Approximative analysis on librational dynamics of deformable satellit

Lingerfelt, J.

Mitchell, T. P.

English

NASA Technical Reports Server

9 
The Librational Dynamics of Deformable Bodies 
1 T. P. Mitchell and J. Li 
A Technical Report to the 
National Aeronautics and Space Administration 
Grant No. NGR 05-010-020 
Mechanical Engineering Department 
University of California 
Santa Barbara, California 93017 
January 30, 1969 
https://ntrs.nasa.gov/search.jsp?R=19690008713 2020-03-12T07:03:44+00:00Z
* 
The Librational. Dynamics of Deformable Bodies 
bY 
T.P. Mitchell  and J. Lingerfel t  
University of Cal i fornia ,  Santa Barbara, Cal i fornia  
ABSTRACT 
The volume average of t he  s t r a i n  tensor i n  a body moving i n  an inverse- 
square force f i e l d  i s  evaluated. The calculat ion i s  car r ied  out assuming the  
s a t e l l i t e  t o  be an i so t ropic  e l a s t i c  body whose center of mass moves i n  a 
planar o rb i t .  An approximate expression, i n  terms of i t s  volume and e l a s t i c  
proper t ies ,  i s  presented f o r  the  s t r a i n  energy i n  the  s a t e l l i t e .  Using t.his 
expression the  equation of planar l i b r a t i o n a l  motion i s  wr i t ten  expl ic i ty .  
This equation i s  discussed f o r  both c i r cu la r  and e l l i p t i c  o rb i t s  and i s  
modified t o  include the  e f f e c t s  of energy d iss ipa t ion  i n  the body. It is  
shown t h a t  the  concept of Adiabatic Invariants  allows one t o  analyze the 
influence of slow changes i n  the mater ia l  volume and e l a s t i c i t y .  
* 
This work was sutmorted by NASA Grant No. NGR O5-OlO-O2O. 
I. INTRODUCTION 
This paper a i m s  t o  study the e f f ec t s  of mater ia l  e l a s t i c i t y  on the 
l i b ra t iona l  motion of an a rb i t r a ry  shaped s a t e l l i t e .  
influence of the e l a s t i c  behavior on the l i b ra t iona l  frequency i s  determined. 
The approach adopted i s  qui te  general i n  t h a t  the spec i f ic  shape of the 
s a t e l l i t e  i s  not prescribed other than t o  assume t h a t  the  o rb i t  plane of i t s  
center of mass coincides with a pr inc ipa l  plane of the  s a t e l l i t e .  Thus the 
in-plane l i b r a t i o n a l  motion i s  considered t o  be uncoupled from the out-of- 
plane mot ions 
In par t icu lar ,  the 
The e l a s t i c  behavior is  assumed t o  be describable within the context of 
the  c l a s s i ca l  theory of inf ini tes imal  e l a s t i c i t y .  Accordingly the body forces,  
i n e r t i a l  and gravi ta t iona l ,  t o  which the  s a t e l l i t e  mater ia l  i s  subjected are  
computed as i f  the  s a t e l l i t e  were r ig id .  However since the mater ia l  i s  actual ly  
deformable it contains s t r a i n  energy of deformation which d i r ec t ly  influences 
the l i b r a t i o n  frequency. 
The desire  t o  avoid the  specif icat ion of the shape of the s a t e l l i t e  removes 
the  analysis of the e l a s t i c  behavior from the normal c lass  of boundary value 
problems i n  the  mathematical theory of e l a s t i c i t y .  Therefore an averaging 
method i s  used t o  compute an approximate s t r a in  energy density the knowledge 
of which enables one t o  write the d i f f e r e n t i a l  equation f o r  the l i b ra t iona l  
motion. This equation, whose form depends upon whether or not allowance i s  
made for energy diss ipat ion,  has been derived previously fo r  per fec t ly  e l a s t i c  
mater ia ls  i n  a special  case [1,2] . For materials w i t h  s ign i f icant  in te rna l  
f r i c t i o n  the  decay time of the  l i b r a t i o n a l  o sc i l l a t ion  can be wri t ten i n  terms 
2 
of a qual i ty  fac tor ,  Q, the  e l a s t i c  constants and the  volume of t he  material .  
Slow changes i n  the  physical p roper t ies  of t he  body can be examined by con- 
s t ruc t ing  an Adiabatic Invariant of the motion. 
11. THE EmRGY OF DEFORMATION 
Assuming the  o r b i t a l  and l i b r a t i o n a l  motions t o  be uncoupled one can 
wr i te  the  equation which determines the planar l i b r a t i o n a l  motion of a r i g i d  
body i n  the  form 
The angle of l i b r a t i o n  i s  denoted by cp, the  pos i t ion  of the center of mass 
by (Rc,e)  and the  g rav i t a t iona l  parameter by K. A t  the  center of mass the 
p r inc ipa l  moments of i n e r t i a  a re  A, about the  l i n e  from which cp i s  measured, 
B,and C about the  axis normal t o  the  o r b i t  plane. 
i s  i l l u s t r a t e d  i n  Fig. 1. To modify equation (1) i n  order t o  make allowance 
f o r  the  e l a s t i c  deformation of the  orb i t ing  body it i s  necessary t o  determine 
the  work done i n  t h i s  deformation by the  body forces .  It can be shown t h a t  
t he  body force,  per unit mass, act ing at  the yos i t ion  (x,y) is 
This geometric configuration 
* 
f = J ( x . P ~ - Y P ~ )  + j (xP  +yp4) 
jLn M 3  
i n  which 
K 2 P1 = (@+Q)* + 3 ( 3  cos cp - 1) 
Re 
3 K  
R 
P2 = - 3 (CC + 1) Sin cp COS cp 
C 
* 
The d e t a i l s  a r e  presented i n  the  Appendix 
3 
p3 = 2j (CI - 1) Sin cp cos cp 
RC 
p4 = (Q + 6 )  2 K  + ( 3  Sin 2 cp + 1) 
and 
CI = (B  -A)/C (7 )  
It follows from the  theory of inf ini tes imal  e l a s t i c i t y  C3] t h a t  the 
s t r a i n  energy density, W, i n  a homogeneous i so t ropic  body i s  
w = 1: A. (eii) 2 + P ei j  e i j  
2 
/ 
where the  Lame 
tensor created by the  body force d is t r ibu t ion  given by equation ( 2 ) .  
Constants a re  represented by h and p and e i s  the s t r a i n  i j  
An 
exact evaluation of W would necessi ta te  the  specif icat ion of the geometric 
shape of t he  body and the  solut ion of the per t inent  boundary value problem. 
Since t h i s  specif icat ion i s  prec ise ly  what it i s  sought t o  avoid i n  the  present 
analysis one has t o  be content with an estimate of the  density W. This es- 
timate can be achieved by using the  Be t t i  Reciprocal Theorm [3) t o  obtain the 
volume average of the s t r a i n  tensor.  A po ten t ia l ly  awkward feature  of the 
application of the  theorem - the  evaluation of surface in tegra ls  - does not 
a r i s e  here because the surface o f t h e  body i s  stress f r ee .  The volume averages 
of the  s t r a in  tensor components are found f o r  symmetrical bodies t o  be 
- e = [(B+C-A)Pl - (5 (C+A-B)P4] /2EV xx 
- e = [(C+A-B)P4 - 0 (B+C-A)PJ /2EV 
YY 
( 9 )  
4 
- 
e =z , = -0 [(B+C-A)P1 + (C+A-B)Pq] /2EV 
- e = [(B+c-A)~~ - (C+A-B)P2] /4pv 
XY 
and 
- - 
e = e  = O  
YZ zx 
where V is  the volume of the  body. The well-known re la t ionships  
have been used i n  the  der ivat ion of t he  averages i n  order t o  simplify t h e i r  
f ina l  form. Direct subs t i tu t ion  i n  equati on (8) and considerable a lgebraic  
reduction produces 
2 2  W = A (b+cp) * 4  + A2( b+b)2 + A Sin 4 cp -i- A4 Sin 2 Cp + A ( e@))  Sin cp (14) 1 3 5 
i n  which 
= ($* + ? - 28yo)/8Ev2 A1 
A = 9 8  [s2+ y2- Pya -8(ltcr) 8 2 2  y /C 2 1 
3 C 
A4 = K 2 [2p2+ 3Y2- 38y0 +- 36( l+0) $ 2 2  Y /C 2 ] /4E$R6 
C 
= 3K(Y2 - D2)/4EV?R3 A5 C 
5 
@ = B + C - A  
and 
y = C + A - B  
If the  center of mass of the  s a t e l l i t e  i s  moving i n  an e l l i p t i c  o r b i t  of 
semi-major axis a and eccen t r i c i ty  e, the  parameter Rc entering i n  equations 
(16), (l?), (18) and (19) ma,y be expressed i n  the  form 
111. THE EQUATION OF LIBRATION 
Adopting the  form (14) f o r  the  s t r a i n  energy density one can wri te  the 
Lagrangian of the  motion as 
( B  - 2A) L ="(h2 + R2 i2) + - ( b  C + 4) 2 K m  + -  - - K 
2 2R: 2 c  c 
2 3K ( B  -A) Cos cp + VW +3
2RC 
(23) 
which leads,  r e s t r i c t i n g  the analysis  t o  O(cp) ,  t o  t h e  bas ic  l i b r a t i o n  equation 
-[(' d t   + 2 K A 6 ) ( 6  + GI)] + [s (13-A) - R3 
C C 
and 
6 
In  wri t ing the  Lagrangian given by equation (23) it i s  assumed t h a t  the 
k i n e t i c  energy of the e l a s t i c  modes of deformation is negl igible  i n  comparison 
with the o r b i t a l  and l i b r a t i o n a l  energies. Furthermore it should be remarked 
tha t  i n  t h e  r i g i d  body l i m i t ,  E 
equation (24) becomes equation (1) t o  O(cp). 
t o  wri te  equation (24) with 8 r a the r  than t i n  the r o l e  of independent var iable .  
This transformation which can be made by invoking Kepler ' s  Second Law 
O0,  both As and tend t o  zero and hence A7 
It i s  convenient i n  some cases 
r e s u l t s  i n  the  d i f f e r e n t i a l  equation 
2 2 
de 
(l+eCos0) [C + 2A6 8 ' ( l+eC0s8)~]  % - 2eSine [C + 5A60 ( l+eC0s8)~]  -$f 
+ [3(B-A) - 2A702(1+eCos8)3] ep = 2eSin0 [C!+SA60 2 (l+eCose) 
w i t h  
Equations (24) and (28) w i l l  be discussed separately fo r  c i r cu la r  and e l l i p t i c  
o r b i t s  respect ively 
N. CIRCULAR ORBITS 
In  the  case e = 0 equation (24) s implif ies  t o  
7 
which corresponds t o  a simple harmonic osc i l l a to r  with a frequency, w, 
determined by 
3(B-A) - 2A7n 2 
2 w2 = [ c f 2A6Q 
Equation (31) provides a simple e a s i l y  applied formula f o r  estimating the 
frequency of planar l i b r a t i o n  of a homogeneous i so t ropic  body of a rb i t r a ry  
shape, volume V, p r inc ipa l  moments of i n e r t i a  A, By and C, Lam; Constants 
1, p and whose center of mass i s  moving i n  a c i rcu lar  o rb i t  with o r b i t a l  
angular veloci ty  R .  The derivation of t h i s  formula was the primary purpose 
f o r  undertaking the  present analysis.  Tha t  W should be l e s s  than the r i g i d  
body frequency R 3(B-A)/C is  t o  be expected. In  f a c t ,  by u t i l i z i n g  
equations (20) , (21) ,, (25) and (26) one can show t h a t  i f  B > A > C then As 
and A are  both necessar i ly  posi t ive.  Accordingly, it i s  c lear  t h a t  the 
effect  of the  mater ia l ' s  e l a s t i c i t y  is  t o  decrease the  frequency of l i b ra t ion  
below the frequency a t t r i bu ted  t o  the  same body were it r ig id .  Two fur ther  
I 
7 
aspects of equation (30) may be remarked upon. 
F i rs t ly , ,  i f  the  body possesses appreciable in t e rna l  f r i c t i o n  the decay 
time of the l i b r a t i o n  can immediately be writ ten,  in ' t e rms  of a mater ia l  
qua l i ty  f ac to r  Q9 i n  t he  form 2Q/W. Here qua l i ty  f ac to r  i s  defined, as usual, 
t o  be 
= 2n energy s tored 
energy l o s t  per period 
and W i s  given by equation (31). 
t he  Solar System are available Secondly, i f  the  analysis i s  t o  be 
Estimates of Q fo r  na tura l  s a t e l l i t e s  i n  
applied t o  na tura l  s a t e l l i t e s  whose physical propert ies  may change over long 
time periods one can wri te  t h a t  
8 
- 
E/M = constant (32) 
where represents the average energy i n  the  l i b ra t ion .  Equation (32) 
r e f l e c t s  the  f a c t  tha t  the  act ion i s  an Adiabatic Invariant of the 
l i b r a t i o n a l  motion f o r  slow changes i n  the bas ic  parameters. Expl ic i t ly ,  
i n  t he  present case one has 
cp 4 2  hz [3(B-A) - 2A7n2] ( C  + 2A6hz 2 ) = Constant 
0 
(33) 
with cp representing the  amplitude of the  osc i l l a t ion .  
0 
V. ELLIPTIC ORBITS 
The form of equation (28) as it stands i s  not par t icu lary  su i ted  t o  a 
discussion of the  l i b r a t i o n  of a body i n  an e l l i p t i c  o rb i t .  The equation 
can be cas t  i n t o  a more convenient form by introducing the transformation 
Y = (1 + e COS 9) cp (34) 
In  terms of the new variable  Y equation (28) becomes 
2 ( l + e C ~ s e ) ~ ]  (l+eCosO) eCos 0 + 6e2A6n2(l+eCose) Sin20 
-l 
+ ( 1 ) [3(B-A) -2A h12(l+eCos8) 
l+eCos 0 7 
9 
The solution of t h i s  equation can be found as follows as a power s e r i e s  
i n  the eccent r ic i ty  e (0 < e < 1). Let 
a3 
Y = C enYn 
n=o 
and equate l i k e  powers of e on both s ides  of equation (35). This process 
leads t o  a s e t  of d i f f e r e n t i a l  equations which can be solved i n  sequence. 
The present discussion w i l l  be confined t o  t h e  f irst  two of these equations 
merely t o  demonstrate t he  method. One f inds  
0 -+ (a1 - a2) Yo = o 
de2 
and 
+ (a +2a2-1) Yo case + 2a4 Sine 1 
where 
a = 3(B-A)/(C + 2Agn 2 ) 
1 
a 2 = 2A7Q2/( C + 2A6Q2) 
"3 = 2A6Q2/(C + 2A6Q2) 
(37) 
(39) 
(40) ' 
10 
The solut ion of equation (37) is  
1 1 
Y = c0 s i n  (al-a2)2 e + D COS (a  -a )" e 
0 0 1 2  (43) 
The homogeneous solut ion of equation (38) is  of similar form and i t s  
pa r t i cu la r  solution, which m a y  be found by the  method of var ia t ion of 
parameters, i s  
\ 1 ( a1+2a2- 1) Y 1 = [  4u + 4 3  U] x {l+p)CoSinr8 -(l-q)CoSins@ -(l+p)DoCosrB +(l-q)D 0 Sins8 r 
+ fa3 { (l-p)CoSinrO -I- (l+q)CoSinsB + (l-p)DoCosrB + (l+q)D 0 COS 
Sin 8 + 2a4 a -a -1 1 2  (44) 
i n  which 
-1 1 2(al - a2)F + 1 = p 
-1 1 -2(al - a2)" - 1 = q 
This solut ion process may be continued s t ep  by s t ep  t o  the  required degree 
of accuracy and the  influence of t he  o r b i t a l  eccent r ic i ty  on the  l i b r a t i o n  
subsequently analyzed. This fur ther  development w i l l  not be presented here 
however 
11 
I V .  CONCLUSIONS 
The l i b r a t i o n a l  frequency of a r b i t r a r i l y  shaped e l a s t i c  bodies can 
be represented approximately i n  a r e l a t ive ly  simple way. 
t a t ion ,  equation (31), i s  based upon a generalization of the c l a s s i c  
McCullagh Formula, f o r  the  grav i ta t iona l  po ten t ia l  of a r i g i d  body i n  a 
Newtonian Force Field, t o  include deformation energy. The approximate 
nature of the  generalization a r i s e s  because the  non-specification of the 
body's exact shape precludes an exact calculation of i t s  e l a s t i c  s t r a i n  
energy. The modification necessary t o  allow f o r  i n e l a s t i c  behavior i s  
immediate i f  a Quality Factor Q i s  available.  Consequently both na tura l  
and a r t i f i c i a l  bodies may be t rea ted .  I n  the  former case the e f f ec t s  of 
slow changes i n  the  physical parameters may be examined by constructing an 
Adiabatic Invariant e In  the l a t t e r ,  where conceptual s a t e l l i t e  designs 
vary widely i n  geometric shape, the  general i ty  of the formulas f o r  the 
l i b r a t i o n  frequency and the s t r a i n  energy, though approximate, should prove 
useful. The extension of the  analysis t o  cover l i b r a t i o n a l  motion i n  t h e  
f i e l d  of two centers of force proceeds i n  a manner d i r ec t ly  similar t o  t h a t  
f o r  a s ingle  center of force presented i n  t h i s  paper. 
This represen- 
12 
REFERENCES 
1. T. P. Mitchell  and H. S. Liu, "The Influence of Material  E l a s t i c i t y  
on S a t e l l i t e  Motions" presented a t  XVIth In te rna t iona l  Astronautical  
Congress, Athens, Greece 1965. Published i n  Proceedings of t he  
Congress. 
2. H. S. Liu and T. P. Mitchell, "Resonant Osci l la t ions of a Flexible 
S a t e l l i t e  i n  an E l l i p t i c  Orbit" AIM Journal, Vol. 5, No. 7, P. 1269, 
1967 - 
3. A. E. H. Love "A Tre t i se  on the  Mathematical Theory of E la s t i c i ty"  
Dover Publications, New York, N.Y. 1944. 
4. P. Goldreich and S. Soter,  "Q i n  the  Solar System", Icarus V o l .  5, 
P. 375, 1966 
13 
. .  
Fig. 1 Angle of Libration and Principal  Axes. 
x 
Fig. 2 Coordinate Systems to Determine Body Force. 
14 
APPENDIX 
T h i s  appendix is  devoted t o  out l inlng the s teps  i n  the  calculat ion 
of the  body force per un i t  mass act ing on the  s a t e l l i t e .  The force is  
composed of the  g rav i t a t iona l  force,  and the react ion forces associated 
with the accelerat ions of the  s a t e l l i t e .  Referring t o  Fig. 2, l e t  XYZ 
be an i n e r t i a l  coordinate system w i t h  or ig in  at  the  focus of the o rb i t ,  
X-axis directed toward the per icenter  and Z - a x i s  normal t o  the o rb i t  
plane. The xyz system of axes is  f ixed i n  the  s a t e l l i t e  w i t h  or ig in  at 
the  center of mass and axes coinciding with the  p r inc ipa l  axes of i n e r t i a .  
The z-axis i s  p a r a l l e l  to t he  Z -  axis. Then 
.. e. .. 
R = R  + z  
w J-JC 
with . .  .. ‘2 + (2Rc0 + R e o )  = (Re - Reo ) .. .. R 
IWC 
and 
.. .. .. 
i n  which 
r e l a t i v e  t o  the  xyz system and rg) i s  the r e l a t i v e  angular ve loc i ty  of the  
and a re  calculated r e l a t i v e  t o  the  XYZ system, zr and Cr 
two systems. Thus 
On using the  f a c t  t h a t  t he  center of mass i s  moving i n  a Keplerian Orbit 
and t h a t  the  problem i s  being considered within Inf in i tes imal  E l a s t i c i t y  
Theory one f inds  
15 
7 
1 W  
.. .. * 2  
+ [- Sin cp - ( e  + cp )  r s i n  5 + ( 0  + cp>r cos 5 j 
R% 
where t an  5 = y/x. Substi tution from t h e  planar l i b r a t i o n  equation 
.. *. 
e + c p = -  3c ' ("-A) s i n  cp cos cp 
C 
R 
produces the  form 
- 2  1 
+ [- -$ Sincp - (6 + S) r Sin 5 - (F)r Cos5 Sincp Coscp' j (A6) 
R3 -1, 
C C 
m e  gravi ta t iona l  force -per un i t  mass i s  
3 0  
R3 
Cos5 + -COS cp COS ( 5  + cp)  Kr Coscp - - 
R3 
C C 
= E- 
C 
30 sincp cos ( 5  + cp)  I d  Kr R3 + sincp - - Sin5 - - R3 
C C C 
+ o($) 
C 
R 
16 
The t o t a l  body force per un i t  mass i s  accordingly 
where the  notation 
P = 3 (a + 1) sincp coscp * R3 
C 
- 1) Sincp cosy 
' 2  K 2 
p4 = ( cp  + 8) + - (3  Sin cp + 1) 
R3 
C 
and 
a = ( B  - A)/C 
i s  used. Equation (A8)  coincides w i t h  equation (2)  of t h e  main t ex t .  


The librational dynamics of deformable bodies

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