missile motion sensitivity to dynamic stability derivatives. · aedc-tr-80-11 missile motion...

AEDC-TR-80-11

Missile Motion Sensitivity to Dynamic Stability Derivatives

T. F. Langham ARO, Inc.

September 1980

Final Report for Period October 1, 1978 - September 30, 1979

Approved for public release; distribution unlimited. 1

ARNOLD ENGINEERING DEVELOPMENT CENTER ARNOLD AIR FORCE STATION, TENNESSEE

AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE

NOTICES

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APPROVAL STATEMENT

This report has been reviewed and approved.

ALVIN R. OBAL, Captain, CF Project Manager Directorate of Technology

Approved for publication:

FOR THE COMMANDER

MARION L. LASTER Director of Technology Deputy for Operations

UNCLASSIFIED REPORT DOCUMENTATION PAGE

'1 REPORT NUMBER ~2. GOVT ACCESSION NO,

AEDC-TR-80-11 L i

l* T I T L E ( ~ d Subtitle)

M I S S I L E MOTION SENSITIVITY TO DYNAMIC STABILITY DERIVATIVES

AUTHOR(s)

T. F. Langham, ARO, Inc., a Sverdrup Corporation Company

S PERFORMING ORGANIZATION ~AME AND ADDRESS

A r n o l d E n g i n e e r i n g Deve lopmen t C e n t e r / D O A i r F o r c e Sys tems Command A r n o l d A i r F o r c e S t a t i o n , TN 37389

11 CONTROLLING OFFICE NAME AND ADDRESS

AFATL/DLJ Eglin Air Force Base, Florida 32542

14 MOqITORING AGENCY NAME ~ AODRESSrtl dlt lerent from Conrrot lml OIhce)

R E A D I N S T R U C T I O N S B E F O R E C O M P L E T I N G FORM

3 RECIPIENT'S CATALOG NUMBER

5 TYPE OF REPORT & PERIOD COVERED

F i n a l R e p o r t - O c t o b e r 1, 1978 t o S e p t e m b e r 3 0 , 1 9 7 9 6 PERFORMING ORG. REPORT NUMBER

B CONTRACT OR GRANT NJMBER(s)

10. PROGRAM ELEMENT. PROJECT, TASK AREA & WORK UNIT NUMBERS

Program E l e m e n t 62602F

12 R£PO RT DATE

S e p t e m b e r 1980 13 NUMBER OF PAGES

110 i5. SECURITY CLASS. ,Iol th,s report)

UNCLASSIFIED

|Sa DECL ASSI FICATION 'DOWNGRADING SC.EDULE N/A

16 DISTRIBUTION STATEMENT (of this Beporl)

Approved f o r p u b l i c r e l e a s e ; d i s t r i b u t i o n u n l i m i t e d .

17. DISTRIBUTION STATEMENT (o( fhe eb~teact entered In Block 20° JI dl(ferenr b~m Reporf)

18 SUPPLEMENTARY NOTES

Available in Defense Technical Information Center (DTIC)

i9 KEY WORDS ( C ~ t | n u e on reverse arde i ! n e c e n e ~ ~ d JdentJ~ ~ bJock numbe~

d y n a m i c s d e g r e e s o f f r e e d o m d e r i v a t i v e s ( m a t h e m a t i c s ) c o m p u t e r p r o g r a m s e n s i t i v i t y s t a t i c s t a b i l i t y c o n f i g u r a t i o n s l e v e l f l i g h t

t u r n i n g flight

20 ABSTRACT (Continue ~ reverae amde I t necea la~ and Iden t l~ by block numbe~

A dynamic derivative sensitivity study was conducted to demonstrate the importance of dynamic derivatives in missile motion simulation studies. Generalized bank-to-turn and yaw-to- turn missile configurations were used with a six-degree-of- freedom linearized stability program. The effects of various dynamic derivatives on missile stability were investigated in both level and turning flight for several Mach numbers and altitude

FORM 1473 ED'TiONOF'MOV6SiSOSSOLETE DD ,JAN?S

UNCLASSIFIED

UNCLASSIFIED

20 . ABSTRACT ( C o n t i n u e d )

c o n d i t i o n s . The r e s u l t s p r e s e n t e d i n a r o o t l o c u s f o r m a t show t h a t t h e l o n g i t u d i n a l a n d 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 moment d e r i v a t i v e s Cm_,q Cm&, Cn r , C~p, C~r , a n d Cnp may s i g n i f i c a n t l y a l t e r t h e r e s p e c t i v e l o n g i t u d z n a l and l a t e r a l - d i r e c t i o n a l s t a b i l i t y m o d e s . A l s o shown i s t h e s i g n i f i c a n t c o u p l i n g e f f e c t b e t w e e n t h e l o n g i t u d i n a l a n d l a t e r a l - d i r e c t i o n a l m o t i o n s r e s u l t i n g f r o m v a r i a t i o n s i n t h e c r o s s - c o u p l i n g d e r i v a t i v e s C~q, Cn , a n d C m . The f o r c e and moment d e r i v a t i v e s C L , C L . , C ,

q P q a Yr Cyu, a n d Cm~ a r e shown t o h a v e l i t t l e o r no e f f e c t on t h e m i s s i l e s t a b i l i t y mbdes a n d , t h e r e f o r e , a r e n o t c o n s i d e r e d i m p o r t a n t t o m o t i o n s i m u l a t i o n s t u d i e s .

A F S C A ~ t d APS T~

UNCLASSIFIED

A E DC-TR-80-11

PREFACE

The work reported herein was conducted by the Arnold Engineering Development Center (AEDC), Air Force Systems Command (AFSC). The results of this research were obtained by ARC), Inc., AEDC Group (a Sverdrup Corporation Company), operating contractor for the AEDC, AFSC, Arnold Air Force Station, Tennessee, under ARO Project No. P34F-31A. The research was sponsored by the Air Force Armament Laboratory. Analysis of the data was completed on September 30, 1979, and the manuscript was submitted for publication on February 1, 1980.

AEDC-TR-80-11

C O N T E N T S

Page

1.0 I N T R O D U C T I O N ........................................................ 7

2.0 M E T H O D OF ANALYSIS ................................................. 7

3.0 T E C H N I C A L D A T A

3.1 Missile Configurations ................................................. 8 3.2 Aerodynamic Data .................................................... 8

4.0 RESULTS A N D DISCUSSION

4.1 General .............................................................. l0

4.2 Bank-to-Turn Configurat ion ............................................ I l

4.3 Yaw-to-Turn Configuration ............................................ 16

5.0 C O N C L U D I N G REMARKS ............................................... 20

R E F E R E N C E S ........................................................... 21

I L L U S T R A T I O N S

Figure

1. Bank-to-Turn Missile Configurat ion ......................................... 23

2. Yaw-to-Turn Missile Configuration .......................................... 24

3. Missile Hight Envelope .................................................... 25

4. Range of Derivative Values ................................................. 26

5. Sample Root Locus Format ................................................ 27

6. Bank-to-Turn Configuration - Locus o f Roots with

Cn~ Variation ............................................................ 28


Cm& Variation ............................................................ 31


CLq, CL~, CTr, Cnq, and Cmr Variation ........................................ 34

9. Bank-to-Turn Configuration - Effect o f Cmq on

Damping Ratio and Time Response .......................................... 37

10. Bank-to-Turn Configuration - Effect o f Cm& on



Cnr Variation ............................................................. 39

12. Bank-to-Turn Configurat ion - Locus o f Roots with

Ctp Variation ............................................................. 42

A EDC-TR-80-11

Figure Page

13. Bank-to-Turn Configuration - Effect o f Cn r on


14. Bank-to-Turn Configuration - Effect o f Ctp on



Ctr Variation ............................................................. 47


Cnp Variation ............................................................. 50


Cyp Variation ............................................................. 53

18. Bank-to-Turn Configuration - Effect o f Ce r on


19. Bank-to-Turn Configuration - Effect o f Cnp on


20. Bank-to-Turn Configuration - Effect o f Cyp on



C ~ Variation ............................................................. 59


Crop Variation ............................................................ 62

23. Bank-to-Turn Configuration - Effect o f Ctq on


24. Bank-to-Turn Configuration - Effect o f Cmp on


25. Yaw-to-Turn Configuration - Locus o f Roots with

Cmq Variation ............................................................ 67


Cmd Variation ............................................................ 69


CLq, Chl , CTr, Cyp, and Cmr Variation ........................................ 71 28. Yaw-to-Turn Configuration - Effect o f Cmq on


29. Yaw-to-Turn Configuration - Effect o f Cm/, on



Cnr Variation ............................................................. 75


Cep Variation ............................................................. 77

4

AEDC-TR-80-11

Figure Page

32. Yaw-to-Turn Configuration - Effect of Cn r o n

Damping Ratio and Time Response ......................................... 79

33. Yaw-to-Turn Configuration - Effect o f Crp on


34. Yaw-to-Turn Configuration - Locus of Roots with

Crr Variation ............................................................ 81


C.p Variation ............................................................ 83

36. Yaw-to-Turn Configuration - Effect o f C~ on


37. Yaw-to-Turn Configuration - Effect of Cnp on

Damping Ratio and Time Response ......................................... 86 38. Yaw-to-Turn Configuration - Locus of Roots with

Crq Varh t ion ............................................................ 8"/


Crop Variation ........................................................... 89

40. Yaw-to-Turn Configurat ion - Locus o f Roots with

Caq Variation ............................................................ 91

41. Yaw-to-Turn Configuration - Effect o f Crq on


42. Yaw-to-Turn Configuration - Effect of Cmp on


43. Yaw-to-Turn Configuration - Effect o f Cnq on


TABLES

1. Back-to-Turn Configuration ............................................... 95

2. Yaw-to-Turn Configuration ............................................... 101

3. Missile Physical and Mass Characteristics .................................... 104

APPENDIX

EQUATIONS DEFINING T H E TOTAL AERODYNAMIC D A T A

IN THE STABILITY AXIS SYSTEM ...................................... 105

N O M E N C L A T U R E ..................................................... 107

A E DC-TR-80-11

1.0 INTRODUCTION

Interest in highly maneuverable, aerodynamically controlled missiles has increased in recent years. Evaluation of mission capability for missiles of this category relies primarily on the missile aerodynamics. It is well known that the missile static aerodynamics are predominant in a mission analysis; what is not known is the importance of the dynamic derivatives in such an analysis, it is necessary to define the effects of these parameters so that in the future, an experimental test apparatus may be designed for their measurement and so that accurate simulations can be performed.

This subject analysis documents the effects of only those dynamic derivatives that may significantly affect the motion of highly maneuverable missiles. The derivatives investigated

are the direct derivatives Cmq, Cm~, CLq, CLg,, Cep, Cn r, and Cyr; the cross derivatives Crr, Cnp, and Cyp; and the cross-coupling derivatives Ctq, Cnq, Crop, and Cmr. The missile configurations used for the investigation are representative of the conventional yaw-to-turn and bank-to-turn missiles. The sensitivity of each configuration is documented with respect to derivative variations Root loci, time response, and damping ratio plots are used.

2.0 METHOD OF ANALYSIS

The sensitivity of flight vehicle motion to variations in aerodynamic and physical characteristics is reflected in the longitudinal and lateral-directional stability modes. The two longitudinal modes are short period and phugoid, and the three lateral-directional modes are -roll, spiral, and dutch roll. Linearized analysis of geometrically and aerodynamically symmetric air vehicles in wings-level, nonrotating flight without sideslip does not require more than a three-degree-of-freedom program since the longitudinal and lateral-directional perturbations are truly uncoupled. However, for asymmetric geometry and nonzero- rotational rates, the perturbations from the reference conditions are all coupled. In many cases, the degree of coupling may be small; hence, a three-degree-of-freedom analysis would be adequate, but when an attempt is made to determine the actual effect of the cross- coupling between the longitudinal and lateral-directional motions, a six-degree-of-freedom analysis is required.

The Arbitrary Degree of Freedom (ADOF) computer program was modified to provide initial trim required for level and steady turning flight conditions. The program is applicable to flight vehicles either with or without feedback control systems and can be used for any set of degrees of freedom. The equations of motion and data input routines used in the program are of a general nature and are not restricted by assuming a symmetric vehicle or an

equilibrium reference condition.

7

AE DC-TR-80-11

The modes of motion are represented by the roots of the characteristic equation formed by linearization of the six equations of motion.

Initially, all dynamic derivatives under investigation were individually assigned a value of unity and then varied over a predetermined range. All other pertinent parameters were

maintained at the required trim values. The stability characteristics were recalculated for each assigned value of the derivative, thereby mapping the effects on the five primary modes of motion.

3.0 TECHNICAL DATA

3.1 MISSILE CONFIGURATIONS

Two missile configurations were selected for the dynamic derivative sensitivity study - - a

conventional yaw-to-turn and a bank-to-turn configuration. The selection of a specific missile for each configuration was made on the basis of current configuration design trends and available wind tunnel data.

For the bank-to-turn missile, one of the early Interlab Air-to-Air Technology (ILAAT) designs was selected and is shown in Fig. I, along with representative full-scale dimensional

characteristics. The basic configurational components consisted of nose, body, wing, and

fins. The body makes a smooth transition from circular to an increasing elliptic cross section to a constant elliptical main body.

A general research model known as the Aerodynamic Data Correlation (ADC) missile

was selected for the yaw-to-turn configuration. Representative full-scale dimensional characteristics are given in Fig. 2. The ADC model consists of an ogive nose section and a cylindrical body with aft mounted cruciform fins.

3.2 AERODYNAMIC DATA

3.2.1 General

The rigid-body aerodynamic data are input to the stability axis system. Data are given in table look-up form as functions of the variables shown in the equations of the Appendix. All

aerodynamic coefficients for both missile configurations are referenced to body cross-

sectional area and diameter.

8

A E DC-TR -80-11

3.2.2 Wind Tunnel Measurements

The static aerodynamic data matrices used in modeling both missile configurations were obtained from wind tunnel tests at AEDC. The static data for the bank-to-turn

configuration (Fig. 1) were obtained for the Mach number range from 0.8 to 3.5 and for an

angle of attack of up to 26 deg. The static data matrices for the yaw-to-turn configuration

(Fig. 2) were formulated from data obtained in previous study at AEDC. The matrices include data for Math numbers from 0.6 to 3.0 and for an angle of attack from -6 through 26 deg.

The matrices for the dynamic aerodynamic data for the yaw-to-turn configuration were formulated from data presented in Refs. 1 and 2 and in other work at AEDC. These

matrices include data in the Mach number range from 0.6 to 3.0 and in the angle-of-attack range from -6 through 26 deg. The roll damping characteristics are documented in Refs. !

and 2, and pitch and yaw damping have been documented.

3,2.3 Estimated

The dynamic stability characteristics for the bank-to-turn configuration have not been determined experimentally. Several prediction methods were reviewed for estimating flight vehicle dynamic derivatives. Both semi-empirical (Ref. 3) and unsteady panel methods (Ref.

4) were reviewed. For the methods available, a semi-empirical method known as the USAF

Stability and Control Datcom (Ref. 3) appeared to be the most extensive and easiest to implement. A portion of Datcom (Digital Datcom; see Ref. 5)'was used in estimating the

bank-to-turn dynamic stability derivatives. The Math number and angle-of-attack ranges

for which the derivatives were estimated were, respectively, from 0.8 to 3.5 and from -6 to 26 deg. It should be noted that Datcom is valid only for the attached flow regime (low angle of

attack).

The missile configuration of Fig. 1 was geometrically modified to conform to the restraints of the Digital Datcom computer code. In the Digital Datcom, flight vehicles are

modeled with the use of body, wing, horizontal, and vertical tail input data. Therefore, for the computer program, fins 3 and 4 of the missile of Fig. 1 were modeled as the horizontal control surfaces, while the areas of fins 1 and 2 (also of Fig. 1), were combined as one upper

vertical surface. In addition, the wing, horizontal, and vertical tail were modeled as straight tapered planforms rather than double delta planforms.

9

A E DC-TR-80-11

4.0 RESULTS AND DISCUSSION

4.1 GENERAL

The dynamic stability analyses for the bank-to-turn and yaw-to-turn missile configurations were conducted in trimmed level and steady turning flight. Turning flight conditions were used to simulate the missiles in maneuvering flight regime. The turning flight condition provides larger values of the angular rate terms (p, q, and r), thereby giving

the aerodynamic cross-coupling moments larger values in the linearized equations of motion. This turning flight condition provides a means for ascertaining the importance of the second-order cross-coupling derivative terms.

The dynamic derivative sensitivity studies were conducted with use of flight envelopes

representative of both missile categories. The specific Mach number, altitude, and g flight conditions at which the investigations were conducted are shown in Fig. 3. The trimmed flight conditions shown for each Mach number and altitude are l-g-level, an intermediate g,

and a maximum g. The maximum g at each Mach number and altitude was limited only by

the control surface authority and by the maximum angle of attack of the data matrix. Tables 1 and 2 summarize the aerodynamics used for the missile flight conditions. Table 3 gives the

mass, inertia, and geometric characteristics used to represent the full-scale bank-to-turn and yaw-to-turn missiles.

Figure 4 gives the range for varying the dynamic derivatives plus the associated maximum and minimum values measured in the wind tunnel and estimated by Digital Datcom. Both

the direct and cross derivative ranges were arbitrarily selected at approximately twice the

maximum of the measured and estimated derivatives. The range for the cross-coupling derivatives was chosen to correspond to the range for the cross derivatives. Experimental

results obtained by Orlik-Ruckemann (Ref. 6) with use of a cone wing model have shown

that the cross-coupling derivatives combined with the acceleration derivatives (Cm~, Cn~,

and Ce&) approach and/or exceed those of the cross derivatives for high angles of attack. It should be noted that the dynamic derivatives for the yaw-to-turn configuration were

experimentally obtained as a combination of the rate and acceleration terms (e.g., Cnr - Cn/~ cos t~). In the sensitivity study, these combination derivative values were used as pure rate terms.

The root locus format shown in Fig. 5 demonstrates the primary method used to show

the effect of derivative variations on the vehicle modes of motion. The resultant roots

(eigenvalues) of the characteristic equation are plotted in the root locus format shown as the

derivatives are varied. The real part of the root represents the damping ratio, whereas the

l0

AEDC-TR-80-11

imaginary part represents the frequency of oscillation (radian/sec). The time required for each mode to damp to one half or to diverge to double amplitude is shown below each plot; the motion period is shown vertically. The roots of the characteristic equation may be either real (aperiodic) or complex (oscillatory) and either stable or unstable.

4.2 BANK-TO-TURN CONFIGURATION

The dynamic derivative sensitivity analysis was conducted for altitudes of 10,000 and 40,000 ft for Mach numbers of 0.8., 1.1, and 3.5 for load factors of up to 25 g. The low Mach number flight conditions correspond to initial maneuvers after launch, whereas the high Mach number corresponds to supersonic terminal flight conditions.

4.2.1 Dynamic Direct Derivative Variations

Longitudinal. The missile motion sensitivity to variations in the direct rate and acceleration derivatives Cmq and Cm~ is shown in Figs. 6 and 7, respectively. The range (+ 1,000 per radian) for which the derivatives were varied was at least twice the magnitude of that obtained from Digital Datcom (see Fig. 4). The influences of altitude and load factor (g's) on the missile's sensitivity to derivative variations are presented for each Mach number. As previously noted, the maximum g's available at each Mach number are limited by the control surface authority and angle-of-attack limit. Only the short period, S/P, mode shows sensitivity to variations in the pitching moment caused by the Cmq and Cm~ derivatives. The sensitivity in the short period mode at each Mach number is essentially unaffected by the load factor, g; there is, however, a significant effect of Mach number on sensitivity of the short period mode with Cmq and Cm~ variations. The short period mode is less sensitive to Cmq and Cm& variations at 40,000 ft than at 10,000 ft. The reduced sensitivity is primarily due to the reduction in density with increased altitude.

The motion sensitivity to the force derivatives CLq and CL& is shown in Fig. 8. The longitudinal and lateral-directional modes of motion were totally insensitive to variations in either CLq or CL~ over the range of + 500 per radian.

Figures 9 and 10 demonstrate, respectively, the influence of variations in Cmq and Cm/, on the vehicle damping ratio and motion response characteristics at Mach 3.5 and 10,000-ft altitude for a 10-g banked turn. Figures 9a and 10a show the sensitivity of the short period and damping ratio, Z, with variations in Cmq and Cm~ over the range of + 1,000 per radian. As can be observed in the root locus plots, the short period damping ratio is quite sensitive to variations in Cmq and Cm&. The time response shown in figures 9b and 10b results from a pitch control doublet and characterizes the changes in the short period motion for three

i l

A E DC-TR-80-11

values each for both Cmq and Cm,;. The changes in the pitch rate response to a unit doublet in pitch control are shown for values of Cmq and Cm~ corresponding to tl~e trim values, values near the point at which the short period is neutrally stable, and values near the maximum or minimum for the experimental ranges shown in Fig. 4. The changes in the time response, again, indicate the senstivity of the short period motion to the direct derivatives Cmq and Cm~ and quantify the significance of the root migration with derivative variation shown in Figs. 6 and 7.,

Lateral-Directional. The missile motion sensitivity to variations ( + 1,000 per radian) in the direct damping derivative in yaw, Cnr, is shown in'Fig. 11. The only significant motion senstivity to Cnr is in the dutch roll mode of motion. The sensitivity in the dutch roll mode due to Cnr increases with increasing load factor (g's) for all Mach numbers at each altitude.

The effects of variations (_+ 1,000 per radian) in the direct roll damping derivative, Ctp, on the longitudinal and lateral-directional modes of motion are shown in Fig. 12. Only the lateral-directional modes are shown here for clarity; the longitudinal modes were insensitive to Cep variations.

The extreme sensitivity of both the roll and dutch roll modes to variations in Cep is associated with the low moment of inertia, Ix, about the roll axis. The sensitivity in the dutch roll mode due to Cep variations increases with increasing load factor (g's) for all Mach numbers at each altitude. For large negative values of Ctp, the dutch roll mode may migrate from a periodic spiral to a roll-spiral coupling (lateral phugoid) to, finally, an aperiodic mode at large positive values. This sensitivity is made apparent by tracing the dutch roll root migration at Mach 0.8 and 40,000-ft altitude for a 5-g banked turn. The roll mode is extremely sensitive to relatively small changes in Ct o as can be seen by tracing the roll root migration at M = 3.5 and altitude = 10,000 ft for l-g level flight. The sensitivity in the roll mode due to Cep increases with increasing Mach number but decreases with increasing altitude and load factor. The roll and dutch roll modes are more sensitive to the same magnitude variations in Cep than to variations in Cnr; this greater sensitivity to Cep can be seen by comparing Figs. 11 and 12. The increased sensitivity of the roll and dutch roll modes to Cep is associated with the previously mentioned low moment of inertia, Ix, about the roll axis.

Figure 8 shows the motion sensitivity to variations in the side force derivative, Cyr, with variations in CLq and CL,~. The longitudinal and lateral-directional modes of motion were totally insensitive to Cy r over the range of _ 500 per radian.

The influence of variations in Cnr on the vehicle lateral-directional damping ratio and motion response is shown in Fig. 13 at M = 0.8 and 40,000-ft altitude for a 10-g banked

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A EDC-TR-80-11

turn. The sensitivity of the dutch roll and spiral mode damping ratio to variations in Cnr is shown in Fig. 13a. As can be noted in the root locus plots of Fig. 11, the dutch roll damping

is very sensitive to Cnr variations. The yaw rate time response shown in Fig. 13b results from a yaw control doublet and characterizes the changes in the dutch roll motion for three values

of Cnr. The sensitivity of the dutch roll mode, as demonstrated by the yaw rate response,

ranges from a highly damped motion for Cnr = -1,000 per radian to a lightly damped motion that approaches a neutrally stable mode at Cnr = 100 per radian. The change in the yaw rate

response for the three values of Cnr corresponds to the changes in the Time-to-Half-

Amplitude, Ti/~, shown in Fig. 11 for the same values of Cnr.

The sensitivity of the dutch roll mode damping ratio and motion response to variations in

Cep over the range of =l= 1,000 per radian is presented in Fig. 14. The flight condition presented corresponds to M = 0.8 and altitude = 10,000 ft for a 10-g banked turn. The damping ratio plot (Fig. 14a) demonstrates the sensitivity of the dutch roll to relatively small

changes in Ctp about zero value. This sensitivity is also apparent in the roll rate time response (Fig. 14b) for Ctp values of -200, -100, and -62.5 (trim value) per radian. The roll rate response attributable to a unit roll control doublet changes from a motion with Ti/2 - 0.1

sec at Ctp = -62.5 to a much less damped motion with TI/2 - 0.04 sec at Ctp = -200 per radian. The roll and dutch roll modes are quite sensitive to small changes in Crp; this can be

seen by tracing the root migrations in Fig. 12a.

Both the roll and dutch roll modes of motion have been shown to be quite sensitive to

changes in Cnr and Crp within the value ranges estimated by Datcom (Fig. 4).

4.2.2 Dynamic Cross Derivative Variations

The missile motion sensitivity to variations in the cross derivative, Ctr, is shown in Fig. 15. Only the lateral-directional modes of motion are affected by a variation in Cer (in the range of _+ 500 per radian). The sensitivity of the dutch roll mode to Cer increases with increasing load factor (g's) with each Mach number, whereas the roll mode sensitivity decreases with increasing g flight at each altitude and Mach number. By comparing the dutch roll root migrations in Figs. 11 and 15, one sees that the sensitivity of the dutch roll

mode to variations in Cer is equal to that for variations in the direct derivative, Cnr.

Variations of the yawing moment caused by the roll rate parameter, Cnp , affect the damping and frequency of the roll and dutch roll modes of motion significantly, as can be

seen by tracing the root migration in Fig. 16. In general, the sensitivity of the roll and dutch roll modes to Cnp increases with increasing load factor (g's) and Mach number. The roll mode may go from aperiodic stable with negative values of Cnp tO unstable with positive

13

AE DC-TR-80-11

values of Cnp. The dutch roll mode may go from an unstable oscillation with negative values

of Cnp to a stable oscillation and finally to a degenerated aperiodic mode wiih positive values

of Cnp. In general, the roll and dutch roll modes of motion are more sensitive to C,p (Fig. 16) than to Cer (Fig. 15) for the flight conditions investigated.

The motion sensitivity to variations in the side force derivative, C.,,p, is shown in Fig. 17.

As with other force derivatives, the effect of Cyp on both the roll and dutch roll modes is not considered significant.

The influence of variations in C~,~ on the vehicle dutch roll damping ratio and on the roll

rate response is shown in Fig. 18 at M = 0.8 and altitude = 40,000 ft for a 10-g-banked

turn. Shown in Fig. 18a is the sensitivity of the dutch roll mode damping ratio to variations

in C,, r over the range _ 500 per radian. The roll rate time responses shown in Fig. ! 8b result

from a yaw control doublet and demonstrate the sensitivity of the dutch roll damping to

Cer. The dutch roll motion, as shown by the roll rate response, ranges from highly damped

motion for C~ = 500 to that approaching a neutrally stable mode at C~, r = -50 per radian. The changes in the roll rate time responses, along with the dutch roll root migrations shown

in Fig. 15, indicate that the missile's motion may be highly sensitive to variations in the cross derivative C~, r.

The sensitivity of the dutch roll damping ratio and yaw rate response to Cnp is presented

in Fig. 19 at M = 1.1 and altitude = 10,000 fl for a lO-g banked turn. The dutch roll damp-

ing ratio (Fig. 19a) is sensitive to variations in Cnp over the range of _+ 500 per radian. The

.yaw rate time response shown in Fig. 19b results from a roll control doublet and

demonstrates the sensitivity of the dutch roll mode motion for three values of Cnp. The

changes in the yaw rate time response with Cn v variations follows trends of the dutch roll

mode root migrations with Cnp variations as shown in Fig. 16. As with Cl'r the bank-to-turn missile motion is sensitive to Cnp variations.

The influence of variations in C~p on the dutch roll damping ratio and yaw rate response

is shown in Fig. 20 at M = 3.5 and altitude = 10,000 ft for a 25-g banked turn. For variation

in C.vp over the range of + 500 per radian, the change in the dutch roll damping ratio of Fig.

20a is not considered large enough to significantly alter the dutch roll motion. This is shown in the minimal change in the yaw rate response of Fig. 20b attributable to a roll

control doublet for range in Cyp of -500 to 500 per radian.

In general, therefore, direct damping derivatives are important in motion analysis. The sensitivity of the missile lateral-directional modes of motion to variations in the cross

derivatives C~, r and Cnp is on the order of the direct derivative Cnr; therefore, Ctr and Cnp are necessary to perform accurate missile motion simulation.

14

A EDC-TR-80-11

4.2.3 Dynamic Cross-Coupling Derivative Variations

In the cross-coupling derivative investigation the derivatives Cmr, Crop, Ctq, and Cnq were

each initially given nominal values of 100 per radian. Then, individual variations in each derivative were performed with .the remaining derivatives at their nominal values. By maintaining nominal (nonzero) values for the cross-coupling derivatives, the possible derivative interaction effects described in Ref. 7 were included.

The motion sensitivity to variations in the cross-coupling derivative Ctq is shown in Fig. 21 for Mach numbers of 0.8, l . l , and 3.5. Including the cross-coupling derivatives in the linearized analysis results in a weak coupling in the longitudinal and lateral-directional axes as indicated by the migration in the short period, spiral, roll, and dutch roll modes. As observed, the sensitivity of the dutch roll mode with variations in Ceq depends on the flight conditions (Mach number, altitude, and load factor). Although the sensitivity of the modes of motion to Ceq is not as significant as it is to Ctr (compare Figs. 15 and 21), errors may occur in missile motion simulation if the Ctq derivative is not modeled when the magnitude approaches the maximum ranges utilized in Fig. 21.

The motion sensitivity with variations in the cross-coupling derivative Crop is shown in Fig. 22. The short period, roll, and dutch roll mode sensitivity to Crop is similar to that experienced with the Cq derivative variations. If it is assumed that the cross-coupling derivative Crop is of a magnitude approaching that of the cross derivatives, then incorrect motion simulation may possibly result if the derivative is not included in the aerodynamic modeling.

The motion sensitivity with variations in the cross-coupling derivatives Cmr and Cnq is

shown in Fig. 8 along with the variations in the force derivatives. The longitudinal and lateral-directional modes of motion are totally insensitive to variations in either Cmr or Cnq

over the range of _ 500 per radian.

The sensitivities of the missile damping ratio and frequency and the roll and pitch rate to variations of Ceq (Fig. 23) and Crop (Fig. 24) are given for M = 3.5 and altitude = 40,000 ft for a 25-g banked turn. At this flight condition, the missile was most sensitive to variations in cross-coupling derivatives. The short period, phugoid, and dutch roll mode damping ratios are shown, in Fig. 23a, to be quite sensitive to variations in Crq over the range of + 500 per radian. The short period and dutch roll damping ratios exhibit similar sensitivity to Crop (Fig. 24b). The sensitivity of the dutch roll mode of motion to Ceq is shown (Fig. 23b) by the changes in the roll rate response resulting from a pitch control doublet, whereas the sensitivity of the short period mode of motion to Crop is shown (Fig. 25b) by the changes in the pitch rate response resulting from a roll control doublet.

15

AE DC-TR-80-11

4.3 YAW-TO-TURN CONFIGURATION

The dynamic sensitivity study for the yaw-to-turn missile was conducted at altitudes of 10,000 and 20,000 ft at Mach 1.3 and at altitudes of 10,000 and 40,000 ft for Mach 3.0. The flight conditions investigated were l-g-level flight at Mach 1.3 and l-g-level and 5- and 10-g turning flight at Mach 3.0. Angle of attack and/or control surface deflection required for trim flight at Mach numbers of less than 1.3 for accelerated g flight conditions exceeded the range in the available aerodynamic data matrix.

4.3.1 Dynamic Direct Derivative Variations

Longitudinal. Figures 25 and 26 show the missile motion sensitivity to variations in the direct rate and acceleration derivatives Cmq and Cm~. The range for which the derivatives were varied (_+ 1,000 per radian) was slightly less than twice the magnitude obtained experimentally (see Fig. 4). Due to the absence of coupling between the longitudinal and lateral-directional planes of motion, only the longitudinal modes are shown. Only the short period mode shows sensitivity to variations in Cmq and Cm~. There is a significant Mach number effect on the sensitivity of the short period mode with Cmq and Cm~ variations. As noted for the bank-to-turn configuration, the short period mode is less sensitive at 40,000 ft to Cmq and Cm~ variations than at 10,000 ft because of the decrease in density with

increasing altitude.

The motion sensitivity to the force derivatives CLq and CL~ is shown in Fig. 27. The missile modes of motion were totally insensitive to variations in either CLq or CL~ over the

range of +_ 500 per radian.

The influence of variations in Cmq and Cm~ on the vehicle longitudinal damping ratio and pitch rate response is shown respectively in Figs. 28 and 29 at M = 3.0 and altitude = 40,000 ft for l-g-level flight. In Figs. 28a and 29a, the short period damping ratio is shown to be sensitive to variations in Cmq and Cm&. The pitch rate time response shown in these figures results from a pitch control doublet and demonstrates the sensitivity of the short period mode to Cmq and Cm~. The changes shown in the pitch rate response correspond to values of Cmq and Cm~ at trim, at values of zero, and at values near which the short period

mode is neutrally stable.

Lateral-Directional. The missile motion sensitivity to variations (__ 1,000 per radian) in the direct damping derivatives in yaw, Cnr, is shown in Fig. 30. The roll, spiral, and dutch roll modes are affected to different degrees by Cnr variation. The most predominant effect is on the dutch roll mode for the flight conditions evaluated. The sensitivity of the roll and spiral

16

AEDC-TR-80-11

modes is more pronounced for the l-g-level trim flight conditions. The dutch roll mode sen-

sitivity due to Cnr variations increases with increasing load factor (g's) and altitude at Mach number 3.0.

The effect of variations ( _+ 1,000 per radian) in the direct roll damping derivative, Cep, on

the lateral-directional modes of motion is shown in Fig. 31. The extreme sensitivity of the

roll and spiral modes to Ctp is associated with the low moment of inertia, Ix, about the roll

axis. The pitch, Iv, and yaw, Iz, moment of inertia are approximately two orders of magnitude larger (see Table 3) than the rolling moment of inertia. The dutch roll mode of

motion is sensitive to variations in Cep, but this sensitivity does not produce an unstable mode for the flight conditions analyzed. The roll and spiral modes are more sensitive to the

same magnitude variations in Ce v than to Cn r as can be seen by tracing the root migrations in Figs. 30 and 31.

The lateral-directional modes of motion are totally insensitive to variations (+ 500 per

radian) in the side force derivative Cyr; these modes are shown with the normal force derivatives CLq and CL& in Fig. 27.

The influence of variations in Cnr on the vehicle spiral and dutch roll damping and yaw rate response is shown in Fig. 32 at M = 3.0 and altitude = 10,000 ft for a 10-g banked turn. The sensitivity of the dutch roll damping ratio with variations ( _ 1,000 per radian) is shown in Fig. 32a. As can be noted from the curve slope, the dutch roll damping is very sensitive to

Cnr variations. The time history shown in Fig. 32b results from a yaw control doublet and

characterizes the effect of changes in Cnr on the dutch roll motion.

The sensitivity of the roll and dutch roll damping and roll rate response to variations in

Cep is shown in Fig. 33 at M = 3.0 and altitude = 10,000 ft for a 10-g banked turn. The

damping ratio plot (Fig. 33a) demonstrates the sensitivity of the spiral, roll, and dutch roll

modes to relatively small changes in Ctp about zero values. This sensitivity in the dutch roll

mode is apparent from the roll rate time response plot (Fig. 33b). For the same magnitude change in the derivatives, the dutch roll mode of motion is significantly more sensitive to Ctp

than to Cnr.

4.3.2 Dynamic Cross Derivative Variations

The vehicle motion sensitivity to the cross derivative Cer is shown in Fig. 34. Only the lateral-directional modes of motion are affected by variations in Cr r over the range of _+ 500

per radian. The sensitivity of the dutch roll mode to Cer increases with increasing load factor (g's), whereas the spiral mode sensitivity decreases with increasing load factor. By

17

AEDC-TR-80-11

comparing the dutch roll root migrations with Fig. 30, it can be seen that the sensitivity of

the dutch roll mode to Clr is greater than that for the direct derivative Cnr.

Changes in the spiral, roll, and dutch roll modes of motion with variation of Cnp are

shown in Fig. 35. The roll and spiral mode sensitivity to Cnp increases with increasing load

factor. The dutch roll mode sensitivity is also highly dependent on the load factor. In most cases, the roll and spiral modes are more sensitive to Cnp than to Cer as can be seen by tracing the root migrations in Figs. 34 and 35.

The lateral-directional modes of motion are insensitive to variations ( + 500 per radian) in the side force parameter Cyp as shown in Fig. 27. For the flight conditions investigated, the

Cyp parameter does not appear to be an important parameter for motion analysis.

The influence of variations in Ctr on the lateral-directional damping ratios and roll rate response is shown in Fig. 36 at M = 1.3 and altitude = 20,000 ft for l-g-level flight. The

damping ratio plot demonstrates the sensitivity of the spiral, roll, and dutch roll modes to

variation in Ctr over the range of + 500 per radian. The roll and spiral modes are quite

sensitive to Ctr over the range of _+ 500 per radian. This is shown by changes in the roots from a coupled root at large negative values to real non-oscillatory roots or modes for values less negative than -100 per radian. The roll rate time response attributable to a yaw control doublet (Fig. 36b) demonstrates the predominance of the spiral and dutch roll roots in the

Ctr range from -100 to 200 per radian.

The sensitivity of the lateral-directional damping ratios and yaw rate time response to Cnp is shown in Fig. 37 for M = 3.0 and altitude = I0,000 ft for a lO-g banked turn.

Changes in the damping ratio of the dutch roll mode shown in Fig. 37a further demonstrate the sensitivity of the dutch roll roots to relatively small changes in Cnp near zero. The

changes in the yaw rate time response (Fig. 37b) with changes in Cnp point up the

significance of the root migrations shown in Fig. 35. The changes in the time response shown

for values of Cnp = -1.2 and 50 demonstrate the dutch roll sensitivity to Cap, whereas the changes shown for values of -1.2 and -30 demonstrate the spiral mode sensitivity.

The sensitivity of the missile lateral-directional modes of motion to variations in the cross derivatives Crr and Cnp is of the same magnitude as for the direct derivatives and

therefore is important and necessary for accurate morion simulation.

4.3.3 Dynamic Cross-Coupling Derivative Variations

As noted for the bank-to-turn configuration, the cross-coupling derivatives Cmr, Crop,

Ceq, and Cnq were assigned nominal values of 100 per radian. Each derivative was varied

18

AEDC-TR-80-11

separately over the range of _+ 500 per radian while the others were fixed at their nominal values.

The sensitivity of the longitudinal and lateral-directional modes of motion to variations in the roll due to pitch rate derivative, Ceq, is shown in Fig. 38. The short period, dutch roll, roll, and spiral modes are shown to be quite sensitive to Ctq over the range +_ 500 per radian for M = 3.0 flight conditions. The dutch roll mode sensitivity to Ceq increases with increasing load factor, whereas the roll sensitivity remains approximately the same with increasing load factor. Although the lateral-directional modes of motion are less sensitive to C~ than to Ctr, inaccurate motion simulation may occur if the Ctq derivative is of a large magnitude and is not included.

The motion sensitivity with variations in the pitch attributable to roll rate derivative, Crop, is shown in Fig. 39. The mode sensitivity to Cmp is almost identical to that shown for Ctq. Neglecting Crop in motion analysis could cause inaccuracies if the derivative is of the magnitudes over the ranges investigated in this study.

The sensitivity of the missile modes of motion to variation in C,q is shown in Fig. 40. The

root locus plot for M = 1.3 flight conditions is omitted because of the insensitivity of any of

the modes to C,q. For the M = 3.0 flight conditions, one can note some movement of the

roots with Cnq variation, but the movement is much less than that associated with Ctq and

Crop variations.

The missile modes of motion are totally insensitive to variations in Cmr and are included in Fig. 27 with the force derivatives.

The sensitivity of the missile damping ratio and roll rate time response to Crq is shown in Fig. 41 at M -- 3.0 and altitude = 10,000 ft for a 10-g banked turn. This flight condition represents the condition for which the mode sensitivity to Ctq was the most pronounced. The short period, phugoid, dutch roll, and spiral damping ratios in Fig. 41a are shown to be quite sensitive to variations in Ctq over the range of _ 500 per radian. The roll rate time response (Fig. 41b) to a pitch control doublet demonstrates the sensitivity of the dutch roll mode to variations in Ceq over the range from -50 to 100 per radian.

The sensitivity of the missile damping ratio and pitch rate time response to Crop variations over the range + 500 per radian is shown in Fig. 42 at M = 3.0 and altitude = 10,000 ft for l-g-level flight. The primary effect shown in Fig. 42a is the sensitivity of the short period damping ratio. Associated with the change in damping ratio is a corresponding change in frequency as can be seen by tracing the root migration in Fig. 39. The pitch rate

19

A EDC-TR-80-11

time response (Fig. 42b) demonstrates the short period sensitivity to Cmp values of -500, 0, and 200 per radian. A point of particular interest is the lack of coupling between the longitudinal and lateral-directional modes for Cmp = 0.0.

The sensitivity of the damping ratio and yaw rate time response to Cnq is shown in Fig. 43

at M = 3.0 and altitude = 10,000 ft for 10-g banked turn. Little change in dutch roll and short period damping ratio is seen in Fig. 43a. The changes in the yaw rate time response

(Fig. 43b) for values of Cnq = -500, 0, and 200 demonstrate the dutch roll sensitivity to Cnq.

The missile modes of motion are much less sensitive to Cnq than they are to Ceq or Crop.

5.0 CONCLUDING REMARKS

Based on this analysis, the following observations and conclusions may be made:

la. Variations in the longitudinal direct derivatives Cmq and Cmd significantly affect the short period mode of motion for both the bank-to-turn and yaw-to- turn missiles. The sensitivity of the short period mode of motion with variations in Cmq and Cm~ increases with Mach number but is relatively unchanged by load factor (g's).

b. Variations in the lateral-directional direct derivatives Cnr and Crp significantly alter the roll, spiral, and dutch roll modes of motion for both the bank-to-turn and yaw-to-turn missiles. The sensitivity of the dutch roll mode increases with increasing load factor (g's) and Mach number. The spiral and roll mode sensitivity may increase or decrease with increasing load factor (g's), depending on the missile configuration.

. Variations in the lateral-directional cross derivatives Cf r and Cnp alter the bank- to-turn and yaw-to-turn roll, spiral, and dutch roll modes of motion in a manner similar to that resulting from variations of the same magnitude in the direct derivatives. The roll and dutch roll motion sensitivity increases with increasing load factor (g's) and Mach number.

. Large variations (_+ 500 per radian) in the cross-coupling derivatives Ceq, Cnq,

and Crop alter the longitudinal and lateral-directional modes of motion for both bank-to-turn and yaw-to-turn missiles. The yawing moment caused by pitch

rate, Cnq, was found to be less important than either Ceq or Crop. The pitching moment associated with yaw derivative, Cmr, appears to be insignificant in missile motion.

20

A EDC-TR-80-11

. Varying the force derivatives CLq, CL~, Cyp, and Cy r did not produce any noticeable changes in the modes of motion for either the bank-to-turn or yaw- to-turn missile.

R E F E R E N C E S

1. Jenke, Leroy M. "Experimental Roll-Damping, Magnus, and Static-Stability Characteristics of Two Slender Missile Configurations at High Angles of Attack (0 to 90 Deg) and Mach Numbers 0.2 through 2.5." AEDC-TR-76-58 (AD-A027027), July 1976.

2. Collins, J. A. "Verification Test of the AEDC High Alpha Roll Dynamics System." AEDC-TSR-78-P50, November 1978.

3. Finck, R. D., et al. "USAF Stability and Control Datcom." McDonnell Douglas Corporation, Douglas Aircraft Division, October 1960. Revised April 1978.

4. Giesing, J. P., Kalman, T. P., and Rodden, W. P. "Subsonic Unsteady Aerodynamics for General Configurations, Part 2, Vol. 2 m Application of the Doublet-Lattice Method and the Method of Images to Lifting-Surface/Body Interference." AFFDL- TR-71-5, April 1972.

5. Williams, John E. and Vukelich, Steven R. "The USAF Stability and Control Digital Datcom, Vol. 1, Users' Manual; Vol. 2, Implementation of Datcom Method." AFFDL- TR-76-45, November 1976.

6. Orlik-Ruckemann, K. J., Hanff, E. S., and Laberge, J. G. "Direct and Cross-Coupling Subsonic Moment Derivatives Due to Oscillatory Pitching and Yawing of an Aircraft- Like Model of Angle of Attack up to 40 ° in Ames 6 x 6 Wind Tunnel." NAE LTR- UA-38, 1976.

7. Butler, R. W., and Langham, T. F. "Aircraft Motion Sensitivity to Variations in Dynamic Stability Parameters." AGARD Conference Preprint No. 235, Paper 35, 1978.

21

MOMENT REFERENCE POINT o . 5 ~

I

0 . 8 6 1.00

L i

- -

L 8 .9

0 . 5 8

T

FIN I FIN 2

0 . 5 0

7

Figure 1. Bank-to-turn missile configuration.

DIMENSIONS IN FEET m 0 0 :4 -n

0

]> m o

.:.1 ~0 & o

/

"--- 3 .5 - ~ (DIAM)

MOMENT REFERENCE POINT o.s~

w

L 14

1 .4

J -I

Fin 1 r e

Fln 4 _ ~ ~ s m ~

Fin 3

Fln 2 T 2.8

L

DIMENSIONS IN FEET

Figure 2. Yaw-to-turn missile configuration.

b~

0 M

BANK-TO-TURN CONFIG 3.5 - - - - - - - .0

YAW - TO- TURN CONFIG 3"0 - - - " I II " / =1 '

I I

{ i . . 30

o

9 20 x

i o

0 0

/_ M--13 l / ~ / / / ; I I /

v/~ V V V ~" I 5 I 0 15 2 0

LOAD FACTOR, g'$

Figure 3. Missile flight envelope.

X

/ X

I /

}> m

(3 0

:0 do o --b

- I 0 0 0 -

RANGES OF VARIABLES RANGE :!: I 000

M A X I M U M OR MINIMUM TABLE VALUES

M E A S U R E O VIii/1/1/1/INk ESTIMATED I I

CONFIGURATION

YAW-TO-TURN

BANK-TO-TURN

I l l

¢3

do 0

. .b

b~ O~

- 5 0 0 -

PER RADIAN

0

5 0 0 -

RANGES OF VARIABLES RANGES OF VARIABLES RANGE __. 500 RANGE ¢ 500

NOMINAL VALUES I ~ I I I I !

DIRECT CROSS CROSS -COUPLING

Cmq, Cn r C~p CLq, Cp, r Cnp Cyr ' Cm& CL& Cyp

Figure 4. Range of derivative values.

Crop Cmr C~q Cnq

t-,3 "-,4

S = (:7+ jO.) (.ROOTS OF CHARACTERISTIC EQ.)

+ IMAG ( j ( J ) )

STABLEI ~ : n

-REAL ~ Z o J n ~

I / / I / I / I /

I . Z 2 = w o

UNSTABLE

+ REAL (,o')

(1 .

RELATIONSHIP OF TERMS

Z= DAMPING RATIO = - o ' l w n = ~ r l ~ c R

Tin/z - -0.693151Gr

P = PERIOD = 2 T / w

J~n(ll2) I I-Z ~ c ~ 2 = ~;- -~ ~

- IMAG

I TIME TO HALF, sec

m

TIME TO DOUBLE, sec

Figure 5. Sample root locus format.

m

o

nn

d0 o

A EDC-TR-80-11

2 0

L6

P" 12 ¢¢ <¢

)- ¢ 8 z m (.9 ¢¢

0

s...~_m

IA~E s ~ o

D

tk

0

,I

IMG~II'I1JOE (1= GA IN

GAIN • 0

GAIN • ] GAIN -*:4:

0<_ ILOGI0(I~INII' < t ] <_ II.OGI0(I~INI). < 2 2< ILOGIoilC.,A1NII' < 3 3 < ILOBIdlB~INII < 4 4<. ILtX;I.dlr~iH, I) < s 5 < ILOGI.OIIGAIHi| < 5 6<_ II.O0101IGAfNI| < f ? < I[O¢,~IGAIHI|; < 8 ,6<_ iLOr.qoIIGAIIIIt, I < ~P

l I LEVEL FLIGHT, Ig (lIT • 1.5 deg _ A L T " I0 k (ZTALT • 40 k =4.5 deg

I i 1

D/R, I0 k

=1 D/R, 4 0 k

= ,-t s, , , ,o: • • t I l K ~ t b t . i

-~ooo o~obo ~ . - S/pc 4'0 k

- tO00 011000 ] ,~---'-S PI R AL, i

R, 40 k ~PHUGOIO. I0 k, 4 0 k

-- "I" I

.5

J 5

. . - p

16

J- 12 oc o.

n- 8 , ( z m

c¢

TURNING FLIGHT, §g

~ T A L T • i 0 k = 5.8 d o g _ _ CLTAL T = 4 0 k -15.1 d i g

lok"~ ! SPIRJ

I t lD l f l , j tO k, 40, k

j - SIP= I0 k e / r l t t ~ K m t = ~ e

- I000 0 I000 ] ~ S / P , 40k

- I 0 0 0 0 I 000 I I

hL, I O k , , 4 o k ~ /-PHUGOID, R , , o ~ - ~ . ~ ¢ ,ok,l,ok

i i

I J5

I 6

1 - 1 2 (]¢

o. >- oc 8 <[ Z m

:~ 4

0 ; i

- 1 4

I 0 . 0 5

TURNING FLIGHT, }Og

__(~TALT i0 k = 9.8 deg (~TALT ~ 4 0 k :24.7 deg

I - 1 2 - I ( ) - 8

IDIR, IO k, 40 k ] -S/P|- IOk

f - I 000 0 I000

| ~ S / P , 40 k

- I'O00 0 IO00

; 1 I SPIRAL, IOk, 4 0 k F-PHLJGOiD,

- 6 - 4 2 0 2 REAL PART

] I . i 1 0.1 0 25 I - I - 0 2 5

TIME TO H A L F , sec

I 45

I i

I

I - I

- 5

4

TIME TO DOUBLE, sec

a. M = 0 . 8 Figure 6 . Bank - to - tu rn conf igura t ion - locus of roots w i t h Cmq var ia t ion.

m

o

UI O.

m

o

W G.

u ¢u m

d c n.- b . O.

28

A E DC-TR-80-11

2 0

1 6

t - 1 2 r~

I1. >- ¢ 8 < Z

=E 4

0

sy_~s LARGE • L~RGE I LARG[ 0

A

0

0

I

MAGNITUDE OF GAIN

GAIN • 0

GAIN • )

GAIN + * m

0<_ ILOGIoqGAINlll < 1 ! <_ fLOGIoIIGAINlll < 2 z <_ II.OGLoIIGAINI)I < 3 -- 3 < ILOGldl~lml+' < 4 4 <. ILOG]dlGAINI)I < 5

. s< ILOG]dlC~INlil < 6 e <. ILOGIoiIGAINIII < 1 - -

, 7 <_ it.OGIoIIGAINII, i < 6 ,, 8< ILQGLOIIGAINI)I < 9

I LEV'EL FLIG'HT, Io

GGTALT no k : I .Odeg : 2 6 dog TALT 40 k

I R,IO k

- - I - -

• I D/R, I0 k

I f S / P , I0 k

e o l t 111[ t t , D e

- I O 0 0 O IOOO

D/R, 40 k . J • / SIP, 40 k

. . . .w- - I O O O O . l O 0 0

I SPIRAL, I0 k, 40 k PHUGOID, R' 40k " I . I I " ~ / Iok ' 4'Ok • '

I

.5

t J°

to

a" o r,- uJ Q.

16

P 12 Qc <[ o. >- r~ 8 < Z (.~ <[ :~ 4

0

16

I- I 2 It- n )- Q: 8

m

¢[ =~ 4

I * * * O l- IOOO - I¢~00 * ~ ' l * ° t L * S I P ' IOR

TURNING FLIGHT, 5g I , i l l ~ S/p , 4 0 k

--aTALT • ,Ok : 3.~ dO0-ID'N, ,Ok, 4 0 ' " IO.OO- o - I O 0 0 - - I

L ' ]TALT '40k 9"3 d ' ' I i

I SPIRAL, IO k, 4 0 k - - ~ PHUGOID,

R, IOk R, 40k_~% L ~ / Iok'140k ~ - ~ - -

m

o" i 2 rr

IJJ Q.

I TURmNG FL,GHT O+ --aTALT ,Ok ". 60 d,g

( / T A L T : 4 0 k 162 deg

I

I R , I O k

o l "t I . , , - ,0 -B

0 .05

I D / R , I0 k , 4 0 k

_ - _ ,,., S/P, I0 k - IUUU I v IAAA

I - I O O O 0 lOOO t S/P, 40 k I

SPINAL, IOk, 40k--~ F PHUGOID' I " + o k ~ . . ~ , / , o k 40J

I - 4 2 0 2

REAL PART n

! I I I I I OI 0 2 5 I - I - 0 2 5

TIME TO HALF, sec TIME TO DOUBLE, sec

i 9 ry"

, UJ l o_ l I

-I 5

J 4

b. M = 1 . 1 Figure 6. Continued.

29

A E DC-TR-80-11

I 0 0

8 0 I -

¢ 6 0 >. n,

Z ~ 4 0

IE

2 0

8 0

I - 6 0 n.,

a.

b- n- 4 0

z (.9

2 0

0

SYld At~GNnUDE OF GAIN LARGE • GAIN - 0 LARGE • GAIN • 1 LARGE m GAIN -",'@

• O <_ II.OGlellr~lNI)l < 1 o t <_ It.OGldlGAINi)I < Z . z <_ ILoGSdlr~ialll < o 3 <_ ILOG~IG, AINI)I < 4 e 4 <_ ILO6IoIIGAINI)I < 5

__ . 5<_ ILO6IQ{I~INIII <6 - - , 5 <_ 11.OGI6{IGAINlil < 1 , ]r <_ it.O61dlr.,AiNI) I < a ,. B< iLO6totIGAINlll < 9

- I 0 0 0

LEVEL FLIGHT, Ig

e T • 0 .2 _ ALT • I0 k _(ZTALT : 4 0 k • 0 .5

i R, I0 k, 40 k

deg deg

TURNING FLIGHT, I O i

a TAL T

a TA LT

R, IO k

: i0 k • I,I deg • 40 k • 3.6 d e g

I . " I - I 0 0 0

R, 40 k w

I SPIRAL,

S / P , I0 k

m ~ i • •

0

S/P, 4 0 k o t O O • l l K e t ] t t o

----" D/R, I0 k

DIRp 40 I0 k , 4 0 k PHUG()ID,

• - ~ . , J , f L 4 0 k

I 0 0 0

S/P, IO k • t ~ t • t m

O IO ;O

S/P, 40 k

- oo;" "1" "o x . " " 1 oo O/R, 40 k

D/R, i01 I I F P H U G O I D ,

l ok ,140k S P I R A l , IO k, 1 0 k ' ~

I

o .2 9

n ,

bJ CL

I

t .2 9

n* jw Q.

i

_J2

8 0

6O p-

¢L

>" 4 0 cc

z

< 2 0 :S B

- 1 4

I 0 . 0 5

I U I ~ N I R I J r L I O I l l l ¢O1~1

C[TAL T • iO k • 2 .6 deg (~TALT__ • 4 0 k , 8 . 4 d i g

l e t t •

- 1000

I

I R, IO k

-12 - I 0 - 8

D/R, 40 k

O I

T IME TO H A L F , sec

S / P , I0 ~

0 I0~)0

S/P, 40 k

x - ' -

- I 0 0 0 0 I 0 0 0 °'" '21" I

i I I S P I R A L , I O k , 4 0 k ~ [

u I ~ I . - - PHUGOID, R, 4O k ] ~1 I / i o k , 4 0 k

w I I ~ i i

- 6 - 4 - 2 0 2

R E A L TIME I

I s 1 I l I 0 . 2 5 I - I -O .25

T IME TO DOUBLE, sec

I

I - I . I

u o

E .2 o_

n- hi CL

i

c. M = 3 . 5 Figure 6. Concluded.

3O J

A E D C - T R - 8 0 - 1 1

2 0

i 6

1 - 1 2 n* x¢ o. >. ¢ 8

:E 4

SYM /~¢~IIiIJDE Of GAIN LARGE z GAIN • 0

LARG[ o GA IN - * m

O < ILOGIo(IGAINIII < ! o t < ILOG10( GAINIII < Z • 2 < iLOG]O4~AINili < 3, O 3 <_ iLOG104iGAINIII <" 4 = 4 <. ILOGI0ilGAINIII < 5

__ 5 <_ ILOGto(IGAIN]II < 6 _ 6 < [LOCtO(IGAINIII < ; 7 < [I.OGIoilGAPIIH < 8

., s<_ ILOGIoIIGAIql)I < 9 I I

LEVEL FLIGHT, Ig

(~TALT ; i o k : 1.5 dog _ _ rWTALT 4 0 k 4 .5 d e g

,0.1 I *----t----

I O/R, I0 k

I " D/R, 4 0 '

X SIP, I0 k . . ~ . , x . . . ~ , .

- IOOO O IOOO

iIOOO _O'~_fO00 _. I ~ - -SPIRAL, PHUGOID,

R, 40 k I " ~ Iok, 4 0 ' • , I , I

~ 5

u ¢D ¢n

o" o rw

IJJ CL

I - n.

n >- n-

=.

¢¢ :E

16

TURNING FL IGHT, Dg

- - O ' T A L T I O k Q~TALT : ~ 5 .8 deq 4 0 k IS . I d e g

R, IO k

I I

TDIR, IO k, 40 k

I S/P, IO k

.... ~'";~oo - IO00 0 - - - ~ S/P, 40k

I ,,4m, I - IOOO O IOOO

I SPIRAL , I 0 k . 4 0 k - ~ LI-PHUGOIO.

I I _ \ # .o, . ;o, R,4--O " I ,

u

m

t-~

o

13.

-,5

J

F-

<X a.

>- tw

Z

t9

4

0

- 1 4

[ 005

' F I I I O,R,,O'. 4o, s,P.,O' TURNING FL IGHT . lOg . , , , , ~ , . , , .

TALT 4 0 k =24.7 d e g - I 0 0 0 v v v [

I i~ ok - I 0 0 0 0 IO00

I , SPIRAL, iok,-~ I" I 4 0 k - . ~ \ / -PHUGOID , ~,.,ok-, 1 ~,4ok-~,~/,ok.,o, ' I

- 1 2 - I 0 - 8 - 6 - 4 2 0 2 4 R E A L P A R T z

[ I I J I I O I 0 2 5 I - I - 0 2 5

T I M E TO H A L F , sec T I M E TO D O U B L E , s e c

r

I - t 5

u ¢D

6 i o

cE bJ O .

- 5

a. M = 0 . 8 Figure 7. Bank - to - tu rn conf igura t ion - locus o f roots w i th Cm,;. var ia t ion .

3]

A E D C - T R - 8 0 - 1 1

2 0

16

~- 1 2

o. ). o: 8 =,

(.9

:Z 4

0

1 6

1 - 1 2

Q. >- n. 8 < z w

: i 4 w

1 6

I -

o .

) -

z

¢.9

IE

SY.~ LAI~ x LARGE 6

LARGE o

6

0

t t

0

I

t

0

Q .

I

! ]

R , I 0 k

MAGNITUDE OF GAIN

GAIN • O

GAIN - I

GAIN - * W

0<_ ILOG]o(IGAINlll < ) ) <_ ILOGIo(IGAINlll < Z' Z < ILOGIOqGAINII < 1 - - ~J <_ ILQi;L~IGAINlu < 4 ' 4 <_ ILOGLo(IGAINI)I < 5 : ~i <_. ILOGIG(IGAINll I < 6 G<_ ILOGIO(IGAINIII < I - - - - I < II.OG!o(IGAINIII < 8 8 <_ IL~ior GAINIH < t

I i LEVEL FLIGHT, Ig

~ TALT = I0 k : I . O d e g TALT • 4 0 k 2.G deg

I SPIRAL

R, 40 k

I I O / R , I 0 k

I S / P I0 k

l a t l l ~ l { t m l ° 0 I 0 0 0

- I O 0 0 ~ . ~ X . - D / R , 4 0 k - ~ I

S/P, 4 0 k ----, I

- I 0 0 0 0 I 0 0 0

, PHUGOID, ,o'.,o'\./,o,.:o, j I

5

u g

o o 0¢ uJ {1.

12

8

4

0

- 1 4

I 0 . 0 5

TURNING FLIGHT, 5 g

__~TALT . iOk : 3 . 5 d . e g _ I D / R ' GTALT = 4 0 k 9 . 3 e g g

l R , I O k

- - - F -

• 1" " " ° ] q " ° ° ° °1 S/P, IO s _ I 0 0 0

- ao00 01 .,41,,, S /p , 4 0 kl

i0 k, 4 o k . - I 0 ~0_0. ._1000

I , I

SPIRAL, IO k, 4.O k -~ /-- PHUGOID~

R , i o k ~ i j . l I ok '140k

'I"

nr~n 0 moo0 .. " ' ~ ' ~ . ~ * ] ~ 4 - • * IS/P, I0 ~

I I / - . m - I TURNING FLIGHT, I0 - I 0 0 . 0 0 I0,00 (2TA . . . . a, ,, 6 0 de,', A I I I S/P, 4 0 k

_ ~ , ~ [ . / = l U " " - " T - I I ,,~TALT • 4 0 k ,,16.2 o e g - / D / R , IO x, 4 0 k !

I I ,

I R, I0 k

- 1 2 - I 0 - 8

i

/ - - PHUGOID, SPIRAL,I R ' 4ok! " - ~ ~ , a l O k , 4 0 k ~ / t O k l ' <'O k

- 6 - 4 2 0 2 R E A L PART i

I I J I I 0.1 0 . 2 5


I - I - 0 . 2 5

TIME TO D O U B L E t sec

.5

- 5 /

u ¢u

o" o n- bJ o.

b. M = 1 . 1 F i g u r e 7 . C o n t i n u e d .

i I

u~

_ i 0 m

bJ

5

32

A E D C - T R - 8 O - I 1

I 0 0

8 0 I - e,,

" 6 0

n-

Z N 4 o

l !

2 0

0

SYM W~GNInl9( OF GAIN

LARGE • GAIN • 0 LARGE 8, GAIN - I LARGE o GAIN - ~ m

• 0 <_ ILO~o(IGAINI)I < ! o I < tLOGIOIIGAINI)I < 2 ~t L¢_ ILOGIOIIr~INI]I < ) o 3 < ILOF, tdlGArNI)I < 4

4 < JLOGI.o[IGAINlll < S , $ <_ EO61dlGAINI)I < e , 6< ILOGI.011GAINI)I < T , ? <_ ILOG~IGAINI)I < II .. 8<_ ILOF, LoIIGAINII, I < 9

|

I--O00 * * * m

L E V E L FL IGHT , Ig

QtTALT • I 0 k " 0 . 2 d e g - - 0 . 5 d e g

. . ~ T A L T , 4 0 k

I k R, I 0 , 4 0 k

S/P, I 0 k

~IK • . •

0 I 0 0 0

t

S / P , 4 0 k I . . . . I * * . . . . . .

- I 0 0 0 / O : IOOO I i - - O I n ' IOk

k PHUGOIO, i D /R . ~ / ~ 4 0 S P I R A L , I 0 k . 4~0 k ~ , , / I o k , 4 0 k

, " , % , - - I

u Q m

c= .2 9

ul a.

..i j

_42

8 0

D.- 6 0 eP

a. >. n, 4 0 < z (.9 ¢[ :~ 2 0

8 0

I-- 6 0 n..

~" 4 0

z

< 2 O If

0

- 1 4

' " lOg I TURNING F L I G H T ,

( ~ T A L T : i 0 k • I . I d e g

G T A L T • 4 0 k : 3 . G d e g

- I (~00 * * *

I R , I O k

R , w 4 0 k

S / P , I 0 k

A • 0 I 0 0 0

I S / P , 4 0 k - I 0 0 . 0 " * " * r O ~ ' O * • , 1 0 0 0

I J w D/R, IO k

i-- PHUGOI° , S P I R A L , I O k , 4 O k - ~ f l o k , 4 0 k

• I i . . I

I

o" .2---

w

I

2

~ r U R N I N G r l . l l i n l ! G;;)I~

CtT 2 . 6 deQ (~ A L T • I 0 k "

TAL T J 4 0 k " 8 . 4 d i g S /P , I 0 k a i

1OO " ° * X • • - I 0

D /R, 4 0 k S / P A I . . . . ,

- I 0 0 0 O/R, I 0 k

"1

t

4 0 k

0

- 8 - 6 - 4 - 2

R E A L T I M E

0 . 0 5 0.1 0 . 2 5

T I M E TO H A L F , s e c

S P I R A L , I0 k , 4 0 k - - ~

- 1 2 - IO

c. M = 3 . 5 Figure 7. Conc luded .

" IO'OO

=~oo

f PHUSOIO, I0 k , 4 0 k

I 0 2 4

I J I 1 I - I - O . 2 5

T I M E TO D O U B L E , s e c

m

a .2 o

r/. LU n

33

A E O C - T R - 8 0 - 1 1

4 0

3O

p- n.. < 2 0 z

- - I O

5YM MAGNITUO[ OF GAIN tAIW( • GAIN " O I, AI~G( • GAIN • I .AIUG[ • GAIN " s o

• O<_ IL~IGIIGAI;;lll • ] o ] < It.OGIOIIGAINIII < i

I * ~ < ILOSIoilGAINI)I < 1

¢ 3 <_ II~]BIIGAINI)I < 4 4<_ I;.O6]I(IGAINIH < |

- - 5<'_ l..Or, llilGAINlll < i i< p,,.OG1:~IGAINIII < i' f < ILOGIQilGAINll I < i

( : ITALT • 40 k .4 .G deg- l O i ~ 4 0 k ~ 'q/P, I0 k

R, 40 k " I ~ S P I R A L , PHUGOIO, I ~ " ~ " Iok* 4 0 k - -

.25 ~

i

J, 2

4 0

3 O .¢[ 0. >-

2o _z <[

I , | TURNING FLI6HT, 6g I _aTA, T . ,o ' ' ..G d . 1

. U T A L T ,, 4 0 k -15.1 d e g . ~ I. ' j

R, IO k

1

D / R , 4 0 k i

D/R, I0 II

SPIRAL, o

I

+o, t

25

w

0 0

W 0 .

5 0

4 o i - n,.

eL 3 0 3. n*

Z

4 I f

I O

0

- 1 4

J 0 0 5

TURNI~IG FLI I ;HT, l o g

~ TAL T • i0 k :' 9 8 d i g , ,247 deg • TAL T 4 0 k 1

I ' D/R, IO N X

- 1 2 - | 0 - 8

0.1

T I M E TO H A L F , sec

I S/P, 4 0 k

S/P, I0 k I

l ~ ' k S / P , 4 0 " SPIRAL. 4 0 k

l

/-.: s P,R,,,., ,o R, IO k R, 40 k / j / ~ - P H U G O I D ,

~. ' -- ~ - iok, 40 k -

- 6 - 4 2 0 2 4 REAL PART

I I I I O 25 - i -O 2 5

j ~

TIME TO DOUBLE, sac

. 28

i i P

r~ .5 o

n*

Figure 8. a. M = 0 . 8

Bank-to-turn configuration - locus of roots with CLq, CL~, Cvr, Cnq and Cmr variation.

34

A E DC-TR-80-11

4 0

~ 3 O

31'-

,~ 2o _z

~- ,o

4 0

3 0 ¢( O. >- ,~ 20 _z

0

5 0

4 D I-

eL 3 0 >- ¢¢ ,4 Z

3 2 O ,4[ :E

ID

o - 1 4

I 0 0 5

SYM IM._.MITtlO[ OF G~llt MRGE x GAIN • 0 IARU • GAIN • 1

• PAIR, - t l . • I ( . ll.Oi~lOAllllll < l

o I <_ ll.OOlollfdilNJil < ! • l< IL~loIl~ml'al < ) O ) <_ IL001ollr.qllil < 4 I I 0 l <.. ILOOlohf.~lilll < $ 1 I S<. ILOGIoIIGAIMII <6

_ _ , * < lu~dl~n.II l < I , 7 < ll,gGlofl~ilMlll < I I I

i~_ Lnn-lOIC~lk,tl ( ! I I I 4 l k - - L E V E L FLIGHT, Ig - - D/R, ID - -

.o,., , i : ! , --~ITALT " IDk .s 2.6 deg $ p, i0 k UTALT • 4Dk n/' R, 40 k

R,ID k , , R

i i i

I I I TURNING FLIOHT, 5g r ,T . • 5 .5 dog _ - ~ L T • iD k a, 9.3 dog

-- QTA,.T • 40 k

R,ID k I

r 0,R. ,D' I I

Jl(~ D/R, 4D k

S,pI, IO k ;1~ I k I / • S I P ' 4 D '

SPIRAL, ID k, 4Dk " t F PHUGDIO, ; R;.D' .D',.D'

25

I* II

0 I o

J, J, -~ .25

I

o , r t

0

.=

i

J , _1 2

i i i i TURNING FLIGHT, IOI I

Q_TAL T _ ,,4D k'D I • S.Od*g I - CITALT -16.2 d i g

J I D/R, 4D k D/R, I0 k I ]1[

t S IP i lOk I ]K SIP, 40 k

] ( / SPIRAL, IO k '14Dk "~ PHUGOID,IDk, 40 k

R.,D' R *D" - . . . ~. I "J ' l i

-12 - I 0 - 8 - 6 - 4 2 0 2 REAL PART l

I i I I I I 0.1 0 .25 I - I -D .25

T IME TO HALF, eec TIME TO DOUBLE, sec

2 5

m ="

.5 o E ==

b. M = 1 . 1 F igure 8 . C o n t i n u e d .

3 5

A E D C - T R - 8 0 - 1 1

SYM MAGNI/IUO[ OF GAIN IAI~( • GAIN - 0

| 0 0 UMI~ i ~ l i = ) LAIIE i GAIN -I~O

i O < ILOGIOI I~ I . IN l l l < I o ] <_ IL, OOIOiIGAINI|I < Z * ! <- ILOGI~W'AINI)I < ) i

8 0 o S ~ ILOGNImlNli. < 4 [ t-lZ, , 4< ILOGLOInGAINlll <S i 4 * | < ItO616liGAIal)l < | I it GO , I< ILO;Ilflr, AINltl <T . ~ L . s / P , IO k .1 )" -- . 1 < ILOIGIOg~AINII' < | / I I (n" .. I ( I.~-#IGAIMLII <9 / - - I

4 0 - LEVEL FLIGHT: Ig --< / O T A L T • i0 k " 0.2 dog J S/P, 40 k . 2 " :E | OTALT = 40 k • 0 § clog I [ " h

2 0 / I i . D/R I0 • I io, R 40k " ' "

R I O k R 40 k I I ' ' " ' ~ = i ~'- PHUGOID, I I " 1 ' ~ " "1 SPIRAL, IOk ,40 k - ~ - - J ll~lk 40 k

8 0

w 6 0

G. ) - n, 4 0

Z

:E 2 0

' I I I I TURNING FLIGHT, lOg QTAI' T • i0 k • I.I d i g

I -- ( ITALT • 40 IL " 3.6 dag S/P, IO k

"1 I 9/P, 40 k

t: O/R, 40 k ~/R, i l I--

R'lOk ~ '| R ' 4 0 k SPIRAi ' IOk ° i o k " ~ ,

i

i0 k

C PHUGOID, IOk,a40k

I

t

I

_:i2

8 0 I I I I

TURNING FLIGHT, 2Sill I CITAL T : i0 k : Z.G dog

; eo --aT.LT-.Ok S 'do0- - S,P~.ok

" I I > 4 0

_ lzo ~

0 ~ - 2 8 - 2 4 - 2 0

0.028

i D/R. 40 k , ,zip. 40 k

x I Iz~ I D/R i l l ~ k

SPINAL : UGh, 4Ot - -~

-16 -12 - 8 - 4 REAL PART

0 05 0125 TIME TO HALF, sac

I PHUGOID, I0 k, 40 k

- - - F - - o 4 iJ

I I I 0 . 5 - 0 5 - 0 . 1 2 5

TIME TO DOUBLE, sec

tt

E - .2 o

Oc =*

_.-12


i

36

AEDC-TR-80-11

o m i - n-

(.9 Z n

0 i

N

i =d, -r n* (.1F" i , - z - - 0 a. ~.)

-I- ( . I

ia.

0.4

0.3

0.2

0.1

0

60 ,000

sY._~M LARGE x LARGE e, L ~ t u

&

0

i t

0 I

t

0.5

MAGNITUDE OF GAIN

GAIN • 0 GAIN - 1 LeA I'~1 ~...IW

0 <_ ILOG[oIIGAiNIH < ]

! < ILOGIoIIGAINI)I < Z

2 < IL(X;LoIIGMNIII < 3 3 <_ ILOGtoIIGAINI)I < 4 4 <_ ILOG[oI GAIN])I <" $

5 < :LOGLO( GAINIli < 6 6 <. iLOGIO( GAINIH < 7 7 < ]LOG[oI GAINIII < 8

8<. ILOGIoIIGAINItl < 9

I

-0.1 - I 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0

a.

- 2 0 0 0 200 4 0 0 Cmq, p e r / r o d

Damping ratio

4 0 , 0 0 0

M = 3 . 5 0 A L T I T U D E = I 0 , 0 0 0 f t TURNING F L I G H T , lOg

~. P DAMPING RATIO

600 8 0 0 I 0 0 0

2 0 , 0 0 0

- 2 0 , 0 0 0

- 4 0 , 0 0 C

-6O, OO0 0 0.2 0.4 0.6 0.8

TIME, sec

b. Time reponse Figure 9. Bank-to-turn configuration - effect of Cmq

ratio and time response.

1.0

on damping

1.2

37

A EDC-TR-8O-11

0 i - n-

(.9 Z n =E

0 I

N

,.,Io a

k - Z --0 Q- 0

n

sy__Ma LARGE z

O, 5 LAR~ • L~Rt~ o

A

13 0.4

I t

O

I

0.5 * #

0.2 i

0.1

0

-0.1 - I000 -800

6 0 , 0 0 0

40,000

2 0 , 0 0 0

0

- 2 0 , 0 0 0

- 4 0 , 0 0 0

LI - 6 o , o o o

o

Figure 10.

M = 3 . 5 0 A L T I T U D E = I 0 , 0 0 0

~GNITUKOrr~m T U R N I N G F L I 6 H T , l O g G A I N - 0

G A I N - 1

G A f N - 2

'0 <_ ILOGIo(IGAINItl < 1 ]. <_ ILOGzdlGAINI)I < Z 2 < ILOGio(IGAINI)I < 3 . 3 <_ ILOGio(IGAtNI)I < 4 4 <_ ILOGio(IGAINI)I < 5 $ < ILOGIoIIGAINIil < 6 6 ( ILOGIo(IGAtetl)I < z ' 7 <_ ILOGI0qGAINI)I < 8 8 < ILOGIo(IGAtNI)I < 9

I J

f t

~ . ~ _ / - - SIP DAMPING RATIO

- 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0

C m ~ p e r / r 0 d

: a. Damping ratio

4 0 0 6 0 0 8 0 0

rl A p.rrOOsOOo,, M i

" ~ . . v " ~ ~ . r -

..//it 'vv

;llV I 0 . 2 0 . 4 0 . 6 0 . 8

T I M E , s e c

b. T ime response Bank-to- turn conf igurat ion - ef fect of Cmd ratio and t ime response.

I.O

on damping

tO00

1.2

38

A E D C - T R - 8 0 - 1 1

4 0

~ 3 0

)-

~,o Z

~- I0

SYM IA~NIilIglE OF GAIN LARGIE • GAIN • 0 LARGE s GAIH • ] lARGE e GAIN - * ¢0

~, O< ILOG1~IGAINlll < 1 o I <. ILI~IG~IGAfNIII < 2

- - ,,, 2 <__ ILOGIoIIGAINIII < ~ --

o 3 <_ ILOG]o(I~I~Illl < 4 n 4 <__ ILOG)~IGAINII, < 5

l * ~ ~ ILOGIo(IGAINI) < 6 , 6 <_ iLOGIoIIGAINI) < 1

1 <_. ILOGIDIIGAIN[): • B ., S <_ ILOGI~IGAINI)I < 9

I I I LEVEL FLIGHT, Ig

~ T A L T • 1.5 d i g I 0 k - - reTALT • 4 0 k " 4 . 5 dog

' I I R, I0 k ---t---

. - , o o o o / m o o D/R. 40" ~ . p ,~--z-t - - .

- I o o ~ IOOO S/P, I?nK JK j [ . ~ S /P , 4 0 k

I R, 40 k I -- SP IRAL, PHUGOID, ,o o o - = - - , o o ~ ,o k, ,ok

- . 2 5

m

0 0

.5 o: w I t

4 0

30 ,¢[ O.

>-

20 _z

• _ a0

I I I

TURNING FLIGHT, 5g

~ T A L T I 0 k . rWTALT ~ ~ :5.8 deg 4 0 k 15.1 d e g ~

i i i oo I o • •

- 1000 O/R, I0 k._

o

i

I .,,o. I i q~rl rl ~ i rl/,L i,~

I i

D / R , 4 0 k 0 IOOC

- - IOC 3 _ _ Q • •

S/P, I 0 k

I aS S / P , 4 0 k

SPIRA k , I0 k , 4 0 k ~ P H U G O I O .

. 2 5

m

0 0

w

5 0

4 0 k- n-

" 3 0 >- n-

Z N 2 0

I 0

0

- 1 4

I 0 0 5

Figure 11.

I

I ! ! I •

; - i o o

TURNING FLIGHT, lOg

~- (~TALT . i o k " 9 .6 d e , i TALT • 4 0 k , 2 4 . 7 d e g

- t O 0 0

DIR, I0 k m • • M E

0

- i 2

I D/R, 4 0 k

I x o

I - I 0 - 8

0.1

T I M E TO H A L F , se¢

* IO(~O

S/P, IO k

G O 0

I

I

I -i .2~

u o m

r t . 5 o

r~ bJ O.

~ S/P ' 4? k I S P I R A L , 40k-~a[r S P I R A L , I0 k

R , IO k R, 40 k , \ % l E PHUGOID,

- 6 - 4 2 0 2 4 R E A L PART

[ I I [ I I 0 . 2 5 I - I - 0 . 2 5

T I M E T O O O U B L E , s e c

a. M = 0 . 8 Bank-to-turn configuration - locus of roots with Cn r variation.

39

A E DC-TR-80-11

4 0

I - 3 0 Q¢

.~ 2 0 Z

z_ no

4 0

5 0 a. >-

2o z_

~ no

SYM kt~GNITUDE OF GAIN LARGE • GAIN - 0 LARGE • GAIN - I LARGE • GAIN - t II

t 0 <_ ILOGIotlGAINlil < l a ] < ILOGLdlGAiNI)I < z

i ~ i <__ ILOGLoIIOAINlil < 1 0 l < ILOGL~tlGAINI)I < 4 , : i <_- 11.OGLotlGllklii < 5 I

* I S<_ LOeidlcJ, Il l l l i < I I - - , I 6< LOGLdlGAINI)I <7

• 7 ,~ ILOGioIIGAIHI): < I .. 6<ILOG~IGAINI) < t I D/R, I 0 k

~ , o I o l - - LEVEL FLIGHT, Ig - - - - - I O 0 0 ~ I O 0 0 - -

GT,, , - • ,,-,k " 1.0 d i g S/P, I0 k D/R, 40 k I ,.,,.. ,v • k o

G T A L T , 4 0 k 2 . 6 d O g I -,ooo ° " ~ 'ooo I i I I I z ' l I-s/P'4°'

R , I O k I ~ S P I R A L , I 0 k , 4 0 k r - P H U G O I D ,

I R O' I ! "O'

, , - T I I- . . . . T'URNING FLIGHT, Sg

[ _ ~TALT : ~S dog "TALT "'Oh " , S d ' ,

4 0 k O/R, I 0 k 0

. . . . . . . . .X~O~I_: : : IO00 - IOOQ D/R, 4 0 k I I s'r !

ilk S / P , 4 0 k

I I SPIRAL, IOk ,n4Ok-~ I R, I0 k I .:o,

.25

¢,i g) iT

0 0 I

W

- ~ 2

1 . 2 5

r , o

.5 ; LO a.

5 0

4 0 k-

<

" 3 0 >- IZ: ,4 Z ~, 2 0 :E i

I 0

0

- 1 4

I ' I I TURNING FLIGHT, log

C~TAL T i0 k • G.O d i g I ~TAL T ~ 40 k - i 6 . 2 dell -

I - I00'0 . f l "=

- I 0 0 O O i D /R , I 0 k

Q i •

D / R , 4 0 k

0 • ~ t • •

tO00

I S / P , I 0 k S / P , 4 0 k

LOGO d •

'R, i~)k i I000 ~ ,r~r~n'

- 1 2 - t 0 - 8

0 0 5

S P I R A L , IO k , 4 0 k " - ~ PHUGOID, ,o,,.o., - 6 - 4 2 0 2 4 REAL PART

I I i I I I 0.1 0 . 2 5 I - I - 0 . 2 5

T I M E TO H A L F , sec T IME TO D O U B L E , sec

.25

in

0 .5o_

~J OL


40

A EDC-TR-80 -11

40

~ 30

n,, < 2 0 Z

<

~-DO

40

3 0 ¢[ G. >-

2 o z_ L~

~ ~o

SYM FA~J,N miD[ OF GAIN LARGE • CAIN • 0 LARGE a, GAIN - I LP~RGE o GAIN - i m

m. 0<_ ILOGIoIIGAINIII < l a I <_ ILOGIoIIG~,INIII < ! . Z < II.~IO(I~iNlll <

o ) <_ ILOQ]O(Ir~INlll < 4 i 4< ILOGIolIGAINI)I < S , 5 <_ iLOGIoHGAINI)I < 6 , 5 <_. iLOGIoIIGAINI)I < 7 , 7 <_ ILOGIoIIGAINi)I < 8 ., e <_. iLOGIoilGAINI)I <

i I. IOO0 ' I • " "

LEVEL FLIGHT, Ig

aT'LT-A • I0 k " 0.2 dog -- nTALT. • 40 k • 0.5 dog

R ' l ok R ' 4 0 k I i SPIRJ

S/P, IO i T

S i P , 4 0 k

D /R, IO k

0 6 0 D/R, 40 k

• a l m w L t m * .

- tOO0 0 IO00 I PHUGOID,

,L. IOk,140k ~ / ION. 40 k I

S/P, I 3 k

GTAL T • 40 k 3.6 d i g

I - IO00

R ' ~ I000

i . . . .

D/R, I0 k I o - I 0 0 0 ~ D /R , 4 0 k I O 0 0

0

' | : 0 k S P I R A L ! JOk| 4 0 k - ~

8OO

PHUGOID, i o k 4 0 k

I

.25

,5

.25

o

5 0

4 0 I - n,.

" 5 0 >- n..

Z 2 0 <[

:E

IO

0

- 1 4

I 0 0 5

TURNING FLIGHT, 25g" . . . . ] f .... I / ~__~ (2TAL T . i0 k • 2 .6 deg I / S,P, IO k

~TAL T = 40 k • 6,4 d i g " - - - ' ~ m ~ S l P , 40k ' -~ --

. . . . . x J . • x . o4 ° - t o o " I o I O

- - - -1 - : - - " " " " - , o ; o - I 0 0 0 D IR, I? k

S P I R A L ~ - ~ I / -

.,o, , , o k - ~ - . ~ - - - ~ - - - - . + - ~ , o o o - . - I - , o o o I ..,. ,

-12 - I o -B - e - 4 2 o 2 , REAL PART

B

i I l i I i O.I 0 .25 I - I - 0 25

TIME TO HALF, sec TIME TO DOUBLE, sec

c. M = 3 , 5 Figure 11. Conc luded .

25

u lip

0 .5 _o

n,. bJ 0.

4 ]

AEIDC-T R-80-11

4 0

~ 30

>-

< 20 Z

- I0

40

~ 3 0 Q.

<~ 20 Z w ¢9

• _ ~ o

50

4O I-- CI: ,lI

a. 30 n-

Z

N 2 0 l i

I 0

SYM IIMG/q'nJIE OF GAIN LARGE • GAIN - 0 LARGE a GAIN • ] I),RGE • GAIN - :1: Co

A O<_. 11.OGIO(IGAINIII < I a I. < ILCGIOIIGAINlll < 2

_ _ , 2 < It.OGIO(IGAIHI)I < O ) ,~ ILOG;O(IGAINIII < 4 , 4 < ILOGIdlGAINlll < 5 + S < ILOGsoIIGAINlll < 6 , 6 <_ ILOG]o(IGAINlll < 7 , 7 <_ ILOGIdlGAINlll < S . . S<_ ILOG]611GAINlll <

I I ~ L E V E L FLIGHT, Ig i CKT • I k " I S dig A L T 0 "

Q T A L T • 4 0 k " 4 . 5 d o g

i R, IO k -

- I S O - I . 0 0 - 5 0 -~_ r l -

I I - -ISO

O/R, I0 k

~ IV"- D/R, 40 k % 1 5 0 , . ~ _ - I 5 0 r S P I R A L , I 0 k ,

I O O O ^ g l O O O . ~ I _v / - , , , IOC

: -:um- -. . - I

R, 40

40 k

, T U R N I N G

Q TAL:T • i O k " S . 8 d i g

_ r l T A L T , 4 0 i , 1 5 . 1 d e g ~

! D/R, I0 k

~ ~ R ' l o k - ~ -

R, 40 k ~

0

~-S pi R A L ,40~k~

" ~ k m

0

- 5 0

TURNING " FLIGHT, lOg' .~TALT • i0 k = 9.8 deg (ITALT • 40 k ,24.7 deg ._ .~ ,

i 1:,2o i •

t

- I000 -~, :

/ /

/ f

- 4 O

I 0.015

TiME TO HALF, sec

D/R, 40 k o I

m t f • • • •

0

D/R, t0 k %

- I000~ mlim

R, iok--~, R ,40 k • ~o~ I ~, I-uooo

SPIRAL, ~ok, 40 - 3 0 - 2 0 - I 0 0

REAL PART I I I I I

0.02 0.03 0.05 o.1

a. M = 0 . 8

Figure 12.

L / ' I O 0 0

\124~:

I

\ \

\

220-~ ~

I000-,

I0 20 30 40

I I l I 0.1 0.05 0 .03 0.02

TIME TO DOUBLE, sec

Bank-to- turn conf igurat ion - locus of roots w i t h C~p variation.

.25

.5

.25

o m

0 0 .5 ~. UJ ,,t

.25

m

.5 _o r~ UJ rt

4 2

AE D C - T R - 8 0 - 1 1

4 0

~ 5 0

< 2 0

0

4 0

~, 3O Q.

20

• _ ~ o

5 0

4 O i - n.,

" 3 0 : l - n *

Z

I 0

SVM LARGE •

~ LARG[ i

LARG[ 0

A

o

~r

O

I

, .

A~GNIZIJOE OF GAIN

GAIN • 0

GAIN • 1

CAIN - * m

0 <. ILOGLoilGAINI)I < I

] ~ ILCGLOIIGAINI)i < 2 <_ ILOGzoIICAIIvl)I < <_ ILOGLoJIG~,IIq)I < 4

4 <_ 1I.O6t0IR, AIHItl < 5 5 < II.OGL0(iGAIHIil < 6 5<_ ILOGI01ic, AOHlll < 1 1 <_ ILOGIOIIGAINlll < S S<_ ILOGIoIIGAINlll < 9

I

30O

- I00


~ T A L T , i0 k • 1.0 d i g

~ I ~ L T • 4 0 k • 2.6 d e g - -

o,~ ,o, I "T

DIR 40 k dl

-,oooJ'~,ooo -~oo . -,oo,", 47k o, , ~

- 50 R, I0 k 0

~- SPIRAL, I0 k, 4 0 k I

IO0 I 200 300 - ; : - ,.. ~. - = i - ,-

50

t I I TURNING FLIGHT, 5(I

aT. • 3 .5 d e g _, ,LT -Iok • s . ~ d * g ~ - - ~ T A L T , 4 0 k

I I - 2 0 0

l

R , 4 0

DIR', I 0 k

-I00_~, r, v!.O 0

, . - ~ /g lOOO % - I 0 0 0

- I O 0 0 r iO00

x,~..~-- D/R, 4 0 k

,'°°l I I I

TURNING FLIGHT, lOng

• 6 . 0 deg Q--TALT " Iok , 16 .2 deg

--U[TAL T , 40 k

°°°/l - I

D/R, 40 k I ] ooooo - 3 o o f " - ~ - "~ " " -

- 4 0 0 * ' * * t

; ,o, i °

~.~PIRAL, 4 0 -k - fO00 I 000

,ooo

~o %*~00

G?O

% • 500

55(: t

- 5 0 - 4 0 - 3 0 - 2 0 - I 0 0

REAL PART i I I t i I

0.015 0 . 0 2 0 .03 0 . 0 5 0.1


I0 2 0 3 0 4 0

i I I I 0.1 0 . 0 5 0 . 0 3 0 . 0 2

TIME TO D O U B L E , see


.25

.s ,.=,

.25

w

r~

o .5 ~.

UJ O.

2 5

u

m

0 . 5 0

O~ i , , G.

43

A E D C - T R - 8 0 - 1 1

4 0

~ 3 0

I1: < 2 0 z ( .9

-- I 0

0

4 0

3O

o. >-

2 0 _z

~ ~o

5 0

4 0

i.- n,,

" 3 0 >. n. < z ~, 2 0 :8

I 0

SY_~M LARGE •

LARGE

~ n u c v

|

o

0

I

t

t

. ,

I - i ~ O

MAGNI11JD[ O f G A I N

, G A I N - 0

G A I N • 1

G<.. ILCG]oilGAINI} I < I ! <_ IL~Io(IC-~mlP < 2 2 <_ L~I~IGAINII < 3 $ <_. LOGIoilGAINIh < 4 4 < ILOGtoJIGAINI) I < 5 S <_ II.O~I.o(iGAIHlll < 6 6 <_. ILOC,164iGAINll I < 1 1<_ ILOGIo(IGAINIll < S 8 < ILOCIo(IGAINi)I < 9

I R ,

- s o J -.1 1


aTALT ,ok " o.2 d,g QTALT : 4 k 0.5 d l g - -

I l "1

DIR 40 k

Q SPIRAL, I0 k , 4 0 k

50

t 2 0 3 0

I I I TURNING FLIGHT. lOg

(~TAL T • iO k • I.I deg --GTALT__ • 40 k • 3.6 d e g ~

I I

- I 0 0 -;

- 3 0

I R , 40 k

. ~ . 1 . -S°L . . . . c . I ~o _ -'1 ' - I . . . . ~Jl" " " "! ~"

. I ~--I - - . , " ~ - - - - [

I0

D / R , 10 k

- I 0 - ~ 1 0

~0~ b O ~ o D / R , 40 k

so , "ioo' - I 0 0 0 J

- S P I R A L , I O k , 4 0 k

] 1 . -

IO0

i

TURNING FLIGHT, 25

n T A L T • i0 k • 2 .6 d i g

-~TAL T • 40 k • 8 .4 d e g ~

I I

- i 5 0

I , / ~ DIR-- I0 k

It -S°- , . _

" , , , - IOO0 J ~-IO00

SPIRAL, I0 k, 40 k R,40k ~ .

7 - , o o . o \ . ,oo I • ,_ _" ~ " - _ -

0 ~ D/R. 40 k

o.,- ~ " ~ - , ~

~',~oo

• 3 0 l )

E

- 5 0 - 4 0 - 3 0 - 2 0 - I 0 0 REAL PART

I I I I i I 0.015 0 . 0 2 0 . 0 3 0 . 0 5 0.1

T IME TO HALF , sac

I0 20 30 40

I I I J 0.1 0 . 0 5 0 . 0 3 0 .02

TIME TO DOUBLE, se¢

c. M = 3 . 5 F i g u r e 1 2 . C o n c l u d e d .

- . 2 5

o t

¢3 0

.5 m b J a .

.25

¢3 g

0 o

b . I r ,

-t I

_i 2

25

m

o" .5 0 m n-

MJ a.

4 4

AEDC-TR-80-11

0

lie

(.9 z O.

¢,1 i

N

I-- bJ

o a

bJ

o re- tw

0 . 5

0 . 4

0 . 3

0 . 2

0.1

0

sY._~M LARGE x

LARGE e,

LARGE 0

4

o

0 I

÷

i

. t

"-..1

NGNITUDEOFGAIN M = 0 . 8 0 A L T I T U D E = 4 0 , 0 0 0 f i GAIN - 0

GAIN • 1

GAIN - + m

0 <_ ILOGL@IGAINql < ]

! <_ ILOGIoIIGAINII I < 2

Z <_ ILOGz011GAINlil < 3 <_. ILOGi011GAINlil < 4 - -

4 <_ ILOGITIGAINIII < 5 5 < !LOGIdlGAINlll • ,~ 6 <. LOGi@IGAINlll < l 1 < ,tOGt@IGAINlil < II ' - -

8 <_ ILOGIoIIGAINlll < 9

I I

- 0 . 1

TURNING F L I G H T , l O g

DAMPING RATIO

- I 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0

Cn r , p e r / r o d

a. Damping ratio

I C n r , p e r / r o d

lO0 / / 7 - 6 0 . 6 (TRIM)

i, t': t It! t, V ' v V v

i

2 0 0 0

1 6 0 0

1 2 0 0

8 0 0

A f\ A

V v

4 0 0

0

- 4 0 0

- 8 0 0

- 1 2 0 0

- 1 6 0 0

- 2 0 0 0

0 . 2 0 . 8 1.0 I . 2

on damping

0 . 4 0 . 6

T I M E , s e c

b. Time response Figure 13. Bank-to-turn configuration - effect of Cnr


I 0 0 0

45

AEDC-TR-8O-1 1

0 I-

z

{1.

i N

n - - J 0

i n . ,

o z e r O

0 . 5

0 . 4

0 . 5

0.2 ~

0.1

0

-0 .1

- 0 . 2

- 0 . 3

- 0 . 4

- 0 . 5 - I 0 0 0 - 8 0 0

4 0 , 0 0 0

3 2 , 0 0 0

2 4 , 0 0 0

I 6 , 0 0 0

8 , 0 0 0

- 8 , 0 0 0

- 1 6 , 0 0 0

- 2 4 , 0 0 0

- 5 2 , 0 0 C

- 4 0 , O O G

0

Figure 14.

i

M = 0 . 8 0 A L T I T U D E = I 0 , 0 0 0 f t SYM

T U R N I N G F L I G H T , l O g LARG[x LARGE

\ .

\ ° !

L-,DIR DAMPING RATIO f~

! " "

t \

- 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0

Cj~p, p e r / r o d

a. Damping ratio

0 . 2 0 . 4 0 . 6 0 . 8

T I M E , s e c

b. Time response Bank-to-turn configuration - effect of C~p ratio and time response.

MAGNITUDE OF GAIN

GAIN - 0

GAIN • !

GAIN - + m

O <_ II.¢GIOIIGAINltl < ] __

t <_ ILOG]0IIGAINIII < 2 2 < ILOGIoI;GAINIII < 3

<_ ILOGto(IGAINlll < 4 - - 4 .< iLOGi01IGAINI)I < 5 5 < ILOGt011GAINlll < 5 - - 6 <. ILOG]0(IGAIN )1 < 7 7 < ILOGt0(IGAINIll < S - - 8 < 'LOGIo(IGAINlll < 9

I

/ /

4 0 0 6 0 0 8 0 0 I 0 0 0

I . 0 I . 2

on damping

46

A E D C - T R - 8 0 - 1 1

4 O

=~ 3o

~ 2 o z_ O

X - - 10

4 0

~n. 3 O

G. >-

20 Z r3 ,¢[ x_ ~o

5 0

4 O I-.

3 0 )- r~ < Z

2 0

=E

i O

SYM f&M;NIIUDE OF GAIN LARGE = GAIN - 0

1 ~ i GAIN, • 1 LAI~ a GAIN "*-m

i 0 <_. ILOG~IGAINI)I < I o 1 < ILO;Lo[IGAINI)I < 2

1 * ~ < ILOGLOIIGAINI)I < 3 o s <.. I~tdlGAml)l < 4 , 4 < ILOGtoilGAHNi)I < 5

~.~ . 5 <. ILOG~IGAINIII < 6 , 6<_ iLOGLOIJGAINI)I < t , ; <_ ILOG~IGAINI)I < 8 .. S<_ ILOG~IGAINIt, I < e

I

R, IO k . m = . - -

1 LEVEL FLIGHT, Ig

TALT : iO k " 1.5 d , , _ _ TALT 4 0 k " 4 . 5 d i g

I D/R, IO k

i O " 5 0 0

I . , , , I ; * - ~ . . . . , : - 5 0 0 15o'o I ~ sin, ,o k I

R, 4() k S IP, 4 0 ! • I - S P I R A L , 4 0 # ~_ PHyGO'O-~E.-SPmAL, 1ok

5oo5o ,o

TUR., O FL,O T. 5g' ! G T A L T . iO k • 5 . 8 deg =(ZTALT • 4 0 k •15.1 d e o - -

i +

5 0 0

O/R, 4 0 k - 5 0 0 • m • m ]1t • • • •

A m , O/R, I0 k

S /P , 10 k

J I I • S /P , 4 0 k ' R , IO k S P I R A L , IOk, 4 0 k ~ I r PHUGOID,

_ _ 9 - 5 0 0 I R, ,o",

0

- 1 4

2 0 0

TURNING

"(~ TA LT TALT

- 1 2

I 0 0 5

Figure 15.

9 •

I D/R, I0 k

O

FLIGHT, lOg

• i 0 k • 9 .8 d e g _ _ • 4 0 k . 2 4 . ? d e g

- I 0 - 8 - 6

i •

I D/R, 4 0 k

-L

I S/P, I 0 k

f S I P , 4 0 k S P I R A L , IOk, 4 0 k ~ . X

R,,O k R, 4 C ~ \ I ~ 5oo.L-5o0 [ ~.~__.L' .q

- 4 2 0

- 5 0 0 a

• - ~ 0 0

PHUGOID, I o k , , 4 0 k

2 4 R E A L P A R T i

I I I i I i 0.1 0 . 2 5 I - i - 0 . 2 5

T I M E TO H A L F , sec T I M E TO D O U B L E ,

a. M = 0 . 8 Bank-to- turn conf igurat ion - locus of roots with C~r variat ion.

. 25

u f) m

o o

.5 E w Q.

. 2 5

(n

Ca 0

tlJ O.

j 2

. 2 5

m

0 . 5 0 m Cc

t~J O.

SeC

47

A E D C - T R - 8 0 - 1 1

4 0

~ 3O

>-

20 Z

¢9 ,¢(

~ no

4 0

3 0

o. 3-

2o Z (.9

0

5 0

4 O I - o-

n. 3 0 >. rv < z

2 0

I 0

0

- 1 4

I 0 0 5

SYM MAGNIIUDE OF GAIN LAItG£ x GAIN - 0

~LARGE I v. , , la - [

LARGE o , GAIN - ± 4 1 .

A O<_.,ILOOIoIIGAINIII < ) o 1 < LOGI01IG A N]ll < 2

- - ~ 2 <_ ILGGIOIIGAINIII < 3 o 3 <. ILOGLoIIGAINrq < 4 n 4 < ILOGLoIIGAIN])I < 5

- - $ < ILOGLglIGAINIII < 6 , 6< ILOGI01IOAINI)I < 1 , I <_. ILOGI0(I~INI|I < II

.. B<_ ILOGIO(IGAINIII < q

' I R, I0 k , P - -

PHUGOID, I0 k ,

R , 40 k i l l ~ |

5 0 0 ~ - S 0

I I 1 L E V E L FLIGHT, Ig

GTALT-- " I ok : 2.61"0 d e g - - r J TAL T = 4 0 k d e G _ _

/O,R.,O; +.,DO S/P, I 0 k - D/R, 4 0 k

z o i - s o o I S O 0 ~ , -41~"1" "

LJKS/P, 4 0 k 40~' -~ I

\ : F S P I R A L ,

- 50

TURN,;G ~ " e ~ T ,

aTALT • iok : ~ . 5 t~ TALT • 4 0 k 9 . 3

I

R , I0 k q ~ .

5 g n

d e g d e g _ _

L L,.,o,, L oo , • . . - - , . I , ,~-T- ' . o ' . ~ . . - L - - ~ t ; . . • • .500

" " ~ - - I ~-01R. 40 k' soo slp, lo~a '

I I S / P , 4 0 k J

I SPIRAL, ~Ok, 40k ~ i f PHueo,o,

I R ' , ~ Ok ~..L ~/ IOk '4Ok '

' ' lOg' TURNING FL IGHT,

= G . O d e g ~TALT , 1 6 . 2 d e g ~ , I0 k

" ~ T A L T . 4 0 k ,

3OO

R,i!k 5 0 0 0 - 5 0 0

- 1 2 - I ' 0 - 8

- . . x ~ - ~ " "

0 D/R, 10 k

S /P

O/R, 4C k 0 Jh" __ . m .

Iok ~ XS/P , 40 k

F - 5 0 0

- - " ' ~ -~'~o

t S P ' R A L , t o k i 4 o k . - ~

I ,..o,,/- - B - 4 2 0 2 4 R E A L P A R T

I ~ t l I I 0.1 0 . 2 5 I - I - 0 . 2 5

T I M E TO H A L F , s e c T I M E TO D O U B L E ,

b. M = 1 . 1 Figure 15. Cont inued.

- - . 2 5

e0

C) 0

.5 O: bJ O.

. 2 5

r~

o .5 ~:

LU O.

-J I

_12

. 2 5

in

o" . 5 0

[ , LLI CL

$ e c

48

A E D C - T R - 8 0 - 1 I

4 0

~ 3O

< 2 O z (.9

:E -- I 0

0

sy...~ LAI~ x LAI~ *, LARGE o

a

m

0 i

+

6

i=

!

I R, IO k , 40 k

iilAGNnIJDi[ GF GAIN CAIN - 0 CAJN - 1 GAIN " i tO

0 <_ ILC6wtl6AINlll <" I ! < It,O6Io(IGAINI)I • 2 • ~ IIAXl~oIIrurAINlll • J ~1 <.. ILOOIOIIGAIMIll < 4

4 <_. ILOG~IGAiNI)I < 5 S< ILOGI6(iGAINIII < 6 6 < It.OGLottGAINlll < T -- ;~ <_ t.O6LotlGABItlII < a s<_ LOGLGIIGAINIII < 9

I LEVEL FLIGHT, Ig

(2TAL T = I0 k " IXTAL T • 4 0 k "

S/p ' l lok !

i SIP, 40 t-

I J0,,. ,o,, ,0o -3oo O/R, 4 0 k 0 . 2 d e g

0 .5 d e g - - 5 0 0 " 0 ~ " S O 0 -

sP,'AL, ,o.. +o, . I [ ~ no , 4o

. 25

w

0 0

.5 " W G.

4 0

3 O

0. 3,-

2o z

0

I I TURNING FLIGHT, lOg

OETAL T • i 0 k • I . I deg • G T A L T ~ • 4 0 k • 3.6 den

I R, IO k

R, 40 k 4 0 0 O

5 0 0

- 5 0 0

I k I S / P , I0

i w S /P , 4 0 k

I-- D/R, I0 k . 5 0 0

O~ee a l

~ 5 0 G ~ - • • t q

0 D/R, 4 0 k

i I S P I R A L , I0 k , 4 0 k " ~

I I

I, - ~ O i

I PHUGOID,

/ 'Okl, 4 o k

. 25

5 0

4 0 I - n,,

" 3 0 I - r,-

z N 2 0

IE

I 0

0

- 1 4

I 0 . 0 5

TURNIN*G FL IGHT, =so I n T A L T , i 0 k • 2 . 6 deg OITALT ,, 4 0 k = 8 . 4 deg

zo~ l "

R, I0 k

- 1 2 - I 0 - 8

D/R, 4 0 k

0

D/R, IO k

t-o" 500

I T S/P , I0 k

I S/P, 4 0 k

=X •

111 - 5~0

: 3 0 0

S P I R A L , I0 k 4ok ' - I i i R, 40 k ~ / - PHUGOID,

15oo O , s o ° / , o k , , + o k I

- 6 - 4 2 0 Z REAL PART

I I I I I I 0.1 0 .25

T IME TO H A L F , sac

I - I - 0 . 2 5

T IME TO D O U B L E , sec

2 5

m ¢3

.5 o

I,g O.


49

A E D C - T R - 8 0 - t 1

4 o

== 50

>.

20 z_

=- tO

0

4 0

~: 3O G. >-

20 Z

< -z i 0

50

4 0 I - n- ,< " 3 0 >. n*

Z 2 0 <

=E

I 0

0 - 2 8

I 0 .025

S~M LAIInE •

~ tARGE 4, LARGE e

4 D lZ ¢

!

i

.g

m~rnJee 0e GAI'~ .25 GAIN • 0 ] GAIN • ) GAIN -±m J

O< II.OGlO(IGAINIII < t t ,~ ILOG]o(IGAINI|I < Z i

) <- ILOGz011GAINII', < 4 4 <" [I.0G)011GAINKI' ( 5 , ~" KOGz0[IGAINI) < 5 "-t . 5 e <_' ILOGtoIIGAINI1 < 1

'<'LOGIoIIGAINi) < S . J J i ~ a<- ILOGtoIIGAINi) < }

/ i I "li~* . ~ / ~ ] ' - D/R, 40 k R,IO k I. S/P, I0 '~ " ' -wL.e ,n ,~nk

--v-.:.~Eol o soo [ 50b 0 500 R, 40 R I --I- ~ . . . . .

, P H U G O I D , I 0 k , k._..~

I ' I TURNING FLIGHT, Sg

TAL T i 0 k " 5.6 d i g TAL T , 40 k - I s . 1 d e g _ _

so o I

i i

S'

D / R , 4 0 k I

0 - 500 500 ~K * ~ ", t

• 5 0 0

D/R, I0 k / "S /P , I0 k f f-SIP, 40 k

I I " ,.-sPi,A;.PHUG, R.,Ok , ,ok 'l .o ' . ,o '

_ _ _ . _ ~_~.(~0"! $oo I - 500 . O 0

-I .25

Q m

0

.5 9

W

)t

i i I TURNING FLIGHT, lOg (~TALT • i0 k • 9.8 deg • TALT • 40 k :24.7 d e g ~

50O

I . m

s'oo

- 2 4 - 2 0 -16

O/R I0 k

0

-12 - 8 - 4 REAL PART

I ]

Figure 16 .

0.05 TIME TO HALF, sec

I D/R, 40 k "1 E • o -'s

• " - 5 0

S/P, I0 k £_S PIRA'L, PHUC

siP, 4ok- - , . F ~ k , 40 k -

R,,O -.soo// .~oo I k _ ~ o j o,~_...:,n, 4o

I 0 4 8

~ I I ! 0.125 0 .5 -0.5 -0.125

TIME TO DOUBLE, sec

a. M = 0 . 8

11 • •

I

Bank-to-turn configuration - locus of roots w i t h Cnp variation.

O 25

m

.5 ~ W O.

I 2

50

A E D C - T R - 8 0 - 1 1

4 0

i 30

2 0

I0

4 0

i-.- ec 3 0 (3- )-

Z

~ to

5 0

4 0 i - n.- <[ " 3 0 n. < z m

2 0 < :E

I0

SYM I~L~GNIIlJ nr OF GAIN

LAE • GAIN -~m • 0 < ILOGI0IIr.,AINlll < ] a 1 < ILOGzo[IGAINIII < Z , 2 <- ILOGzO(IC~INIll < 3

] , 4]ILOG~IGAIHIII <5 | . ~ <_ ILOG~IGAINlll < e I ~ ' 5 < II.O;~lr.~lNI)l < 1 I . 1 .< LO;I~IIGAINI)I < S / "" S<-1II'O6IoIIGAI~Iil <') D/R, I0 k

i I ,I G T k ' l G / I ALT • I0 . dog I Sip , I0~--~ I ------

/ I O*A-T.,O''"' °'°1 ~do.~#_:,~.,ok / IR IO k I I S I P ' ' O k "--~]IK I I ! ' I I I .,.AL.,Ok,.Ok~ ~ P"°GO,O. / / ~ 1 / I O k , 4 0 k / ,oo o ~?o I I ~o o ~oo ,,,

I R,--O k 1 4 'iF' I

I I l TURNING FLIGHT, 5 g

~ T A L T i, iGk = 3 . 5 d e g _O.TALT = 4 0 k = g . 3 d e g I

I I

I R ' l O k 500 ~ I i~00

I / k " =50 :) D/R, I0 • •

• . - J = k . . . . . = so~ , : . / R . 4o hf ~ - 5~G

S/P ~o k X 50"0 ' J I S / P , 40 k

I S P I R A L , I O k 4Gk-~ r PHUGOID,

R'4Gkl ~ / I O k ' i o k -

- 5( ~O '

. 25

0 - 2 8

L T r - ' - T TURNING FLI=GHT, I 0 0

rf'I 'ALT I Iok 6 0 deg - - - - II-'-6,Z d G ~ A L T , 4 0 k e g _ _

~o.o. [

- 2 4 - 2 0 -16 - I REAL PART

; I 0 . 0 2 5

O'Ro'Ok i ~

0 . 0 5 T IME TO HALF, sec

I I I I i 0 . i25 0 .5 - 0 .5 - 0 . i 2 5

TIME TO D O U B L E , sec


a.

I

2

.25

=- w

¢3 .5 o

r,. UJ a.

25

¢3 .5 9 r,-

tkl O.

51

A E D C - T R - 8 0 - 1 1

4 0

~ 5 0

2 0 Z

,4[ Z_ IO

0

4 0

i - a: 3 0 Q.

~ 20 Z

~ ~0

5 0

4 0 [-. *-r

" 3 0 r,.

2 0 :E

I 0

sv. LARGE • LARGE -', LARes o

D

¢ i

+ 4

R, IO k ,----4-

I

MAGNITlJDE OF GAIN GAIN • 0 GAIN • ] GAIN - :l: ,m

0<_ IL(~'~IOIIGAIIIIII < 1 ! ~ IL~I~iGArNlll < l 2 <_ ILOG10(IGAI'alII < ~; 3 < ILOGIO(IGAIqlII < 4

. ; ~=,,~=..l,! <. s,P, ,0 is,P~O 6< =LOGIoIIGAINI) < 7

; ~: ILOGIoIIGAINi) < | i I : , ,500 <- ILOG|oIIGAINI)I < 9

,, °'"i ,o,.../.," OEi'AL T • i0 k • 0 .2 l e g , t-OIR, 40 k aTALT , 4 0 ; 0.5 d e , I " ~ i ~ • " 50(

' l R * 4 0 k I ~oo l" ~ " hi'SPIRAL ' o k , ' O k I ..,~ 44o_,~c4oo II ,-,'.uoo,o.

. ,oo o _ I,o_o ~ '~ ,o ] ? o ~ j ,o*..~0= _~ I - - " I - - I I 1'1" I I "

i

I I TURNING FL leHT, lOg I OTAL T ,, i 0 k . I.I deg .(ZT,,LT_ • 4Ok. = 3.6 d e g ~

f 500

500 R, IO k 4 o o

.0~1 SIP,

D/R, I 0 ~

i 3 0 0 ( . , , I 0 R, 40 k !

S/P, 4 0 k ,,=, j ' - 5 0 0

- 5 0 0

t

PIRAL , I0 e., 4 0 k

/ -PHUGOID, J I0 k , 4 0 k

_,, _ ~-,,~-_ 4 0 0 500

0

- 2 8

I 0 . 0 2 5

CITAL T ,, i0 k " 2 .6 deg

• aTA, T = 4 0 k

_[ . 500

R , I 0 k

= 0 .4 d e g ~

I - I

20O

I * S/P, l o k l

D/R, 40 k ,1[

0

,o/o /

S/P, 4 0 k

J" D/R, I0 k~

SPIRAL, I0 k , 4 O k - ~

I

- 2 4 - 2 0 -16 - 1 2 - 8 - 4 REAL PART

[ I 0 . 0 5 0 125

T IME TO HALF, s,=¢

c. M = 3 . 5

Figure 16. Concluded.

- 4 0 0

4 ; ;

PHUGOID, Iok t ~

I- :5;,ol " I

0 4 8

i l l 0 .5 - 0 . 5 -0 .125


"-1 I 2

.25

0 .5 o n,.

uJ n

2 5

rJ (D

0

5 ° n.- LU r,

.25

ot

0 5 °

UJ G.

. 2 _J

5 2

A E D C - T R - S O - 1 1

4 0

2 O Z

Z_ O0

4 0

3 O

~ , o

z- i o

0

5 0

4 O

L 3 0

Z Z O

<

i,o

SYM At~GHiTUD[ OF CAIN LARG~ • GAIN • 0 LAI;tG[ • GAIN • 1 ~ o r.,Alk - :ii:m

• 0<_ IL'OGIoilGAINITI < t o 1 <_ 'L'OGtoiIGAINIli < ! , 2 < .L06t0flGAINill < 3 O 3 ~ ILOGzo(IGAINIII < 4 , 4 <_ iLOG]6IIGAINIII < 5 . S < ILOGIotlGAINI)I < 6 , 6 <_. It.OGTo(IGAINIII < 7 , 1 <_ ILOG~Ir~INill < It .. s<_ ILO~l~l l l l l l < e

1 , I

R, I0 k - 5GO o 5'GO

TURNIING FLIGHT, sg I

(2TALT • i 0 k - 5 . 8 deg - - (2TALT , 4 0 k ,15 .1 d e g ~

I 1

I I LEVEL FLIGHT, Ig

GTALT i0 k • 1.5 deo (2TALT ~ 40 t ,4 .5 deg

I I I "".'oL

S P I R A L , IO k , 4 o k ~ _ - S / P , 4 0 k R '40 k ~ a r P H U G O I D ,

~ 1 - - ~ F l o k . 40k - 5 o o " - s o 0

D/R, 4 0 I i

s.oo_ o O,R, ;O k ~ - 500

S/P , IO k I I ] - I

S P I R A L , IOk: 4 0 k ' ~ E S I P t 4 0 k I I R 4 0 k \ I ~ PHUGOIO, ' 5oo 2 s o o ' ~ iok. 4ok__

I R, I0 k

. 2 5

. 25

w

.5 _o

W

I TURNING FLIGHT, lOg

_ ~ T A L T • i 0 k • 9 .8 d e 9 _ _ (2TALT • 4 0 k , 2 4 . 7 d e g

i I

- 2 8 - 2 4

I 0.025

- 2 0

I 4o~ O/R,

I "

Figure 17.

5OO

D/R, ,O k ~ .0500

i

: S /P , I0 k i z I

J L ~ IP , 4 0 k S P I R A L , I0 k , 4 0 s ~ l I k R , , l , k ~ " FP,"k"GO'~ '

R,IO ~ t " ~ / 10 ;40 i - 500

-16 -12 - 8 - 4 0 4 8 REAL PART

I I t J 1 i 0.05 0.125 0.5 - 0 . 5 - 0 . 1 2 5

T I M E TO HALF, sec T IME TO D O U B L E , sec

a. M = 0 . 8 Bank-to- turn conf igurat ion - locus of roots w i t h Cyp variation.

.25

m

.5 o ,.J

53

A E D C - T R - 8 0 - 1 1

4 0

~: 3 0

>. 2 o

z (.9 ~( z_ i s

0

4 0

so o. >.

z o ;¢ a

: i - I O

0

5 0

4 0 k- n.

n. 3 0 >-

o:

z o 2 0 <( :E

I 0

SYM MAGNIIUD[ OF GAIN

LARGE • GAIN • 0

LARGE • GAIN - i

LARGE • 6A.IN - i I li 0 < iLOGio(IGAINil < 1 = I <_ LOOioIIGAINiI' < l

- - .,i l ~_ LOGidlGAINII' < ) o t <_ '.LOGIoilGAINIh < l

4 <_ ILOGIoIICAINIil < t ,.. S <_ ILOGIclIGAINI)I < 6 , 5 <_. ILOGIoIIGAINlll < 1 , 7 <_ ,LOGIo[IGAINI)I • I .. e<_. I.OGIitlGAINI)I < 'g

R , IO k

- sos _ sop

TURNII~IG i : i FLIGHT, Sg

~TALT • ~Ok : 3.5 d q UTALT • 4 0 k 9 . 3 d i g _ _

I R , I O k

-s , )o o soo


=(~TAL T • i0 k • 1.0 deg ~ U , TALT = 4 0 k = 2.6 d e g _ _

I ' D/R, I0 II 50 c ai '. sos

S/P, I0 k i

XJI~ =D/R, 40 S/P, 4 0 k I I

SPIRAL, IOk, 4 o k - ~ / - PHUGGID, I o I \ ! , o , l , o -

- 500 ,., 300 •

R, 40 k

D/R, IO k SO0 L~

D/F 4 0 k ~ ' 5 0 0 S/P, I0 k

I i l l S/P,i 4 0 k

SPIRAL, I0 k, 4 0 k I PHU GIOIO,

I

- 2 8

i 0 . 0 2 5

TURNING

- -~TALT • IO k ~TALT , 40 k

FLIGHT, I0 j • 6 . 0 deg • IS . | de9

50O O/R, 40 k I

D/R, I O ~ ' ~ " . 500

w I

- 2 4

S /P , I i X lieS/P. 4 0 k

/ SPIRAL, I0 k, 4 0 k " ~ PHUSOID,

IR,,O' I I \ .,oo o ,o;o .. :o,- . , . , . . / ,o, . .o, , - 2 0 ' -16 -12 - 8 - 4 0 4

REAL PART I I ~ i l I

0 . 0 5 0.125 0 .5 - 0 5 -0 .125 T IME TO HALF, se¢ TIME TO OOUBLE, se¢

b. M = 1 . 1

F i g u r e 1 7 . C o n t i n u e d .

.25

.25

0

.5 o

w 0.

I -4.25

k3

C3 .5 o

n- taJ O.

5 4

4 0

3O

>.

2o _z

~ IO

40

¢P 30 Q. >.

20

< 3E - I O

0

50

4O l - n,,

a. 3 0 >-

Z m 20

=E

IO

sy.._~ L A I ~ • LAIK[ • L A r g o

a

O I

1

R,IO k

A~GNIIUg[ OF GAIN GAIN - 0 GAIN - ! GAIN - ± ¢~

O< ILOGIOIIGAiNi}I < ] ! <_ US]OIIGAINI)I < ! • ~ ILUtI'I~ILIRINI/I • •

3 <- iLOGIoliGAINII[ < 4 il 1 4<__ ItOGIoIIGAINII < | S i P , O k 5 <_ ILO6Io[IGAINI), < 5 S / P , 4 0 k 6 <_ ILOGIOIIr_,AINI)I < ; ~ I T <_ [LO(;IOtlGAINI)I < S I < ILOG]oIIr, AiNll l < 9 O l R ~ I O k

I I LEVEL FLIGHT, ig --'5OO"HIG~- 500

TALT • ,ok : 0.2 deg O/R 14^k TALT" 4 )k O.5, dig . SOO~ u I I I I - ] - 5 0 0 I r k SPIRAL, 10 k, 40 k-', i , - PNUGOID,

,2o ° oo i I

1 I TURNING FLISHT, lOg (2TALT • 10 k • I.I d i g GTALT • 40 k • 3.6 deg

I I 10kl S / P , J S/P, 40 k

I . 00 D/R, IO II

I I sooJ o -SOD D / R , 4 0 k

- 500

SPIRAL, 10 k, 4 o k - ~ FPHUSOID ' R, 40ko 1ok' ~ Ok

_ ~,,'~tl, -m.~ ~ , * , I

AEDC-TR-80-11

. 2 5

u Q m

a" ,5 ~

W

t . 2 5 ! .

0 -28

TURNII~G FLIGIHT, 25g

GTAL T , iO k • Z.6 d i g "~TALT R 4OII • 8.4 d l g - -

f

R,IO k - - -F---

I S/P, 10 k I

I 5 0 0

D/R, 40 k E - 500

I D / R , I 0 k

.oo"'~

S I P ,

?500

I 0.025

40 k

- 24 -20 -16

SPIRAL, ,Ok.,4Ok.~ I F " P H U G O I D ,

R,40 k ~,~ / 1ok, 40 k - 5 0 ~ "1 i .~OO "~

I i

-12 - 8 - 4 0 4 8 REAL PART

i I I I i I 0.05 0.125 0.5 -0.5 -0. I25

TIME TO HALF, see TIME TO DOUBLE, sec


.25

in

.5 o LIJ O.

55

A E D C - T R - 8 0 - 1 1

o I-- ,=¢ i,,-

(9 Z (1.

C3

bJ

SYM MAGNITUOE OF GAIN LARGE x GAIN • 0 LARGE e GAIN - ] LARGE o GAIN - = ~0

0 . 5 ° v~_ ,LU~tO, tu.,r~i,t ~. =

f o I < ILOGt0IIGAINIII < 2 O. 4 * 2 < ILOGt0113AINIII < 3

O 3 ~_ tLOGt011GAINIII < 4 O. '~ ' 4 < ILOG1.0(IGAINIII < .5

, 5 < ILOGI.dlGAINI)I < G , 6< LCGto:IGAINIll < 7

0.2 - 7 < LOGzo,,'IGAINI)I < 8 . . 8 < ILOGIo(IGAINIII < 9

O.I

O

-O.I ~ ~ , J

-0 .2

M = 0 . 8 0 ALTITUDE = 4 .0 ,000 f f

TURNING FL IGHT, lOg

t f ~

j o . . P , . G RAT,o

- 0 . 3

- 0 . 4

o

o

r,.)

>-

- 0 . 5 - 5 0 0 - 4 0 0 " 3 0 0 - 2 0 0 - I 0 0

2 0 0 0

1600

1200

8 0 0

4 0 0

0

- 4 0 0

- 8 0 0

- 1200

- 1 6 0 0

- 2 0 0 0

0

O IOO

C•r, per/rod

a. Damping ratio

2 0 0 3,00 4 0 0

0.2 0 .4 0.6 0.8 1.0 1.2

5 0 0

TIME, sec b. Time response

Figure 18. Bank-to-turn configuration - effect of C~ r on damping ratio and time response.

56

AE D C - T R - 8 0 - 1 1

o I--

iz

z I1.

I - ,,x -J o Iz:

"li

sY.._~ LAqG[ x

LAqGE e,

LAqC~ o

6

0 . 5

0 . 4 o i

,

0 . 3 ,

0 . 2 "

0.1 I

0

-0.1 J

-0 .2

-0 .3

- 0 . 4

- 0 " 5 5 0 0

4 0 , 0 0 0

MAGNITUDE OF GAIN

GAIN • 0

GAIN - 1

GAIN - :l: ,~

0 <.. ILOGI0(IGAINlll < t t <_ ILOGt0(IGAINI)I < Z 2 < ILOGI0IIGAINI'II < 3 3 <. [LOGIoIIGAINJII < 4 4 <. IL,~]0I GAINIfl < 5 5 < ILOG]oI GAINIZl < e 6<. ]LOGIoIIGAIN]II < 7 I < ILOGIo(IGAINlll < 8 8 < LOGI.o(IGAiN'i <

I j j 1 t

M = i . l O A L T I T U D E = I 0 , 0 0 0 f t T U R N I N G F L I G H T , l O g

3 2 , 0 0 0

2 4 , 0 0 0

1 6 , 0 0 0

8 , o o o .

o

- 8 , 0 0 0

- 1 6 , o o o

- 2 4 , 0 0 0

- 3 2 , 0 0 0

- 4 0 , 0 0 0

- 4 o o - 5 o o - 2 o o

a .

- I 0 0 0 I 0 0 C n p , per / rod

Damping ratio

C n p , p e r / r o d

- 2 5 0 - 6 . 6 ( T R I M )

1 i I\

2 0 0 3 0 0 4 0 0

I / \ / \ / V v v

5 0 0

0 0 . 2 0 . 4 0 . 6 0 . 8 1 .0 1.2

Figure 19.

T I M E , sec

b. Time response Bank-to-turn configuration - effect of Cnp ratio and time response.

on damping

57

A E D C - T R - 8 O - 1 1

0 m

<[ o¢

z o. =E <[ P~

I - Q

I -

0 . 5

0 . 4

0 . 3

0 . 2

0.1

0

- 0 . 1

- 0 . 2

- 0 . 3

- 0 . 4

SYM MAGNITUDE OF GAIN

LARG£ x GAiN • 0

LARGE I GAIN - I

LARGE e, GAIN - i r a

a 0 <_ ILQG]0hGAI~;ill < ]

a t < ILOGIoIIGAINIII < 2 , 2 < [LOGI011GAINlil < 3 ¢ 3 < LOGIoIIGAINI|I < 4 : 4<" LOGLotIGAINI)I < 5 • 5 < tOGLo(IGAINrI, I < 6 - , 6<_ LOGL0(IGAINI)I < 7 , 7 < ILOGI0(IGAINI)J < 8 - ,, 8 <. ILOGI0(IGAINIII < 9

M : 3 . 5 0 A L T I T U D E : I 0 , 0 0 0 f t T U R N I N G F L I G H T , 2 5 g

/ - - D / R DAMPING RATIO

~ . . _ . . _ - - - - - -

- 0 . 5 - 5 0 0 - 4 0 Q - 5 0 0 - 2 0 0

2 0 , 0 0 0

I 6 , 0 0 0

1 2 , 0 0 0

8 , 0 0 0

4 , 0 0 0

0

- 4 , 0 0 0

- 8 , 0 0 0

- 1 2 , 0 0 0

- 1 6 , 0 0 0

-2O, O 0 0

- IOO O I O O

C y p , p e r / r o d

a. Damping ratio

200 300 400

C y p , per / rod

- 5 0 0 - 0 . 5 ( T R I M )

5 O O

0 0 . 2 0 . 4 0 . 6 0 . 8 1 .0 1.2

500

Figure 20.

T I M E , sec

b. Time response Bank-to-turn configuration - effect of C v p on damping ratio and time response.

58

A E D C - T R - 8 O - 1 1

4 0

3 0

o. >-

2o Z

4 0

5 0

o.

~ 2 0 z B

< : i - I 0

5 0

4 0

I-- n~

a. 3 0 )- cE

z 2 0

I O

0

- 2 8

I 0 . 0 2 5

$YM 5VIGNIIllD( OF GA IN LARGE x GAIN - 0 LARGE A GAIN • I

A O <_ ILOGL~IGAINI)I < I o I <.. ILOGIoiIr:.AINI)I < l

2 <_ ILOGioIIGAINIll < 3 3 < ILOGIoIIGAINI)I < 4

z 4 <_ ILOGIoIIGAINITI < 5 , 5 <__ iLOGtoIIGAINI) < t , 6 <_ ILOGIoIIGAINIH < 7 , T <. ILOGzofIHGAINI|I < S .. 8 ~ ILO6)o~ GAINIII < 9

] I I L E V E L FLIGHT, Ig

( I T , , • i ^ k • 1.5 d e 0 - -

G ; : L ' T . '40 k . 4 5 d e g

I: o I ! t °/e'41° - ~ 1 1 I i - s o o ~ s o o I

s/P. io k -,,'~.L s/P, 4 0 k I L / "F.CSPmAL, I ?, 4o" : 1 ~ . - - - d ( - , .uoolo.--J

I0 k, 4 0 k

TURNING FLIGHT, g

~ T A L T • i 0 k • 5 .8 deg - - G T A L T , 4 0 k , 1 5 . 1 d i g m

I I D/R, tO k

5 0 0

DIR, I0 k l

- 5 0 0 O

GO I

SIP, IO k I

I " S / P , 4 0 k I

R,~0 k R , 4 0 k ~ I I f S P I R A L , 5 0 0 , ~ PHUGOID,

• - - " w i o k 4 0 k

. 2 5

J I i i

TURNING FLIGHT, I 0 0

~ T A L T , iO k • 9 8 d e g Q

TALT • 4 0 k , 24 .7 d e g

I - 5 0 0 DIR, I0 k

I - 2 4 - 2 0 - 1 6

0 . 0 5

T I M E TO HALF , sec

I 5 0 0

I I DIR, 4 0 k

I x

J S/P, IO k ,= i I

I k I k S / P , 4 0 k t SPIRAL IO 4 0 - ~ - - i I

' ' L \ A I / -PHUGOIO, I R, ,O-\ \ I / , o k 4ok o ,nk.......~,.J. k ~,.?,,,~r ' ,

"r" ' -"m I - 1 2 - 8 - 4 O 4

R E A L PART

I I I I I ', 0 i 2 5 0 . 5 - 0 . 5 - 0 . | 2 5

T I M E TO D O U B L E , sec

a. M = 0 . 8 Figure 21 . Bank- to- turn conf igurat ion - locus of roots

wi th Ct~q variat ion.

m ¢=

.5 o Lkl O.

. 25

o ="

¢2

5 _ ° o~ bJ G.

2 5

t~

¢3 .5 o

r~ bJ

59

A E D C - T R - 8 0 - 1 1

4 0

~ 3 0 a .

20 z

• I0

0

4 0

50 el )-

~,o Z

s- i o

SYM MAGNIllJO( Of GAIN LARGE x GAIN - O

__ LARGE s CAIN " 1 LAI~( o GAIN "t :m

a 0 <_ ILOG~IGAINI)I < I o ] < ILOGI.OIIGAINIH < 2 . z <_ IL(X;t0(IGAINIII < 3 ¢ 3 <_. ILOG.t0( GArN II < 4 i 4 <_ ILOGIO(JGAINlll < 5 , 5 <_ [LOGIoIIGAINI)I < 5 , 5 <_. ILOGIoIIGAINI) <" 7 • 7 <.. It.OGl0tlGAINI)l < I ,, e<_ ILOGtoliOAINI)I < 9

I i i R, IO k 5 0 0 0 -500

i

i I I TURNING FLIGHT, 5g

~TALT • ,Ok " s.5 d,g - - ( [TALT = 4 0 k • 9 . 3 d o g -

[ I

R, I0 kl 500 o -5oo


(2TALT • iO k : 1.0 deg - - U T A L T ~ 4 0 k 2 .6 d e g ~

O/R, I0 k , 111 •

5 0 0 0 " L ' ~ l . _ 5 O 0 L S/P, IOR J " - " ~ J ~ D/R, 40 R

I ~ 1 S I P ' 4 0 k

R, 40kj ~ SPIRAL, PHUGOIO, 5 0 0 0 - 5 0 0 ~!1~ Iok , 40k | ~ i • i

I D/R, ,0 x

"_5~/bO.. ) /R , 40k IS /P , I0 I,

5 0 0 " I S / P , 4 0 ~

I T" SPIRAL, PHUGOID, R, 4Ok-,, \ 1 ink' 40k

[ ~ J . ,

.25

m

m

0

.5 9

(I.

I 2

.25

o to

C) . 5 9

n-

W

5 0 I I I

TURNING FLIGHT, lOg

• 6 . 0 de9 4 0 ~TALT " Iok , 1 6 . 2 d e g

1 QTALT , 4 0 k

a. 3 0 )-

< Z

2 0

I0

0 -28

I

I - 2 4 - 2 0 -16

I 0.025

D/R - 5 0 0 ~ A

D/R, ,0 k b O O

i

-500 S/P, IOk a[ x"

5 0 0

I

4 0 k

- S / P , 4 0 k

SPIRAL, I0 k 4 0 k ~ , , R, |Oh I ' \ : F-PHUGOID,

500 o -5oo R, 40k~.~.L/ ION,, 4°k { . . . . i " , I

- 1 2 - 8 - 4 0 4 8 R E A L PART

I I i J I 0 . 0 5 0.125 0 . 5 -0'.5 -0 .125

T I M E TO HALF, see T IME TO D O U B L E , sec

J25

,5

J 2

0 O: bJ O.

b. M = 1 . 1 Figure 21 . C o n t i n u e d .

6 0

A E D C - T R - 8 0 - 1 1

1 0 0

8 0 I- c¢

" 6 0 ).. e,, ¢[ z N 40

2 0

SY~ I.~RG£ z

LARGE

L A ~ o ,i

iD

Jl

o i

.i.

t

o,

MAGNITUDE OF GAIN

GAIN • O

GAIN • !

GAIN " ~ m

0 < ILOGL~iGAIkl)I <" L

! <_ ILOGTO( GAIN II <~ 2 <_ ILOG!o(IGAINIII < 3 <_ ILOG!oJ'IGAiNlll < 4 4 <__ ILOGIoIIGAINI) I < 5 5 < ILOGIdIGAINIH < e 6 < ILOGLotlGAINlll < T 1 <_ ILOGIO(IGAINll i < 8 S<_ ILOG!o~IGAINlil < 9

i i I LEVEL FLIGHT, Ig

~ TALT • IOk " 0.2 deg TAL T , 4 0 k - 0 . 5 deg

I k I - 5 o o s/P, to I

I,,o. 1 .... °F .... !oo I 1 o -soo / I I

-- ~ ~ = . . . . ',~ ~ r - - - s ~ P , 4 o k - t - - - - - 3 -54 -50 -46 - , 5 o ~ 1 |

SO0 I !ill D/R, I0 k - -

I R , ' O k kO''.'Ok.J ,HO¢O,O, '000 500 "lRA3''O '0',;'0'

• 0 5 0 0

r i:

R , I 0 k . -p

8 0

o- 6 0 d¢ a. P ¢ 4 0 ,¢[ z (.9

: i 2 0 m

0 :~ - 5 8

TURNING FLIGHT, lOg k

QTAL T : i0 k : ' , l d e , - 5 0 0 j " . . OITAL T 3.6 deg

i D/R' 40k I ~lii~l)5/O0 k

I ,oo R,,O 5oo R' ,o*,4ok /

- 5 4 - 5 0 - 4 6

8 0

6O p .

a .

>" 4 0 n.

z B

< 2O Z

0 - 5 4

i 0 . 0 2 5

J I I - - I I TURNING FLIGHT, 25( I I k J aTAL T : i0 k • 2 .8 deg - 5 0 0 S/P, IO I

- - i ~ T , k * 8 . 4 d i g , L ° v . - - I ALT 4 0 " Ji I . o ou¢/ I / • - / I

D/R 40 k / / r S I P , 4 0 k ' "t~,o-50o o / 5 o o

O,R.,O'lSoo. -ooo / I t S ' P I R A L , I O k , 4 0 k ' - , , | r-.PHUGOrO,

1500 R ~ Ok - 5 0 ~ R , 4 0 k ~ / / I O k ' 4 0 k

- 5 0 - 4 6 - 4 2 - 3 8 - 8 - 4 0 4 8 REAL PART

I I I I I I 0 . 0 5 0 125 0 .5 -0 .5 -0 .125

T IME TO HALF, sec TIME TO DOUBLE, sec

c. M = 3 . 5 F i g u r e 2 1 . C o n c l u d e d .

o~

0

.2 _o n * UJ O.

IO

.2 o_ 0¢ W O.

,...,

.2 o 0¢ bJ o.

i

2

61

A E D C - T R - 8 0 - 1 1

4 0

b- == 3 0

~ " 2 0 Z

C9

~ Jo

$ Y.,_..MM MAGNITUDE OF GA IN LARGE • GAIN - O LARGE .s GAIN • I LARGE ~ GAIN - * ,=,

A O< ILO~-IO(IGAINIr < t a I <_ ILOGIdlGAINI)i < 2 , 2 < ILOr.toilGAINI)I < 3 o 3 <_ ILOGIdIGAIN |1 < 4 , 4 <_ ILOGz011GAINnll < 5. . 5 < ILOGI0tlGAINII I < 6 , 6 < ILGE;IdlGAINIII < 1' , 1 <_ [LOGI0(IGAINII I < 8 . . 8 <_ ILOGI0(IGAINIII < 9

I , J

R,IO k I 5(,o, o }. oo . . . . L ~


~ T A L T : i o k " 1 . 5 d e g - - TALT 4 0 k , 4 . 5 dog

I k [ D / R , tOkl

D / R , 4 0 - ~ IlK ; - S / P , I 0 k

-5,ooq=.5oo I SPIRAL, PHUGOID,-~O~ 6"" S/P, 4 0 n ,on.~on I - X . I

! R, 40 k

4 0

1 - 3 0

>-

~ 2 o Z

~- I o

TURNII~G FLIGHT, 5 g '

TAL T • i 0 k • 5 .8 deg TAL T = 4 0 n .15.1 d e g ~

I I O / R , 4 0 k i -?oo ol

o. . . ,on--~oo i is,p, ,on

. l ] I ~ s /P, 4 o n - ~

R,IO ~ I c -SP IRAL , PHUG( 5 0 0 O . 5 0 0 I "='~' I O n ' 4 'Ok

"-r R , 4 0 k J ' I

5 0

4 0

1-

3 0 >. rY < Z

2 0 < =E

I 0

- 2 8 - 2 4

I 0 . 0 2 5

i I TURNING FLIGHT, lOg Q T • 9 .5 deg _ ALT • I0 k O [ T A L T [] 4 0 k : 24.17 deo

i

F i g u r e 2 2 .

i . . . . . . . D / R , 4 0 k ; &

500 ~ ; . = : ]k D / R , I 0 k i

5 0 0 ; = I

i

I ! = j S / P , I 0 k j

: II~ n I ; ~ 5 / P , 4 0 k T ! I ~1 I

S P I R A L , I 0 k , 4 0 k - ~ IK PHUGOID,

I R , 4 o k - ~ , f i O k , 4 0 k Q , nk --.-J, L~ i I

- 1 2 - 8 - 4 0 4 8 R E A L PART =

I I I [ I I 0 . 0 5 0 1 2 5 0 . 5 - 0 . 5 - 0 . 1 2 5

T I M E TO H A L F , se¢ T IME TO D O U B L E , sec

a. M = 0 . 8 B a n k - t o - t u r n c o n f i g u r a t i o n - locus o f r o o t s Wi th Cm p v a r i a t i o n .

i - 2 0 -16

I ,j2

. 2 5

u o

P

0 .5 ~ W (1.

. 2 5

u o m

r~

.5 9

Q .

--I' I ~ 2

_i

2 5

u

ol

d .5 ~

e~

62

AEDC-TR-80-11

4 0

I- ~. 3 0 < n

P

.~ 20 z

~- to

4 0

I- ,,, 3 0 ,< I1, >-

~ 2 0

-~ i o

SYM IIM,GNIIUK OF GA IN

LARGE = GAIN • 0

L P . K I j ~ = ' U F d I N . i

LARGE ,o GAIN - + m

m. o <_ IL~|otlGAINIJl < 1 o I <_ [LOG10(IGAINIIi < 2 . 2 <_ ILOGLO(;GAINIIm < 3 o 3 <_ iLOG1O( GAINIII < 4 n 4<_ ILO6t~ GAINI)I < 5 + 5 <_ ILOGLd r.,AINI|I < 6 , 6<_ ILOG1d ~INI)I < 7

7 <_ ILOGLOI ~INI)I < 8 .. s<_ ILOG1d GAnql)I < 9

R, I0 k

s o o .... o . . . . s o o i

I n i

TURNING FLIGHT, 5g

~2 TALT = tO k • 3 .5 d e g _ TALT • 40 k • 9 .3 d e g . _ _

I R, tO k

500 0 - 500

! I I LEVEL FLIGHT, Ig

~TAL T = i0 k • 1.0 deg ~ U T A L T = 40 k = 2.6 d e ( L - -

I D/R, I0 k

- 500 0 I ~k .~ 500

S/P, I o k - ~ r . - O i R , 40 k

S/P, 4 o k - " ~ SPIRAL, IOk,, 4,0 k ~ PHUGOID,

500..,...- 500 1. f IOk'[ 4 0 k

I R ,'~Oki t

D/R, I0 k -~oolo soo

D/R, 40k - - ~ i

- ,soo o I S/P, I0 k ' ~ * - 500 I

40 k J~ 'SIP, 40 k

SPIRAL, I0 k , I , ~ PHUGOIO, R, 4 7 k - ~ F IO k, 40 k

;

. 2 5

U w

0

.5 ~ ILl O.

. 2 5

u ¢p

0

.5 2 . i

O.

5 0

4O

r,-

o. 3 0 >- n- <[ z

2 0 < :E

I 0

- 2 8

i 0 . 0 2 5

I TURNING FLIGHT, lOg

= T . = 6 . 0 dog_ . - _ - - L T • iok . t s . z deO

U[TAL T . 40 k

- 2 4 - 2 0 -16

TIME TO HALF, sec

I

O/R, 40 k -50%0 =

o/R, ,0 k- 50o

I [ 40 ~j I ;- 500 • IP, Iok j O j ~IK SIP,

I mooo I SPIRAL, IOk,14ok- ~

R, IO k I PHUGOID, so~ o . s~o R. 4 0 k - ~ ~-'ok. 40k ,, -12 - 8 - 4 0 4 8

REAL PART I I I I I

0.05 0.125 0 .5 -0.5 -0.125

TIME TO DOUBLE, sec

-1 I I

- t 2 . J

, 25

u 0

0

.5 2 G~ L,I

b. M = 1 . 1 F igure 2 2 . C o n t i n u e d .

6 3

A E D C - T R ~ O - l t

I 0 0

8 0 I - oc

° ' 6 0 >.

z ~ 4 0

2 0

8 0

0 - 5

~" 6 0

O. >. n= 4 0 <¢ z

~= 2 0

8 0

6 0 I - I¢ c¢ G.

>" 4 0 OC c¢ Z

< 2 0 5

SYM MAGNnlJDE IOF GAIN I~RGE x GAIN • O,

I ' - - LARPS A GAIN • 1

L LAR~ o CAIN - * l a, 0<_ ILOGIoilgAINIll < I o i <_ liOGIoIIGAINIII < l

I * 2 <_ 'I.OGIoilGAINlll < 1 ; 0 ) <_ fLOGIoilGAINlil < 4 I i 4 <. ILOGI.0IIGAINIil < 5,

+ 5 <_ ILOGIoIIGAINlll < 6

r °i ,i , 6 <. 'LOG|O(IGAINlll < T * " * * A ' 0 , 1 <_ ILOG]o(IGAINlll < 8 * * * .. 8 < ILOGtoIIGAINIII < 9

l [ I I ', siP, 40~ ' k ' - 5o0 o I I R, '° I I ' L soo

? o o ! o _ L ,o,= - s e - : 5 4 - :.s'o - - 4e o ' k I

k IR, 4 0 wi

I I i i - 2 8 - 2 4 - 2 0 - 1 6 -12 - 8 - 4 0


~ TALT • IOk " 0 .2 d e 9 TALT , 4 0 k • 0 .5 d l g -

I I o s,P, ,o~ j

PHUGOID, I0 k , 40 k

I 4 8


~ TAL T = i 0 k ,, I.I d e g TALT • 4 0 k • 3.6 d l o

4OO

| - 5

R , I0 k 0

- - w

I I

i - 5 0 0 S/P, IO k

4 , , . . O I . ,

I S / P , 4 0 k J I - soo, ,w, . 5oo

! o , . . 4 o k -~ I o I o I ' ~ " - - I " D / R , I O " I I i- 50O'I'SOOsp,RAL, '0~, ,Ok i R, 40 k :) \ I ~,PHUGOID,

- 5 0 0 1 5 0 0 ~ - 5 0 0 ~ ! J l O k 4 0 k - " ~ r i "I" I

6 - 4 2 - 8 - 4 0 4 II

TURN INIG FL IG~HT, 2G gi ! n T A L T , iO k [ ] 2 " 6 d e g

[aTALT • 40~ 8.4 ==g

500

- 5 4

0 . 0 2 5

1 - 5 0

R,IO k

- 4 6 - 4 2 - 3 8 - 8 R E A L P A R T

I 0 . 0 5

T I M E TO HALF, s e c

i

i - 5 0 0 S/P, IO k * ' -- . O, I -..!...:oo

o~., 4o k-l " 5 ° 0 ~ IP' 4ok ~L.~. :~--_ - ~ . ~ 1 - - - - -

500 500 0 - 5~)0

SPIRAL, fO k, 4 0 k - ~ l ; PHUGOtD,

-5(_. ~ A R ' , 4 o k - ~ l Iok '140k C

v I --I "P I - 4 0 4

I I [ I I 0.125 0 .5 -0 .5 -0 .125


c. M = 3 . 5 F igure 2 2 . C o n c l u d e d .

I

I

- ~ 2

U

b~

o .2 o- n,.

bJ

I

2

I / I

p

.2 ° - ¢c uJ G.

I I

4 . 1

I f J

- . ,2 o_ o¢

I , , , G.

i

A 2

6 4

A E D C - T R - 8 0 - 1 1

o B w

"u

2 >.

0 =, ILl

0 td r~ tl.

I- I¢

Z 0.

I - W

::) , . , o

. j r v '

..J I--

U "l- L)

SYM MAGNITUDE OF GAIN LARGE • GAIN • 0 LARGE i GAIN • ) LARGE o GAIN - * @

,z 0<_ ILOGIO(IGAINIII < 1 o I ~ ILOGIo~IGAINIII < Z . 2 <. ILOG]o(IGAINIII < 3 o ~ <. ILOGi0(Ir:'AiNlil < 4 , 4 <_ ILOGIoIIGAINill <

5 <_ ILOGI.o(k;AINlil < 6 , 6 < ILOGtoIIGAINill < 1

. . I <. ILOGLoilGAINI)I < 9

I ~ PHUGOID FREOUENC~/ I

t I t tt

-4--- i ~-J~" ,-s ip

t t i ~ i - -

0 . 5

0 . 4

0 .3

0 .2

0.1

0

-0.1

- 0 . 2

- 0 . 3

- 0 . 4

- 0 . 5 - 5 0 0 - 4 0 0 - 3 0 0 - 2 0 0 - I 0 0

4 0 , 0 0 0

3 2 , 0 0 0 I

2 4 , 0 0 0 !

I 6 , 0 0 0 ooo} A o "~/

- e , o o o V !

- 1 6 , 0 0 0 ii

- 2 4 , 0 0 0 ! I

- 3 2 , 0 0 0 i I

- 4 0 , 0 0 0 0

Figure 23.

a.

A

J I

0 .2

M = 3 . 5 0 ALTITUDE = 4 0 , 0 0 0 ft TURNING F L I G H T , 2 5 g

LI ,D/R DAMP, .G RAT,O

_|_.t . - I - - . - i - - - - t - ~ D A M P I N G R A T I O

L P H U G O I O , D A M P I N G R A T I O

0 I 0 0 2 0 0 3 0 0 C E q, p e r / r o d

Damping ratio

1 Cj~q, p e r / r o d

I ~ -soo 0 (TRIM) ~

! , tl

0 .4 0.6 o . e I .o

T IME, sec

b. Time response Bank-to-turn configuration - effect of C~q ratio and time response.

4 0 0 5 0 0

1.2

on damping

65

A E DC-TR-80 -11

fJ I ) tJ

"U

>: o ; [ MJ

O llJ M.

a o

i -

z

a .

a

-r

a .

LARGE x

LARGE A

LARGE o

A

o

I t

0 I

0 . 5 , t

0 . 4 , o.

0 . 3

0 . 2

0.1

0

-0 .1

- 0 . 2

- 0 . 3

- 0 . 4

" 0 ' 5 5 0 0

4 0 , 0 0 0

3 2 , 0 0 0

2 4 , 0 0 0

i I 6 , 0 0 0

8 , 0 0 0

0

z~ - 8 , 0 0 0 (.,I

J - 1 6 , 0 0 0 .J

g - 2 4 , 0 0 0

- 3 2 , 0 0 0

- 4 0 , 0 0 0

0

Figure 24.

IMGNITUDE OF GAIN

GAIN - 0

GAIN - |

GAIN ~:kCO

0 < ILOG)OflGAINIII < !

I <. ILOGIo(IGAINIII < 2 2 (_ ILO6IO(IGAINIII < 3

< ILOGt0(IGAINlll < 4 4 < JLOGI011~INI)I < 5 5 < ILOGIoIIGAINI)I < 6 6 <_- ILOGLoIIGAINI)I < l T <_ [LOGIoIIGAiNI)I < 8 8 <. ILOG1011GAINI)I < 9

' t

1 lk I ,

M = 3 . 5 0 ALT ITUDE = 4 0 , 0 0 0 f t TURNING F L I G H T , 2 5 g

i l l -~ PHUGOID FREQUENCY

I

: .~ L.~ .-L ,o J,-'~ i ..... i " • D/R DAMPING RATIO

" P..0o,o

~.-t--~ . 4 - - I

_F 'S IP DAMPING RATIO ~ r , L

DAMPING A 0

- 4 0 0 - 3 0 0 - 2 0 0 - I 0 0 0 I 0 0 Crnp, p e r / r o d

a. Damping ratio

2 0 0 3 0 0 4 0 0 500

VJ. V

IAA

I I I I

A /

V

i

Crnp, p e r / r o d i

~ 4 0 0 - 500

- 0 .0

/ / v /

V

A A

71 .__

V V

0 .2 0 . 4 0 . 6 0 . 8 1.0 1.2

T I M E , sec

b. Time response Bank-to-turn configuration - effect o f Cmp on damping ratio and time response.

66

I -

<[ a .

Z

1 2 -

~

4

0

- 8

SYM LARGE x LARGE LARGE o

A

r ' l

0

I

t

t

I,

MAGNITUDE OF GAIN

GAIN = 0 GAIN = 1 GAIN - + Go

0 < IIOG10(IGAINI)I < 1 I <_ ILOG]0(IGAINI)I < 2 2 < ILOG10(IGAINI)I < 3 3 <_. ILOGIo(IGAINI){ < 4 4 <_ ILOGIo(IGAINI)I < 5 5 <. ILOG10([GAINI)I < 6 6 < ILOGIo(IGAINI)I < 7 7 < ILOGIo(IGAINI)I < 8 8 <_ ILOGt0(IGAINI)I < 9

t t L E V E L F L I G H T , Ig

Or. T ok = 6 . 5 d e g Ct. A L T = I

~ f ~ k 9 . 3 d e g T A L T I[,,,D ~ I F

I I

S / P , I 0 k

- I 0 0 0 0 I 0 0 0 i S/P, 2 0 k

0,,=*~*-~o, - I 0 0 0 0 I 0 0 0

P H U G O I D , I 0 k , 2 0 k

I I "~ - 6 - 4 - 2 0 2

R E A L P A R T

I I . I J I o.t -i

T I M E TO H A L F , s e c T I M E

a. M = 1 . 3 Figure 25. Yaw- to - tu rn con f igu ra t i on - locus o f roots

w i t h Cm q var ia t ion.

. 6

.75

2

5 - I 0

4

I - 0 . 2 5

T O D O U B L E , s e c

U Q (n

w

a o m

n, bJ CL

m

,o d

A E D C - T R - . 8 0 - t I

3 o ~

i - n, i,l

2 0 -~

z

i

Io

0

3 0

I - n-

) -

a: 20 _z

I 0

0

3 0

i - n,

2 0 n,

Z (:1 < ]E - - I 0

0

- 6

s__~ LARGE =

LARGE • t,AR m e

6

Q

O

I

I

6

I °

IIM~IITUDE OF GAIN

GAIN • 0

GAIN • ]

GAIN - ~ m

o<_ ILOGtOIIGAINIh < t I < ILOGI~IGAIHI)I < ! 2 <_ IL~Io(IC, AINIII < 3 3 <_ IL~Io(;GAIN II < 4

4 (_' IL~)o(]6~.INull < 5 5 ,(' ILOGI6116AiNIDI < 6 6 <_ ILO610(IGAINIli < 7 T <__ [LOGIO(IGAINill < 8 e <_ iLO61oII6AINill <' ] " "

, - I 0 0 0

LEVEL FLIGHT, Ig

G T 1.2 deg (2 ALT • I0 k =

T A L T , 4 0 k 3.9 d i g

i i TURNING FLIGHT, 5g G T • 5 . 3 deg n A L T - I 0 k ml3.G cleg ~ T A L T • 4 0 k

I I • m B

- I 0 0 0

S/P , IO k

• • 0 ~ m t •

S/P , 4 0 k

- I O O 0 0 lOOO

[ 0 0 0

.._••__ OVEROA.PE. PHUGOI~, I0 k, 4 0 k

o Q m

a" o

.B • u l eL

i • i

I - I 0 0 0

TURNING FLIGHT, l o g O T k n 9 . 1 d i g _ _ n ALT • I0 ~ = 2 2 . 5 deg ~ T A L T • 4 0 i

S /P , I0 k

0

S /P , 4 0 k e l m a l ] n ~ l e l o

- IOOO 0 I 0 0 0

|

* ' I 0 0 0

~ OVERDAMPEO PHUGOIDu IOk: 4 0 k

I

u

o o

. 5 ; eL

I

0 . I

- 6

/

0 / S/P , I0 k

S /P , 4 0 k • • 1 1 9 J l r ~ D • Q •

- I 0 0 0 0 I 0 0 0

. 4 - 2 R E A L P A R T

I i I 0 . 2 5 I


b. M = 3 . 0 Figure 25. Concluded.

-I . I 0 0 0

I OVEROAMPED

I PHUGOID[ 'O k , 4 0 k

J

o 2 4

I I - I - 0 . 2 5

T i M E TO D O U B L E , sec

to

a 0

• S ; t~l el

6 8

c~

(1.

:>.

Z (.9 =i

12

8

4

0

- 8

SYM

LARGE x

LARGE '~

LARGE u &

D

O

I

.I-

t

MAGNITUDE OF GAIN

GAIN = 0

GAIN = I GAIN - + m

0 < ILOGIo(IGAINI|I < ] <_ ILOGIo(IGAINI) I < 2 < ILOGIo(IGAINI)I < 3 < ILOG]o(IGAINt)I < 4 <_ ILOCIo(IGAINIII < 5 <_ ILOG]o(IGAINI)I < 6 .<_ ILOGIo(IGAINI}I < 7 < ILOGto(IGAINI)I < 8 < ILOGto(IGAINlll <

I- J L E V E L F L I G H T , I g

Q 'T = 6 . 5 d e g C~ A L T = I 0 k 9 , 5 d e g

T A L T 2 0 k

! I

0 . I

I S / P , I 0 k

° ° - - - , ~1 ' - * , , , ,

- I 0 0 0 0 I 0 0 0

o , - I 0 0 0 0 I 0 0 0

F PHUGOI i , I 0 k , 2 0 k

- 6 - 4 - 2 0 R E A L PART

I _ , I 0 . 2 5


2

. 6

- 2

- 5 - I 0

4

I I - I - 0 . 2 5


.75 u G ¢tJ

f .

I a 0

u.I OL

a, M = 1 . 3 Figure 26. Yaw- to- tu rn conf igurat ion - locus o f roots

w i th Cm a var iat ion,

~> m

o ~3

~0 do 0

..L

A E D C - T R - 8 0 - 1 1

n

20

z (.~

: i

I 0

SY._MM LARGE x

LARGE A

3 0 - - U, eG[ o

o It

0

!

0

. .

0

3 0

MAGNITUDE OF GAIN

GAIN • O

GAIN • !

GAIN - = m

O '< 'LOG!oIIGAINltl < ]

I <_ 'LOG]oIIGAINIll < Z 2 <_ ILOGI0(IGAINIII < 3 3 <_ iUX~Io(IGAINIIi < 4 4 <_ ILOGIo(IGAINIII < 5 $ < (LOGIoIIGAINI)i < 5 6 <. [LOGto(iGAINI)I < T 7 <_ ;LOGI0(IGAINI!I < S 8< ILO('.,LGI[GAINI!I < 9

i - IOOO

l • t i

LEVEL FLIGHT, Ig

~ TAL T = I0 k = 1.2 deg _ = 3.9 deg

TALT= 4 0 k

SIP, I0 k I 0 I 0 0 0

SIP, 4 0 k . * • " o Z k ' + ' ° "

- 4 0 0 0 0 IO00 OVERDAMPED PHUGOID, I0 k , 4 0 k

I I I -~

U o o

a" o

.5~. UJ a.

I -

==. >,,.

¢ 2 0 _z (,9

:E

10

3 0

F- r~

2 0 >- n .

z (.9 <¢ X - I 0

0

- 8

I I TURNING FLIGHT, 5g Q~T • 5 . 3 deg n ALT • I0 k =13.6 d i g " T A L T , 4 0 k

I I

- I 0 0 0 S / P , I O k

i o q o

s,P, ;ok" " l ODD 0 I 0 0 0 OVERDAMFED

, ~ , PHUGOID, I0 k, 4 0 k I

! I TURNING FLIGHT, fOg t'YT - k : 9 . t deg

ALT . I U . =22 5 deg TALT = 4 0 K

0.1

S/P, IO k

- 6 - 4

- , 0 0 0

- 2 R E A L PART

I I

- I 0 0 0 • • t •

O IOOO d~ m •" a t1' •

S/P, 4.0 k

1 0 IOOO I

~ - ~ 1 0 V E R D A M P E D ~ h U G O I O , :O i , 4 0 k

0 2

0 . 2 5 I

T i M E TO H A L F , s e c

b. M = 3 . 0

i I - i - 0 . 2 5

Figure 26 . Conc luded .

4


0 m

w Q

0

. 5 ~ b,I O.

u

II}

d o

.5~. W

70

i-- n r

a .

>- e,,.

z m

¢9

12

8

4

0 -8

SYM MAGNITUDE OF GAIN LARGE x GAIN = 0

LARGE a GAIN = 1

LARGE o GAIN -- + m a 0 <_ ILOGIo(IGAINI)I < u 1 <_ ILOGto(IGAINI)I < . 2 <_ ILOGto(IGAINI)I < 0 3 <. ILOGIo(iGAINI)I < , 4 <_ ILOGto(IGAINI)I < . 5 < ILOGIo(IGAINI)I < , 6 < ILOGIo(IGAINi)I < , 7 < ILOGIo(IGAINI)I < . . 8 < ILOG10(IGAINI)I <

I I L E V E L F L I G H T , I g

Q. T A L T = IO k = 6 . 5 d e g

O . T A L T = 2 o k 9 . 3 d e g

i - 6 - 4

I 0,1

TIME

1 2 3 4 5

6 D/R, IO k 7 8 9

S / P , I 0 k t

S / P , 2 0 k

D / R , 2 0 k

'1 pHUGO . ,ok. 2o k

R, k ~ | I - S P I R A L , IO k

- 2 0 2 REAL PART

I I I I I 0 . 2 5 I - I - 0 . 2 5

TO H A L F , sec TIME

.6

J

.75

2

5 I0

TO D O U B L E , sec

a. M = 1 . 3 Figure 27. Y a w - t o - t u r n con f i gu ra t i on - locus o f roots w i t h

C L q , CL, ' , , Cy r , C y p , and Cmr var ia t ion .

0 0 w

0 0

W

~> r n

¢3

;0 do o

A EDC-TR-8O-11

3 0

I-- 0c Q.

>" 2 0 E

z m ¢9 4( :E u

IO

3O

I- n,"

>,.

a: 2 0 ,( z (.9 <( =E

I 0

3 0

2 0

z

:E - 1 0

0

- B


LARGE z GAIN • 0

LARGE m GAIN • 1

L A R ~ o GAIN - ' m

0<_ ILOGIoIIGAINlll < I o ! <_ ILOGI0(IGAINI)I < 2 Jr z < ILOGIoIIGAINIll < 3 o 3 <_. ILGrqdl~lNI)l < 4

4 < ILOGt011~iNI)l < S + 5 ~_ IkOGI~I~,INI)I < 6 , e <_ IL06]OII~, INII,: < ;r , 7 <_ II.~]dl~. jN') I < I • . 8 < II.OGIo(IGAINll l < 9

F t LEVEL FLIGHT, Ig n T k : 1.2 deg (7 A L T , I O - 3 9 d e -

TALT= 40 k" • v

Jr- -I

D/R, 10 k A

D/R, 40 k

SiP, I o k ' ~ _A_.]

I K" S/P, 40 k k A

R , 4 0 - I i F SPIRAL, PHUGOID, R,,o.:-~..j~_ I ,o ' , 4o'

U

I I

o" 0

.5~- eL

- I

5

| " r -

TURNING FLIGHT, 5g a T " 5 .3 d i g

A L T " I0 k =13.6 deg TALT 40 k

DEGENERAT([D D/R, 40 k

_ -,L-=----- t

D/R IO k

s,P, io k OEGENERATeO ----m - s,P, ;ok o,R~ 4o k

~ 1 I - - - - " R,4O k "1l / ' - S P I R A L , PHUGOID,

R, ,o, ~ l V ,o,, ,o,

U

II

w Q 0

, T

TURNING FLIGHT, l o g GT k = 9.1 deg rY ALT TALT" ,~,-!Ok" 2Z S d . . . . . .

i

. . . . . . . . . . i . . . . . . . DEGENERATED

D/R, 40 K

.

S/P, I0 k i i

D/R, 10 k A ~ I S/P, 40 k DEGENERAT~I

. . . . . . . . . . I . . . . . . . . . *f . . . . . . . . . . t I I z-PHUGOID, IO k, 40 k

R,~okR' 40k ~ . . _ S P I R A i ' IOk' 40 k

- 6 - 4 - 2 0 2 REAL PART

I i I 0 . 2 5 I 0.1


b. M = 3 . 0 R g u r e 2 7 . Concluded.

.I

u 41) to

o 0

. 5 ; W n

I

5

4

I I - I - 0 . 2 5

TIME TO DOUBLE, s i c

72

A E D C - T R ~ O - 1 1

0 . 5

0 . 4

o p 0.3 me

0 z 0 . 2 o. =E <[ o

, 0 .1 N

- 0 . 1

_1

-i-

SY.~.M MAGNrRJDE OF GAIN

LARGE z GAIN - O

LARGE .s GAIN • 1

LARGE o GAIN - * o

a 0 <.. ILOGIoAIGAINIII '¢ 1 o I <_ ;LOGL0(IGAINlll < z ~. 2 <_ ILOG10(IGAINIII < 3

O • ¢ ILU~101.1bAINI) I • 4

4 < ILOG]oIIGAINI)I < 5 + 5 <_ ILOGIo~IGAINI)I < 6 , 6 <_ ILOGI0flGAINlll < 7 , ; <_ ILOG]oIIGAINlll < S .. 8 <_ ILOGLoIIGAINlll <' 9

I

~ - - S I P DAMPING RATIO

M= 3 . 0 0 A L T I T U D E = IO,OO0 f t L E V E L FLIGHT, I g

- I 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0

C m q , p e r / r o d

a. Damping ratio 5 0 0 0

4 0 0 0

3 0 0 0

2 0 0 0

I 0 0 0

0

- I O 0 0

- 2 0 0 0

- 3 0 0 0

- 4 0 0 0

- 5 0 0 0

0

4 0 0

%J

// /

0 . 5 1.0

Cmq, p e r / r o d

/ - - 200 f / - - 0 / ~ - - - 2 9 8 (TRIM)

/f 1.5 2 . 0 2 . 5

T I M E , s e c

b. Time response Figure 28. Yaw-to-turn configuration - effect of Crag

damping ratio and time response.

6 0 0 8 0 0

/ - %

S f

o n

I 0 0 0

3 0

?3

A E DC-TR-80 -11

0 , 5

0 . 4

O 1 i - 0 . 3 n, Lg z_ 0.2 O. =E q(

I N 0 , !

- 0 . 1

5 0 0 0 I

4 0 0 0 I

2 0 0 0

,.,r ° I,- a IOOO

;'z J°:_ O ,r.,) I,- i . , . z o - l o o o

l i " z ° ° ° - 3 0 0 0

- 4 0 0 0

- 5 0 0 0

0

m

SYM NAGN |TI.I~ OF GAIN

LAW: GAy,. o LARGE s GAIN - !

LAIIGE • GAIN - "¢. w

a 0 < ILOSLodlIGAINll I < I

o ) <_ iLOGIoilGAINi)I < 2

• 2 ¢ ILOGI011GAINI) i < 3

o 3 <_ ILOGIdlGA;141)I < 4 n 4 <_ ILOGIoIIGAINI) I < 5 , , 5 <. ILOGIaIIGAINIH < 5 , 6 <. iI.0GI01IGAINIH < 1 • 7 <_ iLOG~IGAINI)I < 8 • * o~_ I L t A ~ l l b A l , l l r l • y

I I I

i

/

M = 3 . 0 0 A L T I T U D E = I 0 , 0 0 0 L E V E L F L I G H T , Ig

- S / P

- 1 0 0 0 - 8 O O - 6 0 0 - 4 0 0 - 2 O O 0 2 0 0

Cmc ~ , p e r / r o d

a. Damping ratio i .m& , p w r l r g g

t

~/!'/ i

4 0 0 0 (TR IM)

- 2 9 8

f \ / / !

/ i /

D A M P I N G RATIO

t

4 0 0 6 0 0 8OO

L__I

\/ b

v

0 . 5 1.0 I ,5 2 . 0 2 . 5

T I M E , sec

b. Time Response b. Time response

Figure 29. Yaw-to-turn configuration - effect of Cra a on : damping ratio and time response.

f t

I 0 0 0

3 . 0

74

,-J Lr=

I - rr < 8 (1.

>. n*

Z m (.9

:E 4

0

- 8

SYM N~.GNITUDE OF GAIN

LARGE x GAIN = 0

LARGE '~ GAIN = 1

LARGE o GAIN - - +

A 0 <_ ILOGIo(IGAINI)I < ]

o 1 <_ ILOGIo(IGAINI)I < 2 ,~ 2 <__ ]LOGIo(IGAINI)I < 3 o "~ <. ILOGIo(IGAIN()i < 4 , 4 <_ ILOGIo(IGAINI)I < 5 + 5 < ILOGIo(IGAINi)I < 6

, 6 < ILOG]o(IGAINI)I < 7 ,b 7 < ILOGIo(IGAINI)I < 8 . . 8 < ILOGIo(IGAINI)I < 9

LEVEL FLIGHT, I g

O. TALT = I0 k =6 .gde(J C~ k 9.3 deg

TALT 2 0

I - 6 - 4

! 0.1

@

- I 0 0 0

- I 0 0 0

D/R, I0 k 0

• /I t ~ ] • 11' ,i~ • •

I 0 0 0

0 . 2 5


O / R " 2 0 k l "

? ,ooo R, 4 0 k - ~ I F SPIRA

R, _,ooo

- 2 0 R E A L P A R T

I I | i I - I

T I M E

L , I0 k, 4 0 k

1 2 4

I - 0 . 2 5

TO D O U B L E ,

Figure 30. a. M - - 1 . 3

Yaw-to-turn configuration - locus of roots with Cn r variation.

- . 6

- .75

- I

- 2

- 5 - I 0

S e C

m

11=

0 m

n~ uJ a.

7> m o c)

-n

o

==b

A E D C - T R - 8 0 - 1 1

3 0

i - t.. ,4 G.

2o Z (.-3

I 0

3 0

I,- n.'

>- " 2 0 z_

: E

I 0

0

3 0

i - n-

>- n.,

Z (.9

~E

SY_.~M LARGE x I,.41~£ i LARGE e

a ci t 0 |

,

MAGN rI1LIOF. OF GAIN GAIN • 0

GAIN • ! GAIN -tnD

n:<.. iLOGIDIIGAfNIII < ] 1 <_. ILOG~iGAINI)I < ! 2 (_ It.CGIoffGAJNII~ < ) )<_. ILOGIdiGAINI;t < 4 4 <_. ILOGto(IGAiNIII < 5 S~ i~|Ol i~n, i i l l l l | • • 5 <_ ILOGIotlGAiNI|i < I 1 <_ ili.OGio(iGAINil I < ,11 a< T ILOGIg(IGAINI)I < S

-IOOO

I LEVEL FLIGHT, Ig

~ T A L T , I O k : f.2 deg T A L T , 4 0 k 3.9 dNg

! t-

D/R, I0 k 0

* • " I * * * I 000

D/R, 40 k e l R • 9 ] ~ l e •

- I 0 0 0 0 I 0 0 0

R. 40 ~ 1

R, I0 k'- I ~ ~ t Id'.,,-SP|RAL, 40 k ,© ~ " - , o o o - - - I

g

Q 0

IU (L

i I TURNING FLIGHT, 5g

~ T A L T • 5 .3 deg I0 k TALT ~ 4 0 k =13.6 deg

- I 0 0 0

! 1

D/R, I0 k

DEGE NER ATE D O/R, 40 k . - - j

m Q •

DEGENEI .~_ D/R,

R , 4 0 k l - - S P I R A L ' I0 k

R, IO k ~ . / 'SP IRAL, 40k

,oo04 IOO0 IOO0

8*00

- . I

D

0 • . 5 ; ;

¢L

2 0

tO

0 - 8

TURNING FLIGHT, lOg n T k " 9.1 d i g

3 - ,

- I 0 0 0 m

DEGENERA~ ~D O/R, 40 k

- 6

I 0.1

+

I D/R, I0 x OI [ 4 0 0

R, 40 /-SPIRAl. iok. 40k R , l o k A / - - ' D E G E N E R A T E D

- ~ ~ _ O/R, 4O k .--*. I

- 4 - 2 0 2 4 , REAL PART

I I ~ I I 0 . 2 5 I - I - 0 . 2 5 i

TIME TO H A L F , sec TIME TO D O U B L E , see

- . I

u o w

0 0

- 5 8 :


76

- J

12

i -- nP < 8 G.

>-

Z

( .9 < 4 =E

0

- 8

S YlVl MAGNITUDE OF GAI N

LARGE x GAIN • 0 LARGE "- GAIN • 1 LARGE o GAIN " + (]o

a 0 <_ ILOGto(IGAINI|I < l a I <_. ILOG10(IGAINI)I < Z . Z <_ ILOGt0(IGAINI)I < 3 0 3 <_ ILOGto(IGAINI)I < 4 , 4 <_ ILOG]0(IGAINI)I < 5 ' • 5 <_ ILOGzo(IGAINI)I < 6 , 6 < ILOG]0(IGAINI)t < 7 , 7 <_ ILOGtotIGAINI)I < 8

. . 8 <_ ILOGIo(IGAINI)I < 9

L E V E L F L I G H T , I g

" 0. T = 6 . 9 d e g A L T = I 0 k

T A L T = 2 0 k = 9 . 3 d e g

-30 -ZO

- 4 0 - 3O

o /,ooo D / R , I o k - - O ~ . . - I 0 0 0

I 0 0 0 -.,. - - - 5 0 0

: . ' = b . ~ " - - I 0 0 0 5 0 - - " ~ P - - " 1

D / R , 2 0 k - - - ~ " ~ = a " . . 0

I 0.1

-G

R, I0 k - I0

e - 1

- o R , 2 0 k - I 0

r - I000 ! I 0 0 0 - ~ / rSPIRAL, I0 k

_o t_o [o 20 ~aL :-J-- ~-~-= r. : - -~ -

-~ ~ll~ -b SPIRAl', I0 20 oooJ _.looo 2°k

- 4 - 2 0 2 4 REAL PART

I I J I I 0 . 2 5 I - I - 0 .25

TIME TO H A L F , se© TIME TO OOUBLE, se¢

Figure 31. a. M - - 1 . 3

Yaw-to-turn configuration - locus of roots with C£p variation.

.6

.75 u

i =" o

Q.

2

5 I0

m 0 (3

"n d0 .=

A E D C - T R - 8 0 - 1 1

3 0 - - -

20

<1 z

t O

It,.

_z

Is"

Z

I

0

3 0

2 0

I0

- I 0

2 0

I 0

0

- B

SY.~M MAGNrP.IC( OF GA IN LARGE x GAIN • 0 LARGE s GAIN • I LAI+G£ e GAIN - i m

a 0 < ILRG]OIIGAINlll < t o : ] < ILtX;IoI'GAINI)I < 2 . 2 < [LOGIo(tGAINlll < 3 O 3 <_ ILRGIOIIGAIHI), < 4 , ~ <. ILOGL¢iR, AINI)I < 5 + 5 <. ILOGIo~IGAINll I < 6

+ <_ It.OG1otlGAINPl < 7 , T <_ ILOGIGtlGAINI)I < s ,. S <_ I'0GI0( GAll, Ill < 9

LEVEL FI'IGHT, Ig

T A L T . I O k : 1.2 deg , A L T . 4 0 k 3.9 deg

i R ' Iok - 1_5 ~ - i O - S

0 D/R, I0 k

- I O # O ~ l O 0 0

D/R, 40 k

- I0O0 I0OO

I iooo I000~ f " k o-- I [sv,,,,+.. ,o

- 2 o - , s - , o - s DO s ,o ,s eO R , 4 0 k I 0 0 0 " 1 0 0 0 S P I R A L , 4 0 k

TURNING FLIGHT, 5g j GT - - , : - k " 5 . 3 d e g I J I

A I . T I U =13 6 d i I ! U T A L T : 4 0 k , I Ig I D / R , IO k J

. . . . . . . . . . . . . . . . . . . . . . . . . . . O . . . . . . . J . . . . . . . . I . . . . . . . . .

D/R, 40 k IO0

l ~ d ~ I 000

- I 0 0 0

i I I

~ROLL COUPLED ~ - f O 0 0 . . . . . . . . . . . . I 000 ~ IO00 / SPIRAL, I0 k

~ o ~ \ l ~ l . . . . , _ - - . -*- -~ . . . . . _ - _ - . . . . . . .

-SPIRAL, 4 0 k

. . . . . . . . . . j . . . . . . . .

I TURNING FLIGHT, lOg ~ T 9.1 d i g

ALT • Iok -'-22.5 O dell TALT • 40 k

,o J I I .

5 0 " 0 -- r i

D/R, IO k ~O00 - IOO0

f I " o , o °

I - 1 0 0 0 - 7

/ ,ooo~11 S,,RAL, ,o. -.s R, IO h '~..e, C ,'s

I . . . . I - -- ~ " - - ' A * - - . - , , - . , - ~ - i i i i

R , 4 0 k 0 ~SPIR~AL, 4 0 k 30b

- . I

i i i

0 o

- . B ~- U I O .

- 5

- ~ . 5 ;

I a .

- 1 1

I

J_,

-4.1 I

I I m

I 0 o 'I'5~ l.IJ 12.

1' j s

I 0.1

- 6 - 4 - 2 REAL PART

I I 0 . 2 5

TIME TO H A L F , s i c

0 2 4

- I ' - 0 . 2 5

TIME TO DOUBLE, sec


78

A E DC-TR -80-11

0

,4

Z n

X

o I

N

sY.._~M LARGE •

LARGE a

LARGE 0 dk

0 al

0 !

÷

P

0 .5 _ , , ,

0 . 4

0 .3

0 .2

0.1

-0 .1

M~GNITUD( ~ GAIN

GAIN • 0

GAIN • |

GAIN - ± Q

0<_ ILOG]o(IGAINIII < ]

L (_. ILOGIoflGAINII, I < 2 2 < ILOGI.011GAINI)I < )

< ~L~LoIIGAIHI)I < 4 4 <_ ILOGLotlGAINI)I < 5 5 <_ ILOGI0itGAIIAI)I < 6 6 < JLO(;IO(IGAINIII < 7 T <_. LOG|o4KGAIN'IJ < 8 II <_ ILOGt011GAINlll < 9

I

DIR

. / u

t

- I 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0

a.

0 2 0 0 C n r , p e r / r o d

Damping ratio Cnr , per / rod - 5 0

~ - 2 4 5 (TRIM) !~_ ,oo~ I // A

I"-'7

V

I 0 , 0 0 0

,,,ooo;lll o.oooUlll / ~ 4.0oo ~ , / /

,.,o / ~. l a 2 , 0 0 0

~ o : 0

I ' ° ° ° l / v j~ .,.ooo I/ _.ooo /J - a , o o o

- I 0 , 0 0 0 :

0 0 .5

b.

1.0 1.5

T I M E , sec

Time response Figure 32.

A L T I T U D E = I 0 , 0 0 0 f t M = 3 . 0 0 TURNING FL IGHT, lOg

DAMPING RATIO

4 0 0 6 0 0 8 0 0 I 0 0 0

A /.. /-/

\ / v

A / \

V

2 . 0 2 .5 3 . 0

Yaw-to-turn configuration - effect of Cn r ratio and time response.

on damping

79

AE D C - T R - 8 0 - 1 1

0 I,--

E

z

5

¢3 I

N

j r~

O z o = 0

sv__~m LARGE x LARGE s

0.5 L R K ~ u

a

0

0.4 * 0

!

0 . 3 t

0.2 J

0.1

0

-0.1

F

MAGNITUDE OF G A I N

G A I N - 0

G A I N - !

rJRIR -- Z m

0 <_ ILOGIoIIGAINIII < ] Z <_ ILOGIoIIGAINIfl < Z Z < ILOG]0IIGAINIII < 'Z

< ILOGIoIIGAINI)I < 4 4 <__ ILOGIoHGAINlll < 5 5 < ILOGI011GAINIII < 6 6 < ~LOG[01'GAINI) ,( 7 1' < ILOGI011GAINIII < 8 8 <_ LOGIo~IGAINIII < 9

I I I I

D / R D A M P I N G R A T I O

i I t t

- i 0 0 0 - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0

M = 3 . 0 0 A L T I T U D E = I 0 , 0 0 0 T U R N I N G F L I G H T , l O g

~ d

\

0 2 0 0

C j ~ p , p e r / r o d

a. Damping ratio

I l i l . o o o

,~,ooo ,,~ All -~ ~//lI, A 8 , 0 0 0 -

,,ooo /~/~/ lf:A',.,/ o-' I A ll/ .,,.i. V

,,ooo- !1/Ill I n ',/ " -..o -- ill AI/J ,\: ~ . :,,.o::-- II \/~,~ v*~, ,.ooo r I :

- 2 o , o o o J o

Cf. p, per / rod

- 3.7 (TRIM) - . o °

A

f t

Figure 33.

4 0 0 6 0 0 8 0 0 I 0 0 0

A

~ I ~ I Y I L !/ J l / v

R O L L MODE

2.5

on damping

0 . 5 1 .0 I . 5 2 . 0

T I M E , s e c

b. Time response Yaw-to-turn configuration - effect of Cgp ratio.and t ime response.

3 .0

80

eL

z m

m

12

B

4

ol - 8

r

SYM

LARGE x

LARGE A

LARGE o

6

n

/¢

0

I

+

t

I ,

e t

MAGNITUDE OF GAIN

GAIN = 0

GAIN = 1

GAIN " + co

o < ILOGIo(IGAINI)I < ] I <_ ILOGIo(IGAINI)I < 2 2 <_ ILOGt0(IGAINI)I < 3 3 <_ ILOGz0(IGAINlll < 4 4 <_. ILOGzo(IGAINI)I < P 5 <_ ILOGIoIIGAINI)I < 6 6 <. ILOG10(IGAINI)I < 1 7 < ILOGz0(IGAINI)I < 8 8 <_.. ILOG10(IGAINI)I < 9

Q

9

5 0 0

I I

- 6

0.1

° f

D /R , I0 k 0

I L E V E L F L I G H T , I g

K

DI ~), 2 O k

~ T = 6 . 9 d e g ~ T ~ LT = i 0 k

L T = 2 0 k 9 .3 d e g

Q

F & ~ Q " - - - - .

- 4 0 0

- 4 0 0

~. 5 o o ~ " - - - * - t " - 5 0 0 0 r . S P I R A L 2 0 k

300 200 )

- - 2 4 R E A L P A R T

., I I I I I 0 . 2 5 I - I

T I M E T I M E T O H A L F , s e e

. 6

m

;

- 0 . 2 5

T O D O U B L E , s e c

. 7 5 u 0 m

I " o ix:

n

2

5 I 0

Figure 34. a. M = 1 . 3

Yaw-to- turn conf igurat ion - locus of roots wi th C£ r variation.

m

-n ~o 9 ..b

A EDC-TR-80 -11

3O

i - n*

Q.

eo '4 Z

i

I 0

0

3O

i -

>.

n: 2 0 4( z_ 4( :s

I 0

3 0

F-

2 0

,4 z

:E -- I 0

0

SYM IdAGq I'IUDE OF GAIN

LARG[ • GAIN • 0

LARGE I GAIN • I

LARGE • GAIN " ' ~ ®

• 0 < LOGIO(IGAINIFI < !

o ) <_ LOGIo(IGAIN I ' < 2

, Z < ILOGI~'IGAINIII < 3 o 3 < ILO6IoIIGAINId < 4 , 4<_ ILOGtoII~INI)I < s . S <, ILOClo(IGAI~,III < t , 6<_ II.0n|OI:GAINIII < ;' , 7 <_ ILOG~IGAINIIi < II • . I<. ILOGI611GAINIII < 9

' IS,OO • • • •

LEVEL FLIGHT, Ig

(~TALT. IO k : 1.2de, TALT .40k 3.9 d ig

D/R, I0 k 0 - 5 0 0 a m t -,! • •

D/R. 40 k 600 0 - 5 0 0

-- - • • IlK • t • !

I ~R, 40k rSP IRAL , IO k

~ ' ~ Ok 50(~Op/ / -SPIRAL , 4 ok

. o

I l l

o" o

. 6 ; kd ¢I.

I I TURNING FLIGHT, 5g (:IT • 5 .3 d ig Q ALT • I0 k .13.6 dig

TALT • 40 k l

'200

OEGENERAT D/R, 40 k

-,~oo-joo

D/R, IO k 0 K

:.D

,t *-200

DEGENERAT~C I D/R, 40 k.

I \SFJR,L,,OA R, 40 k - ~

R, , o k ~ - ~ V / - S P I R A L ,O k . ^ ^ . - 5 0 . . . .

D

o . S ;

ILl O.

I TURNING FLIGHT, IOIg (2T k " 9.1 deg -- ALT •10 =22 5 de- nTAL T : 40 k • g

I

150 too e

DEGENERATED D/R, 40 k

• 350 400

. t

0.1

D/R, I0 k

50 0 - 50 j •

k I ~r~PIRAL, 4,ok

R"°\l \ [G.,.L 4.o 500; 0\:500 :' Oi

- 4 2 0 REAL PART

i J 0 . 2 5 I


b. M = 3 , 0 Figure 34. Concluded.

-IOO - ISO t

DEGENERATE| I D/R, 4 . ~ 0 k

' l i k

2 4

O

ld G.

I I - I -O.25

TIME TO DOUBLE, so¢

82

O0

12

t - n,. < 8 G,

>,, n,, <1[ Z (.9 <[ :z

4

0


LARGE x GAIN - 0

LARGE ~ GAIN - 1

LARGE 0 GAIN " + m

a 0 <_ iLOGIo(IGAINI)I < o l <_ ILOGIo(IGAINIfl < . 2 <_ ILOG10(IGAIN])I < o 3 < [LOGIo(IGAINI)I < i 4 <_ ILOGto(IGAIN])I < + 5 < ILOGIo(IGAINI)I < , 6 <. ILOGzo(IGAINI)I < 4 7 < ILOGto(iGAINI)I < .. 8 <_ ILOG10(IGAINI)[ <

L E V E L F ' L I G H T , Ig = 6 .5 d e g

~ T A L T = I o k 9 . 3 d e g Q T A L T 2 0 k

2ooJ ; - -1

- I

" 2 0 0 |

I 0 0

2 3 4 5 6

7 -- D / R , I 0 k 8

- 5 0 0

, oo = -

9 /D/b/k

f t

l i t

- 5 0 0 5 0 0

0 =1D/R ' 2 0 k

R ok \ 5oo S l AL ok / - jpo - ~oo

= ~,, - u . ~ . i 0 0 - R , 2 0 - - J ~ ' - . . . ~ ' ~ . - ~ - n " - - S P I R A L , 2 0 k

0 " ° U V S O 0 -

- 8

l 0.1

- 6 - 4 - 2 0 R E A L P A R T

i i I I L, 0 . 2 5

T I M E TO H A L F , s e c

2 4

I I - I - 0 . 2 5

T I M E T O D O U B L E ,

Figure 35.

a. M = 1 . 3 Yaw- to- tu rn conf igurat ion - locus of roots wi th C .p var iat ion.

- . 6

- . 7 5

- I

- 2

- 5 - I 0

s e c

0 ql) (R

0 m n- t¢l O.

]> m o (-~

-11 do 0

A E D C - T R - 8 0 - 1 1

I - n~

G.

Z

0

i

i - n-

>-

z

< =E

I-- n*

3. n~

¢g

:E

3 0

2 0

I 0

0

3O

2 0

I 0

0

30

0

- 8

2 0

I 0

SYM f ~ H f11[10[ OF GAIN LARGE • GAIN - O UU~.( s GAIN - I bqRGE o GAIN - * ,m

,I 0 <. ILOGtoIIGAINI)I < I u ! ~_ I,I.OGIoflGAINIII < 2 . Z< IL©GIdlGAINIII < 3 o 3 <_ ILOG1011GAINI;I < 4

4<_ ILOG~IGAINI)I < 5 S < ILOGI4tlGAIItill < S

, 6 < ILOG#iGAINIII < 7 , ? < iLOGto(IGAINIII < S ., 8.<_ it.OStoflGAIN:II < 9

1 • t • t

L E V E L FL IGHT, Ig

~ T A L T : I O k : 1 . 2 d e g T A L T n 4 0 k 3.9 deg

- 500

500

DIR, JO k 0

- 5oo

D/R, 4 0 k n I

• § O 0 . w ~ - 5 0 0 /

I i TURNING F L I G H T , 5G

( ~ T A L T " 5 . 3 deg IO k T A L T : 4 0 k : 1 3 . 6 deg

DEGENERATED D/R, 4 0 '

-_2o0 f

ioo I 1

TURNING FL IGHT . l O g

~ T - k ' " 9.1 d e g Q A L T - I u . 2 2 5 deg

T A L T : 4 0 k •

- 5 0 0

- 2 0 0

DEGENERATED D/R, 4 0 k

- 4 0

D/R, I 0 k

~ , . . . , - s o o o 5oo

DEGENERATED D/R, 4 0

. o o

R'~' ~ Y , oo GO0 ""

R.,OkO '-5oo-T"-5oo "~-;'oo _ . . ~ S P I R A L , I 0 k

I - 5 0 0

I . u e ' * t ° e D / R , i 0 k - I o ~ • aJ

i 0.1

- 6

o d

~oo -~ ~R , 4 (

~, o l ~ • J . ,

- 4 R E A L P A R T

I I 0 , 2 5


500

.) lOG DEGE NERATI [D

D/R, 4 0 k

F 5 0 0 J t N / - - S P I R A L , 4 0 k

I ' ~ Q ~" ~GP,R, AL, ,O k . 50 br t- i _m

- 2 0 2 4

I - I - 0 . 2 5

TIME TO D O U B L E , se¢

o , - t

0 .5 ~..

IiJ G.


o

C3 0

bJ a.

¢) ED m

a

0 . S ;

Q.

84

0 I-

z A. IE

o

'"F i - a

Jt~- O' lZ

SYM MAGNInJD[ Of GAIN

LARC.~ • GAIN - 0

LARGIE a GAIN - I

LARG( o GAIN - * @

a 0 < ILOGIo(IGAINIII < l

o I ~_ ILOGldIGAINI) I < 2 ,t 2 < ILOGI01IGAINIIi < 3 O ) <_ iLOGIoIIGAINI)I < 4

4 < ILOGIoilGAINII I < 5

. S < ILOGIo(IGAINII I < 6 6 < ILOGI0(IGAINIII < 7

, 1 < RGGIo(IGAINII I < 8 • . 8<__ LCGLO(IGAIKlll ,¢" 9_~

1.0

0 .8

0 . 6

o.4 ~ ''"0"~--~ ~ ~ ~COUPLED SPIRAL/ROLL j

0.2 - DAMPING RATIO

0 . . ~ i -

-0.2 j r

- 0 . 4 j ~ F "

- o . s .,~V- .jr '\- D,~ OAMP,~O RAT,O

I I ! -I ,0 - 5 0 0 - 4 0 0 - 3 0 0 - 2 0 0 - I 0 0 O

a ,

I 0 , 0 0 0

I 0 0

C j r , p e r / r o d

Damping ratio

M= 1.30 ALTITUDE = 2 0 , 0 0 0 f t

L E V E L FL IGHT, Ig

2 0 0 3 0 0 4 0 0 5 0 0

8 0 0 0

6 0 0 0

4 0 0 0

2 0 0 0

0

- 2 0 0 0

- 4 0 0 0

- 6 0 0 0

- 8 0 0 0

- I 0 , 0 0 0

~ J

C jr r per/rod

(-- SPIRAL

f Y \ r

~ u M S T A B L E DUTCH ROLL

0 0.5 1 . 0 I . 5

TIME, s e c

2.0 2 .5 3 .0

Figure 36. b. Time response

Yaw-to-turn configuration - effect of C£r on damping ratio and time response.

AE DC-TR-80-11

85

A EDC-TR-80-11

_o I-. n.

z [ :E

~3

._1

;o o o

o n-

0 .5

0 .4

0.3

0.2

0.1

O

-0.1

- 0 . 2

- 0 . 3

- 0 . 4

SYM MAGNITUOE OF GAIN LARGE x GAIN • 0

LARGE e, GAIN • | LARGE o GAIN - + ®

e, 0 <_ ILOGIo(IGAINIII < 1 o I <__ ILOGIo(IGAINIII < Z . 2 <. ILOGto('GAINIII < 3 O 3 <_ ILOG'l.Of GAINH < 4

4<_ LOG1(JIGAINIII < ,5

, 5<__ LOG1gl]GAINI)I < 6 , 6 <_ LOGto(IGAIN]ll < 1 , ?<_ LOG10~IGAINIll < 8 . . 8 <. ILOG1~'IGAINlll < 9

I J , I, s

M = 3 .00 ALTITUDE = I 0 , 0 0 0 f t TURNING FLIGHT, lOg

- 0 . 5 i - 5 0 0 - 4 0 0 -3,00 - 2 0 0 - I O 0 0 I 0 0

Cnp, per / rod

a. Damping ratio

DIR DAMPING RATIO

2 0 0 300 4 0 0

J J~

/

Cnp, pe r / rod

5 0

/ / \, \/ v

(TRIM)

A

V

~ UNSTABLE SPIRAL

A

0.2 0.4 0.6 0.8 1.0 1.2

I 0 , 0 0 0

8 , 0 0 0

6 , 0 0 0

4 , 0 0 0

2 , 0 0 0

0

- 2 , 0 0 0

- 4 , 0 0 O'

- 6 ,0 O0

- 8 , 0 0 0

- I 0 , 0 0 0 0

Figure 37.

TIME, sec

b. Time response Yaw-to-turn configuration - effect of Cnp


5 0 0

on damping

86

Go -.4

I- e, <[ ¢L

>* n -

Z m

(.9

:E

12

8

4

0 - 8

$YM MAGNITUDE OF GAIN

LARGE x GAIN = 0

LARGE A GAIN = ]

LARGE 0 GAIN -" • (x)

• ", 0 <_. ILOG10(IGAINI)I < t

o I < ILOG10(IGAINI)I < 2

2 <_ ILOG]0(IGAINI)I < 3

o 3 < ILOGIo(IGAINI)I < 4

, 4 < ILOGI0(IGAIN])I < 5

• 5 < ILOG].o(IGAINI)I < 6

, 6 .<. ILOGt0(IGAINI)I < 7

, 7 <_ ILOGt0(IGAINI)I < 8

. . 8 < iLOG10(IGAINI)I < 9

D/R, I0 k

- 6 - 4

I 0.1

T I M E

LEVEL FLIGHT, Ig O. T = 6 .9 deg (~ ALT = I0 k 9.3 deg

TALT 20 k

- t - soo ~ o ; s,P, ,o"

500 .~- 500

" 5 0 0 l 0 ~ S/P, 2 0 k o F 500

~ _ ~ o o,R, 2o k 5 0 0 - J-'-"

R, I0 k, 20 k '!F/.~-. HuGOID' IOk' 20k 5 0 0 - ] - - 5 0 0 ~ S P I R A L ' I0 k, 20 k

I - " ~ r ' I - 2 O 2

R E A L P A R T

I i I = J 0 . 2 5 I - I - O . 2 5

TO H A L F , s e c T I M E

Figure 38. a. M - - 1 . 3

Yaw-to-turn configuration - locus of roots with C~q variation.

- . 6

-- .75

- I

- 2

- 5 - I O

4

TO DOUBLE, sec

o 6

p

0 I

n , Lu o.

m

o

0

AEDC-TR-80ol I

3 0

I - n* ,¢[ O.

20 < Z (9

: E

I 0

3O

I'-

> -

0= 2 0 z_ ,¢[ :E

I 0

3 0

I,-

2 0 >,,, 0¢

Z (9

:S - - I 0

0 - e

SYM ,MAGNITUDE OF GAIN LARGE x GAIN • O LARGE e, GAIN - 1 LAR~ o GAIN - * ®

a 0<.. iLOGL0(IGAINIII < t o I < ILRGtOIIGAINI)I < 2 , 2 <_ ILOG~IGAtNI)I < 0 3 <_ ILOGt011GAINlil < 4 i 4 <_. tLOGIo(IGAiNlll < 5 • 5 < ILOGI.GIHGAINi)[ < 6 , 6<_ ILOGL~GABNI)I < ? , 1 <_. ILRGtOIIGAINI)I < 8 .. S <_. ILOGIoIIGAINII~ < 9 . . . .

400

i I I L E V E L FL IGHT , I0

o.R,,ok 'ALT:,Ok I I 3.9 deg

]1( T A L T = 4 0 k

- 5 0 0

L ~ D /R . 4 0 k

: z l S / P , I 0 k " S / p , 4 0 k

R ,ok " t ~ OOsPIR''. P,UGO,O. R I0 k ' '500-~" ~ ' ;0 . - -~* '~- Iok, 4ok : ~ o ~ ~ o o ~

,- 5~o~.',;oo '

.I

i , i

TURNING F L I G H T , 5(I

G T • 5 . 3 deg C[ A L T - I 0 k : 1 3 . 6 deg

T A L T ,, 4 0 k I

D/R, I0 k iO

• DEGE NERATI[D ,,, D/R, 4 0 x

4 0 0

- 500" . . I P 500 - 5OO

S/P, !ok. ,~C ' DEGE NERATE I %, w S / P , 4 0 k D/R, 4 0 k

SP IRAL , i O k . ' ~ . I , ~ \ "' . . 500 ! " ~ L : R , 4 0 " . % ,- PHUGOIO,

R, '0"~6. 500\ \ , J - IOk 40k _ J ~,v _ O L - L ' , -

' 5 6 0

T U R N I N G ' F L I G H T , l O g

(2 T • 9.1 d i g (7 A L T : I0 k , 2 2 . 5 deg

T A L T 4 0 k

I - 500

#'Q • I

DEGE NERAT~O D/R, 4 0 I

o'2- ~ 2 2

0.1

- 6

5 0 0 S /P , | 0 k - 5 0 0

. . . . . .

"D /R , I0 k DEGE NE RATI[ ) ~-~ S/P, 4 0 k i n D / R , 4 0 N

"% I I S P I R A L , 40k'..~%'., 5 0 0 / - P H U G O I O , I0 k . 4 0 k

o ,,. ,ok ~ , \ ~--/---spIRA-. ,ok R:,O~-,O ; 5 0 0 ~ V \ 50'0

_ _ |

- 4 - 2 0 2 4 REAL PART

I I I I I 0 . 25 I - I - 0 . 2 5

TIME TO HALF , sec TiME TO OOUBLE. sec


¢) u m

o" o

U.I O.

5

u Q tQ

(.3 0

. 5 ; i.J O.

u l ) at

o . 5 ;

O.

88

O0

12

p.

< 8 0..

> - ,1,,'

Z m

¢.9 < 4 :Z

0 - 8


LARGE x GAIN = 0

LARGE ~ GAIN = ]

LARGE o GAIN " ± (z)

A 0 <_ iLOG]o(IGAINI)I < 1

a ! <_ ILOGto(IGAINI)I < 2

. 2 < ILOGto(IGAINI)I < 3

0 3 <_ ILOGIo(IGAINI)[ < 4

, 4 <_ ILOG10(IGAINIII < 5

, 5 < ILOGzo(IGAINI)I < 6

, 6 <. ILOG10(IGAIN[)I < 7

, 7 < ]LOG]o(IGAINI)I < 8

. , 8 < ILOGIo(IGAINI)I < 9

LEVEL FLIGHT, Ig = 6.9 deg

O..ALTT = i0 k 9.3 deg QTAL T 20 k

- 6 - 4

I 0.1

L

0 .25


~- 500 ] ~ I S / P , I0 k

500 ~--~- 500

0 S/P, k L 20 w

D/R, I0 I~ -~---,-~IIE 500 I~_ 50C i - 500

. 5 0 0 I

O,R, ~o~ #o I I - soo ! I ~-PHUGOID, I0 k, 20 k I " ~ I / -SPIRAL, I0 k

R,IO k, 20k-7 r - - - 5 0 0 / / - S P I R A L , 20k

, oo I " - l r " - - -

- 2 0 2 REAL PART

t 1 I I -o.~s I - I

TIME

4

.6

Figure 39. a. M=1.3

Yaw-to-turn configuration - locus of roots with Cmp variation.

TO DOUBLE, sec

- . 7 5

- I

- 2

- 5 - I 0

(,,t Q

m

0 n . bJ o.

m

o c)

-I1 ~0 0

n,, ,4 G.

>" 2 O a:

z (p

: i

i.- O:

Q.

>.

IE

3 0

I--

Z

SYM 3 0 - L A ~ .

I~ RrJ: a LARr~ o

4

a

O

I

4

I

I 0 - ""

4 0 0 0

2 0

TURNING

G T (2 ALT •

TALT =

I 0

0

3 0

2 0

0

- B

I 0

MAGNITUDE OF GAIN

GAIN - 0

GAIN • I l r

I GAIN - z m

0 < ILOGtoflGAIflIH < 1 [ <_ ILOGIoIIGAINIII < Z D/R, 10 k 2 <. ILOG]dlGAiNlll < ~. ]iK $ <_ ILOGIG(IGAINlll < 4


~z TALT • I0 k == !.2 d i g TALT" 4 0 k = 3.9 deg

4 <_ ILOGz01,~INI)I < 5 I 5 <_ ILOGIoIIGAHNIÁ I < 6 - 5 0 0

I 6 <. ILOGIoIIGAIN II < T 7< iLOGIot,GAINIIi < 8 ~ j / R , 40 k s<_ LOG[6(IGAiNiH < 9 - S / P , iO k t

I o % I • . - 5 0 0

PHUGOJD, ,0 K, 40 k ~ - ' " L , , ~ S /P, 4 0 k

I R 40 k ~ ¢ ' ' ~ . 3 0 0 I

I R, ick 500"'1 ,00 , : i : - , l l , * - ~ i -

- 5 0 0

I I

A E D C - T R - 8 0 - 1 1

FLIGHT, 5g

i 0 k = 5 . 3 deg =13.6 deg

4 0 k - 5 0 0 -

D/R, I0 k

DEGENERATI[D D/R, 4 0 w

5 O 0

I

m l b 5 0 0

- 5 0 0

S/P, 10 k ~,**" i , k DEGENERATED

SPIRAL, I O ' q ' ~ F S/P, 40 ~ O / R . ,O k 1 OVERDAMPED "~ ± / . ~oIk . - - - . / PHUGOID, lO K, 40~. . . .~ '~ ,~ SPIRAL ' 0 [

R. ,ok - ~ \ I ~ " ~ o o / I R, JO' O -'~OO.._"~, / I : : :',-:--~"""-.'.'.'.'.'.'.'.'~5o0 /

- 5 0 0


(2T k = 9.1 deg

DEGENERATED - - D / R , 4 0 k

- 6

0.1

I

5 0 0 S/P, IO k "7 " . ] 500 I -SOD

D/R, IO k O ~ / I ~ w e l p 4,~k D.EGENERATE

OVERDAMPED 1 " , . - - i' ' " I O ' R , 4 0 k P.uGo,o, .o~. 4ok~-"... sO'O l

R, Sp.R,, . R, IO i 500 ,o"

- 4 - 2 0 "- 5 0 0 2 4 R E A L PART

i I 0 . 2 5 I


b. M = 3 . 0 Figure 39 . Conc luded .

I I - I - 0 . 2 5


u

10

d 0

. 5 ; k l a .

u

o o

.5 i ;

O.

¢) u m

m cz o

-57. b,J O.

90

A E D C - T R - 8 0 - 1 1

bJ

n,,

'-r 0 l.- a .

0 . 5

0 . 4

0 . 3

0 0.2 v- rv 0.1

c~ z 0 (1.

:S -0 .1 c3

-o.~

- 0 . 3

- 0 . 4

"0"~500

I 0,000

8 , 0 0 0

SY._MM MAGNITUDE OF GAIN

LARGE z GAIN • 0

L A q ~ s GAIN • I

LARGi[ o GAlE - ± m

A 0 <_ ILOGIoIIGAINI)I < ] a I <_ ILOG[o(IGAIN])I < 2 , 2 <. iLOGI0(IGAINIII < v 3 < ILOG10(IGAINI)I < 4 , 4 <_- ILOGI0rlGAINIII < 5 , 5 < ILOGI~IGAINI)I < 6 , 6<. ILOG|0(IGAI,'~Ifl < '/ , 1< ILOG)0(IGAINIli < 8 .. 8 <_ ILOG]011GAiNlil < 9

[ C S/P

M = 3 . 0 0 A L T I T U D E = I 0 ~ 0 0 0 f t L E V E L F L I G H T , Ig

DAMPING RATIO

O/R DAMPING RATIO

\ I I

I I

\ - 4 0 0 - 3 0 0 - 2 0 0 - I 0 0 O 1 0 0

Crop , per/rod

a. Damping ratio

6 , 0 0 0 ' f ~ .J / _~ 4 , 0 0 0 I

°

"~ 2 , 0 0 0 ' . .J o / n- 0 - '

./ z o - a , o o o !

"-:, - 4 , 0 0 0 j' o / IO : : - 6 , 0 0 0 "

- 8 , 0 0 0

- I 0 , 0 0 0

Figure 42.

2 0 0 3 0 0 4 0 0 5 0 0

J

V

I

C r o p , per/rod

~ - 2 0 0 - 5 0 0

0

0 . 5

b.

1.0 I ,5 T I M E , sec

Time response

2 . 0

Yew-to-turn configuration - effect of Crop ratio and time response.

2 . 5

on damping

3 . 0

93

A E D C - T R - 8 0 - 1 1

O

,E

u 0.5 Q o

0.4

0.3 o z w 0 . 2

O w 0.1

i o

f f -o. i

- 0 . 2

z - O . 3 m

< - 0 . 4 a

-O 5oo

I 0 , 0 0 0

8 , 0 0 0

6 , 0 0 0

4 , 0 0 0

2 , 0 0 0

O

- 2 , 0 0 0

- 4 , 0 0 0

- 6 , 0 0 0

- 8 , 0 0 0

- I O , O O O

O

Figure 43.


LARGE x GAIN - 0

LARGE s GAIN - ]

LARGE o GAIN - • m

a 0 <_ ILCGIOIIGAINI)I < ! o ] <_ ILOGI011GAINI)I < Z ,* 2 <_ ILOGIo(IGAINlll < 3 o .5 <_ 'LOGLo(IGAINIII < 4 , 4<_ LOG1,0(IGAINIII < .5 . 5<_ LOGIGIIGAINI)I < 6 , 6<_ LOG]o(JGAINI)I < 7 - ,, 1 <_ ILOG]oIIGAINI)I < 8 ,, 8 <_ ILOG]oIIGAINIH < 9 -

t J D IR DAMPING RATIO I

t, ~ t , Ir ir 7*

~ S I Z . PHUGOID FREQUENCY P DAMPING RATIO

M = 3 . 0 0 A L T I T U D E = I 0 , 0 0 0 f f T U R N I N G F L I G H T , l O g

,,_m,'W ~

z'

THIRD ~ " OSCILLATORY M 0 D E'-,~ z~

- 4 0 0 - 3 0 0 - 2 0 0 - I 0 0 0 I 0 0

C n q , p e r / r o d

a. Damping ratio

200 300 4 0 0 500

h

C n q , p e r / r o d

, ~ - - 5 0 0 0

2 0 0

r

0 . 5 1.0 I . 5 2 . 0

T I M E , sec

b. Time response Yaw-to-turn configuration -e f fect of Cnq ratio and t ime response.

2.5

on damping

3.0

94

A EDC-TR-80-11

Trim Values

Table 1. Bank-To-Turn Configuration a. M = 0.8, Al t i tude = 10,000 f t

Level Turning Flight, 1 ~ Flight, 5

Turning Flight, i0

e (deg) 1.5 5.8 9.8

#(deg) 0 78.5

~(deg/sec) 0 10.5

84.3

21.3

C m (per rad) q

Cm. (per rad)

J

C (per rad) mp

C (per rad) m r

C L (per rad) q

CL. (per rad)

C n (per rad) r

C n (per rad) P

C n (per rad) q

C £r (per rad)

C% (per rad)

C (per rad)

C (per tad) Yr

C (per rad) Yp

-264.9 -286.6 -301.3

0 0 0

0 0 0

0 0 0

137.4 169.8 186.4

0 0 0

-66.3 -67.0 -66.8

-I.0 -7.4 -13.3

0 0 0

13.4 13.8 14.3

-62.6 -62.8 -62.5

0 0 0

0.3

-0.7

0.2

0.6

0.2

1.8

95

AEDC-TR-80-11

Trim Values

a ( d e g )

Table 1. Continued b. M = 0.8, Al t i tude = 40 ,000 ' f f

Level Turning Flight, 1 g Flight, 5

4.4 15.1

Turning Flight, i0 g

24.7

#(deg) 0 78.5

~(deg/sec) 0 11.8

84.3

23.9

C (per tad) m q

Cm~(per rad)

C m (per rad) P

C (per rad) m r

C L (per rad) q

CL. (per rad)

C n (per tad) r

C n (per rad) P

C n (per rad) q

C£ (per rad) r

C£ (per rad) P

C£ (per tad) q

C (per tad) Yr

C (per rad) Yp

-274.8 -311.2 -330.5

0 0 0

0 0 0

0 0 0

157.6 205.4 231.0

0 0 0

-66.9 -65.4 -60.8

-5.3 -20.9 -33.3

0 0 0

13.7 15.2 17.0

-62.7 -59.7 -58.7

0 0 0

0.3

0.2

0.1

3.5

-0.i

6.3

96

A E D C - T R - 8 0 - 1 1

Trim Values

Table 1. Continued c. M = 1.10, Al t i tude = 10 ,000

Level Turning Turning Flight, 1 ~ Flight, 5 9 Fli~ht, 10 g

u (deg) 0.i 3.5 6.0

(deg) 0 78.5

(deg/sec) 0 7.6

84.2

15.4

C m (per rad) q

Cm. (per rad)

C m (per rad) P

C (per rad) m r

C L (per rad) q

CL~ (per rad)

C n (per rad) r

C (per rad) n P

C n (per rad) q

C£ (per rad) r

C£ (per rad) P

C£ (per rad) q

C (per rad) Yr

C (per rad) Yp

-335.7 -343.3 -367.3

0 0 0

0 0 0

0 0 0

153.0 167.4 184.6

0 0 0

-64.6 -65.1 -65.3

-0.1 -3.3 -6.6

0 0 0

i0.0 10.2 10.5

-55.8 -56.4 -57.1

0 0 0

0.3 0.3 0.3

-i.0 -I.0 0.6

97

A EDC-TR-80-11

Trim Values

(deg)

Table 1. Continued d. M = 1.10, Altitude = 40,000 ff

Level Turning Flight, 1 g Flight, 5 g

2.6 9.3


16.2

(~ (deg) 0 78.5

(deg/sec) 0 8.5

84.2

17.4

C m (per rad) q

C (per rad) m,

C (per ra'd) m P

C m (per tad) r

C L (per rad) q

CL~(per rad)

C (per tad) n r

C (per rad) n P

C n (per rad) q

C£ (per rad) r

C¢ (per tad) P

C~ (per rad) q

C (per tad) Yr

Cyp (per rad)

-339.7 -406.0 -425.8

0 0 0

0 0 0

0 0 0

161.2 203.4 216.3

0 0 0

-65.0 -65.0 -62.7

-2.2 -i0.8 -19.2

0 0 0

i0.i 10.9 12.4

-56.1 -58.0 -53.6

0 0 0

0.3

-0.3

0.2

1.5 3.4

98

A E D C - T R - 8 0 - 1 1

Trim Values

Table 1. Continued e. M = 3.50, Al t i tude = 10 ,000 f t

Level Turning Turning Flight, 1 ~ Fli~ht, 10 g Flight, 25 g

a (deg) 0.2 I. 1 2.6

¢(deg) 0 84.3

~(deg/sec) 0 4.9

87.7

12.2

C (per tad) m q

C (per rad) m-

C m (per rad) P

C m (per rad) r

C L (per tad) q

CL~(per rad)

C (per rad) n r

C (per rad) n P

C (per rad) n q

C£ (per rad) r

C£ (per rad) P

C£ (per rad) q

C (per rad) Yr

C (per rad) Yp

-227.0 -228.1 -230.4

0 0 0

0 0 0

0 0 0

122.0 122.4 123.5

0 0 0

-31.7 -31.8 -32.0

0.7 0.I -0.9

0 0 0

4.5 4.6 4.6

-40.5 -40.6 -40.8

0 0 0

0 .2 0 .2 0 .2

- 0 . 7 - 0 . 6 - 0 . 3

99

AEDC-TR-80-11

Trim Values

Table 1. Conduded f. M = 3,50, Al t i tude = 40 ,000 f t

Level Turning Turning Flight, 1 g Flight, i0 g Flight, 25 9

~ (deg) .5 3.6 8.4

(deg) 0 84.3

(deg/sec) 0 5.5

87.7

13.7

C m (per tad) q

Cm. (per rad)

C m (per rad) P

C m (per rad) r

CLq (per rad)

CL~ (per rad)

C (per rad) n r

C n (per rad) P

C n (per rad) q

C~ (per rad) r

C£ (per rad) P

C£ (per rad) q

C (per rad) Yr

C (per rad) Yp

-227.4 -232.3 -252.2

0 0 0

0 0 0

0 0 0

122.1 124.7 131.4

0 0 0

-31.7 -32.1 -32.4

0.6 -1.6 -4.9

0 0 0

4.5 4.7 4.9

-40.5 -41.0 -42.0

0 0 0

0.2 0.2 0.3

-0.7 -0.i 1.0

100

A E DC-TR-80-11

Table 2. Yaw-To-Turn Configuration a. M = 1 . 3

Trim Values Altitude = I0,000 ft, Level Flight, 1

6.5

Altitude = 20,000 ft, Level Flight, 1 g

9.3

0

0

C m q

C m .

C m P

C m r

C L q

C L .

C n r

C n P

C n q

C£ r

C £ P

C£ q

C Yr

C Yp

-409.7

-389.3

-4.1

-4.9

0

0.8

-398.7

-309.7

- 5 . 1

"~ 0

- 5 . 1

0 . 8

IOl

AE DC-TR-80-11

Trim Values

Table 2. Continued b. M = 3.0, Al t i tude = 10,000 f t

Level Turning Fli~ht, 1 ~ Flight, 5

1.2 5.3


9.1

0 78.5

0 2.8

84.3

5.7

C m q

C m ,

C m

P

C m r

C L q

C L .

C n E

C n P

C n q

C£ r

C£p

C£ . q

Cy r

C Yp

-298.0 -287.7 -302.9

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

-309.7 -307.0 -244.5

-0.8 -0.5 -i. 2

0 0 0

0 0 0

-3.2 -3.6 -3.7

0 0 0

0 0 0

-0.6 -0.2 -0.1

I02

A E DC-TR-80-11

Trim Values

(1

Table 2. Concluded c. M = 3.0, Alt i tude = 40 ,000 ft

Level Turning Flight, 1 9 Flight, 5

Turning Flight, i0 9

3.9 13.6 22.5

0 78.5

0 3.2

84.3

6.4

C m q

C m .

Cmp

Cm r

cL&

C n r

C n P

C n q

C£ r

C~ P

C£ q

C Yr

C Yp

-287.9 -303.8 -301.8

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

-310.0 -182.7 -172.6

-0.5 -3.0 -6.5

0 0 0

0 0 0

-3.5 -3.7 -4.1

0 0 0

0 0 0

- 0 . 2 0 .1 1 .0

103

AEDC-TR-80-11

Table 3. Missile Physical and Mass. Characteristics

Mass

Ix(body axis)

Iy(bOdy axis)

I z (body axis)

IXZ (body axis)

s

d

cg

Bank-to-Turn

5.75 slugs

0.34 slug-ft 2

34.1 slug-ft 2

34.15 slug-ft 2

0.0 slug-ft 2

0.458 ft 2

0.7633 ft

8.9 ft

50 percent body length

Yaw-to-Turn

43.80 slugs

9.4 slug-ft 2

745 slug-ft 2

745 slug- ft 2

0.0 slug-ft 2

I. 54 ft 2

1.4 ft

14 ft

50 percent body length

104

Longitudinal

APPENDIX

AE DC-TR-80-11

EQUATIONS DEFINING THE TOTAL AERODYNAMIC DATA IN THE STABILITY AXIS SYSTEM

F x = ~S[CD(a, 8p, M)]

F Z = yS CL(a, 8p, M) + CLq(~ ~';\2-V/_I

2v / mA2v / J

Lateral-Directional

where

C M / r d \

• r

qd

[CI~(a,M) ~ (pd) M x = ~Scl + C~sy(a , M)Sy + Cl~3r(a'M)Sr - C~p(a,M) ~,

+ Cfp(a,M) ~ + C~,.q

P

~R =

~y =

Pitch Control

Roll Control

Yaw Control

105

AEDC-TR-80-11

Bank-to-Turn Missile

a -6 to 26 deg

M 0.8 to 3.5

6 -10 to 20 deg P

Yaw-to-Turn Missile

Data Matrix Variables

a 0 to 24 deg

M 0.6 to 3.0

-20 to 20 deg P

-I0 to 20 deg P 6 R -20 to 20 deg

-20 to 20 deg Y

Control Authority

-20 to 20 deg P ~R -20 to 20 deg

-20 to 20 deg Y

I06

A E DC-TR-80-11

NOMENCLATURE

(Note: All aerodynamic data are referenced to the stability axis system.)

CD Drag-force coefficient, drag force/~S

CL Lift-force coefficient, lift force/~S

CLq Derivative of lift-force coefficient with respect to pitch rate, 8CL/8(qd/2V), per radian

CL& Derivative of lift-force coefficient with respect to &, 8CL/O(&d/2V), per radian

Ctp Derivative of rolling-moment coefficient with respect to roll rate, 8Ce/O(pd/2V), per radian

Ceq Derivative of rolling-moment coefficient with respect to pitch rate, 8C~/8(qd/2V), per radian

Ctr Derivative of rolling-moment coefficient with respect to yaw rate, #Cr/8(rd/2V), per radian

C~ Derivative of rolling-moment coefficient with respect to ti, 8Ce'8(t~d/2V), per radian

Ce a Derivative of rolling-moment coefficient with respect to angle of sideslip, #Ct/a/3, per radian

ce~ Derivative of rolling-moment coefficient with respect to /~, 8Ce'#(/~d/2V), per radian

Ce/~ R

C%

Cm

Cmp

Cmq

Derivative of rolling-moment coefficient with respect to ~R, 8Cf/86R, per radian

Derivative of rolling-moment coefficient with respect to 6y, #Ce/~96y, per radian

Pitching-moment coefficient, pitching moment/~Sb

Derivative of pitching-moment coefficient with respect to roll rate, 8Cm/8(pd/2V), per radian

Derivative of pitching-moment coefficient with respect to pitch rate, 8Cm/a(qd/2V), per radian

107

AEDC-TR-80-11

Cmr

Cm~

Cm~

Cn

Cnp

Cnq

Cnr

Cn&

Cn~

Cn~

Cn8 R

C 'nBy

C~p

Cy r

Cy~

Cy(~y

Derivative of pitching-moment coefficient with respect to yaw rate, 8Cm/a(rd/2V), per radian

Derivative of pitching-moment coefficient with respect to &, 8Cm/8(&d/2V), per radian

Derivative of pitching-moment coefficient with respect to ~, 8Cm/O(~d/2V), per radian

Yawing-moment coefficient, yawing moment/~Sd about missile cg

Derivative of yawing-moment coefficient with respect to 8Cn/8(pd/2V), per radian

roll rate,

Derivative of yawing-moment coefficient with respect to pitch rate, t3Cn/O(qd/2V), per radian

Derivative of yawing-moment coefficient with respect to yaw rate, 8Cn/8(rd/2V), per radian

Derivative of yawing-moment coefficient with respect to &, 8Cn/8(&d/2V), per radian

Derivative of yawing-moment coefficient with respect to angle of sideslip, aCn/8~, per radian

Derivative of yawing-moment coefficient with respect to/3, 8Ca/a03d/2V), per radian

Derivative of yawing-moment coefficient with respect to 6R, 8Cn/t~R, per radian

Derivative of yawing-moment coefficient with respect to 6y, t~Cn/0~y, per radian

Derivative of side-force coefficient with respect to roll rate, 8Cy/8(pd/2V), per radian

Derivative of side-force coefficient with respect to yaw rate, 8Cy/8(rd/2V), per radian

Derivative of side-force coefficient with respect to angle of sideslip, 8Cy/813, per radian

Derivative of side-force coefficient with respect to 6y, aCy/8~y, per radian

108

cg

D/R

d

Fx ,

Fy

Fz

g

Ix,lv,lz

Ixz

J

M

Mx

My

Mz

m

p,q,r

S

S/P

V

Wn, W

Z

(X

AEDC-TR-80-11

Center-of-gravity location, percent body length

Dutch roll

Reference body diameter, ft

Force acting along X-stability axis, Ib

Force acting along Y-stability axis, lb

Force acting along Z-stability axis, lb

Acceleration of gravity, ft/sec 2

Moments of inertia about X-, Y-, and Z-body axes, respectively, slug-ft 2

Product of inertia, slug-ft 2

Imaginary number, -1

Mach number

Moment acting about X-stability axis, ft-lb

Moment acting about Y-stability axis, ft-lb

Moment acting about Z-stability axis, ft-lb

Mass, slugs

Components of fl about X-, Y-, and Z-body axes, respectively, radian/sec

Dynamic pressure, •V2/2, lb/ft 2

Body reference area, ft 2

Short period

Total velocity, ft/sec

Undamped natural frequency, radian/sec

Damping ratio

Angle of attack, deg

Angle of side force, deg

109

AEDC-TR-80-11

~p

6a

Q

(I

Pitch control deflection, per radian

Roll control deflection, per radian

Yaw control deflection, per radian

Air density, slug/ft 3

Real part of complex variable

Angle of roll, deg

SUPERSCRIPT

Derivative with respect to time

K 1,000 of feet

110

missile motion sensitivity to dynamic stability derivatives. · aedc-tr-80-11 missile motion...

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