Railroad Operations and Control
Train Dynamics: Resistance and Effort
A fundamental problem in railway engineering is determining the size and number of locomotives required to pull a specific train over a specific route. This requires balancing the power provided by the locomotives (Tractive Effort) against the forces opposing the train's motion (Train Resistance).
Tractive Effort (TE)
Train Resistance ()
The sum of all forces opposing the movement of the train. It consists of inherent resistance (on straight, level track) and incidental resistance (due to grades and curves).
The Davis Equation
The classic method for calculating inherent train resistance (rolling friction, bearing friction, and aerodynamic drag) is the Davis Equation, developed in the 1920s but still conceptually valid (though modernized coefficients are used today).
The general form of the equation is:
Davis Equation Terms
- = Resistance (lbs or Newtons)
- = Constant representing journal (bearing) friction and track rolling resistance (independent of speed).
- = Term representing flange friction and wave action of the rail (proportional to speed ).
- = Term representing aerodynamic drag (proportional to the square of speed ). This becomes the dominant force at high speeds.
Important
Grade Resistance is a massive factor for railways. A grade (rising unit vertically for every units horizontally) adds of resistance per ton of train weight. Curve Resistance also adds drag, typically estimated at per ton per degree of curvature.
Interactive Train Resistance Simulator (Davis Equation)
Visualize how the components of train resistance change with speed. Notice how aerodynamic drag () dominates at higher speeds.
Train Resistance (Davis Equation)
Even a 1% grade dramatically increases total resistance for heavy trains.
Total Resistance at 60 km/h
196 kN
Aero ($CV^2$)36
Flange ($BV$)60
Rolling ($A$)100
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Hauling Capacity
The hauling capacity is the maximum load a locomotive can pull. It is dictated by the maximum Tractive Effort (limited by adhesion or horsepower) minus the resistance of the locomotive itself.
When the required hauling capacity for a desired train weight on a specific grade exceeds the capability of a single locomotive, multiple locomotives (a consist) must be coupled together.
Turnouts and Crossings
To move trains from one track to another, specialized trackwork is required.
Turnout (Switch)
Checklist
- Switch: The movable rails that dictate the route.
- Frog: A massive steel casting that permits the wheel flange of the diverging track to cross the running rail of the straight track. The angle of the frog determines how sharp the turnout is (e.g., a No. 10 frog diverges 1 unit laterally for every 10 units longitudinally).
- Crossing: A track arrangement allowing two tracks to intersect at an angle without merging. Also called a diamond.
Signaling and Train Control
Because trains cannot steer and require immense distances to stop (a heavy freight train at may need to stop), their movement must be strictly controlled to prevent collisions.
Block Signaling
Positive Train Control (PTC)
Tractive Effort vs. Speed
The ability of a locomotive to pull a train is not constant; it is highly dependent on how fast it is moving.
Tractive Effort Curve
Key Takeaways
- Locomotive sizing requires balancing the available Tractive Effort against total Train Resistance.
- Tractive Effort is strictly limited by the adhesion (friction) between the steel wheels and rails.
- The Davis Equation models inherent resistance using three components: constant friction, speed-proportional friction, and aerodynamic drag.
- Aerodynamic drag () becomes the dominant resisting force at high speeds.
- Grade and curve resistance are significant incidental forces added to the inherent resistance.
- Visualizing the Davis equation reveals why high-speed rail requires exponentially more power than standard freight operations.
- At low speeds, bearing and rolling resistance make up the bulk of the drag.
- Block signaling prevents collisions by ensuring a strict spatial separation between trains.
- Positive Train Control (PTC) provides an automated safety overlay to enforce signal compliance and speed limits.
- Train Dynamics involves balancing Tractive Effort (power) against Train Resistance (drag).
- The Davis Equation models inherent resistance (), showing that aerodynamic drag () dominates at high speeds.
- Grade Resistance is the most significant hurdle for heavy freight trains, severely limiting capacity or requiring multiple locomotives.
- Block Signaling is the fundamental safety system ensuring only one train occupies a specific track segment.
- Positive Train Control (PTC) is a modern overlay that automatically enforces speeds and signal authority to prevent human-error accidents.
- Available tractive effort is highest at starting speeds (limited by adhesion) and drops off exponentially as speed increases (limited by engine horsepower).