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 (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 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 (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].

Lift-induced 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

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].

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 at: modelica://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.