Quick trim modeling for tiltrotor aircraft in conversion mode

The conversion mode in tiltrotor aircraft is of paramount importance and complexity, necessitating the trim results to ensure stable tilting using the trim model. Quick trim modeling is conducted for tiltrotor aircraft in conversion mode, focusing on the aerodynamics of the rotor using modified lift line theory. Furthermore, we develop the nonlinear flight dynamics of tilt-rotor aircraft. Based on this aircraft model, a double-layer cycle iterations method is employed to establish an efficient and rapid trim model for the tiltrotor in conversion mode at any tilt angle or during any aerial maneuver. Using Boeing Model 222 as an example, trim results for conversion mode are calculated and compared with reference data to validate the effectiveness and precision of our model.


Introduction
In recent years, with the development of Urban Air Mobility (UAM) and Advanced Air Mobility (AAM) concepts, a range of eVTOL vehicle configurations have been designed, and the tiltrotor configuration aircraft has become the preferred configuration in people's vision.According to a recent evtol.news[1] vehicle configuration survey, VFS (2022), there are over 200 vehicle designs with tiltrotor characteristics, such as Joby Aviation S4, Lilium Jet, Bell Nexus, A 3 Vahana, etc.The tiltrotor aircraft includes helicopter mode, conversion mode, and cruise mode.It has excellent performance while having more complex technical problems, especially in conversion mode, which is extremely important and complex.It is an important research topic to complete the dynamic model of the tiltrotor, trim the motion in each mode, and successfully transform from helicopter to cruise mode.
In recent years, there have been plenty of studies on tiltrotors.Most studies focused on the aerodynamic interference between proprotor and fuselage and other flight dynamics performance analyses of tiltrotors [2][3][4].However, there needs to be more information describing the trim problems in these papers.For the research on the trim of tiltrotor, Nabi et al. [5] focused on developing a quasi-Linear Parameter Varying (qLPV) model for the tiltrotor aircraft and then calculated the trim solution.Wang et al. [6] proposed an innovative trim method for tiltrotor based on a genetic algorithm.Thomas et al. [7] calculated the trim envelope for a generic tiltrotor aircraft model for different tilt angles.As an initial, a simple model is established to trim tiltrotor aircraft quickly in this paper, resulting in more timely results.It can provide a real-time reference for pilots and can be applied to the development of the intelligent system of aircraft autopilot technology.

Flight Dynamic Model
The flight dynamic model of the tiltrotor includes the aerodynamic model of the rotor and aircraft (including wing, fuselage, horizontal tail, and vertical tail) and the control system model.

Aerodynamic Model of Rotor.
The rotor aerodynamic model is based on lifting-line theory, firstly determining the blade pitch angle  and the velocity components t u and p u , as shown in Figure 1. is the angle of attack of an airfoil.
where  is air density.In some cases, the rotor with a large pitch is likely to stall, and the calculation of aerodynamic performance using lifting-line theory is not accurate, so it is necessary to correct the values for lift coefficient using the dynamic stall model [8] based on the original two-dimensional airfoil.The increment of the drag coefficient is zero., where   is the pitch rate at the instant of stall.The rotor aerodynamic force can be calculated by integrating the section aerodynamic force and then transforming it into the body coordinate system, as shown in Figure 2. The rotor wake induced velocity i v is obtained from the momentum theory result: where  and f  are the empirical correction factors for the effects of nonuniform inflow, swirl, and blockage. and  are respectively forward ratio and inflow ratio.and moments are six components: three forces ( x F , y F and z F ) and three moments ( x M , y M and z M ): We define the aerodynamic forces and moments of aircraft using the aerodynamic characteristics of the control variables (including flaperon angle f  , aileron angle a  , elevator e  , and rudder angle r  ), angle of attack  and sideslip angle  by experimental data or CFD simulation.Then, we obtain the aircraft aerodynamics of the wing-body: About the horizontal tail loads: And about vertical tail loads: , , , , 2.1.3Control system.The control variables of the tiltrotor included the rotorcraft controls consisting of the collective and cyclic pitch of the rotors and the aircraft controls, as described in Section 2.1.2.The control vector is thus: The pilot's control variables include collective stick 0  , lateral cyclic stick c  , longitudinal cyclic stick s  , pedal p  and throttle t  : A linear relation between the pilot's control inputs and the rotor aircraft control variables is assumed: where 0 v is the value of control variables when all the pilot's control sticks are in the center position.

Quick Trim Method
The trim iteration is a loop (shown in Figure 3).The Newton-Raphson method is used to solve this trim model.The control variables and attitude of the aircraft under different environmental conditions and different aerial maneuvers are determined using an inner motion trim model so that the net force and moment of the tiltrotor aircraft is 0. Based on motion trim iteration convergence, executing an outer load trim iteration to convergence is necessary.The motion trim solution at a specified flight speed and rotating speed (possibly has manoeuvres with a specified turn rate F  ) is obtained by the following criterion: ( ) The induced velocity distributed throughout the disc must be considered in rotor motion.The relationship between the induced velocity and rotor thrust is obtained by Equation (4).
The induced velocity will be updated with the rotor thrust changes.This procedure continues until the change in load from one iteration to the next iteration is less than a specified tolerance parameter  .

Results and Analysis
The Boeing Model 222 tiltrotor aircraft is chosen as an instance to calculate the trim solution.It has two three-bladed hingeless rotors and some basic parameters of the tiltrotor aircraft, which are shown in Table 1 (detailed information about tiltrotor aircraft is in [9] and [10]).4 shows the variation of the operating parameters along the flight path and the data [11].In helicopter mode, the speed extends to 80 knots from a hover.The pylon begins to tilt at 80 knots, and the tilt angle goes from 90 to 0, with the velocity going from 80 knots to 140 knots.The rotor speed is reduced for conversion mode from 551 to 386 rpm.
The aircraft tilts from the helicopter forward flight mode when the forward flight speed reaches a certain value.There's no need to verify the hover mode, and the verification starts from the helicopter's forward flight mode.Figure 5 shows the variation of rotor pitch in helicopter mode with the velocity.The pitch changes slightly during the forward flight in helicopter mode, about 5 degrees.Comparing the literature data with calculated results, it can be seen that the error is about 2 degrees.Figure 6 shows the variation of rotor thrust coefficient in helicopter mode and the calculated results are with errors about 0.005 compared with the reference data.Figure 5 and Figure 6 can this model is successful in helicopter mode.Figure 6.Rotor thrust coefficient in helicopter mode.In cruise mode, the pitch of the rotor will increase with the increase of the forward speed.The fixed rotor speed is 386 rpm, and the trim solution at different forward flight speeds will be calculated.The pitch is generally large in cruise mode, up to 56 degrees.The thrust coefficient of the rotor hardly changes which may be related to the amount of power that the engine can provide.Figure 7 shows the rotor pitch with the errors between the calculated results and the reference data of 3 degrees.Figure 8 shows the variations of the rotor thrust coefficient with an error of 0.02.The tiltrotor starts to tilt from helicopter to cruise mode at 80 knots.With the increase of the tilt angle, the forward speed increases while the rotor speed gradually decreases.According to the transition corridor in Figure 4, each tilt angle corresponds to the forward and rotor speeds, and then we calculate the trim solution for each.In conversion mode, the pitch of the rotor varies widely from about 5 to about 30 degrees.It is the opposite of increasing the pitch.The rotor thrust coefficient gradually decreases from about 0.07 to 0.02.As shown in Figure 9 and Figure 10, the errors of rotor pitch and rotor thrust coefficient between the calculated results and reference date are within 4 degrees and 0.005, respectively.From the figure, we can see that the calculated results fit well with the literature data, and these results can verify the trim model established in this paper.In actual flight, the variables like the tilt angular speed of the pylon, forward speed, and rotor speed are changing rather than a constant value.So, we choose to trim the motion of the aircraft in each condition to obtain the solution close to the truth independent of the tilt angular speed of the pylon, forward acceleration, and rotor angular acceleration.The calculating results are by the reference data through the above analysis in any mode, and the model is reasonable.It can study the trim problem in conversion mode and provide a reference value for the operation in conversion mode.

Conclusion
A reasonable and accurate quick trim model of tiltrotor aircraft is established, which can trim the tiltrotor motion in any mode and working condition, and the trim solution can be obtained within 2 s.The calculation results are coincidental with the real variables in the reference, and the model is verified.The solution is close to the truth independent of the tilt angular speed of the pylon, forward acceleration, and rotor angular acceleration.This model, independent of the tilt angular speed of the pylon, forward acceleration, and rotor angular acceleration, can calculate for any mode, working condition, and maneuver to obtain the solution closer to the truth.It references flight vehicle attitude, pilot's controls, and autonomous control technology in conversion mode.

Figure 2 .
Figure 1.Rotor blade section aerodynamics.Figure 2. Rotor aerodynamics 2.1.2Aerodynamic Model of Aircraft.The aerodynamic model considers the aircraft aerodynamic forces acting on the wing-body (WB), horizontal tail (HT), and vertical tail (VT).Specifically, the aerodynamic loads needed are three forces, three moments of wing-body, and the lift and drag of horizontal and vertical tail.The generalized forces for the aircraft produced by the aerodynamic forces

Figure 5 .
Figure 5. Rotor pitch in helicopter mode.Figure6.Rotor thrust coefficient in helicopter mode.In cruise mode, the pitch of the rotor will increase with the increase of the forward speed.The fixed rotor speed is 386 rpm, and the trim solution at different forward flight speeds will be calculated.The pitch is generally large in cruise mode, up to 56 degrees.The thrust coefficient of the rotor hardly changes which may be related to the amount of power that the engine can provide.Figure7shows the rotor pitch with the errors between the calculated results and the reference data of 3 degrees.Figure8shows the variations of the rotor thrust coefficient with an error of 0.02.

Figure 7 .
Figure 7. Rotor pitch in cruise mode.Figure 8. Rotor thrust coefficient in cruise mode.

Figure 8 .
Figure 7. Rotor pitch in cruise mode.Figure 8. Rotor thrust coefficient in cruise mode.

Figure 9 .
Figure 9. Rotor pitch in conversion mode.Figure 10.Rotor thrust coefficient in conversion mode.

Figure 10 .
Figure 9. Rotor pitch in conversion mode.Figure 10.Rotor thrust coefficient in conversion mode.