Theodore W. Wright

Glenn Research Center, Cleveland, Ohio

A Raster Based Approach to

Solar Pressure Modeling

NASA/TM—2014-218318

June 2014
NASA STI Program . . . in Prole

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientic and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Ofcer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signicant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signicant 

scientic and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientic 

and technical ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientic and 

technical ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientic and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientic, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientic and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI 

Information Desk at 443–757–5803

• Phone the NASA STI Information Desk at

 443–757–5802

• Write to:

 STI Information Desk

 NASA Center for AeroSpace Information

 7115 Standard Drive

 Hanover, MD 21076–1320
Theodore W. Wright

Glenn Research Center, Cleveland, Ohio

A Raster Based Approach to

Solar Pressure Modeling

NASA/TM—2014-218318

June 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identication 

only. Their usage does not constitute an ofcial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 


	

		

	

	

	 	

	



		
	

	



	 

		

	



				
	
				

		 
	! 
	


	
	
 	"# 
	



		 $ 	!
	



	 	
	


 	



					%			
			


	
		

 
			

			


		&	
'
		
	

	
	

				 
				
		

	

	%	

A Raster Based Approach to Solar Pressure Modeling 

Theodore W. Wright 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

NASA/TM—2014-218318 iii

	



 

 


 
 


 




 

"( )		$                            *

 
 

! "#
 $

 "
	
 $

+( ,	-		-)	
	                          .

+/ ),
$0                            (1

+" 2	)	
	,34	                         (1

$ 	

%&

 



' 
 


5( 6		

                               (/

5/ $

	
	

                                   (*

5" $		

                              (*

( 	%) "
**+ 
!


, 	
-. 
(

(1( -7	                                    (8

(1/ ,0
	,#	                               (8



 "/ 


 "
 

0-1--
. !

2 	%) 
0

	 "	

  

NASA/TM—2014-218318 v


 



	

		

	

	 

	 




9
9$$:;<
	=	)4< 	0-)<! 



4	 ,	 
		=		 	




		

		



		
 0



			



		

		
'				

	
	=




	

	

	%		 
 
! 
	=	

	
	

	






	
 					
)4< 	

)4< 

 	

	
			


		

	
	 			

	
		 $ "# 
)4<

		/# 	

	
	
 	

	

	
	
	 4	
		

	%		 

		

%		

	

	
	
	

 


;			
			
			


0
	
 	
		

			

	
	
	

	>

? 		/# 		!			
!

? 		
	


? 
		&
	&

? 	
@ A 	!	B !
	
	

? 
	

	

	
		


'	



	
		

7		
	 	'	
		

		

$ =

	

	

	
		
'		

	
	=		 
	
	
		' 
	

NASA/TM—2014-218318 1
, (> )4< 

		
	
	

 






,	



	 	)4< 	

	

			
	
		;7C 	 

				
	;7C 
			
 


 
! 		/111
) 		


, ( 	
	
)4< 	 	, / 	

				&		

			



! 		

		

	&
	
	 , " 	
)4< 
	




 		
	
	 D=	 	


	
	
		

	
 		


	
	

	

	)4< E3F 
	

)4<,	

	)4< 
	
	 	"# 	

	)$#  2 	

	

	

	


		

	

	

NASA/TM—2014-218318 2
, /> )4< 			
	


, "> )4< 




	

NASA/TM—2014-218318 3
$	 			%		


	
	


	 %		
		
	
	

	
	
	





	D	
$; @ 	C	

	! 
	
	

&
		 


	
	
	 ;

&	
	&


 


		
	 	 
	


	
&		

&		

	 $


&				
	
&!			

$ $!	

			
	

	

				

	
		



	
		! 		

	


	
		

		
			 		=

			

	

		
	 2 	

	

		 
			

		=
	=			







;7C 	
 

			
	

		
 	
 $					




	
	
		



 ;7C 
	 



[x, y, z] 
		
		
 	&	

&

		

4
 	

"/-


		



	> 

	

	

	

 
	!		



	

	

 
	!( 	
	

	

	
:	 
			 		

	


		1
 	 	! 
		



	
		
)4< 		
	

	
	
=> "51" m 
	
	
=G
	

	

! 	
	


	






B 		


	 



! @ 	
 A 	

($			


	;)0 
	




	
$ H

NASA/TM—2014-218318 4
, *> <
	



	


 	

	
		 	

B 		! 				@ 	 

					@ 	

, * 


	 	

	
			1

					 				

	
	


B 
! 
	
			 

	


 [x, y, z] ;
	
	
 [−y, −z, x] 	


	

	

	
	
	
!	
		
&"4	> 
 	 

	 		 9			:	 
 	 	

		
 -
				:

@ 	 $ 	


	1
 	 	!	

	

	
	

 A					

:A 	
						
! $ 		



			

	
	

 3

				:B 	
					

	
	I		! $ 

				

	





=	
 -
			81	

J81 A			
	


1"+1! , K )4<

	



	 [x, y, z] 			

	 [x

, y, z] 		
 	 	

				
			 		



			 
	
		!		

	


   /K   	
=	 	

	
    *1  	

	!  	 	:	 	 *1 J/K    	 	


	
			 
	
1"+1

6	

	

	 		


	


NASA/TM—2014-218318 5
, K> )4< 				 [x

, y, z]

 		


		

	% 




&
	
=		
	
		

6
	%
	

	'	
	 

%

	
'	
		
			=

	


	
							%




	 [xm, ym, zm] >

	 = arcsin(−ym)

 = arctan xm

−zm



⎡

⎣

xm

ym

zm

⎤

⎦ =

⎡

⎣

sin()cos(	)

−sin(	)

−cos()cos(	)

⎤

⎦

 


4	
	

	

	
	
	 




	%
		! 




%
	
		

! ;

	

 
	
	=		





		= 	
	




		
  H;C! )4< 	E3F
		%


NASA/TM—2014-218318 6



 


	%
&! )


)4< 


	 ,%		
	

	
	 


		

'		
	 

%
 	
	%


$	

	
					' $ 	


	

 	
'

	
		
		

	
'	
		


%

	


 


					L
 #


	L
	



	

! "#



	
	
	'
	
		 
		
	


=	"# 
	
				

	'			
E"F 
	
	
		

				

		

		
	

		

	

	
! 6
		 



	



	=
	
	
		
	 	

	




	

		 6

= 




	
	

 "
	


;
 		 		

	
	= 	&


					

	
	 	

				

	%		 ,
	
	
	
	
	
	 	





	

	


 4
 5"
	


	

	
	%				%		:

					

>

fN =

r0

r

	2 J0

c



 r0 4	:	

 r 	

	
:	

 J0 

			%	 r0 	 c 
 



	

NASA/TM—2014-218318 7
	%
 				%
! $ 
	

%
	

 		

%
	
	



 2/3

	


	
	%		
		 θ 
		

%
	E( /F> 	



		 F⊥ 	

	


		 F

	



		
	 $




 γ 

%

		
				 	 (1−γ) 



	 $


 β 
	

%


		
	  βγ 

%

	 	 (1 − β)γ 



%


 

	 

%
	

	
	 	

%


(11M

	



		>

f⊥ = fN




(1 + βγ) cos2 θ +

2γ

3 (1 − β) cos θ



	


		>

f = fN (1 − βγ) sin θ cos θ

	






 	
		


		%
 



				

		%

		 θ 
		! 



	


	%
	
		 





	

%
			
	


 6	 	




	%
	>

θR = θ − arctan 

(1 − βγ) sin θ

2γ

3 (1 − β) + (1 + βγ) cos θ




, + 	   
	!
)4< 

		
	%		

		 	



 

	%

 
	! 	

		
 



		%	

	



 			=	
		

					
		
	
	

	!

6



	 		

	
	  



			
! 6
%			
		

	

	%		 θ 	
								

cos θ θ 			 90  
 θ 		
 90 	
			

	
	

NASA/TM—2014-218318 8
, +> ,


		%


NASA/TM—2014-218318 9
 4 	
5



3	&						
	
	!


 %	 	  fi 	
	  	 	 ri 


	
	
	L
	> 	'	



	
	
		!>

ftotal = fi

mtotal = (ri × fi)

6
	

	 rcm =
	! 
:	



	
	
	
	
	>

mcm = mtotal − (rcm × ftotal)

$ 

		

	&	
		'	 	

 
 
		 
	 $ 	 
 
  	 
	

 (ri × fi) & 2 
	

 fi 	
	

	
 

	"

 4 6	
5

-.%

$ 	
	
	
	


				
	


	

) 	)4< 	4	 	 
  r0/r = 1! 

				
	

	 			

	
	

		"

> 
	
			 	

	
=	

	
			

! 	
	
			"1

m 		
	
	/. m 					>

 = 2πr2

array + πr2shell

= 2π(3.0 [m])2 + π(2.7 [m])2

= 79 [m2]

	%	4		 J0 ("+. J/m2s '	%


 c 
 2.998 · 108 m/s>


 = J0/c

= 1367 [J/m2s]

2.998 · 108 [m/s]

= 4.56 · 10−6 [N/m2

]

NASA/TM—2014-218318 10
0%				

>

	 = 
 · 

= 4.56 · 10−6 [N/m2] · 79 [m2]

= 0.000360 [N]

 	 	
 
	 
=
		 			

 

	
			 
	
= 				




 	=
	
=	


		 


	
						N %

	
	



		

$ 	

%&



$;7	
9			)			
	

=	


	
	 ,
	 ;	

	
	"	!	

		
			*	!		= 	

	 		

	

=	
	
		 

	
			=



	

	

 	

	

			
 	&!		

	
 <	 		

		K					'		

			

	 " 		 *!

	&'
=

				 ,	 
	




		81	J81 	
	

	 	

				
		
1"+1 						


*1J/K 				K
 

 (180/5 + 1) · (360/5)4 · (65/5 + 1)2 = 194, 889, 203, 712 	 6
	

  	&	


	5
%	 

	
	' 3 · 2 · 8 = 48 
	
 	'		
5KH


 6
	K(/K(/	
	
		=1"

 

(511	'
		 )	 	

		

	

;									

	  			

	
	 
 	! 
	


						 			 
K 

	
 (180/5 + 1) · (360/5)2 = 191, 808 	 '**

0H 
 	
	 (+ 
 , 			  
	
 	

	


			
	
	 
O	O
	L

			
 			

	



! 
	
		=


NASA/TM—2014-218318 11
$
				
	 3
		







	'	
	 

	




		
	 	
			

		 	(1
									

		 (180 + 1) · 360 = 65, 160 	 '"10H 
	


	K


$						
	
		
	
		


	
	
			 		


				

	 			  
 	
	   %
	
	 



	
						

' 


'4
 -




	

	


	
 $ 

		

	 			=
	
	

$ 	
			

	'	

	

	
	
	

$		 
 
				


 N 			

	
	
	
	 

	
		


  			 	 0		

		
	

	 ( 	&
		L
		

	
 

	
	
		

	
	

	!
	
	

%			=		
	
*'	 		





 		


	
	/5"K1K

9B 
! 6	 '& O-	O

	
%		!

	
	

	


	
			



B 	! 		 
	

			 / 
			

	

	&
	

L
		

		

H		 ( 
	




		


K(/K(/ 	&
( 

	
	"


	
	
	 			

"M

/

					30	'		 	

	
		 
		
			 		

NASA/TM—2014-218318 12
	 (> -
	
 

  3 $	 -	 ,
 0

& 3 s! m2! 4N! 4N · m!

K +*+* 11118 "..K.*M! "1 (+"1+K.*M! "*/1+

* +*+* 11118 "8*(*+M! *8 *(K1.(*+M! /511+

" +*+* 111(/ *1.(51M! 81 K(/1.(51M! //81+

/ +*+* 111(5 "8*(*+M! (8+ *(K1.(*+M! 5KK//

( +*+* 111*5 "8*(*+M! .5* *(K1.(*+M! (*11+

K (/5(/5 111/K *(K"+8M! ("/ (1K1+"+8M! /(K1+

* (/5(/5 111/+ "8*(*+M! (8+ *(K1.(*+M! (*11+

" (/5(/5 111*1 *15/18M! "+( K8/1./18M! (+*//

/ (/5(/5 111+/ "8*(*+M! .5* *(K1.(*+M! "(+//

( (/5(/5 11(5+ "8*(*+M! "("+ *(K1.(*+M! .111.

K /K+/K+ 11188 "51*8KM! *5* (*11+*8KM! ""51.

* /K+/K+ 11188 "8*(*+M! .5* *(K1.(*+M! .111.

" /K+/K+ 11(K1 "851+1M! (*1+ (.11.1+1M! KK81.

/ /K+/K+ 11/*. "8*(*+M! "("+ *(K1.(*+M! **.//

( /K+/K+ 11.+/ "8*(*+M! (/K** *(K1.(*+M! "K11.

K K(/K(/ 11"1( "851K5M! /1/K (+K1.1K5M! (*+/(

* K(/K(/ 11*18 *1(1"1M! "(8/ 5*(151"1M! "K+1.

" K(/K(/ 11+"* "851K5M! K+/K (+K1.1K5M! ((*/(

/ K(/K(/ 1(1/+ *1(1"1M! (/.+8 5+"151"1M! "K+1.

( K(/K(/ 1"("" *1(1"1M! K(1.+ 5+"151"1M! (.51.

K (1/*(1/* 1(//+ *1/1K/M! 5(81 (*51.1K/M! //51.

* (1/*(1/* 1(+/5 *1(1"1M! (/.+8 5+"151"1M! (.51.

" (1/*(1/* 1/K+. *11115M! //+K1 /"(15115M! (*(1.

/ (1/*(1/* 1*(+" *1(1"1M! K(1.+ 5+"151"1M! /+5//

( (1/*(1/* (/KK( *1(1"1M! /1*"1* 5+"151"1M! 58(15

NASA/TM—2014-218318 13
'4 /


		

	
					
 $ 		

		 	'	&	







	 / 	




		L 6	 	

	
		%	


	 2 

	
	
		

		

	 		
	K(/K(/	
	%		 

	&
(	! ;		=


			@ 	A 	
	
!

					'	 

!		1"M 

				
1"/M

	 /> P	

  $	 $	4 -	 ,
4 04

( 1"(* *1(/ 1"1M K(1.+ 5+"15 (.51.

/ 1"(/ "88* 1(*M K15K1 "8K15 "K(1.

" 1"(* *1(/ 1"1M K(1.+ 5+"15 /5/1.

* 1"(/ "88* 1(*M K15K1 "8K15 "K51.

      

8. 1"(/ "88* 1(*M K15K1 "8K15 ".K1.

85 1"(" *1(/ 1"1M K(1.+ 5+"15 /*+1.

88 1"(" "88* 1(*M K15K1 "8K15 //(1.

(11 1"(( "8.. 1K5M K1+/K (+K1. *551.

0	 1"(8 "888 1"1M K181+ 51K15 /K51.

# 111(/( 11(/8 1"/M (+*+ *"815 (((1.

'4 -




	

				=	 	


	)4< 					" /K	 



OO 	
			

		/K	

	
			K (1 	/1	 

		

			
			

" ,	 	
1"+1/K 2 
		

					
		



π/64 /5(/K

NASA/TM—2014-218318 14
	 " 			 

				

		K (1 	/1				
/K K 	(1

	 						

	 "> 3	

$	 ,
0	 00	

	
 5 ◦ 1(+M 1(+M 155M

	
 10 ◦ 1/8M 1/5M (/KM

	
 20 ◦ (1+M 185M "11M

	 * 			 

				


				 π/32 		K+/K! π/16 		((/K! 	 π/8 		//K

!				 
 π/64 		 /5(/K! π/32 		

K+/K! 	 π/16 		((/K!	

	 *> -
A		

$	 ,
0	 00	

	
 5.625 ◦ 1*KM "++M 1*5M

	
 11.25 ◦ (//M 515M (/8M

	
 22.5 ◦ /..M (5"M /5"M

	
	
		
				

	 $		

 π/32 		K+/K!

			(1




=			(M 

	
"++M


	

( 	%) "
**+

/K	
			

	
	

	
	)4< 			 			 

 	'


K81((/'
	

 		

	! 

&					 	


	%


			


"/"

, . 		
			
		 	

			." m2 = $ 
			

	.1 m2 		"/ m2 0				
.5 m2 

		51 m2 .8 m2 
	
	
		


:	

NASA/TM—2014-218318 15
, .> 2	
			
		

30 40 50 60 70 80

illuminated area (square meters)

0

10000

20000

30000

40000

50000

60000

70000

80000

number of attitudes

Area Distribution

, 5 	
	


	!		




	 

	

		 ,
		

/11	*11 μN 
 	!
		
	
		



	
	 		
	




, 8 	
	'
	 		



		!		


	 

	


 
 	 	 '	 	  1 	 ((11 μN · m 0	 
 	!


		
	
		'	
	 

1+11 μN · m 	 2 	
 	!
		

'*11 μN · m 	((11 μN · m 
	

#
				&		
'

		
	 6	 		
	
	
		

	
					 
'






			

, (1  	  

	 
	    


	   

		

	
: 
	
 		


&! 

		

	
	81	


			


, (( 	'	
			



NASA/TM—2014-218318 16
, 5> 	

	 			

 	!

0 45 90 135 180 225 270 315 360

roll (degrees)

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

0.00040

solar pressure force magnitude (Newtons)

Solar Pressure Force vs. Attitude (2.5 degree roll, pitch, yaw sweep)

, 8> 	'	 			

 	!

0 45 90 135 180 225 270 315 360

roll (degrees)

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

solar pressure torque magnitude (Newton meters)

Solar Pressure Torque vs. Attitude (2.5 degree roll, pitch, yaw sweep)

NASA/TM—2014-218318 17
, (1> 	

	
	

, ((> 	'	
	

NASA/TM—2014-218318 18
		! 
	 '		&!

	

	
:
	
		
&! '	

	

	
		+1 	
		



, 	
-.


,4
 
+1


		
	

	




		




 		&	
	



	
			

		





$				
	

		
	
	

	 watts/m2 &
					 

			

	

	
	 	

		



				 		


			
	

		





	
			%
	
 1367 watts/m2 		

		




(.M 3
	
		
	
	


	





,4 5
5
+7-

6

	

		

 

		
	
 

	%	 6 

	%	


 			
	
	
 %

	
	' 	&	


				

 	

	%

				

 	
		

$
	&	 	
			%			 				


HE*F 		!


	%	

	%	
	

		 A

	>

NASA/TM—2014-218318 19
Cn =

(1 + )

√π

s sin α + (1 − )

2

 Tr

T∞



e−s2 sin2 α

+



(1 + )(1

2 + s2 sin2 α) + (1 − )2

 Tr

T∞

√πs sin α



(1 + erf(s sin α))
/s2

Cp = (1 − )

√π

s cos α



e−s2 sin2 α + √πs sin α(1 + erf(s sin α))

/s2

	

	


	%					


			


 

	
 	

	
>

F ̄ =

ρV 2

2



i

Ai(Cnin ̄i + Cpip ̄i)

			


 !"! 

	%	



	



03) 

		r ̄!>

M ̄ =

ρV 2

2



i

r ̄i × Ai(Cnin ̄i + Cpip ̄i)

α 	



 	 α = 0 					 

α = 90 			
	
 	 α = −90 			
	
		


 ,

 α < 0 	& 				


	

! 9		



	
	
	

		



	
	
	α = 90 − θ!

  			#	 G


		
	%
 	(

!
				

  
	


	
		

	 
			
 0 


!
			 
	

E+F 

	
				
 0.03  0.13

s 
				
	! 6

	'	
			&
			L
 H	


		(1	
 6
	
	>

Runiversal = 8.3144621

Rspecif ic = Runiversal/(0.001 · mw)

Tr = (1 − )Twall +  T∞

s = V

2Rspecif icT∞

NASA/TM—2014-218318 20
Runiversal D	7	)	 Rspecif ic 

	
	
	


	 mw 

	 24.97578 kg/kg − mole! T∞ 
		 



	 746.22 K Twall 	

	
		 

			



	
 300 K ρ 
	 

	 4.4 × 10−9 kg/m2 V 


	 
	 7000 m/s


		
 mw T∞ Twall ρ V 	  

			


		;$#		H=  mw T∞ 	 ρ 			


	 

	

 $!% &'( 				
 480, 000 f t  V 	

	
	
4	 			



	
 		
4	:	 				

				
73$01.!

 Tr

T∞

	

	=	 !! 	


	
		
	




 7	

		

				!





		
		
 s ! mw T∞ Twall ρ V  	

Aref 

		!	
OO		

	

 F ̄ 	 M ̄ 
	

	


		


	
ρV 2/2!	

			
Aref )>

F ̄ =

ρV 2

2



Aref (Cxx ̄ + Cyy ̄ + Czz ̄)

M ̄ =

ρV 2

2



ArefLref (Mxx ̄ + Myy ̄ + Mzz ̄)

Lref 
	
			
	

 Aref 		> Lref = 2Aref /π

6	:
	


  −z 
 	 −Cz




 Cd  	#	 	


	

C2

x + C2y ! Cl  	#	

	
		;)0 
	
	





$ H!>

? @ 		
=!



? A 						



? B 	:	!



? 03

)
	
)4< 	
=


	

	
!

6;)0 
	
	


>

NASA/TM—2014-218318 21
? Cx  		#	

? Cy  		#	

? Cz  		#	

? Mx  	#	

? My  		#	

? Mz  	#	

6


		 Aref 

			





		! 


 
 
 
	 


	
	 	
	
 Aref 


;$#		H=		  		!	


 Aref 
 19.86 m2  30791 in2!	 	  
 O 
	
O 



						! OO		
 Aref 
		




	

				 Lref 
	
	
 Aref !



 "/

$ 	 			

	
	 	  
 	
 
 
 	

	
	=	

	




 $ 

	
			

	


	 		
/# 				%		
	
		



$ 



			

	
	
	

	

			


 			
				
	




		



 $ "# 	
	:	


	
	L



		

 	
			
	



	
	


			
			
					 4		

 		

	
			1"M

;	/1(1	=
 
	
		
	
		

	K(/K(/1"
 -
	
		
		

	
	

		=			 


		
	
	 ;
		

							=					 


		

	
		

	
	 

NASA/TM—2014-218318 22
	 		
			


	

	

		* 		


(1
	
		
 $	(1

	

	

3
			
	


	 	

				=
	


-	
=>

? 0	=				
	)$# 	


;7C 
	

? 6
	
	
	=			
		 ,	
	


	=	





!

*3
	
	
	
	

NASA/TM—2014-218318 23

 "


E(F 307
 	 )!*	+,
%	

")		

9$$ 

		""*8*

;
( (8.(

E/F -)2 )		%
-	

#-
	 6
 0	9A

/11*

E"F 0
	#
=& *!

>II=
	I
I	I		I		


P		/* /11.

E*F 7$H 	
$-	-	)
$*

;
D-

(88*

EKF P$
 %	-
.	)		*& 	
*

$	
)3/11"""8.!(

/11/

E+F $
	

	-C)	 *!#$

-
D-

(8K5

NASA/TM—2014-218318 24
, (/> +

	
	
		

, ("> 		
	
				





0-1--
.

$ 				
			


	

				 		
	
		OO

			
	

		

							
	, (/!		


8 		
	
	 6 
 4	

 6 
		

 	!
	


	

	
	

	
	
	 
	
		
	
	




 		
	
	 4

	, ("!

#
	


	
		



 π/2  π/4 		  18 	

 3
	


	 66, 258, 1026, 4098, ... 22n + 2 	

 
	

	
	

 2 	
	

			

	 
	


	
		&		 



	
	
'		
=

	&

NASA/TM—2014-218318 25
2 	%) 
0

	 "	







		

		

	'


	
	
	 [xs, ys, zs] ;	
	:	


	
	 [xm, ym, zm] 	
	=
	






C
	
			
 	

"51">

⎡

⎣

xs

ys

zs

⎤

⎦ =

⎡

⎣

3.803 − zm

−xm

ym

⎤

⎦

,


	
	

	>

Ftotals =

⎡

⎣

Fxs

Fys

Fzs

⎤

⎦ =

⎡

⎣

−Fzm

−Fxm

Fym

⎤

⎦

'	
	

	
	


	
	>

M

totalm = Mtotalm − (rorigins × Ftotalm)

⎡

⎣

M

xm

M

ym

M

zm

⎤

⎦ =

⎡

⎣

Mxm

Mym

Mzm

⎤

⎦ −

⎛

⎝

⎡

⎣

0

0

3.803

⎤

⎦ ×

⎡

⎣

Fxm

Fym

Fzm

⎤

⎦

⎞

⎠

	
	
	
	>

Mtotals =

⎡

⎣

Mxs

Mys

Mzs

⎤

⎦ =

⎡

⎣

−M

zm

−M

xm

M

ym

⎤

⎦

6


	
	
	 '
	
	
	>

Mcms = Mtotals − (rcms × Ftotals )

NASA/TM—2014-218318 26