Determining railway super-elevation at equilibrium velocity

This post will discuss the physics of the railroad.

In modern railway systems track geometry is such that all railroads are composed of a series of straight lines and curves. The straight lines are called tangents and the curves are of various radii. In order to allow the trains to travel at higher speeds the curves are canted in order to balance the centrifugal force of the train with the x-component of the gravitational force. The measure of this cant is called super-elevation. Depending on the train speed and the radius, each curve will be canted in order to achieve an equilibrium state where these forces are balanced. The transition section of track from the tangent to the curve body of a specific super-elevation, is called a spiral. The higher the super-elevation the longer the spiral.

The mass of the train need not be known, as it will be cancelled out of the equation. Therefore the only two factors that determine the required super-elevation for a particular curve are it’s radius and the speed of the train. The radius for the curve can be accurately determined given the length of a chord and the distance from it to the arc. The following image is exaggerated for clarity, but the actual chord length would be much smaller compared to the curve’s radius in a real world situation.

The following will demonstrate how the radius of this curve can be calculated given the chord length (DE) and the distance to the arc (BC).

let AB = a, BC = b, BE = c, DE = d, AC = r and AE = r.

We begin with the following equations:

(1)    \begin{equation*} a &= r - b \end{equation*}

(2)    \begin{equation*} c &= \frac{d}{2} \end{equation*}

(3)    \begin{equation*} r^2 &= a^2 + c^2 \end{equation*}

Substitute equations 1 and 2 into equation 3 and solve for the r:

     \begin{align*} r^2 &= (r - b)^2 + \frac{d^2}{4} \\ r^2 &= r^2 -2rb + b^2 + \frac{d^2}{4} \\ 2rb &= b^2 + \frac{d^2}{4} \\ r &= \frac{b^2 + \frac{d^2}{4}}{2b} \\ \end{align*}

The distance between the two inside heads of the rail where the wheel flanges sit is called the track gauge (G). The height of the high rail also known, as the super-elevation, is h and the angle of the cant is θ. The following images illustrate the gravitational forces acting on the train when is sits on a super-elevated curve.

Using the diagram above we can derive the following equation for the sine of the angle θ.

     \begin{align*} sin\theta &= \frac{h}{G} \end{align*}

A break down of the gravitational force components is as follows:

The gravitation force acting on an object is equal to the product of the mass and the acceleration due to gravity, approximately 9.8 m/s2 on earth.

     \begin{align*} F_g &= mg \end{align*}

The x-component of this gravitational force can be determined using trigonometry as follows and by substituting in the equation above.

     \begin{align*} sin\theta &= \frac{F_x}{F_g} \\ F_x &= F_g \cdot sin\theta \\ F_x &= mg \cdot sin\theta \end{align*}

The centrifugal force of the train traveling around the curve is calculated using the following formula:

     \begin{align*} F_c &= \frac{mv^2}{r} \end{align*}

At the equilibrium velocity these two forces should be equal in order to ensure safe travel for the train. In order to determine the appropriate super-elevation we equate these two forces and solve for h. It should be immediately obvious how the mass cancels in this equation.

     \begin{align*} F_x &= F_c \\ mg \cdot sin\theta &= \frac{mv^2}{r} \\ g \cdot sin\theta &= \frac{v^2}{r} \\ sin\theta &= \frac{v^2}{rg} \\ \end{align*}

We can now substitute the value for the sine of θ to solve for height of the high rail or super-elevation.

     \begin{align*} \frac{h}{G} &= \frac{v^2}{rg} \\ h &= \frac{Gv^2}{rg} \\ \end{align*}

Depending on the situation, the frequencies of the various types of trains (passenger or freight trains), and their different speeds, the actual super-elevation that is implemented in the real world may be either slightly higher or lower than that of the equilibrium velocity. The term used to describe this is called cant excess and cant deficit.

That concludes this post on railway curve super-elevation. The next article on this topic will discuss the role of the Harsco Mark IV tamper and how it implements these formulas to perform its operations.

Dan Roy, Founder and Creator of Universal Physics.

Category(s): Physics
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