The second and final step in the development of first-order secular solutions to rotational motions of triaxial bodies is presented

Fitzpatrick, P. M.

Liu, J. J. F.

English

NASA Technical Reports Server

FURTHER COMMENTS ON THE APPLICATION OF THE METHOD
OF AVERAGING TO THE STUDY OF THE ROTATIONAL MOTIONS
OF A TRIAXIAL RIGID BODY, PART 2
March 27, 1973
Joseph J. F. Liu
Philip M. Fitzpatrick
Department of Mathematics
Auburn University
Auburn, Alabama
NASA-CR- 131878 ) FURTHER COMMENTS ON THE N73-2255
APPLICATION OF THE METHOD OF AVERAGING
TO THE STUDY OF THE ROTATIONAL MOTIONS
OF A TRIAXIAL RIGID BODY, PART 2 Unclas
(Auburn Univ.) 25 p HC $3.25 CSCL 12A G3/19 .17552(-Aul~~~~~~urG/1 175.; 52 / 
https://ntrs.nasa.gov/search.jsp?R=19730013828 2020-03-23T02:22:01+00:00Z
TABLE OF CONTENTS
1. INTRODUCTION ............... ................... 1
2. FIRST-ORDER SECULAR SOLUTION ................... 2
BIBLIOGRAPHY ................... ........... 23
1FURTHER COMMENTS ON THE APPLICATION OF THE METHOD OF
AVERAGING TO THE STUDY OF THE ROTATIONAL MOTIONS OF A
TRIAXIAL RIGID BODY, PART 2
1. Introduction
In [A.R.,1971], we described some of the results which wE
have obtained in applying the averaging technique described in
[F.R.,1971] to the variational equations which arise in treat-
ing the perturbations of the free rotational motions of a tri-
axial rigid body. In [A.R.,1971] we carried out the first step
of the averaging procedure and derived the averaged differential
equations for a set of canonical variables ( k, 3k) for the
problem of a triaxial body in a precessing, elliptic orbit
about an attracting center. The development was carried out
to the point that the averaged differential equations [A.R.,
1971,(5.31)] are in a form which can readily be integrated if
it is so desired. The second step of the averaging procedure
was not carried out for these canonical variables (o( ki, k)
because we were planning to use the averaging technique to
develop first-order secular solutions for an alternative set
of noncanonical variables. In [F.R.,1972] we began our dis-
cussion of the development of these secular solutions by carry-
ing out the first step of the averaging procedure for a con-
venient set of noncanonical variables. In the present report,
we complete the second and final step in the development of
these first - order secular solutions.
References to our earlier reports of June 16, 1970,
February 19, 1971, August 2, 1971 and February 21, 1972 are
indicated by [J.R.,1971], [F.R.,1971], [A.R.,1971] and
[F.R.,1?72], respectively.
22. First-Order. Secular Solution
Equations (4.24)' may be integrated numerically with a
much longer integration step time than may be used with equations
(4-.3) In this way, we can obtain the first-order secular solu-
ttfia for the rotational motion under the influence of the
gravity-gradient torque. We can also, howeveri integrate the
averaged system (4.24) analytically with the aid of the inte-
gral (4.30). We address ourselves to this problem in the re-
mainder of this section.
To begin with, we attempt to integrate (4.24(b)), noting
that -s iis constant. We first express cH (and hence s 
H
)
in terms of 6H by substituting (b) and (c) of (4.25) into
(4.30). We find that, to first order,
2.
c = - (cl h c ca h (2.1)
It follows from (2.1) that
1/2 4 c3 + 2 1/2(AX) (-c + A3 + A2 c + Ale + A (2.2)
where
A = c''h2
4b't
All equations designated (4.iJ) i,j, nonnegative integers,
refer to equations given in [F.R.,1972].
3
2 2
I* 4b' (1-C)
A = h
o l 2 
c' I LI
8a 'b' 'C h
C" h 2
- 4b' 1 2 a 112 _
2h bI2
c'hCh - 1)
bI I I -1) /
a"
c h
If (2.2) is substituted into (4.24(b)), we can write
dcH
4 * 3 c2 +A ) 1/2(-C e. +A 3C +A2CeH +Alel +A
= -(A*)1/2s .2dt. (2.3)
.+ GP
In order to integrate (2.3), we first note that the bi-
quadratic equation
4 Ac 3 X-2
+ A3 H + A2CH + AlcH+ A (2.4)
has the following allowable roots:
(i) four distinct real roots.
(ii) four real roots with two identical roots,
3
A = -
3
(iii) two distinct real roots with a pair of complex roots,
(iv) two identical real roots and a pair of complex roots.
In what follows, we consider each of the four cases.
Case (i): Four distinct real roots
In this case, we can write equation (2.3) in the Jacobian
normal form for real roots [see (BF250.04) and (BF250.06) W
dce
1/2
[( al-C ) ( a2-Hel ) ( a-C ) ( ce-a4 )
-(A ) / a n +.. 
+ U 
(2.5)
1/2
r(al-a 3)(a2 -a4)1
if al, a2, a3 and a4 are the real roots of the biquadratic
equation (4.4), and it is assumed that a1 >a2 >a3 >a4 . The
variable co is related to u through the equation
H
co = A1 +A2 sn
'
u.
A3 +A4 sn2 u"
, 0:< U < K .
du
g du =
where
(2.6)
$ Reference to equations in H and Book of Elliptic Integrals
for Engineers and Physicists, Byrd,P.F. and Friedman, M.D.,
Springer,Berlin, 91954, are prefixed by the notation BF.
5Here K is defined by the integral
a/2
K =d
(l-kJ (1 k} 11/2
0 (1-k(" sins )
where ki is the modulus of Jacobian elliptic functions and in-
tegrals. Explicit values of k* and A i, i=1,2,3,4 will be given
in the study of the special cases which follows.
Since the time rate of change of cg is real, the value of
CH must either lie between al and a2 inclusive or between a3
and a4 inclusive. We will analyze (2.5) in the subcases which
follow.
(1) alCHZ > a2 : The A i, i=1,2,3,4, have the values
A1 = al(a2- a4) , (a)
A2 = a4( a l - a2 ) , (b)
(2.7)
A3 = a2 - a4 , (c)
A4 = a1- 2 (a)
and kf' has the value
k* = (al - a2)(a - al) (2.8)
(al- a 3 )( a 2 - a4 )
6for this and the remaining three possiblities under Case (i).
If we integrate (2.5) with respect to u from 0 to u (i.e.,
integrating with respect to time t from tl to t), we find
[see (BF257.00)] that
u =+ 1(A) s2 ( t-t, (2. 9)
where tl is the value of t at which cH = al
Substituting (2.9) into equation (2.6), we obtain the
first-order secular solution for co and thus the first-order
secular solution for H 
(2) a > CH > a2 : The Ai, i=1,2,3,4, have the values
A
1
= a2 (a3 - a1 ) , (a)
A1 = a 2 ) , (b)
(2.10)
A
3
= a3 - a (c)
A4 = al - a2 . (d)
If we integrate (2.5) with respect to time from t2 to t, we
find [see(BF256.00) ] that
U2 =+ 1 .(A)1/2 s?. (t-t , (2.11)
where t2 is the value of tat which = a2
where t2 is the value of t at which CeH = a2
7Substituting (2.11) into equation (2.6), we obtain the
first-order secular solution for OH.
(3) a3 < C6H a4 : The Ai, i=1,2,3,4, have the values
A1 = a3 (a4 - a2 ) (a)
A2 = a 2 (a3 - a4) , (b)
(2.12)
A3 a4 - a2 , (c)
A4 a3 a (d)
If we integrate (2.5 ) with respect to time t from t3 to t,
we have [see (BF253.00)) that
u3 + 1 (A+) s1/2 (t-t3) (2.13)
where t3 is the value of t at which cg = a3
Substituting (2.15) into equation (2.6), we obtain the
first-order secular solution for H'.
(4) a3 c ce < a4 ; The Ai, i=1,2,3,4, have the values
A1 = a4(al - a3) , (a)
A2 = al(a3 - a4) , (b)
(2.14)
A3 = 1 - a3 , (c)
A4 = 3 -a 4 , (d)
8If we integrate (2.5) with respect to time t from t
4
to t,
we find (see(BF252,00)] that
u + (A ) -s ( t-t (2.15)
where t
4
is the value of t at which co = a4.
Substituting (2.15) into equation (2.6), we obtain the
first-order secular solution for eH.
The associated first-order secular solution for the var-
iable Y H for each of the subcases described above, is readily
obtained from (2.1). This eliminates the need to integrate
(4.24(a)) directly. Four more integrals of (4.24) remain to
be determined. We consider next the variable .
H
In order to integrate equation (4.24(c)), we first re-
write it in the form
c 2
x3 (sl-r) =VH - (xs1 3X) 31'H)c -X3' . (2.16)
H HS~H
We note that if we can integrate c I/s. and c2 with respect
to time, to first-order, we can then obtain the solution for CH.
From relations (4.25(b)) and (4.30), it is found, to
first-order, that /
2
c hYH= h = h0 (c" 'I a''I hzo - Ch)
CH=- h ° = - =-2b'' b'; (2.17)
sH e hn-hc h -hh
1H H zo
9If we introduce the identities
= 1 ( 1
2h h+hzO
= ( 1
2 h- hz o
2 h+hZ-
Zo
h-hzo
zo
equation (2.17)
C H = D1
sH
takes the separated form
1
h-c
aH
I + D3
th+c 
&11
where
D1 = - 1 (a''h2 -
2
D
2
= - 1 (a''h2 +
2
b'"h + c")
D = a''h .
We can express the right hand side of (2.19) in terms of
u if we replace c through the use of (2.6). We use (2.5)
to replace t by u" as the variable of integration. We then
obtain the differential form
S (a)1
h2-hzo
zo
hz
K2-h2o
h 
z 0
h -hz O
(b) (2.18)
(c)
(2.19)
b''h + c'')
10
CYH dt = -
SH
g
(A) 1/29 5LID 1h~'A3-A 1 ) A3+A4 sn2u1-r 1 sn-u'
+ D2 A3+AI sn2 u
h3 +A 1- ' 2 sn2 u
2
I A -A
-a = 4 
-2 A3 - A1
2 A +A
2 =A + A1
du (2.20)
( a)
(2.20) '
(b)
Upon integrating (2.20) from t
i to t, we find, from (BF336.01)
and (BF337.01) 
. that
t
Y H dt =
SeH s8 o
D 1
h(A3-A1)
(2.21)
+ D2
h( A3 +Al ) LA3 (V1 (Ui u2) _ V1 (0° 23)
- A4 (W1 (U 2) - W1( , ))] 3 u i
+
where
Ili=t
t i
A3 (Vl(ut 2) -V1(0l1(
-
- A W (Ui, y 1) - W (0O I I),
11
where i=1,2,3,4, and
V1( u:i -j )
Wl( U i ¥' j)
= II(u i j)
- II(u., vj
(-J =1,2)
= 1 I(Ui T2)-F(u
i ) , (b)
i
Here II(ui, , ~, a positive integer is Legendre'S incom-
plete integral of the third kind and ; 1 or y J k i .
For the special cases where = or j = k the reader
may refer to (BF11li.06) for appropriate formulas. It can be
evaluated by using Formulas (BF430) through (BF440).
If, next, we square (2.6), we have that
CH 1
2 A2
- 3 A
where
,2 2 U 2
sn u
sn2=4
2 A
4 A3
Repeating the procedure used to obtain Ili , we find that
(use BF340.02)
t
I2i= 
ti
2 I A 4
c dt=- 1 1 ui +
%H (A") s 2 0 3 
(2.25)
+ 2¥3( 4 -y2 ) EVl(u 4r) - V
v
l ( O° )7
(a)
(2.22)
(2.23)
(2.24)
12
2 fV 4 (* 2)
where
V1(ui , r214 = II(*Ui ) $ (a)
_ 2 4E(ui )+ (k -
4 u
k4- 2 > - 4
14
- 3 S) II(u 4)
ns n uiS n cu i dn u i 
1-4 sn ui
(2.26)
(b)
If we take the unperturbed solution (c H)o as our initial
value of OH , the initial value of x3 will be zero and we can
write
X3= H- (O H) o= ( s e oL) I l i -(Xl'' - 3x3 ')I2 i - X 3xt 
(2.27)
The remaining three integrals of (4.24) follow easily. They
are explicitly
x, = 6' - (')_ = 0 (2.28)
4 0o
x5 = ' - (41) =- x5 (t-31 2i) (2.29)
+ (7 (2.25)
V2( ui 1. 
2(y4-1) (k*2_ g
- V 2(,4) 2
- 4) ui
13
X
6
= h - (h)o = . (2.30)
while the unperturbed solutions are taken as initial values
of the relevant variables.
Summarizing, we have the secular, first-order solution
for Case (i):
t=-_ c' h C a' C h+  (a)PH sG 2b ' b h
2 x
A +A sn u(
+ A2sn ui , 0 < u i c K (b)
H A3 + A4sn2 u* -
(2.31)
( H)o + (s )Ili-(Xx3 2i 3
=H i 1( - 3x 3 'I)I 2 i-x''t , (c)
e, = I ( )o ' - (d)
= ( fb)c - x5' 5 '(t-3I2i) (e)
h =(h) , (f)0
where ui~ is given by whichever of (2.9),(2.11),(2d13) or
(2.15) applies to the appropriate subcase.
Case(ii): Four real roots with identical roots
This case can be treated as a special case of (i) and can
be further grouped into two subcases.
/
(1) al= a2 A CH or a3= a4 CH: It is seen from (2.8) that
k = 0. Thus all the elliptic functions reduce to trigonometric
functions (i.e., sn u = sin u ,etc.). In either case, equa-
tion (2.6) becomes
c =A 1 +A2 :in u , O < u*< R/2 (2.32)
H A3+A4 sinaui
where the A
i
, i=1,2,3,4, are given by either (2.12) or (2.14)
if a a2 . They are given by either (2.7) or (2,10) if a= a4.
Then ui is given by (2.13) or (2.15) if al= a2 and it is given
by (2.9) or (2.11) if a3 = a4
We note that if we replace (2.31(b)) by (2.32) and if,
for this case (k*= 0), we can evaluate Ili and I2i which
correspond to (2.21) and (2.25). Then equations (2.31) will
give us the first-order secular solutions. It is also seen
from equations (2.21) ,(2.22), (2.25) and (2.26) that if we
evaluate both II(ui, ;j), j=1,2,3,4 and F(u) at k= 0 then
these equations will determine bdth Ili and I2i. Formulas
2 can be foun
for the elliptic integrals II(ui ) and F(u can be found
in [1]. Explicitly, they are [see (BFlll.01) and (BF121.01)3.
F(u*) = ui
*IXuiv 2
II(ui, ) = ut if j = o (2.33)
14
15
tan- [(1- y ) 1tan u1]
tan 1/ , if < 1,
(1 a-l1)
(2.33)
tan-1 )) /2 tantan t(' X.-1) tan u(] 2
a__ _1/2__i_ , if > 1.
1/2
(y j -1)
If Y = 1, we can use (BFlll.01), (BF121.01) and (BF11l.06),
and write
II(uil) = tan ui. (2.34)
Thus, the integrals Ili and I2i are determined and equations
(a),(b),(c),(d),(e) and (f) of (2.31), together with (2.32),
give the first-order secular solutions for the six variables
of interest.
(2) a2 = a3: It follows from (2.8) that ks = 1. Thus all
elliptic functions reduce to hyperbolic functions (i.e.,
sn u = tan h u, cn u = sec h u.). Equation (2.6) takes the form
c = Al +A 2tan h ui, 0 < ui _ (2.35)
A3 +A4tan h u*
where the Ai, i=1,2,3,4, are given by either (2.7), (2.10),
(2.12) or (2.14) and u. is given by the associated relation
(2.9), (2.11), (2.13) or (2.15).
16
The analysis proceeds as in the preceding subcase, and
we have, from (BFlll.04 ), that if k = 1
F(u±) = -n(tan sec i = am Uj (a)
(2.36)
II(ui, ') = 1 I Cn(tan±i + seci)
(b)
I l+ ' )1/2 2 1)
- j yn J ) ] v).Yj"·i 1 - j soi
.[Here y J cannot take on the value one since II(ui'l) = ] .
Thus the integrals Ili and I2i can be determined by (2.21),
(2.22) and (2.25),(2.26), respectively, and therefore equa-
tions (a),(b),(c),(d),(e),(f) coupled with (2.35), give the
first-order secular solutions.
Case (iii): Two distinct real roots and a pair of complex roots
Let al,a2 be the real roots and let a3, and its complex
conjugate a3 be the complex roots and assume that al > a2.
We can write equation (2.3) in the Jacobian normal form for
complex roots [see (BF250.05) and (BF250.06)]
g du= daH -(A s)1 /2s A dt.
c1,2 + 0O
[(al-ceH) (celH-a2 ) (celH-a3 )( H-a3 1/2 +
(2.37)
17
where g is given by the equation
g 1 1
(Al B')
and
A' = [(al - b)2 2]1/2
B' = [(a2- b ) + a ]I= a --~ 2+ a d /
a
b
= - (a3- a) 2
3
= 1 (a + a
2 3 3
The variable c is now related to u through the equa-
tion
A + A cn up
CH =1 2H A3+ A4cn u-)3 4
(2.40)
where K has the same definition
new modulus of Jacobian elliptic
Since the time rate of change of
ce must lie between al and a2.
A1 = alB' + a2A' 9 
A2 = a2A' - alB' ,
as in Case (i) and k- is the
functions and integrals.
c is real, the value ofH
If a1 > c0 > a2 , we have
(a)
(b)
A
3
= A' + B'
A4 = A' - B'
(2.41)
( c)
(d)
(23.8)
(a)
(b)
(2.39)
(c)
(d)
18
and k* 2 has the value
k*= (al-a2) - (A'-B') (2.42)
4A'B,
If we integrate (2.37) with respect to time t from t2 to t,
we obtain Csee (BF259.00)]
u =- 1 (A*)1/2 sa (t-t2 ) , (2.43)
where t2 is the value of t at which co = a2 . Substituting2 6 H 2. II
(2.43) into equation (2.40) we obtain the first-order secular
solution forceH. With the time dependence of coH known,
equation (2.1) gives the first-order secular solution for H.
Proceeding as in Case (i), and using equations (2.40)
and (2.43) in conjunction with equation (2.19), we find that,
to first-order,
VCH dt = - r 1 A 3 l+'3 cn u
9aH (A*)1/2s Ad h(A3-Al) 1 +' ln up "
o 1+ +1 lcn uu
+ D2A3 7 d (2.44)
h(A3+A1) 1+ Y2cn u 3 J
19
where
lY, A -A2,
3 1
A4 + A 2
A3 + A1
A
3
(a)
(b) (2.45)
(c)
To integrate equation (2.44) with respect to time t, we can
use (BF361.62) and' rearrange it in the form
' H dt = -
sa H
Y3 +
(Aw)
1- I3'Y1 U
1- ylcn u*,
+ D2A3 (
h( A3+A1)
1 - r 3/ ) 
1 +2 cn u*/
+ D3] du.'
If Y 1, and X 2 1,
1 2
(BF341.03) ], that
we find, from (BF361.54) [or from
t
ii =i
ti
D1A33 +
h(A3 -Ai1)
D2 A3
h( A3 +A 1)
j3 + D3 ) u+
(2.44)
_YH dt
s
.g
1/2-
(A ) i 2 Q T(
20
+ D1 A
h( A3 -A1
D2A3
h(A3+Al
*(1- r3
1 3
¥2 ' 3
Y2
[Rl(UC,' ) - R l ( O , - )
[R1 (u 9 2 ) -R1(0,Y2)]t
R1 (u ' Ž)
2f = 1- y ,1/2 1
9- 2i tanK)
kk +k '1 i)
2
1 -2 - 1i i
- yi' fl(U-')-T2I
I1 - 1r u I s) d 
, (a)
19
2
if .- I
-1
- sd u ,
2
if ¥i
ai-1
=2
= k
2 2&i)
(b) (2.47)
1/2
dn u t-
=( yi-l T
k 2 +k"4 ' 2(2
1/2
-1) sn u 
l
( i- 1) sn u~
:k2-+k*, 2 12)
if Yi > k 2
2-1i3
where
(2.46)
1/2
dn u -
21
2
If either Yi - 1 or y 2 1, the integral in (2.46) is to be
replaced with the integral given by either (BF361.51) or
(BF341.53),
Equation (2.40)
t
I2i =
can be used to write
C dt
aH
t2
=
( A)1 /2s o 44
A1
A
3
+ 1
(1- Y4 cn
4 =A24 A2
(2.48)
(a)
(2.49)
(b)
A3
Integrating with respect to time, we find from (BF341.03) and
(BF341.04) that
Z Y2 ( 4al'- 4) VR(u , ) 
4 4 4
1
1+ Y cn un
+
where
I =
2
2
A1
2
3
(A-) / 2' -ASC1(0
u 2
o id4
du 2
22
-R ( 2) +2 )4(2k -1)-2k 2 R (ui) _
2- k r J R 4 , -2 n (2.5 0)u
whaere
-2(u ,t4) = z (()2_2k_ * p )
+ k 2yi( 4 Ucos 1a' lln u )dn u,
-I o = 1, the integrals in (2.48) are to be replaced
i i te 
-
integralsn 
. Thus
equations (a(b(c,(,(e() o (2.31) together with (2),
er
, y(U u + ¥k cos-(dn u (a)
(2.51)
+ 2¥4 k cos- dn u
and Y4 ~
If , 4 
=
l  the integrals in (2.48) are to be replaced
with the integrals given by (BF341.53) and (BF341.54). Thus
equations (a),(b),(c),(d),(e) ,(f) of (2.31) together with (2,40),
(2.46) and (2.50) give the associated first-order secular
solutions for the six variables.
23
Case (iv): Two identical real roots and a pair of complex roots
This ia a special case of Case (iii). It can be shown in
a straightforward manner from (2.40) that if al = a2 then
cH = al and 9H is a constant of the motion. Consequently
YH' GHe ' and h are all constants of the motion and H and
' are linear functions of time.
BIBLIOGRAPHY
I1] Byrd,P.F. and Friedman,M.D., Handbook of Elliptic Integrals
for Engineers and Physicists, Springer,Berlin, 1954.
/


Further comments on the application of the method of averaging to the study of the rotational motions of a triaxial rigid body, part 2

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