WOLFRAM SYSTEM MODELER
VerticalTailContribution from vertical tail to aerodynamic forces 
SystemModel["Aircraft.Physical.FixedWing.Parts.Surfaces.Components.VerticalTail"]
This model calculates the aerodynamic forces due to the vertical tail and rudder, estimates the mass properties of the vertical tail and solves the consequent Newton—Euler equations in the bodyVerticalTail component 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. Stall due to high sideslip angles is not modeled for the vertical tail.
All other parameters, including the vertical tail and rudder 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 location of the vertical tail. The location of the aerodynamic center along the body x axis is defined by the parameter lVTwingAC, 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 vertical tail span, root chord, tip chord and sweep angle.
The equations to estimate the aerodynamic center location as well as any other estimated vertical tail parameter can be bypassed by entering the known value in the Vertical Tail group in the Derived Properties parameter tab in the AircraftBase model.
This section describes how the magnitude and orientation of the lift force are calculated in this model. For the vertical tail, the lift force acts perpendicular to the wind on the body xy plane.
The parameter for the lift curve slope of the 2D airfoil (ClAlphaVT2D) is used together with other vertical tail properties and the Mach number to derive the lift curve slope for the entire vertical tail (C_{Lα,VT}) such that the air compressibility effects are also considered. The complete method to derive C_{Lα,VT} is described in detail in section 3.3.2 in Reference [1]. Furthermore, the lift curve slope of the vertical tail with respect to its rudder deflection angle (C_{Lδr,VT}) is calculated as the product of C_{Lα,VT} and rudder effectiveness parameter, which is derived from the rudder and vertical tail surfaces according to the method presented by Nelson [2].
However, observing the forces and orientation from an entire aircraft perspective, the lift force acting on the body xy plane on the vertical tail actually is a function of the sideslip angle rather than the angle of attack. Thus, the lift curve slope for the vertical tail with the contribution from the sidewash angle due to the sideslip angle (sigmaBeta) is denoted as C_{Y,β,VT} or CyBeta, specifying more clearly the direction of the force and the cause of it. The sigmaBeta is calculated based on a formula presented in section 3.3.4 in Reference [1] and originally presented in USAF DATCOM [6].
Thus, the lift coefficient (or in other words, the Yforce coefficient) of the vertical tail with contribution by the rudder deflection is solved by
,
where β_{eff,VT} is the effective sideslip angle seen by the vertical tail, such that the induced sideslip angle due to yaw rate is considered according to the method presented by Nelson [2].
The dimensionless Yforce coefficient for vertical tail with contribution by the rudder deflection is multiplied by the global dynamic pressure and vertical tail reference area to get the magnitude of the Yforce force acting on the aerodynamic center. The Yforce is oriented such that it is perpendicular to the free stream rather than acting only along the body y axis, as shown in Figure 1.
Figure 1: Orientation of the aerodynamic forces acting on the vertical tail aerodynamic center.
For solving the parasite drag coefficient (C_{D,0}) of different components in the aircraft, including the vertical tail, the component buildup method presented by Raymer [3] is used, defined as
,
where the form factor (FF_{VT}) and the area (S_{wet,VT}) are functions of the geometry of the vertical tail. The skin friction coefficient (C_{f,VT}) is a function of the surface roughness height of the vertical tail, Mach number and Reynolds number for the flow over the mean aerodynamic chord of the vertical tail. Thus, the air compressibility effects are included in the drag calculations. The complete derivation of the C_{D,0,VT} can be found in section 3.3.1 in Reference [1].
Liftinduced drag is also considered in the complete drag coefficient for the vertical tail (C_{D}_{,VT}), which reads as
,
where K_{VT} is an empirical factor, and its value is based on the vertical tail geometry and calculated through a method given by Cook [4].
The C_{D}_{,VT} coefficient is multiplied by the global dynamic pressure (q) and the wing reference area (S_{ref,w}) to get the magnitude of the drag force acting on the aerodynamic center of the vertical tail. The drag force is also oriented such that it is always parallel with the free stream, as shown in Figure 1.
If the weight estimation method is used and no mass properties are entered by the user, the vertical tail mass properties are estimated by using an empirical relationship based on the given parameters describing the vertical tail geometry, its position and the design variables in the AircraftBase model. The vertical tail mass is estimated by a method presented by Nicolai and Carichner [5], and it is further described in section 3.4.3 in Reference [1].
The center of mass location is solved by using an empirical relationship describing its location as fractions of the spanwise and chordwise lengths. The derivation of the vertical tail center of mass location and the equations to solve for its coordinate with respect to the fuselage reference point are described in detail in section 3.5.2 and in Appendix A.1 in Reference [1], respectively.
The vertical tail moments of inertia are estimated by a method presented in USAF DATCOM [6], and it is also converted to be used with SIUnits in Appendix A.2 in Reference [1].
[1] 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.
[2] 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.
[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.
weightEst 
Value: Type: Boolean Description: true, if weight estimation method is used for masses, center of mass location and inertia tensor 

zCMdry 
Value: Type: Length (m) Description: Aircraft center of mass zcoordinate w.r.t. fuselage reference point (with total mass for electric aircraft and gliders, positive zaxis towards ground) 
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 
wFus 
Value: Type: Length (m) Description: Fuselage maximum width 
hFus 
Value: Type: Length (m) Description: Fuselage maximum height 
bWing 
Value: Type: Length (m) Description: Main wing span 
xWingRootLE 
Value: Type: Length (m) Description: Main wing root leading edge xcoordinate w.r.t. fuselage reference point (positive xaxis towards nose) 
SrefWing 
Value: Type: Area (m²) Description: Main wing reference area 
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) 
lambdaWingLE 
Value: Type: Angle (rad) Description: Main wing leading edge sweep angle 
zHTrootLE 
Value: Type: Length (m) Description: Horizontal tail root leading edge zcoordinate w.r.t. fuselage reference point (positive zaxis towards ground) 
bVT 
Value: Type: Length (m) Description: Vertical tail span 
cVTroot 
Value: Type: Length (m) Description: Vertical tail root chord 
cVTtip 
Value: Type: Length (m) Description: Vertical tail tip chord 
tVTroot 
Value: Type: Length (m) Description: Vertical tail root thickness 
zVTroot 
Value: Type: Length (m) Description: Vertical tail root zcoordinate w.r.t fuselage reference point 
lambdaVT 
Value: Type: Angle (rad) Description: Vertical tail sweep angle at 1/4 chord 
Srdr 
Value: Type: Area (m²) Description: Rudder area 
SrefVT 
Value: Type: Area (m²) Description: Vertical tail reference area 
ARvt 
Value: Type: Real Description: Aspect ratio of vertical tail 
TRvt 
Value: Type: Real Description: Taper ratio of vertical tail 
cVTmean 
Value: Type: Length (m) Description: Vertical tail mean chord 
lVTcm 
Value: Type: Length (m) Description: Vertical tail arm length (from aircraft center of mass to vertical tail 1/4 chord) 
lVTwingAC 
Value: Type: Length (m) Description: Vertical tail arm length (from wing aerodynamic center to vertical tail aerodynamic center) 
zVTac 
Value: Type: Length (m) Description: Vertical tail center of pressure zcoordinate w.r.t. fuselage reference point 
SwetVT 
Value: Type: Area (m²) Description: Vertical tail wetted area 
lambdaVTle 
Value: Type: Angle (rad) Description: Vertical tail leading edge sweep angle 
FFvt 
Value: Type: Real Description: Vertical tail form factor 
sdVT 
Value: Type: Real Description: Fuselage drag factor for vertical tail 
kdVT 
Value: Type: Real Description: Empirical constant for Oswald efficiency factor for vertical tail 
tauRdr 
Value: Type: Real Description: Rudder effectiveness parameter 
sigmaBeta 
Value: Type: Real Description: Change in sidewash due to beta 
kSkinVT 
Value: Type: Length (m) Description: Vertical tail surface roughness height 
ClAlphaVT2D 
Value: Type: CurveSlope (rad⁻¹) Description: Change in the section lift coefficient of the vertical tail airfoil (2D) due to alpha 
deltaRdrMax 
Value: Type: Angle (rad) Description: Maximum rudder deflection 
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 
mVT 
Value: if machDes < 0.4 then if compMat then 0.75 * (98.5 * (MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.87 * (SrefVT * 10.764 / 100) ^ 1.2 * (bVT * 3.281 / (tVTroot * 39.37)) ^ 0.5) / 2.2046 else 98.5 * (MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.87 * (SrefVT * 10.764 / 100) ^ 1.2 * (bVT * 3.281 / (tVTroot * 39.37)) ^ 0.5 / 2.2046 else if compMat then 0.75 * 0.19 * ((1 + min(zHTrootLE  zVTroot, 0) / (bVT)) ^ 0.5 * (MTOMdes * 2.2046 * nMax * 1.5) ^ 0.363 * (SrefVT * 10.764) ^ 1.089 * (sqrt(2 * qMax / rho0) / a0) ^ 0.601 * (lVTwingAC * 3.281) ^ (0.726) * (1 + Srdr / SrefVT) ^ 0.217 * ARvt ^ 0.337 * (1 + TRvt) ^ 0.363 * cos(lambdaVT) ^ (0.484)) ^ 1.014 / 2.2046 else 0.19 * ((1 + min(zHTrootLE  zVTroot, 0) / (bVT)) ^ 0.5 * (MTOMdes * 2.2046 * nMax * 1.5) ^ 0.363 * (SrefVT * 10.764) ^ 1.089 * (sqrt(2 * qMax / rho0) / a0) ^ 0.601 * (lVTwingAC * 3.281) ^ (0.726) * (1 + Srdr / SrefVT) ^ 0.217 * ARvt ^ 0.337 * (1 + TRvt) ^ 0.363 * cos(lambdaVT) ^ (0.484)) ^ 1.014 / 2.2046 Type: Mass (kg) Description: Vertical tail mass 
rCMvt 
Value: {xWingRootLE  tan(lambdaWingLE) * (yWingAC  wFus / 2) + xWingAC  lVTwingAC, 0, zVTac} + {xVTcm, 0, bVT * (1 + 2 * TRvt) / (3 + 3 * TRvt) + zVTcm} Type: Length[3] (m) Description: Vertical tail center of mass coordinates w.r.t. fuselage reference point 
IxxVT 
Value: 0.000293 * (mVT * 2.205 * (bVT * 39.37) ^ 2 * k1VT) / 18 * (1 + 2 * cVTroot * 39.37 * cVTtip * 39.37 / (cVTroot * 39.37 + cVTtip * 39.37) ^ 2) Type: MomentOfInertia (kg·m²) Description: Vertical tail roll moment of inertia 
IyyVT 
Value: IxxVT + IzzVT Type: MomentOfInertia (kg·m²) Description: Vertical tail pitch moment of inertia 
IzzVT 
Value: 0.000293 * 0.771 * (i0VT  wvtxVT ^ 2 / (mVT * 2.205)) Type: MomentOfInertia (kg·m²) Description: Vertical tail yaw moment of inertia 
xVTcm 
Value: 0.25 * cVTmean + tan(lambdaVTle) * bVT * (1 + 2 * TRvt) / (3 + 3 * TRvt)  tan(lambdaVTle) * abs(zVTcm)  0.42 * cVTcm Type: Length (m) Description: Vertical tail center of mass xcoordinate w.r.t. Vertical tail aerodynamic center (positive xaxis towards nose) 
zVTcm 
Value: (min(zHTrootLE  zVTroot, 0) / (bVT) * (0.55  0.38) + 0.38) * (bVT) Type: Length (m) Description: Vertical tail center of mass zcoordinate w.r.t. its root at fuselage (positive zaxis towards ground) 
cVTcm 
Value: (cVTtip  cVTroot) / bVT * abs(zVTcm) + cVTroot Type: Length (m) Description: Chord length at vertical tail center of mass 
caVT 
Value: min({bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTtip * 39.37 + bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTroot * 39.37}) Type: Real Description: Factor for calculating vertical tail moment of inertias 
cbVT 
Value: sum({bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTtip * 39.37 + bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTroot * 39.37})  caVT  ccVT Type: Real Description: Factor for calculating vertical tail moment of inertias 
ccVT 
Value: max({bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTtip * 39.37 + bVT * 2 * tan(lambdaVTle) / 2 * 39.37, cVTroot * 39.37}) Type: Real Description: Factor for calculating vertical tail moment of inertias 
k1VT 
Value: 0.988158 + 2.20444 * (abs(zVTcm) / (bVT / 3 * (cVTroot + 2 * cVTtip) / (cVTroot + cVTtip))) ^ 1.1 Type: Real Description: Factor for calculating vertical tail moment of inertias 
rhoiVT 
Value: mVT * 2 * 2.205 / (caVT + cbVT + ccVT) Type: Real Description: Factor for calculating vertical tail moment of inertias 
wvtxVT 
Value: rhoiVT / 6 * (caVT ^ 2 + cbVT ^ 2 + ccVT * cbVT + ccVT ^ 2) Type: Real Description: Factor for calculating vertical tail moment of inertias 
i0VT 
Value: rhoiVT / 12 * (caVT ^ 3 + cbVT ^ 3 + ccVT ^ 2 * cbVT + ccVT * cbVT ^ 2 + ccVT ^ 3) Type: Real Description: Factor for calculating vertical tail moment of inertias 
flightData 
Type: FlightData Description: Global flight data variables 

flightData 
Type: FlightData Description: Global flight data variables 


liftVerticalTail 
Type: WorldForce Description: Vertical tail and rudder lift force 

verticalTailShape 
Type: FixedShape Description: Visualization of vertical tail 

rudderShape 
Type: FixedShape Description: Visualization of rudder 

rudderDynamics 
Type: CriticalDamping Description: Simplified model of the rudder actuator dynamics 

deltaRdrLimiter 
Type: Limiter Description: Limits rudder deflection to its assigned limits 

dragVerticalTail 
Type: WorldForce Description: Vertical tail drag force 

translVT 
Type: FixedTranslation Description: Position of vertical tail aerodynamic center w.r.t fuselage reference point 

bodyVerticalTail 
Type: Body Description: Mass and inertia of vertical tail 

translVT0 
Type: FixedTranslation Description: Translation to bypass bodyVerticalTail if no weight estimation is used 
Aircraft.Physical.FixedWing.Parts.Surfaces Model of system of surfaces in conventional wing configuration 