Theory of coplanar two-bar zero gravity simulato

Byars, E. F.

Shreeves, R. W.

Yuan, H.-F.

English

NASA Technical Reports Server

FINAL REPORT 
on 
NULL GRAV I TY S I MU LATOR 
NASA - NsG 533 
Hs i-Fang Yuan 
Edward F. Byars 
R o b e r t  W. Shreeves 
N67 1 2 2 1 8  
Y a 
s ITHRUI (ACCESSION NUMBER1 
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https://ntrs.nasa.gov/search.jsp?R=19670002889 2020-03-16T16:58:13+00:00Z
) -  
' e  
F i n a l  Report 
N u l l  Grav i t y  S imulator  
Hsi-Fang Yuan 
Edward F. Byars 
Robert W. Shreeves 
As was pointed o u t  i n  t h e  proposal, a c o n d i t i o n  of zero g r a v i t y  pro- 
duced by mechanical means on t h e  surface of t h e  ea r th  i s  impossible. The 
ques t ion  a r i s e s  as t o  whether or no t  one could approximate t h e  c o n d i t i o n  
o f  zero  g r a v i t y  by t h e  u t i l i z a t i o n  of  some simple mechanism. 
/!?I 
q.2 Figure I shows a coplanar two-bar me- 
chanism w i t h  f i x e d  cam a t  A and movable 
cam a t  B. From such a mechanism one 
hopes t h a t  t h e  s t a t e  o f  zero g r a v i t y  
can be approximated f o r  mass m by ap- 
p rop r ia te  p r e s c r i p t i o n s  o f  t h e  motions 
\ 
\ 
, I  ir XI 
, ,, I ,  // / ' ,' I' 
Figure  I of  t h e  two bars. The k inemat ic  con- 
s i d e r a t i o n  of m leads t o  t h e  expressions for t h e  v e l o c i t y  
and t h e  acce le ra t i on  
w i t h  F igure  2 
2 
v R  
/ I  
The dynamic equation of motion of rn 
gives 
The weightless state fur m requires 
that no surface force, i.e., R and T 
acts on it and that the resultant in- 
ertial force acting on it balances 
Figure 2 that of the gravitational one. 
Hence for total weightlessness equation ( 4 )  reduces to 
or, after substituting aml and am2 from equation ( 3 )  
aJ;  shy  - aji2cosy - b 6 2 + $ 5 ~ g  = o 
uy C O S J  + &f2Jh9  t bb'+$CosO = 0 
which has a lso  been confirmed by Hamilton's Principle. 
It can be shown from equation (6) that 
I ett i ng 
(6) 
one has 
where 
Equation (7) can then be transformed to 
As no general solution for equations (61, (71, and ( I O )  has been found, 
it is commn to simplify equation ( I O )  for certain particular cases of 
constant angular velocities. 
Adopting the new initial configuration, t = 0, $o = eo = f o = o  
as shown in Figure 3, equation ( I O )  becomes 
il The initial condition requires that 
m 1.
A7 For a= zb equation (12) leads to 
2 Figure 3 3 = b 2 d  
from wh i ch 
where . 
Case A 6 = w = const. 
Equation ( 1 1 )  is then reduced to 
0 
4 
One manages, first of all, to linearize it by assuming 
whose validity will be verified with the solution one gets therefrom. 
Equation ( 1 5 )  becomes, after linearization 
The solution for the first branch has been obtained as 
f = -  K* (18) 
[ cos f - s I..'? / O j  - 2 G s&U=fq 
9/s, where y 
for t = o ,  Y = O .  
i s  an arbitrary constant and evidently g-= KO = 0 
A computer program has been set u p  to evaluate q for K~ = I and 
arbitrarily assigned y s under the condition of equation (16). It does 
not provide any output, because of the breakdown of the assumption. The 
second branch of equation (17 )  is also an equation of the Riccati type 
whose solution depends on the solution of 
I 
0 
y" - $ ( I 8  
The solution of this equat 
mentioned above. 
Case B = wo = -
on was not investigated because of the reason 
const. Equation (7 )  becomes 
. . 2  . 4  
No solution to equation (20) has been found. 
Case C Small angle changes. One tends to use Taylor's ex- .. -
pansion to approximate equation (6) and (7).  It does not, however, 
pear t o  ease t h -  non- l inear  s i t u a t i  
l i n e a r i z a t i o n  of equation (201 g ives 
6 2  g + e 4  = do4 - z d 0 t  
n o f  t h e  equations. For example, 
(21 1 
I t  i s  s t i l l  non- l inear  and t h e  s o l u t i o n  o f  t h i s  equat ion i s  no t  ex- 
pected t o  y i e l d  very p r a c t i c a l  information. 
A t  t h i s  p o i n t  one might f e e l  t h a t  t h e  reac t i ng  fo rces  R and T could 
be minimized instead o f  r e q u i r i n g  them t o  vanish. Such i s  n o t  t h e  case, 
:r, t h e  c l a s s i c a l  sense, s ince t h e  min imiz ing p r i n c i p l e s  requ i re  R and T 
to vanish i d e n t i c a l l y  ( t h i s  i s  no t  surpr is ing) .  
Since t h e  above approach f a i l s  t o  a r r i v e  a t  any f r u i t f u l  resu l t ,  a 
f u r t h e r  i n v e s t i g a t i o n  i n t o  t h e  equation (14) i s  t he re fo re  necessary. 
The i n i t i a l  cond i t i ons  a t  t = 0 . ( y o =  #o=y=O) requi re ,  from 
equations (121, (131, and (141, t h a t  
I t  
and q = i n d e f i n i t e .  
0 
Solv ing  q i n  terms o f  q' and assuming JI being c lose  t o  i t s  i n i t i a l  
value, t h e  f i r s t  two terms o f  equation (14) o f  second o rde r  of magnitude, 
can be neglected. Hence as a r e s u l t  o f  t h e  approximation equat ion (14)  
becomes 
6 
ana 
substituting equation (24) into aqiiation (141, one has, after lineari- 
zation, the second iteration for q 
Tne third iteration can be obtained following the same procedure as 
Substitution of equations (23) and (24) into the left hand of equation 
( 1 4 )  leaves 3~~ sin JI which vanishes only at the initial position. 
Though equation ( 2 3 )  does not satisfy equation ( 1 4 )  exactly, it 
2 2  
does satisfy the condition of equation (16). Equations (25)  and (261, 
nowever, are not as such, for equation (25)  makes 
t-ience equation (23)  should be the best approximation among all the solu- 
tions obtained by the above mentioned iteration procedure. To ascertain 
such a claim, numerical evaluations of  R and T according to equations (231, 
(251, and (26) respectively are necessary. 
The radial and tangential components of the surface force acting on 
the point mass are given respectively by 
For 6 = uo = const., 0 = toot  and 0 
equations (12)  and (13) has been applied. 
Computer programs are then devised t o  evaluate t h e  L%.. and L' 5 mq 
J J 
from equat ion (6)'. 
For t h e  f i r s t  i t e r a t i o n  
where 
0 
then 
-7- The val i ies of -< and " ~ 9  a r e  computed f o r  a t o t a l  range of 45O f o r  /n3 
t h e  u n i t  i n t e r v a l  of  lo. 
The computations f o r  the second and t h i r d  i t e r a t i o n s  a r e  c a r r i e d  out 
!n a s i m i l a r  way u,sing IBM 7040 computer. 
' 8  
T h e i r  r e s u l t s  follow from Table I: 
. ' 
- 'I 
I 
I R/mg i T/mg 
1 + ( " i  - 
I 
j First Second T h i r d  First Second Th i td 
0 
I 
2 
3 
4 
5 
6 
7 
8 
9 
I O  
! I  
12 
13 
14 
15 
i6 
17 
1 8  
14 
20 
2i 
22 
25 
2L; 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
0 
-.OOO I54 
-.0006 14 -. 00 I 38 I -. 002453 -. 003827 
-.005502 
-, 007476 -. 009744 -. 0 I 2303 -. 0 I5 I49 - . 0 I8278 
-.02 I683 
-.02536 I -. 029303 -. 033506 
-, 037960 -. 042659 -. 047595 
-.052759 
-.058142 -. 063736 -. 06953 I 
-.0755 I6 -. 08 I680 -. 0880 I 2  - . 09450 I -. 101 134 -. 107899 -. I 14783 -. 121772 -. I2885 I -. I36007 -. I43223 -. I50486 -. I57777 -. I6508 I -. I72379 -. 179656 -. I86890 -. I94065 
-.201 I60 
-.208 I54 -. 2 I5027 
-.22 I757 
-.226322 
C 
-.0006 I5 
-.002456 -. 005520 -. 009796 
-.O I5274 
-.02 I939 
-.029772 
-.038754 
-.048858 
-.060059 -. 072326 -. 085626 -. 099922 -. I15175 -. I3 I343 -. I48380 -. I66239 -. I84868 -. 2042 I4  
-.224219 - .244825 .-. 265968 - 287584 -. 309603 
-.33 I955 -. 354564 -. 377355 -. 400245 -. 423 I52 -. 445986 - . 468658 
-.49 107 I 
-.5 131 25 - . 5347 I 7 -. 555735 - . 576064 
-.595581 
-.6 I4 I55 
-.63 I647 -. 647907 
w.662773 -. 676066 - . 687593 
-.697136 - . 704452 
0 
-.001884 
-.007535 - . 0 I 6953 -. 030 I34 -. 047070 -. 067749 
-.092 I55 -. 120268 -. I52062 -. I87506 -. 226563 
-.269 I88 -. 3 I5329 -. 364924 
-.4 I7905 -. 474 192 -. 533694 -. 5963 I 2  -. 66 193 I -. 730427 
-.801662 
-.875483 - ,95 I 726 
- I  .030213 
- I .  I10749 
-I. 193131 
-1.277138 
- I .  362540 
-I .449094 
-I . 536549 
- I .  624648 
-1.713133 
- I  .801752 
- I  '890276 - 1 .978523 
-2 . 06644 I 
-2. I54395 
0 . 035063 
.0701 I I 
.IO5131 
.I40109 . 175032 . 209886 . 244658 
,279336 
.3 I3905 . 348352 
.382665 
.4 I 6830 
.450835 . 484667 
.518312 . 55 I759 
.584995 
.6 I 8007 
.650784 
.6833 I4 
.7 I5583 
,747582 . 779297 
.8 I07 I8 
.84 I833 
.872632 
.903 IO2 
.933234 . 9630 I 7 
.992439 
I .02 I492 
I .050164 
I .078446 
I . 106328 
I . I33800 
I . 160853 
I .  I87476 
I .2l3662 
I .264682 
I .289499 
1.313841 
I .33770l 
I .36l068 
I -383934 
1.239400 
0 
.06 I 354 . 122646 
.I83819 
,2448 I6 
.305579 
.366052 
.426 I79 . 485904 
.545 I73 . 603932 
.662 I 30 
.7 I97 I5 
.776638 
.a32848 
.88830 I . 942949 
.996750 
I .04966 1 
I . IO1642 
I . I52655 
I .202664 
I .25 I633 
I .299532 
I .346329 
I .39l997 
I .4365IO 
I .479844 
I .52 I978 
I . 562893 
I .602572 
I .640999 
I .678163 
I .714052 
I .748657 
I .781972 
I .8l3990 
I ,844707 
1,8741 19 
I .go2225 
I .92902 I 
I .954504 
I .97867 I 
2.001513 
2 . 02302 I 
2.043 I79 
0 . I33637 
.266934 
0399544 
.531 I I8 
.66 I3 I3 
.789784 . 9 I6 193 
I .040203 
1.161483 
1.279707 
I .394553 
I .505707 
I .6 I2863 
1.715719 
I .8l3986 
I .go738 I 
1,995632 
2 . 078476 
2. I55662 
2 . 22695 I 
2.2921 I7 
2 . 350944 
2.403233 
2.448797 
2 . 487466 
2.5 I908 I 
2.543503 
2.560605 
2.570278 
2.572425 
2 . 566964 
2.553823 
2.532934 
2.504228 
2 . 476607 
2 . 422903 
2.369692 
I 
8 
It is obvious then that the first iteration gives the best fit for 
weightless state for m as predicted. 
Conc I us i on 
I .  Exact ,alution of the null gravity state for m has not been found 
except at its initial configuration shown in Figure 3. It is un- 
likely that a prolonged exact solution for null gravity exists for 
the simple mechanism shown in Figure I or, possible, any mechanism 
one can construct on the earth's surface. 
m happens either to be a satellite of the earth or to be subjected 
to a free falling condition. 
It i s  possible only when 
2. No minimization of either R or T uf the surface force is possible. 
3. An approximate solution for the motion of the mechanism close to its 
initial configuration (Figure 3 )  has been found in the form of equa- 
tion (IO)'. The angular displacement as a function of time t can be 
S 
t 
on 
found, numerically, using a digital computer. The deviations of th 
equation from the ideal null gravity state are listed in Table I .  
can be seen that the deviation will reach about one-g after the mot 
carries on for 0.5 seconds from its initial configuration or for a 
corresponding angular displacement J, L 30'. 
4. No better approximate methods'have been found. 


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