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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Review Article Volume 2 Issue 4

Undercarriage heat power in aircraft landing 

Weiji Wang

University of Sussex, England

Correspondence: Weiji Wang, University of Sussex, Brighton BN1 9QT England

Received: June 06, 2018 | Published: July 17, 2018

Citation: Wang W. Undercarriage heat power in aircraft landing. Aeron Aero Open Access J. 2018;2(4):215-221. DOI: 10.15406/aaoaj.2018.02.00053

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Abstract

This paper analyse the energy flow which generates heat and causes wear during the acceleration period of undercarriage wheels after aircraft touchdown on the runway. In this acceleration period, a sliding between the wheel tyre and runway surface takes place. The sliding friction causes high temperature which emits pollution smoke and produces excessive tyre wear. A model based on mechanical dynamics is established for analysing the high temperature, heat transfer rate to tyre and concrete runway. The effect of adopting a pre-rotation device is also analysed in order to lower the temperature and reduce the tyre wear. The width of tyre is shown also having significant effect on the raised temperature. A blackened runway is believed to have changed the heat conduction therefore affecting the landing quality.

Introduction

During aircraft landing, significant high tyre temperature rises when the undercarriage is just touching down the runway. The sudden acceleration of wheels is accompanied by smoke and excessive wear, not just emitting pollution particles to surrounding air but depositing carbonized rubber on the runway. The high temperature weakens the tyre strength, even melts the rubber and brings about more rubber losses. With the number of landing increases, the touchdown area on the runway is usually severely blackened with deteriorated surface friction and heat dissipation properties. Normal practice is to use torch to burn off the rubber in scheduled intervals, which incurs extra airport operating cost and reduces airport operating efficiency. On the other hand, the tyre life is reduced due to such intensive wear, bringing other additional costs from high frequency of changing tyres. A number of invented designs for pre-rotating aircraft landing wheels have been proposed since the 40s of 20th century, which promisingly solve such problem effectively.1‒15 The power source for spinning the wheel has been proposed from side wind turbines, electric motors, etc. However, the real application is generally lacking and has to be looked forward into the future. Some attention to the problem still exits mainly in aviation engineering research and environment quality areas.16‒18 Recently, the science and technology behind the pre-rotation have been systematically investigated and some solutions have been developed.19‒24

With current static wheel in landing, the ground friction has to spin the wheel to match the landing speed with the same tangential speed. The limited friction is always below the demand of instant acceleration which in theory needs an infinitely strong force, therefore a sliding inevitably takes place, which grinds rubber away from tyre surface softened and burnt by high temperature at the touch area. The reported research in the references19‒24 has covered finding the effectiveness of the technique of pre-spinning the wheel to reduce the tyre tread heat and wear, and choosing the initial wheel rotation speed that prevent the tread rubber from melting temperature. For achieving this, a coupled structural–thermal transient analysis in well-known software package ANSYS has been used to model a single wheel main landing gear as a mass-spring system. The model has been chosen to analyse the wheel’s dynamic behaviour and tyre tread temperature and wear during the short period from static to a matching free-rolling velocity in which the wheel is forced to accelerate by the friction between the tyre and ground. The tyre contact surface temperature and wear have been calculated for both the initially static and pre-spun wheels in order to compare the temperature and wear levels for different initial rotation speeds.

The required torque to spin the aircraft wheel to the required angular speed at approaching speed has been calculated using ANSYS CFX, which is used to determine the wheel aerodynamic forces developed by simulation of fluid flows in a virtual environment. Several types of side wind turbines have been simulated using ANSYS CFX in order to optimize with regard to the geometry, target rotation speed, and required acceleration. In one type of the wind turbines, air flow is being guided positively to increase the torque. Simulation and application of Savonius turbines25 have been taken example. This paper is addressing the effect of sliding displacement and mechanical work done by the ground friction. The heat due to wheel sliding is transferred to both tyre and concrete ground. Relatively higher heat conductivity and mass capacity of runway can actually ease the high temperature by taking a portion of the sliding energy. Both the wheel tyre and runway play the role in conducting and absorbing the heat during the acceleration period of undercarriage wheels. However, the energy intensity of the temperature rise which causes smoke and wear to the tyre is the main focus.

Total wheel tangential displacement and actual ground displacement during acceleration

For modern aircraft, 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 between a fraction of a second and a couple of seconds. The acceleration time is depending on the wheel inertia and geometry, ground friction and other resistance such as from its shaft, surrounded air viscosity and ground rolling resistance. The runway is normally a concrete rough surface. No matter how great the friction can be provided, the torque by the ground friction is always limited. It needs some time to accelerate the wheel from static to a tangential speed to match the required aircraft landing speed. During the acceleration, the tyre has to slide before reaching the landing speed.

Let v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ be the aircraft landing speed and ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHjp WDaaa@3852@ the angula0072 speed of the landing gear wheel. When acceleration is completed after touchdown, the tangential speed of wheel at touch point relative to shaft becomes also v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ . Assuming the friction provided by the runway ground F= μ W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb Gaeyypa0JaeqiVd0wcfa4aaSbaaSqaaaqabaqcLbsacaWGxbaaaa@3C31@ , where µ is the sliding friction coefficien t, depending on type of tyre, contact area and runway condition, and assumed to be constant for simplicity; W is the vertical force share on the wheel shaft due to weight of aircraft, which normally has multiple wheels, as shown in Figure 1. For a usual soft landing, W is much lighter during the wheel acceleration period than the full weight share of a wheel.

After the touch instant, using the classical Newton’s second law of motion for rotational circumstances, with a resistance torque Tr which includes those friction in shaft, air viscosity and ground tyre rolling resistances, the equation can be described as

Fr T r =I dω dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaeyyXICTaamOCaiabgkHiTiaadsfajuaGdaWgaaWcbaqcLbmacaWG YbaaleqaaKqzGeGaeyypa0JaamysaKqbaoaalaaakeaajugibiaads gacqaHjpWDaOqaaKqzGeGaamizaiaadshaaaaaaa@480C@     (1)

where I is the moment of inertia of the whole wheel, r is the radius. This radius is regarded as the effective radius, having taken into account of tyre deflection. It is the friction between the concrete runway and rubber tyre that accelerates the wheel to a matching landing rotational speed within ts by limited torque Fr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaeyyXICTaamOCaaaa@3A91@ . During this acceleration time, sliding exists as at the beginning the wheel is accelerated from static. In the late stage of acceleration, sliding approaches 0 as the wheel’s tangential speed is matching the aircraft speed, which is also the speed of wheel shaft. The wheel needs a period of time to accelerate to v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ as the ground can only provide a limited, not unlimited, friction F. Although the ground friction is changing according to the sliding speed, usually the faster the smaller, for simplicity, a flat average constant can be assumed. The sliding time ts = 0 only if the ground could provide infinitive friction to generate an infinitive acceleration from static to v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ , which is impossible. Therefore, a sliding is inevitable. During this acceleration period with part wheel sliding can be assumed to be accelerated uniformly for simplicity. The resistance torque Tr also is regarded as being averaged to a constant. The wheel angular acceleration becomes:

α= Fr T r I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGgbGaeyyXICTaamOCaiab gkHiTiaadsfajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaaGcbaqcLb sacaWGjbaaaaaa@4484@    (2)

The angular velocity of wheel can be calculated from the integration: ω= 0 t α dτ=αt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHjp WDcqGH9aqpjuaGdaWdXaGcbaqcLbsacqaHXoqyaSqaaKqzadGaaGim aaWcbaqcLbmacaWG0baajugibiabgUIiYdGaamizaiabes8a0jabg2 da9iabeg7aHjabgwSixlaadshaaaa@4B79@ . Then the tangential velocity is

v t ' =rαt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aa0baaSqaaKqzadGaamiDaaWcbaqcLbmacaGGNaaaaKqzGeGa eyypa0JaamOCaiabeg7aHjabgwSixlaadshaaaa@43B4@     (3)

The sliding velocity is the difference relative to the required tangential velocity as

v d ' = v 0 rαt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aa0baaSqaaKqzadGaamizaaWcbaqcLbmacaGGNaaaaKqzGeGa eyypa0JaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqabaqcLbsacq GHsislcaWGYbGaeqySdeMaeyyXICTaamiDaaaa@48C8@    (4)

Let ts be the required acceleration time, also assumed the angular acceleration is uniform from static. Eventually the acceleration is completed, i.e. there is no sliding any more: v 0 rα t s =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiabgkHiTiaadkha cqaHXoqycqGHflY1caWG0bqcfa4aaSbaaSqaaKqzadGaam4CaaWcbe aajugibiabg2da9iaaicdaaaa@46BC@ , the required acceleration time is:

t s ' = v 0 rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aa0baaSqaaKqzadGaam4CaaWcbaqcLbmacaGGNaaaaKqzGeGa eyypa0tcfa4aaSaaaOqaaKqzGeGaamODaKqbaoaaBaaaleaajugWai aaicdaaSqabaaakeaajugibiaadkhacqaHXoqyaaaaaa@45E6@    (5)

Considering a pre-rotation is applied. Letting   ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzaaa@382C@ ( 0ε1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa GaeyizImQaeqyTduMaeyizImQaaGymaaaa@3D0B@ ) be the pre-rotation ratio, i.e., the percentage of landing speed v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ being pre-rotated by a whatsoever pre-rotation device, and ts will be the required acceleration time from 0 or a slow spinning speed, to the matching landing speed v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ . With the pre-rotation, the tangential velocity and the sliding velocity are, respectively,

v t =ε v 0 +rαt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaajugibiabg2da9iabew7a LjaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaS IaamOCaiabeg7aHjabgwSixlaadshaaaa@489A@    (6)

v d =(1ε) v 0 rαt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaajugibiabg2da9iaacIca caaIXaGaeyOeI0IaeqyTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLb macaaIWaaaleqaaKqzGeGaeyOeI0IaamOCaiabeg7aHjabgwSixlaa dshaaaa@4B96@     (7)

Therefore, with pre-rotation, the required acceleration time is:

t s = ( 1ε ) v 0 rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqyTdugaki aawIcacaGLPaaajugibiaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaa leqaaaGcbaqcLbsacaWGYbGaeqySdegaaaaa@4A15@      (8)

From the above equation, a shorter t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A69@ can be expected if a pre-rotation ε>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcqGH+aGpcaaIWaaaaa@39EE@ as v d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@3A5C@ is between v d =(1ε) v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaajugibiabg2da9iaacIca caaIXaGaeyOeI0IaeqyTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLb macaaIWaaaleqaaaaa@4441@ at t=0 and 0 at t=ts. The average sliding velocity over the acceleration time t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A69@ is:

v ¯ d = 1 t s 0 t s v d dτ= (1ε) v 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b GbaebajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaeyypa0tc fa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWG0bqcfa4aaSbaaS qaaKqzadGaam4CaaWcbeaaaaqcfa4aa8qmaOqaaKqzGeGaamODaKqb aoaaBaaaleaajugWaiaadsgaaSqabaqcLbsacaWGKbGaeqiXdqNaey ypa0daleaajugWaiaaicdaaSqaaKqzadGaamiDaSWaaSbaaWqaaKqz adGaam4CaaadbeaaaKqzGeGaey4kIipajuaGdaWcaaGcbaqcLbsaca GGOaGaaGymaiabgkHiTiabew7aLjaacMcacaWG2bqcfa4aaSbaaSqa aKqzadGaaGimaaWcbeaaaOqaaKqzGeGaaGOmaaaaaaa@5FC4@  (9)

As described in the introduction section, several types of devices have been proposed for the pre-rotation. For example, a side Savonius wind turbine is suggested to fit on the wheel, on which an air dynamic torque is generated to accelerate the wheel to achieve a tangential speed reached or close to the landing speed before the touchdown. After the touchdown, no energy or just smaller energy (depending on the pre-rotation ratio) is required to accelerate the wheel to the matching speed v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A2D@ . Even partial pre-rotation is implemented; the temperature can be significantly reduced. The high temperature which causes rubber melting or burning off can be avoided. If the pre-rotation level ε=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcqGH9aqpcaaIWaaaaa@39EC@ , t s = v 0 rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaa GcbaqcLbsacaWGYbGaeqySdegaaaaa@440C@ ; it usually takes as long as a fraction of second for a real aircraft to accelerate. If ε=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcqGH9aqpcaaIXaaaaa@39ED@ , t s =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiabg2da9iaaicda aaa@3CB8@ . i.e. no acceleration time is needed, no sliding exists nor heat and smoke will be generated. Certainly, pre-rotating the wheel will reduce the acceleration time t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A69@ , during which there is a sliding. The total acceleration distance, or aircraft travelled distance with sliding is

s s = v 0 t s = ( 1ε ) v 0 2 rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiabg2da9iaadAha juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaamiDaKqbaoaaBa aaleaajugWaiaadohaaSqabaqcLbsacqGH9aqpjuaGdaWcaaGcbaqc fa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabew7aLbGccaGLOaGaay zkaaqcLbsacaWG2bqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajuaG daahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaamOCaiabeg7aHb aaaaa@5669@    (10)

The acceleration distance s s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A68@ contains sliding as the wheel is gradually increasing its tangential speed from ε v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcaWG2bqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3BD4@ . Since the wheel is in the state of sliding, the actual tangential displacement of wheel at contact point pushed by ground friction, s r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@3A67@ , is less than the acceleration distance s s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A68@ . With the average speed during the acceleration ( 1+ε ) v 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqbaoaabmaakeaajugibiaaigdacqGHRaWkcqaH1oqzaOGaayjk aiaawMcaaKqzGeGaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqaba aakeaajugibiaaikdaaaaaaa@4228@ and acceleration time t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A69@ , it can be obtained the tangential displacement of wheel relative to the wheel shaft, or the tangential displacement is:

s r = ( 1 ε 2 ) v 0 2 2rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqyTduwcfa 4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPaaajugibiaa dAhajuaGdaqhaaWcbaqcLbmacaaIWaaaleaajugWaiaaikdaaaaake aajugibiaaikdacaWGYbGaeqySdegaaaaa@4F5F@     (11)

It can be defined a follow up ratio to describe how much percent the rolling distance the wheel has caught up the acceleration distance:

η= s r s s = 1+ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGZbqcfa4aaSbaaSqaaKqz adGaamOCaaWcbeaaaOqaaKqzGeGaam4CaKqbaoaaBaaaleaajugWai aadohaaSqabaaaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGym aiabgUcaRiabew7aLbGcbaqcLbsacaaIYaaaaaaa@4A31@    (12)

Also a sliding ratio can be defined to describe the level of sliding as

ρ= s s s r s s =1η= 1ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCcqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGZbqcfa4aaSbaaSqaaKqz adGaam4CaaWcbeaajugibiabgkHiTiaadohajuaGdaWgaaWcbaqcLb macaWGYbaaleqaaaGcbaqcLbsacaWGZbqcfa4aaSbaaSqaaKqzadGa am4CaaWcbeaaaaqcLbsacqGH9aqpcaaIXaGaeyOeI0Iaeq4TdGMaey ypa0tcfa4aaSaaaOqaaKqzGeGaaGymaiabgkHiTiabew7aLbGcbaqc LbsacaaIYaaaaaaa@5409@    (13)

It describes how much percent of acceleration distance the wheel has slid for. If without pre-rotation, ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzaaa@382C@ =0, it can be obtained η=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3B62@ , and ρ=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCcqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3B76@ , i.e. the wheel is 50% of followed-up and 50% of sliding. If ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzaaa@382C@ =0.5, it can be obtained ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzaaa@382C@ , and ρ=0.25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCcqGH9aqpcaaIWaGaaiOlaiaaikdacaaI1aaaaa@3C32@ , i.e. the wheel is 75% of followed-up and 25% of sliding. In the case of ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzaaa@382C@ =1 with full pre-rotation, η=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpcaaIXaaaaa@39F2@ and ρ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCcqGH9aqpcaaIWaaaaa@3A05@ , the wheel is 100% of followed-up and 0 of sliding.

Figure 1 Aircraft undercarriage wheel at landing.

Kinetic work and sliding work

Only part of the friction is contributed to push the wheel to gain the angular rotation and stored as kinetic energy, that is F s r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb Gaam4CaKqbaoaaBaaaleaajugWaiaadkhaaSqabaaaaa@3B32@ , named kinetic work, is

E r = ( 1 ε 2 ) v 0 2 F 2rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqyTduwcfa 4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPaaajugibiaa dAhajuaGdaqhaaWcbaqcLbmacaaIWaaaleaajugWaiaaikdaaaqcLb sacaWGgbaakeaajugibiaaikdacaWGYbGaeqySdegaaaaa@508B@       (14)

In the hypothetical cases of friction torque T r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9iaaicda aaa@3C97@ , combining Eq.(2), the kinetic work is E r = 1 2 ( 1 ε 2 )I ( v 0 r ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaqadaGcba qcLbsacaaIXaGaeyOeI0IaeqyTduwcfa4aaWbaaSqabeaajugWaiaa ikdaaaaakiaawIcacaGLPaaajugibiaadMeajuaGdaqadaGcbaqcfa 4aaSaaaOqaaKqzGeGaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqa baaakeaajugibiaadkhaaaaakiaawIcacaGLPaaajuaGdaahaaWcbe qaaKqzadGaaGOmaaaaaaa@545F@ , i.e. the kinetic energy gained by the wheel from the ground friction work. Without a full pre-rotation, s r < s s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabgYda8iaadoha juaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaaa@3FDD@ . The acceleration displacement caused by the forces acted on the touch point during the whole period ( 0t t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa GaeyizImQaamiDaiabgsMiJkaadshajuaGdaWgaaWcbaqcLbmacaWG Zbaaleqaaaaa@3F86@ ) is s=ε v 0 t+ 1 2 αr t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb Gaeyypa0JaeqyTduMaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqa baqcLbsacaWG0bGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcba qcLbsacaaIYaaaaiabeg7aHjabgwSixlaadkhacaWG0bqcfa4aaWba aSqabeaajugWaiaaikdaaaaaaa@4C01@ . Therefore, the total work done by the ground friction at any time 0t t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa GaeyizImQaamiDaiabgsMiJkaadshajuaGdaWgaaWcbaqcLbmacaWG Zbaaleqaaaaa@3F86@ , Fs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb Gaam4Caaaa@3848@ , is

e r (t)=F( ε v 0 t+ 1 2 rα t 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiaacIcacaWG0bGa aiykaiabg2da9iaadAeajuaGdaqadaGcbaqcLbsacqaH1oqzcaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiaadshacqGHRaWk juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdaaaGaamOCai abeg7aHLqbakabgwSixNqzGeGaamiDaKqbaoaaCaaaleqabaqcLbma caaIYaaaaaGccaGLOaGaayzkaaaaaa@5660@    (15)

At t=ts, e r ( t s )= F( 1 ε 2 ) v 0 2 2rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiaacIcacaWG0bqc fa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiaacMcacqGH9aqpju aGdaWcaaGcbaqcLbsacaWGgbqcfa4aaeWaaOqaaKqzGeGaaGymaiab gkHiTiabew7aLLqbaoaaCaaaleqabaqcLbmacaaIYaaaaaGccaGLOa GaayzkaaqcLbsacaWG2bqcfa4aa0baaSqaaKqzadGaaGimaaWcbaqc LbmacaaIYaaaaaGcbaqcLbsacaaIYaGaamOCaiabeg7aHbaaaaa@5677@ . The rate of work done by the ground friction, F ds dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSaaaOqaaKqzGeGaamizaiaadohaaOqaaKqzGeGaamizaiaa dshaaaaaaa@3CE3@ , i.e. the power, becomes

e ˙ r (t)=F ds dt =F( ε v 0 +rαt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaaiikaiaa dshacaGGPaGaeyypa0JaamOraKqbaoaalaaakeaajugibiaadsgaca WGZbaakeaajugibiaadsgacaWG0baaaiabg2da9iaadAeajuaGdaqa daGcbaqcLbsacqaH1oqzcaWG2bqcfa4aaSbaaSqaaKqzadGaaGimaa WcbeaajugibiabgUcaRiaadkhacqaHXoqycqGHflY1caWG0baakiaa wIcacaGLPaaaaaa@55CB@  (16)

At t=0, e ˙ t (0)=Fε v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWG0baaleqaaKqzGeGaaiikaiaa icdacaGGPaGaeyypa0JaamOraiabew7aLjaadAhajuaGdaWgaaWcba qcLbmacaaIWaaaleqaaaaa@4426@ , and at t=ts, e ˙ t ( t s )=F v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWG0baaleqaaKqzGeGaaiikaiaa dshajuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaKqzGeGaaiykaiabg2 da9iaadAeacaWG2bqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@4638@ .

 The instantaneous rate of sliding work, or the instantaneous sliding power, is F v d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamODaKqbaoaaBaaaleaajugWaiaadsgaaSqabaaaaa@3B27@ , i.e.

e ˙ d (t)=F( (1ε) v 0 rαt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaaiikaiaa dshacaGGPaGaeyypa0JaamOraKqbaoaabmaakeaajugibiaacIcaca aIXaGaeyOeI0IaeqyTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLbma caaIWaaaleqaaKqzGeGaeyOeI0IaamOCaiabeg7aHjabgwSixlaads haaOGaayjkaiaawMcaaaaa@5165@     (17)

During the acceleration period ( 0t t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa GaeyizImQaamiDaiabgsMiJkaadshajuaGdaWgaaWcbaqcLbmacaWG Zbaaleqaaaaa@3F86@ ), the aircraft displacement at time t is v 0 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiaadshaaaa@3BB5@ , and the tangential displacement of wheel is the average speed multiplied by the time, i.e. ( 1+ε ) v 0 2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqbaoaabmaakeaajugibiaaigdacqGHRaWkcqaH1oqzaOGaayjk aiaawMcaaKqzGeGaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqaba aakeaajugibiaaikdaaaGaamiDaaaa@4321@ , where the average tangential velocity of wheel is v ¯ = ( 1+ε ) v 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b GbaebajuaGdaWgaaWcbaaabeaajugibiabg2da9Kqbaoaalaaakeaa juaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaeqyTdugakiaawIcaca GLPaaajugibiaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaaGc baqcLbsacaaIYaaaaaaa@4619@ . Therefore, the sliding distance is s d (t)= v 0 t ( 1+ε ) v 0 2t t= 1ε 2 v 0 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaajugibiaacIcacaWG0bGa aiykaiabg2da9iaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaK qzGeGaamiDaiabgkHiTKqbaoaalaaakeaajuaGdaqadaGcbaqcLbsa caaIXaGaey4kaSIaeqyTdugakiaawIcacaGLPaaajugibiaadAhaju aGdaqhaaWcbaqcLbmacaaIWaaaleaaaaaakeaajugibiaaikdacaWG 0baaaiaadshacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaGaeyOeI0 IaeqyTdugakeaajugibiaaikdaaaGaamODaKqbaoaaBaaaleaajugW aiaaicdaaSqabaqcLbsacaWG0baaaa@5E93@ . The sliding velocity then is d s d dt = 1ε 2 v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadohajuaGdaWgaaWcbaqcLbmacaWGKbaaleqa aaGcbaqcLbsacaWGKbGaamiDaaaacqGH9aqpjuaGdaWcaaGcbaqcLb sacaaIXaGaeyOeI0IaeqyTdugakeaajugibiaaikdaaaGaamODaKqb aoaaBaaaleaajugWaiaaicdaaSqabaaaaa@48EE@ . The average sliding power can be obtained as

E ˙ ¯ d =F ( 1ε ) v 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGfb GbaiGbaebajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaeyyp a0JaamOraKqbaoaalaaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaey OeI0IaeqyTdugakiaawIcacaGLPaaajugibiaadAhajuaGdaqhaaWc baqcLbmacaaIWaaaleaaaaaakeaajugibiaaikdaaaaaaa@48E9@     (18)

The total sliding work is E ˙ ¯ d t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGfb GbaiGbaebajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaamiD aKqbaoaaBaaaleaajugWaiaadohaaSqabaaaaa@3EBE@ , i.e.

E d = ( 1ε ) 2 v 0 2 F 2rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqyTdugaki aawIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaa dAhajuaGdaqhaaWcbaqcLbmacaaIWaaaleaajugWaiaaikdaaaqcLb sacaWGgbaakeaajugibiaaikdacaWGYbGaeqySdegaaaaa@507D@  (19)

Alternatively, using Eq. (10) and (11), the total sliding work at the touch point E d =F( s s s r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaajugibiabg2da9iaadAea juaGdaqadaGcbaqcLbsacaWGZbqcfa4aaSbaaSqaaKqzadGaam4Caa WcbeaajugibiabgkHiTiaadohajuaGdaWgaaWcbaqcLbmacaWGYbaa leqaaaGccaGLOaGaayzkaaaaaa@4886@ is , Eq.(19) can also be obtained. Using the average sliding power E d t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamyraKqbaoaaBaaaleaajugWaiaadsgaaSqabaaakeaa jugibiaadshajuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaaaaaa@3F50@ , Eq.(18) can also be obtained. This part of energy, E d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@3A2B@ , generates heat and smoke, and causes tyre wear. During the acceleration period, the wheel has gained kinetic energy from the ground friction as E r = 1 2 ( 1 ε 2 )I ( v 0 r ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaqadaGcba qcLbsacaaIXaGaeyOeI0IaeqyTduwcfa4aaWbaaSqabeaajugWaiaa ikdaaaaakiaawIcacaGLPaaajugibiaadMeajuaGdaqadaGcbaqcfa 4aaSaaaOqaaKqzGeGaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqa baaakeaajugibiaadkhaaaaakiaawIcacaGLPaaajuaGdaahaaWcbe qaaKqzadGaaGOmaaaaaaa@545F@ if Tr=0. In the case of zero pre-rotation, ε=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcqGH9aqpcaaIWaaaaa@39EC@ , E r = 1 2 I ( v 0 r ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaacaWGjbqcfa4aae WaaOqaaKqbaoaalaaakeaajugibiaadAhajuaGdaWgaaWcbaqcLbma caaIWaaaleqaaaGcbaqcLbsacaWGYbaaaaGccaGLOaGaayzkaaqcfa 4aaWbaaSqabeaajugWaiaaikdaaaaaaa@4B22@ . After the acceleration period, this energy will be eventually dissipated in the subsequent braking in the late stage of running after the acceleration period, which is outside this topic. The total work done by ground friction force in the wheel acceleration period is

E= E r + E d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0JaamyraKqbaoaaBaaaleaajugWaiaadkhaaSqabaqcLbsa cqGHRaWkcaWGfbqcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@4120@     (20)

Using Eq.(14) and (19), the total ground friction work is:

E= ( 1ε ) v 0 2 F rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0tcfa4aaSaaaOqaaKqbaoaabmaakeaajugibiaaigdacqGH sislcqaH1oqzaOGaayjkaiaawMcaaKqzGeGaamODaKqbaoaaDaaale aajugWaiaaicdaaSqaaKqzadGaaGOmaaaajugibiaadAeaaOqaaKqz GeGaamOCaiabeg7aHbaaaaa@49B1@    (21)

If without pre-rotation, ε=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1o qzcqGH9aqpcaaIWaaaaa@39EC@ , the total energy will be E= v 0 2 F rα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamODaKqbaoaaDaaaleaajugW aiaaicdaaSqaaKqzadGaaGOmaaaajugibiaadAeaaOqaaKqzGeGaam OCaiabeg7aHbaaaaa@43A8@ . In the case that Tr is small and can be ignored, E=I ( v 0 r ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0JaamysaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWG 2bqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGaamOCaa aaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaa @4462@ . i.e. in this case, the total work done by the ground friction is the twice of the kinetic energy gained by the wheel during the acceleration period. Only the second term in Eq.(20), i.e. the energy E d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@3A2B@ , causes the high temperature, then smoke and wear problems.

Heat transfer at the contact area of tyre and runway

Sliding friction is a complicated subject. Two surfaces of bodies are interacting with physical and chemical processes. In the interfacial media, tribo-chemical reaction, such as smoke in the intensive cases like aircraft landing, takes place and tribo-particulates are also generated. In the adjacent solids of two contacted bodies, there exist deformation, fatigue and fracture.26 In aircraft landing cases, due to high speed, the sliding friction causes significant temperature rise, smoke and excessive tyre wear. The sliding friction originally causes substance losses but the high temperature intensifies such losses, partly burned off as smoke and partly torn off as particulate matter deposited on the runway surface. The rubber tyre is relatively thin, the heat is transferred to the whole depth of circumference but the runway is normally thick, it gives too short time to transfer heat to its whole depth. The heat generated by the sliding also dissipates to the runway, normally made of concrete with a higher heat transfer coefficient than the tyre rubber. Consequently, the heat affected depth is greater than the rubber tyre.

Assuming the width contact is b, which is regarded as the tyre width. The average sliding power is Δ ¯ E ˙ d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuqHuo argaqeaiqadweagaGaaKqbaoaaBaaaleaajugWaiaadsgaaSqabaaa aa@3BB2@ , which is the heat generating rate by ground friction, contributing to the tyre and runway heat. This heat will give a significant temperature rise to both rubber tyre and runway concrete body. The heat transfers to the rubber and to runway of contact area are, respectively,27

Q ˙ r = k r A r T h T 0 δ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGrb GbaiaajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0Ja am4AaKqbaoaaBaaaleaajugWaiaadkhaaSqabaqcLbsacaWGbbqcfa 4aaSbaaSqaaKqzadGaamOCaaWcbeaajuaGdaWcaaGcbaqcLbsacaWG ubqcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabgkHiTiaads fajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaaGcbaqcLbsacqaH0oaz juaGdaWgaaWcbaqcLbmacaWGYbaaleqaaaaaaaa@5316@    (22a)

Q ˙ c = k c A c T h T 0 δ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGrb GbaiaajuaGdaWgaaWcbaqcLbmacaWGJbaaleqaaKqzGeGaeyypa0Ja am4AaKqbaoaaBaaaleaajugWaiaadogaaSqabaqcLbsacaWGbbqcfa 4aaSbaaSqaaKqzadGaam4yaaWcbeaajuaGdaWcaaGcbaqcLbsacaWG ubqcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabgkHiTiaads fajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaaGcbaqcLbsacqaH0oaz juaGdaWgaaWcbaqcLbmacaWGJbaaleqaaaaaaaa@52DA@    (22b)

where k r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@3A5F@ and k c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb qcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaaaaa@3A50@ are the heat conduction coefficients of rubber and concrete respectively, T h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaaaaa@3A3E@ and   T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGGa GaamivaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaaaa@3AAE@ are the higher and lower temperature sides, or hotter and colder sides, respectively.   T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGGa GaamivaKqbaoaaBaaaleaajugWaiaaicdaaSqabaaaaa@3AAE@ is also the surrounding temperature. A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@3A35@ and A c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaaaaa@3A26@ are the contact area between tyre and runway surface, δ r  and  δ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaaeiiaiaabgga caqGUbGaaeizaiaabccacqaH0oazjuaGdaWgaaWcbaqcLbmacaWGJb aaleqaaaaa@4425@ are the thickness of tyre and affected concrete runway surface. Using Eq.(22a) and (22b) the heated depths for the rubber tyre and concrete runway are respectively,

δ r = k r A r T h T 0 Q ˙ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0Jaam4A aKqbaoaaBaaaleaajugWaiaadkhaaSqabaqcLbsacaWGbbqcfa4aaS baaSqaaKqzadGaamOCaaWcbeaajuaGdaWcaaGcbaqcLbsacaWGubqc fa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabgkHiTiaadsfaju aGdaWgaaWcbaqcLbmacaaIWaaaleqaaaGcbaqcLbsaceWGrbGbaiaa juaGdaWgaaWcbaqcLbmacaWGYbaaleqaaaaaaaa@5316@    (23a)

δ c = k c A c T h T 0 Q ˙ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaWGJbaaleqaaKqzGeGaeyypa0Jaam4A aKqbaoaaBaaaleaajugWaiaadogaaSqabaqcLbsacaWGbbqcfa4aaS baaSqaaKqzadGaam4yaaWcbeaajuaGdaWcaaGcbaqcLbsacaWGubqc fa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabgkHiTiaadsfaju aGdaWgaaWcbaqcLbmacaaIWaaaleqaaaGcbaqcLbsaceWGrbGbaiaa juaGdaWgaaWcbaqcLbmacaWGJbaaleqaaaaaaaa@52DA@    (23b)

If assuming Q ˙ r = Q ˙ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGrb GbaiaajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0Ja bmyuayaacaqcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaaaaa@3F9D@ and A r = A c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9iaadgea juaGdaWgaaWcbaqcLbmacaWGJbaaleqaaaaa@3F6B@ , it can be arrived

δ c = k c A c k r A r δ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaWGJbaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaam4AaKqbaoaaBaaaleaajugWaiaadogaaSqaba qcLbsacaWGbbqcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaaaOqaaKqz GeGaam4AaKqbaoaaBaaaleaajugWaiaadkhaaSqabaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaqcLbsacqaH0oazjuaG daWgaaWcbaqcLbmacaWGYbaaleqaaaaa@539C@    (24)

It is evident that the blackening of runway will reduce the heat conductivity of runway surface therefore may intensify the touchdown temperature. To find the contact side temperature of tyre, from Eq. (22a), the highest temperature of contact area is

T h = T 0 + Q ˙ r δ r k r A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabg2da9iaadsfa juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSscfa4aaS aaaOqaaKqzGeGabmyuayaacaqcfa4aaSbaaSqaaKqzadGaamOCaaWc beaajugibiabes7aKLqbaoaaBaaaleaajugWaiaadkhaaSqabaaake aajugibiaadUgajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGa amyqaKqbaoaaBaaaleaajugWaiaadkhaaSqabaaaaaaa@539A@     (25)

Assuming Q ˙ r = Q ˙ c = 1 2 e ˙ d (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGrb GbaiaajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0Ja bmyuayaacaqcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaajugibiabg2 da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaaceWG LbGbaiaajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaaiikai aadshacaGGPaaaaa@4B29@ , the instantaneous temperature is

T h = T 0 + δ r F 2 k r A r ( (1ε) v 0 rαt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabg2da9iaadsfa juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSscfa4aaS aaaOqaaKqzGeGaeqiTdqwcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaa jugibiaadAeaaOqaaKqzGeGaaGOmaiaadUgajuaGdaWgaaWcbaqcLb macaWGYbaaleqaaKqzGeGaamyqaKqbaoaaBaaaleaajugWaiaadkha aSqabaaaaKqbaoaabmaakeaajugibiaacIcacaaIXaGaeyOeI0Iaeq yTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqz GeGaeyOeI0IaamOCaiabeg7aHjabgwSixlaadshaaOGaayjkaiaawM caaaaa@63B7@     (26)

At t=0, T h = T 0 + δ r F 2 k r A r ( 1ε ) v 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabg2da9iaadsfa juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSscfa4aaS aaaOqaaKqzGeGaeqiTdqwcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaa jugibiaadAeaaOqaaKqzGeGaaGOmaiaadUgajuaGdaWgaaWcbaqcLb macaWGYbaaleqaaKqzGeGaamyqaKqbaoaaBaaaleaajugWaiaadkha aSqabaaaaKqbaoaabmaakeaajugibiaaigdacqGHsislcqaH1oqzaO GaayjkaiaawMcaaKqzGeGaamODaKqbaoaaBaaaleaajugWaiaaicda aSqabaaaaa@5B98@ . At t=ts, T h = T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabg2da9iaadsfa juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaaaa@3F59@ . i.e. There is no temperature rise. An average temperature rise can be obtained using average sliding power during ts. Thus Q ˙ r = Q ˙ c = 1 2 E ˙ ¯ d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGrb GbaiaajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0Ja bmyuayaacaqcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaajugibiabg2 da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaaceWG fbGbaiGbaebajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaaaa@483F@ . Considering strong forced convection happens during the landing, the heat flow 1 2 E ˙ ¯ d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaiqadweagaGagaqeaKqb aoaaBaaaleaajugWaiaadsgaaSqabaaaaa@3D03@ is largely reduced, a factor, 0<φ<1, must be introduced, therefore, the hotter side temperature of tyre is

T h = T 0 + ϕ δ r F(1ε) v 0 4 k r A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiAaaWcbeaajugibiabg2da9iaadsfa juaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSscfa4aaS aaaOqaaKqzGeGaeqy1dyMaeqiTdqwcfa4aaSbaaSqaaKqzadGaamOC aaWcbeaajugibiaadAeacaGGOaGaaGymaiabgkHiTiabew7aLjaacM cacaWG2bqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGa aGinaiaadUgajuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaam yqaKqbaoaaBaaaleaajugWaiaadkhaaSqabaaaaaaa@5B72@    (27)

From outer surface to inner surface of tyre, the temperature ranges from Th to T0 linearly according to Newton’s conduction law. Figure 2 illustrates the temperature gradient of tyre and runway contact surfaces. The bottom layer of highest temperature easily reaches a smoke point. The thickness of lost rubber depends on Th and temperature gradient, i.e. the depth σ which has been softened or reached the smoke point. Since the thermal conductivity of concrete is about twice as much as the rubber and the contact area is larger, this tends to deepen the heated concrete surface. However, the heat on the runway can be spread to surroundings. And during the whole wheel acceleration period, the total contact area of runway is larger than that of tyre. Considering that the contact area is the whole circumference of tyre, the damaged rubber mass is

m r =2 π rbσ ρ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9iaaikda cqaHapaCjuaGdaWgaaWcbaaabeaajugibiaadkhacaWGIbGaeq4Wdm NaeqyWdixcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@4803@    (28)

where ρ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaaaa@3B2F@ is the rubber density. All aircraft suffer from such losses for each landing except a pre-rotation is applied. A part of damaged mass is likely to be melt or burned off, and the other part may be torn off to print on the runway.

Figure 2 The temperature gradient in tyre and runway contact areas.

Landing interpretation of numerical example

The touchdown temperature against several parameters has been calculated based on meaningful Boeing passenger aircraft undercarriage data.28 The following figures are used in the following example: A soft landing friction force F=4.49×104N and the landing speed v0=80.8m/s. The heat conduction coefficients for rubber and concrete k r = 0.293 W/m C o , k c = 0.80 W/m C o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9iaaicda caGGUaGaaGOmaiaaiMdacaaIZaqcfa4aaSbaaSqaaaqabaqcLbsaca WGxbGaai4laiaad2gacqGHflY1juaGdaahbaWcbeqaaKqzadGaam4B aaaajugibiaadoeacaGGSaqcfa4aaSbaaSqaaaqabaqcLbsacaWGRb qcfa4aaSbaaSqaaKqzadGaam4yaaWcbeaajugibiabg2da9iaaicda caGGUaGaaGioaiaaicdajuaGdaWgaaWcbaaabeaajugibiaadEfaca GGVaGaamyBaiabgwSixNqbaoaaCeaaleqabaqcLbmacaWGVbaaaKqz GeGaam4qaaaa@5E6F@ are . The moment of inertia of wheel is I=46.19kg-m2, the wheel with tyre radius r=0.622m. The density of tyre rubber is ρ r =1185kg/ m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqzGeGaeyypa0JaaGym aiaaigdacaaI4aGaaGynaiaadUgacaWGNbGaai4laiaad2gajuaGda ahaaWcbeqaaKqzadGaaG4maaaaaaa@45E2@ . With 10% of resistance torque Tr applied, without a pre-rotation, it can be worked out the acceleration time t s =0.795 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaajugibiabg2da9iaaicda caGGUaGaaG4naiaaiMdacaaI1aaaaa@3FAD@ s, the angular acceleration is 163rad/s.

Figure 3 shows the tangential velocity and sliding velocity change with the time. The tangential velocity v t =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaajugibiabg2da9iaaicda aaa@3CBB@ is corresponding to the initial touch instant t=0. The sliding velocity v d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@3A5C@ at t=0 is at the maximum of 80.8m/s, i.e. the aircraft landing speed, while at the end of the acceleration, the v d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamizaaWcbeaaaaa@3A5C@ =0, i.e. the sliding is finished, the wheel has reached a required velocity, exactly equal to the touchdown speed.

Figure 4 is the acceleration time against the pre-rotation ratio. This required acceleration time is decreasing linearly with the pre-rotation ratio from 0.8 seconds to 0 seconds. It can be seen that a 50% pre-rotation ratio can halve the acceleration time. To shorten this acceleration time ts, according to Eq. (8), the acceleration can be increased by a harder landing or a pre-rotation is applied. Since the friction coefficient between the rubber tyre and concrete runway is limited, a landing with harder suspension spring will generate more heat. Obviously, a pre-rotation will be ideal to reduce the acceleration time so as to reduce the heat.

Figure 5 displays the acceleration distance against the pre-rotation ratio ε. It decreases linearly with an increased pre-rotation ratio. The wheel tangential displacement is a half of the total acceleration distance without pre-rotation ε=0, which means a sliding exists. The follow-up ratio is 0.5 without a pre-rotation but reaches 1 with a full pre-rotation ε=1. A follow-up ratio of 0.5 indicates half of the distance is reached by rotating and half is by sliding. Since the wheel is in the state of sliding, the actual tangential displacement of wheel at contact point pushed by ground friction, s r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@3A67@ , is less than the acceleration distance s s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaaa@3A68@ . The sliding ratio is 0.5 without a pre-rotation but reaches 0 with a full pre-rotation. Similarly, a sliding ratio of 0.5 means the distance is reached by half sliding and half rotating.

Figure 6 shows the kinetic work and sliding work done by the ground friction, respectively. The kinetic work is absorbed by the wheel as the form of kinetic energy, which will not generate heat during the acceleration stage. It is the sliding work that is the source of heat of raising temperature of the tyre and runway. They share a half each of the total friction energy at the pre-rotation ε=0, as high as 433kJ. The total work done by the ground friction force is 865KJ. The average sliding power at ε=0 is 545KW, which is a powerful source to generate high temperature. This is equivalent to the engine’s power of 2 large trucks. However, this high power generated by the ground friction is mitigated by strong wind blowing over the tyre, i.e. a strong forced convection is applied. Hence the actual power is much lower in heating up the tyre.

Figure 7 shows the highest temperature of tyre outer layer. But it reduces with an increased pre-rotation ratio ε. There will be no temperature rise at all if a full pre-rotation is applied. Obviously, a certain degree of pre-rotation is likely to drop the highest tyre temperature below smoke point. This has just demonstrated the significance of fitting a pre-rotation device, either a side-wind turbine, a pneumatic or electric motor. Of course, in latter cases, power sources and added weight must be considered.

Figure 8 tells the temperature rise is related to the tyre width. The temperature rise is reduced when tyre width is increased because the area of contact increases. This can be explained by Eq. (27). The actual contact area between tyre and runway surface, A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaaaaa@3A35@ , has been increased so that the ground friction generated heat has been shared by a wider area.

Figure 3 Wheel’s tangential velocity and sliding velocity.

Figure 4 Wheel’s acceleration time against pre-rotation ratio.

Figure 5 The acceleration distance, wheel tangential displacement against pre-rotation ratio, follow-up ratio and slip ratio against pre-rotation ratio.

Figure 6 The kinetic work and sliding work done by the ground friction, the total ground friction work against pre-rotation ratio, and the average sliding power against pre-rotation ratio.

Figure 7 The touchdown highest temperature on tyre against pre-rotation ratio.

Figure 8 The touchdown highest temperature against tyre width.

Discussion

The depth of damaged tyre layer by intense heat during landing depends on the maximum temperature and its gradient, i.e. it depends on how deep the rubber molecules have reached the breaking down point. In the range of 250~400℃, tyre rubber will generate smoke and decompose oil. Higher than 400℃, more carbon particulate is formed. If in each landing, 0.2mm depth of rubber is lost, a total rubber loss is approximately, using m r =2πrbσ ρ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaSbaaSqaaKqzadGaamOCaaWcbeaajugibiabg2da9iaaikda cqaHapaCcqGHflY1caWGYbGaamOyaiabeo8aZjabeg8aYLqbaoaaBa aaleaajugWaiaadkhaaSqabaaaaa@4904@ , 440grams. The runway ground normally is very thick and under it there is infinite layer of mud and rocks. The heat affected depth of concrete by a single landing will be limited to a certain depth. Even ordinary cement concrete can withstand as high as up to 1000oC, much stronger and heat resistant than tyre rubber. However, when the touchdown area of a runway is coated with large quantity of rubber particulate matter coming off from the tyre, both the heat conductivity and friction coefficient of runway surface will be reduced, leading to higher tyre temperature. The stiffness of suspension spring will determine the friction of touchdown instant. According to Eq. (17), e ˙ d (t)=F( (1ε) v 0 rαt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaaiikaiaa dshacaGGPaGaeyypa0JaamOraKqbaoaabmaakeaajugibiaacIcaca aIXaGaeyOeI0IaeqyTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLbma caaIWaaaleqaaKqzGeGaeyOeI0IaamOCaiabeg7aHjabgwSixlaads haaOGaayjkaiaawMcaaaaa@5165@ , a relatively small friction reduces the sliding power at t=0, i.e. e ˙ d (0)=F( (1ε) v 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGLb GbaiaajuaGdaWgaaWcbaqcLbmacaWGKbaaleqaaKqzGeGaaiikaiaa icdacaGGPaGaeyypa0JaamOraKqbaoaabmaakeaajugibiaacIcaca aIXaGaeyOeI0IaeqyTduMaaiykaiaadAhajuaGdaWgaaWcbaqcLbma caaIWaaaleqaaaGccaGLOaGaayzkaaaaaa@49D1@ . The hottest instant power is reduced, so is the temperature. An initial softer non-linear spring in the suspension of landing gear may help to achieve this.

Conclusion

Dynamic modelling of acceleration period of undercarriage wheels after aircraft touchdown has been approached and developed, giving fundamental relationship between the forces and motion of aircraft undercarriage. The total acceleration distance, sliding displacement and sliding velocity have been derived. The sliding power has also been derived, which is generated during the wheel acceleration period, producing heat, smoke, and causing excessive tyre wear. The analysis shows the sliding work is a part of the work done by the ground friction. During the sliding period, the wheel has also gained kinetic energy from the ground friction in order to match the landing velocity. The maximum temperature rise formula is established based on thermodynamics. The heat transfer to both tyre and runway has been analysed. Ideally, the touchdown temperature will decrease with the increase of the pre-rotation ratio linearly as effective solution for landing smoke, tyre wear and blackening of runway. The tyre width will also affect the touchdown temperature. Increase of the tyre width will reduce the temperature. Relative to tyre, the heat transferred to runway is spread wider and deeper, therefore the damage is less serious apart from caused by rubber removal activities. The knowledge in this research unveils the mechanism of landing smoke and tyre particulate emission hopefully can be used as a new attempt to find the solution for the large-scale and world-wide landing smoke problem. Further desirable move should be in finding out the required torque by side-wind turbine or other auxiliary power source, electrical or pneumatical, to drive the undercarriage wheels in laboratory and operating conditions. On-board experiment is most desirable, which will involve large scale of funding and regulatory issues, leading to significant future improvement of aircraft landing and airport operation.

Acknowledgements

None.

Conflicts of Interest

The author declares that there is no conflict of interest.

References

  1. Beazley RH. Aircraft wheel spinner and control. Washington, DC: U.S. Patent and Trademark Office; 1947.
  2. Khal S, Khal A. Apparatus for pre-rotating aircraft tires. Washington, DC: U.S. Patent and Trademark Office; 2013.
  3. Sweet RM, Gilleran N, Edelson JS, et al. Integrated electric motor and gear in an aircraft wheel. Washington, DC: U.S. Patent and Trademark Office; 2014.
  4. Didey A. Landing gear drive systems. Washington, DC: U.S. Patent and Trademark Office; 2014.
  5. Karl W. Free Spinning Wheel for Airplanes. Washington, DC: U.S. Patent and Trademark Office; 2014.
  6. Khal S, Khal A. Apparatus for Pre-Rotating Aircraft Tires. Washington, DC: U.S. Patent and Trademark Office; 2013.
  7. Ushiyama I, Tawara A. Driving device for wheel of airplane. Washington, DC: U.S. Patent and Trademark Office; 2013.
  8. Schmitz RH. Method and System to Accelerate Aircraft Wheels Before Touch-Down. Washington, DC: U.S. Patent and Trademark Office; 2009.
  9. Zha G. Landing Gear Wheels with Side-Mounted Air Foils. Washington, DC: U.S. Patent and Trademark Office; 2007.
  10. Horvath V, Szoke B. Airplane tire saver by protrusion airfoils. Washington, DC: U.S. Patent and Trademark Office; 2006.
  11. Suzuki T. Airplane Wheel Unit. Washington, DC: U.S. Patent and Trademark Office; 2002.
  12. Lyons DE. Method of Rotating Aircraft Wheels. Washington, DC: U.S. Patent and Trademark Office; 2002.
  13. John M Opitz. Airplane Wheel Rotator. Washington, DC: U.S. Patent and Trademark Office; 1975.
  14. Almen Michael D. Wheel Rotation Device. Washington, DC: U.S. Patent and Trademark Office; 2002.
  15. Gannatal JP. Space Shuttle Wheel Acceleration System. Washington, DC: U.S. Patent and Trademark Office; 1993.
  16. Gooding D, Singh K, Wilson S. Development of a Windmill Machine Pre-Rotation Device for an Aircraft Nose wheel. The Journal of the Association of Professional Engineers of Trinidad and Tobago. 2011;40(2):49–56.
  17. Bennett M, Christie SM, Graham A, et al., Composition of Smoke Generated by Landing Aircraft. Environ Sci Technol. 2011;45(8):3533–3538.
  18. Broitzman T. Self-Rotating Aircraft Tire that Spins up Prior to Landing; 2009.
  19. Alroqi AA, Wang WJ. The prevention of aircraft tires overheating by pre-rotating the wheels. London: 7th International Conference on Mechanical and Aerospace Engineering; 2016. 443–447 p.
  20. Alroqi AA, Wang WJ. Reduction of Aircraft Tyre Wear by Pre-Rotating Wheel Using ANSYS Mechanical Transient. Advanced Engineering Forum. 2016;17:89–100.
  21. WJ Wang. Dynamics and Energy Conversion of Aircraft Landing Gears at Touchdown. UK: 3rd International Conference on Mechanics and Mechatronics Research; 2016. 4 p.
  22. Abdurrhman AA, Weiji Wang. Comparison of Aircraft Tire Wear with Initial Wheel Rotational Speed. International Journal of Aviation, Aeronautics, and Aerospace. 2015;2(1).
  23. Alroqi AA, Wang WJ. Pre-Rotation Wind Turbine Design and Optimization for Aircraft Landing Gear. Advanced Engineering Forum. 2017;23:88–103.
  24. Abdurrhman AA, Weiji Wang, Yong Zhao. Aircraft Tire Temperature at Touchdown with Wheel Pre-rotation. Journal of Aircraft. 2017;54(3):926–938.
  25. Altan BD, Atilgan M. A study on increasing the performance of Savonius wind rotors. J Mech Sci Technol. 2012;26(5):1493–1499.
  26. Peter J Blau. Friction Science and Technology: From Concepts to Applications. 2nd ed. Florida: CRC Press; 2008. 432 p.
  27. Yunus AC, Michael AB. Thermodynamics: An Engineering Approach. 7th ed. New York: McGraw-Hill Higher Education; 2014. 1024 p.
  28. Approach speeds for Boeing airplanes. Chicago: Boeing Commercial Airplane Co; 2011. 4 p.
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