WOLFRAM SYSTEM MODELER
HorizontalTailContribution from horizontal tail to aerodynamic forces 
SystemModel["Aircraft.Physical.FixedWing.Parts.Surfaces.Components.HorizontalTail"]
This model calculates the aerodynamic forces due to the horizontal tail and elevator, estimates the mass properties of the horizontal tail and solves the consequent Newton—Euler equations in the bodyHorizontalTail 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.
All other parameters, including the horizontal tail and elevator 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 horizontal tail. Instead of having two aerodynamic centers, i.e. one on each side of the horizontal tail, here only one aerodynamic center is modeled in the middle of the horizontal tail along the body y axis. The location of the aerodynamic center along the body x axis is defined by the parameter lHTwingAC, 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 horizontal tail span, root chord, tip chord and sweep angle. On the body z axis, the aerodynamic center is at the given z coordinate for horizontal tail root chord leading edge (zHTrootLE), as no dihedral angle is modeled for the horizontal tail.
The equations to estimate the aerodynamic center location as well as any other estimated horizontal tail parameter can be bypassed by entering the known value in the Horizontal 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.
The parameter for the lift curve slope of the 2D airfoil (ClAlphaHT2D) is used together with other horizontal tail properties and the Mach number to derive the lift curve slope for the entire horizontal tail (C_{Lα,HT}) such that the air compressibility effects are also considered. The complete method to derive C_{Lα,HT} is described in detail in section 3.3.2 in Reference [1]. Furthermore, the lift curve slope of the horizontal tail with respect to its deflection angle (C_{Lδe,HT}) is calculated as the product of C_{Lα,HT} and the elevator effectiveness parameter, which is derived from the elevator and horizontal tail surfaces according to the method presented by Nelson [2]. Thus, the lift coefficient of the horizontal tail with contribution by the elevator deflection is solved by
,
where α_{eff,HT} is the effective angle of attack seen by the horizontal tail, such that the contribution of the horizontal tail incidence angle, horizontal tail zerolift angle, the downwash angle generated by the main wing and the induced angle of attack due to pitch rate are considered according to the method presented by Nelson [2]. Currently, the horizontal tail incidence angle is fixed and cannot be trimmed during flight.
The dimensionless lift coefficient for the horizontal tail with contribution by the elevator deflection is multiplied by the global dynamic pressure and horizontal tail reference area to get the magnitude of the lift force acting on the aerodynamic center. The lift force is oriented such that it is perpendicular to the free stream, as shown in Figure 1.
Figure 1: Orientation of the aerodynamic forces acting on the horizontal tail aerodynamic center.
For solving the parasite drag coefficient (C_{D,0}) of different components in the aircraft, including the horizontal tail, the component buildup method presented by Raymer [3] is used, defined as
,
where the form factor (FF_{HT}) and the area (S_{wet,HT}) are functions of the geometry of the horizontal tail and fuselage. The skin friction coefficient (C_{f,HT}) is a function of the surface roughness height of horizontal tail, Mach number and Reynolds number for the flow over the mean aerodynamic chord of the horizontal tail. Thus, the air compressibility effects are included in the drag calculations. The complete derivation of the C_{D,0,HT} can be found in section 3.3.1 in Reference [1].
Liftinduced drag is also considered in the complete drag coefficient for the horizontal tail (C_{D}_{,HT}), which reads as
,
where K_{HT} is an empirical factor, and its value is based on the horizontal tail geometry and calculated trough a method given by Cook [4].
The C_{D}_{,HT} 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 horizontal tail. The drag force is also oriented such that it always is parallel with the free stream, as shown in Figure 1.
Stall, which occurs at high angles of attack due to flow separation on the wing upper surface, is modeled for the horizontal tail up to α_{eff} = 90°, but not for large negative angles of attack.
The horizontal tail lift coefficient (C_{L}_{,}_{HT}) follows the linear lift curve slope, i.e. the C_{Lα,HT}, until the vicinity of achieving the maximum value for horizontal tail lift coefficient (C_{L}_{,}_{max,HT}), 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}_{,}_{HT} needs to be differentiable everywhere.
The parameter for C_{L}_{,}_{max,HT}, which is derived from the maximum lift coefficient of the 2D airfoil (ClMaxHT2D), is used together with C_{Lα,HT} 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. [1]
When modeling the stall for the horizontal tail, the lift contribution by elevator deflection is first omitted as the C_{L}_{,}_{HT} value is solved during stall, and it is later added on top of the solved C_{L}_{,}_{HT}.
The horizontal tail drag coefficient (C_{D}_{,}_{HT}) follows the equation defined in the previous section until the C_{L}_{,HT} coefficient achieves C_{L}_{,}_{max,HT}, after which it starts following a sine curve toward the maximum drag coefficient value (C_{D}_{,}_{max,HT}) at α_{eff} = 90°, as shown in Figure 3. The value of the C_{D}_{,}_{max,HT} coefficient is calculated based on the drag coefficient of a flat plate, which has same aspect ratio as the horizontal tail and is perpendicular to the wind. The function for C_{D}_{,}_{HT} is currently not differentiable where it transits into the sine curve. However, it is continuous everywhere.
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 horizontal tail mass properties are estimated by using an empirical relationship based on the given parameters describing the horizontal tail geometry, its position and the design variables in the AircraftBase model. The horizontal 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 such that the rotation around the horizontal tail incidence angle is also considered. However, as the horizontal tail is modeled as one rigid body, it has only one center of mass located along the fuselage centerline. Furthermore, as no dihedral angle is modeled for the horizontal tail, its center of mass is at the given z coordinate for the horizontal tail root chord leading edge (zHTrootLE).
The derivation of the horizontal 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 horizontal tail moments of inertia are estimated by a method presented in USAF DATCOM [6], and they are also converted to be used with SI units 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 

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 
nMax 
Value: Type: Real Description: Maximum load factor 
wFus 
Value: Type: Length (m) Description: Fuselage maximum width 
dFusHT 
Value: Type: Length (m) Description: Fuselage diameter at horizontal tail 1/4 chord 
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) 
iWing 
Value: Type: Angle (rad) Description: Main wing incidence angle 
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 
bHT 
Value: Type: Length (m) Description: Horizontal tail span 
cHTroot 
Value: Type: Length (m) Description: Horizontal tail root chord 
cHTtip 
Value: Type: Length (m) Description: Horizontal tail tip chord 
tHTroot 
Value: Type: Length (m) Description: Horizontal tail root thickness 
zHTrootLE 
Value: Type: Length (m) Description: Horizontal tail root leading edge zcoordinate w.r.t. fuselage reference point (positive zaxis towards ground) 
lambdaHT 
Value: Type: Angle (rad) Description: Horizontal tail sweep angle at 1/4 chord 
iHT 
Value: Type: Angle (rad) Description: Horizontal tail incidence angle 
Selv 
Value: Type: Area (m²) Description: Elevator area 
SrefHT 
Value: Type: Area (m²) Description: Horizontal tail reference area 
ARht 
Value: Type: Real Description: Aspect ratio of horizontal tail 
TRht 
Value: Type: Real Description: Taper ratio of horizontal tail 
cHTmean 
Value: Type: Length (m) Description: Horizontal tail mean chord 
lHTcm 
Value: Type: Length (m) Description: Horizontal tail arm length (from aircraft center of mass to horizontal tail 1/4 chord) 
lHTwingAC 
Value: Type: Length (m) Description: Horizontal tail arm length (from wing aerodynamic center to horizontal tail 1/4 chord) 
SwetHT 
Value: Type: Area (m²) Description: Horizontal tail wetted area 
lambdaHTle 
Value: Type: Angle (rad) Description: Horizontal tail leading edge sweep angle 
FFht 
Value: Type: Real Description: Horizontal tail form factor 
sdHT 
Value: Type: Real Description: Fuselage drag factor for horizontal tail 
kdHT 
Value: Type: Real Description: Empirical constant for Oswald efficiency factor for horizontal tail 
tauElv 
Value: Type: Real Description: Elevator effectiveness parameter 
alpha0Wing2D 
Value: Type: Angle (rad) Description: Zerolift angle of attack of the main wing airfoil (2D) 
kSkinHT 
Value: Type: Length (m) Description: Horizontal tail surface roughness height 
ClAlphaHT2D 
Value: Type: CurveSlope (rad⁻¹) Description: Change in the section lift coefficient of the horizontal tail airfoil (2D) due to alpha 
alpha0HT2D 
Value: Type: Angle (rad) Description: Zerolift angle of attack of the horizontal tail airfoil (2D) 
CLmaxHT3D 
Value: Type: Real Description: Maximum lift coefficient of the horizontal tail (3D) 
CDmaxHT3D 
Value: Type: Real Description: Maximum drag coefficient of the horizontal tail (3D)*(SrefHT/SrefWing) 
deltaElvMax 
Value: Type: Angle (rad) Description: Maximum elevator deflection 
CADshapes 
Value: Type: Boolean Description: true, if external CAD files are used for animation 
mHT 
Value: if machDes < 0.4 then if compMat then 0.75 * 127 * ((MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.87 * (SrefHT * 10.764 / 100) ^ 1.2 * (lHTwingAC * 3.281 / 10) ^ 0.483 * (bHT * 3.281 / (tHTroot * 39.37)) ^ 0.5) ^ 0.458 / 2.2046 else 127 * ((MTOMdes * 2.2046 * nMax * 1.5 / 10 ^ 5) ^ 0.87 * (SrefHT * 10.764 / 100) ^ 1.2 * (lHTwingAC * 3.281 / 10) ^ 0.483 * (bHT * 3.281 / (tHTroot * 39.37)) ^ 0.5) ^ 0.458 / 2.2046 else if compMat then 0.75 * 0.0034 * ((MTOMdes * 2.2046 * nMax * 1.5) ^ 0.813 * (SrefHT * 10.764) ^ 0.584 * (bHT * 3.281 / (tHTroot * 3.281)) ^ 0.033 * (cHTmean * 3.281 / (lHTwingAC * 3.281)) ^ 0.28) ^ 0.915 / 2.2046 else 0.0034 * ((MTOMdes * 2.2046 * nMax * 1.5) ^ 0.813 * (SrefHT * 10.764) ^ 0.584 * (bHT * 3.281 / (tHTroot * 3.281)) ^ 0.033 * (cHTmean * 3.281 / (lHTwingAC * 3.281)) ^ 0.28) ^ 0.915 / 2.2046 Type: Mass (kg) Description: Horizontal tail mass 
rCMht 
Value: {xWingRootLE  tan(lambdaWingLE) * (yWingAC  wFus / 2) + xWingAC  lHTwingAC, 0, zHTrootLE} + {xHTcm, 0, 0} Type: Length[3] (m) Description: Horizontal tail center of mass coordinates w.r.t. fuselage reference point 
IxxHT 
Value: 0.000293 * (mHT * 2.205 * (bHT * 39.37) ^ 2 * k1HT) / 24 * (cHTroot + 3 * cHTtip) / (cHTroot + cHTtip) Type: MomentOfInertia (kg⋅m²) Description: Horizontal tail roll moment of inertia 
IyyHT 
Value: 0.000293 * 0.771 * (i0HT  whtxHT ^ 2 / (mHT * 2.205)) Type: MomentOfInertia (kg⋅m²) Description: Horizontal tail pitch moment of inertia 
IzzHT 
Value: IxxHT + IyyHT Type: MomentOfInertia (kg⋅m²) Description: Horizontal tail yaw moment of inertia 
xHTcm 
Value: 0.25 * cHTmean + tan(lambdaHTle) * (bHT / 2  dFusHT / 2) * (1 + 2 * TRht) / (3 + 3 * TRht)  tan(lambdaHTle) * (yHTcm  dFusHT / 2)  0.42 * cHTcm Type: Length (m) Description: Horizontal tail center of mass xcoordinate w.r.t. horizontal tail aerodynamic center 
yHTcm 
Value: 0.38 * bHT / 2 Type: Length (m) Description: (Half) Horizontal tail center of mass ycoordinate w.r.t. fuselage centerline 
cHTcm 
Value: (cHTtip  cHTroot) / (bHT / 2  dFusHT / 2) * (yHTcm  dFusHT / 2) + cHTroot Type: Length (m) Description: Chord length at half horizontal tail center of mass 
caHT 
Value: min({bHT * tan(lambdaHTle) / 2 * 39.37, cHTtip * 39.37 + bHT * tan(lambdaHTle) / 2 * 39.37, cHTroot * 39.37}) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
cbHT 
Value: sum({bHT * tan(lambdaHTle) / 2 * 39.37, cHTtip * 39.37 + bHT * tan(lambdaHTle) / 2 * 39.37, cHTroot * 39.37})  caHT  ccHT Type: Real Description: Factor for calculating horizontal tail moment of inertias 
ccHT 
Value: max({bHT * tan(lambdaHTle) / 2 * 39.37, cHTtip * 39.37 + bHT * tan(lambdaHTle) / 2 * 39.37, cHTroot * 39.37}) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
k1HT 
Value: 1.06793 + 1.99535 * (yHTcm / (bHT / 6 * (cHTroot + 2 * cHTtip) / (cHTroot + cHTtip))) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
rhoiHT 
Value: mHT * 2 * 2.205 / (caHT + cbHT + ccHT) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
whtxHT 
Value: rhoiHT / 6 * (caHT ^ 2 + cbHT ^ 2 + ccHT * cbHT + ccHT ^ 2) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
i0HT 
Value: rhoiHT / 12 * (caHT ^ 3 + cbHT ^ 3 + ccHT ^ 2 * cbHT + ccHT * cbHT ^ 2 + ccHT ^ 3) Type: Real Description: Factor for calculating horizontal tail moment of inertias 
flightData 
Type: FlightData Description: Global flight data variables 

aircraftRP 
Type: Frame_b Description: Connector to aircraft reference point 


yLiftHT 
Type: RealOutput Description: Lift due to horizontal tail 

deltaElvCmd 
Type: RealInput Description: Elevator deflection command 

uEpsilonAlpha 
Type: RealInput Description: Change in downwash due to alpha 
flightData 
Type: FlightData Description: Global flight data variables 


liftHorizontalTail 
Type: WorldForce Description: Horizontal tail and elevator lift force 

horizontalTailShape 
Type: FixedShape Description: Visualization of horizontal tail 

elevatorShape 
Type: FixedShape Description: Visualization of elevator 

elevatorDyanimcs 
Type: CriticalDamping Description: Simplified model of the elevator actuator dynamics 

deltaElvLimiter 
Type: Limiter Description: Limits elevator deflection to its assigned limits 

dragHorizontalTail 
Type: WorldForce Description: Horizontal tail drag force 

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

bodyHorizontalTail 
Type: Body Description: Mass and inertia of horizontal tail 

rotHT 
Type: FixedRotation Description: Rotation of horizontal tail around incidence angle 

rotHTforces 
Type: FixedRotation Description: Rotation of aerodynamic forces acting on horizontal tail around incidence angle 

translHT0 
Type: FixedTranslation Description: Translation to bypass bodyHorizontalTail if no weight estimation is used 
Aircraft.Physical.FixedWing.Parts.Surfaces Model of system of surfaces in conventional wing configuration 