# Full text of "Study on using a digital ride quality augmentation system to trim an engine-out in a Cessna 402B"

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I: i! I (N&SA-CB- 177272) SIOOX ON OSIIIC i nrnTw.T HC A03/HP iOI ' '^ CSCI OIC G3/08 1 I — J ' ''! ...I 1186-26342 Dnclas 43603 I I ThE UNlvERsliy of Kansas Center For ResearcN, Inc. 2291 Irving Hill Drive-Campus West Lawrence. Kansas 66045 ■ft ■■ V i \ t ■ ■:\ STUDY ON USIH6 K DIGITAL ftlDE QUALITY AUGMENTATION SYSTEM TO TRIM AN ENGlNE-OUT IN A CESSNA 40 2B XU-FtU.-6132-3 by: K«nt E. Donaldson Supervisad by: Dr. Jan Roskam THE UNIVERSITY OF KANSAS CENTER POR RESEARCH, INC. Plight Raatarch Laboratory 'i Lawranca, Kansas t . .i.;.' STUDY ON USING A DIGITAL RIDE QUALITY AUGMENTATION SYSTEM TO TRIM AN ENGINE-OUT IN A CESSNA 402B ABSTRACT A linear model of the Cessna 402B was used to determine if the control power available to a Ride Quality Augmentation System was adequate to trim an engine-out. Two simulations were completed: one using a steady-state model, and the other using a state matrix model. The amount of rxidder available was not sufficient in all cases to completely trim the airplane, but it was enough to give the pilot valuable reaction time. The system would be an added measure of safety for only a relatively small amount of development. f\v(' \ r -M I r./\V' • \' TABLE OF CONTENTS ABSTRACT i LIST OF FIGURES iii LIST OF TABLES iv LIST OF ABBREVIATIONS AND SYMBOLS v 1. INTRODUCTION 1 2. RIDE QUALITY AUGMENTATION SYSTEM 2 3. LINEAR STEADY-STATE MODEL 2 3.1 STEADY-STATE EQUATIONS 5 3.2 COORDINATE REFERENCE SYSTEM 6 3.3 DEVELOPMENT 6 3.4 RESULTS B 4. SMALL PERTURBATION MODEL 10 4.1 EQUATIONS OF MOTION 10 4.2 COORDINATE REFERENCE SYSTEM 12 4.3 DEVELOPMENT 12 5. DISCUSSION OF RESULTS 15 r 6. CONCLUSIONS AND RECOMMENDED RESEARCH .... 18 6.1 CONCLUSIONS 18 6.2 RECOMMENDED RESEARCH 19 I REFERENCES 20 APPENDIX A Cessna 402B Stability Derivatives . . A.l [• ii ■4 LIST OF FIGURES 1 Cessna 402B Three-view 3 2 Airplane Coordinate Systems •» 3 Rudder required for an engine-out on a Cessna 402B 9 4 Time history of an engine-out in a RQAS controlled Cessna 4C2B 14 5 RQAS Block Diagram 1"^ A.l Approximate Cessna 402B Lift-Curve A. 2 A. 2 Propeller Efficiencies A. 4 iii LIST OF TABLES 1 Control Surface and Actuator Requirements . . 4 2 Small Perturbation Matrices li A.l Lateral-Directional Non-dimensional Stability Derivatives A. 3 A, 2 Variation of Derivatives with Vertical Tail Size A, 6 A. 3 State Matrices A. 7 A. 4 Lateral-Directional Dimensional Stability Derivatives A. 8 A. 5 Modified Lateral-Directional Dimensional Stability Derivatives A. 9 iv LIST OF ABBREVIATIONS AM) SYMBOLS ABBREVIATIONS PCI Flight Condition #1 ICAD Interactive Control Augmentation Design RQAS Ride Quality Augmentation System SYMBOLS A Continuous State Matrix b wing Span, ft B Continuous Control Matrix Cig Variation of Rolling Moment Coefficient with Sideslip Angle, rad"^ Ci Variation of Rolling Moment Coefficient with "DP Differential Flap Deflection, rad"^ Cig Variation of Rolling Moment Coefficient with Rudder Deflection, rad"^ Cnp Variation of Yawing Moment Coefficient with Sideslip Angle, rad^^ Cn. Variation of Yawing Moment Coefficient with "DP Differential Flap Deflection, rad"^ Cn. Variation of Yawing Moment Coefficient with "R ♦ Rudder Deflection, rad"^ Cyp Variation of Side Force Coefficient with Sideslip Angle, rad'^ Cy Variation of Side Force Coefficient with "DP Differential Flap Deflection, rad'^ P R 9 variation of Side Force Coefficient with Rudder Deflection, rad" Side Force due to Asynunetrlc Thrust, lb Acceleration of Gravity, ft/sec^ L^ Rolling Moment due to Asymmetric Thrust, ft lbs Airplane Mass, slugs Yawing Moment due to Asymmetric Thrust, ft lbs Yawing Moment due to Drag of Inoperative Engine, ft lbs Perturbed Roll Rate, rad/sec Dynamic Pressure, lb/ft Perturbed Yaw Rate, rad/sec Reference (Wing) Area, ft^ u Control Vector Vmc Minimum Control Speed, ft/sec Vs Stall Speed, ft/sec^ X State Vector GREEK SYMBOLS Sideslip Angle, deg, rad Glide Path Angle, deg, rad 5jjp Differential Flap Deflection, deg, rad iJ^ Rudder Deflection, deg, rad ^ Bank Angle, deg, rad m Nt ANd p q r S If vi 1. UtTRODUCTIQN Due to the large increase in the commuter airline industry, with the federal deregulation of major carriers in 1978, more people are riding in small, short-haul, propeller-driven aircraft. This has caused an increased effort to make riding in such an aircraft as pleasant as possible. One project undertaken has been the development of a Ride Quality Augmentation System (RQAS). This system uses acceleration feedback to lessen the efscts of turbulence by counteracting the undesirable accelerations with appropriate separate surface control deflections. It was felt by the developers of this system that since the control power was available, it would also be beneficial to use this system to trim the airplane in the event of an engine-out. This was the basis for the following investigation into using a RQAS to control a Cessna 402B in an engine-out flight condition. The investigation was conducted as part of NASA grant NAGl- 345. Mr. Lou Williams is the grant monitor. Chapter 2 of this report describes the proposed RQAS for the Cessna 402B. The models used to evaluate the system in an engine-out are then given: the steady- state model in Chapter 3 and the small perturbation model in Chapter 4. The results of using these models for an engine«*out are then discussed in Chapter 5. 2> RIDE QOALITT AUGMEWTATION SYSTEM Reference 1 documents the proposed control surface modifications for the RQAS in the Lateral-Directional mode (see Figure 1): 1. Replace the outboard section of the split flap with a differentially deflecting plain flap that can deflect -*-15 to -45 degrees. 2. Use the entire existing rudder (limiting the RQAS range of deflections to +5 degrees). The outboard flaps were computed to have 67% of the control power of the ailerons in the nonlinear model. Table 1 summarizes the control surface and actuator requirements for the RQAS of reference 1. 3. LINEAR STEADY-STATE MODEL A nonlinear simulation model of the Cessna 402B (Ref . 2) was used to obtain a linear model about its most critical condition for an engine out. That is: 1. sea level; 2. flaps down; 3. full throttle; 4. maximum landing weight. X-BOOY AXIS SASdOM) IJ2 PIGURS 1 csiana 402B Thr«a-vitiw I Table 1 Control Surface and Actuator Requiremonta Conti'ol Surface Deflection and Rate Limits Surface Deflection Range (deg) Maximum Rate(deg/sec) SS Elevator ±5 50 Flap (inboard) fl5 to -45 120 Differential Flap ♦15 to -45 120 Rudder^ +32 50 ■1 Actuator Requirements Surface Max Load (lbs) Speeed (in/sec) SS Elevator 65 3.50 Flap (inboard) 750 8.50 Differential Flap 380 8.50 Rudder 520 3.50 Stroke (in) 0.75 4.25 4.25 4.50 ^ The deflection is for the standard rudder, uses a deflection range of +5 degrees. The RQAS I II I H PIII t !! ]■ This will be called Flight Condition »1 ^FCl) and is defined as: One Engine Out V « 130 fps .. - ft Full Throttle Xcg - 0.25 W - 6200 lbs Full Flaps The stability and control derivatives given in Appendix A, Table A.l are for the maximum landing weight. Because the maximum landing weight differs from thi' maximum takeoff weight by less than 2%, the 7^ alues were used without correction. Pl'ni, \iiditior. #1 would be the condition in an emerge: ^y go-around. 3.1 STEADY-STATE EQUATIONS The basic assumption made to determiae needed control surface deflection was that the airplane motion could be modelled about a steady-state point as a set of first-order differential equations, as shown in Bqn. 2.1 for an engine-out flight condition. "Cyg Cy« DF Clg Ci DF Cl,. ^DPr «R ^- (mqsin»cosY ♦ FyT^ qSb (2.1) - (Nt » ANp) qsD B ©DP 6^ These equations have been uncoupled from the full 6 degree of freedom equations by choosing bank angle, ^. They are written in the stability axis system (see p ' section 3.2). Their derivation can be found in reference 3. 3.2 COORDINATE REFERENCE SYSTEM The body-axis system is an orthogonal, right-hand set of axes with its origin at the airplane's center of mass. The X-axis lies along the center line of the body. The X- and Z-axes lie in the airplane plane of symmetry, while the Y-axis is pointed out the right ving of the airplane. This can be seen in Figure 2. The non-dimensional derivatives listed in Appendix A, Table A.l are given in the stability-axis system. This system is also an orthogonal, right-hand set of axes with its origin at the canter of mass of the airplane. The difference between the body- and the stability-axis systems is that the stability X-axis is oriented in the direction of the steady-state velocity of the airplane on its XZ-plane. They both share the same Y-axis as seen in Figurf 2. 3.3 OEVfiLOPMENT The thrust was calculated using an engine model and » its average propeller efficiency. The average propeller efficiency is greater than the actual propeller efficiency by approximately 6%; therefore, the actual pitching moment and yawing moment during engine-out Body Axis Coordinate System Stability AmU Coordtnat* Syttam FIGURE 2 Airplane Coordinata Systams I ■ '\ would be slightly less. The propeller efficiencies are given in Appendix A, By assuming a weight and flight condition, the angle of attack was obtained from the airplane lift- curve slope and intercept as shown in Appendix A. Some of the derivatives are functions of angle of attack. Once the angle of attack was calculated, the non- dimensional derivatives were obtained from reference 2. They are given in Appendix A. By varying the speed and the vertical tail size and solving Equation 2.1 as shown in Appendix A, the sideslip, rudder, and differential flap deflections were determined. Figure 3 shows how rudder deflection varies with flight condition. 3.4 RESULTS It can be Seen from Figure 3 that at speeds below approximately 125 fps, there is not enough rtidder to keep the airplane in straight-line flight. This is the minimum control speed, Vmc* This is 15 fps less than the minimum control speed' given in the operating handbook; therefore, a minimum control speed of 130 fps is still conservative. At 130 fps, it was found that to fly straight with a bank angle, ^, of -5* required 30* of rudder deflection, 7.5* of differential flap deflection, with a sideslip of -3*. The pedal force 8 CESSNA 402SC9AS/C) Ws 4500 It X..3»o.i5 SEh LEVEL FULL THROTTLE 0N£ ENC^INEaur /WAX. RUDDER ^EFLfCT/ON FACTbfi. TlfhES EKivriN^ y.T SIZE ' o.so MO fho ISO 200 Sy«42.0ft2 PI6URS 3 Ruddtr rsquirtd for an angi no-out on a Caasna 40 2B ■.A required was also found to be within federal regulations. It can also be seen from Figure 3 that if one- fourth of the vertical tail were removed, the airplane would no longer meet PAR 23 requirements. PAR 23 requires that the minimum control speed, Vmc be greater than 1.2Vs. This model does not account for any transient phenomena of the airplane in reaching its steady-state condition. 4. SMALL PERTt»BATlON MODEL The non- linear simulation model was also used to develop a linear model to study the dynamic behavior of the Cessna 402B in open- and closed-loop simulations. This was only done for FCl. 4.1 EQUATIONS OF MOTI(»f From the assximptions made in the steady-state model, a new set of matrix equations can be written: X - A X ♦ B u (3.1) where X' - {B» P, r, ♦}, and u' - {6dp» «r}. Derivations of this equation can be found in reference 4. These matrices are defined in Appendix A. The matrices used for FCl are given in Table 2. 10 ir''" Table 2 small Perturbation Matrices Cessna 402B (PCI) f I B •f 0.105 -0.000867 -0.9899 0.245 1.329 -1.752 0.483 0.0 1.408 -0.0428 -0.299 0.0 0.0 1.0 0.149 0.0 0.0 0.030r •0.968 0.206 0.0603 -0.835 0.0 0.0 __ 0.0 " -0.0468 0.44S 0.0 11 Equation 3.1 is written with the following assumptions: 1. perturbations are small, and 2. initial condition is a straight-line trimmed flight condition. 4.2 COORDINATE REFERENCE SYSTEM All Of the dimensional and non-dimensional derivatives were calculated in the stability-axis system of Figure 2. The instrxjments onboard the airplane will sense the body-axis motion. This can be simply transformed to the stability-axis by rotating about the Y-axis by the airplane angle of attack. 4.3 DEVELOPMENT To simulate an engine-out situation using the state matrices, a disturbance matrix, D, was added to the state and control matrices in Equation 3.1 yielding Equation 3.2. This disturbance matrix was made up of the constant angular accelerations imparted on the airplane in the pitch and yaw directions due to the engine-out. In all the cases, this matrix was commanded to "turn on" at one second into the simulation. I x-Ax + Bu-fDw (3.2) where w » {0 or 1} The open-loop response of the airplane with the addition of the disturbance matrix was calculated using 12 1 I [■ l lip I I II lyipTjppUpWWi^PW^WI I I "'f "" - ^" "" " ■■■ .!,. ■,. . ^ .^ m ... i m ^pi p nn n i^ ^ M " ipw the Interactive Control Augmentation Design program (ICAD)[5]. This was done on the University of Kansas School of Engineering's Harris computer system. The closed-loop response was also done on ICAD. As allowed by FAR 23, a bank angle of -5^ was used to lower the rudder required as much as possible. This turned out to be a difficult task on ICAD as currently written, tdeally, bank angle, sideslip, and yaw rate should be driven to -5®, 0, and simultaneously. To get these, only bank angle was commanded and all the variables except roll rate were weighted heavily. This gave large transient values for control positions and rates, but in an actual system this would not happen. Figure 4 shows an example time history. The average values of control deflections and control rates have been drawn to show an approximation of what would actually happen. It can be seen that to trim the airplane requires all 32^ of the rudder to be available to the RQAS. As proposed, only 5^ of rudder deflection is available to the RQAS; therefore, time histories were also simulated allowing only 5 and 15 degrees of rudder deflection. With 15^ of rudder, the RQAS would be able to trim an engine out at speeds above about 170 fps. Five degrees 13 Time t, (Seconds) Time t, (Seconds) FIGURE 4 Time history of an engine-out in a RQAS controlled Cessna 402B 14 ,,j ♦ only slows the rates and could not trim the airplane below its "never exceed" speed. 5. DISCUSSION OP RESULTS The control deflections obtained for FCl were: Model B 6d 6nF (deq) (deq) (deq) Steady-State -2.9 30.4 7.5 Small Perturbation -5.0 30.0 5.0 These values were obtained for a steady-state velocity of 130 fps. The difference in the values between the two models is due to the fact that in the small perturbation model, yaw and pitch rate are not exactly zero. The bank angle is also not quite -5". It can also be noted from Figure 4 that the RQAS has the airplane trimmed in a fraction of a second, much faster than a human pilot could react. This result is obtained assuming the servo is modelled as a first-order lag. The slowing of the yaw rate due to only 5* of rudder deflection would give the pilot valuable time to » react to the increasing sideslip. In addition, the differential flaps have far more control power than is needed to control roll and bank angle. The quicker the airplane reaches a bank angle of -5*, the longer the pilot has to respond to the yaw. 15 The RQAS has two obvious advantages. First, if the dynamic pressure is high enough and the available rudder is adequate, the RQAS could control an engine-out without the pilot touching the controls. Second, the RQAS would give the pilot more time to react to the engine-out even if the RQAS was not able to trim the airplane entirely. In addition, the RQAS wii.l introduce digital con^uters into commuter airplanes which could then be used for many other jobs which of themselves do not warrant the expense of a computer. If only 5* of rudder deflection is deemed adequate, no modifications need to be made in addition to those proposed for the RQAS. The RQAS block diagram is shown in Figure 5. This system would treat the engine-out as a large disturbance. The gains chosen in the normal use of the RQAS, however, might not be suitable for the engine-out. In that c&se, engine-out sensors would be required to tell th« system when to change gains. Giving more control power to the system for an engine-out could be done in two ways. First, the RQAS could be given the authority to move the rudder more than 5*. This would require a proportionately stronger actuator, but it would not change the stroke given in Table 1 which is for 32». Secondly, an engine-out 16 Q^ Hi -J -4 5 o h 0^ ^^ g «.« Q a VJ i u U o s 3 in 17 n^ I ■•.***■ sensor could trigger a change In the amount of rudder I allowed so that the RQAS Is still only allowed 5* while j the englne-out Is allowed more. ' NO additional software will be required If gain \ scheduling Is not needed or Is already Incorporated In ' the RQAS. The components needed for this system are a hydraulic pump to be mounted on one of the engines, an •: accumulator, and a set of electro-hydraullc actuators. :' The design of the actuator and Its Implementation will n be contracted to Cessna, in addition, a set of englne- out sensors will be required. Reliability, ease of [ Implementation, and cost of these should be r Investigated. 1 T fi. CONCLUSIONS AND RECOMI^IDSD RESBARCH 6.1 CONCLUSIONS j The following conclusions were reached: 1. The RQAS with control of all 32- of rudder ) deflection can trim an englne-out in any steady- f state flight condition.^ /' 2. With only the proposed 5* of rudder deflection, • i the RQAS can slow the divergence of the airplane i. ■' \. significantly. {■'' 3. using only the proposed control surface V i modifications. Implementing the englne-out i i ■li n 18 i ': 1: ■.it" capability would require a set of englne-out sensors and the Incorporation of gain scheduHng. 4. If It Is deemed necessary to use more than the 5 degrees of rudder needed by the RQAS, the actuator would need to be proportionately stronger. 6.2 RECOMMENDED RESEARCH Further development of this project must Include the following steps: 1. Perform a preliminary failure analysis for rudder deflections greater than 5 degrees. 2. Use the gains required at the most critical condition to see their effect In other conditions, 3. Examine and evaluate options available to sense an englne-out, 4. Investigate the possibility of Increasing the amount of rudder deflection available to the RQAS, 5. Design and build the necessary hardware, 6. Perform a flight test of the system on the Cessna 402B. 19 Referenceat 1. Downing, D.R.j Davis, D.J.; Llnse, D.J.? Bntz, D.P.: "Prjllmlnary Control Law and Hardware Designs For a Ride Quality Augmentation System For Commuter Aircraft", NAGl-345, Feb 1986. 2. Hoh, R.H.; Mitchell, D.G.; Myers, T.T.: "Simulation Model o£ Cessna 4028". NASA CR 152176, July 1978. 3. Roskam, J.j Airplane Flight Dynamics and Automatic Flight Controls, Part I . Roskam Aviation ana Engineering Corp. 1979. 4. Roskam, J.: Airplane Flight Dynamics and Automatic Flight Controls. Part II . Roskam Aviation ano Engineering Corp. 1979. 5. Hammond, T.A.; Amin, S.P.; Paduano, J.D.; Downing, D.R. ! Design of a Digital Ride Quality Augmentation System For Commuter Aircraft, NASA CR 172419, Oct 1984. 6. Davis, D.J., "A Comparison of Two Optimal Regulator Design Techniques for the Weighting of Output Variables Which are Linear Combinations of States and Controls", M.S. Thesis, The University of Kansas, Lawrence, KS, 1986. 7. Hoak, D.B. et al? USAF Stability and Control DATCOM Wright Patterson Air Force Base, Ohio, 45433, April, 1976. 20 ; 1 APPENDIX A Cessna 402B Stability Derivatives This appendix gives the method used for estimating the airplane angle of attack and determining its stability and control derivatives. Because weight, wing area, and dynamic pressure are known, the airplane lift coefficient can be found from the equation: W - Cl q S Prom this lift coefficient, the AC^ due to flaps from reference 2 was subtracted. This lift coefficient was then found on Pigure A.l and its corresponding angle of attack was read. The airplane angle of attack was then used to obtain the non-dimensional derivatives from reference 2. These are listed for an angle of attack, a, of 8.5* in Table A.l. The engine model was taken from reference 2. This gave the maximum power of a C402B engine as 300 brake horsepower. The propeller efficiencies, average and actual, are given in Pigure A. 2. Thrust was then calculated using the follbwing equation: t T - 550 BHP np (A.l) V where np " Propeller Efficiency V ■ Airplane Speed BHP * Engine Brake Horsepower A.l i : Figure A.l ApproxinaU Caisna 402B(Tail-offf) Lift-Curv« A. 2 Table A.l Lateral-Directional Non-dimensional Stability Derivatives : Li One Engine Out Pull Throttle l^ill Flaps Cessna 402B V - 130 fps teg 0.25 h • ft W » 6200 lbs ! I Derivatives Cyg a -0.670 rad~' Cyp » -0.00063 CyfiR ' • 0.195 rad"^ Cyr = 0.42 Cy6DP ' • 0.0 rad~^ Cnp = -0.084 Cn0 » 0.129 rad"^ Cnr = -0.170 CnfiR - • -0.0795 rad"^ Clp • -0.81 CnfiDP " • 0.0057 rad"^ Clr « 0.216 Cl0 - . -0.0888 rad"^ ClfiR - C1*DP ■ > 0.0146 > -0.0685 rad"^ rad-^ I; \i A. 3 soo tta* AifSpMtf (tt/«M) Propeller Efficiency at Several Plight *^ Conditions I / _^ Cstrepolotim fram Pig. IS4 in Oi«M /-"^ in owa ■ f4h 'lOO So 20 ISO 200 280 300 TrMAirapttdtft/Me) Assumed Average Propeller Efficiency for All Plight Conditions Plgure A.2 Propeller Efficiencies A.4 I ") r From Figure 1, it can be seen that the thrust from one engine creates a moment about the airplane center of [ gravity with a moment arm of 7^50 feet* The orientation i of the thrust line of one of the engines is given in I reference 2. By finding the X and 2 components of the thrust y the moments, % and L^, can be found from: ' % « 7,50 Txf and f Lt « 7.50 T^. Reference 2 also gives the ACp due to an engine-out. ANd in Equation 2.1 is the yawing moment due to drag on the inoperative engine and is: ANd * 7.50 ACd q S Equation 2.1 was then solved to obtain the sideslip, differential flap deflection, and rudder deflection required. To account for changing the vertical tail size, the non-dimensional derivatives were recalculated and are sxommarized in Table A. 2. The dimensional derivatives were then calculated as shown in Table A. 3. These were calculated using: Ixx • 11100 slug ft^ Izz « 14900 slug ft^ 1x2 ■ -5B3 Slug ft^ To use these in the state matrices. Table A. 4, they must be in the form shovn in Table A. 5, where, Ai - Ixz/Ixx and Bi « Ixj/Izz • A. 5 J^ Table A. 2 variation of Derivatives witft Vertical Tail Size Cessna 402B (FCl) Relative V.T. Size ="6 deg"^ dec"* 1 ^"*^. i dec - ! ^ 0.50 0.00085 Q.CCiaE ; -C. 000-0 i I * jQ^iSr 0.75 0.00165 0.0024E i -C. 00104 ■ i . 000232 1.00 0.00240 C.0023C 1 -0.0013? C-00C3I: 1.25 0.00216 0.00413 1 -0.00174 C. 000286 Note: C^ ' *^ tJP « -0.00062 deg~i vas calculated froxr ref. A. 6 Table A. 3 state Matrices :1 ',1 B L*J I ^i V Yr' y/ LS' Lp' Lr X " P Nfi' Np Nr r 1 tanQi ♦ 7 B 6df 'sr i i A.7 I «i Table A. 4 Lateral-Directional Dimensional Stability Derivatives [31 q.SC ^l I. ^ ^& -2 *A -•> Y, • ' (ft sec *) L. • = — ~ (sec * or * ' ^ ^xx . -2 . -I L. ■ , (sec " or Ng--j— i (sec 2) ^0. (ft sec" or "^i'—r- ftsec-^deg'h ^l^^S N- - -: ' (sec ^) ^ ^ij^ (ft sec ^ or ^6j^ ' S ft sec'^ deg'b ^l^^^^n «p--2I-uf ^"^"'^ ^ 22 1 q^SbC 3 , -2, h ' TT' ^"*'" ^ q,Sb2c XX q,Sb2c 1 % -1 22 1 ^ \ ,-..-i. 'A -5 L • -T (sec" ) r 21 U, ^'*^ ' iiSbC ^xx"l ^1-" ng N. - —7 ^ (sec*^ or '^ '« sec-2 deg'S A.8 Table A. 5 Modified Lateral-Directional Dimensional Stability Derivatives [6] Yb - Yg/Ui Lj^j • (AiN6^j+Lfi^j)/(l-AiBi) Yp - Yp/Ui Yp - (Yp/Ui)-1 Y^ » gcos9i/Ui ^«df " ^«df/"l ^«sr • ^«s/"l Lg - (AiNj+L3)/(l-AiBi) Lp' • (AiNp+Lp)/(l-AiBi) Lr' - (AiMr+Lr)/(l-AiBi) Wgr - <AiN5^^*Lfi^^)/(l-AiBi) Ne • (BiLg*Kg)/(l-AiBi) V* (BlLp+Np)/(l-AiBi) Kp » (BiLr^Nr)/(l-AiBi) N«^f - (BiLfi^^*Nfi^^)/(l-AiBi) Nfi^^. - (BlL6^^+N«^^)/(l-AiBi) A.9