Dynamics and Energy Conversion of Aircraft Landing Gears at Touchdown

Dynamics of aircraft landing gears with and without pre-rotation has been analysed. The ground friction work is been formulated which raises the temperature of touch point to the rubber smoke point. The sliding time, distance, power and energy released by the friction at touch point have been presented. Simulation of target speed has been shown for different levels of pre-rotation torque.


Introduction
Hundred thousands of aircraft take off and make their landing everyday worldwide.Up to date, during the touchdown, all aircraft generate smoke, blacken the runway and cause significant tyre wear.No doubt, this behaviour of landing applies damage to environment and increases costs on aircraft operation.Since 1940s, some inventors have made attempt to develop pre-rotation devices and a number of them have been patented [1][2].However, so far after 80 years past, surprisingly, none of such devices has been put into practical use.Some of the landing gear manufacturers concern about the complicity and additional weight by adding such a device.The landing touchdown smoke is still a standard scene to which most people have got accustomed.
With increasing pressure on environment in this 21st century, action apparently is needed.Aircraft operators provided experience and data for investigators [3].Rubber tires for vehicles and their friction behaviours are still attractive topics [4][5][6][7][8][9].Some have intensively investigated rubber combustion [10].However, fundamental understanding to the touchdown problem and the measure to reduce the landing smoke are seriously lacking.In recent years, the authors have proposed some fundamental approach to analyse and simulate the dynamic behaviours [11][12] for aircraft touchdown processes.This paper is reflecting the novel but approach to the problem by the author, aiming for eventually getting rid of the landing smoke, increasing the tyre life cycle and protecting the environment.

Landing gear sliding with prerotation
Instantly after the landing gear touches the runway, it is for the ground friction to provide a torque to accelerate the static wheel to match the landing speed within a fraction or a couple of seconds.where µ is the friction coefficient, depending on type of tyre, contact area and runway condition; W is the vertical force on the wheel shaft due to weight of aircraft as shown in Fig. 1.Normally there are a number, from several to dozens, of wheels for an aircraft.After touchdown, using Newton's second law of motion for rotational circumstances, with a resistance torque T r which includes those in shaft, air viscosity and ground rolling resistances, where I is the moment of inertia of the wheel, r is the radius of wheel.This radius can be the effective radius, taking into account of tyre deflection.Letting H be the It is the friction between the ground and tyre that accelerates the wheel to a landing rotational speed within t s by limited torque r F .The wheel needs a period of time to accelerate to 0 Z as the ground can only provide a limited, not unlimited, friction F. Therefore, a sliding is inevitable.The sliding time t s = 0 only if the ground can provide infinitive friction to generate an infinitive acceleration from static to 0 Z , which is impossible.
During this sliding period the wheel is assumed being accelerated uniformly.side wind turbine is suggested as in the patents to fit on the wheel, on which an air dynamic torque is generated to accelerate the wheel to achieve a tangential speed before the touchdown.After the touchdown, no energy or just smaller energy is required to accelerate the wheel to the matching speed 1) and ( 2) Therefore, the wheel sliding time is Since the wheel is in the state of sliding, the tangential motion of wheel at touch point pushed by ground friction r s is less than s s .
The tangential displacement caused by the forces acted on the touch point during the whole sliding period ( The rate of work done by ground friction, or the friction power, is 2 It is the power density during the sliding generated by the ground friction on a single tyre.In the case of no pre-  5) and ( 6), the ground friction work at the touch point is According to the first law of thermodynamics, the energy is conserved.This work will be totally converted to heat energy during the sliding.Using the public domain data from Boeing 747-400 aircraft without prerotation, the moment of inertia of real wheel I=46.19 kgm 2 , the tire radius r=0.622 m, the landing velocity v 0 =80.8 m/s, g E ' will be 433 kJ within the sliding time t s =0.74 seconds.Most of it will raise temperature of the rubber near local contact area to its smoke point, generating a large quantity of smoke.The remaining part will be dissipated against the friction T r , which is a relatively small number.A tyre is made up of carbon polymer.Primarily, the sliding between tyre and runway surface causes rubber loses.In addition, since the conduction of rubber is poor, this energy will be converted to heat to rise local area temperature at the tyre-ground interface significantly.As a result, the high temperature is to burn the tyre circumference layer to cause the tyre rapid wear and landing smoke.The rubber heat capacity is in average 0.48KJ/KgK.This energy can cause a high temperature up to 500 C o on the rubber beyond its ignition point.As a polymer, the flammability depends on oxygen supply and ignition source.Although self-ignition usually does not occur due to strong air flow, it has already entered a temperature range high enough to start chemical oxidization reaction on rubber, causing a large quantity of oxidization smoke.On the other hand, the wheel has gained energy through the sliding from part of the friction work.The kinetic energy gained by the whole wheel is In the case of zero pre-rotation, . This energy will be eventually dissipated in the subsequent

H
, the total energy will be In the case that T r is small and can be ignored, . i.e. in this case, the work done by the ground friction is the twice of the kinetic energy gained by the wheel during the sliding.

Pre-rotating torque before touchdown
After the landing gear is dropped and locked into place, with a pre-rotation device fitted, it is designed to produce a constant torque T, when the aircraft is still in air, Newton's law of motion, again, for the wheel is applicable: where Z a T is the resistance torque from the shaft bearing due to wheel's aerodynamic drag and weight, and the viscosity from surrounding air.The task of prerotation torque T is to accelerate the wheel from 0 to  12) The above can be deformed to a different appearance as This non-linear pre-rotation system in Eq.( 14) can be solved by SIMULINK modelling in MATLAB as shown in Fig. 2. Using the data from Boeing 747-400 aircraft, the moment of inertia of real wheel I=46.19 kg-m 2 .The tire radius r=0.622 m, the landing velocity is v0=80.8m/s.Pre-rotating torques T=200 Nm and 500 Nm, respectively, are applied.In Fig. 3, for a relative low pre-rotating torque, the wheel velocity has not reached the matching speed when the resistance has reached the maximum.In the case of T=200 Nm, the final velocity of wheel can reach to 63% of the landing speed.i.e. the pre-rotation ratio When applying a higher torque T=500 Nm, the wheel will reach the matching speed 100% within approximately 35 seconds as shown in Fig. 4. Further increasing the torque will be over pre-rotated and is unnecessary.with the increase of H .In addition, the friction work at the touch point, wheel kinetic energy increment and total friction work against pre-rotation ratio H are shown in Fig. 6.It can be seen, if the pre-rotation is applied by even only 50%, the temperature raising energy caused by the friction work will be reduced by 75%.

Conclusion
The dynamic model for simulating landing sliding process is established to analyse the phenomenon of aircraft landing smoke and successive tyre wear.100% pre-rotation ratio will cancel totally the landing smoke.A partial pre-rotation will have an enlarged effect on the friction heat reduction.The interpretation of the results show that the findings are reasonable and meaningful for understanding the dynamics and energy conversion during the landing, and also for helping the design of the pre-rotation devices.Up to date, the work represents a fundamental approach for understanding the landing smoke problem and an effective solution to it, which is novel and worth further investigation.

Figure 1 .
Figure 1.Landing gear wheel at touchdown Let v be the aircraft travel speed and Z angular speed of the landing gear wheel.For a landing speed 0 v v , by a 100% matching, the tangential speed of wheel at touch point is also 0 0 Z r v .Assuming the friction provided by the runway ground e. no sliding exists nor smoke will be generated.The sliding distance, or aircraft travelled distance during sliding is 4 N and v 0 =80.8 m/s, E ' =585 kW.This is very powerful figure.In the case of full pre-rotation, e. the ground friction does not need to do any work on tyres.Using the displacement Eqs. (

Figure 2 .
Figure 2. Simulink model for the motion during approaching with non-linear resistance torque

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Wheel's tangential velocity has matched the landing speed as resistance torque reaches the maximum (η=0.03,T=500 Nm) The sliding time, distance and friction work under different levels of pre-rotation are presented.Numerical examples with different pre-rotation torques are given.Ideally, a