WOLFRAM SYSTEM MODELER

HorizontalTail

Contribution from horizontal tail to aerodynamic forces

Diagram

Wolfram Language

In[1]:=
SystemModel["Aircraft.Physical.FixedWing.Parts.Surfaces.Components.HorizontalTail"]
Out[1]:=

Information

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.

Lift due to Horizontal Tail and Elevator

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 (CLα,HT) such that the air compressibility effects are also considered. The complete method to derive CLα,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 (CLδe,HT) is calculated as the product of CLα,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 αeff,HT is the effective angle of attack seen by the horizontal tail, such that the contribution of the horizontal tail incidence angle, horizontal tail zero-lift 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.

Drag Due to Horizontal Tail

For solving the parasite drag coefficient (CD,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 (FFHT) and the area (Swet,HT) are functions of the geometry of the horizontal tail and fuselage. The skin friction coefficient (Cf,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 CD,0,HT can be found in section 3.3.1 in Reference [1].

Lift-induced drag is also considered in the complete drag coefficient for the horizontal tail (CD,HT), which reads as

,

where KHT is an empirical factor, and its value is based on the horizontal tail geometry and calculated trough a method given by Cook [4].

The CD,HT coefficient is multiplied by the global dynamic pressure (q) and the wing reference area (Sref,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

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 αeff = 90°, but not for large negative angles of attack.

The horizontal tail lift coefficient (CL,HT) follows the linear lift curve slope, i.e. the CLα,HT, until the vicinity of achieving the maximum value for horizontal tail lift coefficient (CL,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 CL,HT needs to be differentiable everywhere.

The parameter for CL,max,HT, which is derived from the maximum lift coefficient of the 2D airfoil (ClMaxHT2D), is used together with CLα,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 CL,HT value is solved during stall, and it is later added on top of the solved CL,HT.

The horizontal tail drag coefficient (CD,HT) follows the equation defined in the previous section until the CL,HT coefficient achieves CL,max,HT, after which it starts following a sine curve toward the maximum drag coefficient value (CD,max,HT) at αeff  = 90°, as shown in Figure 3. The value of the CD,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 CD,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].

Mass Properties

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].

References

[1]  Erä-Esko, N. (2022). "Development and Use of System Modeler 6DOF Flight Mechanics Model in Aircraft Conceptual Design."
      Available atmodelica://Aircraft/Resources/Documents/EraeEskoThesis.pdf.

[2]  Nelson, R. C. (1998). Flight Stability and Automatic Control. 2nd ed. McGraw-Hill.
      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.

Parameters (57)

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 take-off 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 x-coordinate w.r.t. fuselage reference point (positive x-axis 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 x-axis towards nose)

yWingAC

Value:

Type: Length (m)

Description: Main wing aerodynamic center from fuselage centerline (y-coordinate 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 z-coordinate w.r.t. fuselage reference point (positive z-axis 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: Zero-lift 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: Zero-lift 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 x-coordinate w.r.t. horizontal tail aerodynamic center

yHTcm

Value: 0.38 * bHT / 2

Type: Length (m)

Description: (Half) Horizontal tail center of mass y-coordinate 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

Inputs (1)

flightData

Type: FlightData

Description: Global flight data variables

Connectors (4)

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

Components (12)

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

Used in Components (1)

Conventional

Aircraft.Physical.FixedWing.Parts.Surfaces

Model of system of surfaces in conventional wing configuration