# MATHEMATICAL MODEL OF PROPELLER CONTROL SYSTEM

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1 18 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation 2015 MATHEMATICAL MODEL OF PROPELLER CONTROL SYSTEM Jaroslav BRAŤKA 1 Jozef ZAKUCIA 2 Abstract: This contribution deals with the development of a dynamic mathematic model of an aircraft turboprop engine variable pitch propeller system. The presented model follows a real physical principle in which a hydromechanical governor regulates propeller speed by varying the pitch of the single acting propeller of pressure to decrease pitch type. They are considered both main operating modes of propulsion: speed control in range of forward thrust and control of the propeller pitch in reverse mode in which the pitch is controlled directly via position feedback. Model function verifying is presented as dynamic behavior in feedback loop compared with a record of real engine behavior. All model parameters are working with real physical units. The model was built in the Matlab Simulink program platform. The resulting model is applicable to all similar propeller systems commonly employed throughout the aerospace industry including ESPOSA engines BE1 and BE2. Keywords: Mathematical Model, Propeller Control System, Hydromechanical Propeller Governor 1. INTRODUCTION Historically, a hydromechanical constant speed governor has been employed in conjunction with a variable pitch propeller on free turbine turboprop engines. Presently, hydromechanical propeller speed governors are generally employed for turboprop engines with power rate up to 1000 kw. The hydromechanical propeller speed governor, which is analyzed in this contribution, in connection with single acting reversing propeller is schematically presented in Fig. 1. Figure 1 Propeller speed control system with hydromechanical governor This system works in two basic operational modes; so - called ALFA mode and BETA mode. In ALFA mode, a governor regulates propeller speed by continually varying the pitch of propeller blades to keep constant speed. The sensing element of the governor is a set of flyweights driven mechanically by reduction gearbox. Flyweights actuate the pilot valve, which in turn supplies hydraulic pressure to modulate the propeller pitch actuating mechanism. The movement of the flyweight is opposed by the force of an adjustable speeder spring. The load exerted by the speeder 1 - Ing. Jaroslav Braťka, Aerospace Research and Test Establishment, Beranových 130 Praha Czech republic, 2 - Ing. Jozef Zakucia, Aerospace Research and Test Establishment, Beranových 130 Praha Czech republic, T:

2 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation spring determines the required RPM. The beta valve serves as minimum flight pitch lock in this operational mode. In BETA mode, a propeller pitch is set in range below In-flight-low-pitch only by the Beta Valve according to engine power. The reversing operation is controlled in the cockpit by the power lever and is initiated by moving it aft of the idle position. The power lever is linked to a propeller reverse lever thus the power lever adjusts both engine power and propeller blade angle. The pilot valve is disconnected by the 3-way valve in this mode. 2. MATHEMATICAL MODEL The model was developed in the Matlab Simulink program platform and tasks are solved in physical units which is important for easier imagination of system behavior and comparison with real test data. Model inputs: - Nc, nc speed of propeller [RPM] - Nd, demanded propeller speed (in ALFA mode) [RPM] - X travel of beta lever (in BETA mode) [mm] - IFLP In flight low pitch [º] - Ps input oil pressure [MPa] - Re Reverse enable signal Model outputs: - β angle of propeller blade [º] - y travel of propeller actuator piston [mm] - Pv pressure in the actuator [MPa] - Qv oil flow [m 3 /hr] Mathematical models of each component are described with the aid of non-linear one-dimensional equations and steady-state components characteristics. The mathematical model comprises several algebraic loops for solution of flow through throttling elements. The system depicted on Fig. 1 was converted to a substitute hydraulic scheme Fig. 2, for purpose of modeling. Figure 2 Substitute hydraulic scheme 2.1 Speed sensor Function of a speed sensor is based on the force developed on the speeder spring due to the angular momentum of the rotating weights witch is equal to the square of speed. Result of forces comparison is movement in rotation axis. The speed sensor is expressed by formula: ^2 2, Where are: nc - flyweights rotation speed [1/min] m - mass of flyweights [kg] Rk - corrected radius of rotation [m] K - spring constant [N/m] Z - travel [m], (1)

3 20 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation Pilot valve The pilot valve is a hydraulic distributor actuated by the speed sensor. The valve directs boosted oil flow to or releasing oil from the propeller actuator according to its + or - position. Flow through throttle orifice is described by formula: Where: K R orifice coefficient S R flow cross-section [m 2] Ρ density [kg/m 3 ] P pressure difference [MPa] Considered travel of the pilot valve slider is ±2 mm and edges overlap 0,1 mm. 2.3 Beta valve The beta valve (also called a hydraulic lock) is a special valve which prevents a propeller from travel of blades to a position below the in-flight low-pitch position in speed control (ALFA mode). Its slide is mechanically linked with the actuator piston and closes oil feeding into a hydraulic actuator and simultaneously opens release from an actuator in case the propeller pitch equals the in-flight lowpitch. In the Beta Mode, the Beta Control Valve operated by the power lever linkage and directs oil pressure generated by the governor boost pump to the propeller actuator or relieves oil from the propeller to change the blade angle. The mechanical linkage between the actuator piston and the slide of beta valve (mechanical feedback loop) ensure proportional response to demand from the power lever. 2.4 Gear pump It was used standard characteristic of a gear pump described by function Qp = f(nc, pc). 2.5 Propeller hydraulic actuator It is used a single acting actuator in which sum of centrifugal force from counterweights, a spring and centrifugal twisting moment on blades is balanced by oil pressure. The oil pressure is varied by the governor as necessary to adjust the blades to the desired angle. Static and coulomb friction of the propeller drive train are neglected as they are assumed to be small compared to the viscous wind loading of the propeller so this pitch control mechanism is modelled as a perfect fluid integrator.! (2) Figure 3 Single Acting Propeller Actuator

4 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation Resulting model The resulting model of the propeller control system is in a form of a Simulink block diagram (see Fig. 4). The model consists of two blocks: Propeller governor and Propeller actuator. An advantage of this arrangement is that we can easily replace each of them by another type of the Propeller governor or Propeller actuator. Figure 4 Model of the Propeller Control System 3. SIMULATION Simulations of the propeller control system were realized in the open loop and also in the close loop configuration. A model of a free turbine, used in the closed loop configuration, was provided by a third party (it is not described in this article). 3.1 Simulation in open loop An example of the simulation in the open loop is presented in Fig. 5. The model of the propeller control system was operated in following regimes: - ALFA mode, propeller pitch is set in the range above the In-flight-low-pitch. - BETA mode, propeller pitch is set in the range below the In-flight-low-pitch. A response of the pitch to a step in the lever position is also simulated. Inputs which were changed during the simulation are listed in Tab. 1. Table 1 Inputs and output of the propeller control system in the open loop Inputs Nd demanded propeller speed Nc actual speed of the propeller Re reverse enable signal X lever position Output Pitch angle β

5 22 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation Nc vs. Ncd Nc Ncd RPM 3800 Lever position (for Reverse Mode) [mm] Lever position with reverse enable signal Beta [ ] Pitch Figure 5 Simulation in the open loop Reverse Enable Lever position [mm] Simulation and model verification in close loop Dynamic behaviour in ALFA mode of the resulting propeller control system model was verified in feedback control loop using a mathematical model of dynamic and power characteristics of both a power turbine and a propeller. The model of this propulsion part is not described here because it is not an object of this article. The model block diagram of closed loop which was implemented for verification is depicted in Fig. 6 below. Figure 6 Block diagram of the closed loop model As an input of model was used record of real change of engine power in range from flight idle to take of power at ground condition and zero velocity. Response of propeller speed was output of simulation. Comparison of transient characteristics, model output compared to real propeller speed record, is depicted in following Fig. 7.

6 Proceedings of the International Scientific Conference Modern Safety Technologies in Transportation Real NP vs. Simulated NP Simulated NP Real NP RPM Figure 7 Comparison of both simulated and real transient characteristics 4. CONCLUSION In this article we have presented a derivation of the mathematical model of the aircraft propeller control system. The model of the propeller control describes nonlinear dynamic behavior of this system within its both operational modes (ALFA and BETA mode). We should also mention that very important part of the modeling of such hydromechanical system is to eliminate occurrences of algebraic loops. Occurrences of the algebraic loops in the Simulink model could interrupt the simulation. In our model we have eliminated algebraic loop successfully, hence the model works in whole operational area without limitation. Simulations of the proposed model in the open loop yielded expected results in both operational modes. Simulation of the model in the close loop was possible only in ALFA mode, because we have had the model of the controlled system only in the range above the IFLP. In this mode we could also perform validation of the model with the measured data from the engine. The model well matched the real system behavior in this mode. REFERENCES [1] Delp, F. Aircraft Propellers and Controls. Jeppesen Sanderson Training Products [2] Braťka, J. Simulační model regulační smyčky vrtule. Zpráva VZLÚ, R-3936, 2006 [3] Kerlin, T; Bratka, J. CP-CS Modelling and Simulation Results Report. CESAR Deliverable document CE-UNIS-T3.2-D Rev. 0; 2008 [4] Technical and design documentations of propeller governor EA Avia Propellers a.s., 2006 [5] NOSKIEVIČ, P. Modelování a identifikace systémů. Ostrava: Montanex, pp. ISBN [6] Kulikov, Gennady G., and Haydn A. Thompson. Dynamic Modelling of Gas Turbines: Identification, Simulation, Condition Monitoring, and Optimal Control. London : Springer, pp.

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