What happens at the Wheel-Rail contact surface area?

Co-ordinate systems & tasks of Wheel/Rail system

Before getting into technicalities it is important to understand the coordinate system. Imagine a right-hand thumb rule with the palm facing downwards, then the index finger shows the longitudinal direction (x-axis), the thumb shows the lateral direction (y-axis) and the middle finger shows the vertical direction (z-axis). There is a total of three coordinate systems that have to be considered while understanding Wheel-Rail effects i.e. Rail fixed, Wheel fixed and the one at the contact surface.

When a wheel is on the rail, It has three following tasks.

i. To provide guidance to the car body in the lateral direction.

ii. To provide support in the vertical direction.

iii.?Driving/braking condition in the longitudinal direction.

Hertzian theory & assumptions

The contact surface area of the wheel and rail is just the size of a thumbnail. Yes, in that little area lots of physics happens. To study a condition where two bodies come in contact, we have a theory called Hertzian theory is given by German physicist Heinrich Hertz. The statement is “If two bodies come into contact at a point with the contact pressure acting perpendicular to the contact plane, as a general rule they will deform producing an elliptic contact area.”

This theory is applicable with the following assumptions.

i.?????????The shapes of both bodies in the vicinity of the contact point can be described by means of at most second-order polynomials (sphere, plane, cylinder, ellipsoid, paraboloid, hyperboloid).

ii. The material of both bodies is homogeneous and isotropic.

iii.?Hooke‘s law is valid.

???? iv.?The deformation is small compared to the bodies’ dimensions.

?????v.?Only normal stresses are present on the contact surface.

How this theory is applicable to Wheel-Rail contact? Are the assumptions valid? Let's discuss that. The first assumption is fulfilled, we can describe the shapes of wheel and rail as above mentioned shaped depending upon their profiles. The second one is also fulfilled as both of them are made of steel. Hooke’s law is only partially fulfilled here because the deformation or wear caused is not directly proportional to the load. The fourth assumption is also partially fulfilled. There are shear stresses present at the contact surface So, which makes the fifth assumption invalid.

But why do we need to follow this theory? What is the outcome? We get to know the contact area, maximum contact pressure, and approach the distant points that are useful for calculating adhesive force and tractive effort needed.

Slip-Adhesion curve

What is a slip? When does it occur? When the longitudinal velocity of a wheel is less than its circumferential velocity, drive slip is observed. And when it's another way around, brake slip is observed. In the condition where both velocities are equal, we get the pure rolling condition. If one thinks that we should achieve pure rolling condition then please have a look at the graph below.

In the above graph, you see Adhesion co-efficient on the y-axis and slip on the x-axis. At the zero value of slip, the adhesion co-efficient is also zero. That means no adhesive force and no forward motion. So, slip is important to set the wheel in motion. If we go on increasing slip we get more adhesion but after some point, it decreases and stays constant. The maximum value of adhesive co-efficient in the first quadrant is the stability limit, that’s the point where Traction Control System (TCS) comes into the picture. Similarly in the third quadrant considering brake slip, the Stability limit is for Wheel Slide Protection (WSP). The value of adhesion we get is reduced when the tracks are wet and is unstable when there is sand present on the tracks. To keep the tracks dry, a sandblaster is used.

Rolling Resistance

From all the wheel resistances I will discuss only the origin of Rolling resistance. At the contact point when plastic and elastic components of pressure are considered, the hysteresis is shifted a bit in the rolling direction. That distance is called longitudinal displacement of the center of the contact patch. And due to this eccentricity, a reaction force is generated opposing the motion. The only way to eliminate this resistive force is to levitate the car body. Oh!! Did I just mention Hyperloop technology?

Wheel-Rail effects

There are the following four effects observed at wheel-rail contact.

? i.?Longitudinal Adhesion force - This force is responsible for making the forward or backward motion of a wheel. It depends on adhesion co-efficient and wheel load.

ii. Lateral Adhesion force - This force comes into the picture when the wheelset enters a curve. It depends on the angle of attack (the angle between wheel and rail in top view.)

iii.?Spin Moment - The spin moment is observed at the contact surface plane. A moment is created which steers the wheel in the direction of the rail. This happens due to the conical wheel profile and as both the wheels are connected with the axle, moments of both wheels are nullified.

iv.?Geometric lateral wheel force - This lateral force plays a vital role in the self-centering ability of rail vehicles. It depends on wheel load and the angle between wheel and rail profile at the point of contact.

The first three effects cause wear to both wheel and rail but the geometric lateral force is not responsible for any wear. The calculation of these forces is done using Kalker Theory.

Does wear affect the self-centering ability of the wheelset? A big YES, wear changes the wheel and rail profile, therefore the profile inclination angle is varied. That’s where Equivalent conicity comes into the picture. The kinematic property of the wheelset is similar to the wheelset with conical tread and the tangent of the cone angle is Equivalent conicity. If I go through the wheel and rail profile there is never a steady contact between them. It changes from flange face to outer tread and from rail gauge corner to railhead. And due to that task of the suspension system is very complex, It has to reduce hunting motion and the vibration caused by track irregularities too.

Now, during your next train journey, you know what is happening down there. Thank you for reading.

Jinay Bhagat