APPENDIX A: SEPARATE SURFACE CALCULATIONS
APPENDIX A: SEPARATE SURFACE CALCULATIONS FOR LONGITUDINAL DYNAMICS The purpose of this Appendix is to present a summary of the method and results used to determine t h e elevator area and gain requirements for a SSSA system to achieve the commonality design goals.
A.1 Annie of Attack and Pitch Rate Gain Requirements From Section 6.2.3 o f Reference 13, the 2-Dimensional short period approximation was found to be:
W n = Z a M q / U l - M a (A. 1 )
SP = -(Mq + z a / U l + M a ) / 2 h S p ( A . 2 )
3 SP
Where M a is the dominant term for s h o r t period frequency and M q is the dominant term for short period damping.
From Table 6.3 of Reference 13, -2 M a = S Z Cmlu / ~ y y (sec (A.3) - 1 c .
Mq = q S 7 5 2 Cmq / 2 Iyy U l (sec 1 ( A . 4 ) The relationships for angle of attack and pitch rate gains were found in Reference 13 to be: K a = A C m a / Cm6E (A.5) K q = (ACmq 1 Cm6E) E/2Ui (A.6) where Cma was determined as: ( A . 7 ) and + Z a/Ul + M u > (A. 8 ) w h e r e '7sp=fspdes -7spbasic when either is The inter-related nature of W n s p and ljsp modified was ignored for simplicity of the model.
These gains were calculated based upon the normal control surface sizes and must be multiplied to account for the ratio of Separate Surface sizes to the primary control surface sizes.
A . 2 SSSA Longitudinal Surface Sizing Requirements The minimum required surface areas were determined for one percent probability and thunderstorm gusts. Using the VonKarman scales in Section 9.8.1 of Reference 14, the root- mean-square gust intensity and the resulting change in angle of attack due to gust perturbation were determined to be: TABLE A.1: Longitudinal Gust and Perturbations Clear A i r Thunderstorm
* *
Cruise Min.Contro1 Cruise Min.Contro1 u (fps) 4.6 6.6 21 21 W' a gust' (rad) .0066 .0318 .0302 . l o 1 2 it At 500 ft. altitude It is obvious that the critical flight condition that will sire the surface required f o r the SSSA system is for a thunderstorm gust at min. control speed. The required elevator area was determined according to the method of balancing moments in the longitudinal axis as presented in Section 6.6.5 of Reference 13.
C m o c bagust = Cm 6E A 6E ( A . 9 ) where: A E m a x = i 20 deg.
C m a = Cmabasic + C m ' d e s F r o m Reference 13, Section 4.1.4, the relationship of the elevator to the affected horizontal tail area was determined to be:
-
Cm6E = - C L d n H Sh/S ('jlach - Xcg) t, (A. 10)
From this, i t is obvious that the percentage of required elevator area that must be dedicated to SSSA is:
Percent S = - ( CmdEreq 1 (Sh/S)/C (Fach-xcg)(CLaH) ( Q H ) (Z,) 3 (A. 1 1 )
E where: CmSE = C m a ( ~ r X ~ ~ ~ ~ / A 6E) req For a chosen elevator size for the SSSA control surface', the maximum gust intensity that the system can overcome was found as: Q ( *6Emax/hCm R
1 ( -CL M)flH ( Sh/S 1 (XacH-Tcg 1& ( A . 21 1
wMax req From these relationships, a spreadsheet analysis was defined to show simultaneously the effect of design choices on the requirements for all of the airplanes. This facilitated the trade study shown in Section 2.1 of this report, from A sample spreadsheet is which the design point was chosen.
presented for the design point at the critical min. control speed in Table A . 2 .
' TABLE A . 2 Sample Spreadsheet f o r Longitudinal Dynamics
4774 I ;4w . 1 3 -1,5551 -.m ,5968 I.:.! .m4 1*=341
d.ha.des 3455
K-a -.3142 -.2607 - . 1 X -.0710 .M20 ,3121 -.:c25 -.m -.DE - . w 2
d.h. a.req -.76r"J -.lFA -1.1255 -.5131 -2.3417 -1,1307 -.74¶? ,9651 -!.23?5 ,701
APPENDIX B: SSSA CALCULATIONS FOR
APPENDIX B: SSSA CALCULATIONS FOR LATERAL-DIRECTIONAL DYNAMICS The purpose of this Appendix is to present a'summary of the method and results used to determine the rudder area and gain requirements for a SSSA system to achieve the commonality design goals.
B . 1 Yaw Rate Gain Requirements From Section 6.3.5 of Reference 13, the Dutch Roll approximation f o r Lateral-Directional dynamic stability was found to be:
W n = d i l u 1 ( Y B N r t N p U 1 - NpYr)l (B. 1 )
D ( B . 2 ) = -i/2dnD (Nr t Y W I ) 3 D Because of the inter-related nature of the Dutch roll damping and frequency and the rather common usage of yaw rate sensors, it was decided to choose a design damping ratio and to allow the Dutch roll frequency to result from the nature of the equations.
With the yaw rate as the measured quantity, its relationship to Dutch roll damping and frequency through the dimensional derivative, N r , was found in Table 6.8 of Reference 13 to be: N r = S b 2 Cnr / 2 Izz U i (B, 3) The relationship for the yaw rate gain was found in Reference 13 to be: K r = (hCnr / CnhR) ( b 2 u i ) (B.4) where: A C n r = -2 Izt U l / q S b 2 ( 2 W n 63 t y p / u i ) (8.5) D D
-
where:
' ' J D - 3Ddesign - 3Dbasic
This gain was calculated based upon the normal control surface sizes and must be multiplied to account for the ratio of the Separate Surface size to the primary control surface size.
The change in the dimensional derivative, N r , required & recalculation of the resulting Dutch roll frequency by Equation ( B . 1 ) . The relationships of D, n and nD were then checked to insure all airplanes met Level 1 handling requirements at all flight conditions and C.G. locations for the chosen design point.
B.2 SSSA Lateral-Directional Surface Sirinn Requirements The minimum required surface area for the rudder was determined for one percent probability and thunderstorm gusts. Using the VonKarman scales of Section 9 . 8 . 1 of Reference 14, the root-mean-square gust intensity and the resulting change in sideslip due to gust perturbation was found to be: TABLE B . 1 Lateral-Directional Gusts and Perturbations Thunderstorm Clear A i r
*
Cruise Min.Control* Cruise Min. Contro 1 0"' ( f ps 1 4 . 6 0.71 21 21 (rad) .0066 .0419 .0302 . l o 1 2 Pgust'
* at 500 ft. altitude
It is obvious that the critical flight condition that w i l l size the surface required for t.he SSSA system is for a thunderstorm gust at min. control speed. The required rudder area was determined according to the method of balancing moments in the Lateral-directional axis.
C n g Aggust = C n 6 ~ A & R , , , ( B . 6 )
or CnSR = Cng ( A g g u s t / A 6 R m a x ) where: = i 40 deg A sRmax From Reference 15, Sections 12.1 and 12.3, the relationship o'f the rudder to the affected vertical tail area was determined to be:
= -
Cn6R )ASv(Lvcos cx+Zvsindb) (B. 7 ) 6Rbasic/Svbasic From this i t is obvious that the percentage of the required rudder area that must be dedicated t o SSSA is: X S R = ( - I / C Y ~ R 1 (b/Lvcos a+Zvsina)(CnpApgust / 4 6 R m a x ) ( B . 8 ) basic For a chosen rudder size for the S S S A control surface, the maximum gust intensity that the system can overcome was found (B.9) as : U 6R /CnP) (-CydRbasic/SVbasic 1 (Sv) (Lvcos a+Zvsin#/b) vmax max From these relationships, a spreadsheet analysis was defined to show simultaneously the effect of design choices on the requirements for all of the airplanes. A sample spreadsheet is presented for the design point at the critical min. control speed in Table B . 2 .
I TABLE B . 2 S a m p l e S p r e a d s h e e t for L a t e r a l - D i r e c t i o n a l D y n a m i c s !
H I W . M m M Z-Fm a - A f t S-Fm % - A f t 5 0 - F W 3 - M t 75-F0n 75-Aft 1 0 0 - F ~ r l00-Afk Lat-Dirut lkl 2 0 7 . 5 0 0 0 2 0 7 . 5 0 0 0 2 0 7 . 5 0 0 0 207.5000 2 0 7 . 5 0 0 0 2 0 7 . 5 0 0 0 2 0 7 . 5 0 0 0 207.5000 2 0 7 . 5 0 0 0 207.5000 S 5 9 z . o o o o 592.oooo 592.oooo 5 9 z . m 592.oooo ! m o o 0 0 1182.m lltp.oo00 1 1 e z m 1182.m sv 170.oooO 1 7 0 . o o o O 1 7 0 . o o o O 1 7 0 . o o o O 1 7 0 . o o 0 0 1 7 0 . o o o O 34O.oooO 34O.oooO 34O.oooO 34O.ooO b 84.3OOo 81.3ooo 81.3OOo M.3OOo M . 3 I 1 0 0 M.3OOo 132.500 1 3 2 , 5 9 0 0 132.5ooo 132.5ooo 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 5 1 . 1 7 0 0 n l p b c d e g ) 9.oooo 9 . o o o O 9.oooo 9 . m 9 . o o 0 0 9.oooo 9 . o o o o 9 . o o o O 9 . o o o o 9 . oooo 23.9500 2 3 . 9 5 0 0 2 8 . 3 9 0 0 2 8 . 3 9 0 0 3 7 . 4 7 0 0 3 7 . 4 7 0 0 26.8600 2 6 . 8 6 0 0 3 4 . 8 2 0 0 3 4 . 8 2 0 0 Lv,Zv,Alpha 2 4 7 3 9 . o o o O 2 3 3 6 1 . o o O O N334.oooO 28574.oooO 4 3 1 4 1 . o o o O 2 5 9 7 8 . o o o O 7 1 4 1 9 . o o o O 148M.oooO 85044.ooOO 50666.oOO Height r 7 6 8 . 2 9 1 9 7 2 6 . 1 1 8 0 942,0497 8 8 1 . 3 9 1 3 1 3 3 9 . 7 8 2 6 8 0 6 . 7 7 0 2 2 2 1 7 . 9 8 1 4 1391,4286 2 6 4 1 . 1 1 8 0 1 5 7 3 . 4 7 8 3 122, 1 7 7 0 6 6 . o o o O 1 8 0 6 3 ) . o o O O 2BiuZ4.oooO 3 1 0 3 6 1 . o o o O 5BOO46.oooO 4 5 1 1 1 3 . o o o O 1779161,oooO 1 3 2 4 8 7 1 . o o o O 2328189.oooO 1 4 S ? S O 5 . o o o O N,B.basic 1 . 4 1 2 0 1 . 3 8 4 0 1 . 6 2 3 0 1 . 4 8 7 0 1 . 2 3 1 0 1 . 5 6 2 0 ,3210 s o 7 0 .w .m
N.r.basic -.w - . 4 m - 3 1 0 -.m -. 3340 -. 4240 -. 1120 -. 1770 - , 1 4 3 0 -.a
- 4 2 . 0 2 1 0 - 2 9 . m - 4 7 . 3 8 1 3 0 - 2 4 . 5 3 2 0 -41.1780
Y.B.basic - 4 6 . 5 s i o -51340 - 3 9 . 5 0 3 0 -n.m -46.2200
4.3040 4 . m 4.1m 4 . 4 1 6 0 3.8600 6 . 4 1 0 0 3.330 5.3290 3 . 6 0 8 0 6.0510 Y.r.basic
Cn. B. basic .m ,0380 ,1810 ,1810 ,2800 ,2800 ,0710 ,0710 ,1200 . lzoo
Cn.r.basic - . 1 5 3 0 - . 1 5 3 0 - . 2 1 5 0 - . 2 1 5 0 - . 3 7 4 0 - . 3 7 4 0 -I 0780 - . O m -. 1310 -. 1310 Cy.dR.basic - . 3 2 4 0 - . E 4 0 - . 3 2 4 0 - a 4 0 - . E 4 0 - . 3 2 4 0 - . 3 2 4 0 - . 3 2 4 0 - . 3 2 4 0 -.m
.w ,0650
C n . a ,0920 ,0920 I lop0 ,1090 ,1440 ,1440 ,0660 .m
. m o .m .m ,2220 ,2770 .m .2#10
Zeta.D. basic ,2260
1,2900 1.23bO 1 . 1 1 9 0 1 . 2 6 8 0 .57M ,7310 .M50 . a z o
n . D. basic
Zeta. M . D ,2915 .m ,2339 .m ,1279 .na 1 3 0 9 ,2137
Basic Class h e U l i t i e s
2eta.D - rin yes yes
yn yes yes yes yes yes yes yes
n . D - rin yes yes yw y# yes ycs yes
yes yes yes no I
Zetailh.Dlin yes y# w yes
yes Yc5 yes yes
Zeta. D. des ,2900 .m ,2900 .m .zwo ,2900 ,2900 ,2900 ,2900
d . Z e t a . D ,0110 .ooLo .wo ,0630 ,0810 ,0350 0 0 6 9 0 ,0130 .@70
d . C n . r ,0707 .OB11 ,0141 .m -.m ,1181 ,0451 .m ,0056
N. r . m l t - . 2 4 1 2 - . i W -. 3665 - . 3 1 2 6 - 3 1 6 -304 -. 0473 ,0319 -. l3ao
Kr ,1561 ,1791 ,0262 .05p - . O M 4 . lbbb ,2184 .us2 .oms 3 9 1
.m ,2900 .2900 .m .m ,2900
Zeta. 0. des ,2900 ,2900 ,2900 ,2900
Lh.D.cc 1 . 1 9 9 6 1 . 1 8 5 2 1.2886 1 . 2 3 2 3 1 . 1 2 2 2 1.m ,5680 ,6976 ,6444 .m
Zeta. Iwn. D 3 7 9 ,3137 .m ,3574 ,3254 . w 3 .1M7 .m ,1869 ,2331
s1\6 - Class One U l i t i e s
Zeta.D - rin yes yes yes yes yes yes yes Y E
yes F -
h.D - rin yes yes
yes yes yes yes yes yes yes F S ZetaihD-rin yes yes yes Yes yes r# yes yes yn F Gust Response 6ust speed 2 1 . o o o o 21.oooo 21.oooo 21.oooo 2 1 . o o o o 2 1 . o o o o 2 1 . o o o o 21.oooo 2 1 . o o o o 21.oooo ,1012 d.B. gust-rad ,1012 ,1012 ,1012 ,1012 ,1012 ,1012 ,1012 1 0 1 2 ,1012 40.oooo 4o.oooo
D . d R . n x d e g 4Q.oooO 40.oooO 4 0 . oooo 4 0 . oooo 4o.oooo 4o.oooo lo.w 1o.oooo
D. S v . req 2 6 . 2 3 7 1 2 6 . 2 3 7 1 4 0 . 8 7 9 4 0 . 8 7 9 8 4 7 . 9 1 4 8 4 7 . 9 1 4 8 53.28(13 53.2803 6 9 . 1 6 5 1 6 9 . 1 6 5 1 3o.oooo 3o.oooo
Percent - sv 3 0 . m 3o.oooo 3 o . w 3o.oooo 50,ooOo 3o.oooO 3o.oooo 3o.oooo
Sv.nx 51.oooO 5 1 . o o o O 51.ooOO 51.oooO S 1 . W 51.oooO 102.W 102.W 1 0 2 , o o o O 1 0 2 . o o o O 6USt - M X 4 0 . 8 2 0 0 4 0 . 8 2 0 0 2 6 . 1 9 8 8 2 6 . 1 9 8 8 22.3522 2 2 . 3 5 2 2 40.X'5 10.2025 3 0 . 8 3 5 6 3 0 . 8 3 5 6
APPENDIX C: CALCULATIONS FOR ROLL MODE DYNAMICS
APPENDIX C: CALCULATIONS FOR ROLL MODE DYNAMICS The purpose of this Appendix is to present a summary of the method and results used to determine the aileron area and gain requirements f o r a SSSA system to achieve the commonality design goals.
From Section 6.6.3 of Reference 13, the Rolling approximation was found to be: TR = - 1 / Lp (C. 1 ) And, LPt Phi(t) = -L6A 6A/Lp t + L6A 6A/Lp2 (e
- 1 ) (C.2)
The roll rate and the roll acceleration were also calculated for all airplanes, and the lateral acceleration for the twin-bodies was determined.
P(t) = -L6A 6A/Lp ( 1 - eLPt)
(C.3) LPt P.dot(t) = L ~ A &A e (C.4) and the lateral acceleration was: Lat.acc = (y)CP.dot(t)I ( C . 5 ) where y = fuselage distance from Centerline y = 289 in.
Due to the nearness of the grouping of time constants and roll rates within each group of similar planform, and the magnitude that these values exceeded the m i n i m u m Level I requirements, and augmentation system was not designed f o r the Roll mode.
These calculations were made in a spreadsheet analysis.
A sample spreadsheet demonstrating Level I requirements is demonstrated in Table C.1.
4 3 TABLE C . 1 Sample Spyoadsheet f o r R o l l Mode Dynamics I
CRUISE * m o r e 25dft &-Fore 36-Aft 5o-fore %-Aft 75-Fore 75-Aft 1OMore lOOdft
----------
IcH"I -1.8765 -3.3409 -1.5435 -2.8611 -6.9196 -4.5205 -6.5766 -3.7529 -3.2809 -6.3444 L.da 88.4574 57.788 84.0728 47.976 41.9419 81.1047 15.1456 26.9658 12.1659 23.1075 5 5 5 dr (deg) s S 5 5 5 5 5 Time(sec) 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 55.7787 64.6199 51.9564 62.6734
F a 7 112.2070 107.3068 111.7255 104.4271 102.0010 111.3701
Lwel O n e
Y - yes Y e yes y# Y K YK yes Y e s rn
P (rad/sec) 1.1155 1.1153 1.1155 1.1146 1.1133 1.113 .6844 ,7031 ,6672 .7017 P . h t ,00002 ,00094 .oooO3 .0033S .00718 .00004 ,05138 ,00412 ,05793 ,00878 Lat k c e l (ft/s#"2) ,9003 ,0992 1.3952 .2116 llPPRmm* 25-Fore ZSdft &Fore 36-w 50-Fon M t 75-Forr M t 1 W O r e 1OOdft
----------
-
-4.4867 -2.9311 -4.2643 -2.4334 -2.1274 -4.1138 -1.1936 -2.125 -.9816 -1.8196 4.6632 2.1551 3.996 L. da 17.2738 11.2847 16.4176 9.3687 8.1903 1 J . m 2.619l' 15 15 15 da (deg) 10 10 10 10 10 10 15 1.8 1.8 1.8 1.9 1.8 1.8 1.8 1.8 1.8 T i r e 1.8 clllc Phi (deg) Level me P (rd/s#) ,6717 .&E5 ,6716 .b#X ,6513 ,6715 .SO74 ,5619 ,4767 .SZQ P.&t ,00094 .01007 ,00133 ,02048 .03105 .00168 .07999 .02663 .09642 , 0 3 9 2 3 5 Lat k c e l (ft/s#"2) 1.9265 ,6414 2.3222 .9525