Development of simulation models of the hydraulic system of passenger and transport aircraft in normal and failure situations

. The paper presents a simulation model of the hydraulic system of an aircraft containing a block supply with a variable flow pump, the discharge line of which is presented in the form of a certain container with liquid, the modulus of elasticity of which is constant, and a hydraulic accumulator is installed in the discharge line a certain capacity and pre-charge pressure, resistance of the discharge and drain lines. The developed simulation model allows failure situations, such as: violation of the internal and external tightness, changing the parameters of the accumulator, reducing the pressure of zero flow due to a change in the stiffness of the springs of the control spool and the servo piston of the regulator, jamming regulator in an intermediate position, etc. In addition, the model allows you to calculate the heat losses associated both with the aircraft flight mode and with changes in the hydrodynamic parameters of the system in regular and failure modes.


Introduction
In the process of developing and creating a software simulation module, it is necessary to develop mathematical models of the units and the aircraft hydraulic system as a whole, which allow: -provide visualization of the initial state of the hydraulic system and the corresponding functional systems before and after starting the engines; -provide visualization of the normal (fail-safe) operation of the hydraulic system and the corresponding functional systems in all flight modes: takeoff, cruising, refueling, landing; -provide visualization of the operation of the hydraulic system and the corresponding functional systems in all flight modes: takeoff, cruising, refueling, landing in failure situations; -to carry out simulation modeling of the main elements of the hydraulic system that determine its functioning (pumps, hydraulic accumulators, pressure relief valves, actuators, network resistance); -develop models of functional subsystems, the actuators of which affect the characteristics of the aircraft as a whole; -to determine the dependence of the position of the output links of the actuators on the internal parameters of the hydraulic system (pressure, flow, etc.) and the flight mode of the aircraft (chassis, spoilers, wing mechanization), which will allow to analyze their influence on the characteristics of the aircraft, including in failure situations.

Model and method
The theoretical basis for simulation modeling of an aircraft hydraulic complex is a graphical-analytical method for analyzing and calculating the operating modes of hydraulic systems [1]. When implementing this method, it is required to determine the load (expenditure) characteristics of the elements of the hydraulic system and the magnitude of the loads acting on the output links of the actuators (△ ( )). The flow characteristics of the elements of the hydraulic system are quite complex dependencies and it seems extremely important to find a reasonable degree of complexity, since, as practice shows, too detailed formation of the design problem, including the requirements for the mathematical description of design objects, has a strong impact on the complexity of design, but in in many cases leads to a slight refinement of the design results obtained on the basis of simplified models [2].
The intersection point of the characteristics of the power supply unit and the total characteristics of the consumers of the hydraulic system (point of joint operation) determines not only the total consumption of the power supply unit, the costs in each actuator, but also the nature of the change in time of the position of the output link of any consumer. The joint work point is found by successive iterations with any required accuracy [3].
When creating a model that implements the graphical-analytical method, the problem of modeling such failure (extraordinary) situations as internal and external leaks, failure of individual elements of the hydraulic complex is solved.
The solution of this problem provides the necessary conditions for visualizing the operation of the aircraft hydraulic complex and determining the effect of the position of the actuators on the aircraft performance characteristics, which, in the final case, is the goal of simulation modeling of the aircraft hydraulic complex. The power supply unit of each of the systems includes a pump start-up system, in fact, pumps, a tank, a pressure relief valve, a check valve and hydraulic accumulators. In addition, the corresponding control and visualization system will be assigned to the power supply unit [4].
An important task in modeling the hydraulic complex and the power supply, in particular, is the creation of a simple simulation model for the occurrence and growth of functional failures, which makes it possible to investigate the process of the influence of functional failures on aircraft characteristics [5].
The reason for the change in the flow characteristic may be external and internal leaks and a decrease in the zero flow pressure. Failures of many units included in the power supply unit and hydro complex lead to such consequences, and there is no need to consider the impact of each failure separately due to the significant complication of the simulation model.

Research and results
For uniform consideration of the hydraulic systems of aircraft, regardless of the absolute values of pressures, flow rates, specific feeds, etc. we will carry out the modeling process in relative terms [6].
In relative terms, consider the mathematical model of the pump, its flow characteristic ( ) where -pump discharge pressure, -gas pressure in the accumulator, -engine speed, -flow rate of the working fluid in the discharge line, -specific theoretical effective pump flow, 0 -rated pump discharge pressure (zero flow pressure). The design scheme of the hydraulic system is shown in fig. 1  After starting or even during starting the engine, consumers of hydraulic energy can operate, the flow rate from which we estimate as The flow rate of the working fluid in the accumulator is The volume of the gas cavity of the accumulator depends on the flow rate and volume of the liquid received during charging (discharging) The gas pressure in the accumulator is , where 0 -pre-charge pressure of the gas chamber of the accumulator, It should be noted that the modeling process is somewhat complicated by the presence of a pressure relief valve, especially in difficult situations, when a relatively large pressure drop across the pressure relief valve [7].
The pressure relief valve model can be considered as an aperiodic link (regulation is carried out according to the integral error between the valve setting pressure and the current pressure in the discharge line up to the pressure relief valve) [8]. Another model (simplified) of a pressure relief valve is the dependence of its resistance on the pressure difference between the valve setting and the current pressure in the discharge line up to the pressure relief valve). On fig. 2 shows a block diagram for determining the parameters of the hydraulic system, where engine speed, 1accumulator pressure, 2brake accumulator pressure, Qtconsumption in the braking system, Qokflow through check valve, Qspspoiler subsystem consumption, Qitotal consumption of all subsystems, except for spoilers, pnkrelief valve pressure, Wtspecific theoretical pump flow On transport aircraft, the main power sources are driven by a sustainer engine, emergency pumping stations are either electric or turbo driven, and the operation of emergency stations begins some time after the control signal. The connection time is from two to ten seconds. The aircraft usually uses up to three independent hydraulic systems isolated from each other.
As the main sources of pressure, hydraulic pumps of variable flow installed on the gearboxes of propulsion engines are used. As a backup source, the same type of electric pumping stations of variable capacity are used, powered by an alternating current network of variable frequency [9]. Pumping stations are switched on in flight in case of pressure drop in the corresponding hydraulic system or in case of failure of the left (right) engine.
As a backup source of pressure in one of the hydraulic systems, a turbopump unit operating from the oncoming flow is installed.
If the main engine fails, the main hydraulic pump, which is driven by this engine, in the autorotation mode reduces the speed in proportion to the decrease in the speed of the main engine. When the main engine reaches (decreases) idle speed and after 10 ... 15 seconds, the pump is disconnected from the main engine to reduce the load during manual restart.
The connection of a backup pumping station with variable flow occurs in the presence of an alternating voltage from any side, after the pressure in the hydraulic system drops below 12 MPa. The pumping station is turned on and its output to the nominal operating mode occurs within 1 ... 2 seconds. With a decrease in the speed of the propulsion engines on the starboard and port sides at the same time below the idle speed, the pumping stations of variable flow are switched off due to a power outage [10].
The release of the turbopump installation is carried out with a decrease in the speed of the two propulsion engines below the idle speed. The total release time and reaching the nominal operating mode is 5 ... 10 seconds, during this time the hydraulic supply pressure in the system is maintained by a hydraulic accumulator.
Engine speed = ( ) are an external input parameter that does not depend on the operation of the hydraulic system. To control the work of consumers, we set the keys (0initial position, 1 -work), during operation we will consider the costs constant at the first stage, and the operating time is set by sending a signal to the corresponding key. Since the analysis is carried out in relative terms, the costs of consumers during their operation are also considered relative as a proportion of the corresponding maximum flow from the pump. The pressure in the pump discharge line is maintained (during normal operation of the hydraulic system) not lower than a certain value . The input signals of all functional subsystems also change from zero to one in the case of relay systems and from -1 to +1 in the case of servo control systems for control surfaces (ailerons, elevators and rudders, turning the front leg of the landing gear). The parameters of the output links (displacement, rotation angle, etc.) also change from 0 to +1 in the case of relay systems and from -1 to +1 in the case of servo systems [11].
For the functional subsystem of braking, the displacement of the output link (plungers of the brake cylinders) is considered an internal parameter of the hydraulic system, while it is convenient to consider the output values as the pressure in the brake cylinder, which determines the braking torque along with the condition and quality of the runway. The second output parameter determines the flow rate of the working fluid, which affects the total flow rate in the system and, accordingly, the value of the discharge pressure [12].
A large number of consumers of hydraulic energy and actuators of the hydro complex (taking into account duplication and redundancy, we have, in fact, about fifty actuators) forces us to use relatively simple models of their functioning. The most effective method for simplifying models and their relationships between themselves and the power supply is the linearization of characteristics, consideration and analysis of the relative values of hydrodynamic and design parameters. This method can significantly reduce the probability of errors in modeling. On Fig. 3…7 shows the results of simulation modeling in the Mathlab (Simulink) system of some processes in the hydraulic system of a passenger aircraft [13].
As an example, in fig. 3 shows the process of changing the hydrodynamic parameters of the power supply when starting the engine. The differences in pressure values are associated with the hydraulic resistance of the supply lines, the capacity and pre-charge pressures of the main and brake accumulators. Rapid increase in pressure up to = 0,3 associated with the compressibility of the working fluid, the elastic modulus of which is sufficiently large = (1. . .2)10 4 , and with a further increase in discharge pressure > 0,3 gas is compressed in the accumulator during = 3. . .4 .
The mathematical description of non-stationary thermal regimes in the systems of transport aircraft is usually carried out on the basis of the lumped capacity method. The following can be distinguished as thermal capacities: air inside the compartments, enclosing structures, structural elements of the hydraulic system, working fluid. The calculation of the system of thermal control and cooling of the working fluid of the hydraulic system usually begins with the cruising mode [14].
The process of heat transfer along the path of the hydraulic system is described by a system of differential equations in partial derivatives [15].
As objects with lumped parameters, in which the absorption or release of heat takes place: -a pump with appropriate fittings and valves, -pressure and drain pipelines with local resistances, -hydraulic motor with distribution and regulation system, -heat exchanger -working fluid tank.
To obtain a mathematical description of the thermal mode of operation of the hydraulic system, the heat balance equations for these objects are considered [16]. The model of thermal loading of the hydraulic system is considered taking into account thermal losses in the power supply unit, network, drives, heat exchange with the environment, the refrigerant of the heat exchanger, thermal capacity of the hydraulic system structural elements depending on the flight mode, loading and consumption of hydraulic energy.
During the operation of the hydraulic system, heat flows in various blocks change, which leads to the dependence of the temperature of the working fluid on the flight time, the operating mode of hydraulic energy consumers and external conditions (Fig. 4). If the pump regulator fails, the pressure rises to the pressure setting of the safety valve, which leads to an increase in losses for throttling the working fluid and an increase in its temperature to a maximum value [17]. The simulation results of the considered process of retracting and extending the landing gear in the event of failure of the main pump and connection of an emergency pumping station with an electric drive are shown in fig. 5 and fig. 6. To solve the problem in general terms, the initial state of the chassis is considered in the lock of the released position, after the command to clean, the speed of the pumping station increases linearly to the nominal( = 1).
The value of the discharge pressure due to the presence of leaks increases with an increase in the speed of the pumping station and when the pressure is reached = 0,9 the pump regulator comes into operation. After retracting the chassis, the pump pressure and the pressure in the rod end become the same and equal = 2 ≈ 1,0. A feature of the slat and flap control systems is the hot redundancy of the drive operating simultaneously from two systems, and when operating from one system in the operating range of flight speeds, the available force is sufficient to transfer the actuator from one extreme position to another. The speed of movement of the output link in this case is reduced by half [18].
Another feature of the drive is its tracking nature and the ability to stop in any intermediate position. In this, the operation of the flap and slat drives is identical to that of the spoilers.
Leaks between the cavities of the hydraulic motor lead to a decrease in the speed of movement of the rod with an obstructive load and an increase with an assist load.
In relative terms, we get the rod speed ( ) hydraulic motor and, accordingly, the flow rate from the input signal ( )  On fig. Figure 7 shows the results of simulation of the operation of the servo control system for spoilers.
On the chart in Fig. Figure 7 shows the phase shift between the input and output links, and when working with an obstructive load, the shift is significantly greater than when working with an assisting load. This is due to the dependence of the quality factor of the control loop on the magnitude of the load The situation is similar with the amplitude of the output link. During the operation of the hydro complex, the characteristics of the system for changing the pump delivery (flow regulator) change. Two types of feed regulator failures can be considered: -jamming of the regulator in an intermediate position; -decrease in zero flow pressure due to a change in the stiffness of the springs of the control spool and servo piston of the regulator [19].
Regulator jamming in the intermediate position leads to the operation of the safety valve, which practically does not change the discharge pressure, and a decrease in the maximum pump flow is equivalent to an increase in internal leakage. Therefore, as the main malfunction, we take into account the decrease in the zero flow of the pump. When modeling, the simulation of these faults is carried out by connecting the corresponding values of the quantities 0 , 1 , through switches that can serve to simulate an aircraft main engine failure.
The main parameters of the hydraulic accumulator are the pre-charge pressure. The failure mode of the accumulator is a decrease in the pre-charge pressure. 0 and time constant , the change of which is a failure of the accumulator.
Violation of the tightness of one of the hydraulic systems is identical to the failure of all pumps of the corresponding hydraulic system. The simulation can look like this: the flow rate of external leaks is set, integrated and subtracted from the nominal volume of liquid, which is fixed by the liquid level indicator in the tank. In the common injection line of the model, we set a multiplier proportional to the liquid level in the tank. A gradual decrease in the amount of liquid in the tank of the corresponding hydraulic system leads to a decrease in the supply of pumps and an increase in the response time of hydraulic drives [20][21][22].
Leaks between the cavities of the hydraulic motor lead to a decrease in the speed of movement of the rod with an obstructive load and an increase with an assisting load, therefore, imitation of internal leaks is carried out by introducing some resistance between the cavities of the hydraulic motor.

Conclusion
Thus, the simulation model of the aircraft hydraulic system should contain a power supply unit with a variable flow pump, the discharge line of which is presented in the form of a certain container with liquid, the modulus of elasticity of which is constant, and a hydraulic accumulator of a certain capacity and pre-charge pressure, resistance of the injection and drain lines is installed in the discharge line.
To simulate failure situations, blocks for changing the parameters of the power source and hydraulic system are needed: violation of internal and external tightness, changing the parameters of the hydraulic accumulator, reducing the zero flow pressure due to changes in the stiffness of the springs of the control spool and servo piston of the regulator, jamming of the regulator in an intermediate position.
The model of thermal loading of the hydraulic system considers thermal losses associated with both the aircraft flight mode and changes in the hydrodynamic parameters of the system in normal and failure situations in the power supply, network, drives.