Train Dynamics

…or “What do all those parameters in the file header do?” — a very quick, basic run of through train physics.

On this page…

  1. 1 Parameters
    1. 1.1  Weight
    2. 1.2  Tractive Effort
    3. 1.3  Power
    4. 1.4  Maximum speed (Max)
    5. 1.5  Gradients
  2. 2 Adhesion
  3. 3 Links

1 Parameters

1.1  Weight

The Weight parameter actually specifies the Mass (M) of each vehicle. The mass affects how difficult it is to accelerate or brake the train; The weight is obtained by multiplying the mass by the gravitational acceleration (g=9.8 m/s2), and affects how hard it is to pull the train up a hill.

1.2  Tractive Effort

Tractive Effort (TE) is the force that a locomotive exerts on the rail to accelerate the train. It is measured in force units (Newtons or pounds-force). In a steam locomotive the force comes from steam pressing on a piston to turn the wheel, via a connecting rod; in an electric or diesel-electric it comes from the torque of a motor applied to the axle. The train accelerates with an acceleration (a) according to Newton’s Second Law:

TE = M.a

Rail3D uses the te parameter to represent the te that a loco can exert when starting from rest. The te of a steam loco is essentially the same at any speed (as long as the boiler can generate steam fast enough to supply it to the cylinders), but other types of loco have motors that supply a constant power, so the te decreases as the speed increases.

1.3  Power

Power is the rate of doing work. In railway terms, it is the product of tractive effort and speed, and is measured in kilowatts or horsepower.

P = TE.v

If we have a locomotive with ideal motors that produce a constant power output P, then the equation of motion becomes

a = P/(M.v)

Thus the train keeps on accelerating as the speed increases, but the rate of acceleration becomes smaller and smaller. Conversely, if you switch the motor off (set P = 0), the train would keep going at constant speed.

1.4  Maximum speed (Max)

In real life there are various factors that slow the train down, in particular mechanical losses in wheels and bearings, and air resistance. These losses act as a force decelerating the train.

M.a = P/v - Resistance

Since the air resistance, in particular, increases strongly with speed, there will be a certain speed on level track at which the acceleration becomes zero. This is the train’s effective maximum speed.

In real life, the train is usually restricted to a lower maximum speed, determined by factors like safe braking distance and collision resistance. This permitted speed is defined by the Max parameter in Rail3D.

1.5  Gradients

When the train is on a gradient, one component of its weight acts along the track, giving the train a tendency to accelerate down the gradient. For the small gradients normally encountered on railway lines, this component is approximately the weight M.g multiplied by the slope s (expressed as a fraction). So we can add a new factor to the equation:

M.a = P/v - Resistance - M.g.s

Notice that the train will “stall” when

P/v = M.g.s

s = P/(M.g.v)

For example, if a 2000kW electric locomotive is pulling a 1000 tonne train at 36km/h (= 10m/s), it can manage a maximum slope of 0.02 (1 in 50).

2 Adhesion

The force the train can apply to the rails when braking or accelerating is limited by the friction between wheel and rail (except for rack railways…). When two surfaces are pressed together by a force R, the frictional force preventing the surfaces from slipping is given by µR, where µ is a coefficient that depends on the type of surfaces we are using. For steel-on-steel, µ can theoretically be up to 0.4, but 0.2 is more like what you can realistically expect under railway conditions.

For our loco, if all wheels are powered, the total force between wheel and rail is Mg, so the maximum te the loco can apply is limited to µMg.

3 Links