Train services for trapezoidal and quadrilateral curves

A trapezoidal speed-time curve (acceleration, free run, braking) represents main-line (long-distance) service, while a quadrilateral curve (acceleration, coasting, braking) represents urban and suburban service.

Trapezoidal curve: main-line service

• Stations are far apart (often more than 10 km), so the train runs most of the time at constant (crest) speed.
• The acceleration and braking periods are short compared with the free-run period.
• So the actual curve is closely approximated by three straight lines: constant acceleration, constant speed and constant retardation. Together with the time axis these form a trapezium.
• It is used for main-line and long-distance express trains.

Quadrilateral curve: urban and suburban service

• Stops are close together (about 1 km for urban and 1–8 km for suburban), so the train never runs long at constant speed.
• After the initial constant acceleration, the speed rises slowly along the motor curve; then power is switched off, the train coasts, and finally it brakes.
• In the simplified curve, the speed-curve running is merged into the acceleration line, so there are three straight lines: acceleration, coasting and braking. Together with the time axis these form a quadrilateral.
• Used for city and suburban trains, metro and tram services, where high acceleration and braking matter more than top speed.

 Speed                       Speed
  |   ________                |    /\
  |  /        \               |   /  ``--._
  | /          \              |  /         \
  |/            \             | /           \
  +--------------\-- t        +/-------------\-- t
  Trapezoidal (main line)     Quadrilateral (urban)