WOLFRAM SYSTEM MODELER

VerticalTail

Contribution from vertical tail to aerodynamic forces

Diagram

Wolfram Language

In[1]:=
SystemModel["Aircraft.Physical.FixedWing.Parts.ControlSurfaces.Components.VerticalTail"]
Out[1]:=

Information

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.

Lift (or Y-force) due to Vertical Tail and Rudder

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 x-y 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 (CLα,VT) such that the air compressibility effects are also considered. The complete method to derive CLα,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 (CLδr,VT) is calculated as the product of CLα,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 x-y 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 CY,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 Y-force coefficient) of the vertical tail with contribution by the rudder deflection is solved by

,

where βeff,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 Y-force 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 Y-force force acting on the aerodynamic center. The Y-force 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.

Drag due to Vertical Tail

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

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

,

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

The CD,VT 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 vertical tail. The drag force is also oriented such that it is always parallel with the free stream, as shown in Figure 1.

Mass Properties

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 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 (58)

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 x-coordinate w.r.t. Vertical tail aerodynamic center (positive x-axis 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 z-coordinate w.r.t. its root at fuselage (positive z-axis 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

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 z-coordinate w.r.t. fuselage reference point (with total mass for electric aircraft and gliders, positive z-axis towards ground)

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

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 x-coordinate w.r.t. fuselage reference point (positive x-axis 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 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

zHTrootLE

Value:

Type: Length (m)

Description: Horizontal tail root leading edge z-coordinate w.r.t. fuselage reference point (positive z-axis 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 z-coordinate 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 z-coordinate 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 sea-level

a0

Value:

Type: Velocity (m/s)

Description: Speed of sound at sea-level

Inputs (1)

flightData

Type: FlightData

Description: Global flight data variables

Connectors (2)

aircraftRP

Type: Frame_b

Description: Connector to aircraft reference point

deltaRdrCmd

Type: RealInput

Description: Rudder deflection command

Components (10)

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

Used in Components (1)

Conventional

Aircraft.Physical.FixedWing.Parts.ControlSurfaces

Model of system of surfaces in conventional wing configuration