Stability and Control Derivative Estimation and Engine-Out Analysis. by Joel Grasmeyer Graduate Research Assistant January, 1998 VPI-AOE-254

Transcription

1 Stability and Control Derivative Estimation and Engine-Out Analysis by Joel Grasmeyer Graduate Researh Assistant January, 1998 VPI-AOE-254 This work is supported under NASA Grant NAG W.H. Mason, Faulty Advisor Multidisiplinary Analysis and Design Center for Advaned Vehiles Department of Aerospae and Oean Engineering Virginia Polytehni Institute and State University Blaksburg, VA 24061

2 ii Contents List of Figures... iii List of Tables... iii Nomenlature... iv Symbols... iv Subsripts... v 1. Introdution Control Surfae Sign Conventions Engine-out Methodology Required Yawing Moment Coeffiient Maximum Available Yawing Moment Coeffiient Why an t the vertial tail ahieve its maximum lift oeffiient? Stability and Control Derivative Estimation Angle of Sideslip Derivatives Sidefore Coeffiient Rolling Moment Coeffiient Yawing Moment Coeffiient Lateral Control Derivatives Sidefore Coeffiient Rolling Moment Coeffiient Yawing Moment Coeffiient Diretional Control Derivatives Sidefore Coeffiient Rolling Moment Coeffiient Yawing Moment Coeffiient Validation Boeing Input Output...15 Referenes...17 Appendix: Code Listing for Stability Subroutine (stab.f)...18

3 iii List of Figures 1-1 Control surfae sign onventions Engine-out geometry Engine windmilling drag validation... 3 List of Tables 4-1 Comparison of stability and ontrol derivatives for Stability and ontrol derivative orretion fators...14

4 iv Nomenlature Symbols A b b vtail C Dewm C L C Lαhtail C lαvtail C lαvtail eff C Lαwb C navail C nreq C y β C lβ C nβ D ewm d fuse d fusevtail d i d naelle l l e L ext k Cy βv K ' K b K H K MΓ K N K Rl K MΛ K wb K wbi l tv l vtail aspet ratio wing span vertial tail span drag oeffiient due to windmilling of failed engine lift oeffiient lift urve slope of the horizontal tail setion lift urve slope of vertial tail effetive lift urve slope of vertial tail lift urve slope of the wing and body available yawing moment oeffiient at the engine-out flight ondition required yawing moment oeffiient at the engine-out flight ondition variation of sidefore oeffiient with yaw angle variation of rolling moment oeffiient with yaw angle variation of yawing moment oeffiient with yaw angle drag due to windmilling of failed engine maximum fuselage diameter depth of the fuselage at the vertial tail quarter-hord position engine inlet diameter naelle diameter horizontal distane between CG and vertial surfae buttline of outboard engine external rolling moment empirial fator for vertial tail sideslip derivative estimation empirial orretion fator for large ontrol defletions flap span fator fator aounting for the relative size of the horizontal and vertial tails ompressibility orretion to dihedral empirial fator for body and body + wing effets Reynold's number fator for the fuselage ompressibility orretion to sweep fator for fuselage loss in the lift urve slope wing-body interferene fator horizontal distane between CG and engine nozzle horizontal distane between CG and aerodynami enter of vertial tail

5 v M N engines N req N max q eo S htail S o S ref S vtail T T o V n V Y ext z tv z vtail C L Mah number number of engines required yawing moment maximum attainable yawing moment dynami pressure at the engine-out flight ondition horizontal tail area ross-setional area of fuselage wing referene area vertial tail area maximum available thrust at given mah and altitude stati thrust at sea level ratio of mean nozzle exit veloity to freestream veloity external sidefore vertial distane between CG and engine nozzle vertial distane between CG and aerodynami enter of vertial tail hange in vertial tail C L due to irulation ontrol α angle of attak (rad) β sideslip angle (positive with relative wind from right) β M ompressibility fator = 1 - M 2 δ a δ r η htail φ Γ κ Λ /2 Λ /4 σ Subsripts avail bs eff fuse htail req aileron defletion (positive for right up, left down) rudder defletion (positive right) dynami pressure ratio at the horizontal tail bank angle (positive right roll) dihedral angle (deg) ratio of atual lift urve slope to 2π half-hord sweep angle quarter-hord sweep angle ratio of density at a given altitude to density at sea level available body side irulation ontrol effetive fuselage horizontal tail required

6 vi tv vtail wb wing thrust vetoring vertial tail wing-body wing The FORTRAN ode variable names and definitions are given in the Appendix.

7 1 1. Introdution This report desribes the estimation of stability and ontrol derivatives using the method of Referene [1] (whih is essentially DATCOM [2]), and the establishment of the engine-out onstraint based on the required yawing moment oeffiient. The use of thrust vetoring and irulation ontrol to provide additional yawing moment is also desribed Control Surfae Sign Conventions The ontrol surfae sign onventions are defined suh that a positive ontrol defletion generates a positive roll or yaw moment aording to the right hand rule with a onventional body axis oordinate system, as shown in Figure 1-1. A positive aileron defletion is defined with the right aileron up and the left aileron down. The aileron defletion is the average defletion of the two surfaes from the neutral position. A positive rudder defletion is defined with the trailing edge to the right, as viewed from above. + Rudder Rear View, Looking Forward + Aileron + Aileron Figure 1-1: Control surfae sign onventions 2. Engine-out Methodology The engine-out onstraint is established by onstraining the maximum available yawing moment oeffiient (C navail ) to be greater than the required yawing moment oeffiient (C nreq ) for the engine-out flight ondition: C navail C nreq (2-1)

8 Required Yawing Moment Coeffiient The required yawing moment oeffiient is the yawing moment oeffiient required to maintain steady flight with one failed outboard engine at 1.2 times the stall speed, as speified by FAR The remaining outboard engine must be at the maximum available thrust, and the bank angle annot be larger than 5. Figure 2-1 shows the engine-out geometry for a twin-engine onfiguration. The yawing moment oeffiient required to maintain steady flight with an inoperative engine is given by: C nreq = T + D ewm l e qs ref b (2-2) where T is the maximum available thrust at the given Mah number and altitude, and D ewm is the drag due to the windmilling of the failed engine. Figure 2-1: Engine-out geometry The drag due to the windmilling of the failed engine is alulated using the method desribed in Appendix G-8 of Torenbeek [3]. D ewm = qs ref C Dewm (2-3)

9 3 C Dewm = d 2 i + 2 π d 2 V n M 2 4 i V 1 - V n V (2-4) S ref where: d i is the engine inlet diameter M is the Mah number V n is the nozzle exit veloity V n V 0.92 for high bypass ratio engines S ref is the wing referene area Torenbeek s windmilling drag equation was validated against the flight test data of the 747. As shown in Figure 2-2, Torenbeek s equation shows relatively good agreement with the flight test data over a range of Mah numbers. 8,000 Engine Windmilling Drag (lb) 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 Atual 747 Data [4] Torenbeek Equation Mah Number Figure 2-2: Engine windmilling drag validation

10 Maximum Available Yawing Moment Coeffiient The maximum available yawing moment oeffiient is obtained at an equilibrium flight ondition with a given bank angle (φ) and a given maximum rudder defletion (δ r ). The bank angle is limited to a maximum of 5 by FAR , and the airraft is allowed to have some sideslip (β). The sideslip angle is found by summing the fores along the y-axis: Sidefore Equation: C y δ a δ a + C y δ r δ r + C y β β + C L sin φ - T sin ε qs ref - C L S vtail S ref = - Y ext qs ref (2-5) In a onventional ontrol system, the vertial tail is the dominant ontroller for generating a yawing moment. However, thrust vetoring and irulation ontrol an be used to generate additional yawing moments. Sine the engine-out ondition is a ritial onstraint for a truss-braed wing with tip-mounted engines, the apability to model thrust vetoring and irulation ontrol on the vertial tail was added to the ode. The fifth term in the equation above ( T sin ε qs ref ) is due to the thrust being vetored at an angle ε to the enterline, and the sixth term ( C S vtail L S ref ) is due to the hange in C L at the vertial tail due to irulation ontrol. Sine the external sidefore (Y ext ) is zero, and C y δ a is assumed to be zero, this equation an be simplified and solved for the sideslip angle: - C y δ δ r r - C L sin φ + T sin ε + C S vtail qs L β = ref S ref (2-6) C y β The aileron defletion required to maintain equilibrium flight is obtained by summing the rolling moments about the x-axis: Rolling Moment Equation: C lδ a δ a + C lδ r δ r + C lβ β - T sin ε qs ref z tv b - C S vtail z vtail L = - L ext S ref b qs ref b (2-7) By setting the external rolling moment (L ext ) equal to zero, this equation an be solved for the aileron defletion: - C δ lδ r r - C lβ β + T qs sin ε δ a = ref C lδ a z tv b + C S vtail z vtail L S ref b (2-8)

11 5 The rudder defletion is initially set to the given maximum allowable steady-state value, and the sideslip angle and aileron defletion for equilibrium flight are determined by Eqs. (2-6) and (2-8). The maximum allowable steady-state defletion is typially This allows for an additional 5 of defletion for maneuvering. A warning statement is printed if the alulated defletion exeeds the maximum allowable defletion. The maximum available yawing moment is found by summing the ontributions due to the ailerons, rudder, and sideslip: Yawing Moment Equation: C navail = C nδ a δ a + C nδ r δ r + C nβ β + T sin ε qs ref l tv + C S vtail l vtail b L S ref b (2-9) This value of the available yawing moment oeffiient is then onstrained in the optimization problem to be greater than the required yawing moment oeffiient, as shown in Eq. (2-1) Why an t the vertial tail ahieve its maximum lift oeffiient? The Output setion shows the results of the above methodology for a 747 with no thrust vetoring and no irulation ontrol. The maximum available yawing moment is ahieved with a bank angle of 5 and a sideslip angle of 3. This orientation would be used for a failure of the left engine. The pilot or automati flight ontrol system would roll the airraft 5 in the diretion of the operating engine and yaw slightly away from it. Note that in this flight ondition, the vertial tail is only flying at an angle of attak of 3, whih is far below the angle of attak orresponding to the maximum lift oeffiient of a typial vertial tail. One might expet that the maximum available yawing moment is obtained when the vertial tail is flying at its maximum lift oeffiient, but this is not true beause the equilibrium equations above must always be satisfied for steady flight. To illustrate this point, Eq. (2-5) has been solved for the bank angle with no thrust vetoring and no irulation ontrol: φ = sin -1 - C y δ r δ r + C y β β C L (2-10) Aording to Referene [5], the angle of attak orresponding to the maximum lift oeffiient for a NACA 66(215)-216 airfoil setion with 15 of flap defletion is 15. Therefore if the vertial tail in the 747 example mentioned above were flying at the maximum lift oeffiient, the rudder defletion (δ r ) would be 15, and the vertial tail angle of attak (β) would be at least 15 (3D effets would require an even larger angle).

12 6 If these values are plugged into Eq. (2-10) with a C L of 1.11 and the 747 values for the stability and ontrol derivatives (as given in Nelson [6]), the bank angle required to maintain equilibrium flight is Sine this bank angle is muh larger than the maximum allowable bank angle of 5 speified in FAR , the vertial tail annot fly at the maximum lift oeffiient and maintain equilibrium flight. This brief analysis shows the need for irulation ontrol or thrust vetoring. Sine both of these mehanisms an generate a larger side fore at the vertial tail without requiring a hange in β, they an reate a larger yawing moment oeffiient at the same flight ondition. 3. Stability and Control Derivative Estimation The stability and ontrol derivatives are estimated using the method of Roskam [1], whih was adapted from the USAF Stability and Control DATCOM [2]. MaMillin [7] used a similar approah for the High-Speed Civil Transport. In MaMillin s work, however, the baseline stability and ontrol derivatives were estimated using a vortex-lattie method, and the DATCOM method was only used to augment these baseline values with the effets due to hanging the geometry of the vertial tail. The Fortran soure ode for the stability subroutine is shown in the Appendix Angle of Sideslip Derivatives Sidefore Coeffiient The variation of sidefore oeffiient with sideslip angle has ontributions from the wing, fuselage, and vertial tail. Note that all of the stability and ontrol derivatives have units of rad -1. C y β = C y β wing + C y β fuse + C y β vtail (3-1) The wing ontribution is a funtion of the dihedral angle (in deg). C y β wing = Γ 180 π (3-2) The fuselage and naelle ontributions are estimated by: where: C y β fuse = -2K wbi S o S ref (3-3)

13 7 K wbi is the wing-body interferene fator, whih is determined from a urve fit to Figure 7.1 in Roskam: K wbi = z wing d fuse /2 + 1 for z wing d fuse /2 < 0 (3-4) K wbi = 0.5 z wing d fuse /2 + 1 for z wing d fuse /2 > 0 (3-5) and S o π d fuse Nengines π d naelle 2 2 (3-6) The ontribution of a vertial tail in the plane of symmetry is found from: where: C y β vtail = -k C y βv C l αvtail eff 1 + dσ dβ η v S vtail S ref (3-7) k Cy βv is determined from a urve fit to Figure 7.3 in Roskam: k Cy βv = 0.75 for b vtail d fusevtail < 2 (3-8) k Cy βv = 1 6 b vtail d fusevtail for 2 < b vtail d fusevtail < 3.5 (3-9) k Cy βv = 1 for b vtail d fusevtail > 3.5 (3-10) C lαvtail eff = 2 + 2πA A 2 2 β M κ tan2 Λ /2 β M (3-11) κ = C l αvtail 2π (3-12) C lαvtail is assumed to have a value of 2π.

14 8 β M = 1 - M 2 (3-13) S vtail 1 + dσ dβ η v = S ref z w A (3-14) 1 + os Λ /4 d Note that the effetive aspet ratio of the vertial tail must be used in plae of A in Eqs. (3-11) and (3-14). A vtaileff = A V B A V A vtail 1 + K H A V HB A V B - 1 (3-15) where: A V B A V is the ratio of the aspet ratio of the vertial tail in the presene of the body to that of the isolated panel, whih is determined from the following urve fit to Figure 7.5 in Roskam, with the taper ratio assumed to be less than or equal to 0.6: A V B A V = b vtail b vtail b vtail b vtail b vtail (3-16) d fuse vtail d fusevtail d fusevtail d fusevtail d fusevtail A V HB is the ratio of the vertial tail aspet ratio in the presene of the horizontal A V B tail and body to that of the tail in the presene of the body alone. It is assumed to have a value of 1.1, based on Figure 7.6 in Roskam. This is valid for the 747 and 777 tail geometries. K H is a fator aounting for the relative size of the horizontal and vertial tails, whih is determined from the following urve fit to Figure 7.7 in Roskam: K H = S htail S vtail S htail S vtail S htail S vtail S htail S vtail (3-17) Rolling Moment Coeffiient The variation of rolling moment oeffiient with sideslip angle has ontributions from the wing-body, horizontal tail, and vertial tail. C lβ = C lβ wb + C y β htail + C y β vtail (3-18)

15 9 The ontribution from the wing-body is estimated by: C lβ wb = C L C lβ C L Λ/2 K MΛ K f + C l β C L A + Γ C lβ Γ K M Γ + C l β Γ + C lβ Z w 180 π (3-19) where: C lβ C L Λ/2 is the wing sweep ontribution, obtained from the following urve fit to Figure 7.11 in Roskam for λ = 0.5: C lβ = Λ /2 C L Λ/ π (3-20) K MΛ is the ompressibility orretion to sweep, assumed to have a value of 1.0, based on Figure 7.12 in Roskam. This is valid for the 747 and 777 geometries at low Mah numbers. K f is the fuselage orretion fator, assumed to have a value of 0.85, based on Figure 7.13 in Roskam. This is valid for the 747 and 777 geometries. C lβ C L A is the aspet ratio ontribution, assumed to have a value of 0, based on Figure 7.14 in Roskam for λ = 0.5 and a high aspet ratio. This is valid for the 747 and 777 geometries. C lβ is the wing dihedral effet, obtained from a urve fit to Figure 7.15 in Roskam Γ for λ = 0.5, low sweep, and a high aspet ratio. Note that for extremely high aspet ratios, the urve fit is an extrapolation from the plot in Roskam. K MΓ is the ompressibility orretion to dihedral, assumed to have a value of 1.0, based on Figure 7.16 in Roskam. This is valid for the 747 and 777 geometries at low Mah numbers. C lβ Γ = A d b 2 (3-21)

16 10 d = π d fuse (3-22) C lβ Z w = /π A z wb 2d (3-23) b The ontribution from the horizontal tail is approximately zero, sine it has a small lift oeffiient, small dihedral, and small area relative to the wing. C lβ htail = 0 (3-24) The ontribution from the vertial tail is estimated by: C lβ vtail = C y β vtail z vtail os α - l vtail sin α b (3-25) The fuselage angle of attak is the ratio of the lift oeffiient to the lift urve slope minus the effetive wing inidene angle. The effetive wing inidene angle with 20 of flap defletion is approximately 5. α = C L C Lα π 180 (3-26) The airraft lift urve slope is alulated by: where: C Lα = C Lαwb + C Lαhtail η htail S htail S ref (3-27) C Lαwb = K wb C Lα w (3-28) C Lα w and C Lαhtail are found using the following equation with the appropriate values ofaspet ratio and sweep. C Lα = 2 + 2πA A 2 2 β M κ tan2 Λ /2 β M (3-29) β M = 1 - M 2 (3-30)

17 11 The dynami pressure ratio at the horizontal tail is assumed to be η htail = 0.95 (3-31) Yawing Moment Coeffiient The variation of yawing moment oeffiient with sideslip angle has ontributions from the wing, fuselage, and vertial tail. C nβ = C nβ wing + C n β fuse + C n β vtail (3-32) The wing ontribution to the yawing moment oeffiient is negligible for small angles of attak. C nβ wing 0 (3-33) The fuselage ontribution to the yawing moment oeffiient is determined by: C = -K S nβ fuse NK bs l fuse Rl S ref b 180 π (3-34) where: K N is an empirial fator for body and body + wing effets, assumed to have a value of , based on Figure 7.19 in Roskam. This is valid for the 747 and 777 geometries. K Rl is a Reynolds number fator for the fuselage, obtained from a urve fit to Figure 7.20 in Roskam, based on the alulated fuselage Reynolds number. The fuselage side area is approximated as 83% of the fuselage length times diameter. This is a good approximation for the 747 and 777 geometries. S bs = 0.83l fuse d fuse (3-35) The ontribution from the vertial tail is estimated by the following equation, where α is defined in Eq. (3-26). C nβ vtail = -C y β vtail l vtail os α + z vtail sin α b (3-36)

18 Lateral Control Derivatives Sidefore Coeffiient The variation of sidefore oeffiient with aileron defletion is assumed to be zero. C y δ a = 0 (3-37) Rolling Moment Coeffiient The first step in the estimation of the rolling moment oeffiient is to estimate the rolling moment effetiveness parameter βc' lδ /κ from Figure 11.1 in Roskam. For 747 and 777- like onfigurations with λ = 0.5 and M = 0.25, it is approximately The rolling effetiveness of two full-hord ontrols is estimated by: ' C lδ = κ β M ' βc lδ κ (3-38) where the setion lift urve slope is assumed to be 2π/β M, and κ is the ratio of the atual setion lift urve slope to 2π/β M. The aileron lift effetiveness is estimated from Roskam s Figures 10.5 and 10.6 with f / = 0.20 and t/ = These assumptions result in a value of 3.5 from Figure 10.5, and a value of 1.0 from Figure 10.6 The aileron effetiveness is given by: C lδ = C lδ C lδ Theory C lδ Theory (3-39) α δ = C l δ C lα (3-40) The rolling effetiveness of the partial-hord ontrols is estimated by: C lδ = α δ C lδ (3-41) The δ in the equation above refers to the sum of the left and right aileron defletions. Sine we define the aileron defletion (δ a ) as one half of the sum of the defletions, the variation of rolling moment oeffiient with aileron defletion is given by: C lδ a = C l δ 2 ( 3-42)

19 Yawing Moment Coeffiient The variation of yawing moment oeffiient with aileron defletion is given by: C = KC nδ a LC lδ a (3-43) where K is estimated from Figure 11.3 in Roskam with λ = 0.5, A = 8, and η i = Diretional Control Derivatives Sidefore Coeffiient The variation of sidefore oeffiient with rudder defletion is given by: where: C y δ r = C l αvtail eff α δ C L α δ C l K ' K b S vtail S ref (3-44) α δ C L α δ C l is the ratio of the 3D flap-effetiveness parameter to the 2D flapeffetiveness parameter. It is estimated with a pieewise urve fit to Figure 10.2 in Roskam with an assumed value of f / = K b is the flap span fator, whih is estimated to be 0.95 from Figure 10.3 in Roskam with η = K ' is an empirial orretion fator for large ontrol defletions. It is estimated with a urve fit to Figure 10.7 in Roskam with f / = Rolling Moment Coeffiient The variation of rolling moment oeffiient with rudder defletion is given by: C lδ r = C y δ r z vtail os α - l vtail sin α b (3-45) Yawing Moment Coeffiient The variation of yawing moment oeffiient with rudder defletion is given by: C nδ r = -C y δ r l vtail os α + z vtail sin α b (3-46)

20 14 4. Validation 4.1. Boeing The stability and ontrol derivatives were validated with the Table 4-1 shows a omparison of the predited stability and ontrol derivatives with the flight test derivatives presented in Nelson [6]. Note that the sign differenes in the last three values are due to a different sign onvention for the rudder defletion. Table 4-1: Comparison of stability and ontrol derivatives for Derivative Flight Test Predition Error C y β C lβ C nβ C lδ a C nδ a C y δ r C lδ r C nδ r A orretion fator was applied to eah of the derivatives to inrease their auray. Eah orretion fator shown in Table 4-2 is the ratio of the atual value to the predited value for the for the M = 0.25 flight ondition given in NASA CR-2144 [8]. These orretion fators may have to be realibrated if the onfiguration is signifiantly different from the 747. Table 4-2: Stability and ontrol derivative orretion fators Derivative Corretion Fator C y β C lβ C nβ C lδ a C nδ a C y δ r C lδ r C nδ r

21 15 5. Input The following listing is a sample input file for the Boeing The input variables are given in the Appendix. This set of inputs was used to reate the orretion fators shown in the Validation setion. input file for stab boeing dihedral_wing (deg) 6.2 z_wing (ft) 23.0 dia_fuse (ft) sref (ft^2) 33.5 hspan_vtail (ft) 14.4 depth_fuse_vtail (ft) 36.4 _vtail_root (ft) 11.5 _vtail_tip (ft) 0.25 mah_eo 45. sweep_vtail_1_4 (deg) 33.5 sweep_wing_1_2 (deg) 97.8 hspan_wing (ft) 36.4 hspan_htail (ft) sweep_htail_1_2 (deg) 1.11 l 26. z_vtail (ft) 100. l_vtail (ft) length_fuse (ft) 4 new 0 nef 8.4 dia_naelle (ft) sh (ft^2) e-3 rho_eo (slug/ft^3) a_eo (ft/s) e-7 mu_eo (slug/(ft-s)) 15. dr_max (deg) 25. da_max (deg) 0. thrust_tv (lb) 0. angle_tv (deg) 122. l_tv (ft) 7. z_tv (ft) 0.0 l_ir_trl 6. Output The following listing is the output file for the Boeing The definitions of the variables are given in the Appendix. Note that the stability and ontrol derivatives in this file represent the orreted values for the alibration ase shown above.

22 16 stab output file boeing747 Input 1 = write_flag = dihedral_wing (deg) = z_wing (ft) = dia_fuse (ft) = sref (ft^2) = hspan_vtail (ft) = depth_fuse_vtail (ft) = _vtail_root (ft) = _vtail_tip (ft) = mah_eo = sweep_vtail_1_4 (deg) = sweep_wing_1_2 (deg) = hspan_wing (ft) = hspan_htail (ft) = sweep_htail_1_2 (deg) = l = z_vtail (ft) = l_vtail (ft) = length_fuse (ft) 4 = new 0 = nef = dia_naelle (ft) = sh (ft^2) = rho_eo (slug/ft^3) = a_eo (ft/s) E-06 = mu_eo (slug/(ft-s)) = dr_max (deg) = da_max (deg) = thrust_tv (lb) = angle_tv (deg) = l_tv (ft) = z_tv (ft) = l_ir_trl Output = y_beta (rad-1) = l_beta (rad-1) = n_beta (rad-1) = y_da (rad-1) = l_da (rad-1) = n_da (rad-1) = y_dr (rad-1) = l_dr (rad-1) = n_dr (rad-1) = beta (deg) = phi (deg) = da (deg) = dr (deg) = ar_vtail_eff = n_avail

23 17 Referenes [1] Roskam, J., Methods for Estimating Stability and Control Derivatives of Conventional Subsoni Airplanes, Roskam Aviation and Engineering Corporation, Lawrene, Kansas, [2] Hoak, D.E. et al., USAF Stability and Control DATCOM, Flight Control Division, Air Fore Flight Dynamis Laboratory, WPAFB, Ohio, [3] Torenbeek, E., Synthesis of Subsoni Airplane Design, Delft Univ. Press, Delft, The Netherlands, [4] Hanke, C.R., The Simulation of a Large Jet Transport Airraft, Vol. I: Mathematial Model, NASA CR-1756, Marh [5] Abbott, I.H., and von Doenhoff, A.E., Theory of Wing Setions, Dover, New York, [6] Nelson, R. C., Flight Stability and Automati Control, MGraw-Hill Co., New York, [7] MaMillin, P.E., Golovidov, O.B., Mason, W.H., Grossman, B., and Haftka, R.T., Trim, Control, and Performane Effets in Variable-Complexity High-Speed Civil Transport Design, MAD , July, [8] Heffley, R.K., and Jewell, W.F., Airraft Handling Qualities Data, NASA CR- 2144, Deember, 1972.

24 18 Appendix: Code Listing for Stability Subroutine (stab.f) /////////////////////////////////////////////////////////////////////// subroutine stab This subroutine alulates the maximum available yawing moment oeffiient of a given airraft onfiguration at a given flight ondition. Note that right rudder defletion is defined as positive, and right aileron up, left aileron down is defined as positive. Both of these ontrol defletions generate positive moments about their respetive axes. This is the onvention used by Roskam. The thrust vetoring angle (angle_tv) is also defined as positive for a right defletion. Inputs outfile output filename title title of airraft onfiguration write_flag write flag (0 = no output file, 1 = output file written) dihedral_wing wing dihedral angle (deg) z_wing distane from body enterline to quarter-hord point of exposed wing root hord, positive for the quarter-hord point below the body enterline (ft) dia_fuse fuselage diameter (ft) sref wing referene area (ft^2) hspan_vtail vertial tail span (ft) depth_fuse_vtail fuselage depth at the fuselage station of the quarter-hord of the vertial tail (ft) _vtail_root root hord of vertial tail _vtail_tip tip hord of vertial tail mah_eo mah number sweep_vtail_1_4_deg vertial tail quarter-hord sweep angle (deg) sweep_wing_1_2_deg average wing half-hord sweep angle (deg) hspan_wing wing half-span (ft) hspan_htail horizontal tail half-span (ft) sweep_htail_1_2_deg horizontal tail half-hord sweep angle (deg) l lift oeffiient z_vtail vertial distane from CG to AC of vertial tail (ft) l_vtail horizontal distane from CG to AC of vertial tail (ft) length_fuse fuselage length (ft) new number of engines on the wing nef number of engines on the fuselage dia_naelle naelle diameter (ft) rho_eo density at engine-out flight ondition (slug/ft^3) a_eo speed of sound at engine-out flight ondition (ft/s) mu_eo visosity at engine-out flight ondition (slug/(ft-s)) dr_max maximum allowable steady-state rudder defletion (deg) da_max maximum allowable steady-state aileron defletion (deg) thrust_tv maximum available thrust of the aft engine (lb) angle_tv horizontal angle between the fuselage enterline and the effetive thrust vetor (deg, positive to the right) l_tv horizontal distane between CG and thrust vetoring nozzle (ft)

25 19 z_tv vertial distane between CG and thrust vetoring nozzle (ft) l_ir_trl hange in lift oeffiient due to irulation ontrol (nondimensionalized by q and the vertial tail area) Outputs ar_vtail_eff effetive aspet ratio of vertial tail n_avail maximum available yawing moment oeffiient Internal Variables alpha angle of attak (rad) alpha_d setion lift effetiveness alpha_d_l setion flap effetiveness (from Figure 10.2) ar wing aspet ratio ar_vtail atual aspet ratio of vertial tail ar_htail atual aspet ratio of horizontal tail avb_av ratio of the aspet ratio of the vertial panel in the presene of the body to that of the isolated panel (from Figure 7.5) avhb_avb ratio of the vertial panel aspet ratio in the presene of the horizontal tail and body to that of the panel in the presene of the body alone (from Figure 7.6) bld_kappa rolling moment effetiveness parameter (from Figure 11.1) beta sideslip angle, positive from the right (rad) beta_m square root of (1 - mah_eo)**2 f_ ratio of flap hord to wing or tail hord f_fator flap hord fator (from Figure 10.2) l_alpha lift-urve slope of entire airraft (rad^-1) l_alpha_2d 2-dimensional lift-urve slope at MAC (rad^-1) l_alpha_h lift-urve slope of horizontal tail (rad^-1) l_alpha_vtail original lift-urve slope of vertial tail (rad^-1) l_alpha_vtail_eff effetive lift-urve slope of vertial tail (rad^-1) l_alpha_w lift-urve slope of wing (rad^-1) l_alpha_wb lift-urve slope of wing-body ombination (rad^-1) l_beta variation of rolling moment oeffiient with sideslip angle l_beta_or orreted value of l_beta l_beta_htail horizontal tail ontribution to l_beta l_beta_vtail vertial tail ontribution to l_beta l_beta_wingbody wing-body ontribution to l_beta l_d rolling effetiveness of partial-hord ontrols l_da variation of rolling moment oeffiient with aileron defletion l_da_or orreted value of l_da l_dr variation of rolling moment oeffiient with rudder defletion l_dr_or orreted value of l_dr lb_l_a aspet ratio ontribution to l_beta_wingbody (from Figure 7.14) lb_l_lambda wing sweep ontribution to l_beta_wingbody (from Figure 7.11) lb_gamma dihedral effet on l_beta (from Figure 7.15) ld_prime rolling effetiveness of two full-hord ailerons (Equation 11.2) ld_ratio empirial orretion for plain TE flaps (Fig. 10.6) ld_theory theoretial lift effetiveness of plain TE flaps (Fig. 10.5)

26 20 n_beta variation of yawing moment oeffiient with sideslip angle n_beta_or orreted value of n_beta n_beta_fuse fuselage ontribution to n_beta n_beta_vtail vertial tail ontribution to n_beta n_beta_wing wing ontribution to n_beta n_da variation of yawing moment oeffiient with aileron defletion n_da_or orreted value of n_da n_dr variation of yawing moment oeffiient with rudder defletion n_dr_or orreted value of n_dr y_beta variation of side fore oeffiient with sideslip angle y_beta_or orreted value of y_beta y_beta_fuse fuselage ontribution to y_beta y_beta_vtail vertial tail ontribution to y_beta y_beta_wing wing ontribution to y_beta y_da variation of side fore oeffiient with aileron defletion y_dr variation of side fore oeffiient with rudder defletion y_dr_or orreted value of y_dr d d in Equation 7.10 (estimated from Equation 7.11) da aileron defletion, positive for right aileron up, left aileron down (rad) dlb_gamma body-indued effet on wing height (from Equation 7.10) dlb_zw another body-indued effet on wing height (from Equation 7.12) debug_flag printing flag for debugging output (0 = no debugging info printed, 1 = debugging info printed) dr rudder defletion, positive for right defletion (rad) eff_vtail vertial tail effetiveness fator estimated by Equation 7.5 eta_h dynami pressure ratio at the horizontal tail f_y_beta orretion fator for y_beta f_l_beta orretion fator for l_beta f_n_beta orretion fator for n_beta f_l_da orretion fator for l_da f_n_da orretion fator for n_da f_y_dr orretion fator for y_dr f_l_dr orretion fator for l_dr f_n_dr orretion fator for n_dr flap_eff_ratio flap effetiveness ratio (from Figure 10.2) i index k empirial fator for estimating the variation of yawing moment oeffiient with aileron defletion k_b span fator for plain flap (from Figure 10.3) k_y_beta_v empirial fator from Figure 7.3 k_f fuselage orretion fator (from Figure 7.13) k_h fator aounting for relative size of horizontal and vertial tails (from Figure 7.7) k_m_lambda ompressibility orretion to wing sweep (from Figure 7.12) k_m_gamma ompressibility orretion to dihedral effet (from Figure 7.16) k_n fator for body and body + wing effets (from Figure 7.19) k_prime empirial orretion for lift effetiveness of plain flaps at high flap defletions (from Figure 10.7) k_r_l Reynold's number fator for the fuselage (from Figure 7.20)

27 21 k_wbi wing-body interferene fator from Figure 7.1 k_wb fator for loss in lift urve due to body kappa ratio of the atual lift-urve slope to 2*pi phi bank angle, positive to the right (rad) q dynami pressure (lb/ft^2) re_fuse fuselage Reynolds number sbs body side area (ft^2) sh area of horizontal tail (ft^2) sv area of vertial tail (ft^2) sweep_htail_1_2 horizontal tail half-hord sweep angle (rad) sweep_vtail_1_2 vertial tail half-hord sweep angle (rad) sweep_vtail_1_4 vertial tail half-hord sweep angle (rad) sweep_wing_1_2 average wing half-hord sweep angle (rad) x temporary variable for urve fits Created by: Joel Grasmeyer Last Modified: 03/01/98 /////////////////////////////////////////////////////////////////////// subroutine stab(outfile,title,write_flag,dihedral_wing,z_wing, & dia_fuse,sref,hspan_vtail,depth_fuse_vtail,_vtail_root, & _vtail_tip,mah_eo,sweep_vtail_1_4_deg,sweep_wing_1_2_deg, & hspan_wing,hspan_htail,sweep_htail_1_2_deg,l,z_vtail,l_vtail, & length_fuse,new,nef,dia_naelle,sh,rho_eo,a_eo,mu_eo,dr_max, & da_max,thrust_tv,angle_tv,l_tv,z_tv,l_ir_trl,ar_vtail_eff, & n_avail) impliit none harater*72 outfile, title integer i, write_flag, unit_out, new, nef, debug_flag real pi, dihedral_wing, z_wing, dia_fuse, sref, hspan_vtail, ar, & depth_fuse_vtail, _vtail_root, _vtail_tip, mah_eo, sv, sh, & sweep_wing_1_2, hspan_wing, y_beta, ar_vtail, k, thrust_tv, & y_beta_wing, y_beta_fuse, y_beta_vtail, ar_vtail_eff, alpha, & l_alpha_vtail, beta_m, eff_vtail, kappa, l_alpha_vtail_eff, q, & l_beta, l_beta_htail, l_beta_vtail, l_beta_wingbody, sbs, & l, k_wbi, avb_av, avhb_avb, k_h, lb_l_lambda, n_da, l_tv, & k_m_lambda, k_f, lb_l_a, lb_gamma, k_m_gamma, dlb_gamma, & d, dlb_zw, l_alpha, z_vtail, l_vtail, n_beta_fuse, da, & n_beta_vtail, n_beta, n_beta_wing, k_n, k_r_l, phi, angle_tv, & length_fuse, re_fuse, l_da, y_da, bld_kappa, ld_prime, l_d, & alpha_d, ld_theory, ld_ratio, l_alpha_2d, l_dr, dr, n_dr, & y_dr, f_fator, k_prime, alpha_d_l, flap_eff_ratio, k_b, & f_, beta, rho_eo, a_eo, mu_eo, n_avail, k_y_beta_v, da_max, & dia_naelle, dr_max, f_y_beta, f_l_beta, z_tv, l_ir_trl, & f_n_beta, f_l_da, f_n_da, f_y_dr, f_l_dr, f_n_dr, x, & sweep_vtail_1_4, sweep_vtail_1_2, sweep_wing_1_2_deg, k_wb, & sweep_vtail_1_4_deg, y_beta_or, l_beta_or, n_beta_or, & l_da_or, n_da_or, y_dr_or, l_dr_or, n_dr_or, & hspan_htail, sweep_htail_1_2_deg, sweep_htail_1_2, eta_h, & ar_htail, l_alpha_h, l_alpha_w, l_alpha_wb pi = aos(-1.) Initialize value of debug_flag debug_flag = 0 Convert sweep angles from degrees to radians sweep_wing_1_2 = sweep_wing_1_2_deg*pi/180.

28 22 sweep_htail_1_2 = sweep_htail_1_2_deg*pi/180. sweep_vtail_1_4 = sweep_vtail_1_4_deg*pi/180. Append extension to idrag output filename i = 1 do while (outfile(i:i).ne. '.') i = i + 1 end do outfile(i+1:i+5) = 'stab' outfile(i+6:) = '' Write input data to output file for onfirmation if (write_flag.eq. 1) then unit_out = 171 open(unit_out,file=outfile) write(unit_out,"('stab output file')") write(unit_out,"(a72)") title write(unit_out,*) write(unit_out,"(a5)") 'Input' write(unit_out,*) write(unit_out,101) write_flag, '= write_flag' write(unit_out,100) dihedral_wing, '= dihedral_wing (deg)' write(unit_out,100) z_wing, '= z_wing (ft)' write(unit_out,100) dia_fuse, '= dia_fuse (ft)' write(unit_out,100) sref, '= sref (ft^2)' write(unit_out,100) hspan_vtail, '= hspan_vtail (ft)' write(unit_out,100) depth_fuse_vtail, '= depth_fuse_vtail (ft)' write(unit_out,100) _vtail_root, '= _vtail_root (ft)' write(unit_out,100) _vtail_tip, '= _vtail_tip (ft)' write(unit_out,100) mah_eo, '= mah_eo' write(unit_out,100) sweep_vtail_1_4*180./pi, & '= sweep_vtail_1_4 (deg)' write(unit_out,100) sweep_wing_1_2*180./pi, & '= sweep_wing_1_2 (deg)' write(unit_out,100) hspan_wing, '= hspan_wing (ft)' write(unit_out,100) hspan_htail, '= hspan_htail (ft)' write(unit_out,100) sweep_htail_1_2*180./pi, & '= sweep_htail_1_2 (deg)' write(unit_out,100) l, '= l' write(unit_out,100) z_vtail, '= z_vtail (ft)' write(unit_out,100) l_vtail, '= l_vtail (ft)' write(unit_out,100) length_fuse, '= length_fuse (ft)' write(unit_out,101) new, '= new' write(unit_out,101) nef, '= nef' write(unit_out,100) dia_naelle, '= dia_naelle (ft)' write(unit_out,100) sh, '= sh (ft^2)' write(unit_out,100) rho_eo, '= rho_eo (slug/ft^3)' write(unit_out,100) a_eo, '= a_eo (ft/s)' write(unit_out,103) mu_eo, '= mu_eo (slug/(ft-s))' write(unit_out,100) dr_max, '= dr_max (deg)' write(unit_out,100) da_max, '= da_max (deg)' write(unit_out,100) thrust_tv, '= thrust_tv (lb)' write(unit_out,100) angle_tv, '= angle_tv (deg)' write(unit_out,100) l_tv, '= l_tv (ft)' write(unit_out,100) z_tv, '= z_tv (ft)' write(unit_out,100) l_ir_trl, '= l_ir_trl' end if Calulate stability and ontrol derivatives via Roskam's methods Sideslip angle derivatives

29 23 y_beta_wing = *abs(dihedral_wing)*180./pi Estimate k_wbi from Figure 7.1 (urve fit) if (z_wing/(dia_fuse/2.).le. 0.) then k_wbi = 0.85*(-z_wing/(dia_fuse/2.)) + 1. elseif (z_wing/(dia_fuse/2.).gt. 0.) then k_wbi = 0.5*z_wing/(dia_fuse/2.) + 1. end if Estimate the side fore oeffiient due to the fuselage and naelles y_beta_fuse = -2.*k_wbi*( pi*(dia_fuse/2.)**2 + & (new + nef)*pi*(dia_naelle/2.)**2 )/sref Estimate k_y_beta_v from Figure 7.3 (urve fit) x = hspan_vtail/depth_fuse_vtail if (x.le. 2.) then k_y_beta_v = 0.75 elseif (x.gt. 2..and. x.lt. 3.5) then k_y_beta_v = x/ /12. elseif (x.ge. 3.5) then k_y_beta_v = 1. end if Estimate avb_av from Figure 7.5 (urve fit for taper ratio <= 0.6) x = hspan_vtail/depth_fuse_vtail avb_av = 0.002*x** *x** *x** *x**2 + & *x Fator from Figure 7.6 is for zh/bv = 0. avhb_avb = 1.1 Estimate k_h from Figure 7.7 (urve fit) sv = hspan_vtail*(_vtail_root + _vtail_tip)/2. x = sh/sv k_h = *x** *x** *x** *x - & Estimate the effetive aspet ratio for the vertial tail ar_vtail = hspan_vtail**2/sv ar_vtail_eff = avb_av*ar_vtail*(1. + k_h*(avhb_avb - 1.)) Assume the setion lift-urve slope is 2.*pi l_alpha_vtail = 2.*pi Estimate the effetive lift-urve slope for the vertial tail kappa = l_alpha_vtail/(2.*pi) beta_m = sqrt( 1. - mah_eo**2 ) sweep_vtail_1_2 = atan( (_vtail_root/4. + hspan_vtail* & tan(sweep_vtail_1_4) + _vtail_tip/4. - & _vtail_root/2.)/hspan_vtail ) l_alpha_vtail_eff = 2.*pi*ar_vtail_eff/( 2. + & sqrt( ar_vtail_eff**2*beta_m**2/kappa**2* & ( 1. + tan(sweep_vtail_1_2)**2/ & beta_m**2 ) + 4. ) ) Estimate the third term in eqn. 7.4 from eqn. 7.5 eff_vtail = *sv/sref/(1. + & os(sweep_vtail_1_4)) + 0.4*z_wing/dia_fuse + & 0.009*ar_vtail_eff y_beta_vtail = -k_y_beta_v*l_alpha_vtail_eff*eff_vtail*sv/sref

30 24 Calulate total variation of side fore oeffiient with sideslip angle y_beta = y_beta_wing + y_beta_fuse + y_beta_vtail Fator from Figure 7.11 is approximated by a urve fit for lambda = 0.5 lb_l_lambda = /45*sweep_wing_1_2*180./pi Fator from Figure 7.12 is approximated for 747 and 777 onfigurations at low Mah numbers k_m_lambda = 1.0 Fator from Figure 7.13 is approximated for 747 and 777 onfigurations k_f = 0.85 Fator from Figure 7.14 is approximated for lambda = 0.5 and high AR lb_l_a = Fator from Figure 7.15 is approximated by a linear urve fit for lambda equal to 0.5, low sweep, and high AR ar = (2.*hspan_wing)**2/sref lb_gamma = /10*ar Fator from Figure 7.16 is approximated for 747 and 777 onfigurations at low Mah numbers k_m_gamma = 1.0 Estimate body-indued effet on wing height from eqns. 7.10, 7.11, and 7.12 d = sqrt(pi*(dia_fuse/2.)**2/0.7854) dlb_gamma = *sqrt(ar)*(d/(2.*hspan_wing))**2 dlb_zw = -1.2*sqrt(ar)/(180./pi)*z_wing/(2.*hspan_wing)* & 2.*d/(2.*hspan_wing) Wing-body ontribution to l_beta (wing twist effet is negleted) l_beta_wingbody = ( l*(lb_l_lambda*k_m_lambda*k_f + & lb_l_a) + dihedral_wing*(lb_gamma*k_m_gamma + dlb_gamma) + & dlb_zw )*180./pi Sine the horizontal tail has a small lift oeffiient, small dihedral, and small area relative to the wing, it is negligible. l_beta_htail = 0. Calulate the lift urve loss fator due to the fuselage x = dia_fuse/(2.*hspan_wing) k_wb = *x** *x Assume the 2D lift-urve slope is 2*pi/beta_m l_alpha_2d = 2*pi/beta_m kappa = l_alpha_2d/(2.*pi/beta_m) Calulate the lift urve slope of the wing alone and wing-body ombination l_alpha_w = 2.*pi*ar/( 2. + sqrt( ar**2*beta_m**2/kappa**2* & ( 1. + tan(sweep_wing_1_2)**2/beta_m**2 ) + 4. ) ) l_alpha_wb = k_wb*l_alpha_w Calulate the lift urve slope of the horizontal tail ar_htail = (2.*hspan_htail)**2/sh l_alpha_h = 2.*pi*ar_htail/( 2. + sqrt( ar_htail**2*beta_m**2/ & kappa**2*( 1. + tan(sweep_htail_1_2)**2/beta_m**2 ) & + 4. ) ) Assume the dynami pressure ratio at the horizontal tail is 0.95 eta_h = 0.95

31 25 Calulate the lift urve slope of the total airraft l_alpha = l_alpha_wb + l_alpha_h*eta_h*sh/sref Calulate the angle of attak of the fuselage enterline. The wing inidene angle is assumed to be 5 deg. alpha = l/l_alpha - 5.*pi/180. Estimate the vertial tail ontribution to l_beta l_beta_vtail = y_beta_vtail*( z_vtail*os(alpha) - l_vtail* & sin(alpha) )/(2.*hspan_wing) Calulate total variation of rolling moment oeffiient with sideslip angle l_beta = l_beta_wingbody + l_beta_htail + l_beta_vtail Wing ontribution to n_beta is negligible for small angles of attak. n_beta_wing = 0. Estimate empirial fator for body and body + wing effets from Figure 7.19 Constant value assumed for 747 and 777-like onfigurations k_n = Calulate fuselage Reynolds number at the engine-out flight ondition re_fuse = rho_eo*mah_eo*a_eo*length_fuse/mu_eo Estimate fuselage Reynolds number effet on wing-body from Figure 7.20 k_r_l = /log(350.)*log(re_fuse/ ) Estimate fuselage ontribution to n_beta sbs = 0.83*dia_fuse*length_fuse n_beta_fuse = -180./pi*k_n*k_r_l*sbs/sref* & length_fuse/(2.*hspan_wing) Estimate vertial tail ontribution to n_beta n_beta_vtail = -y_beta_vtail*( l_vtail*os(alpha) + & z_vtail*sin(alpha) )/(2.*hspan_wing) Calulate total variation of yawing moment oeffiient with sideslip angle n_beta = n_beta_wing + n_beta_fuse + n_beta_vtail Assume variation of sidefore oeffiient with aileron defletion is zero y_da = 0. Estimate the rolling moment effetiveness parameter from Figure 11.1 for lambda = 0.5, and for 747 and 777-like ailerons at mah 0.25 bld_kappa = 0.18 Estimate the rolling effetiveness of two full-hord ontrols by Eqn ld_prime = kappa/beta_m*bld_kappa Estimate aileron effetiveness by assuming f/ = 0.20 and t/ = 0.08 ld_theory = 3.5 ld_ratio = 1.0 l_d = ld_ratio*ld_theory alpha_d = l_d/l_alpha_2d Determine the rolling effetiveness of the partial-hord ontrols by Eqn Note that this is the hange in l with respet to a hange in the sum of the left and right aileron defletions (d). l_d = alpha_d*ld_prime

32 26 Estimate variation of rolling moment oeffiient with aileron defletion by negleting differential ontrol effets. Sine the aileron defletion (da) is defined as half of the sum of the left and right defletions, l_d from the equation above must be divided by 2. l_da = l_d/2. The method in Roskam for estimating n_da does not aount for the effet of differential ailerons and the use of spoilers for roll ontrol on the yaw moment. Therefore, the fator k is estimated based on the ratio of n_da to l_da from the 747 flight test data presented in Nelson. Note that the effet of l is absorbed into the fator k. k = / Estimate variation of yawing moment oeffiient with aileron defletion n_da = k*l_da Estimate the flap hord fator from Figure 10.2 for f/ = 0.33 The flap effetiveness ratio is estimated with a pieewise urve fit f_ = 0.33 alpha_d_l = -sqrt( 1. - (1. - f_)**2 ) if (alpha_d_l.ge. -0.5) then flap_eff_ratio = *alpha_d_l elseif (alpha_d_l.ge. -0.6) then flap_eff_ratio = *alpha_d_l elseif (alpha_d_l.ge. -0.7) then flap_eff_ratio = *alpha_d_l else flap_eff_ratio = alpha_d_l end if flap_eff_ratio = 1. + flap_eff_ratio/( ar_vtail_eff - & 0.5*(-alpha_d_l - 2.1) ) f_fator = flap_eff_ratio*alpha_d_l Estimate empirial orretion for lift effetiveness of plan flaps at from Figure 10.7 for f/ = x = dr_max if (x.lt. 15.) then k_prime = 1. else k_prime = 4e-7*x**4-7e-5*x** *x** *x + & end if Estimate span fator for plain flap from Figure 10.3 for delta eta = 0.85 k_b = 0.95 Estimate variation of sidefore oeffiient with rudder defletion y_dr = l_alpha_vtail_eff*f_fator*k_prime*k_b*sv/sref Estimate variation of rolling moment oeffiient with rudder defletion l_dr = y_dr*( z_vtail*os(alpha) - l_vtail*sin(alpha) )/ & (2.*hspan_wing) Estimate variation of yawing moment oeffiient with rudder defletion n_dr = -y_dr*( l_vtail*os(alpha) + z_vtail*sin(alpha) )/ & (2.*hspan_wing) Multiply empirial estimates by their respetive orretion fators The orretion fators are the ratio of the atual 747 derivatives to the 747 derivatives predited by the method above at the M=0.25 flight

33 27 ondition defined in NASA CR-2144 and Nelson. The rudder defletion was 15 deg for this alibration. y_beta_or = *y_beta l_beta_or = *l_beta n_beta_or = *n_beta l_da_or = *l_da n_da_or = *n_da y_dr_or = *y_dr l_dr_or = *l_dr n_dr_or = *n_dr Calulate the dynami pressure q = 0.5*rho_eo*(mah_eo*a_eo)**2 Set the rudder defletion to 20 deg, and the bank angle to 5 deg dr = dr_max*pi/180. phi = 5.*pi/180. Solve for the sideslip angle and aileron defletion beta = ( -y_dr_or*dr - l*sin(phi) + & sign( thrust_tv*sin(angle_tv*pi/180.)/(q*sref), & angle_tv ) + l_ir_trl*sv/sref )/y_beta_or da = ( -l_dr_or*dr - l_beta_or*beta + sign( thrust_tv* & sin(angle_tv*pi/180.)*z_tv/(q*sref*2.*hspan_wing), & angle_tv ) + l_ir_trl*z_vtail/(2.*hspan_wing)* & sv/sref )/l_da_or Chek if the aileron defletion is greater than the max allowable value if (da.gt. da_max) then print*,'warning from stab.f: Required aileron defletion is ', & 'greater than the maximum allowable value.' end if Calulate the maximum available yawing moment oeffiient n_avail = n_da_or*da + n_dr_or*dr + n_beta_or*beta + & sign( thrust_tv*sin(angle_tv*pi/180.)*l_tv/ & (q*sref*2.*hspan_wing), angle_tv ) + & l_ir_trl*l_vtail/(2.*hspan_wing)*sv/sref Write output data if (write_flag.eq. 1) then write(unit_out,*) write(unit_out,"(a6)") 'Output' write(unit_out,*) This setion is normally ommented out. It an be used to print the unorreted values of the derivatives for debugging purposes. if (debug_flag.eq. 1) then write(unit_out,100) y_beta_wing, '= y_beta_wing (rad-1)' write(unit_out,100) y_beta_fuse, '= y_beta_fuse (rad-1)' write(unit_out,100) y_beta_vtail, '= y_beta_vtail (rad-1)' write(unit_out,100) y_beta, '= y_beta (rad-1)' write(unit_out,*) write(unit_out,100) l_beta_wingbody, & '= l_beta_wingbody (rad-1)' write(unit_out,100) l_beta_htail, '= l_beta_htail (rad-1)' write(unit_out,100) l_beta_vtail, '= l_beta_vtail (rad-1)' write(unit_out,100) l_beta, '= l_beta (rad-1)' write(unit_out,*) write(unit_out,100) n_beta_wing, '= n_beta_wing (rad-1)' write(unit_out,100) n_beta_fuse, '= n_beta_fuse (rad-1)'

34 28 write(unit_out,100) n_beta_vtail, '= n_beta_vtail (rad-1)' write(unit_out,100) n_beta, '= n_beta (rad-1)' write(unit_out,*) write(unit_out,100) y_da, '= y_da (rad-1)' write(unit_out,100) l_da, '= l_da (rad-1)' write(unit_out,100) n_da, '= n_da (rad-1)' write(unit_out,*) write(unit_out,100) y_dr, '= y_dr (rad-1)' write(unit_out,100) l_dr, '= l_dr (rad-1)' write(unit_out,100) n_dr, '= n_dr (rad-1)' write(unit_out,*) end if This setion prints the orreted values of the derivatives write(unit_out,100) y_beta_or, '= y_beta (rad-1)' write(unit_out,100) l_beta_or, '= l_beta (rad-1)' write(unit_out,100) n_beta_or, '= n_beta (rad-1)' write(unit_out,*) write(unit_out,100) y_da, '= y_da (rad-1)' write(unit_out,100) l_da_or, '= l_da (rad-1)' write(unit_out,100) n_da_or, '= n_da (rad-1)' write(unit_out,*) write(unit_out,100) y_dr_or, '= y_dr (rad-1)' write(unit_out,100) l_dr_or, '= l_dr (rad-1)' write(unit_out,100) n_dr_or, '= n_dr (rad-1)' write(unit_out,*) write(unit_out,100) beta*180./pi, '= beta (deg)' write(unit_out,100) phi*180./pi, '= phi (deg)' write(unit_out,100) da*180./pi, '= da (deg)' write(unit_out,100) dr*180./pi, '= dr (deg)' write(unit_out,100) ar_vtail_eff, '= ar_vtail_eff' write(unit_out,100) n_avail, write(unit_out,*) lose(unit_out) endif 100 format(f11.4, 1x, a) 101 format(7x, i4, 1x, a) 102 format(f11.0, 1x, a) 103 format(g11.4, 1x, a) return end '= n_avail'