cg/stability map for the reference h cycle 3 supersonic ... reference mission profile was taken from...

NASA/CR-1999-209527

Cg/Stability Map for the Reference H Cycle 3

Supersonic Transport Concept Along the

-High Speed Research Baseline Mission Profile

Daniel P. Giesy and David M. Christhilf

Lockheed Martin Engineering & Sciences Company, Hampton, Virginia

December 1999

https://ntrs.nasa.gov/search.jsp?R=20000021227 2018-06-09T12:45:33+00:00Z

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NASA / CR- 1999-209527

Cg/Stability Map for the Reference H Cycle 3

Supersonic Transport Concept Along theHigh Speed Research Baseline Mission Profile

Daniel P. Giesy and David M. Christhilf

Lockheed Martin Engineering & Sciences Company, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research CenterHampton, Virginia 23681-2199

Prepared for Langley Research Centerunder Contract NAS1-96014

December 1999

Available from:

NASA Center forAeroSpace Information (CASI)

7121 Standard Drive

Hanover, MD 21076-1320

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171(703) 605-6000

Abstract

A comparison is made between the results of trimming a High Speed Civil Transport

(HSCT) concept along a reference mission profile using two trim modes. One mode uses

the stabilator. The other mode uses fore and aft placement of the center of gravity. A com-

parison is made of the throttle settings (cruise segments) or the total acceleration (ascent and

descent segments) and of the drag coefficient. The comparative stability of trimming using the

two modes is also assessed by comparing the stability margins and the placement of the lateral

and longitudinal eigenvalues.

Introduction

The study reported in this document is a follow on study to the work reported by Chowdhry

and Buttrill in their memorandum [1]. In that report, they documented "a study performed to

assess the bare airframe stability characteristics of a HSCT configuration and to investigate the

benefits, if any, of using an active fuel management system to reduce trim drag." The study consists

of trimming the aircraft at each point along a reference mission profile (1) holding the x-cg at

its nominal position and using the stabilator and (2) holding the stabilator at zero deflection and

moving the x-cg to trim. Stability and performance measures of the aircraft are compared at the

two trim conditions at each point of the reference mission profile.

Ref. [1] studied the Reference H, cycle 1 HSCT concept. The present study uses the

Reference H Cycle 3 airplane (Ref. H-3) [2].

[3, pp.

Seg. 1

Reference Mission Profile

The reference mission profile was taken from a TCA Configuration Description Document

A32 - A43]. It is described by a sample of 79 flight conditions divided into five segments:

Climb, 30 flight conditions (numbers 1 - 30).

Seg. 2 Supersonic cruise, 13 flight conditions (numbers 31 - 43).

Seg. 3 Supersonic descent, 16 flight conditions (numbers 44 - 59).

Seg. 4 Subsonic cruise, 13 flight conditions (numbers 60 - 72).

Seg. 5 Subsonic descent, 7 flight conditions (numbers 73 - 79).

This mission profile was designed for the Technology Concept Airplane (TCA) which is

heavier than Ref. H-3. To correct for this weight differential, the aircraft weights from the TCA

mission profile have been multiplied by the factor 649914/740600 which is the ratio of the maxi-

mum taxi weight of Ref. H-3 to that of TCA. Each flight condition of the reference mission profile

is then defined by its Mach number, its altitude, its (scaled) weight, and its rate of climb. These are

shown in Figs. 1 - 4.

Modeling and Analysis Software

The model of Ref. H-3 and much of the analysis software was contained in an integrated

MATLAB/SIMULINK HSCT simulation [4]. The software module refhsim3_trim, m in [4]

performed an analysis of the Ref. H-3 aircraft at each flight condition using specific values of

the trim variables. For this analysis, the ALrrOCG_FLAG and the AUTOWT_b'LAG were turned

off; weight was scheduled by the reference mission profile as already explained and the center

of gravity was determined by interpolation using the five standard mass sets as explained in the

next section. The AUTOFLAP._FLAG was on, so that flaps followed their automatic schedule, and

RIGID_FLAG was off, so Quasi-Steady AeroElastic (QSAE) increments were incorporated. The

MATLAB Optimization Toolbox constrained optimization routine constr, m was used to vary

the trim variables to achieve the trim conditions. Once trim was achieved, the derivatives cOCM/aa

and OCz/c3a were estimated using a central difference derivative approximation with Aa = . 1

deg., further utilizing refhs im3_trim, m. Stability margin (as a per cent of mean aerodynamic

chord) was then calculated to be -100(,9C_/Oa )/(OCL/0o_).

The trimmed aircraft was then linearized using 1inearraodel.m and

refhsim3_lin.m from [4]. Lateral and longitudinal eigenvalues were calculated by ap-

plying MATLAB built-in function eig to subblocks of the linear system matrix.

Trim Details

In all trim runs, the thrust multiplier was set to 1.09, so that maximum power available

was 109% of nominal maximum. This was suggested by Dr. Chris Gracey of LaRC based on his

experience with calculating fuel optimal trajectories.

At all flight conditions, nominal X and Z positions of the center of gravity were determined

by linear interpolation using gross weight as the independent variable and using the simulation

mass sets from [2, Table 3.1-1, Page 3-2]. The relevant data are (Z-CG data come from a Boeing-

supplied FORTRAN subroutine DATA statement):

Mass Set Gross Weight (lb) XCG BS(in) ZCG WL(in)

M01 279,080 2153.5 201.50

MFC 384,862 2139.2 207.46

MCR 501,324 2155.7 203.15

MIC 614,864 2132.2 200.59

M13 649,914 2086.0 199.56

At each flight condition, the aircraft was trimmed in two different ways. In the Stabilator

trimmed case, the X-CG was held at its nominal value, and trim was achieved using the coupled

stabilizer and elevator (with DEr,EV1 = DELEV2 = 2 *DSTAB). In the CG trimmed case, the

stabilator was held at 0 degrees deflection, and the X-CG was moved to achieve trim.

3

In all trim casesthebodyaxisX-velocity,u, the body axis Z-velocity, w, and the Euler pitch

angle, 0, were used as trim variables. The parameters gross weight and altitude were scheduled by

the flight condition.

In all trim cases, the trim constraints included constant angle of attack (& = 0) and constant

pitch rate (0 = 0). Mach and rate of climb were constrained to values scheduled by flight condition.

During climb (Seg. 1), throttle was set to max (100%) and during descent (Segs. 3 and 5),

throttle was set to idle (3%). During the cruise segments (Segs. 2 and 4), an additional constraint

was imposed; total velocity was to be constant (lit = 0). The throttle setting then became an

additional trim variable.

After trim was calculated, the stability margin was determined as previously noted. The

trimmed aircraft was then linearized. The linearized states are:

1. Total Velocity, VT,

2. Angle-of-attack, a,

3. Pitch rate, q,

4. Euler angle 0,

5. Altitude, h,

6. Roll rate, p,

7. Yaw rate, r,

8. Bank angle _b,

9. Sidslip angle r,

10. Euler angle _b,

11. Latitude, and

12. Longitude.

Linearization of the trimmed aircraft produces a 12 by 12 state matrix, A. The longitudinal

eigenvalues are found by taking the eigenvalues of the sub-matrix of A consisting of the first 5 rows

and columns of A. The lateral eigenvalues are found by taking the eigenvalues of the sub-matrix

of A consisting of rows and columns 6 - 9 of A.

4

Trim Results

Trim values of trim variables are shown in Figs. 5 - 10. Fig. 5 shows that, with the X-CG

held to its nominal position, trim can be established throughout the reference mission profile with

stabilizer deflections of no more than about -t-1.5 ° (with coupled elevator deflections of up to about

+3°). Fig.. 6 shows the nominal X-CG schedule (circles) and the X-CG position needed to trim

in the absence of stabilator deflection (x's). The horizontal dash-dot lines show the forward (48%)

and aft (54%) X-CG limits as given in [2, Page B-8]. The nominal X-CG schedule takes it outside

these limits during part of the supersonic cruise segment (Seg. 2). The trim position of the X-CG,

however, exceeds the aft X-CG limit both more often and by a greater amount than the nominal.

The elapsed time to fly from flight condition 8 to 15 (the last condition before the X-CG exceeds

its aft limit until the first condition where the X-CG has returned to its limit) is 10 minutes 22

seconds. Besides the question of whether one would want to exceed the limit by so much, there is

the question of whether one would want to include high enough capacity fuel pumping systems to

move the X-CG that rapidly. The X-CG trim position throughout the entire descent and subsonic

cruise phase (Segs. 3 - 5) falls outside the aft limit, the excess becoming as much as 6% of mean

aerodynamic chord.

The value of trim variable u to trim the aircraft is shown in Fig. 7. It seems to be quite

insensitive to whether the aircraft is trimmed by stabilator deflection or X-CG positioning. The

values of trim variables w and 0, shown in Figs. 8 and 9, show only slightly more sensitivity to thetrim mode.

In the cruise segments, the power lever setting is used as an additional trim variable, sup-

plemented by adding the constraint that the total acceleration should be zero (VT = 0). Fig. 10

shows that the X-CG trimmed aircraft uses slightly less power in these segments. In the climb and

descent segments, Segs 1, 3, and 5, where the power setting is programmed to a fixed setting, the

X-CG trimmed aircraft experiences slightly more acceleration (Fig. 11). Both of these phenomena

are attributable to the reduction in drag coefficient, Co, which is the result of trimming by varying

the X-CG position as opposed to using the stabilator. This is shown in Fig. 12.

Stability Results

Stability margins for each trimmed condition are shown in Fig. 13. Although the results

between the two trim paradigms are mixed over the reference mission profile, the X-CG trimmed

case seems to have more tendency to go to a negative stability margin than the stabilator trimmed

case,

The lateral eigenvalues are well behaved, considerately separating into the conventional

spiral mode, roll mode, and Dutch roll pair. The spiral mode remains stable (Fig. 14), as does

the roll mode which is shown in Fig. 15 and the Dutch roll pair whose real and imaginary parts

are plotted as a function of flight condition number in Fig. 16. Dutch roll damping also seems

adequate.

5

The longitudinaleigenvalueswerenot soreadily identifiable.Fig. 17showsthereal partof the mostunstablelongitudinaleigenvalue.Fig. 18givesa scatterplot of all the longitudinaleigenvalueswith thetwo trim paradigmsplottedseparately.Theraweigenvaluedataaretabulatedat theendof this report.

Acknowledgement

The authors are grateful to Dr. Christopher Gracey of NASA Langley Research Center for

his help in locating data for this study and in formulating the approach.

References

[ 1] Rajiv Singh Chowdhry and Carey Buttrill, "Reference-H Assessment: Bare Airframe Stability

and Trim Drag Reduction," Memo to GFC ITD, September 13, 1995.

[2] "High Speed Civil Transport Reference H - Cycle 3 Simulation Data Base." A Boeing report

for NASA Contract NAS1-20220, Task Assignment No. 36, WBS 4.3.5.1.2.1.

[3] "High Speed Research Program HSR II- Airframe task 20; Task 2.1 - Technology Integration;

Sub-task 2.1.1.1 Refine Technology Concept Airplane: Configuration Description Document;

Deliverable Report." Approved by J. B. Coffey, BCAG; J. K. Wechsler, MDC; P. F. Sweetland,

BCAG; H. R. Welge, MDC. Prepared for NASA Langley Research Center, NASA Contract

NAS1-20220. April 1, 1996.

[4] Rajiv S. Chowdhry, "User's Manual for MATLAB Reference_H Cycle 3 Simulation Software",

December, 1996.

6

r,-o

Parameter defining nominal mission: Mach

I I I I t I I H I

J S_g. 3 I: Seg. 4i Seg. 5_.s............_grn.ent.1.............i',....Seg-_....,......._...........,_........................X :1 X " I: •

X :1 X " I: .

x il x " ai •:1 I:•

2 ........................i_x ......:i ......:..............×! ...........,i.......................it x: I]

X :1 ._ I:

x: il :X I:x il x li

1.5 ................... X ............... -.'t ............... t ....... ;..-X ....... I-: ............... I ........

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X :1 : X I: I

x :l • X ii I

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-.x ................................ :r .................................. r:............... , .... x-]t t:. i x;,:1 I: I

!1 i! I:1 I: I

0 I I1 I 11 I

0 10 20 30 40 50 60 70 80Nominal mission flight condition number

0.5

Figure l: Reference Mission Profile: Mach

7

4=

¢f4"I3

.,..,<

3

2

00

x 104 Parameter defining nominal mission: AltitudeI I I I = I q I I

Segment i[ Seg.._. ' seg. 3 i seg. 4i ',seg. 5............ I ....... : ...... K ............ :- ..........

il I • Ii " I:1 I . I: . I

:1 I . I: . I

........... :............ . ........... !_.......... ::_x_,xg .... :........... _!............ :..I .......: : ? .vxxxX_ K_! If ! I

i ; _xxX,, i % :: :-',..........._............:......x_;::...........:..........i"_......[;............": .....

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......................... "it........................ ; "• X . :1 I: X• X "

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........... i---x" ...... - ........... if ................................. i:

X: " :1 I: X:1 I:

....................... :............ ; ............ ..... • ....... ; ............ ;........................

x " it J::1 1: Xil li

.... _ .................. X I:X :1 I: .

il liX :1 I: " X

×_K II I II I

10 20 30 40 50 60 70 80Nominal mission flight condition number

Figure 2: Reference Mission Profile: altitude

8

6.5

6

.= 5.5

t-.__

5.o(.9

4.5

3.50

x 10s Parameter defining nominal mission: Gross weightI I I I ! I II t

Segment I ? Seg. ;_ S_g. 3 I,i Seg. 41 Seg 5............ :............ :............. :| ........... : ....... ; ........... _:............ :........ ;._

X_x.,¢o%_i ! il i i li i

_x'/_<Xx.. i i', i i ',i i....... _ "T_,_ _, _..... i ........... !i............ _.....

• I!:1

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................... L: ................ 1 .......

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..................................... :I ........... :.... I ................... f: _XX_ " : ..........!l i i li 'Xx_x,il i I li i _O<X_:X:K

:1 : I I: II II I I h I I


Figure 3: Reference Mission Profile: Gross Weight

100

8O

60

0

40.5E

._

._ 20

0

-2O

-40

-600

Parameter defining nominal mission: Rate of climb

l I I I = I It I

iSegment := Seg. 2. i Seg. 3 =: Seg. 4 Seg. 5........... :........................ i_................ ' ................... _i............ :-..........

"1 I I:

: :l I I: •

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........... i ............ • ........

il

:1

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10 20 30 40 50

......... ri..............xl!

XX_X I:ti i

60 70

Nominal mission flight condition number

X

I

I

80

Figure 4: Reference Mission Profile: rate of climb

10

1.5

• 0,5"0

-dt""

E 0E0

_ -0.5e,J

m

-1

-1.5

-2o

Trim variable "stabilator"I I I1 I I I II I I

Segment1 _' Seg.2 ' ......S_g:3......',....Seg'"....',S_g:S.........................:............il...............I .-

! o I I! : IOC_ : 0 I I: I

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o i_ i li

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:j o li

•o.i.°......................i_...................._ .................._OO :I 0 0 I:

:1 : I: I

:1 : I: II I . il I II I I


Figure 5: Trim settings of the stabilator

)1

56

r_u I× 54

Trim variable "X_CG"

I I II i | i II I I

_Segmentl :l Seg. 2 ' S..eg :! : I62 ............................ it. •I. :.3 Seg. 4 Seg. 5........ . ......... . ........................... | ........

I I! _ IZ

I : I i : I

I :1 i :

60 .... x .... CGtrimm_17 ............ >:r................ i ....... _....... x_ .......

o Stabilator tdrhmed x liI:

r . Xo .... I,i .............

58 ............ ............ "1 ...... I" : ...........X I: :

X. : I: . "X X"

x it x_Y.x_ ...... i:x i ! _ . ,.._ ,i i

....... x- ......... -+ ....... -_ _ .... i-_ ........

52 ............ :.... x o .... :............ _r........ _: 'l ....... ! ........... r7............ ?f ........

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O0" :1 : I li I

48 ................. "_...... -"J ........ t:.......... t .......I I I


Figure 6: Trim settings of the X-CG

12

2400

2200

2000

1800

o 1600

Z 1400

1200

1000

800_

600

400 I_0

Trim variable 'u"

I I I I I o I q I u

_Segment 1 :r Seg. 2 = Seg. 3 o! Seg. 4 1 IiSeg. 5.................................... .:I............ :.... I ....... _........... I-:............ ............

' ' l! i-ITI I I'. , I

................................. l_l!r ................ r ® ..... : ........... r!............ :"r .......[] !u I[] : Ji I

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i i II I II ' I , I_j

10 20 30 40 50 60 70 80


Figure 7: Trim settings of the x-velocity

13

Trim variable "w"

I , ,, , , ! ,, ! ,:Segment1 :' Seg. 2 ' Seg. 3 ..' Seg. 41...'.S.eg.:5160 ............ :............ :............ iL ........... :.... i ................. . . .-

il I L t:1 : I I: I

140 ................................. _'Ci .... t ................... _:............ :.-t .......i i : ! I! i I

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............... _ ............... ii"................................ "b"Fx,T........................

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I

100

80

40 ..................................... ii ........... !.... i .................................,: ! i

il : I li:1 I i; -

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8O

Figure 8: Trim settings of the z-velocity

14

16

14

Trim variable "Theta"I I I I I * I ml I m

• "1 i; Sag 2 ' Seg. 3 ;! Seg. 4i ISe.g:5

_i :1 I: :

:1 £

:1 I:

:i I:

12

_10

:1 I:

•_ .................................. !t ........................ ! ........... ti......................:1 - I:

it ....................... . ........... ti...............'' "_ ................................ i[ ........

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:1 . I • I-

6 .......... _ ....................... i_ -I ....... i ........... t.:............... j .......

i_:l;_ ;' , : ':1 I - I

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,:1 . I

2 .................................... !t ........... i.... r....... i ,,_ ......il i J ,_:............ : l .......

0 I I II I I I II I [


Figure 9: Trim settings of the Euler pitch angle

15

Throttle: Fixed in segments 1, 3, 5; variable in segments 2, 4I I II I .....I' I =I I J

Sement 1' !" Seg. 2 ' Seg. 3 i: Seg. 4i iseg " 5100 _ ..........."....I......._...........l.!............:... :.-il : =i -:

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80 ............:......................... . "......!...........(i........................il i li

E .;t....... i. "...........L:."5 70 ............!............:.............................................

:I k

: il . li :60 ............:............:..............'t............:...........i..........._ .......

.C : : :I : :

: :I : I: i

" sol-.l..........:............i...........I_,...........i....f.......::...........,i............---;........= / I × CG trimmed! Ill ! t i li ! =

i

jl ,- i i i ' .........a. 30FI .......... : ........................ _ ................................... t ............ i.-._-

I :I I: : I

!i li i i20 ........................•............':r............•.....................t:............:'"r........

i i i, i li _ ,! i i, i Ji ! t

10 ...........:............:"...........:i...........:............:...........(:............: i ........

0 i I _ , f"'_"0 10 20 30 40 50 60 70 80


Figure 10: Trim settings of the throttle

: z

16

2ro

"lO

I -2

-4

-6

Total acceleration

I I I= I J t II I I

Segment 1 :' Seg. 2 ' S_g. 3 !i Seg. 4 i ,seg " 5_6 ............ :............ :............ i_........... :.... i ....... , ........... _............. :: il : I I! .." J

:1 1 I: I

:1 I I: I

4 ...... "................ " ........... :_........ " I ................... _i............ :..I ......._ ::, , ,: ,

;I I I: I

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i il i!

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ii 'Y ..............................• [iI:

--8 I I II I I I h I I


Figure 11: I?r

17

Drag coefficient

T I_ I J ! ,I 1

::S.egment :1 Seg. 2" i Seg. 3 I: Seg. 4i ISe.g 50.04o ............. - .... i_ .I ................... _i........... :.. :x il i! ;

0 : :1 I: ::1 I:

X :1 I:

0.035 ............ i........................ !f .................................... t'!............ : ..........

0.03

E3

°t 0.025

0.02

0.015

0.010

:1

X Tiim with x__cg :'

o .... T_m.with DSTAB ...................

:1

:1

I:

i!.. I ................... li.

liI:

I:

it li

- • O ................................. :1".................................... I:

X :1 1:

:l OO l:

.................................... :l ................,.............. L:............... l ........o " i O:'_b_i i i co

...._ ....... "........... li[:............ i'"lI ........

• ,.l - I:

: :1 i: : I

I II l It i I


Figure 12: Co

18

15

10

O<

¢- 5

E

o

-5

-100

Stability margin of trimmed aircraft! H , T I J ! tl I i

_Segment :' Seg. 2 ' S_g 3 =_.i seg. 4i seg. 5

3 : a]

OO :1 . I:

;o ,x ;i i : i; iO; _ X" :I : ; I: :

.......... : .... c_ .... : ........... "( ........... :.... i ....... : ............ :............ :"; .......• xO _ : : : ': " ': 0 _X,. I : I • _ ' : I

: o o., : ."y.,_:, : ' io°""-, o ;i i: O: : . _ I: : I

_-o°i x _ :' :,,oC°_... ix,, ': : io.... _ .........-;_bo_ ......_o_x_, ......_ .......

X: : _O : I X __1• ; : X I:

o ' !t x_ li jxxx_ . !i xl! l

.... i :I .................. l.i............... !.q_....._:I

:I

........._....cG._mm_d......i',.• o Stabilator tdmmed

;I

I I II

10 20 30

t......... f

I

I

: I

: I

i f

40 50

0I:

x liI:

........... I:ri........... •........ x,_I:

I:

I:

II I

60 70 80Nominal mission flight condition number

Figure 13: Stability Margin

19

0

-0.01

=¢-0.02>t-(9

"_ -0.03

8E_ -0.04

09

-0.05

-0.06

-0.070

Lateral Eigenvalues:spiral modeI I I= I I I it I

Segment i :i Seg. 2 f Seg. 3 i: Seg. 4: I Se,g. 5_........... :............ :............ ? ........... :.... =................... _;............ :...•I - I I: : I

....... : ............ - ................... _:............ :,., ........

Oi " Ii ! I

0 0 :I I . I, I__

^ x :l i x I! I "x2.................................. !r ................ _................ , r!................ f .... _"

il ti:1 1::1 I:

,0 ............................ _i........................ ;........... ri........................_X :1 : I:

:1 : I:

:1 " I:

........... i............ : ........... !fi • Ili ............ ! " i ........! i !,x CG!trimmed ! i! ! ,

ii, i"l o Stabilalortrimmed I1: : ,

:1 I I:

l t II l I I II I

10 20 30 40 60 60 70 80Nominal missionflightconditionnumber

Figure 14: Lateral Eigenvalues: Spiral mode

2O

-0.4

-0.6

-0.8

¢1>f- --1

(!)(I)-o -1.2oEon--I.4

Lateral Eigenvalues: rollmodeI I' | I I II I

Segment :' Seg. 2 ' Seg. 3 J ....Seg..4 .....tiSeg. 5........... ............. ............. i_ ........... ..... I ....... , ............

:1 I;

....._...;_.._.._._......................i...........!i............;............................._ i fi "

......................._...........!_................i..®i :J l

:I I

....................._ ..............it................i..:I I

"® ..........._i........................i® l::® I:

...i..=5.......ti......................

:1 I I_1_1: I ._......................................!_............:....f.......:.......®f!...............i "[]

il 1: I

......... ._ ..................... ? ........... :........... i ........... l:............... I .......E1 181 l_® :1 : : h

:li : : i:-I.6 ..... _ ............................ilx ....CG!tHRih_di ........... ,i.......................

io Stabila{ortrimmed i!I : ili

; :1 : I • : 1i il i I f i j

--2 i il i I II i I

0 10 20 30 40 50 60 70 80Nominal missionflight conditionnumber

Figure 15: Lateral Eigenvalues: Roll mode

21

-0.05

-0.1n

rr

-0.15

-0.20

Lateral EIgenvalues: Dutch roll pair

;Segment 4 ....... i[... Seg: .2. i .S.;g, 3 ;i Seg: 41 i.S.eg,5_........... _ ........ :..... :, : , : ,: : I I

oo i i, _ i ,i i _ I

eo........i e_.....iJl_ .....i l_,::............:.........ii £_'i I ;O!1,Stabilator: tn_med i I!i iI I W, i i .I II I I

10 20 30 40 50 60 70 80


• / ......... 0:0,_ .... :"......... :' : , : t: : ,i YA-,_'?_,-_6 i, i 1 i =i i i

__1.1 I'- .... ",._--:q:_:__,; ...... :1"........... :.... I ....... : ........... r:............ : r .......

1Fo ......... : ............ : ..... x ..... :l ........... :.... 1....... : .... J.L-r: ............ :r .......•-= IoX,", : : c_ : : :_._..,dj::['_°...I: : i_0 9 L_' X........ ! ............ i....... cK...ivs,,_ :....i .... i .... _._i ............ i..., ......._E • I- o i i _'"'__ _ , ..._"':" -....¢,____i ,_

o_x........,............,........._- __--_0.7/ i ; i, i _ ',:, ;,_


Figure 16: Lateral Eigenvalues: Dutch roll pair

22

0,3

0.25

0.2

_. o.15

13:

0,1

0.05

-0.050

Maximum real part of Ion! titudinal eigenvalueI I I I I _1 I I '

:l Seg, 2 Seg, 3 i: Seg, 4: IISeg.5............ s ?grr,en!............... i_.................................. _,i............ :...........il : li .: l:1 : I: : Iil x ii i

x CGtrimmed i . ii J: I

O Stabilator tdrfimed il J: : iI: I x

.......... :............ :.................. :.... _" ti ............ i...t ......: il I: : I

x il Ii : I:1 I: : I X XX

............x :........................ ':li• ................ ' ................... _""_ Xrl............ :"........... :1 I: :

: :1 I:• : X

....................... :............ _l..................................... fi ............ ?...........

O_ : :1 I: :, O:1 I;

° 'x! i_ _ oo

X :il li i

• " : I:

I

:1 : I I: ;• . I

:1 I I: : I

I I , fl I I II [ I


Figure 17: Longitudinal Eigenvalues: Real part of most unstable

23

Q.

E_0

t-

--E_1

-2-1.6

2

E"0

t--

¢1

-E_ 1

-2-1.5

Longitudinaleigenvalues, CG trimmed1 ! ! " I I I I I I

:_xx

I1_ × ×

)0< XX )_ )_ _>_XX

X x X

1 i I ,,I I 1 I i f

-1.4 -1.2 -1 -0,8 -0.6 -0.4 -0.2 0 0.2Real part

Longitudinal eigenvalues, Stabilator trimmedI I I

goO

_) 00D 0 0 0 (l) 00_0

0

&

I l I

-1 -0.5 0Real part

Figure 18: Longitudinal Eigenvalues: All

0.4

0.5

24

fe#

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

293O

3132

33

34

35

36

37

38

39

4O41

42

43

Seg1

1

1

1

1

1

1

1

1

1

1

11

11

1

1

1

11

1

1

1

1

1

'1

I1

1

1

2

2

2

2

2

2

2

2

22

2

2

2

Table

Eig 1-0.0630

-0.0428

-0.0431

-0.0264

-0.0188

-0.0173

-0.0173

-0.0172

-0.0174

-0.0200

-0.0205

-0.0228

-0.0271

-0.0226

-0.0196

-0.0174

-0.0157

-0.0144

-0.0136

-0.0130

-0.0125

-0.0122-0.0122

-0.0125

-0.0128

-0.0118

-0.0107

-0.0094

-0.0093

-0.0094

-0.0082-0.0079

-0.0076

-0.0074

-0.0071

-0.0069

-0.0066

-0.0064

-0.0062

-0.0060-0.0058

-0.0056

-0.0054

Eig 2-0.1493+0.9023i

-0.1509+0.9580i

-0.1329+0.8018i

-0.1265+0.9223i

-0.1260+ 1.0645i

-0.1264+ 1.0663i

-0.1239+1.0697i

-0.1106+1.1023i

-0.0827+ I. 1024i

-0.0618+ 1.0538i

-0.0616+1.0540i

-0.0863+1.1031i

-0.1442+1.1536i

-0.1365+1.1651i

-0.1452+1.1675i

-0.1559+1.1521i

-0.1683+ 1.1366i

-0.1831+1.1271i

-0.1761+I.1244i

-0.1680+1.1181i

-0.1591+1.1091i

-0.1544+1.1038i

-0.1544+ 1.1038i

-0.1472+ 1.0714i-0.1399+1.0004i

-0.1304+0.9612i

-0.1208+0.9172i

-0.1110+0.8687i

-0.1110+0.8687i

Eig 3-0.1493-0.9023i

-0.1509-0.9580i

-0.1329-0.8018i

-0.1265-0.9223i

-0.1260-1.0645i

-0.1264-1.0663i

-0.1239-1.0697i

-0.1106-1.1023i

-0.0827-1.1024i

-0.0618-1.0538i

-0.0616-1.0540i

-0.0863-1.103 li

-0.1442-1.1536i

-0.1365-1.1651i

-0.1452-1.1675i

-0.1559-1.1521i

-0.1683-1.1366i

-0.1831-1.1271i

-0.1761-1.1244i

-0.1680-I.1181i

-0.1591-1.1091i

-0.1544-1.1038i

-0.1544-1. I038i

-0.1472-1.0714i-0.1399-1.0004i

-0.1304-0.9612i

-0.1208-0.9172i

-0.1110-0.8687i

-0.I 110-0.8687i

-0.1059+0.8542i -0.1059-0.8542i

-0.1036+0.8968i

-0.0996+0.8916i-0.0970+0.8875i

-0.0946+0.8835i

-0.0924+0.8794i

-0.0902+0.8753i

-0.0876+0.8674i

-0.0851+0.8535i

-0.0828+0.8427i

-0.0806+0.8320i

-0.0787+0.8213i-0.0769+0.8103i

-0.0754+0.7999i

-0.1036-0.8968i

-0.0996-0.8916i-0.0970-0.8875i

-0.0946-0.8835i

-0.0924-0.8794i

-0.0902-0.8753i

-0.0876-0.8674i

-0.0851-0.8535i

-0.0828-0.8427i

-0.0806-0.8320i

-0.0787-0.8213i-0.0769"-0.8103i

-0.0754-0.7999i

1. Lateral eigenvalues, CG-trimmed

Eig 4-1.0335

-1.0804

-1.3392

21.4956

-1.6232

-1.6235

-1.5972

-1.4659

-1.4031

-1.3915

-1.3914

-1.3939

-1.4499

-1.4847

-1.3590

-1.2302

-1.0990

-0.9739

-0.9027

-0.8336

-0.7728

-0.7429

-0.7429

-0.6984-0.6324

-0.5832

-0.5414

-0.5091

-0.5091

-0.4949

-0.5002

-0.5030-0.5066

-0.5097

-0.5123

-0.5144

-0.5196

-0.5288

-0.5380

-0.5470

-0.5558-0.5643

-0.5730

ease

25

fc# Seg Eig 144 3 -0.0070

45 3 -0.0079

46 3 -0.0087

47 3 -0.0094

48 3 -0.0103

49 3 -0.0105

50 3 -0.0110

51 3 -0.0113

52 3 -0.0116

53 3 -0.0144

54 3 -0.0174

55 3 -0.0199

56 3 -0.0224

57 3 -0.0268

58 3 -0.0240

59 3 -0.0232

60 4 -0.0210

61 4 -0.0214

62 4 -0.0213

63 4 -0.0212

64 4 -0.0211

65 4 -0.0210

66 4 -0.020967 4 -0.0208

68 4 -0.0208

69 4 -0.0208

70 4 -0.0207

71 4 -0.0207

72 4 -0.0207

73 5 -0.023274 5 -0.0243

75 5 -0.0262

76 5 -0.0280

77 5 -0.0318

78 5 -0.0387

79 5 -0.0389

Table 1 (concluded).

Eig 2 Eig 3 Eig 4-0.0762+0.7364i -0.0762-0.7364i -0.5683

-0.0809+0.7791i -0.0809-0.7791i -0.5983

-0.0856+0.8147i -0.0856-0.8147i -0.6296-0.0902+0.8439i -0.0902-0.8439i -0.6617

-0.0951+0.8664i -0.0951-0.8664i -0.6968

-0.1024+0.8804i -0.1024-0.8804i -0.7537

-0.1097+0.8895i -0.1097-0.8895i -0.8101-0.1152+0.8994i -0.1152-0.8994i -0.8704

-0.1203+0.9095i -0.1203-0'9095i -0.9300

-0.1214+0.9226i -0.1214-0.9226i -0.9977

-0.1231+0.9220i -0.1231-0.9220i - 1.0594-0.1271+0.9247i -0.1271-0.9247i -1.1136

-0.1325+0.9218i -0.1325-0.9218i -1.1579

-0.1425+0.9011i -0.1425-0.9011i -1.2064

-0.I075+0.8418i -0.1075-0.8418i -1.1486

-0.0920+0.7933i -0.0920-0.7933i -1.0663

-0.0939+0.8088i -0.0939-0.8088i -1.0650

-0.0941+0.8009i -0.0941-0.8009i -1.0507

-0.0940+0.8000i -0.0940-0.8000i -1.0518

-0.0941+0.7989i -0.0941-0.7989i -1.0524

-0.0941+0.7978i -0.0941-0.7978i -1.0531

-0.0941+0.7967i -0.0941-0.7967i -1.0538

-0.0941+0.7957i -0.0941-0.7957i -1.0545

-0.0941+0.7941i -0.0941-0.7941i -1.0552

-0.0941+0.7924i -0.0941-0.7924i -1.0560

-0.0941+0.7907i -0.0941-0.7907i -1.0568

-0.0938+0.7898i -0.0938-0.7898i -1.0549

-0.0931+0.7900i -0.093 i-0.7900i -1.0491

-0.0924+0.790ii -0.0924-0.7901i -1.0434

-0.0903+Q,7742i -0.0903-0.7742i -1.0447-0.1043+0.7918i -0.1043-0.7918i -1.0980

-0.1070+0.7750i -0.1070-0.7750i -1.1856

-0.1081+0.7643i -0.108I-0.7643i -1.3206

-0.1115+0.7452i -0.1115-0.7452i -1.5272

-0.1216+0.7253i -0.12i6-0.7253i -1.8309-0.1236+0.7263i -0.1236-0.7263i -1.8739

Lateral eigenvalues, CG-trimmed case

26

fc# Seg Eig 11 1 0.1682

2 1 0.1335

3 I 0.0732

4 1 0.0038+0.0634i

5 1 0.0017

6 1 0.0026

7 1 0.0019

8 1 0.0264

9 1 0.0374

10 1 0.0007+0.0752i

11 I 0.0006+0.0752i

12 1 -0.0016+0.0587i

13 1 -0.0039+0.0815i

14 1 -0.0015+0.0676i

15 1 -0.0003+0.0571i

16 1 -0.0001+0.0556i

17 1 0.0030

18 1 0.0010

19 1 0.0033

20 1 0.0023

21 1 0.0008+0.0475i

22 1 0.0009+0.0474i

23 1 0.0009+0.0474i

24 1 0.0009+0.0476i

25 1 0.0008+0.0470i26 1 0.0007+0.0462i

27 1 0.0003+0.0447i

28 I 0.0002+0.0434i

29 1 0.0002+0.0434i

30 1 0.0002+0.0438i

31 2 0.0000+0.0441 i

32 2 0.0000+0.0444i

33 2 -0.0000+0.0448i

34 2 -0.0000+0.0449i

35 2 -0.0000+0.0446i36 2 0.0000+0.0443i

37 2 0.0001+0.0438i

38 2 0.0001+0.0432i

39 2 0.0002+0.0426i

40 2 0.0002+0.0420i

41 2 0.0003+0.0415i

42 2 0.0003+0.0410i

43 2 0.0002+0.0409i

Eig 2 Eig 3•.0.0021 -0.0209+0.1220i

0.0042 -0.0367+0.1312i

-0.0052 -0.0574+0.0950i

0.0038-0.0634i -0.0034

0.0006+0.0426i 0.0006-0.0426i

0.0005+0.0429i 0.0005-0.0429i

0.0012+0.0458i 0.0012-0.0458i

-0.0121+0.0258i -0.0121-0.0258i

-0.0180+0.0154i -0.0180-0.0154i

0.0007-0.0752i -0.0058

0.0006-0.0752i -0.0060

-0.0016-0.0587i -0.0094

-0.0039-0.0815i -0.0054

-0.0015-0.0676i -0.0026

-0.0003-0.057 li -0.0012

-0.0001-0.0556i -0.0010

0.0016+0.0569i 0.0016-0.0569i

0.0008+0.0521i 0.0008-0.052 li

0.0006+0.049 li 0.0006-0.049 li

0.0007+0.0480i 0.0007-0.0480i

0.0008-0.0475i 0.0005

0.0009-0.0474i 0.0001

0.0009-0.0474i 0.0001

0.0009-0.0476i -0.0007

0.0008-0.0470i -0.0019

0.0007-0.0462i -0.0021

0.0003-0.0447i -0.0012

0.0002-0.0434i -0.0029

0.0002-0.0434i -0.0029

0.0002-0.0438i -0.0029

0.0000-0.0441i -0.0021

0.0000-0.0444i -0.0022

-0.0000-0.0448i -0.0023

-0.0000-0.0449i -0.0022

-0.0000-0.0446i -0.0021

0.0000-0.0443i -0.0021

0.0001-0.0438i -0.0021

0.0001-0.0432i -0.00220.0002-0.0426i -0.0022

0.0002-0.0420i -0.0023

0.0003-0.0415i -0.0024

0.0003-0.0410i -0.0023

0.0002-0.0409i -0.0022

Eig 4-0.0209-0.1220i

-0.0367-0.1312i

-0.0574-0.0950i--0.6039+0.2934i

-0.6206+0.2855i

-0.6209+0.2866i

-0.6133+0.3017i

-0.5714+0.3881i

-0.6071+0.6670i

-0.6225+0.7989i

-0.6223+0.7984i-0.5890+0.9098i

-0.4855+1.0626i

-0.4910+1.2908i

-0.4681 + 1.3885i

-0.4422+1.4448i-0.4131+1.4752i

-0.3823+ 1.5067i

-0.3626+1.4479i-0.3424+1.3677i

-0.3281 + 1.3679i

-0.3212+1.3674i

-0.3212+1.3674i

-0.3094+1.3440i

-0.2878+1.3070i

-0.2703+1.2564i

-0.2549+1.2098i

-0.2425+ I. 1705i

-0.2425+ 1.1705i

-0.2360+1.1348i

-0.2357+ I. 1014i

-0.2348+1.0948i

-0.2345+1.0902i

-0.2343+1.0859i

-0.2341+1.0816i-0.2339+1.0772i

-0.2332+ 1.0708i

-0.2319+1.0624i

-0.2305+1.0524i

-0.2292+1.0422i

-0.2278+1.0315i

-0.2264+1.0205i-0.2250+1.0095i

Table 2. Longitudinal eigenvalues, CG-trimmed case

Eig 5-1.1338

-1.0580

-1.0364

-0.6039-0.2934i

-0.6206-0.2855i

-0.6209-0.2866i

-0.6133-0.3017i

-0.5714-0.3881i

-0.6071-0.6670i

-0.6225-0.7989i

-0.6223-0.7984i

-0.5890-0.9098i

-0.4855-1.0626i

-0.4910-1.2908i

-0.4681-1.3885i

-0.4422-1.4448i

-0.4131-1.4752i

-0.3823-1.5067i

-0.3626-1.4479i

-0.3424-1.3677i

-0.3281-1.3679i

-0.3212-1.3674i

-0.3212-1.3674i

-0.3094-1.3440i

-0.2878-1.3070i-0.2703-1.2564i

-0.2549-1.2098i

-0.2425-1.1705i

-0.2425-1.1705i

-0.2360-1.1348i

-0.2357-1.1014i

-0.2348-1.0948i

-0.2345-1.0902i

-0.2343-1.0859i

-0.2341-1.0816i

-0.2339-1.0772i-0.2332-1.0708i

-0.2319-1.0624i

-0.2305-1.0524i

-0.2292-1.0422i

-0.2278-1.0315i

-0.2264-1.0205i

-0.2250-1.0095i

27

fc# Seg Eig 1 Eig 2 Eig 3 Eig 444 3 -0.0000 -0.0022+0.0406i -0.0022-0.0406i -0.2235+0.8686i

45 3 0.0001 -0.0030+0.0438i -0.0030-0.0438i -0.2307+0.8743i

46 3 -0.0002 -0.0031+0.0456i -0.0031-0.0456i -0.2385+0.8752i

47 3 -0.0000 -0.0034+0.0472i -0.0034-0.0472i -0.2461+0.8712i

48 3 -0.0000 -0.0036+0.0489i -0.0036-0.0489i -0.2556+0.8455i

49 3 -0.0004 -0.0032+0.0488i -0.0032-0.0488i -0.2657+0.8205i

50 3 -0.0019 -0.0020+0.0472i -0.0020-0.0472i -0.2762+0.7972i

51 3 -0.0013+0.0458i -0.0013-0.0458i -0.0028 -0.2908+0.8599i

52 3 -0.0018 -0.0031+0.0501i -0.0031-0.0501i -0.3037+0.8989i

53 3 -0.0054 -0.0054+0.0580i -0.0054-0.0580i -0.3235+0.8706i

54 3 0.0062 -0.0038+0.0696i -0.0038-0.0696i -0.3418+0.7570i

55 3 0.0006 -0.0057+0.0707i -0.0057-0.0707i -0.3613+0.6660i

56 3 0.0002 -0.0065+0.0795i -0.0065-0.0795i -0.3791+0.5439i

57 3 0.2657 0.0459 0.0069 -0.0478

58 3 0.0069+0.0259i 0.0069-0.0259i -0.0333 -0.4568+0.2841i

59 3 0.1301 0.0017 -0.1040+0.1323i -0.1040-0.1323i

60 4 0.1336 -0.0025 -0.1098+0.1347i -0.1098-0.1347i

61 4 0.1367 -0.0024 -0.1067+0.i365i -0.1067-0.1365i

62 4 0.1371 -0.0024 -0.1056+0.1365i -0.1056-0.1365i

63 4 0.1377 -0.0023 -0.1044+0.1367i -0.1044-0.i367i

64 4 0.1383 -0.0023 -0.1032+0.1369i -0.1032-0.1369i

65 4 0.1388 -0.0022 -0.1021 +0.1370i -0.1021-0.1370i

66 4 0.1393 -0.0022 -0.1010+0.1371i -0.1010-0.1371i

67 4 0.1399 -0.0021 -0.1000+0.1372i '0.1000-0.1372i68 4 0.1405 -0.0021 -0.0990+0.1373i -0.0990-0.1373i

69 4 0.1411 -0.0020 -0.0980+0.1374i -0.0980-0.1374i

70 4 0.1418 -0.0019 -0.0969+0.1375i -0.0969-0.1375i

71 4 0.1426 -0.0019 -0.0960+0.1377i -0.0960-0.1377i

72 4 0.1433 -0.0019 -0.0950+0.1378i -0.0950-0.1378i

73 5 0.1412 0.0013 -0.0898+0.1340i -0.0898-0.1340i

74 5 0.2171 0.0002 -0.0438+0.1206i -0.0438-0.1206i

75 5 0.1599 -0.0005 -0.04 14+0.1053i -0.0414-0.1053i

76 5 0.1076 -0.0007 -0.0696+0.1025i -0.0696-0.1025i

77 5 0.1286 -0.0011 -0.0474+0.1043i -0.0474-0.1043i

78 5 0.1553 -0.0001 -0.0297+0.1025i -0.0297-0.1025i

79 5 0.1635 -0.0005 -0.0315+0.1077i =0.03 i5-0.1077i

Eig 5-0.2235-0.8686i

-0.2307-0.8743i

-0.2385-0.8752i

-0.2461-0.8712i

-0.2556-0.8455i

-0.2657-0.8205i

-0.2762-0.7972i

-0.2908-0.8599i

-0.3037-0.8989i

-0.3235-0.8706i

-0.3418-0.7570i

-0.3613-0.6660i

-0.3791-0.5439i

-1.0780

-0.4568-0.2841i

-0.8140

-0.7984

-0.7962

-0.7973

-0.7984

-0.7994

-0.8003

-0.8011-0.8019

-0.8028

-0.8036

-0.8052

-0.8084

-0.8116

-0.8265

-1.0078

-1.0066

-0.9879

-1.1805

-1.4020

-1.4251

Table 2 (concluded). Longitudinal eigenvalues, CG-trimmed case

28

fc# Seg Eig 1 Eig 2 Eig 3 Eig 41 1 -0.0603 -0.1550+0.9428i -0.1550-0.9428i -1.0277

2 1 -0.0417 -0.1566+0.9948i -0.1566-0.9948i -1.07333 1 -0.0387 -0.142!+0.8476i -0.1421-0.8476i -1.3315

4 1 -0.0253 -0.1322+0.9450i -0.1322-0.9450i _1.4914

5 1 -0.0190 -0.1326+1.0786i -0.1326-1.0786i -1.6236

6 1 -0.0174 -0.1330+1.0809i -0.1330-1.0809i -1.6239

7 1 -0.0174 -0.1304+1.0850i -0.1304-1.0850i -1.5976

8 1 -0.0174 -0.1168+1.1183i -0.1168-1.1183i -1.4668

9 1 -0.0155 -0.0914+1.1634i -0.0914-1.1634i -1.4064

10 I -0.0176 -0.0707+I.1226i -0.0707-1.1226i -1.3950

11 1 -0.0181 -0.0705+1.1227i -0.0705-1.1227i - 1.3949

12 1 -0.0207 -0.0932+1.1618i -0.0932-1.1618i -1.3982

13 1 -0.0255 -0.1486+1.1968i -0.1486-1.1968i -1.4545

14 I -0.0224 -0.1374+1.1734i -0.1374-1.1734i -1.4853

15 1 -0.0198 -0.1444+1.1613i -0.1444-I.1613i -1.3586

16 1 -0.0179 -0.1539+1.1344i -0.1539-1.1344i -1.2293

17 1 -0.0164 -0.1651+I.1085i -0.1651-1.1085i -1.0977

18 1 -0.0152 -0.1791+1.0910i -0.1791-I.0910i -0.9723

19 1 -0.0145 -0.1727+1.0888i -0.1727-1.0888i -0.9007

20 1 -0.0138 -0.1652+I.0837i -0.1652-1.0837i -0.8312

21 1 -0.0133 -0.1566+1.0722i -0.1566-1.0722i -0.7698

22 1 -0.013 ! -0.1520+1.0652i -0.1520-1.0652i -0.7397

23 1 -0.0131 -0.1520+I.0652i -0.1520-1.0652i -0.7397

24 1 -0.0129 -0.1440+1.0521i -0.1440-1.0521i -0.697725 1 -0.0140 -0.1386+0.9588i -0.1386-0.9588i -0.6278

26 1 -0.0130 -0.1295+0.9199i -0.1295-0.9199i -0.5784

27 1 -0.0120 -0.1202+0.8749i -0.1202-0.8749i -0.5364

28 1 -0.0108 -0.1107+0.8228i -0.1107-0.8228i -0.5037

29 I -0.0107 -0.1107+0.8228i -0.1107-0.8228i -0.5037

30 I -0.0107 -0.1055+0.8124i -0.1055-0.8124i -0.4902

31 2 -0.0091 -0.1029+0.8615i -0.1029-0.8615i -0.4968

32 2 -0.0088 -0.0988+0.8546i -0.0988-0.8546i -0.4997

33 2 -0.0086 -0.0961+0.8485i -0.0961-0.8485i -0.5033

34 2 -0.0083 -0.0937+0.8424i -0.0937-0.8424i -0.5063

35 2 -0.0081 -0.0913+0.8362i -0.0913-0.8362i -0.5089

36 2 -0.0079 -0.0890+0.8299i -0.0890-0.8299i -0.5110

37 2 -0.0076 -0.0865+0.8240i -0.0865-0.8240i -0.5165

38 2 -0.0072 -0.0840+0.8191i -0.0840-0.8191i -0.5265

39 2 -0.0068 -0.0818+0.8139i -0.0818-0.8139i -0.5362

40 2 -0.0065 -0.0798+0.8088i -0.0798-0.8088i -0.5456

41 2 -0.0061 -0.0780+0.8035i -0.0780-0.8035i -0.5548

42 2 -0.0059 -0.0764+0.7967i -0.0764-0.7967i -0.5636

43 2 -0.0056 -0.0750+0.7894i -0.0750-0.7894i -0.5725

Table 3. Lateral eigenvalues, Stabilator trimmed case

29

fc# Seg Eig 1 Eig 2 Eig 3 Eig 444 3 -0.0067 -0.0765+0.7499i -0.0765-0.7499i -0.5691

45 3 -0.0075 -0.0814+0.7940i -0.09i4-0.7940i -0.5991

46 3 -0.0083 -0.0863+0.8312i -0.0863-0.8312i -0.630447 3 -0.0090 -0.0912+0.8624i -0.0912-0.8624i -0.6625

48 3 -0.0098 -0.0965+0.8876i -0.0965-0.8876i -0.6976

49 3 -0.0100 -0.1040+0.9039i -0.1040-0.9039i -0.7546

50 3 -0.0104 -0.1115+0.9142i -0.1115-0.9142i -0.8112

51 3 -0.0108 -0.1169+0.9218i -0.I169-0.9218i -0.8714

52 3 -0.0111 -0.1220+0.9300i -0.1220-0.9300i -0.9308

53 3 -0.0137 -0.1236+0.9471i -0.1236-0.9471i -0.9986

54 3 -0.0162 -0.1264+0.9597i -0.1264-0.9597i -1.0609

55 3 -0.0185 -0.1314+0.9676i -0.1314-0.9676i -1.1150

56 3 -0.0212 -0.1378+0.9604i -0.1378-0.9604i -1.1566

57 3 -0.0242 -0.1485+0.9598i -0.1485-0.9598i -1.2061

58 3 -0.0219 -0.1148+0.8910i -0.1148-0.8910i -1.1476

59 3 -0.0213 .-0.1001+0.8385i -0.1001-0.8385i -1.0623

60 4 -0.0205 -0.1015+0.8365i -0.1015-0.8365i -1.0603

61 4 -0.0205 -0.1014+0.8312i -0.1014-0.8312i -1.0457

62 4 -0.0205 -0.1015+0.8300i -0.1015-0.8300i -1.0466

63 4 -0.0204 _0.i01"7+0.8286i -0.1017-0.8286i -1.0472

64 4 -0.0204 -0.1018+0.8273i -0.1018-0.8273i -1.0476

65 4 -0.0203 -0.10i9+0.8260i -0.1019-0.8260i -1.048266 4 -0.0203 -0.1021+0.8246i -0.1021-0.8246i -1.0488

67 4 -0.0202 -0.1022+0.8233i -0.1022-0.8233i -1.0494

68 4 -0.0202 -0.1023+0.8220i -0.1023-0.8220i -1.0501

69 4 -0.0201 -0.1025+0.8207i -0.1025-0.8207i -1.0508

70 4 -0.0200 -0.1022+0.8199i -0.1022-0.8199i -1.0489

71 4 -0.0200 -0.1014+0.8199i -0.10i4-0.8199i -1.0433

72 4 -0.0200 -0.1007+0.8200i -0.1007-0.8200i -1.0377

73 5 -0.0211 -0.0991+0.8220i -0.0991-0.8220i -1.0398

74 5 -0.023I -0.1092+0.8169i -0.1092-0.8169i -1.0933

75 5 -0.0251 -0.1116+0.7961i -0.1116-0.7961i -1.180876 5 -0.0270 -0.11i8÷0.7811i -0.1118-0.7811i -1.3167

77 5 -0.0305 -0.116i+0.7646i -0.1161-0.7646i -1.5226

78 5 -0.0368 -0.1281+0.7503i -0.1281-0.7503i -1.8241

79 5 -0.0367 -0.1309+0.7556i '0.1309-0.7556i -1.8665

Table 3 (concluded). Lateral eigenvalues, Stabilator trimmed case

30

fc# Seg Eig 11 1 0.0523

2 1 0.0717

3 1 0.0014+0.0604i

4 1 0.0033+0.0645i

5 ! 0.0031 +0.0507i

6 I 0.0034+0.0508i7 I 0.0035+0.0513i

8 1 0.0035

9 1 0.0033

I0 1 0.0002+0.0542i

11 I 0.0001+0.0542i

12 1 -0.0008+0.0407i

13 1 -0.0029+0.0693i

14 I -0.0013+0.0651i

15 1 -0.0004+0.0589i

16 1 -0.0004+0.0602i17 1 0.0027

18 1 0.0005+0.0622i

19 I 0.0029

20 1 0.0020

21 1 0.0005+0.0577i

22 1 0.0005+0.0580i23 I 0.0005+0.0580i

24 1 0.0005+0.0580i

25 1 0.0003+0.0566i

26 1 0.0003+0.0553i

27 1 -0.0002+0.0536i

28 1 -0.0000+0.0522i

29 1 -0.0000+0.0522i

30 1 -0.0000+0.052 li

31 2 -0.0001+0.0522i

32 2 -0.0001+0.0536i

33 2 -0.0002+0.0553i

34 2 -0.0002+0.0565i

35 2 -0.0001+0.0570i

36 2 -0.0001+0.0573i37 2 -0.0000+0.0554i

38 2 0.0000+0.0518i

39 2 0.0001 +0.0489i

40 2 0.0002+0.0466i

41 2 0.0003+0.0447i

42 2 0.0003+0.003 li

43 2 0.0002+0.0024i

Eig 2 Eig 3-0.0012 -0.0803+0.0798i

0.0068 -0.0670+0.1107i

0.0014-0.0604i -0.0038

0.0033-0.0645i -0.0036

0.0031-0.0507i -0.00180.0034-0.0508i -0.0013

0.0035-0.0513i -0.0013

0.0006+0.0386i 0.0006-0.0386i

-0.0015+0.0294i -0.0015-0.0294i

0.0002-0.0542i -0.0052

0.0001-0.0542i -0.0054

-0.0008-0.0407i -0,0098

-0.0029-0.0693i -0.0055

-0.0013-0.0651i -0.0025

-0.0004-0.0589i -0.0012

-0.0004-0.0602i -0.00120.0015+0.0655i 0.0015-0.0655i

0.0005-0.0622i 0.0005

0.0005+0.0590i 0.0005-0.0590i

0.0005+0,058 li 0.0005-0.058 li

0.0005-0.0577i 0.0003

0.0005-0.0580i -0.0002

0.0005-0.0580i -0.00020.0005-0.0580i -0.0008

0.0003-0.0566i -0.0021

0.0003-0.0553i -0.0024

-0.0002-0.0536i -0.0015

-0.0000-0.0522i -0.0030

-0.00_-0.0522i -0.0030

-0.0000-0.0521i -0.0030

-0.0001-0.0522i -0.0023

-0.0001-0.0536i -0.0024

-0.0002-0.0553i -0.0025

-0.0002-0.0565i -0.0024

-0.0001-0.0570i -0.0024

-0.0001-0.0573i -0.0024

-0.0000-0.0554i -0.0025

0.0000-0.0518i -0.00250.0001-0.0489i -0.0025

0.0002-0.0466i -0.0025

0.0003-0.0447i -0.0025

0.0003-0.043 li -0.0024

0.0002-0.0424i -0.0023

Eig 4-0.0803-0.0798i

-0.0670-0.1107i

-0.5650+0.4259i-0.6236+0.7279i

-0.6544+0.8184i

-0.6512+0.8201i-0.6424+0.8317i

-0.6005+0.8960i

-0,6349+1.2812i

-0.6494+1.4243i

-0.6491+1.4242i

-0.6128+1.4306i

-0.5003+ 1.2222i

-0.4939+1.3527i

-0.4660+I .3386i

-0.4373+ 1.3149i

-0.4065+ 1.2609i

-0.3747+1.2288i-0.3560+1.1741i

-0.3367+1.I015i

-0.3223+ 1.0893i

-0.3153+1.0801i

-0.3153+1.0801i

-0.3038+1.0592i-0.2827+1.0341i

-0.2660+0.9927i

-0.2510+0.9481 i

-0.2392+0.8980i

-0.2392+0.8980i

-0.2330+0.8842i-0.2330+0.8645i

-0.2320+0.8432i-0.2315+0.8223i

-0.2313+0.8013i-0.2310+0.7794i

-0.2307+0.7564i

-0.2302+0.7662i

-0.2293+0.8067i

-0.2284+0,8420i

-0.2274+0.8729i-0.2264+0.8997i

-0.2253+0.9235i

-0.2243+0.9443i

Eig 5-0.9530

-0.9995

-0.5650-0.4259i

-0.6236-0.7279i

-0.6544-0.8184i

-0.6512-0.8201 i-0.6424-0.8317i

-0.6005-0.8960i

-0.6349-1.2812i

-0.6494-1.4243i

-0.6491-1.4242i

-0.6128-1.4306i

-0.5043-1.2222i

-0.4939-1.3527i

-0.4660-1.3386i

-0.4373-1.3149i

-0.4065-1.2609i

-0.3747-1.2288i

-0.3560-1.1741i

-0.3367-1.1015i

-0.3223-1.0893i

-0.3153-1.0801i

-0.3153-1.0801i

-0.3038-1.0592i

-0.2827-1.0341i

-0.2660-0.9927i

-0.2510-0.9481 i-0.2392-0.8980i

-0.2392-0.8980i

-0.2330-0.8842i

-0.2330-0.8645i

-0.2320-0.8432i

-0.2315-0.8223i

-0.2313-0.8013i

-0.2310-0.7794i-0.2307-0.7564i

-0.2302-0.7662i

-0.2293-0.8067i

-0.2284-0.8420i

-0.2274-0.8729i

-0.2264-0.8997i-0.2253-0.9235i

-0.2243-0.9443i

Table 4. Longitudinal eigenvalues, Stabilator trimmed case

31

fc# Seg Eig 144 3 -0.0001

45 3 -0.000i

46 3 -0.0003

47 3 -0.0002

48 3 -0.0004

49 3 -0.0008

50

51

52

53

54 3 0.0061

55 3 -0.0003

56 3 -0.0015

57 3 -0.0033

58 3 0.0001+0.0339i59 3 0.0390

60 4 0.0533

61 4 0.053962 4 0.0535

63 4 0.0531

64 4 0.0528

65 4 0.0525

66 4 0.0522

67 4 0.0517

68 4 0.0513

69 4 0.0509

70 4 0.0508

71 4 0.0511

72 4 0.0514

73 5 0.0367

74 5 0.0838

75 5 0.0472

76 5 0.0340

77 5 0.0286

78 5 0.0045

79 5 0.0006

Eig 2 Eig 3 Eig 4 Eig5-0.0021+0.0390i -0.0021-0.0390i -0.2245+0.9566i -0.2245-0.9566i

-0.0028+0.0412i -0.0028-0.0412i -0.2320+0.9771i -0.2320-0.9771i

-0.0028+0.0424i -0.0028-0.0424i -0.2400+0.9950i -0.2400-0.9950i-0.0031+0.0435i -0.0031-0.0435i -0.2480+1-.0096i -0.2480-1.0096i

-0.0031 +0.0439i -0.003 I-0.0439i -0.2581 + 1.0166i -0.2581-1.0166i-0.0028+0.0434i -0.0028-0.0434i -0.2686+1.0147i -0.2686-1.0147i

3 -0.0018+0.0420i -0.0018-0.0420i -0.0023 -0.2793+1.0082i -0.2793-1.0082i

3 -0.0013+0.0418i -0.0013-0.0418i -0.0031 -0.2938+1.0500i -0.2938-1.0500i

3 -0.0024 -0.0026+0.0452i -0.0026-0.0452i -0.3071+I.0761i -0.3071-1.0761i

3 -0.0044+0.0498i -0.0044-0.0498i -0.0065 -0.3280+1.0833i -0.3280-I.0833i

-0.0028+0.0548i -0.0028-0.0548i -0.3489+1.0788i -0.3489-1.0788i-0.0047+0.0531i -0.0047-0.0531i -0.3700+1.0654i -0.3700-1.0654i

-0.0060+0.0552i -0.0060-0.0552i -0.3953+0.9568i -0.3953-0.9568i

-0.0065+0.0552i -0.0065-0.0552i -0.4098+0.5658i -0.4098-0.5658i

0.0001-0.0339i -0.02050.0044 -0.0476

-0.0061 -0.0444

-0.0059 -0.0451

-0.0058 -0.0447

-0.0058 -0.0445

-0.0058 -0.0442

-0.0057 -0.0439

-0.0056 -0.0436

-0.0056 -0.0433

-0.0055 -0.0430

-0.0055 -0.0426

-0.0054 -0.0427

-0.0052 -0.0432

-0.0052 -0.04350.0046 -0.0459

0.0008 -0.1140

0.0005 -0.0693

0.00013 -0.0495

-0.0023 -0.0581

-0.0070+0.0579i -0.0070-0.0579i-0.0083+0.0480i -0.0083-0_0480i

-0.4705+0.9422i -0.4705-0.9422i

-0.4832+0.6328i -0.4832-0.6328i

-0.4849+0.6366i -0.4849-0.6366i-0.4785+0.6342i -0.4785-0.6342i

-0.4775+0.6355i -0.4775-0.6355i

-0.4765+0.6368i -0.4765-0.6368i

-0.4754+0.6380i -0.4754-0.6380i

-0.4743+0.6392i -0.4743-0.6392i

-0.4732+0.6403i -0.4732-0.6403i

-0.4721+0.6414i -0.4721-0.6414i

-0.4711+0.6424i -0.4711-0.6424i

-0.4700+0.6433i -0.4700-0.6433i

-0.4692+0.6420i -0.4692-0.6420i

-0.4693+0.6386i -0.4693-0.6386i

-0.4694+0.6352i -0.4694-0.6352i

-0.4676+0.6315i -0.4676-0.6315i

-0.4353+0.1319i -0.4353-0.1319i

-0.3120 -0.6170

-0.3303 -0.6919

-0.1644 -0.9728

-0.1213 - 1.2059

-0.1204 -1.2215

Table 4 (continued). Longitudinal eigenvalues, Stabilator trimmed case

32

Form ApprovedOMB No. 07040188REPORT DOCUMENTATION PAGE

Publicreportingburdenforthis collectionof informationis estimatedto average1 hourperresponse,inducing the timefor reviewinginetnctions,searchinge,_s_ngdata sources,gatheringand maintainingthe data needed,andcompletingand reviewingthe collectionofinforma'_on.Send _ts regar_ng this burdenas_mate or any otheraspectof ff_iscollectionof information,inctu_ng suggestionsfor reducingthisburden,toWashingtonHeadquartersServices,Directoratefor InformationOperationsand Reports,1215 JeffersonDavisHighway,Suite1204, Arlington,VA 222(_-4302, and to fhe Offee of ManagementandBudget,Pa0erworkReductionProject(0704-0188),Washington,DC 20503.

1. AGENCY USE ONLY (Leave blank) 12.REPORT DATE 3. REPORT TYPE AND DA.E,_ COVERED

I December 1999 Contractor Report

4. I_ITLE AND SUBTITLE 5. FUNDING NUMBERS

Cg/Stability Map for the Reference H Cycle 3 Supersonic Transport ConceptAlong the High Speed Research Baseline Mission Profile

6. AUTHOR(S)

Daniel E Giesy and David M. Christhilf

7. PERFORMING'_)RGANIZATION NAME(S) AND ADDRESS(ES)

Lockheed Martin Engineering & Sciences Co.Langley Program OfficeHampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESSEES)

National Aeronautics and Space Administration

Langley Research CenterHampton, VA 23681-2199

C NAS1-96014WU 537-07-24-21

8. PERFORMING ORGANIZATIONREPORT NUMBER

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA/CR- 1999-209527

11. SUPPLEMENTARY NOTES

Langley Technical Monitor: Carey S. Buttrill

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 08 Distribution: NonstandardAvailability: NASA CASI (301) 621-0390

12b. DI.'_TRiBUTION CODE

13. ABSTRACT (Maximum 200 words)

A comparison is made between the results of trimming a High Speed Civil Transport (HSCT) concept along a ref-erence mission profile using two trim modes. One mode uses the stabilator. The other mode uses fore and aft place-ment of the center of gravity. A comparison is make of the throttle settings (cruise segments) or the totalacceleration (ascent and descent segments) and of the drag coefficient. The comparative stability of trimming usingthe two modes is also assessed by comparing the stability margins and the placement of the lateral and longitudinal

eigenvalues.

14. SUBJECT TERMS

High Speed Civil Transport; High speed supersonic transport; Longitudinal trim;Variable center of gravity; Stability

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

18. SECURITY CLASSIFICATIONOF THIS PAGE

Unclassified

NSN 7540-01-280-5500

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Unclassified

15. NUMBER OF PAGES

3716. PRICE CODE

A03

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UL

Stand,==d Form 298 (Rev. 2-89)Prescribedby ANSI Std.Z39-18298-102

cg/stability map for the reference h cycle 3 supersonic ... reference mission profile was taken from...

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