Tire Heating on Landing

When an airplane first touches the runway, there is an interesting moment of drama because the ground (relative to the airplane’s wheel) is moving past quite rapidly, while the wheel just at the instant of touchdown, is not moving at all.  Of course, there is a great deal of friction between the non-rotating wheel and the moving ground, so the wheel is accelerated rapidly in rotation until it “catches up” to the ground speed and rolls without sliding.  However, the wheel has some rotational inertia and so requires a little bit of time to get up to speed.  During that time, the tire is sliding across the ground and generating a lot of heat. Today we’ll talk about a rough approximation for the heat generated and transferred to the tire during an airplane landing.

Before we start talking about the heat generation, it is worth noting that you would have a similar situation if you tried to move a car off of, or onto, a moving ramp.  The problem in all of these cases is that the tire rotational speed does not (at first) match the speed of the surface that it is contacting.
 
There is a lot going on in that very brief period (typically for an airplane landing it would be
somewhere between a few 10’s  to a couple 100’s of milliseconds) while the speeds get matched.  The wheel, of course, is rapidly accelerating rotationally.  The airplane is slowing down because of the drag from the wheels sliding across the runway as well as the aerodynamic drag from the wings and possibly reverse thrust from the engines.  The tires are heating up from the frictional heat generation which causes the mechanical and thermal material properties of the rubber to change.  Often, some rubber tire material is being mechanically eroded away by the sliding contact.  (Have you ever noticed puffs of smoke coming from an airplane’s tires at touchdown?)

It would be a very involved problem to take all of those things into account simultaneously, but we could make a first (rough) approximation to the heating problem by making some simplifying assumptions.
(1)    If someone could measure (by instrumenting an actual landing gear) or calculate (using a mechanical dynamics model of the landing) the instantaneous horizontal force on the landing gear, the instantaneous horizontal velocity of the airplane, and the instantaneous rotational velocity of the tire, we could use conservation of energy to determine the total rate of heat being generated at the tire/runway interface. 
(2)    We would have to use some kind of model of the tire flexure along with the weight of the airplane in order to estimate a sliding contact area between the tire and the runway.
(3)    Using the total heat generation rate and the contact area, we could estimate a total heat flux at the tire/runway interface which we would need to divide between a part going into the tire and a part going into the runway.  We could make an estimate for that division by using the ratio of (k*ρ*c)^0.5.  We would neglect any heat carried away by mechanical erosion of the rubber and assume that all of the heat generated would be conducted into either the rubber or the runway.
(4)    As soon as the tire touches the runway, it starts to roll and the rotational velocity increases rapidly.  So the tire is both rolling and sliding during the period of interest.  Since we know both the forward velocity of the airplane, and the rotational velocity of the wheel, we can calculate the time that any particular point on the tire is in sliding contact with the ground, and hence, has heat being forced into the tire.
(5)    We can approximate the temperature profile inside the rubber tire material by using the solution for transient conduction into a semi-infinite solid with an imposed surface heat flux.  It turns out that over the very small time scale involved for any particular point on the tire, the conduction is confined to a very small depth, and the semi-infinite solution provides a good approximation.

By the time that the tire is rolling freely, the horizontal force at the landing gear will be effectively zero (relative to the force while the tire is sliding and accelerating) and the heat generation rate will become zero as well.   This approximate analysis only considers heat generated by sliding friction and does not include heat generated by tire flexion which would not be insignificant.

In the next post, we’ll look at notional results of such an analysis.