3 = a, x a2 are fixed in A and mutually perpendicular unit vectors bl , b, and b3 = b, x b, are fixed in B parallel to principal axes of inertia of B for B*, with b3 parallel to the symmetry axis of B. The moments of inertia of B with respect to lines passing through B* and parallel to bi , b2 and b3 have the values I, I and J, respectively. f
A
h
FIG. 1. Model of an axially symmetric
rigid body.
Motion of a symmetric rigid body subjected to a specialized body-fixed force
417
It is presumed that B is subjected to the action of a force F of constant magnitude and fixed direction in B. This force is applied to B at a fixed point Q which is located relative to B* by a vector q. Furthermore, it is assumed that both F and q are perpendicular to b,. Hence, if the nine scalar quantities xi, Fi and qi(i = 1,2, 3) are defined as xi P X .ai
(1)
Fi 4 F . bi
(i = 1,2,3)
(2)
qi ’ Q. bi
(3)
then F3 = q3 = 0. When the force F exerted on B at Q is replaced with a force applied at B* together with a couple of torque T, then T=qxF=
7h,
(4)
where T is defined as T s qlF2-q2F1.
ATTITUDE
MOTION
The work that follows is facilitated by introducing a neither in the body B nor in the inertial reference frame such a way that a unit vector c3, fixed in C, remains at all which is fixed in B. The angular velocity of B relative to necessarily parallel to c3 and can be expressed as
reference frame C which is fixed A, but is constrained to move in times equal to the unit vector b,, C, to be denoted by %I?, is then
ceP = SC3
(6)
where s is a function of time t. As will be seen presently, it is the choice of s which furnishes the key to the solution of the problem at hand. However, no matter how s is chosen, the angular velocity of C relative to A, AceC,can be expressed as A
”
C-
-
Plcl
(7)
+P2cZ+P3c3
where pi(i = 1,2,3) are functions oft, and c 1 and c2 are unit vectors fixed in C, perpendicular to each other and such that c1 x c2 = c3. It then follows that the angular velocity of B relative to A is given by AeP =
plc,+p2c2+(p3+s)c3.
(8)
As b3 = c3, it follows from equation (4) that the moment about B* of the force F can be expressed as T = Tc,
(9)
and, in accordance with the angular momentum principle, the quantities pi, p2, p3 and s are thus related to each other as follows : rp1 +p,[Js-(I-J)p,]
zp2 -
p1 [Js
-
(I
-
= 0
(10)
J)p,] = 0
(11)
Jo3+S)
= T.
(12)
418
T. R. KANE and B. S. CHIA
Equations (10)-(12) contain four unknown quantities, p, , p2, p3 and s. However, s was introduced solely for the purpose of facilitating the analysis and may, therefore, be chosen at will; and a choice which, indeed, simplifies the subsequent analysis is s = Lp,
(13)
I-J J
(14)
where L is defined as L
p
-.
Now, if pi0 denotes the initial value of pi(i = 1,2,3) and A is defined as A-‘;
(15)
thensubstitutionfromequation(l3)into(l0)-(12),followed yields Pl = PlO,
Furthermore,
by integration oftheseequations, P3 = At+P3o.
P2 = P20,
(16)
from equations (13) and (16), s = L@t + p30)
(17)
and the angular velocity of B in A can now be found by substitution into equation (8). However, as it is advantageous to express ‘aB in terms of b, , b2 and b, , rather than cl, q, and cg , a brief digression will prove useful. If c, is taken to be equal to bl at t = 0 and 8 is defined as 84
* s dt = L(@t2 fp30t) s0
(18)
then 8 is the radian measure of the angle between bl and c1 ; and if Oi(i = 1,2,3) is now defined as Wi “oB.bi
(i =
1,2,3)
(19)
it follows from equation (8) that w1 = p1 cos 8+p, w2 = -pl
sin 8
(20)
sin@+p,cos@
(21)
w3 = p3+s.
(22)
Hence, if oiO denotes the initial value of Oi(i = 1,2,3), then Pl = WlO,
s=L
and
i
P2 = w209
At+% l+L
I
pa = At+%
l+L
(23) (24)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
419
so that, substituting into equations (20H22), one finds o1 = or0 cos @+w,e sin = J(o:, w2
=
+ O&J cos e + tan [ +(-:)I
--~,,sin8+~,,cos8
=
= J(f$, 03
e
+ &)
sin e + tan [ -‘( -~)I
(27)
(1 +L)&+o,,.
(28)
Equations (25j(28) constitute a complete solution of the dynamical equations of rotational motion for the problem at hand. In order to describe the instantaneous orientation of B in A, one must integrate yet one more set of differential equations, the so-called kinematical equations. Conceptually, the simplest way to obtain a complete description of the attitude motion of B in A would be to solve the nine first-order differential equations governing the elements of the direction cosine matrix relating the body-fixed unit vectors b, , b,, b, to the spacefixed unit vectors a,, a*, a3. However, since these equations are strongly coupled and involve the time-dependent functions Wi(i = 1,2,3) [see equations (26)-(28)], there appears to be little hope of carrying such a solution to a successful conclusion. Alternatively, one may take advantage of the relative simplicity of the differential equations governing the elements of the direction cosine matrix relating c, , q, c3 to a 1, a2, a, and of the fact that, once the orientation of C in A is known, the orientation of B in A can be found easily, because B performs a simple rotational motion in C and the time-dependence of the angle 0 associated with this rotational motion is already known [see equation (251-J. Accordingly, one may begin by defining aij(i, j = 1,2,3) as aij A ai. cj
(i, j = 1,2,3)
(29)
and then take advantage of the fact that the first time-derivative of ai in A can be expressed as Adai = Cdai -~+ROCXtIi dt
(i =
1,2,3).
Since ai is fixed in A, this derivative is equal to zero. Using equation (7), one is thus led to ci,lCl+ (ii*C* +cii3C3 =
(P3%2
-p2@3)ct
+(plai3-_p3ail)c2+(p,ai,
-plai&3 (i
=
1,2,3).
(31)
The quantities pi(i = 1,2,3) appearing in equation (31) are known functions of t [see equations (2311; furthermore, pZ may be set equal to zero, for this implies only that @20
=
0
(32)
that is, that AeP is initially perpendicular to bz, which can always be arranged by choosing the orientations of b, and b2 in B suitably. Consequently, substitution from equations (16)
420
T. R. KANE and B. S. CHIA
into (31), and subsequent rearrangement,
results in
Liii-P~eUiz = ItUi, lif2_Pi@i3+P3oUil
=
-AlUi$_
(33) (i =
1,2,3)
(34)
cii3+Pioaiz = 0.
(35)
The initial value of ajj, to be denoted by Uij(O),depends on the initial orientation
of
ci , c2 and cg relative to a,, a2 and a3. If ci = ai at t = 0, which can always be arranged by
choosing aj(i = 1,2,3) properly, then [see equations (29)J aij(O) = 6ij
(i, j = I, 2,3)
(36)
where I&,p 1
(i = 8,
dij k 0
(i # j).
(37)
These initial conditions also imply that bl , b, and b3 are respectively paralfel to a,, a2 and a3 at t = 0, since cr , c2 and c3 initially coincide with bit b2 and b,, respectively. Despite their relative simplicity, equations (33)-(35) do not appear to admit of an exact solution. Hence we shall resort to the following method of successive approximations: integrate equations (33)-(35) with I = 0, and require the solution to satisfy equation (36). Next, substitute the resulting expressions for a,, and ui2 into the right-hand members of equations (33H35), integrate these equations and again require the solution to satisfy equation (36). Finally, repeat this process once more. Since the expressions for aij(i, j = 1,2,3) obtained in this way are quite lengthy, it is advantageous to cast the associated approximate description of the attitude motion of C in A in a simpler, equivalent form. This can be accomplished by defining pot Pi and P3 as PO 9 J(P:o + P30),
pi c ~lO~P0,
(38)
p3 4% P30/~0
and then introducing a dextrai set of orthogonal unit vectors, e, , e2 and e3, fixed in reference frame A in such a way that e, has the same directions as the initial angular velocity of C in A, e, = a2 and e3 = e, xe,. Keeping in mind that ci = ai at t = 0, one can then express the relation between ei and ai(i = 1,2,3) as
(39)
from which it follows by reference to equations (29) that
(40)
where ~1@11+~3a3, a21
i -P3a,,+Pla3,
~P%2+~3a32 a,, -P3a,2+Pla32
PA3
+P3a33
(41)
a23 -P3a13+Pla33
I
Motion of a symmetric rigid body subjected to a specialized body-fixed force
421
and
(43)
smpot+P, +P~P3(~)(~) . 2(2i)2(P~+P~SP,-3P~P~)(1-cospot)
e21
M P3 sin pot+PT
+zpzp3
-Pi
1 3 0 _& P$
f
$
sin pot - 2P:F,
z (1 0& At
-’
3
cosp,t-~P:
At 2
PO sin pot + f Pi -“, ( I 0PO 0 -;i Iz PO
1
(I-cosp,t)+~P:P,
i
I
i
-2
0 PO It
4
sin
e22
x
cos --
I
_z PO
j$
(
g
Ii
-I
a* ~0
sin pot I
0PO
cos
pot
2 It -z cospot i POI( PO1
*sinpOt-_P:P, I
(45)
sin pot-
1 A -= At 4 0 8pot p2“20- 5P3 00po cospot+P,
- ;P:
Rt =
pot
Pi Ii ;It PO1 +p*p1 3 ( A_
1
-I
W)
1
:Pf
1 -l
T
0
it3
I
0PO
2~~,~-~~~)~cos~~pot g
(44)
422
T. R. KANE and B. S. CHIA
(47)
e31 x
-P,cosp,t-Pi
(48) e32 x sinpot+-P,
1 2
1 -I
-i 0PO
1 At2 cospot+gPf 0PO
;1 -I At 3 cos pot 7 0PO 0PO
(49)
sin pot
(50) It is worth noting that, if pSo = 0, so that P, = 1 and P3 = 0 [see equations (38)], then eij is equal to aij(i, j = 1,2,3) because ei is then equal to a,(i = 1,2,3).
Motion of a symmetric rigid body subjected to a specialized body-fixed force
423
Equations (42)-(50) have been written in such a way that t is always multiplied by Aand divided by p. when it appears outside of the argument of a trigonometric function and Ais divided by pg when it is not multiplied by t. This representation of e&i, j = 1,2,3) permits one to carry out a systematic simp~i~cation of the equations and thus to arrive at approximate results which, as will be seen in the sequel, can furnish an entirely satisfactory description of the motion under consideration. Specifically, dropping from equations (42)-(50) all terms involving (A/&” with n > 1 and all terms involving (It/pJ” with m 2 3, replacing At/p0 and (&‘po)z in equations (59H67) with sin(~~/~~) and 2[1 -cos(&,/po)], respectively, and introducing new dimensionless p~ameters x and z as (51) one arrives at e,, % k~-~~~~(l-cosz~)(~-cosx)
(52)
cl2 R -PIP&--coszx)sinx
(53)
e13 3 k, +P:P,(l
(54)
-cos zx)(l -cos x)
ezf w k,sinx-t-k,cosx
@22 --
= ~~k~+k~)cos
-k5 sinx+kscosx
= ,/(kffkz
ez3 RJ -k,sinx-k,cosx
e31
x
[
xi-tan
-‘( -~)I
cos
= ~(k~~k~~cos
(56)
[
x-!-tan
-‘(-~)]
= q’(kg-bki) sin[x+tan-‘(
-k,cosx$k,+sinx
-~)I
e32 x k, cos x + k, sin x = J(k: c kz) sin
(57) (58) (59)
e33 z k, cos x - kz sin x = ,/(kf -t-k$ sin x + tan
[
where kl,...,
(55)
-‘( -:)]
k6 are defined as k, 4 PI --PIP, sin 2x--P,(l-4F$)(l k2 g F&z-‘(1
-cos
-cos zx)
2x)
(61) (62)
k, k P3 + Pi sin 2x - 4PfP3( 1 - cos 2x)
(63)
k4 A Pp-“(1 -cos 2x)
(64)
k5 A+P3z-l(1 -cos
(65)
k, k 1.
zx)
(66)
Now that approximate expressions for the direction cosines eij (i,j = I, 2, 3) are at hand, approximate expressions for angles describing the orientation of the unit vectors
424
T. R. KANE
and B. S. CHIA
and c3 relative to e, , e, and e3 can be constructed. One set of such angles can be generated by aligning ci with ei (i = 1,2,3) and then performing successive right-handed rotations of C of amount @I about an axis parallel to c3, O2 about an axis parallel to cz , and e3 about an axis parallel to ca. These angles are then related to eij (i,j = 1,2,3) as follows :
cl, c2
sin81=
J(e{l;e;3).COS@l = J(e;:;;3)
(67)
-1
sin8, =
cos0,
(e13e3d-e23e32e33),
=
e33
(68)
e22-J(ei3+43)
To determine the motion of the body-fixed unit vectors bl, b, and b3 relative to e, , e2 and e3, it is only necessary to observe that equations (18), (38) and (51) permit one to express 0, the angle between b, and c, , as 0 =
(70)
L($zX2fP3X)
and that three angles, say ‘pl, (p2 and (p3, analogous to f?,, e2 and 8,, but governing the orientation of bl, b, and b3 relative to el, e2 and e3, are given by 91
=
4,
402 =
825
93
=%,-I-&
(71)
The parameters P, and P3, introduced in equations (38) for purposes of analytical convenience, can be expressed in a physically more meaningful form after introducing R, and Sz, as *l p &;L:,)’
n3 A J(o;L:O).
(72)
Referring to equations (16), (23) and (72), one then obtains (73) Evidently, ifeither Q, = 1 (so that Q3 = 0) or 52r = 0 (so that S23 = I), then either P1 = Q, and P3 = 0 or PI = 0 and P3 = &, regardless of the value of L. This means that the initial angular velocity of C is equal to that of B if B initially has a pure tumbling or pure spinning motion. In summary, given the dimensionless parameters Q, , S& , L, z and x, one proceeds as follows to evaluate (pl, (p2 and (p3 : 1. 2. 3. 4. 5. 6.
compute P1 and P3 from equations (73); use equations (61H66) to evaluate k, , . . . , k6 ; find eij (tj = 1,2,3) from equations (52x60); evaluate Bi, 8, and e3 by reference to equations (67)(69); find 8 from equation (70) ; determine ql, p2 and pp3by using equations (71).
Motion of a symmetric rigid body subjected to a specialized body-fixed force
425
A measure of the utility of the procedure described above may be obtained by comparing values of 01, Oz and 8, found by means of this procedure with values resulting from a numerical integration of equations (33)-(35), subsequent evaluation of eij (i,j = 1,2,3) by reference to equation (41), and determination of 01, 13~and e3 by inversion of equations (67H69). Figures 2-5 each show 01, f12 and 0, as functions of x, the values corresponding to the digital computer solution of the exact equations being represented by solid curves, whereas the approximate solution is represented by crosses ; and each figure applies to a different combination of values of Q, , Q3, L and z. The accuracy of the approximate solutions is seen to be relatively insensitive to the numerical values of R, , R, (see Figs. 2-4) and L (see Figs. 3 and 4), but to decrease both
*
Exact solution I
x
Approximote
solution
-100
c
0
2.5
5-o
7.5
10-O x
12.5
15.0
17.5
20.0
FIG.2. Comparison of solutions for orientation angles. R, = 1.0, R3 = 0, z = 0.01.
x
x
Exact
a
solution
Approximate
solution
-100 0
2.5
50
7.5
10-O
12.5
15.0
17.5
20.0
x
FIG. 3. Comparison of solutions for orientation angles. Q, = 0.70711, f& = 0.70711, z = 0.01, L = 9.0.
426
T. R. KANE and B. S. CHIA
0
2.5
50
75
10.0 Y
12.5
15.0
17.5
20.0
FIG. 4. Comparison of solutions for orientation angles. R, = O-70711, Q3 = 0.070711, z = 0.01, L = 1.0.
when z increases (cf. Fig. 3 with Fig. 5) and when x increases future reference that for z = 0.01 (see Figs. 24) the exact and (i = I, 2,3) are nearly indistinguishable from each other so than 15.00 and that similarly good agreement obtains for z =
TRANSLATIONAL
(see Figs. 2-5). We note for the approximate value of oi long as x remains smaller 0.05 (see Fig. 5) for x < 2.5.
MOTION
To study the translational motion of B, that is, the motion of the mass center B* of B, we recall the definitions of m,X and F; and, after defining Xi (i = 1,2,3) andA (i = 1,2) as Xi 4 X.ei
(74)
A B Fi/m
(75)
express Newton’s Second Law of Motion in the form 8, = [err(fr cos O-fi sin O)+e,,(fI
sin O+fz cos 0)]
(76)
X, = [ezI(fi cos 0-f*
sin f3)+e,,(fi
sin
e+f2 cos e)j
(77)
ifs = [e3,(fi cos S-f2
sin 8)+e,,(fl
sin e+f,
cos @I.
(78)
Equations (76)-(78) are coupled to the rotational motion of B by the quantities 8 and eij (i, j = 1,2,3). If these are eliminated by using equations (70) and (52x60) and if new dimensionless quantities Di, 6, i and pi are defined as (i = 1,2,3)
f;:
d
J(2Jlm)
(i = 1,2,3)
(79)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
0
7.5
-
Exact soluticm
x
X * App~ximote solution
KI.0
12.5
15.0
17.5
427
20.0
X
FIG. 5. Comparison of solutions for orientation angles. a, = O-70711, as = 0.70711, z = 0S05,L = 9.0.
4
4% +Lz
p1 A &LP,
& p (LP,+z)/2 83 k (LP,--2)/Z j.Q A (LP,- q/2 /?s h (LP,+1)/2 & A (LPs-z- 1112 87 A (LP,-z-U)/:! & k (LP,+z--1)/2 &I e (LP,+z+1)/3 then, when primes are used to denote differentiation found to be governed by
with respect to x, Di (i = 1,2,3) are
p3
0’; w FIPI
(4P$ -P$ cos(W + 2/31x)--Z_[sin(&2 -I-2/$x)-
sin@* 4 2/&x)
i + cos(W
+ 2&x)
+ cos(~x2 +2&x)]
f P3cos(rx2 + 2#&x)
+ $(l - 4Pi?,-f-PJ [cos(Tx2 + 2&x) -t cos(l;x2 + 2&x>] I -&PI
- P3) sin@* + 2B,x) +[cos(~x2
+ 2/&x) - cos(ex2 f Z&X)
428
T. R. KANE and B. S. CHIA + sWx2 + $(I -
+2&x)
+
sin(ix’ f 2&c)] + P3 sin([x2
-t-2p5x)
4P5+ P3)[sin(ix* + 2B2x) + sin([x* + 2p3x)]
(91)
i D’i X FI +(P, - 4PfP,)[sin([x* i
- cos(jx2 +
+ 2&x) - sin(@ + 2/3,x) + sin(cx’ +
-sWx2+2&x)lj
- 6
I
+2/&x)
28,x) - cos(jx2 + 2&x) + cos(
+ PfPJsin(ix’
-i[cos(ix2
+2/3,x) - sin([x2 +2/3*x)] +:[cos(cx’
2&x) i
+s;P:,--‘rt[c0s(lx2+2~~~)+coe(jx’+2~~x~~ + 2&x)
+ costixz +2&x)
+ cos(~x2 +2/53x) + cos(gx2 + 2/.&x)]
%P, - 4P_fP3)[cos(ix2 + 2/34x)- cos(~xZ + 2&x)]
+ 2 [sin(Cx2 + 2&x) - sin(5x2 + 2&x) - sin(ix2 + 2/&x) +
sin(ix* +2&x) + P~P3[cos(~x2 + 2/3,x) -
+ cos([x2
+
+
COS(~$~ + 2p,~)
2&$) - cos({x2 + 2&x)] 1 - 9 2 P2z 3 -
sin(ix* + 2p5x)] -$[sin(ix2
‘{~bin(~x2+2~4x)
+2&c) + sin(
+ sinKx2 + 2&x) + sin(ix2 + 2&x)] + .FI +Jsin(cx2 + 2j14x) I i fsin(ix*
+~~,x)]-~z-~[cos(~x~
+Zg,x)-COS([X*
+2p5x)3
+~z-1~~os(~x~+2~~x)-cos(~x*+2~~x)+cos{~x*+2~~x) -cos(~x2 +2&x)]
1
+ 35 f[cos(yx2 + 2B4x) +cosgx* +2/35x)] i
-$z-‘[sin(ixZ+2/l,x)-sin([x’
+2~~x)J+~~-‘[sin(~x*
-sin(ix* + 2&x) + sin(jx* + 2p,x) - sin([x2 +2&x)] D”3 * &
1
$-
I
1 +4P:)[cos(~x*+2p4X)+COS(5X*+2&X)]
-~[sin(jx2+2&x)+sin(bc2
-I-2&x)-sin(&x2 +2/3,x)
- sWx* f 2/&x) - PiP3[cos(~x2 + 2/3,x) + cos(~x2
+
2&x)
+2&x) z#O
(92)
Motion of a symmetric rigid body subjected to a specialized body-fixed force + cos(@ -‘
sin(ilx2
+ +
2/3,x) + cos(5x2 -I-2&x)] 1 + 9 I P2z 3 -
’ {%sWx’
429
+2&x)
2f14x)]- $[sin(cx’ +2&x) - sin(&? +2/&x) + sin(cx2 + 2/&x) - sz
- sWx2 +2&x)] I
i
?( - 1 -t-4Pf) [sin(W +2&x) + sin([x* +2&x)]
-~[c0s(5x~+2j!I&+c0s(~x~+2~,x)-c0s(~x~+2~.& -c&ix2
+2/&x)] - P:Ps[sin@Y + Z&x) + sin(cx’ + Z&x) - SzP$z- l {$[cos([x2 + 2f14x)
+ sin(5x’ + 2&x) +sin(5x2 +2&x)] I -cos(gx2 + 2/I&x)]-$[cos(ix2
+ 2/$3x)- cos([x2 + 2p,x)
-I-C0S(jX2 + 2fi6x) -cos(@
$.z- ‘[sin([x2 f 2fi4x)
+ sin(ix2 + Z/Q)] +~[cos(~x2 +2&x) -cos([x2 - ?z-
+2/Q)]
‘[sin([x2 f 2&x) + sin([x’ +2&x) + sin@? +2&x)
+ sin([x2 f 2/&x)] + Fz FzI
+f[sin(5x2 +2&x)-sir@?
l [cos(Cx*
f
2&x) f c0s(l;x2 + 2pSx)]
i
+2p4x)] -?z-
‘[cos(
+2&x)
+ cos(Yx2 +2&x) + cos([x2 f 2&x) + cos(gx2 + 2/3,x)] 1 Furthermore,
2 # 0.
(93)
Di(X) (i = 1,2,3) must satisfy the initial conditions Di(0)= 0,
D;(o) = f$,
(i = 1,2,3).
(94)
[The reason for taking Oi(O) (i = 1,2,3) equal to zero is that no loss of generality results from regarding B* as initially coincident with point 0 (see Fig. l), since the choice of point 0 is arbitrary.] The solution of equations (91)--(93) can be expressed in terms of certain functions related to Fresnel integrals. The functions to be employed are defined as follows :
(95)
(96)
T. R. KANE and B. S. CHM
430
9
C(a,b,x)
[~~(~)(uX+b)]{coS(~)C*[~(ux+b)i]
+sin(~)S,[$ax+b)‘]}--ksin(ax’+?bx)
a > 0
(97)
a > 0
(98)
S(a,b,x) A [;J(;)‘aX+b)]{cos(5JS2[$2X+b,2] -sin(~)C2[~(ax+b)‘]}+&cos(ax2+2hn) C,(a, b, x)
e J(%)
S,(a, b, x) p J(E) x C*(a, b, x)
A
”
ss0 0
{cos(~)C2[&zx+b)2]+sin(~)S2[$ax+b)2]~
a > 0
(99)
{cos(~)S2[~(ax+b)2]
a > 0
(100)
-sin(‘)
cos(au2 + 2bu) du du
= C(a, b, x) - C,(a, b, 0)x - C(u, b, 0)
a>0
(101)
a > 0.
(102)
ss x
S*(a, b, x) p
C2[i(ax+b)2]j
”
sin(au2 + 2bu) du do
0 0
= S(a, b, x) - S,(u, b, 0)x - S(a, b, 0)
Two successive integrations of equations (91H93) and use of equations (94) now lead to the following expression for D, , D, and D, : D, = &f’~
+
(4P:--p,)C*K,P,,x)-
c*(L
+ s*ti,
89
7
x)1+p&*(I, B53x) - 3u-4p: + P3) [C*(L 83,x)
P9, xl + pJ*(I,
+v1:ox
~[~~(~,B~.x)--S*(I,PJ,X)~C*(~,L,,X)
P5 7 xl ++u
-4p:
+ P3) P*(i,
+ c*K-,
P2 9 x) + s*tr,
82,
83 9
xl
41
[>O
D 2 = 9% t(P3-4P:P3)[S*(i,PJ,X)--*(5,84,41
(103)
+~[C*(i,B*,x)-C*(~,Bg, 4
- c*c, P6,XI+ c*K, 87,x)1+ p~p,[s*(i, 8794 - s*a 86,X)+ s*Ki B934
+ c*ti, 87, XI- c*(r, Ps7x)+ c*K B9941
w3 - 4p:p,)[C*(L P4, XI
Motion of a symmetric rigid body subjected to a specialized body-fixed force
431
432
T. R. KANE and B. S. CHIA
which permits one to express the solution to equationi (91~(93) satisfying equations (94) as follows :
--Y~P”,z- l
-~[s*(i,84tX)+S*(%,85,X)l+ars*cr,P,,X)+S*(r,Bs,X)
+s*(l,87tx)+s*(r,86,x)1
I
+4
Ii
-~~s*ce,8~,X~+S*~i,B5rX11-~z-*~C*tbr84rX~
-c*~~,P,.~~l+~~-‘~c*~~,8.,~~-c*~i’,~~,x~+c*~i.P~,~,-c*ciB,,x~lj +% f~r~~,84,X)fC*(ir85,Xll+~z-1rS*~~,B5,X~-S*(lr84rX~l i -~z-1[s*~~,~,,~~-s*i;.~~,~~~s*r~,B~,x)-s*~e.B,,x~l~ +v*,x
r-co
(1W
D3 = eFl ~~-1+4P:~rc*~i,8,,x~+c*~~,P,,x~l+~~s*~~,8,,x~+s*~i^,B,,x~l i - s*c, 87 9XI- s*G 86 >$I- mw*(L
86 3XIf c*G 87, XI+ c*cb, A37XI
Motion of a symmetric rigid body subjected to a specialized body-fixed force
+
mw*(B, &?f x) + s*K,Bs, xl + s*
-szpsz -“kw*(%,
84,-4 -
+c*(5,D6,x)--*(%,87,~)1
433
I
c*tc, 85,x)1-arc*cr, B*,x1- c*(L 89,x) I
+&
i
-~z-‘[s*~b,Bl,x)+s*(T,B~,n)l
+f[c*(i,84,X)--C*(I,85,X)1+~z-1rs*c~,P,,n,+s*(i,Bs.x)+s*(T,87,X) +~*uL~)l
+v,,x
I
+e
~z-‘[c*(i,BJ,x)+c*(!,85,x)l-t[s*(i,B5,1)
1
5 < 0.
W@t
Finally, for c = 0, the solution to equations (91)-(93) that satisfies the initial conditions in equation (94) is
- c*G%3x) + W.&f XI+ s*w9 3a + p$*&,
XI+ $0 - 4G + PA @*(A 3-4 (110)
D2 - c*vb >4 +” c*w7 3$I+ f9w*(B7
3XI- s*(AT,XI+ s*@g 2XI- s*u&, x)1
!dc*vL x) + c*w5 7xl1- $[c*&i, x) + c*uh , XI+ C*u&l,x)
- s*& 9XI- s*uL XI+ s*u&3t XII+
mw*aj ,XI-
c*fB,, XI+ c*tAj, XI
T. R. KANE and
434
B. S. CHIA
--c*(ir,~x)!) -WW’ ~~iSI(~~,x~+S*~Pr.x~l-~~s~~~
I 1 +6
+S*(B7,x)+S*(&,x)l
:[s*(~,,r)+s*(Bs,x)l-~z-1[C*(a4,X)
i
-c*(P,,*)1+~z-'~c*(P*,x)-c*(II9,x)-cc*(~~,~~-~*~~~,x~1
+@? it~C*(Bd,X)fc*(PS,x)]-~z-1[s*(~,,X)-S*(~~rX)] I
-~*(P,,x)+~*(P,,x)--*(~s,~,l
+&ox
z#O
(fll)
D, = .Fl ~(-l+4P:)ic*~~~,x)+c*~~5,x)]-~ts*(~~,x)+s*(~~,x)-s*(~~,x) i -
s*uL
3
PlPp,Z 2 -i
x)1-
Q~s*cBs ?4 - s*(84,x)l- tD*(p,, xl - s*ub 4 + s*(& -4 3
i
-s*~Ps,x)l
+
I
mw*(B6 >4 + CYB,, 4 + c*(P,, 4 + c*@-J,41
7
I-e i
~(-l+4~f)~S*(B,,xf+s*(P,,x)l-~~C*(B,,x)
c*@7 4 - c*(P*,x) - c*uL XII- ww*uL 7
+s*u?6,x)l} -*F2P:z-l { w*uL,
-4 + s*&. ,x1+ s*(fb, 4
xf- c*w,,41 -w*(P*? xl-
c*uh 4
+95 ~~-l~c*(alrx,+c*~~,,X)!+i~S*~~S,X)-S*(P~,X)i i -~z-1~c*(B~,x)+c*~B7,x)+c*(P8,x)+c*(po,x)1
i
+vj,x
z # 0.
(112)
where the functions C*(pi, x) and S*(8,, x) are defined as C*(p,, X) ’ j: l: COS(2piU) du do
= ~[l z
-cOS(2PiX)]
(i =
1,. . . ,9)
(113)
Motion of a symmetric rigid body subjected to a specialized body-fixed force
435
and S*(8i, X)
r J”
+&
sin(2fiiu) du dv
0 0
= --J-sin(2/?ix)+z
4Piz
(i
=
1,. . . ) 9).
(114)
2Bi
In summary, given the dimensionless parameters SZ!r, R,, I,, z, x, PI, S2 and I&, (i = 1,2,3), one may proceed as follows to evaluate D,, D, and D, : 1. determine P1 and P3 by reference to equations (73) ; 2. use equations (81~90~ to evaluate I and /Ii (i = 1,. . . ,9); 3. if 5 > 0, evaluate D, , D, and D, by reference to equations (103~105) together with the definitions in equations (101) and (102); 4. if [ > 0, form [ and pi (i = I,. . . , 9) by using equations (106), then find D,, D, and D3 from equations (107~(109), using the definitions in equations (101) and (102); 5. if i = 0, determine D,, D, and D, by using equations (110)-(112) together with the definition in equations (113) and (114). Once again we turn to the computer to obtain a measure of the accuracy of the solution just obtained. In choosing parameter values for this purpose it is well to keep in mind that the approximate solution was generated by using approximate expressions for eij (i,j = 1,2,3), which suggests that the magnitudes of z and x bear directly on the accuracy of I), , D, and D, ;that is, one must expect the discrepancies between the approximate and the exact solution to grow as z and x increase. Figures 6-10 each show D, , I),and D3 plotted as a function of x. Numerical solutions of the exact equations of motion are represented by solid curves, while values obtained by using the approximate solution are represented by crosses. In all cases, B* is presumed to be at rest initially. Cursory examination of Figs. 6-10 leads to the immediate conclusion that the approximate solution furnishes qualitatively correct results in all cases and that, as expected, quantitative discrepancies increase as x increases. The effect of increasing z can be assessed by comparing Fig. 7 with Fig. 9. As for the effect of other parameters on accuracy, this appears to be negligible; that is, for a given value of z, the accuracy of the approximate solution is independent of the values of St 1, Qt,, c&, F2 and L. Figures 6-10 also permit one to explore the effect of parameter values on the nature of the translational motion. Consider, for example, the parameter L,which characterizes the inertia ellipsoid of the body. Figures 6 and 10 suggest that the mass centers of bodies possessing differently shaped inertia ellipsoids must be expected to follow markedly different trajectories. The force level, on the other hand, plays a relatively minor role as regards the shape of the trajectory, as may be seen by reference to Figs. 7 and 8, which show that, although the values of Pi and FS used to generate Fig. 8 are twice as large as those for Fig. 7, the trajectories followed by the mass center are almost identical. This is not to say, however, that the force level is totally irrelevant to the motion. On the contrary, as shown by equations (103)--(105), (107H109) and (110)-(112), D,,D2 and D, are directly proportional to the magnitude of the applied force F, since P1 and & are proportional to this quantity ; and this proportionality is, in fact, the reason underlying the similarity of Figs. 7 and 8.
T. R. KANE and
436
IO
x
FIG. 6. Comparison
FIG. 7. Comparison
Exact
-
x
x
B. S. CHIA
solution
Approximate
solution
of solutions for position coordinates of mass center. RI = 1.0, R, = 0, z = 0.01, L = 9.0, FI = 0.2, 4 = 0.2, & = 0 (i = 1,2,3).
of solutions for position coordinates of mass center. R, = 0.70711, a3 = 0.70711, z = 0.01, L = 9.0, 9, = 0.2, Fz = 0.2, &:., = 0 (i = 1,2,3).
Motion
of a symmetric
rigid body subjected
to a specialized
body-fixed
force
241 22..
-
20..
I
Exact solution * x Approximate
solution
16.. 16.. 14.. 12,. 10.. 8.. 6.. 4.. 2..
-6 .. -6, 0
FIG. 8. Comparison
.
.
.
.
. 5
.
.
.
. IO x
.
.
.
. 15
...+ 20
of solutions for position coordinates of mass center. R, = 0.70711, R3 = 0.70711, z = 0.01, L = 9.0, .FI = 0.4, Sz = 0.4, I& = 0 (i = 1,2,3).
6 Exact
-
solution
5 x
x
): Approxtmote
solution
4
FIG. 9. Comparison
of solutions for position coordinates of mass center. R, = 0.70711, RS = 0.70711, z = 0.05, L = 9.0, SI = 0.2, .Fz = 0.2, I& = 0 (i = 1,2,3).
431
438
T. R. KANE
and B. S.
CHIA
40
35
-
30
x
Exact salutiofl x
x Approximate
solutiin
25..
FIO. 10.
Comparison
of
solutions for position coordinates of mass center. Q, = l*O,Q, = 0, z = 0.01, L = -0.25, FJ f 0.2, .Ez = 0.2, 6, = 0 (i = I, 2,3).
In conclusion, it can be said that the approximate expressions for Di (i = 1,2,3) describe all essential features of the motion of the mass center and that the agreement between the approximate and exact solution is such as to justify considerabte confidence in the former. Furthermore, it may be worth pointjng out that this paper applies to a somewhat broader class of problems than that covered by the title. ~pe~~~a~~y, the approximate solutions derived here produce good results for certain gyrostats, as will be shown in a subsequent paper.
REFERENCES LEIMANIS, The General Problem of the Motion of Coupled Rigid Bodies About a Fixed Point. SpringerVerlag (196sj. [Z] R. LIZ The motion of a self-excited asymmetric rigid body. Arc&.+ration. Mech. Analysis 22, 129-142 (1966). [33 R. GRAMMEL, Der Selbsterregte Unsymrnetrische Kreisel. J. appf. Mu&. Me& 18, 73-97 (1954). [4] U. I’. I%DEWADT~ Der Symmet~~he Kreisel bei Zeitfester Drehkraft. Mark 2. 55, 310-320 (1952).
[I] E.
/5] R. R, A~EL~~~, On the Motion of a Symm~t~~I Rigid Body with Constant Body-Fixed Torques, Private Commun~cafion. [6J K. G. VALEEY, OR the probIem of the motion of an axially symmetrical body under the action of a constant moment. J. appl. M&h. Mech. 29, 162-165 (1965). [7’j B. S. CHIA and T. R. KANE, Motion of a Symmetric Rigid Body Under the Action of a Body-Fixed Force, Technical Report No. 203, Department of Applied Mechanics, Stanford University (1971). (Received 8 September 1971)
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