1 Simulation and Optimization of Control 2 of Selected Phases of Gyroplane Flight 3

The optimization methods are increasingly used to solve challenging problems of 7 aeronautical engineering. Typically, the optimization methods are utilized in design of aircraft 8 airframe or its structure. The presented study is focused on an improvement of 9 aircraft-flight-control procedures through the numerical optimization approach. The optimization 10 problems concern selected phases of flight of light gyroplane a rotorcraft using an unpowered 11 rotor in autorotation to develop lift and an engine-powered propeller to provide thrust. An original 12 methodology of computational simulation of rotorcraft flight was developed and implemented. In 13 this approach the aircraft-motion equations are solved step-by-step, simultaneously with the 14 solution of the Unsteady Reynolds-Averaged Navier-Stokes equations, which is conducted to 15 assess aerodynamic forces acting on the aircraft. As a numerical optimization method, the BFGS 16 algorithm was adapted. The developed methodology was applied to optimize the flight-control 17 procedures in selected stages of gyroplane flight in direct proximity of the ground, where properly 18 conducted control of the aircraft is critical to ensure flight safety and performance. The results of 19 conducted computational optimizations proved qualitative correctness of the developed 20 methodology. The research results can be helpful in design of easy-to-control gyroplanes and also 21 in the training of pilots of this type of rotorcraft. 22


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Optimization methods are widely considered to be a very effective tool that can significantly 26 improve the performance and exploitation properties of contemporarily designed and constructed 27 aircraft. Typically, the optimization methods are utilized in design of aircraft airframe or its 28 structure that may be optimized simultaneously using a multi-disciplinary approach [1][2][3]. Fast 29 development of both computational methods and computer hardware offers opportunities to 30 expand the range of applications of optimization methods. As part of this trend, the application of 31 modern computational methods to optimize aircraft-flight-control procedures is presented in this 32 paper. The developed methodology for optimization of flight control procedures is discussed on the 33 example of flight of a light gyroplane.

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A gyroplane is an aerodyne equipped with unpowered main rotor and engine-powered 35 propeller, generating a thrust force necessary to move the aircraft forward. During the gyroplane 36 flight the air flowing around rotating blades of main rotor generates aerodynamic reaction whose 37 vertical component balances the aircraft weight, while the aerodynamic moment is driving the main 38 rotor that rotates in autorotation. However to induce the autorotation phenomenon, the rotor should 39 be initially pre-rotated, which is usually done by the engine driving the propeller. Before the 40 gyroplane loses contact with the ground, the drive of the rotor must be disconnected because this 41 type of rotorcraft does not have any anti-torque devices.

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• changeable collective pitch of rotor blades that may be used for torque reduction in pre-rotation 48 and is necessary to conduct so called "jump takeoff".

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The classic takeoff of a gyroplane is similar to typical takeoff of an airplane. The gyroplane, with 50 pre-rotated main rotor, starts accelerated run along the runway. The rotor generates more and more 51 thrust. When the thrust exceeds the weight of aircraft, the gyroplane takes off. Gyroplanes usually 52 need short runway to conduct the classic takeoff and they mostly belong to STOL (Short Takeoff and 53 Landing) type of aircraft.

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In the case of so called "jump takeoff", the gyroplane takes off directly from the ground, without 55 a run along the runway. To perform this maneuver, the rotor head design should allow changing an 56 angle of blade collective pitch during the flight. After initial pre-rotation of the rotor, the drive is 57 disconnected and simultaneously the higher angle of the blade collective pitch is established. The 58 inertia-driven rotor generates high thrust, which makes that the gyroplane jumps upwards. The

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propeller starts driving the gyroplane in horizontal direction, which makes that the horizontal 60 velocity starts growing and after some time the rotor starts rotating in autorotation, similarly as it is 61 in a case of the classic takeoff.

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All studies presented in this paper, have been conducted for a gyroplane presented in Figure 1.

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The gyroplane is equipped with a teetering main rotor, three-bladed tractor-type propeller, front 64 landing gear and v-tail that also serves as a rear landing gear. Like in a case of most gyroplanes, the 65 main rotor of the presented gyroplane is characterized by a simple design. It is two-bladed, teetering 66 rotor. Its blades have a rectangular planform, uniform spanwise distribution of airfoil and are not 67 twisted. To enhance the controllability of the gyroplane its rotor-head design enables to control the 68 collective pitch of rotor blades.
69 Figure 1. Model of the gyroplane being a subject of the research discussed in the paper.

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An effective and safe takeoff of the gyroplane requires accurate and rapid deflections of 71 flight-control devices. This especially concerns the control of the rotor-pitch angle that has to be 72 changed in time optimally so as to enhance the autorotation effect as much as possible. Additionally,

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during the jump takeoff, the dynamic changes of the rotor-pitch angle have to be synchronized 74 optimally with dynamic changes of the collective pitch of the rotor blades.

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The main idea of the presented research was to search for possibly optimal procedures of 76 control of the gyroplane flight through application of a numerical-optimization methodology. To 77 realize this idea, the following three main goals of the research undertaken were established:

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Advanced numerical optimization methods coupled with Navier-Stokes solvers, are actually 105 used mostly for an optimization of aircraft external shapes (aerodynamic design) or its structure.

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In

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In comparison to previously used computational tools supporting the design and optimization 115 of rotorcraft flight control, the approach discussed in this paper is not directly geared to industrial 116 applicability but rather to explore new solutions that could in future be applied in the practice of

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To solve the defined above optimization problem, the appropriately adapted BFGS

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In quasi-Newton methods, including the BFGS Algorithm, the Hessian matrix of second 179 derivatives does not need to be evaluated directly. Instead, the Hessian matrix is approximated 180 using updates specified by gradient evaluations or approximate gradient evaluations. The latter 181 approach has been applied in the presented optimization studies.

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Like in all cases of gradient-based methods, the optimal solution has been searched for, in 183 sequential iterative steps. In each step, the components of the gradient vector (partial derivatives of 184 the objective function with respect to the unknown design parameters) were evaluated by means of     procedure during the classic takeoff has been defined by graphs presented in Figure 5, where:

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In this case, the optimization problem consisted in determination of optimal values of unknown 214 parameters D1, F1 and F2. The optimization aimed at maximization of the altitude (H) reached by the 215 gyroplane after traveling the distance X=200 m from the takeoff place, which is explained in Figure 6.

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The optimization problem was formulated in mathematical terms as a search for the set of the design 217 parameters D1, F1 and F2 maximizing the following function Φ: taking into account the following constraints: where: λ1 -limit of angular speed of change of ϕR, ϕRmax -maximum of rotor pitch and θ0max -220 maximum of blade collective pitch.

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The optimization problem has been solved iteratively using the BFGS Algorithm, discussed 225 in section 2. The assumed initial classic-takeoff-control procedure is presented by the graphs shown taking into account the following constraints: F2 ≤ θ0max , where: λ1, λ2 -limits of angular speed of changes of ϕR and θ0 respectively, ϕRmax -maximum of rotor

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The initial flight-control strategy was assumed in the form presented on a left side of Figure 17.

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The optimization process consisted in gradual improvement of this strategy, so as to increase as

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The research results can be helpful in the process of design of easy-to-control gyroplanes and 357 also in the training of pilots of this type of rotorcraft. However, the presented methodology, seems to 358 have much wider potential of future applications. These possible applications may concern not only 359 other gyroplanes or in general: rotorcrafts but also may be utilized for optimization flight-control 360 procedures of any aircraft, e.g. taking off or landing airplane.