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There are quite a few phenomena related to the theoretical physics of the tire that can be modeled by quadratic forms on its surface. First, one needs to account for the surfaces delimiting the tire. These are basically two: the internal surface of the carcass delimiting, together with the metallic part of the wheel (the rim), the tire cavity, and sustaining the internal pressure. While rolling, this surface vibrates, producing a sound which is amplified by the tire cavity. The problem is to describe these vibrations and the phenomena related to them. Another surface delimiting the tire is the external surface of the rubber, comprising roughly two regions: the main external (crest) region of the tread, and the lateral regions of the sidewalls. While rolling, these surfaces vibrate too, producing the exterior sound, but this is not the only phenomenon to be taken into consideration here. There are also phenomena related to the footprint determined by the dead weight of the vehicle, the most important of which is the rolling loss of energy. This is the phenomenon mostly responsible for the consumption of fuel. The two main surfaces of the tire are connected by design, and this connection is provided by the tire bead, whose reinforcement is instrumental in the so-called phenomenon of flat spotting. However, in order to properly understand the flat spotting phenomenon we need to take into consideration the fact that the two surfaces of the tire are also naturally connected by the material structure of the tire itself, whose main physical component is the rubber. It is through this material structure that the dead weight and its variations are transmitted to the footprint. It is also through this material structure that the high frequency vibrations of the internal surface of the tire cavity are transmitted to the environment. By the cycling thus induced, this material structure produces heat which in the short term is dissipated, but in the long run contributes to the aging of the tire rubber.
     These are, in broad lines, the phenomena related to a ‘working’ tire, selected starting from the criterion that the tire communicates with the environment through the surface of its physical structure. When talking here of ’surface’ we have in mind mainly its geometrical definition, and a certain way to introduce the physics in this definition. To start with, it is quite obvious that the overall process through which a surface ‘works’ is a deformation process. When a surface is deformed, one can recognize this phenomenon locally, through the variation of the two fundamental forms of the surface, and this is how we came up with the subject matter of the present work: both of them are quadratic forms. The first fundamental form, i.e. the metric of surface, gives the infinitesimal distance around a point of the surface, as measured in the tangent plane in that point. According to the common geometrical wisdom, the second fundamental form represents the curvature of the surface in a point, taking as reference the tangent plane in that point (Struik, 1988). Practically, however, the second fundamental form is the height of the surface above or beneath tangent plane in a point. This is the interpretation to be considered, for instance, when the intimate profile of the road is to be accounted for, or when this profile penetrates the rubber of the tire tread, producing the tiny local cycling spots of the rubber in rolling. The physical properties of a surface are thus to be embedded somehow in the six coefficients of the two fundamental forms, and more importantly, in their variations. In what follows we describe the channels through which this embedding can be done.
     Let’s consider the deformation of a surface in its utmost generality. This can be described by the variation of both fundamental forms of the surface, and by the relation of these variations. Therefore, we have to consider in general the variation of a quadratic form, which can be the first or the second fundamental form, or some other physical or geometrical properties for that matter, and show how this is described in connection with the surface. Let’s denote this quadratic form by

(1)

and assume that it is defined in any point of the surface. This ‘definition’ means that we know the coefficients X, Y, Z as functions of the coordinates on the surface, and also the differentials s¹ and s² of the position vector of the points on surface, which play the part of coordinates in the tangent plane. These differentials are the main tool in problems of the local theory of surfaces. They allow for instance the definition of the curvature matrix (Guggenheimer, 1977): this is the transition matrix between the vector (s¹, s²) from the tangent plane and the components of the variation of the normal vector to surface.
     When there is a variation of this quadratic form, it is described by both the variation of the coefficients: X, Y, Z → X+dX, Y+dY, Z+dZ, and by the second variation of the position vector itself. The total variation of the quadratic form can then be obtained according to the known rules, by differentiating it to obtain

(2)

One can then decide if the variation of the quadratic form is strictly due to the variation of its coefficients, by a simple test: the second part of the equation (2) should be zero. Thus the variation of the quadratic form is strictly due to the variation of the coefficients along the curves from the tangent plane in a point, given by the differential equation

(3)

Along these curves the equation (2) becomes:

(4)

One can say that along these curves the variation of the quadratic form is ‘perceived’ only through the variation of its coefficients. Therefore, along these curves, the quadratic form is constant only if its coefficients are constant. The equation (3) represents a Hamiltonian motion in the tangent plane of a point of surface. If the quadratic form is defined, and its coefficients are constant, the equation (3) can represent two harmonic oscillators on the surface. Indeed, assuming that we discovered a time parameter with respect to which we can characterize the motion continuously, the differential equation (3) can be written in the form of a Hamiltonian system in the ‘phase plane’ of coordinates x ≡ s¹ and y ≡ s²:

(5)

Now, either by direct exponentiation or by finding the second order differential equation for each component, one can see that indeed, we have to deal with two harmonic oscillators having the same frequency. We choose the exponentiation, because it makes more obvious the fact that the quadratic form (1) is both the generator of motion and is conserved along motion. Indeed, denoting ω² ≡ XZ – Y², the solution by exponentiation is

(6)

where E is 2×2 identity matrix, and the index ʻ0’ marks the ‘initial conditions’ at t = 0. One can directly verify the equality

(7)

which is the clear expresion of the conservation of the generator along the motion it generates.
     If the coefficients are not constants, but vary with the point of surface, in order to integrate equation (3) they must satisfy some integrability conditions, amounting to the fact that they can be expressed as functions of the local coordinates in the tangent plane, or that these coordinates can be expressed with respect to them or their variation. In these integrability conditions it is implicitly understood that the differentials ds¹ and ds², i.e. the second order differentials with respect to position on surface, are taken as fundamental, while the first order differentials s¹ and s², which are fundamental in the regular theory of surfaces, should be expressed with respect the differentials of the second order. Practically the integrability condition amounts to the vanishing of the exterior differential of the left hand side of equation (3):

(8)

Then, by one of the Cartan lemmas one has

(9)

where λ, μ, ν are three external parameters, the entries of a matrix ‘assisting’ in integrability. Therefore, assuming that the matrix here is nonsingular, s¹ and s² must satisfy the system of differential equations

(10)

Performing the matrix multiplication, we get

(11)

The elementary ‘second order’ area – the so-called symplectic form – is given by equation

(12)

This quadratic form is algebraically conjugated to the variation due to coefficients of the original quadratic form (1)

(13)

and to the quadratic form introduced by the matrix ‘assisting’ in integrability:

(14)

It is this quadratic form that accounts for the physical conditions determining the integrability.
     Now, in order to solve (11) the matrix of evolution is essential. It can be written as

(15)

where by ω_1,2,3 we denoted the differential forms

(16)

In cases where the quadratic forms (13) and (14) are algebraically apolar, the matrix of evolution (15) reduces to

(17)

The parameters λ, μ, ν may represent new conditions of a geometrical or physical nature, affecting the deformation process of the surface. These conditions may be external (for instance the road surface) or internal to the surface (air pressure variation), reflecting its geometry (the first and second fundamental forms) or physics (for instance the deformation matrix, or the material condition of the tire).

Conclusions

Two main points are worth fixing in mind, as a conclusion of the previous exercise.
     First, is the fact that the frequency of vibration of the tire is a surface phenomenon, and needs to be treated as such. The usual wisdom puts first a simple model (Kelvin-Voigt or Maxwell) describing it together with the material underneath, in a time that has nothing to do with the real phenomenon, and then trying to fit experimental data to results. These are then improved by complicating the things with series of simple models. This philosophy is mainly entertained, in the prediction ‘industry’, by the preexisting commercial calculational instruments. It has, however an essential shortcumming: the frequency model is one-dimensional, and there is no way to take into consideration the complicate structure of the spectrum which is mainly a statistical process in a real time. In other words, the phase space associated to the classical constitutive models is actually four-dimensional, not two-dimensional, as the usual wisdom holds true. 
     Secondly, the variations of a quadratic form, be it the first or the second fundamental form, needs an external quadratic form, ‘assisting’ in integrability, in order to clarify physically the mechanics of deformation of the surface. The integrability is not just a mathematical process, but mainly a physical one. We think that, in order to understand this point, an example will do best. Consider the tread of the rolling tire on the rough surface of the road. First we have a stationary deformation due to the load of vehicle. This is a first-order process of deformation, according to which we need to ‘update’ the first and second fundamental forms of the tire surfaces snd thus describe the tire footprint shape. The road surface can be modeled by such a quadratic form which, naturally, is external to tire tread surface. Then the coefficients of this quadratic form go, in the manner shown above, into the variation of the first and second fundamental forms of the tire tread from the footprint region, thus imprinting a statistical variation of the frequency of vibration of the tire. This statistical variation is characteristic to the road surface.

References

Guggenheimer, H. W. (1977): Differential Geometry, Dover Publications, New York
Struik, D. J. (1988): Lectures on Classical Differential Geometry, Dover Publications, New York