CHAPTER 3 QUARTER AIRCRAFT MODELING


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1 30 CHAPTER 3 QUARTER AIRCRAFT MODELING 3.1 GENERAL In this chapter, the quarter aircraft model is developed and the dynamic equations are derived. The quarter aircraft model is two degrees of freedom model to deal with a single landing gear. The most commonly and widely used model for representing any vehicle suspension system is a quarter model. A quarter model is a simple model with two translational degree of freedom, which can depict the basic principle involved in a ride problem. This model essentially consists of a proper representation of controlling the aircraftbody and landing gear variations during landing impact and taxing on the surfaces with undulations. The advantage of using this model is that it allows in controlling or modifying the landing gear parameters in a simple manner, since it does not take into account any complex dynamics. However, this model contains no representation of the geometric effects of an aircraft and hence the effects of longitudinal and lateral interconnections cannot be studied. Initially in this chapter, two system models representing the passive and active landing gear system were modeled with the runway bump input for parametric analysis of landing gear and for numerical simulations to compare the dynamic response of passive and active landing gear. Passive control of vibration is relatively simple, reliable, robust and economical but it has its limitations. The control force generated in the passive device, depends entirely on the natural dynamics. Once the device is designed, after choosing the values of mass, stiffness coefficient, damping coefficient, location, it is not possible to adjust the control forces that are
2 31 naturally generated in real time. Further, there is no supply of power from an external source. Hence even the magnitude of the control forces cannot be changed from their natural values. Then PID controller was designed and implemented in the system model individually and, series of the simulation runs were carried out for the system for different runway excitations. The results which were obtained were then analyzed and a comparative study was done to compare the system responses of the passive system with the active system using PID controller. During a series of simulations, the effectiveness of the controllers were validated and the controllers optimum tuning values are obtained. 3.2 MODEL FORMULATION OF PASSIVE LANDING GEAR Passive landing gear system consists of upper and lower chambers. These two chambers are connected by an orifice. The upper chamber is filled with pressurized nitrogen or air and the remaining upper and lower volume is filled with hydraulic oil. The oil flow is regulated in the orifice area by metering pin. This absorber produces spring and damping characteristics. During the aircraft landing, the shock strut experiences compression and extension. This motion forces the oil to pass through the orifice which dissipates the large amount of energy created by landing impact. The oil flows from the lower to upper chamber, compressing the nitrogen that stores the remaining impact energy. This stored energy extends the shock strut and the oil flows from the upper chamber thus dissipating the residual impact energy. This compression and extension oscillation continues until all landing impact energy dissipates. The schematic diagram of passive landing gear is as shown in Figure 3.1.
3
4 33 Figure 3.2 Mathematical model of passive landing gear Linear Model Assumptions 1. All motions are in the vertical direction. 2. Vehicle seat cushioning is neglected. 3. Small motions. 4. Gravity ignored (measurement from equilibrium position). 5. Ground contact is maintained. 6. Rigid suspension linkages and vehicle body. 7. Damping in the tire. 8. Linear tire behavior. 9. No friction in suspension members. 10. Linear springs and dampers.
5 Dynamic Equation of Motion By Newton s second law of force equilibrium method the dynamic equations of motion are derived by using the free body diagram concept of sprung mass and unsprung mass.the second order differential equation describing dynamics of the passive landing gear is written by using free body diagram. Figure 3.3 Free body diagram of sprung mass From the Figure 3.3, the equation of motion of sprung mass representing the aircraft body is written as Equation (3.1) + ( ) + ( ) = 0 (3.1) Figure 3.4 Freebody diagram of unsprungmass
6 35 From the Figure 3.4, the equation of motion of un sprung representing the wheel components is written as Equation (3.2) + ( ) + + ( ) + = 0 (3.2) The equations of motion for the two degree of freedom system can be written in matrix form as Equation (3.3) y = 0 (3.3) Quarter Aircraft Model Parameters The model parameters taken from the Fokker aircraft for numerical simulation are given in Table 3.1 to analyze the vertical vibration levels such as acceleration, displacement and shock strut travel of the passive landing gear. Table 3.1 Quarter aircraft model parameters Description Value sprung mass ( ) 8800 kg unsprung mass ( ) 260 kg sprung mass stiffness rate ( ) N/m sprung mass damper rate ( ) N.s/m unsprung mass tire stiffness rate ( ) N/m unsprung mass tire damper rate ( ) N.s/m
7 36 Assuming sprung mass damping ratio = 0.35, The landing gear damping coefficient is = 2 = / The unsprung mass damping ratio and the tire damping coefficient as c = 2 (k + k )m c k 2 m= 37411Ns m the wheel mode. These parameters will yield the frequency for the body mode and 2 1 = = = Bump Model A widely used method to construct fortified runways is the casting of large plates using liquid concrete. These plates are separated from each other by gaps filled with rubber. Aging of concrete runways causes the plates to settle unevenly, leading to long wavelength bumps, ramps and steps at the gaps. Figure 3.5 illustrates an assumed half sine type runway bump of height 0.06 m and wave length 44 m (0.8*55 m/s) over which the airplane travels. Figure 3.5 shows the runway profile, generated as a function of time for simulations based on the relation Time=Distance/velocity. The ride dynamic behavior of the aircraft
8 37 due to a sinusoidal excitation is investigated. The excitation frequency based on the vehicle speed and the wavelength is computed as approximately 1.25 Hz (7.85 rad/sec) (frequency=velocity/wave length). The generated bump input is used for the analysis of model parameters Runway bump input time(s) Figure 3.5 Bump input for simulation 3.3 SIMULINK MODELING OF PASSIVE LANDING GEAR As MATLAB is a high level technical computing language and interactive environment for algorithm development, data visualization, data analysis, numerical computation and control system. Tool box software provides tools for systematically analyzing, designing, and tuning linear control systems. Simulink software is closely integrated with the MATLAB environment. It requires MATLAB to run, to define and evaluate model and block parameters. Simulink can also utilize many MATLAB features. There are six steps to modeling any system in the Simulink: 1. Defining the System 2. Identifying System Components
9 38 3. Modeling the System with Equations 4. Building the Simulink Block Diagram 5. Running the Simulation 6. Validating the Simulation Results The first three steps of this process is performed outside of the Simulink software before building the model. Appendix 1.5 represents the Simulink model in the block diagram form of the sprung mass equations of motion given in Equation (3.1) and unsprung mass equations of motion given in Equation (3.2). The model assumes that the sprung mass is free to move through vertically and the unsprung masses have contact with the runway surface. Thus, the vertical acceleration, velocity and displacement of the aircraft center of gravity are functions of the vertical displacement of the quarter aircraft model. The vertical acceleration, velocity, and vertical displacement from the sprung mass dynamics and un sprung masses dynamics are calculated by simulations.the developed simulink model of the passive landing gear is validated with the active suspension using PID controller (Mouleeswaran Senthil kumar 2008).The simulation of simlink model for parametric analysis of passive landing gear system is done through the matlab programme in Appendix 1.1. The simulink block diagram of the passive landing gear system for the simulation is shown in Appendix PARAMETRIC ANALYSIS OF PASSIVE LANDING GEAR Unsprung to Sprung Mass Ratio Effect The analysis has been performed by maintaining the aircraft parameters constant, as specified in the Table 3.1 and changing the unsprung mass. The quarter model un sprung mass changes from 50% of its nominal value and through its nominal value and to a final 200% of the nominal mass. These changes correspond to an unsprung mass to sprung mass change of 400%.The changing of un sprung mass is given in the Table 3.2.
10 39 Mass Table 3.2 Unsprung mass range for analysis Value(kg) Un sprung to sprung mass ratio 50% of unsprung mass Base line % of sprung mass The response to an excitation of a bump input with amplitude of 0.06 m is used to simulate the sprung mass and unsprung mass system. Figures 3.6 and 3.7 show that the body response is marginally changed by the variation in the unsprung to sprung mass ratio and the results are tabulated in Table Fuselage acceleration for different unsprung mass 50% of unsprungmaas Base line 200% of unsprung mass time(s) Figure 3.6 Acceleration response for different unsprung mass
11 Fuselage displacement for different unsprung mass 50% of unsprung mass Base line 200% of unsprung mass time(s) Figure 3.7 Displacement response for different unsprung mass Table 3.3 Dynamic response for change in unsprung mass Characteristic outcome Body mode frequency(rad/s) 50% of unsprung mass Base line 200 % of unsprung mass Body response is slightly affected Body settling time(s) Body peak displacement(m) Body peak acceleration(m/s²)
12 Landing Gear Spring Stiffness Effect The analysis was done by varying the spring stiffness while maintaining all other parameters constant, as shown in Table 3.1. The quarter model spring stiffness changes from 50% of its value through the original value and to a final 200% of the stiffness value. These changes correspond to a stiffness change range of 400%.The sprung mass response to a bump input shows that as the stiffness increases, the body mode is less damped and the natural frequency value increases. Figure 3.8 shows the higher amplitude of vibration when the stiffness value increases and Figure 3.9 shows the increases in displacement. The results are tabulated in the Table Fuselage acceleration of different landing gear stiffness 50% of stiffness Base line 200% of stiffness time(s) Figure 3.8 Acceleration response for different landing gear stiffness
13 Fuselage displacement for different landing gear stiffness 50% of stiffness Base line 200% of stiffness time(s) Figure 3.9 Displacement response for different landing gear stiffness Table 3.4 Dynamic response for change in landing gear stiffness Characteristic Outcome Body mode frequency(rad/s) 50% of spring stiffness Base line 200 % of spring stiffness Body response is less damped.higher natural frequency and higher acceleration Body settling time(s) Body peak displacement(m) Body peak acceleration(m/s²)
14 Landing Gear Damping Coefficient Effect The analysis was performed by maintaining all the aircraft parameters constant, as specified in Table 3.1, and changing damping coefficient. The quarter model damping coefficient was changed from 50% of its nominal value through the nominal value to a final 200% of the nominal damping coefficient. The sprung mass response to a bump input shows that as the landing gear damping increases, the body mode is more damped, with no change in natural frequency, and lower amplitude. From the Figure 3.10 and Figure 3.11 base line is suitable, there is too much overshoot for lower values of damping coefficient and the system gets too fast for higher values of damping coefficient and tabulated in Table Fuselage acceleration for different landing gear damping coefficient 50% of damping coefficient Base line 200% of damping coefficient time(s) Figure 3.10 Acceleration response for different landing gear damping coefficient
15 Fuselage displacement for different landing gear damping coefficient 50% of damping coefficient Base line 200% of damping coefficient time(s) Figure 3.11 Displacement response for a change in landing gear damping coefficient Table 3.5 Dynamic response for change in damping coefficient Characteristic Outcome Body mode frequency (rad/s) 50% of sprung mass damping coefficient Base line 200 % of sprung mass damping coefficient Less damping time with the damping increases, no change in body frequency Body settling time (s) Body peak displacement(m) Body peak acceleration(m/s²)
16 MODELING OF ACTIVE LANDING GEAR SYSTEM Figure 3.12 shows the active landing gear system consisting of low pressure reservoir, hydraulic pump, high pressure accumulator, servo actuator and electronic controller (Howell et al 1991). The passive system does not include servo actuator, transducers and electronic controllers. When an aircraft lands, the shock absorber stroke is influenced by the aircraft s payload and varies depending on runway excitations. The generation of active control energy is to attenuate the vibrations to improve the ride comfort. Figure 3.12 Schematic diagram of active landing gear system Active landing gear is mathematically modeled (Irwin Ross & Edson 1983, Horta et al 1999) and the active force is controlled by electronic controller which is activated by the sensors fitted in the landing gear. Energy is supplied through the hydraulic fluid to the landing gear system and also withdrawn from the system depends on load requirements by the servo system. In the active landing gear system, the stroke is measured by the
17 46 transducers and their signal input into the PID controller. This controller directs the servo valve to regulate the oil flow into or out of the shock absorber, hence producing the active control force to reduce the vibration level and also the force transferred to the airplane (Freymann & Johnson 1985, Freymann 1987, 1991). The mathematical modeling of the active landing gear system is as shown in Figure By Newton s second law, the dynamic equation of motion is derived, is the active control force. The equations of motion are written by using free body diagrams. Figure 3.13 Mathematical model of active landing gear system Figure 3.14 Free body diagram of sprung mass
18 47 From the Figure 3.14, the equation of motion for sprung mass is written as Equation (3.4) + ( ) + ( ) = 0 (3.4) Figure 3.15 Free body diagram of un sprung mass From the Figure 3.15, the equation of motion for unsprung mass is written as Equation (3.5) + ( ) + + ( ) + + = 0 (3.5) Dynamic equations can be written in matrix form as Equation (3.6) y = 0 (3.6) The general equation is written as Equation (3.7) [ ]{ } + [ ]{ } + [ ]{ } = { } (3.7)
19 48 where, [ ] is the mass matrix given by [ ] = 0 0 [ ] is the damping matrix given by [ ] = + [ ] is the stiffness matrix given by [ ] = + { } is the displacement vector given by { }= { } is the velocity vector given by { }= y { } is the acceleration vector given by { }= and { } is the force vector given by
20 49 { } = + The governing equation can be simplified as Equation (3.8) { } = [ ] { } [ ] [ ]{ } [ ] [ ]{ } (3.8) 3.6 SHOCK STRUT FORCES During operation of oleo pneumatic shock strut, damping effect is created by compressing the oil through metering orifice whose area is varied by the metering piston on various loading conditions. The air/nitrogen in the pneumatic chamber area is compressed by the hydraulic oil which provides air cushion spring effect throughout its operation. Sliding movement of parts in the system induces frictional forces adding to the shock strut forces. The gear forces are obtained as follows Air spring force is the force simulating the pressure of nitrogen gas in the upper chamber of the cylinder (Jayarami Reddy et al 1984). It is assumed that the pressure and volume of the gas satisfies the state of polytrophic equation of gas (3.9). = (3.9) where = pressure in the cylinder = area of the piston = Initial volume of the cylinder = stroke of the piston n = polytrophic constant
21 50 Damping force is provided by oil flow forced through an orifice by vertical strut position. The hydraulic oil flow is controlled by means of metering pin, The equation is written as Equation (3.10) = (3.10) where =Density of hydraulic fluid = area of the piston = velocity of the piston stroke = orifice coefficient =area of the orifice Friction force: The friction is proportional to the velocity and the air spring force developed in the system. This friction model is accurate in dynamic loading circumstances. The equation is written as Equation (3.11). y where = coefficient of friction =air spring force = velocity of the piston stroke The total axial force in the shock absorber = + + (3.11)
22 CONTROLLER DESIGN PID Controller ProportionalIntegralDerivative controller (PID) is a generic control loop feedback mechanism widely used in industrial control system. It is commonly used feedback controller. The error value is calculated as the difference between a measured variable and reference point. The controller has got three control parameters called the proportional, the integral and derivative values. It is denoted as P, I and D.P depends on the present error, I is the accumulation of past errors and D is a prediction of future errors based on the current rate of change. The weighted sum of these three actions is used to adjust the active control force by controlling the servo valve PID Controller Theory The PID control is named by three terms viz, the proportional (P), integral (I) and derivative (D) (Shinners 1964) are summed to calculate the output of the PID controller. It is written as Equation (3.12) = ( ) + ( ) + ( ) (3.12) Proportional term The proportional term is used to change the output. The output is proportional to the current error value. The proportional value is calculated by multiplying the error by a constant.this is called proportional gain. The proportional term is given by the Equation (3.13) = ( ) (3.13)
23 52 If the proportional gain is too high, the controller system will become unstable. If the proportional gain is too low, the control will be very small and will not respond to the system disturbances Integral term The integral term is proportional to the magnitude of the error and the duration of the error. The integral is the sum of instantaneous error over time.which is known as accumulated error. The integral term is calculated by multiplying the accumulated error and the integral gain( ).It is given by the Equation (3.14) = ( ) (3.14) The integral term accelerates the movement of the process towards reference point and eliminates the residual steady state error that occurs with a pure proportional controller Derivative term It is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain.the derivative term is given by the Equation (3.15). = ( ) (3.15) The derivative term slows the rate of change of the controller output. Derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the controller stability.
24 Tuning Method Tuning a control loop is the adjustment of its control parameters to the optimum values for the desired control response. PID tuning is a difficult problem, because it must satisfy complex criteria within the limitations of PID control. Generally stability of response is required and the process must not oscillate for any combination of process conditions and set points, though sometimes marginal stability is accepted or desired. There are several methods (Datta et al 2000) of tuning of PID controller. The most effective methods generally involve the development of some form of process model, with appropriate P, I and D based on the dynamic model parameters. The different methods are manual method, Ziegler Nichols method, Cohen coon method and software tools as given in the Table 3.6. Manual tuning methods can be relatively inefficient, particularly if the loops have response times on the order of minutes or longer. In this work, Ziegler Nichols method is simple and often used. Table 3.6 Various tuning methods Method Advantages Disadvantages Manual tuning ZieglerNichols Software tools Cohencoon No mathematical knowledge required. Online method Proven method. Online method Consistent tuning, online or offline method. May include valve and sensor analysis. Allow simulation before downloading. Can support non steady state tuning. Good process models Requires experienced personnel Process upset some trial and error. Very aggressive tuning Some cost and training involved. Some math, offline method. Only good for first order process.
25 ZieglerNichols method This method is introduced by John Ziegler & Nathaniel Nichols (1940). First the and gains are set to zero. The gain is increased until it reaches the ultimate gain at which the output of the loop starts to oscillate. and the oscillation period are used to tune the gains as shown in Table 3.7. Table 3.7 Ziegler Nichols method Control type P 0.5 PI / PID / /8 Equation (3.16) The PID controller design Haitao Wang et al (2008) is defined by = ( ) + ( ) + ( ) (3.16) is the current input from the controller. is the proportional gain, and is the integral and derivative gain of the PID controller. ( ) represents a reference signal and is the feedback signal measured from the sensors fitted in the landing gear. The simulink modeling of PID controller is shown in Figure The error function is written as Equation (3.17) ( ) = ( ) ( ) (3.17)
26 55 Figure 3.16 Simulink model of PID controller The output signal of the controller gives the displacement of the servo valve as Equation (3.18) ( ) = { ( ) [ ( ) ( )]} + { ( ) ( ) ( )} + { ( ) [ ( ) ( )]} (3.18) The feedback coefficients viz,, are adjusted by Ziegler Nichols tuning rules to obtain the best control over the servo valve. 3.8 HYDRAULIC POWER SUPPLY SYSTEM The following subsections comprise a brief description of the Principal Hydraulic Elements that make up a typical position controlled system. The block diagram of the hydraulic system is shown in Figure Low Pressure Reservoir A hydraulic reservoir is a tank or container designed to store sufficient hydraulic fluid for all conditions of operation. Reservoirs have additional storage place to have a reserve of fluid for the emergency operation of the landing gears, flaps, etc. The reservoir is pressurized to provide a
27 56 continuous supply of fluid to the pumps. The reservoir may be pressurized by spring pressure, air pressure or hydraulic pressure. The desired pressure to be maintained ranges from 10 psi to 90 psi approximately Hydraulic Gear Pump Gear pump is commonly used in the hydraulic system. It is a positive displacement pump. The gears of the pump are driven by the power source, which could be an engine drive or electric motor drive. The fluid trapped in the clearance between the gears and casing is forced through the out port High Pressure Accumulator An accumulator is basically a chamber for storing hydraulic fluid under pressure. It can serve one or more purposes. It dampens pressure surges caused by the operation of an actuator. It can aid or supplement the system pump when several units are operating at the same time and demand is beyond the pump capacity. An accumulator can also store power for limited operation of a component if the pump is not operating. Finally it can supply fluid under pressure for small system leaks that would cause the system to cycle continuously between high and low pressure. The accumulators are of the diaphragm, bladder, and piston types. The pressure in the accumulator is approximately 3000 psi Servo Actuator Servo actuator is designed to provide hydraulic power and it includes an actuating cylinder, a multiport flow control valve, check valves and relief valves together with connecting linkages. The movement of the
28 57 piston in the servo actuator depends on the control signal from the electronic controller. Figure 3.17 Block diagram of hydraulic servo system Active Control Force Active control force is a function of the flow output of the servo valve. The servo valve displacement ( ) is controlled by the PID controller. The controller actuates the servo valve by the velocity signal, measured by the transducers. There is no exact relationship between the active control force and the flow quantity from the servo valve (Sharp 1988).It is often determined through experiments or by empirical formula. It is assumed that the active control force (Haitao Wang et al 2008) is described by Equation (3.19) = (3.19) The flow quantity is calculated by Equation (3.20)
29 58 = (3.20) when the displacement of the servo valve ( ) > 0, the hydraulic oil would have positive flow from the accumulator in to the landing gear system and a positive control force > 0. When ( ) < 0, oil is drawn from the landing gear in to LP reservoir so that < 0, where ( ) is the displacement determined from the controller as in Equation (3.18). 3.9 SIMULINK MODELING OF THE ACTIVE LANDING GEAR SYSTEM Simulink model of the single active landing gear is the block diagram form of the equations of motion given in Equation (3.8).The model assumes that the sprung mass is free to move in the vertically and the un sprung masses have contact with the runway surface. Thus, the vertical acceleration, velocity and displacement of the aircraft center of gravity are functions of the vertical displacement of the quarter aircraft model. The simulation of this simulink model is done through the mat lab program in Appendix 1.2. The Simulink block diagram of the active landing gear system for the simulation is shown in Appendix BUMP MODEL An assumed half sine type runway bump of height 0.10 m and wave length 44 m (0.8*55 m/s) were generated over which the airplane travels. The runway profile is generated as a function of time for simulations based on the relation Time=Distance/velocity. 'The ride dynamic behavior of the aircraft due to a sinusoidal excitation is investigated (Catt et al 1992). The excitation frequency based on the vehicle speed and the wavelength is computed as
30 59 approximately 1.25 Hz (7.85 rad/sec) (frequency=velocity/wave length).the equation is written as Equation (3.21) = 100(1 cos ) otherwise (3.21) The half sine wave bump model with a height of 0.1m is designed in Matlab/Simulink. The model is generated based on the above equation. The product of step block and sine wave block is used in the bump model generator. The profile generator of bump input for simulation is shown in Figure Step1 Step Constant Product 1 Output Sine Wave Figure 3.18 Simulink model of bump input
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