WOLFRAM SYSTEM MODELER
WingBodyContribution from wingbody combination to aerodynamic forces 
SystemModel["Aircraft.Physical.FixedWing.Parts.ControlSurfaces.Components.WingBody"]
This model calculates the aerodynamic forces due to the wingbody (combination of fuselage and main wing), estimates the mass properties of the main wing and solves the consequent Newton—Euler equations in the bodyShapeLeftWing and bodyShapeRightWing components if the weight estimation is used. If any of the mass properties are known when the weight estimation method is used, their value can be entered directly in the Estimated Mass Properties tab, thus bypassing the equation to estimate their value.
All other parameters, including the wingbody specific parameters, are propagated to this model from the AircraftBase model, and therefore their values should not be changed here but only in the complete aircraft model. Additionally, the variables of the global flight conditions are propagated here from the AircraftBase model inside the flightData record.
All forces in this model act on the aerodynamic center locations on both sides of the main wing, including the aerodynamic forces due to the fuselage. The location of the aerodynamic center is defined by the parameters xWingAC and yWingAC, which will assume the location to be at quarter chord at the mean aerodynamic chord (MAC). For calculating the location of the MAC, a simple trapezoidal wing shape is assumed, with the given wing span, root chord, tip chord and sweep angle. On the body z axis, the aerodynamic center location is solved from the given z coordinate for main wing root chord leading edge (zWingRootLE), dihedral angle (gammaWing) and yWingAC.
The equations to estimate the aerodynamic center location as well as any other estimated main wing parameter can be bypassed by entering the known value in the Main Wing group in the Derived Properties parameter tab in the AircraftBase model.
The contribution to the aerodynamic forces by ailerons is not included in this model but in its dedicated Ailerons model.
This section describes how the magnitude and orientation of the lift force are calculated in this model.
The parameter for the lift curve slope of the 2D airfoil (ClAlphaWing2D) is used together with other main wing properties and the Mach number to derive the lift curve slope for the entire main wing (C_{Lα,w}) such that the air compressibility effects are also considered. Furthermore, an empirical relationship is used to derive the lift curve slope of the wingbody combination (C_{Lα,wb}) from the main wing lift curve slope and the properties of the fuselage. Thus, the lift coefficient of the wingbody is solved by
,
where α_{eff,w} is the effective angle of attack seen by the main wing, such that the contribution of the wing incidence angle, wing zerolift angle and the induced angle of attack due to wing dihedral in sideslip (Δα) are considered. The Δα is estimated here as the product of the sideslip angle (β) and wing dihedral angle (Γ), as given by Nelson [1]. Due to the differential contribution by Δα to the α_{eff,w} value, the lift coefficient of the wingbody have different values on different sides of the wing (if Γ ≠ 0). All steps to derive the wingbody lift coefficient are described in detail in section 3.3.2 in Reference [2].
The dimensionless lift coefficient for wingbody is multiplied by the global dynamic pressure and half of the wing reference area to get the magnitude of the lift force acting on the aerodynamic center on one side of the main wing. The lift force is oriented such that it is perpendicular to the free stream. Additionally, components along the body y axis are added to the lift force to account for the wing dihedral angle, as shown in Figure 1. Thus, the resultant lift force is actually greater than L_{wb} but the components acting on the body xz plane still always equals to L_{wb}.
Figure 1: Orientation of the aerodynamic forces acting on the main wing aerodynamic center.
In addition to the ordinary lift force, the induced lift forces acting along the body z axis are also modeled and added to the forces acting on the main wing aerodynamic centers. They are the induced z force due to yaw rate and the induced z force due to roll rate, and their values are calculated as presented in Nelson [1].
For solving the parasite drag coefficient (C_{D,0}) of different components (c) in the aircraft, including the main wing and fuselage, the component buildup method presented by Raymer [3] is used, defined as
,
where the form factor (FF_{c}) and the area (S_{wet,c}) are functions of the geometry of the main wing and fuselage. The skin friction coefficient (C_{f,c}) is a function of the surface roughness height of main wing and fuselage, Mach number and Reynolds number for the flow over the mean aerodynamic chord of the main wing and over the fuselage length. Thus, the air compressibility effects are included in the drag calculations. The complete derivation of the C_{D,0,c} can be found in section 3.3.1 in Reference [2].
For both the fuselage and the main wing, liftinduced drag is also considered in the complete drag coefficient (C_{D}_{,w} and C_{D}_{,}_{fus}). For the main wing, the complete drag coefficient reads as
,
where K_{w} is an empirical factor, and its value is based on the wing geometry and calculated through a method given by Cook [4]. For the fuselage, the liftinduced drag is calculated as a function of the maximum fuselage drag coefficient that is achieved at α = 90° and the current angle of attack. The complete derivation of the equations to solve for C_{D}_{,}_{fus} is described in section 3.3.1 in Reference [2].
The C_{D}_{,w} coefficient is multiplied by the global dynamic pressure (q) and half of the wing reference area (S_{ref,w} / 2) to get the magnitude of the drag force acting on the aerodynamic center on one side of the main wing. To solve the drag force due to fuselage, the C_{D}_{,}_{fus} coefficient is multiplied by q and S_{ref,w}. The magnitude of the drag force due to the wingbody acting on the aerodynamic center on either side of the main wing is
.
This drag force might be different on each side of the main wing due to the contribution by the main wing liftinduced drag, as the lift can differ between the different sides of the main wing. The drag force is also oriented such that it always is parallel with the free stream, as shown in Figure 1.
In addition to the ordinary drag force, the induced drag force acting along the body x axis due to roll rate is also modeled and added to the forces acting on the main wing aerodynamic centers. Its value is calculated as presented by Nelson [1].
Stall, which occurs at high angles of attack due to flow separation on the wing upper surface, is modeled for the wingbody up to α_{eff} = 90°, but not for large negative angles of attack.
The wingbody lift coefficient (C_{L}_{,}_{wb}) follows the linear lift curve slope, i.e. the C_{Lα,wb}, until the vicinity of achieving the maximum value for main wing lift coefficient (C_{L}_{,}_{max,w}), after which it starts following different quadratic equations. The quadratic equations are calculated based on the constraints shown in Figure 2 to generate a generic stall model for a wing. The constraints also include that the function for C_{L}_{,}_{wb} needs to be differentiable everywhere.
The parameter for C_{L}_{,}_{max,w}, which is derived from the maximum lift coefficient of the 2D airfoil (ClMaxWing2D), is used together with C_{Lα,wb} to define the constraints for the different quadratic equations, which are highlighted with different colors in Figure 2.
Figure 2: Model for wing lift coefficient in stall. [2]
The main wing drag coefficient (C_{D}_{,}_{w}) follows the equation defined in the previous section until the C_{L}_{,w} coefficient achieves C_{L}_{,}_{max,w}, after which it starts following a sine curve toward the maximum drag coefficient value (C_{D}_{,}_{max,w}) at α_{eff} = 90°, as shown in Figure 3. The value of the C_{D}_{,}_{max,w} coefficient is calculated based on the drag coefficient of a flat plate, which has same aspect ratio as the main wing and is perpendicular to the wind. Additionally, in order to make the function for C_{D}_{,}_{w} differentiable where it transits into the sine curve, a cubic function is fitted between 1° before and 1° after the intersection.
Figure 3: Model for wing drag coefficient in stall. [2]
Further explanation about stall and how it is modeled in the library can be found in section 3.3.3 in Reference [2].
If the weight estimation method is used and no mass properties are entered by the user, the main wing mass properties are estimated by using an empirical relationship based on the given parameters describing the wing geometry and the design variables in the AircraftBase model. The main wing mass is estimated by a method presented by Nicolai and Carichner [5], and it is further described in section 3.4.2 in Reference [2].
The center of mass location is solved individually for both sides of the main wing by using an empirical relationship describing their locations as fractions of the spanwise and chordwise lengths such that the rotations around the wing incidence and dihedral angles are also considered. The derivation of the main wing center of mass locations and the equations to solve for their coordinates with respect to the fuselage reference point are described in detail in section 3.5.2 and in Appendix A.1 in Reference [2], respectively.
The main wing moments of inertia are estimated by a method presented in USAF DATCOM [6], and it is also converted to be used with SI units in Appendix A.2 in Reference [2].
[1] Nelson, R. C. (1998). Flight Stability and Automatic Control. 2nd ed. McGrawHill.
Available at: http://home.eng.iastate.edu/~shermanp/AERE355/lectures/Flight_Stability_and_Automatic_Control_N.pdf.
[2] EräEsko, N. (2022). "Development and Use of System Modeler 6DOF Flight Mechanics Model in Aircraft Conceptual Design."
Available at: modelica://Aircraft/Resources/Documents/EraeEskoThesis.pdf.
[3] Raymer, D. P. (1992). Aircraft Design: A Conceptual Approach, 2nd Ed. American Institute of Aeronautics and Astronautics.
[4] Cook, M. (2012). Flight Dynamics Principles. 3rd ed. Elsevier.
[5] Nicolai, L. M. and G. E.Carichne., (2010). Fundamentals of Aircraft and Airship Design, Volume 1–Aircraft Design.
American Institute of Aeronautics and Astronautics.
[6] Finck, R. D. (1978). USAF (United States Air Force) Stability and Control DATCOM (Data Compendium).
MCDONNELL AIRCRAFT CO ST LOUIS MO
Available at: https://apps.dtic.mil/sti/citations/ADB072483.
mWing 
Value: if machDes < 0.4 then if compMat then 0.8 * 96.948 * ((MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.65 * (ARwing / cos(lambdaWing)) ^ 0.57 * (SrefWing * 10.764 / 100) ^ 0.61 * ((1 + TRwing) / (2 * (tWingRoot / cWingRoot))) ^ 0.36 * sqrt(1 + sqrt(2 * qMax / rho0) * 1.9438 / 500)) ^ 0.993 / 2.2046 else 96.948 * ((MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.65 * (ARwing / cos(lambdaWing)) ^ 0.57 * (SrefWing * 10.764 / 100) ^ 0.61 * ((1 + TRwing) / (2 * (tWingRoot / cWingRoot))) ^ 0.36 * sqrt(1 + sqrt(2 * qMax / rho0) * 1.9438 / 500)) ^ 0.993 / 2.2046 else if compMat then 0.8 * 0.00428 * (SrefWing * 10.764) ^ 0.48 * (ARwing * (sqrt(2 * qMax / rho0) / a0) ^ 0.43 * (MTOMdes * 2.2046 * nMax * 1.5) ^ 0.84 * TRwing ^ 0.14) / ((100 * (tWingRoot / cWingRoot)) ^ 0.76 * cos(lambdaWingHC) ^ 1.54) / 2.2046 else 0.00428 * (SrefWing * 10.764) ^ 0.48 * (ARwing * (sqrt(2 * qMax / rho0) / a0) ^ 0.43 * (MTOMdes * 2.2046 * nMax * 1.5) ^ 0.84 * TRwing ^ 0.14) / ((100 * (tWingRoot / cWingRoot)) ^ 0.76 * cos(lambdaWingHC) ^ 1.54) / 2.2046 Type: Mass (kg) Description: Main wing mass 

rCMwingRight 
Value: {xWingRootLE + tan(lambdaWingLE) * wFus / 2  cWingRoot / 4, 0, zWingRootLE  tan(gammaWing) * wFus / 2} + {xWingCM, yWingCM, tan(gammaWing) * yWingCM} Type: Length[3] (m) Description: Right wing center of mass coordinates w.r.t. fuselage reference point 
rCMwingLeft 
Value: {xWingRootLE + tan(lambdaWingLE) * wFus / 2  cWingRoot / 4, 0, zWingRootLE  tan(gammaWing) * wFus / 2} + {xWingCM, yWingCM, tan(gammaWing) * yWingCM} Type: Length[3] (m) Description: Left wing center of mass coordinates w.r.t. fuselage reference point 
IyyWing 
Value: 0.000293 * 0.703 * (i0Wing  wvxWing ^ 2 / (mWing * 2.205 / 2)) Type: MomentOfInertia (kg⋅m²) Description: (Half) wing pitch moment of inertia 
IxxWing 
Value: 0.000293 * (mWing / 2 * 2.205 * (bWing / 2 * 39.37) ^ 2 * k1Wing) / 18 * (1 + 2 * cWingRoot * 39.37 * cWingTip * 39.37 / (cWingRoot * 39.37 + cWingTip * 39.37) ^ 2) Type: MomentOfInertia (kg⋅m²) Description: (Half) wing roll moment of inertia 
IzzWing 
Value: IxxWing + IyyWing Type: MomentOfInertia (kg⋅m²) Description: (Half) wing yaw moment of inertia 
xWingCM 
Value: if lambdaWing < 0 or lambdaWing > 0 then cWingRoot / 4  tan(lambdaWingLE) * (0.35 * bWing / 2  wFus / 2)  0.5 * ((cWingTip  cWingRoot) / (bWing / 2  wFus / 2) * (0.35 * bWing / 2  wFus / 2) + cWingRoot) else cWingRoot / 4  tan(lambdaWingLE) * (0.4 * bWing / 2  wFus / 2)  0.4 * ((cWingTip  cWingRoot) / (bWing / 2  wFus / 2) * (0.4 * bWing / 2  wFus / 2) + cWingRoot) Type: Length (m) Description: Main wing center of mass xcoordinate w.r.t. wing root quarter chord (inside fuselage) 
yWingCM 
Value: if lambdaWing < 0 or lambdaWing > 0 then 0.35 * bWing / 2 else 0.4 * bWing / 2 Type: Length (m) Description: (Half) wing center of mass ycoordinate w.r.t. fuselage centerline 
caWing 
Value: min({bWing * tan(lambdaWingLE) / 2 * 39.37, cWingTip * 39.37 + bWing * tan(lambdaWingLE) / 2 * 39.37, cWingRoot * 39.37}) Type: Real Description: Factor for calculating wing moment of inertias 
cbWing 
Value: sum({bWing * tan(lambdaWingLE) / 2 * 39.37, cWingTip * 39.37 + bWing * tan(lambdaWingLE) / 2 * 39.37, cWingRoot * 39.37})  caWing  ccWing Type: Real Description: Factor for calculating wing moment of inertias 
ccWing 
Value: max({bWing * tan(lambdaWingLE) / 2 * 39.37, cWingTip * 39.37 + bWing * tan(lambdaWingLE) / 2 * 39.37, cWingRoot * 39.37}) Type: Real Description: Factor for calculating wing moment of inertias 
k1Wing 
Value: 0.988158 + 2.20444 * (yWingCM / (bWing / 2 / 3 * (cWingRoot + 2 * cWingTip) / (cWingRoot + cWingTip))) ^ 1.1 Type: Real Description: Factor for calculating wing moment of inertias 
rhoiWing 
Value: mWing * 2.205 / (caWing + cbWing + ccWing) Type: Real Description: Factor for calculating wing moment of inertias 
wvxWing 
Value: rhoiWing / 6 * (caWing ^ 2 + cbWing ^ 2 + ccWing * cbWing + ccWing ^ 2) Type: Real Description: Factor for calculating wing moment of inertias 
i0Wing 
Value: rhoiWing / 12 * (caWing ^ 3 + cbWing ^ 3 + ccWing ^ 2 * cbWing + ccWing * cbWing ^ 2 + ccWing ^ 3) Type: Real Description: Factor for calculating wing moment of inertias 
weightEst 
Value: Type: Boolean Description: true, if weight estimation method is used for masses, center of mass location and inertia tensor 
xCMdry 
Value: Type: Length (m) Description: Aircraft center of mass xcoordinate w.r.t. fuselage reference point (with total mass for electric aircraft and gliders, positive xaxis towards nose) 
MTOMdes 
Value: Type: Mass (kg) Description: Design maximum takeoff mass 
machDes 
Value: Type: Real Description: Design Mach number 
compMat 
Value: Type: Boolean Description: true, if composite materials are used in structures 
qMax 
Value: Type: Pressure (Pa) Description: Maximum dynamic pressure 
nMax 
Value: Type: Real Description: Maximum load factor 
lFus 
Value: Type: Length (m) Description: Fuselage length 
wFus 
Value: Type: Length (m) Description: Fuselage maximum width 
SwetFus 
Value: Type: Area (m²) Description: Fuselage wetted area 
FFfus 
Value: Type: Real Description: Fuselage form factor 
bWing 
Value: Type: Length (m) Description: Main wing span 
cWingRoot 
Value: Type: Length (m) Description: Main wing root chord (where wing intersects with fuselage) 
cWingTip 
Value: Type: Length (m) Description: Main wing tip chord 
tWingRoot 
Value: Type: Length (m) Description: Main wing root thickness 
xWingRootLE 
Value: Type: Length (m) Description: Main wing root leading edge xcoordinate w.r.t. fuselage reference point (positive xaxis towards nose) 
zWingRootLE 
Value: Type: Length (m) Description: Main wing root leading edge zcoordinate w.r.t. fuselage reference point (positive zaxis towards ground) 
lambdaWing 
Value: Type: Angle (rad) Description: Main wing sweep angle at 1/4 chord 
gammaWing 
Value: Type: Angle (rad) Description: Main wing dihedral angle 
iWing 
Value: Type: Angle (rad) Description: Main wing incidence angle 
SrefWing 
Value: Type: Area (m²) Description: Main wing reference area 
ARwing 
Value: Type: Real Description: Main wing aspect ratio 
TRwing 
Value: Type: Real Description: Main wing taper ratio 
cWingMean 
Value: Type: Length (m) Description: Main wing mean chord length 
xWingAC 
Value: Type: Length (m) Description: Main wing aerodynamic center from wing leading edge at mean chord (positive xaxis towards nose) 
yWingAC 
Value: Type: Length (m) Description: Main wing aerodynamic center from fuselage centerline (ycoordinate w.r.t. fuselage centerline of mean chord) 
SwetWing 
Value: Type: Area (m²) Description: Main wing wetted area 
lambdaWingLE 
Value: Type: Angle (rad) Description: Main wing leading edge sweep angle 
lambdaWingHC 
Value: Type: Angle (rad) Description: Main wing halfchord sweep angle 
FFwing 
Value: Type: Real Description: Main wing form factor 
sdWing 
Value: Type: Real Description: Fuselage drag factor for main wing 
kdWing 
Value: Type: Real Description: Empirical constant for Oswald efficiency factor for main wing 
kSkinFus 
Value: Type: Length (m) Description: Fuselage surface roughness height 
CDmaxFus 
Value: Type: Real Description: Maximum drag coefficient of the fuselage 
kSkinWing 
Value: Type: Length (m) Description: Main Wing surface roughness height 
ClAlphaWing2D 
Value: Type: CurveSlope (rad⁻¹) Description: Change in the section lift coefficient of the main wing airfoil (2D) due to alpha 
alpha0Wing2D 
Value: Type: Angle (rad) Description: Zerolift angle of attack of the main wing airfoil (2D) 
CLmaxWing3D 
Value: Type: Real Description: Maximum lift coefficient of the main wing (3D) 
CDmaxWing3D 
Value: Type: Real Description: Maximum drag coefficient of the main wing (3D) 
CADshapes 
Value: Type: Boolean Description: true, if external CAD files are used for animation 
rho0 
Value: Type: Density (kg/m³) Description: Air density at sealevel 
a0 
Value: Type: Velocity (m/s) Description: Speed of sound at sealevel 
flightData 
Type: FlightData Description: Global flight data variables 

aircraftRP 
Type: Frame_b Description: Connector to aircraft reference point 


yLiftWingbody 
Type: RealOutput Description: Lift force due to wingbody 

uLiftAC 
Type: RealInput Description: Total lift force 

yEpsilonAlpha 
Type: RealOutput Description: Change in downwash due to alpha 

yCLalphaWingbody 
Type: RealOutput Description: Change in the lift coefficient of the wingbody due to alpha 
flightData 
Type: FlightData Description: Global flight data variables 


liftRightWingbody 
Type: WorldForce Description: Right wingbody lift force 

liftLeftWingbody 
Type: WorldForce Description: Left wingbody lift force 

dragRightWingbody 
Type: WorldForce Description: Right wingbody drag force 

dragLeftWingbody 
Type: WorldForce Description: Left wingbody drag force 

bodyShapeRightWing 
Type: BodyShape Description: Mass and inertia tensor of right wing 

translWing 
Type: FixedTranslation Description: Position of the main wing root quarter chord (inside fuselage) w.r.t fuselage reference point 

rotRightWing 
Type: FixedRotation Description: Rotation of right wing around dihedral and wing incidence angle 

rotLeftWing 
Type: FixedRotation Description: Rotation of left wing around dihedral and wing incidence angle 

bodyShapeLeftWing 
Type: BodyShape Description: Mass and inertia tensor of left wing 

rotRightWingForces 
Type: FixedRotation Description: Rotation of aerodynamic forces acting on right wing around dihedral and wing incidence angle 

rotLeftWingForces 
Type: FixedRotation Description: Rotation of aerodynamic forces acting on left wing around dihedral and wing incidence angle 

rightWingShape 
Type: FixedShape Description: Visualization on right wing 

leftWingShape 
Type: FixedShape Description: Visualization on left wing 

translRightWing 
Type: FixedTranslation Description: Translation from the main wing root quarter chord (inside fuselage) to right wing aerodynamic center 

translLeftWing 
Type: FixedTranslation Description: Translation from the main wing root quarter chord (inside fuselage) to left wing aerodynamic center 
Aircraft.Physical.FixedWing.Parts.ControlSurfaces Model of system of surfaces in conventional wing configuration 