Question: What part of the train moves backwards when the train moves forward?

Question: What part of the train moves backwards when the train moves forward?

Answer: Suppose the center of mass of the train moves in the direction of longitudinal feed a distance xtrain. According on the non-slip rolling condition, the movement is governed by:

xtrain = rwheel·θ

being θ the angle rotated by the wheel in its feed and rwheel the radius of the wheel at the point of contact.

Taking as a reference system a static point on the track, and referring exclusively to the x-axis, any point that is part of the wheel will undergo a movement composed of linear feed and angular rotation. Analyzing the case of the points of the rolling zone remains:

xwheel = xtrain - rwheel·sin(θ) = rwheel ·θ - rwheel·sin(θ)

xwheel = rwheel (θ - sin(θ))

In this expression, it can be clearly seen that (θ - sin(θ)) will always be greater than or equal to zero, so that xwheel ≥ 0.

Rail vehicles are self-guided, so the wheel has a flange on the inside that interacts with the rail to provide that guidance. Now let's look at the points that are part of the flange and whose rflange > rwheel.

Developing in the same way as the previous case,

xflange = xtrain - rflange·sin(θ) = rwheel·θ - rflange·sin(θ)

Therefore, xflange will be negative whenever the condition is met:

rflange·sin(θ) > rwheel ·θ

inequation that can only be solved by numeric methods.

To illustrate these results, you will solve the problem with common metal rolling data in metropolitans. Being rwheel 840/2 mm, and the thickness of the flange 28 mm, which makes rflange  896/2 mm.

The results obtained are:

The flange moves in the opposite direction to the train while the wheel rotates an angle of 35.42o and the train travels a distance of 259.66 mm, the maximum reverse displacement occurs at approximately 20 degrees. The figure below presents the above results.

I hope you've aroused interest!

Best regards. CR.