STUDY OF PRESLIP AS APPLIED TO THE ROLLING OF A WHEEL ON A RAIL

I. R. Konyakhin

(Presented by the scientific seminar of the Department of Strength of Materials)

Displacement of a Stationary Wheel

To study the fatigue wear of the surfaces of wheels and rails, it is necessary to know the magnitude of their preliminary displacement relative to one another.

According to the experiments of N. A. Karpov [1], the value of this microslip as a function of the normal load is expressed by the formula

\[ \delta = \sqrt{\frac{N}{K}}, \tag{[[unclear: formula number]]} \]

where \(\delta\) is the magnitude of the preliminary displacement, \(N\) is the normal load, and \(K\) is a certain coefficient, equal for steel to \(8.3 \cdot 10^8\ \text{kg}/\text{cm}^2\).

In the formula given, the static coefficient of friction is not included; it is incorporated into the value \(K\). We carried out experiments with microdisplacement of specimens on an apparatus that is a certain improvement of the apparatus of A. V. Verkhovsky [2]; the experimental results confirmed the dependence given by N. A. Karpov. In addition, we investigated the relation between the magnitude of the preliminary displacement and the coefficient of static friction \(\mu_s\), by repeating the same experiments at different values of the latter.

The results are given below in the form of a table.

If, from these data, one constructs a graph in the coordinates \(\sqrt{K}\); \(\dfrac{1}{\mu_s}\), then a linear dependence is observed,

\[ \sqrt{K} = \sqrt{R \cdot \frac{1}{\mu_s}}, \]

where \(\sqrt{R}\) is the coefficient of proportionality. Hence

\[ K = \frac{R}{\mu_s^2}. \]

Substituting this into expression 1, we obtain

\[ S_{\mathrm{k}}=\mu_s \sqrt{\frac{N}{R}}, \tag{2} \]

\(S_{\mathrm{k}}\) is the magnitude of the elastic preliminary displacement,
\(R\) is a coefficient depending on the material and characterizing the stiffness of the contact in the tangential direction.

Since the microdisplacement in the present case is an elastic shear deformation of the materials in contact, and the surface roughness in all experiments was approximately the same (7 μm), an attempt should be made to relate the coefficient \(R\) to the shear modulus \(G\), by investigating several materials for this purpose.

In all subsequent experiments the upper specimen used was one and the same, made of hardened steel; the lower specimens were made of different materials; their contact areas were treated with the same emery paper. In the experiments the magnitude of the preliminary displacement \(S_{\mathrm{k}}\) was measured; the coefficient of friction “\(\mu_s\)” was determined, and then the value of \(R\) was found from formula 2.

The table below gives the values of \(R\) and of the shear moduli for certain materials.

From these data the graph in Fig. 1 was constructed.

Fig. 1. Some correlation between the coefficient \(R\) and the shear modulus \(G\).

With some approximation, we shall assume that the graph in Fig. 1 shows a linear dependence between \(G\) and \(R\); then

\[ G=\Psi^2 \cdot R. \tag{3} \]

Here \(\Psi\) is the proportionality coefficient.

As was found from experiments with surfaces having microroughnesses from 1 to 12 microns, the magnitude \(\Psi\) does not depend on roughness.

For all the materials tested it has, on average, the value \(\Psi = 0.02\), deviating from experiment to experiment sometimes by as much as 12% in both directions, and in an indeterminate manner.

From the simultaneous solution of (2) and (3) one obtains

\[ S_k = \Psi \mu_s \sqrt{\frac{N}{C}} . \]

Preliminary Displacement in the Rolling of a Wheel on a Rail

When a wheel is pressed against a rail by a force \(N\), a contact area in the form of an ellipse, with its major axis transverse to the path, is formed at the places where they touch. During motion this area moves along the rail.

In view of the difference in normal pressure at different points of contact, caused by the difference in the heights of the microasperities and by the curvature of the contacting surfaces, the pressure over the microasperities is distributed nonuniformly. Since the microelements of the contact are pressed down differently, the force of static friction on them is also different [3].

Let us take on the rail a stationary area equal to the contact area, and let the wheel move over it. The moving contact ellipse will first touch the stationary ellipse chosen by us, then coincide with it, and then pass off it. Entering a given place, the wheel gradually loads it up to a maximum by the force \(N\), since the greatest pressure occurs at the center of the moving contact area; the wheel then unloads it to zero. At the same time, tangentially to the area of the rail, a tractive force acts, directed relative to the rail head in the direction opposite to the motion.

Since the friction force is directly proportional to the normal force, on this stationary area it, together with the normal load, first increases and then decreases, remaining proportional to the normal load. In view of the validity of Coulomb’s law of friction for each microasperity of the rough surface, the friction on them disappears together with the pressure, and while the moving contact area passes over the stationary one, both the number of acting microasperities and the force of static friction increase gradually and then decrease together with the normal load.

Analogously to this, we carried out experiments on steel-on-steel specimens. During their displacement, a normal load and a displacing force were applied simultaneously in such a way that they increased and decreased while remaining proportional to one another.

One of the experiments was carried out at a constant load \(N = 45\) kg and, for comparison, another at a varying pressure, with the ratio of the forces

\[ \mu_c = \frac{P}{N} = 0.6 . \]

The results of these experiments are shown in the following two tables:

First experiment

Second experiment

If, from the data in the tables, one constructs graphs in the coordinates \(P; S_k\), then in the first case one obtains a curve with its convexity upward, while in the second case the opposite is true. In the second case, with a gradual reduction of both loads, keeping them proportional to one another, i.e., with \(\mu_c = \dfrac{P}{N}=0.6\), the preslip does not occur; the upper specimen is immobile relative to the lower one. The microprotrusions, elastically displaced by the preslip, in the course of the indicated unloading gradually slip one after another and, owing to microfriction, convert their elastic-potential energy into heat. This is also the reason for the loss of energy in the contact when a wheel rolls on a rail. The areas of the graphs correspond to the magnitude of the energy losses in the contact.

Dependence of the magnitude of preslip on the ratio
of the displacing force to the normal load \(\mu_c=\dfrac{P}{N}\)

To determine the magnitude of elastic microfriction from the ratio of the displacing force to the normal load, experiments were conducted on the same specimens. Loading by the normal force and by the displacing force was carried out simultaneously, at their constant ratio \(\mu_c = 0.1;\ 0.2;\ 0.3;\ 0.4;\ 0.5;\ 0.6\), with a coefficient of static friction \(\mu_s=0.67\). The displacement was each time continued until the force reached the value \(N=45\) kg.
The results of the experiments are shown in the table.

The table shows that the greater the ratio \(\dfrac{P}{N}\), the greater the magnitude of the elastic displacement of the contact material. These experiments, as was said, were conducted at a constant coefficient of friction \(\mu_s\), and therefore its influence has not been determined. To clarify this, as a measure of displacement we selected the quantity \(a=\dfrac{\mu_s}{\mu_c}\); this was done on the basis of some additional experiments.

Earlier we had found the coefficient \(R\) as a constant quantity for displacement at constant \(N\). In the present case, however, when we are dealing with a variable load \(N\), the quantity \(R\) does not remain constant.

On the graph, an approximately linear dependence is obtained between the stiffness \(R\) and the displacement coefficient \(a=\dfrac{\mu_s}{\mu_c}\); moreover, independently of \(\mu_s\), the points lie on one and the same straight line—for example, the points marked by the numbers 0.2, 0.4, and 0.58.

From the slope of the straight line of the graph we take

\[ \sqrt{R}=\sqrt{b}\,a, \]

where \(\sqrt{b}\) is the proportionality coefficient; in other words, \(R=b\cdot a^2\).

In the experiments with steel it was obtained, at constant \(N\), that

\[ R=19.7\cdot 10^8\ \text{kg}/\text{cm}^2, \]

and from the last graph, also for steel,

\[ b=10.5\cdot 10^8\ \text{kg}/\text{cm}^2; \]

their ratio is

\[ \frac{R}{b}=1.8. \]

Approximately we take

\[ \frac{R}{b}=2. \]

We assume that the deviation of \(\dfrac{R}{b}\) from 2 occurred because of the difference in the roughness of the contact areas in the experiments of the first and second series.

Thus, we take

\[ R=2b. \]

Fig. 2. Approximate linearity of the dependence of \(\sqrt{R}\) and \(a\).

Earlier we had

\[ S_{\mathrm{k}}=\mu_s \sqrt{\frac{N}{R_c}}; \]

from the last graph

\[ R=a^2\cdot b, \]

substituting the latter expression into the preceding one, we obtain

\[ S_{\mathrm{п}}=\mu_s \sqrt{\frac{N}{a^2 b}}; \]

earlier the coefficient was represented as

\[ \Psi=\frac{G}{R}, \]

therefore

\[ \Psi=\frac{G_c}{2b}, \]

whence

\[ b=\frac{G}{2\Psi}. \]

Replacing, for \(S_{\mathrm{п}}\), the quantity \(b=\dfrac{G}{2\psi}\), we find

\[ S_{\mathrm{п}}=\mu_s \sqrt{\frac{2N\Psi}{G a^2}}. \]

In the last expression we replace \(\alpha = \dfrac{\mu_s}{\mu_c}\), as was adopted earlier; then

\[ S_{\mathrm{p}}=\mu_s \sqrt{\frac{2N\psi^2\mu_c^2}{G\mu_s^2}} \]

or, finally,

\[ S_{\mathrm{p}}=\psi\mu_c \sqrt{\frac{2N}{G}}, \tag{5} \]

Using this formula, one can calculate the magnitude of the preliminary displacement in the rolling of a wheel on a rail in the presence of tractive effort.

REFERENCES

1. A. V. Andreev. Transmission by Friction. Mashgiz, 1963.
2. V. D. Kuznetsov. Physics of the Solid State. Vol. IV, Tomsk, 1947.
3. I. R. Konyakhin. Static microdisplacement of one body over another. Izvestiya TPI, vol. 114, 1964.