SIMULATION MODEL FOR THE PIPER PA-30 LIGHT MANEUVERABLE AIRCRAFT IN THE FINAL APPROACH

(along stability Z axis when no subscript) Xaccel distance from center of gravity to accelerometer location measured along X fuselage axis, positive forward ft. XE X-axis in local vertical coordinate frame - YE Y-axis in local vertical coordinate frame ZE Z-axis in local vertical coordinate frame ZT pitching moment arm of the thrust vector positive downward ft. a angle of attack = tan (w/u) rad. a angle between X-stability axis and r fuselage axis rad. aT angle between X-stability axis and thrust axis rad. 5 angle of sideslip = sin (v/VT) rad. S glideslope error, localizer error rad. 6 deflection of throttle positiont (6 = 1 = full throttle) 6 0 deflection of control surface rad. e,,ip Euler angles referenced to stability axis rad. P atmospheric air denisity slug/ft 3 ') derivative with respect to time Subscripts a aileron B fuselage reference frame E local vertical coordinate frame e elevator or stabilizer f flaps ix ' o eauilibriuim or reference condition r rudder s aircraft stability coordinate frame t throttle u,a,q,2,r~p zea'r as defined above ' x 1.0 INTRODUCTION The primary objective of this research effort is to derive a light, maneuverable aircraft-autopilot model as one extreme of aircraft type for final approach simulation studies. This model is to serve as a high priority vehicle in two FAA projects: developing requirements for a Scanning Beam Microwave Instrument Landing System and developing an all-encompassing generalized set of equations of an aircraft during approach and landing for all-weather landing system studies on the NAFEC hybrid computa- tion facility. In general, no such model existed for a common and repre- sentative light aircraft at the beginning of this effort because light aircraft are not generally designed by analysis and simula- tion. Similarly, light aircraft generally do not have automatic landing systems as standard equipment. The definition and identification of a light maneuverable aircraft is treated in Reference (1). The Piper PA-30 "Twin Comanche" is selected as the light maneuverable aircraft for modeling primarily because of the availability of data from wind tunnel and actual flight tests and because of an existing, partially useful, simulation model at the NASA Edwards Flight Research Center. The final aircraft model is derived from this simulation model and NASA TN D4983. The flight condition is based upon a high wind environment (headwind and sidewind of approximately 24 feet per second). This condition was selected to represent an extreme case for the simulation studies. The autopilot description is based on a report (Reference 6) prepared for TSC by Dr. Kohlman of the University of Kansas. The final configuration (i.e., gain values, gain scheduling and logic) was determined by simulation at TSC. 2.0 VEHICLE DESCRIPTION 4 The Piper PA-30 is a light twin engine low-wing monoplane. Figure 1 gives the principal dimensions. The airplane has a 2wing span of 35.98 ft., a wing area of 178 ft, an aspect ratio of 7.3, and a mean aerodynamic cord of 5 feet based on projec- tion of the outboard leading edge of the wing through the fuse- lage. The wing airfoil section is a modified NACA64 2 A215 air- foil with the trailing-edge cusp faired out. The wing has 50 of dihedral with no twist and is at 20 positive incidence with respect to the fuselage reference line. The airplane has the standard three-control system. The horizontal tail is of the all-movable type with a control deflection range of 40 to -14". The tail has a trailing-edge tab which moves in the same direc- tion as the tail with a deflection ratio (tab deflection to tail deflection) of 1.5. The control deflection range on each aileron is from 140 to -180. The rudder control deflection range is +270. 2.1 Vehicle Model A final approach model is presented for the Piper PA-30 aircraft based on data available from the NASA Edwards Flight Research Center simulation model and NASA TND 4983. The model c-nsists of rigid body, six-degree-of-freedom aircraft equations of motion which are basically linear perturbation equations in the stability axis system (some of the nonlinear cross coupling terms :Lave been included). 2.2 Axis Systems The stability axis frame (s) is depicted in Figure 2. The definition of airplane angles and sign convention is described in Figure 3. The stability axis is fixed to the aircraft and rotates and translates with the aircraft. Its origin is the center of mass of the aircraft. The X-axis is in the direction of motion of the airplane in a reference condition of steady -2- 12- (81) L75 (0.53) 3.40 WSW STA.J60 1,10.97) O Z=__ G~.0 01. 83) D1AM 5DIHEDRAL 4.25 Figure 1. Piper Dimensional Data- -3- CCL C C, C m9 Figure 2. System of Axes anid Positive4 Sense of Angles, Forces and Moment-s -4 - XE XSX TRIM CONDITION Sic-1 tonventio:=> 2'Stick Right, Right Aileron uP 1> a S Stick- Pak. Trai Iifloedge O => -6e Left ped.al! que~er to left -> "r )> ZSzE Figure 3. Definition of Airplane Angles and Sign Convention syn~etric flight. The Y-axis is normal to the aircraft's plane of symmetry (positive to the right), and the Z-axis is in the plane of symmetry (positive downward) and orthogonal to the X- and Z-axes. The aerodynamic stability derivatives are all re- ferenced to this axis system. A fuselage referenced body coordinate frame (B), is defined for determining the angular rates and velocities in body axis, and the aircraft normal acceleration. This axis system is similar to the stability axis system except that the X-axis is directed along the fuselage. The angle ar relates the two axis systems. A local vertical coordinate frame (E) is defined for determining the velocities of the aircraft with respect to the air mass. This coordinate frame has its origin at the center of mass of the aircraft with the X-axis pointing North, and Y-axis pointing East, and the Z-axis pointing down. The velocities from this coordinate frame can be converted to ground velocity by adding the various components of the steady wind. 2.3 Assumptions in Using Aircraft Equations The derivation of the aircraft equations involved the following assumptions: 1. Aircraft mass is constant. 2. The earth can be considered an inertial frame. - The aircraft is a rigid body. 4. The aircraft is symmetrical about its X-Z plane. 5. The aircraft is initially in equilibrium flight with no linear or angular accelerations, no angular rates, and no initial roll angle or lateral velocity. 6. Small disturbance (perturbation) theory is used. Motions and forces are referred to the equilibrium flight condition. 7. Hinge moments are insignificant. -6- 2.4 Equations of Motion Draq Equation M + (cos T)AT Lift Equation 2 CL 0 cmu 0 \ / C q UV 2Uc L&V / \+ c- + Co a + L qfs - + mq mUo m qsi CT A ++- 0 qu .- CL6 et e 7S Pitching Moment Equation C I Z cmu - a - c 6 + - z LT u U m- m - q U 2 m q - m. e sc o oo e Sideforce lauation mU o b mU b pC C - g + - r --- c r + -ru -sO p qqo CYr Cy 6 r6r + C6a6a Rolling Moment Equation XXs b C x z s b- C r =C 6r + C 6a + +f I-- kU 2U 0oY r £6r 6 2U p Yawinq Moment Equation I I -c b ZZ s b C r=C 6r+C 6 a n qS 2U0 n pP+ 2U-- nr n rn CnsB- - P- 2- Pr n7 Body Axis Rates and Velocities q B q PB = p cos(r) - r sin( r ) r = r cos(at ) + p sin (ar u = (UO + u) cos(a r ) -Uoa sin (at) wB = U a cos(ar) + (U + u) sin(a B o r 0 r vB 0 Accelerations Vertical acceleration at center of gravity wB = Uo cos(t ) + u sin(a r ) Vertical acceleration at accelerometer * Wacce I = w7B - Xacce 1 Measured Normal Acceleration a= Wace I + PBVB - q UB Euler Anle Rates = cos p - r sin D = p + (q sin D 4- r cos P) tan e S= (q sin P + r cos D) sec 0 8 Translation Equations With Respect to Moving Frame XE = (Uo+U) cos Cos + Uo(sin sin 6 cos 0-cos sin Y) + U a(cos P sin 6 cos + sin sin )0 YE = (Uo+u) cos 6 sin Y + Uo (sin I sin e sin '+cos cos Y) + U0 (cos 4 sin e sin tb -sin ¢ cos ') ZE = (Uo+u)(-sin e) + Uoa sin cos S + Uo0 cos ¢ cos . 2.5 Vehicle Data Geometry b = 35.98 ft. c = 5. ft. S = 178. ft. 2 ZT = -. 75 ft. a T = 0 deg. Weight and Inertias m = 111.9 slugs IxxB = 2800 slug-ft 2 Ixx s = 2801.7 slug-ft 2 I = 1900 slug-ft 2 I = 1900. slug-ft 2 IyyB yys Izz B = 4500 slug-ft 2 1zz s = 4513.7 slug-ft 2 IxzB = 80 slug-ft2 Ixz s = - 7.9 slug-ft 2 Trim Flight Condition - Final Approach(High Wind Environment) U 0 = 176. ft/sec o = .002378 slugs/ft 3 CLo = .55 q = 36.8 lbs/ft 2 CDo = .034 a = .0515 rad (2.95 ° ) o = 0.0 CT (thrust coeff.)= .034 YO = 0.0 6 (trim elevator setting)= 0.40 center of gravity at 10% .LAC ( egear down 6f (trim flap setting) = 0.0 9 Nor-Dimensional Derivatives - Final Approach CD =0. CL =0. Cm = 0. C D =.275 C = 5.04 Cm = -1.147 D = 0. rL. 5.3 CM. = -14.55 CD = 0. CL = 9.12 Cm = -25.0 q q q CD; e = 0. CL6 e = 1.05 Cri e = -2.87 C = -.086 C .0756 - -.494 Xang -a C = .11 Cn = -.16 Cy = 0. r r r C. = -.50 C = -.063 C = 0. p np yP CZ6 r = .01147 CnS r = -.0573 Cy r = .143 Ct = -.0803 C = .00573 Cy = -.00916 6a n6a 6a 2.6 Control Wheel and Pedal Characteristics Aileron: Gearing constant at zero control deflection, 0.80 radian/ft.; wheel deflection, ±900; max. force at end of wheel, 15 lb. Elevator: Gearing constant at zero control deflection, 0.42 radian/ft.; wheel throw, 4 inches forward, 5 inches aft; force differential 40 lbs. (detent position is 4 inches from firewall) Rudder: Gearing constant at zero control deflection, 0.93 radian/ft.; pedal deflection, 4 1/2 inches; maximum force at full deflection, 120 lbs. -10- 3. ENGINE THRUST DATA Two 160 h.p. Lycoming IO-320-B four-cylinder air-cooled en- gines power the PA-30. Figure 4 presents the maximum available manifold pressure from each engine as a function of standard tem- perature altitude. This curve should be mechanized so that he pilot's indicator shows the maximum value as a function of alti- tude at full throttle. Figure 5 presents the power available per engine as a func- tion of r.p.m., MAP, and altitude. This curve should be mechan- ized so that power can be obtained from the r.p.m., MAP, and alti- tude. Once the power is obtained, thrust can be calculated from the following equation: T (lbs.) =325 N BHP VT (knots) where Np (power efficiency) = 0.74 Alternatively, a simplified algebraic expression for BHP has been derived from Figure 5 for the final approach configuration and can be used in place of Figure 5. The expression is given by BHP = .0024(h) + .0028(r.p.m.) [ t (29.2-.000989(h))] -8.0 1 where h = altitude in feet 6t = 1 = full throttle An engine lag of about 0.1 second can be used in completing the engine transfer function (i.e. throttle-thrust). -11- 0F 0 0 0 06 oo 0 0 4-i to 0 0-. . -0 0 0 -0 0 In 0 0 6H N 3unS38d010.1NVY 31O~lVA wnrsxv Ew C. > aw E W D a . X U 0 0 fw LL -Ja. 0 w ,0 -J 0 : 0 J WJo -J o- J 0 o .. ='-> 0 :az 0w -J < -J(f) r I- 0 L<E 0 -- I - 0 U1 _ u.~ 0:u 0 0 0 00 oz 4-() U) - 0 0 U o-J CD' SS8 OD W ISI 3~3S0 X8 & 0. 00 U'0 ____ _ __ ____ ___ ____ ____ ___ ____ ____ ___ -2~3- 4. AUTOPILOT This section describes a representative PA-30 three-axis autopilot with stabilization system and with an automatic ILS ap- proach mode available down to decision heiqht. Block diagrams of the longitudinal and lateral axis are shown in Figures 6 and 7 respectively. Damping is provided about all three axes, and inputs to the system come from a vertical gyro (0, ),yaw rate gyro (r), altitude sensor, and navigation receiver. Limiters are installed in the lateral channel to prevent exces- sive roll angles in response to large error signals. Gain schedul- ing is provided in the longitudinal channel to desensitize the system to glidepath errors as the runway threshold is approached. Automatic ILS approaches are possible down to decision height. The pilot must take over manually at that point because there is no automatic flare capability. The auto-trim system, which actuates the elevator trim tab to unload the elevator servo, is not included in the block diagrams since it has no measurable influence on the performance or dynamic response of the airplane-autopilot system. It is a time-delayed, slow, integrating actuator and as such does not respond to tran- sients. The output of the autopilot system is expressed in control surface deflections. Because of the dependence of deflections on dynami pressure, the final gain of 1/a represents the aerodynamic gain of the control surfaces. The glideslope system is engaged by approaching the glidepath at constant altitude (usually in the altitude hold mode) on the localizer beam. Initially the glideslope error signal will be a strong nose up command but switch B will be open. The "glideslope engage logic" circuit continuously monitors the signal at point B, which will gradually decrease in strength as the glidepath is ap- proached. When the signal at point B reaches a 0.044 rad. (2.50) nose up command, switch B is closed, and the altitude hold mo&e is -14- 0' 0 . 0 0 3 U) U a- P cc 0 3-cI 0a 0 i) IX 1 S1 ~ 0 ~j Li ~ I0 .CI! ~ 3I C, 0 S1 ~t+ ~ -wI I- WI + ILII 0+0 Z i,7 0 0 cr > 0 -15- -71- A 60 - vi - -- -; A z X ~. r 1 aLe hij z ) x C! IF-- I- +z 01 F- I < IJ - 1 2 Fl1 17 ILI -'< 0 +'4 ,lsengaged. To prevent a sudden pitchup at this point a 0.044 rad. nose down cowand is biased into the attitude circuit, to cancel the input frrcm switch B. Thus, after engagement, the air- plane will continue in level flight. As it moves closer to the glidepath, the signal at B will continue to decrease, and the bias signal will gradually pitch the nose down. The gains and logic switches are set so that the airplane gradually approaches the correct glideslooe with minimal cvershoot. Speed is controlled manually at all times in the representative autopilot system since auto-throttles are not typically provided. l'weve, a preliminary, auto-throttle was derived at TSC by simulation and zan be used for completely automatic approach studis. The auto-throttle sys- tem is shown in Figure 8. I -17- 0 0 0W Wcn~in~41 U) -U0 00 La 0 m W3 Ir w w0 5. INSTRUMEW.T PANEL 5 The instrument panel layout is shown in Figure 9. It is de- signed to accommodate the customary advanced flight instruments on the left side in front of the pilot and engine instruments on the right side. -19- a 0 0 2~2 *:51-a 161~ , ~ Qw* 20 2 40~ 4 > 00 t 4 J 0 42... 10111 z Hw 00 C 2 - 2 -j: a,:: 2 2 - oz0 z 44 I Z C; aa :; V 0 02C am xi - -~~- o 0 0 04) 0 4 n0.-0~ I* ~....' 3,44o0. 2 I I-00020 REFERENCES 1. Koziol, Joseph, Manueverability of Conventional Fixed Wina Aircraft in the Landinq Approach, in-House Memorandum, December 15, 1970 2. Koziol, Joseph, Selection of a Representative Liaht Maneuver-- able Aircraft for Instrumented Approach and Landinq Studies, In-House Memorandur August 13, 1970 3. Koziol, Joseph, A Comparison Between the Cessna 310 and the Piper PA-30 Twin Comanche, In-House Meioran- dum, October 6, 1970 4. Fink, P. Marvin, and Delma C. Freeman, Jr., Full-Scale Wind-Tunnel Investiaation of Static Longitud- inal and Lateral Characteristics of a Light Twin-Engine Airplane, NASA TN D-4983, January, 1969. 5. Piper Twin Comanche Owner's Handbook, Part No. 753 773, January, 1970 6. Kohlman, David L., An Analytical Description of Typical Light Airplane Autopilot and Control Systems, Prepared for TSC under Purchase Order No. TS-578, December 14, 1970. - -21- I