UNIVERSITÀ DI PISA
Scuola di Ingegneria
Dipartimento di Ingegneria Civile ed Industriale
Tesi di Laurea Magistrale in Ingegneria
Aerospaziale
Aircraft directional dynamics with differential
brakes: landing gears modelling and anti-skid
control with gear-walk suppression
Relatori Candidato
Prof. Ing. Gianpietro Di Rito Antonino Franco Pizzino
Prof. Ing. Roberto Galatolo
Ing. Francesco Schettini
Alla mia famiglia e tutti coloro che mi hanno accompagnato e sostenuto
Abstract
This thesis is part of the research activities carried out at the University of Pisa, Department of Civil and Industrial Engineering, for the development of the landing roll-out simulator of a light jet aircraft. The main objective of the simulator is to investigate the potentialities and the limitations of implementing automatic control functions for the directional control of the aircraft on ground through the combined use of conventional commands (steering and rudder) and differential brakes. An accurate simulation of landing gears and brakes plays a key role in this context, because the additional loads requested to brake can emphasize structural vibrations on landing gears. With this objective a model of main landing gears with variable wheel track and gear walk mode vibration is developed and integrated in the aircraft simulator. The analysis aims at evaluating the effectiveness of directional control laws using differential brakes and nose-wheel steering only, especially if anti-skid control includes gear walk suppression capabilities. In particular, part of the work is focused on the anti-skid control design to prevent instability of main landing gear vibration during the braking phase.
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Table of contents
List of Figures………..………iii
List of Tables………ix
Introduction………..1
1. Stability and control issues in aircraft braking dynamics……….2
1.1 Landing roll-out: a critical mission phase ………....2
1.2 Differential brakes for system safety enhancement………...7
1.3 Gear-walk phenomena……….….9
1.4 Ground dynamics simulator: previous version………..12
1.4.1 Aircraft model features………12
1.4.2 The reference aircraft ………..13
1.4.3 Directional control architecture……….15
2. Main landing gear modelling………..20
2.1 Aircraft landing gear configuration……….20
2.2 Enhancement of the kinematic model ……….……….26
2.3 System dynamics at ground impact……….33
2.3.1 Simplified model with rigid tyre……….33
2.3.2 Equilibrium point definition……….………37
2.3.3 Effect of shock absorber dynamics. ……….…40
2.3.4 Effect of tyre vertical deformation………45
2.3.5 Effect of tyre lateral forces……….47
2.4 Gear-walk modelling………...55
3. Preliminary design of anti-skid control with gear-walk suppression……….58
3.1 Reduced models of the longitudinal braking dynamics……….58
3.1.1 Simplified two-state model transfer functions………..……….58
3.1.1.1 Open-loop………...62
3.1.1.2 Closed-loop………64
3.1.2 Incremental modelling for system analysis………..…64
3.1.2.1 Effects of brakes transfer function………..64
3.1.2.2 Effects of gear-walk mode………68
3.2 Gear-walk suppression technique ……….82
4. Aircraft directional control performances ……….88
4.1 Test campaign definition ……….88
4.2 Commands and disturbances time histories ………...89
4.3 Ground impact and symmetrical braking dynamics………90
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4.3.2 Effects and compensation of gear-walk phenomena……….…96
4.3.3 Effects of runway conditions……….…99
4.4 Ground path control during braking………..102
4.4.1 Lateral bump gust response ………..102
4.4.2 Effects of runway irregularities. ……….110
5. Conclusions and future developments ……….114
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List of Figures
Figure 1.1 - Fatal Accidents and Onboard Fatalities by Phase of Flight [1] ………...3
Figure 1.2 Fatalities by CICTT Aviation Occurrence Categories [1] ……….4
Figure 1.3 – A nose wheel steering tiller located on the flight deck [24] ……….6
Figure 1.4 - Standard actions that can be taken by the flight crew to regain control during landing ground roll[4].…6 Figure 1.5 - Influence of nose wheel steering, differential braking and rudder on ground directional control [2] …….7
Figure 1.6 - Working scheme of Kilner and Warren’s directional control [3]………8
Figure 1.7 - Vibration modes of a landing gear [13] ……….………...10
Figure 1.8 - Instabilities of landing gear ………..………..….10
Figure 1.9 – Shimmy and Gear Walk [15] ………...11
Figure 1.10 - Example of tripod-type main landing gear model ………12
Figure 1.11 – Reference Aircraft details of [21] and [22] ……….…………13
Figure 1.12 – Eclipse 500, "Very Light Jet" aircraft manufactured by Eclipse Aviation ……….……….... 14
Figure 1.13 – Scheme of the nested loops……….………...…16
Figure 1.14 - Top level of the nonlinear model ………..17
Figure 1.15 - Simplified brake system [6] ………..…18
Figure 1.16 - Detail of scheme in Simulink………..……….……….…19
Figure 2.1 – Tricycle type landing gear on a Learjet(left) and a Cessna 172(right) [24] ………..………… 20
Figure 2.2 – Tripod type landing gear with non-shock absorbing struts [24] ………21
Figure 2.3 - Example of the tripod type MLG configuration studied in the present thesis work [26] ………21
Figure 2.4 - How technologies help meet challenges [25] ………22
Figure 2.5 – Simplified model of the vertical shock strut used in previous works ………23
Figure 2.6 - Oleo-pneumatic shock absorber ………23
Figure 2.7 – Shock absorber Load deflection curve………...24
Figure 2.8 – Scheme for the calculus of landing gears reaction in function of their distance from CG ………..24
Figure 2.9 – Initial configuration of reference MLG………..26
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Figure 2.11 - 𝛽𝑤 and 𝛽𝑎 as function of the vertical displacement 𝒛𝑨− 𝒔𝒘 ………...31
Figura 2.12 - ∆𝒂 as function of the vertical displacement 𝒛𝑨− 𝒔𝒘 ………...32
Figura 2.13 - 𝒚𝒘 as function of the vertical displacement 𝒛𝑨− 𝒔𝒘 ………..………32
Figure 2.14 - Simplified model of the single mass dynamics ………..……33
Figure 2.15 – Dynamics of the block Mm∗ ……….34
Figure 2.16 - Rotation dynamic of the component AB of the MLG ……….35
Figure 2.17 – 1 DOF simplified model with a spring damper system ………36
Figure 2.18 - 𝐹𝑎 necessary to guarantee equilibrium for each possible configuration ∆𝑎_𝐸𝑄 ……….37
Figure 2.19 – Required stiffness 𝐾𝑎_𝐸𝑄 that guarantees equilibrium for each configuration as function of ∆𝑎_𝐸𝑄……38
Figure 2.20 -Intersections between 𝐹𝑎_𝐸𝑄𝑠𝑝𝑟𝑖𝑛𝑔 and 𝐹𝑎_𝐸𝑄 as a function of ∆𝑎_𝐸𝑄 , decreasing stiffness constant...…39
Figure 2.21 - 𝑧𝐴_𝐸𝑄 as function of ∆𝑎_𝐸𝑄 ……….…39
Figure 2.22 - Shock absorber model ………...…40
Figure 2.23 - Pressure jumping through the orifice ………41
Figure 2.24 - Classical polytropic compression of a gas [27] ……….42
Figure 2.25 – Plot of static load 𝐹𝑔 (Eq. 2.46) ……….44
Figure 2.26 – Intersection between functions of Eq.2.27 and Eq.2.46 in the desired equilibrium configuration …..…..44
Figure 2.27 – Simplified model of vertical tyre deformation ……….45
Figure 2.28 – Effects of Tyre deformation over shock absorber pressure 𝑃ℎ ………..46
Figure 2.29 – Effects of Tyre deformation over vertical loads 𝐹𝑣𝑤 ……….47
Figure 2.30 – System dynamic considering tyre lateral viscous forces ………..…….47
Figure 2.31 - 𝐹𝑎 necessary to guarantee equilibrium for each desired configuration ∆𝑎_𝐸𝑄………48
Figure 2.32 - Intersection between functions of Eq.2.53 and Eq.2.55 in the desired equilibrium configuration …….…49
Figure 2.33 - Temporal evolution of 𝑦𝑤 ……….50
Figure 2.34 – Temporal evolution of Fy_w as function of the chosen 𝑉𝑦∗ ………..51
Figure 2.35 – Effects of 𝐹𝑦_𝑤 over vertical displacement ………52
Figure 2.36 - Effects of 𝐹𝑦_𝑤 over lateral displacement of the wheel ………..52
Figure 2.37 – Effects of 𝐹𝑦_𝑤 over angle 𝛽𝑤 ………53
Figure 2.38 - Effects of 𝐹𝑦_𝑤 over angle 𝛽𝑎 ………...53
Figure 2.39 – Effects of 𝐹𝑦_𝑤 over shock absorber compression ………..54
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Figure 2.41 – Effects of 𝐹𝑦_𝑤 over vertical load 𝐹𝑣_𝑤 weighing on wheel ……….55
Figure 2.42 - Model used to study MLG longitudinal vibration ………55
Figure 2.43 – Temporal evolution of longitudinal wheel deflection in the nonlinear model ………57
Figure 3.1 - Simplified model of the longitudinal aircraft dynamics during braking………..….58
Figure 3.2 – Open loop of the simplified 2 states system ………..……62
Figure 3.3 – Scheme of the slip ratio control by using the technique of loopshaping ………63
Figure 3.4 - Closed loop unit step response of Sx ………63
Figure 3.5 – Insertion of the brakes actuation dynamics transfer function in the open loop ………..64
Figure 3.6 – Root locus of the transfer function S̃x(s) T̃ (s)b Hbrake(s) ………..64
Figure 3.7 – Bode diagram of the transfer function S̃x(s) T̃ (s)b Hbrake(s) ……….65
Figure 3.8 – Effect of poles and zeros introduced by Hbrake(s) on the frequency response ………..65
Figure 3.9 – Insertion of a lag-lead compensator in the open loop ………..66
Figure 3.10 - Closed loop of the 2-states system considering brakes actuation dynamics ……….66
Figure 3.11 – Root locus of the open loop transfer function GOL= R(s) Hbrakes(s) TS̃x(s) b ̃ (s) ………..67
Figure 3.12 - Closed loop unit step response of Sx considering brakes actuation dynamics ……….67
Figure 3.13 – Schematic representation of gear walk vibration ……….68
Figure 3.14 – Schematic representation of forces acting on wheel ………..….69
Figure 3.15 - Open loop of the reduced 4 states model ………...72
Figure 3. 16 – Position of gear walk dynamic poles and zeros for ζw= 0.1 ………73
Figure 3.17 - Position of gear walk dynamic poles and zeros for ζw= 0.05 ………..74
Figure 3.18 – Insertion of the transfer function of brakes actuation dynamic in the open loop of the 4-states model….………..74
Figure 3.19 - Effect of poles and zeros introduced by gear walk dynamics for ζw= 0.1 ………75
Figure 3.20 - Effect of poles and zeros introduced by gear walk dynamics for ζw= 0.05 ……….75
Figure 3.21 - Closed loop of the 4-states system ……….76
Figure 3.22 – Frequency response of the slip ratio open loop transfer function GOL(s) of the 4-states system for ζw= 0.1………..76
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(with closed loop poles evidenced) ………...……77 Figure 3.24 - Closed loop unit step response of Sx of the 4-STATES system for ζw= 0.1 ………77
Figure 3.25 – Effects of Gear walk dynamic over closed loop unit step response………..78 Figure 3.26 - Unit step response of the longitudinal deflection of the wheel δ̃w for each value of ωw
and ζw= 0.1……….…79
Figure 3.27 - Unit step response of wheel longitudinal acceleration for ζw= 0.1 ………..79
Figure 3.28 - Frequency response of the slip ratio open loop transfer function GOL(s) of the 4-states system
for ζw= 0.05………80
Figure 3.29 - Root locus of the open loop transfer function GOL for ωw= 30 Hz and ζw= 0.05
(with closed loop poles evidenced) ………...…80 Figure 3.30 - Closed loop unit step response of Sx of the 4-STATES system for ζw= 0.05 ………80
Figure 3.31 – Effect of the damping factor ζwon the δ̃wunit step response ……….81
Figure 3.32 – Effect of the damping factor ζw on the unit step response oh the wheel longitudinal acceleration …….81
Figure 3.33 – Insertion of a notch filter inside the open loop of the 4-states model……….82 Figure 3.34 – Slip ratio control of the 4-states system by using a lag-lead compensator and a notch filter………82 Figure 3.35 - Frequency response of the slip ratio open loop transfer function 𝐺𝑂𝐿(𝑠) of the 4-states system
with 𝜁𝑤= 0.05……….83
Figure 3.36 - Root locus of the slip ratio open loop transfer function 𝐺𝑂𝐿 of the system for 𝜔𝑤= 30 𝐻𝑧
and 𝜁𝑤= 0.05(with closed loop poles evidenced) ………83
Figure 3.37 - Closed loop unit step response of 𝑆𝑥 of the 4-STATES system for 𝜁𝑤= 0.05
with a notch filter applied………...….84 Figure 3.38 – Damping effect of notch filter on the closed loop unit step response of the 4-states model………84 Figure 3.39 – Damping effect of notch filter on the closed loop response of 𝛿̃𝑤for each value of 𝜔𝑤
and 𝜁𝑤= 0.05………85
Figure 3.40 - Effect of the notch filter on the closed loop response of the wheel longitudinal acceleration………..85 Figure 3.41 - Effect of the notch filter on the closed loop response of the wheel longitudinal acceleration
for 𝜔𝑤= 60 𝐻𝑧………..…86
Figure 3.42 - Effect of the notch filter on the closed loop response of the wheel longitudinal acceleration
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Figure 4.1 – Temporal evolution of the thrust………..………89
Figure 4.2 – Temporal evolution of the gust ……….……89
Figure 4.3 – Temporal evolution of the load factor on the right wheel ………..…90
Figure 4.4 – Temporal evolution of right tyre deformation ………91
Figure 4.5 - Temporal evolution of right main landing gear vertical displacement ………..…91
Figure 4.6 - Temporal evolution of the load factor on the nose wheel ………91
Figure 4.7 - Temporal evolution of vertical tyre deformation of the nose wheel………...…92
Figure 4.8 - Temporal evolution of nose landing gear vertical displacement ………92
Figure 4.9 - Temporal evolution of aircraft pitch angle……….…93
Figure 4.10 - Temporal evolution of the right wheel slip ratio………..…93
Figure 4.11 - Temporal evolution of the right wheel braking torque ………..…94
Figure 4.12 - Temporal evolution of the right wheel braking pressure ………..…94
Figure 4.13 - Temporal evolution of the aircraft landing distance ………95
Figure 4.14 - Temporal evolution of right wheel slip ratio ……….95
Figure 4.15 - Temporal evolution of 𝑎𝑤………96
Figure 4.16 – Effect of notch filter on the temporal evolution of 𝑎𝑤………..97
Figure 4.17 – Effect of notch filter on the temporal evolution of 𝑎𝑤 for 𝜔𝑤= 60 𝐻𝑧 ………..…97
Figure 4.18 – Effect of notch filter on the temporal evolution of 𝑎𝑤 for 𝜔𝑤= 60 𝐻𝑧 and 𝜔𝑤= 60 𝐻𝑧 ……….98
Figure 4.19 – Temporal evolution of nose landing gear load factor………..98
Figure 4.20 - Temporal evolution of nose landing gear vertical displacement………..99
Figure 4.21 - Temporal evolution of the aircraft pitch angle……….99
Figure 4.22 - Temporal evolution of right wheel slip ratio ………..100
Figure 4.23 – Temporal evolution of the braking torque………...100
Figure 4.24 - Temporal evolution of the aircraft speed ………..101
Figure 4.25 - Temporal evolution of aircraft landing distance ……….101
Figure 4.26 - Temporal evolution of landing gears load factor ………102
Figure 4.27 - Temporal evolution of landing gears vertical displacement……….103
Figure 4.28 - Temporal evolution of the right wheel slip ratio ………..103
Figure 4.29 - Temporal evolution of the left wheel slip ratio ………..104
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Figure 4.31 - Temporal evolution of the left braking torque ………..………105
Figure 4.32 - Temporal evolution of the right braking torque after the instant of the brakes activation……….………105
Figure 4.33 - Temporal evolution of brakes pressure ………..………106
Figure 4.34 - Effect of notch filter on the temporal evolution of 𝑎𝑤 for 𝜔𝑤= 30 𝐻𝑧 ……….………106
Figure 4.35 - Temporal evolution of nose wheel steering ……….……107
Figure 4.36 - Temporal evolution of yaw angle ………...…107
Figure 4.37 – Temporal evolution of the sideslip angle and the yaw angular velocity ……….…108
Figure 4.38 – Temporal evolution of the roll angle and the roll angular velocity ………109
Figure 4.39 - Temporal evolution of the aircraft lateral displacement ………..…110
Figure 4.40 - Temporal evolution of the right wheel slip ratio ………..…110
Figure 4.41 - Temporal evolution of the left wheel slip ratio ………..…111
Figure 4.42 - Temporal evolution of the right braking torque ………..…111
Figure 4.43 - Temporal evolution of the left braking torque ……….….112
Figure 4.44 - Temporal evolution of the wheels brake pressure ………..…112
Figure 4.45 – Effect of notch filter on temporal evolution of a_w of the right wheel ………..113
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List of Tables
Table 1.1 - Most important causal factors for landing veeroffs [2] ………..5
Table 1.2 - Aircraft Aeromechanical characteristics ………..……..14
Table 2.1 - 𝑀𝑚∗ , 𝑀𝑛∗, 𝑏𝑚, 𝑏𝑛 in function of the chosen ɛ𝑙𝑔 ………..…25
Table 2.2 – Parameters numerical values of MLG initial configuration ……….27
Table 2.3 - Initial values of kinematic variables ……….28
Table 2.4 – Comparison between initial and equilibrium values of kinematic variables ………31
Table 2.5 - Input data for the shock absorber sizing in the desired equilibrium condition………43
Table 2.6 – Shock absorber parameters obtained from data of Table 2.4 ………43
Table 2.7 -Parameter used to model dynamics of vertical tyre deformation ………..45
Table 3.1 - Values considered the transfer functions of the longitudinal braking dynamic linearized model ………… 62
Table 3.2 - Damping factor ζz of “gear walk” zeros varying ………..…73
1
Introduction
The aircraft ground dynamics and the related vibration issues are one of the key factors of a successful aircraft design. This thesis analyses the performances of a light jet aircraft during the landing roll-out phase (i.e. from the touchdown to the taxi condition) through a flight simulator. This model includes the 6-DOFs dynamics of the aircraft and detailed models of hydraulic brakes, shock-absorbers tyre loads and the nested architecture of the four closed loop controls for the directional control of the aircraft, through the combined use of standard commands (steering and rudder) with hydraulic differential brakes.
This work is divided in 4 chapters. In the first chapter is showed why landing is a critical mission phase and it is explained the importance of development of automatic systems for the directional control of an aircraft during the ground portion of the landing procedure. The purpose of the second chapter is to develop a model of main landing gears with a tripod-type configuration and variable wheel track to show how it increases the directional control capabilities of differential brakes, allowing to minimize the use of rudder. On the other hand, the additional loads requested for the differential brake can emphasize structural vibrations on main landing gears. In particular, tripod type configuration is known to suffer from gear walk problems also in normal braking conditions.
For this reason, the aim of the chapter 3, is to get a linearized model of the longitudinal braking dynamics, in which it is considered the wheel gear walk dynamics. A new slip ratio control is designed in order to prevent instability of landing gear vibration during braking phase. In chapter 4 the model of main landing gears with variable wheel track and gear walk mode vibration is integrated in the aircraft simulator. Through the nonlinear analysis of Simulink model, the aim is to investigate the potentialities and limitations of the updated aircraft directional control through the combined and continuous use of only nose wheel steering and differential brakes and to verify the effectiveness of the antiskid control design for gear walk suppression.
2
Chapter 1 - Stability and control issues in
aircraft braking dynamics
Crosswinds and adverse weather conditions, that reduce the runway friction, typically interfere with or limit the operation of aircrafts. In particular, depending on the crosswind severity, the runway conditions, the width of the runway and the type of aircraft, landing may be impossible at an intended destination, maintaining adequate safety margins. In such a situation, the aircraft is often diverted to an alternate landing site or its landing must be delayed until wind and/or runway conditions improve to a point that a landing attempt is safe, within the operational capabilities of the aircraft and its crew. Needless to say, the resulting delays and diversions are costly in terms of fuel and time expended and can cause interruptions in scheduled aircraft operation [3], so the possibility to land in critical conditions can be an enormous advantage from an economic point of view.
In this perspective, it assumes a great importance the development of automatic systems for the directional control of the aircraft during the ground portion of the landing procedure. For this purpose and for safety reasons, this control is realized through the combined use of conventional commands (steering and rudder), and differential brakes, so that the control can be maintained also in case of failure of one the axis command.
1.1 Landing roll-out: a critical mission phase
Aircraft landing is a difficult manouvre. To execute it safely, the pilot must achieve many goals according to the phase. The lack of some devices (brakes, anti-skid, reverse thrust, lift-dump, etc.) or critical weather conditions (wet runway, crosswind, etc.) can considerably influence the landing.
The landing has a small percent duration on a complete mission (only 1%), but a high number of fatal accidents occur during this phase (24%). Figure 1.1 shows its high level of criticality.
Figure 1.2 highlights that Runway Excursion (RE) during landing is the most relevant cause of accidents, i.e. the event in which an aircraft veers off or overruns the runway surface during landing. Runway Excursion is the third reason of accidents, so the studying and the improving of the aircraft behavior during the ground portion of landing procedure are very important to avoid accidents.
Chapter 1 - Stability and control issues in aircraft braking dynamics
3
Chapter 1 - Stability and control issues in aircraft braking dynamics
4
Chapter 1 - Stability and control issues in aircraft braking dynamics
5
The present and previous thesis works focus on automatic control systems and methods utilized to maintain an aircraft within the lateral boundaries of a runway when a crosswind landing is executed on a runway that exhibits reduced frictional properties because of environmental factors such as rain, snow or ice or because of other deleterious runway conditions.
A statistical analysis of most important causal factors in landing veeroffs (which occurred worldwide) is shown in Table 1.1 and puts into evidence the importance of such an automatic control system.
Table 2.1 - Most important causal factors for landing veeroffs [2]
Wet/contaminated runways and crosswind appear to be dominating causal factors. It follows crosswinds exceeding the capabilities of the aircraft or inadequate compensation by the pilots (see the factor “Aircraft directional control not maintained” in Table 1.1). It is important to note that only crosswind operations on dry runways conditions are certified. Aircraft manufactures only give advisory information on crosswind limits for wet/contaminated runways.
Another important factor in landing veeroffs is nose wheel steering issues. There are several reasons for the nose wheel steering issues, however improper maintenance seems the more dominant and also the incorrect use of the steering system. Directional control can be maintained on ground by rudder deflection (through the rudder pedals), nose wheel steering (through the rudder pedals and/or the nose wheel steering control handle or “tiller”), differential braking and/or through differential (reverse) thrust.
However, during a normal landing roll in which the aircraft is decelerated, only the rudder pedals are used to steer the aircraft on the runway centreline. The rudder pedals deflect the rudder and, once the nose wheel is on the ground, have limited authority over the nose wheel deflection (a maximum of 5 to 7 degrees nose wheel deflection with maximum rudder pedal deflection is a common value [24]). At lower speed the rudder becomes ineffective and the tiller is then used e.g. for exiting the runway, for turns during taxiing and for apron movements.
Chapter 1 - Stability and control issues in aircraft braking dynamics
6
Figure 1.3 – A nose wheel steering tiller located on the flight deck [24]
The tiller, shown in Figure 1.3, can command a much larger nose wheel deflection pedals (up to 65-75 degrees nose wheel deflection for full tiller deflection [24]). Flight crews are advised not to use the nose wheel steering tiller until reaching taxi speed. The use of the nose wheel steering tiller at higher speeds can introduce directional control problems. On the Airbus aircrafts, the tiller authority in terms of maximum steering angle is progressively reduced above a groundspeed of around 20 m/s (to avoid usually unsuccessful attempts of correcting aircraft path with the tiller at high speed), and rudder pedals authority is progressively reduced above 50 m/s of ground speed (to avoid excessive inputs at high speed).
If the aircraft track deviates from the runway centreline during landing ground roll there are a number of standard actions, shown in Figure 1.4, that can be taken by the flight crew to regain control. The pilot must reduce reverse thrust to reverse idle (if reverse thrust was selected) or even to forward idle, release the (auto-) brakes, use rudder pedals and if necessary use differential braking to correct back to the runway centerline.
Figure 1.4 - Standard actions that can be taken by the flight crew to regain control during landing ground roll [4]
When re-established and re-aligned on the runway centreline, the pilot should resume normal braking techniques by applying brakes and reverse thrust as required. Figure 1.5 illustrates how rudder, differential braking and nose wheel steering influence the directional control as function of ground speed from touchdown to a full stop [2].
Chapter 1 - Stability and control issues in aircraft braking dynamics
7
It is worth to be noted that, in the landing roll-out simulator developed in these thesis works, the aircraft braking is always active.
Figure 1.5 - Influence of nose wheel steering, differential braking and rudder on ground directional control [2]
Directional control must be maintained during all the phase of landing roll-out, until normal taxi speed is reached and the aircraft is brought to a stop. In particular, the attention of previous and present thesis works are focused on directional control of a light jet aircraft during the ground portion of the landing procedure (see Figure 1.4) wherein the aircraft decelerates from the touchdown velocity to a taxi or turn-out speed.
1.2 Differential brakes for system safety enhancement
Several factors contribute to the problem of maintaining directional control from touchdown to taxiing speed. One of these factors is the capability of aircraft pilot of anticipate perturbations and effects of crosswind. Because of the relatively high landing speed of most modern aircraft, the pilot has little time to implement a control action during the initial phase of landing rollout (primarily operation of the rudder) and little time is available to evaluate the results of an asserted control action and implement any required corrective measure. This implied several restrictions and limitations that have prevented modern high speed aircraft from attaining true all-weather operating capability [3]. Since the 50s, the landing was studied and a lot of patents were developed to allow landing in critical conditions. A. W. Blanchard [7] created a control over the aircraft ground roll trajectory. It was based on rudder control, when the aircraft had high speed, and on nose
wheel steering control, when the aircraft had low speed. Such idea demonstrated to be very effective in normal conditions (dry runway), but less suitable with critical weather conditions (wet runway or crosswind).
Another patent by Kilner J.R. and Warren S.M. [3] was based on a control system for maintaining an aircraft within the lateral boundaries of a runway during landing roll-out, under conditions of
Chapter 1 - Stability and control issues in aircraft braking dynamics
8
combined crosswind and low runway friction. In this patent, the aircraft position and velocity signals were generated by an inertial navigation system, and rudder, steering and braking systems assisted in maintaining the aircraft on the runway. The three command axes worked together with the following sequence:
• only rudder from touchdown speed to a threshold intermediate speed V1
• both rudder and nose wheel steering from V1 to taxing speed V with differential braking in addition at low speed, if rudder and nose wheel steering were not able to give the necessary corrective action.
The working scheme is depicted in Figure 1.6:
Figure 1.6 - Working scheme of Kilner and Warren’s directional control [3]
As the aircraft decelerates the aerodynamically generated lift forces rapidly decrease and the weight of the aircraft is transferred to the landing gear. As aircraft velocity decreases, the yawing moment and lateral force produced by a given amount of rudder deflection decreases thereby lessening
the directional control available through operation of the rudder until a speed is reached wherein the rudder is substantially ineffective in steering the aircraft along the runway. Since the transfer of weight to the aircraft landing gear that accompanies deceleration increases
the directional control capability of the aircraft nose wheel, directional stability and control can be maintained by simultaneously controlling the rudder and aircraft steering
This directional control system generates rudder, steering and braking signals that cooperatively interact so that: (a) the aircraft rudder is the primary lateral control mechanism at touchdown
and during the initial portion of the landing rollout sequence wherein the aircraft travels at a relatively high speed; (b) both rudder and steering control signals are generated during an intermediate speed portion of the landing rollout sequence wherein the rudder begins to lose effectiveness as a means of
Chapter 1 - Stability and control issues in aircraft braking dynamics
9
lateral control and the aircraft steering system becomes more effective as the weight of the aircraft is transferred to the aircraft landing gear; and (c) differential braking signals are generated to
supplement the rudder and steering control signals if the aircraft is travelling at a relatively low speed and the rudder and steering control signals are not sufficient to cause the necessary corrective
action[3].
The innovative idea, studied in this thesis work, to solve the problem of maintaining directional control from touchdown to taxiing speed is to show how it is possible to minimize the use of rudder for the control direction through the combined and continuous use of only nose wheel steering and differential brakes. This can be done thanks to the particular configuration of main landing gears studied in this work. The tripod-type configuration with variable wheel track should increase the directional control capabilities of differential brakes, allowing not to use the rudder.
Furthermore, thanks to the ground path control, the aircraft directional control can rely on the self-alignment within the lateral boundaries of a runway during landing roll-out, allowing the pilot control the landing with the simple use of a joystick. This can find easy application for drones.
1.3 Gear-walk phenomena
Landing gears are the connecting members between aircraft and the ground. They act as the load bearing members between the fuselage and the ground. The term landing gear indicates one of the main functions of the gear, namely the containment of the landing impact, but it fails to describe the other main functions, namely the provision of means for the aircraft to maneuver on the ground, taxi and take off. [9]
The aircraft ground dynamics and the related vibration issues are one of the key factors of a successful aircraft design. The modelling and the adequate control of landing gear dynamics, especially shimmy and gear walk vibrations, is a very important topic for the aircraft safety enhancement. Though this phenomena are not directly catastrophic, they can lead to fatal accidents due to excessive wear. Both instabilities are low frequency vibrations and can also shorten the gear life and cause discomfort to the pilot and passengers. The main reasons are slender fuselages that frequently arise from the stretching of existing aircraft, see [10], and the use of new, light-weight structures and materials that influence the vibrational properties of fuselage and wings. Not only unsuitable combination of structural stiffness, damping, and pneumatic tire characteristics but also an unlucky combination of brake system design with the tire physics can produce a serious vibration problem [11].
The aircraft landing gear is a complex multi-degree-of-freedom dynamic system, which may encounter vibration modes that are influenced by brake frictional characteristics and design features. The braking performance requirements include normal landing / rejected takeoff braking distances limits, thermal requirements on the landing gear components, durability of friction material and overall weight consideration. Due to superior performances of carbon, increasing numbers of airplanes are using carbon brakes [12]. Although carbon is lighter, has higher specific heat capacity, higher friction coefficient, better wear rate compared to steel, it is more sensitive to vibrations. Brake friction acts in the pitch-plane of the landing gear system, and so it affects the stability of the three pitch-plane modes of vibration as shown in Figure 1.7.
Chapter 1 - Stability and control issues in aircraft braking dynamics
10
Figure 1.7 - Vibration modes of a landing gear [13]
The gear walk, squeal and chatter are friction induced vibrations, which are caused by characteristics of friction between the rotating and non-rotating parts of the brake. Squeal refers to the high frequency rotational oscillation of the brake stator assembly whereas chatter and gear walk refer to the low frequency fore and aft motion of the gear (chatter is typically above 50 Hz and coupled with the squeal mode). [14]
From the point of view of frequency, mainly there are two types of instabilities. As shown in Figure 1.8, they are low frequency and high frequency vibrations. High frequency vibrations include squeal and chatter. Low frequency vibrations include gear walk and shimmy.
Figure 1.8 - Instabilities of landing gear
Shimmy and gear walk are low frequency vibration of the strut and they are schematically represented in Figure 1.9. Shimmy may be caused by a number of conditions such as low torsional stiffness, excessive free play in the gear, wheel imbalance or worn parts.
Gear walk is defined as the cyclic fore and aft motion of the landing gear strut assembly about a normally static vertical strut center line.
Landing gear vibrations
Strut vibrations (Low frequency vibrations) Gear Walk (10-50 Hz) Shimmy (10-50Hz) Brake vibrations (High frequency vibrations Squeal (100-1000Hz) Chatter (above 50 Hz)
Chapter 1 - Stability and control issues in aircraft braking dynamics
11
Figure 1.9 – Shimmy and Gear Walk [15]
In this thesis, a specific attention is dedicated to the “gear walk” mode, i.e. the fore-and-aft low frequency vibration of the landing gear primarily due to the coupling of the brake anti-skid dynamics with the leg structural deformation. Basically, as the brakes are applied and the horizontal ground load develops, the leg flexes rearwards; when an incipient skid is detected by the system, the brakes are released and the leg springs forwards. This accelerates the wheel rapidly, tricking the anti-skid system into applying the brake pressure again. If this working cycle repeats with increasing loads/ deformation amplitudes, an instability can occur.
The frequency range of the phenomenon depends on the type of landing gears: some authors place it in the 10–20 Hz range [16], some indicate ∼10 Hz [17], while others generally speak of a range between 10 and 50 Hz [18]. The effects of this instability can be felt throughout the aircraft structure as a low frequency shudder, for example 12 Hz on the MD80, that can reach very high amplitudes. This instability is known to be affected by factors such as tire inflation pressure and the presence of air in the brake hydraulic system [19]. It is potentially dangerous and can lead to complete landing gear structural failure [18,19].
While well-known instabilities, such as shimmy, have been extensively investigated through the years, the vibration phenomena such as gear walk, chatter and squeal, are only recently being addressed.
The case study presented in this thesis work regards an aircraft with a tripod-type main landing gear (MLG) which is known to suffer from gear walk in normal braking conditions. A tripod landing gear is peculiar from a kinematic and dynamic standpoint, as it increases the gear track during compression (Figure 1.10).
A symmetrical approach is adopted under the assumption that the time scale of the aircraft yaw dynamics strongly differs from that of the deformable landing gear longitudinal dynamics [18].
Chapter 1 - Stability and control issues in aircraft braking dynamics
12
Figure 1.10 - Example of tripod-type main landing gear model
Both the nose landing gear (NLG) and the MLG are modeled with oleo-pneumatic shock absorbers and direction-dependent fixed orifice areas. The nonlinear oleo-pneumatic shock absorber model implemented in Simulink uses the following constitutive law:
• the nonlinear elastic forces are modelled using the classical polytropic compression • viscous forces are generated by velocity-squared damping equations.
Interesting to be said is that:
-
the deceleration applied during braking induces a pitch in the aircraft attitude, causing a vertical load transfer between the MLG and the NLG. Although occurring at relatively low frequencies, the vertical load variation can influence the behavior of the anti-skid control system.-
The tripod-type configuration with variable wheel track increase the effectiveness of path control for differential brakes, and this, combined with an increase of vertical load on the NLG, may involve the possibility to realize the aircraft directional control on ground only through the combined and continuous use of nose wheel steering and differential brakes.1.4 Ground dynamics simulator: previous version
1.4.1 Aircraft model features
The starting point of this thesis is related to the model developed in [21]. In the first part, a 6 DOFs model was developed according to the following hypothesis:
• rigid body; • constant mass;
Chapter 1 - Stability and control issues in aircraft braking dynamics
13 • negligible effects of rotating mass;
• symmetric thrust;
• negligible structural deformations.
The model allowed the study of aircraft stability and it had 6 DOFs:
• aircraft translation along the longitudinal axis; • aircraft translation along the vertical axis; • aircraft translation along the lateral axis; • aircraft rotation along the pitch axis; • aircraft rotation along the yaw axis; • aircraft rotation along the roll axis.
Once defined the model, a shock absorber model was developed with a heading control and, in the last part, a nonlinear simulation was realized. A new transfer function of the nose wheel steering actuator has been received from a parallel activity about the control of an electro-mechanic actuator. A new model was developed in [22], with a more detailed models of hydraulic brakes, shock-absorbers tyre loads (via Dugoff formulae), and digital signal processing controllers. The work was focused on the design and verification of the aircraft simulator control laws, with particular reference to pressure control, heading control and ground path control.
1.4.2 The reference aircraft
The aircraft used as reference for the simulator development is light business jet aircraft with a T-tail,
similar to the Cessna 510 Citation Mustang. The reference article about the aircraft is in [23], the geometric features are depicted in Table 1.2 and in Figure 1.11
Chapter 1 - Stability and control issues in aircraft braking dynamics
14
Table 1.2 - Aircraft Aeromechanical characteristics
In the present work it will be used the same reference aircraft, but with a tripod type main landing gear configuration. An example is given by Eclipse 500, shown in Figure 1.12.
Chapter 1 - Stability and control issues in aircraft braking dynamics
15
1.4.3 Basic architecture of the directional control laws
The basic concept for the directional control laws is that rudder, nose wheel steering and differential brakes simultaneously work to align the aircraft to the demanded runway path, while the vehicle is decelerating. A scheme of the nested loops is depicted in Figure 1.13. The model has many closed loop, listed from the innermost one to the most external one:
• brakes pressure control; • anti-skid control
• heading control; • ground path control
The pressure loop has to follow the requested pressure coming from the block Antiskid and Heading regulators. The anti-skid loop has to permit a good rolling of the wheel without any blocking of the tyre; the heading loop, through the three command axes, has the task to realign the aircraft. The ground path loop gives the right signal to the heading loop to reduce the distance to the center line.
Moving the attention to the nonlinear model implemented on Simulink, a list of the hypothesis on which previous model and the present one are based is presented as follows
• Constant mass
• Rigid body with six degrees of freedom
• Negligible structural deformations, except for longitudinal vibration of main landing gear. • Negligible gyroscopic effects of rotating mass(engine)
• Symmetric thrust
The model has four inputs (thrust, lateral gust, the symmetrical desired slip ratio, and the signal 𝑦𝐴𝐶𝑖
setted to zero). Moreover, with some parameters, it is possible to change the runway conditions, in particular it is possible to choose a wet runway or a small area with a local low friction coefficient. In the section “aircraft and tires” is inserted a more realistic dynamic of main landing gears and longitudinal vibrations of main landing gear.
The goals of this model are:
• to evaluate the effects of tripod-type configuration and the variable track of the wheel on the directional control of the aircraft
• To evaluate the effectiveness of directional control laws using differential brakes and nose-wheel steering only
• To realize a Slip Ratio Control by taking into account the gear walk dynamics; • to prevent instability of main landing gear vibration during the braking phase;
At the top level of the nonlinear model, depicted in Figure 1.14, there are four principal blocks:
• Brakes and Pressure regulator • Antiskid and Heading regulator • Ground path controller
Chapter 1 - Stability and control issues in aircraft braking dynamics
16
• Aircraft and Tires
Chapter 1 - Stability and control issues in aircraft braking dynamics
17
Chapter 1 - Stability and control issues in aircraft braking dynamics
18
Most of the stopping power for transport aircraft during landings and rejected takeoffs is provided by the wheel brakes. The brake pressure (which essentially determines brake torque) is controlled by an electrohydraulic pressure valve. Hydraulic power is almost exclusively preferred for actuators in aircraft because it offers high power to weight ratio and reliable, self-lubricating operation. On the other hand, dynamics of hydraulic actuators are predominantly nonlinear and depend on factors which are difficult to measure or estimate on-line, such as oil bulk modulus, viscosity, and temperature. While the actuator itself may offer fast response because of its low inertia, delays caused by the
connecting lines may be a limiting factor on the available control bandwidth. Antiskid brake control can be improved by using an inner control loop around the valve to achieve faster and linearized behavior [5].
Figure 1.15 - Simplified brake system [6]
The simulated airplane has a hydraulic brake system composed by four small pistons pulling on the pressure plate, which acting on rotor and stator disks, gives a deceleration to the wheel. Four springs, one for each small piston, bring again the small pistons to the rest position. When the brake is
requested the pressure in the chamber pressure increases and rotors scrape on stators. The generated torque is opposite to the angular velocity of the wheel, so the wheel decelerates. The hydraulic brake system with a single small piston is schematically depicted in Figure 1.15.
The value of the pressure deriving by the block "Antiskid and Heading regulators" is a reference value that must be chased by the brake system, so a closed loop system is necessary. A detail of the Simulink scheme of this loop is depicted in Figure 1.16.
Chapter 1 - Stability and control issues in aircraft braking dynamics
19
Figure 1.16 - Detail of scheme in Simulink
The error signal of the pressure multiplied by the transfer function of the pressure controller gives the reference signal of the spool displacement of the proportional servo valve (the real displacement of the spool depends on the dynamic of the servo valve itself). The movement of the spool determines the value of the area of the orifices then the values of pressure.
20
Chapter 2 - Landing Gear Modeling
On the following chapter, main landing gear dynamic is simulated considering real geometry and kinematics of a tripod type configuration involving variable wheel track and gear walk mode vibration. In Chapter 4, the model is developed and integrated in the aircraft simulator of Simulink and it will be shown the effects on directional control of the aircraft during the ground portion of the landing procedure.
2.1 Aircraft landing gear configuration
Aircraft landing gear supports the entire weight of an aircraft during landing and ground operations. They are attached to primary structural members of the aircraft. The type of gear depends on the aircraft design and its intended use.
Three basic arrangements of landing gear are used: tail wheel type landing gear (also known as conventional gear), tandem landing gear, and tricycle-type landing gear. The most commonly used landing gear arrangement is the tricycle-type landing gear. It is comprised of main gear and nose gear. [Figure 2.1]
Figure 2.1 – Tricycle type landing gear on a Learjet(left) and a Cessna 172(right) [24]
Tricycle-type landing gear is used on large and small aircraft with the following benefits:
1. Allows more forceful application of the brakes without nosing over when braking, which enables higher landing speeds
2. Provides better visibility from the flight deck, especially during landing and ground maneuvering.
3. Prevents ground-looping of the aircraft. Since the aircraft center of gravity is forward of the main gear, forces acting on the center of gravity tend to keep the aircraft moving forward rather than looping, such as with a tail wheel-type landing gear.
In addition to supporting the aircraft for taxi, the forces of impact on an aircraft during landing must be controlled by the landing gear. This is done in two ways: 1) the shock energy is altered and transferred throughout the airframe at a different rate and time than the single strong pulse of impact, and 2) the shock is absorbed by converting the energy into heat energy.
Many aircraft utilize flexible spring steel, aluminum, or composite struts that receive the impact of landing and return it to the airframe to dissipate at a rate that is not harmful. The gear flexes initially and
Chapter 2 - Landing Gear Modeling
21
forces are transferred as it returns to its original position. [Figure 2.2] The most common example of this type of non-shock absorbing landing gear are the thousands of single-engine Cessna aircraft that use it.
Figure 2.2 – Tripod type landing gear with non-shock absorbing struts [24]
True shock absorption occurs when the shock energy of landing impact is converted into heat energy, as in a shock strut landing gear. This is the most common method of landing shock dissipation in aviation. It is used on aircraft of all sizes and it will be used also for the main landing gear configuration studied in this chapter. An example of the tripod type MLG configuration, with shock absorption, has already been given in Figure 1.10. A more detailed design of the tripod MLG configuration studied in this thesis work cannot be showed in this work for reasons of secrecy. However, in Figure 2.3, an example of light jet that uses this type of MLG configuration is shown.
Figure 2.3 - Example of the tripod type MLG configuration studied in the present thesis work [26]
One of the purpose of the landing gear in an aircraft is to provide a suspension system during taxi, take-off and landing. It is designed to absorb and dissipate the kinetic energy of landing impact, thereby reducing the impact loads transmitted to the airframe. The landing gear also facilitates braking of the aircraft using a wheel braking system and provides directional control of the aircraft on the ground. It is often retractable to minimize the aerodynamic drag on the aircraft while flying.
Chapter 2 – Landing gear modelling
22
The landing gear design takes into account various requirements of strength stability, stiffness, ground clearance, control and damping under all possible ground attitudes of the aircraft. Landing gear is a heavily loaded structure. Its weight varies from 3% of aircraft all-upweight for a fixed type to about 6% for a retractable type landing gear. The challenge is to reduce the weight of the landing gear without compromising on its functional, operational, performance , safety and maintenance requirements [25]. As said in chapter 1, the use of new, light-weight structures and materials that influence the vibrational properties of landing gear components, along with a combination of structural stiffness, damping, and pneumatic tire characteristics and an unlucky combination of brake system design with the tire physics can produce a serious vibration problem [11] (see paragraph 1.3). Further, landing gear should occupy minimum volume in order to reduce the stowage space requirements in the aircraft and the service life of the landing gears should be same as that of the aircraft.
To sum up, there are many challenges to face during landing gear design and development. How technology help meets these challenges? Figure 2.4 shows how it is possible.
Figure 2.4 - How technologies help meet challenges [25]
It is evidenced how electronically controlled antiskid brake management system and computer simulations helps improving performances, life and cost of the product, and how computer simulations give positive contribute to meet all challenges in question.
Dynamic simulation helps to predict the performance of a component or assembly. The results of this simulations will be more accurate if compared to hand calculations. These simulations help in
Chapter 2 – Landing gear modelling
23
handling large number of studies in short time. For example, the landing gear shock absorber performance is evaluated by a dynamic simulation of the landing. This takes into account the hydraulic damping, air spring characteristics. Using the computer models developed for this purpose the shock absorber parameter are enhanced to maximize its efficiency [25]. This helps in preliminary estimation of impact and ground loads, which influence landing roll out dynamic.
All the forces involved in ground dynamic works together inside a very complex nonlinear model which is implemented in Simulink, letting understand how each parameter can influence the ground dynamic and loads that must be controlled. In particular, the attention of the work is focused on the positive influence of the reference MLG configuration on the directional dynamic control.
• Previous model
In the previous model it is used a vertical shock strut with an oleo-pneumatic shock absorber.
Figure 2.5 – Simplified model of the vertical shock strut used in previous works
A typical design of an oleo-pneumatic shock absorber is shown in figure 2.6.
Figure 2.6 - Oleo-pneumatic shock absorber
The space above the oil is pressurized with dry air or nitrogen (an inert gas). When the aircraft lands, oil is forced from the lower chamber to the upper chamber through the orifice. Although this need only be a hole in the orifice plate, the hole area is often controlled by a varying-diameter metering pin to
Chapter 2 – Landing gear modelling
24
maximize efficiency by obtaining a fairly constant strut load during dynamic loading similar to that shown in the drop test curve of Figure. 2.7. A 100% efficient strut would have a rectangular-shaped drop test curve, but in practice the obtained efficiency is usually between 80 and 90%.
Figure 2.7 – Shock absorber Load deflection curve
The first thing to be done is to choose the simulation reference sink speed . In the present work has been considered the same sink speed of the previous work simulations ( 𝑣𝑠=3ft/s).
The second thing to be done is to define the weight percentage load over the shock absorber. After touchdown, it is the load that weigh on the MLG, reduced by the component of aerodynamic forces, which are function of the dynamic pressure.
In Figure 2.8 is shown the scheme for the calculus of landing gears reaction in function of their distance from the center of gravity, where it is applied the total weight of aircraft
𝑊 = 𝑀
𝑎𝑐𝑔
.Figure 2.8 – Scheme for the calculus of landing gears reaction in function of their distance from CG.
For the translational equilibrium along vertical axis and rotational equilibrium about the center of gravity CG, it can be written the following equation system:
𝑭
𝒗𝑳≡ 𝑭
𝒗𝑹𝐅
𝐧Chapter 2 – Landing gear modelling
25{
𝐹
𝑛+ 𝐹
𝑣𝐿+ 𝐹
𝑣𝑅= 𝑊 = 𝑀
𝑎𝑐𝑔
− 𝐹
𝑛𝑏
𝑛+ 𝐹
𝑣𝑅𝑏
𝑚+ 𝐹
𝑣𝐿𝑏
𝑚= 0
𝐹
𝑣𝐿= 𝐹
𝑣𝑅{
𝐹
𝑛+ 2 𝐹
𝑣𝑅= 𝑀
𝑎𝑐𝑔
𝐹
𝑛= 2
𝑏 𝑏𝑚 𝑛𝐹
𝑣𝑅By substituting the second equation in the first one, it can be derived
2 (
𝑏𝑚𝑏𝑛
+ 1) 𝐹
𝑣𝑅= 𝑀
𝑎𝑐𝑔
, from which it follows the expression of landing gear vertical reactions.{
𝐹
𝑣𝑅=
12(
𝑏𝑛 𝑏𝑚+ 𝑏𝑛) 𝑀
𝑎𝑐𝑔 = 𝑀
𝑚 ∗𝑔
𝐹
𝑛= (
𝑏𝑚 𝑏𝑛+ 𝑏𝑚) 𝑀
𝑎𝑐𝑔 = 𝑀
𝑛 ∗𝑔
with{
𝑀
𝑚∗=
1 2(
𝑏𝑛 𝑏𝑚+ 𝑏𝑛) 𝑀
𝑎𝑐𝑀
𝑛∗= (
𝑏𝑚 𝑏𝑛+ 𝑏𝑚) 𝑀
𝑎𝑐In other words, the percentage of weight over landing gears is function of 𝑏𝑚 and 𝑏𝑛, that are the longitudinal distances of nose and main landing gears from the center of gravity CG. If
𝑙
𝑔𝑒𝑎𝑟is
defined and the dimensionless parameter ɛ
𝑙𝑔(𝑤𝑖𝑡ℎ ɛ
𝑙𝑔∈ [0,1] ) is introduced as follow
{
𝑙
𝑔𝑒𝑎𝑟= 𝑏
𝑛+ 𝑏
𝑚𝑏
𝑚= ɛ
𝑙𝑔𝑙
𝑔𝑒𝑎𝑟𝑏
𝑛= (1 − ɛ
𝑙𝑔) 𝑙
𝑔𝑒𝑎𝑟{
𝑀
𝑚 ∗= (1 − ɛ
𝑙𝑔)
𝑀2𝑎𝑐𝑀
𝑛∗= ɛ
𝑙𝑔𝑀
𝑎𝑐The last equation of the system explicitly expresses the mathematical meaning of
ɛ
𝑙𝑔 , as the weight dimensionless percentage over the nose landing gear. By taking into account that:
{
𝑀
𝑎𝑐= 4536 Kg
𝑙
𝑔𝑒𝑎𝑟= 4.93 𝑚
Table 2.1 shows values of 𝑀
𝑚∗, 𝑀
𝑛∗
, 𝑏
𝑚, 𝑏
𝑛in function of the chosen ɛ
𝑙𝑔. The reference
value, used in the previous and the present thesis work, is ɛ
𝑙𝑔= 0.15
𝒃𝒎 [𝒎] 𝑴𝒎∗ [𝑲𝒈] 𝒃
𝒏[𝒎] 𝑴𝒏∗ [𝑲𝒈]
ɛ𝒍𝒈= 𝟎. 𝟏𝟓 0.74 1927,8 4.19 680,4
Table 2.1 - 𝑀𝑚∗ , 𝑀𝑛∗, 𝑏𝑚, 𝑏𝑛 in function of the chosen ɛ𝑙𝑔
Once defined 𝑀𝑚∗ , a simplified model of oleo-pneumatic shock absorber is done, with a fixed area of
the orifice and single floating piston. that is generally referred to as oleo-pneumatic. The
The compression stroke of the shock strut begins as the aircraft wheels touch the ground. As the center of mass of the aircraft moves downward, the strut compresses, and the lower cylinder or piston is forced upward into the upper cylinder. At the end of the downward stroke, the compressed air in the lower cylinder is further compressed, therefore the compression stroke of the strut is absorbed with minimal impact. Energy stored in the compressed air in the lower cylinder causes the aircraft to start moving upward as the strut tries to rebound to its normal position. The snubbing of fluid flow across the orifice area, during the compression and the subsequent extension, dampens the strut rebound and reduces oscillation caused by the spring action of the compressed air.
In an equivalent form
⇔
it follows⇒
Chapter 2 – Landing gear modelling
26
2.2 Enhancement of the kinematic model
The starting point of the landing gear modelling is the kinematics of MLG as the aircraft touch the ground. The aim of the following analysis is to replace the geometrical layout of the real model with a simplified one, which allows the kinematic analysis of the system.
The first attempt to modelize the geometrical layout is shown in the model of figure 2.9, which rapresents the initial configuration at the time of the impact on the ground.
Figure 2.9 – Initial configuration of reference MLG
The scale factor between lengths of the real and the simplified model has been calculated, so that effective lengths of segments AB and AC has been obtained, as well as the horizontal and vertical distance (“a” and “b”) between the hinges A and D. Numerical values of these lengths are summered in Table 2.2. The initial value of the angle 𝛽𝑤 between AC and the horizontal direction, has been
Chapter 2 – Landing gear modelling
27
Table 2.2 – Parameters numeriacal values of MLG initial configuration
From values of table 2.2, through trigonometric consideration, it can be obtained 𝑐0, the initial length of the DC, as expressed in Eq 2.1
𝑐0 = 𝑙𝑎 sin 𝛽𝑤0− b
sin 𝛽𝑎0 = 0.57 𝑚 (2.1)
The segment DC is the variable length line where the shock absorber is collocated (trough the link to hinges C and D). In Eq. 2.2 it has been derived the value of 𝛽𝑎0 ,which is the initial value of the angle between DC and the horizontal direction.
𝛽𝑎0 = arctg (
𝑙𝑎 sin 𝛽𝑤0− b
𝑙𝑎 cos 𝛽𝑤0− a ) = 78,88° (2.2) Because of the variable length of the segment DC, it is defined 𝛥𝑎 as
𝛥
𝑎= 𝑐
0− 𝑐
(2.3)where 𝑐 = 𝐷𝐶̅̅̅̅ in a generic instant 𝑡 = 𝑡∗.
• Kinematic analysis
As shown in the Figure 2.9, the reference system is the classical reference system: body- axes. It is integral with the aircraft and rotated by three angles, the Euler angles. The hypotesis of the model are the following:
- The system has 2 DOF (degrees of freedom) in the z-direction(translational DOF in vertical direction of the point A ,indicated with “𝑧𝐴”
and compressionof wheel tyre, indicated with “
𝑠
𝑤” ) Points A and D are hinges that move together in vertical direction as a unique block(
𝑧
𝐴= 𝑧
𝐷).- The rigid beam AB moves with 2 DOF (vertical translation 𝑧𝐴 and rotation of angle 𝛽𝑤 about
the hinge A)
- The point B is an hinge vincolated to move only in y- direction(lateral displacement of the tyre)
PARAMETER VALUE
a (horizontal distance of A and D) 0.2518 m
b (vertical distance of A and D) 0.0488 m
𝒍𝒂 (length of segment AC) 0.7107 m
𝒍𝒘(length of the segment AB) 1.09 m 𝜷𝒘𝟎 ( angle between AC and the horizontal
Chapter 2 – Landing gear modelling
28
From a kinematic point of view, vertical translation of point A implies:
-
compression(or extension) of c- translation in y-direction of the point B (𝑦𝐵)
The point B can be modelized as an horizontal trail , which allows only lateral displacement 𝑦𝐵 of the wheel. Tyre compression of the wheel 𝑠𝑤 is modelized as a vertical deformation of the horizontal trail. The kinematic model is better explained by Figure 2.10 ,
Figure 2.10 – Kinematic model of MLG in the y-z plane
To sum all, there are 5 kinematic variables: 𝑧𝐴− 𝑠𝑤 is the independent variable,
𝛽
𝑤, 𝛽
𝑎, ∆𝒂 and 𝑦𝐵are the dependent variables, functions of 𝑧𝐴− 𝑠𝑤 . In table 2.3, their initial values are summered:
At the generic instant 𝑡 = 𝑡∗. Eq. 2.4 shows trigonometric relation between 𝑧
𝐴− 𝑠𝑤 and the resulting
variation of the angle 𝛽𝑤
𝑧
𝐴− 𝑠
𝑤= 𝑙
𝑤(𝑠𝑖𝑛 𝛽
𝑤0− 𝑠𝑖𝑛 𝛽
𝑤)
(2.4)So, 𝛽𝑤 is expressed as a function of the independent variable 𝑧𝐴− 𝑠𝑤, as follows:
𝒛𝑨− 𝒔𝒘 [𝒎] 𝜷𝒘 [𝒅𝒆𝒈] 𝜷𝒂 [𝒅𝒆𝒈] 𝜟𝒂 [𝒎] 𝒚𝑩 [𝒎]
𝒕 = 𝟎 0 59,35° 78,88° 0 0
Table 2.3 - Initial values of kinematic variables
