Trains and mathematics

Summary: Trains and railways present interesting mathematical problems related to force and load, scheduling, and geometry.

Railroads influenced nearly every aspect of nineteenth and early twentieth century U.S. society. Companies building infrastructure for railroads (and railroads themselves) dominated the U.S. economy as more goods and people were transported via rail. Investors clamored to profit from the railway boom, inspiring engineers and mathematicians to improve the technology used in the railway system. As more people traveled by train, punctuality and reliability needed to improve. Time zones in the United States were established primarily because competing rail companies used different standard times for their schedules. In addition, Christophorus Buys-Ballot and others conducted experiments using trains to explore the Doppler effect, named for mathematician and physicist Christian Doppler. At the start of the twenty-first century, wooden and electric railway sets remain popular toys with children of all ages, while railroad enthusiasts design elaborate model train layouts in various scales reflecting the days when towns were centered around train stations.

Locomotives

Locomotives are classified using the Whyte system, named for mechanical engineer Frederick Whyte, which utilizes numbers to describe the wheel arrangement of the engine. For example, a 4-8-4 type locomotive has four wheels in the front, 8 driving wheels in the middle, and 4 wheels in the rear. The capacity of a locomotive depends on the amount of friction the driving wheels have with the track and the weight of the engine over the driving wheels. These quantities are related by the equation F=MW, where f represents the maximum pulling force of the train, M represents the coefficient of friction between the wheels and the track, and W is the portion of the weight of the locomotive over the driving wheels. While this relationship indicates that heavier trains can pull larger loads, more power is needed to move the train, leading to higher fuel costs. Increasing the coefficient of friction gives the train better traction and thus more pulling force, so most locomotives have a sandbox on the front from which sand is sprayed onto the track when the rails are slippery. Though friction is needed to get the train started, reducing M increases efficiency once the train is in motion, lowering operating costs.

Modern diesel-electric locomotives use high-tech designs to achieve more horsepower while reducing engine weight significantly. Equipped with a sophisticated array of sensors, onboard computers, and control systems, twenty-first-century trains maintain their hauling capacity while reducing fuel consumption and emissions. The future may see more magnetic levitation (Maglev) trains, which use magnetic fields to suspend the train above the track. The first commercial Maglev train opened in 1984 in Birmingham, United Kingdom, but ceased operations in 1995 in part because of design problems. A Maglev train in Japan recorded a maximum speed of 581 kilometers per hour (361 miles per hour) in 2003, the highest ever speed for a Maglev transport.

Passengers and Timetables

Commercial trains, whether passenger trains or freight trains, follow carefully written schedules. Composing these intricate timetables is a daunting task. Railways must ensure that trains do not collide on the tracks, and that goods and people are transported in a timely and efficient manner. In 2006, the Netherlands introduced a new railway timetable for all trains and mathematical modeling played a key role in developing the timetable. To determine how a set of trains should be routed through a station, researchers listed all feasible routes through the station for every train. Each combination of a train and a feasible route is represented by a node on a graph. Nodes on this large graph are connected if they belong to the same train or if there is a routing conflict between the train/route combinations. Presenting the scheduling problem in graph form enables sophisticated computer programs to generate a usable timetable. Additional modifications improve the efficiency of the timetable in the case of unexpected delays.

Railway passengers expect trains to be on time and to have sufficient space for a comfortable ride. Timetables can be fine-tuned to meet these customer demands using another type of mathematical modeling called “peak load management.” Consultants work with railways to determine when trains are the most crowded and when passenger demand is highest. Mathematicians quantify the notion of “attractiveness,” a measure of how satisfied a rider on a given train will be as a function of the journey time on the train, the time the passenger would like the train to arrive at its destination, and the actual arrival time. Another constant is added to the equation to determine how much attractiveness is reduced for each minute the actual arrival time differs from the customer’s ideal arrival time. More terms can be added to measure the crowding on the train—overcrowding having a significant impact on attractiveness. Using this model, railways can develop timetables that increase the probability that a customer will ride on an “attractive” train. Further refinements to the model attempt to minimize the chance that a passenger will need to stand while riding.

Track Geometry

Freight yards use combinations of switches, sidings, and turnaround loops to sort railway cars, assembling them into trains bound for various destinations. The fact that trains cannot pass each other on a single track leads to many challenges. The optimal arrangement of freight cars in the most efficient manner is another problem for mathematical modeling, but these fascinating switching systems have inspired mathematicians to investigate interesting questions involving train track layouts and railway switching puzzles.

A switch (also known as a “turnout,” or “point”) is a Y-shaped structure used to split tracks into two lines or to combine two lines into one. The directional nature of a switch makes the dynamics interesting: trains entering at the “top” of the Y will always exit through the bottom branch, but trains entering through the bottom have the option of traveling on the left branch or the right branch. Switches are used to sort cars in freight yards, enable locomotives to move onto a siding to allow a train traveling the opposite direction on the track to pass, and make it possible via a turnaround loop for a train traveling one direction to reverse direction.

How can two trains traveling in opposite directions, say eastbound and westbound, pass one another? If there is a siding long enough to contain one of the trains, the problem is easy. But what if only one car can occupy the siding at a time? Variations on this train-passing puzzle have been around for over a century. The trains can still pass each other through clever use of the siding. The eastbound train leaves its cars behind, moves onto the siding, and waits for the westbound train to pass through. After the eastbound engine emerges from the siding, the westbound train backs through the siding, bringing along one of the eastbound train’s cars and leaving that car on the siding. After the westbound train has pulled forward past the siding, the eastbound train can pick up its car, and the process repeats until the entire eastbound train is through.

Imagine a child playing with a toy railroad. Given a set of switches and plenty of track, how many different layouts can the child make? To determine whether two track layouts are different, the structure is transformed into a graph, with nodes representing lengths of track. Nodes are connected if there is a switch allowing a train to travel from one length of track to another. Layouts are said to be different if their graphs are the same. A child with two switches can make five distinct layouts. Using more switches and combinations of other types of switches, like the three-way pitchfork-shaped variety, even more layouts can be made and counted using mathematics.

Bibliography

England, Angela. “Train Math Lesson Plan.” http://www.suite101.com/content/train-math-lesson-plan-a45144.

Gent, Tim. “Model Trains.” http://plus.maths.org/content/model-trains.

Hayes, Brian. “Trains of Thought.” American Scientist 95, no. 2 (2007).

Kroon, Leo. “Mathematics for Railway Timetabling.” ERCIM News 68 (2007).

Lynch, Roland H. “Locomotives.” Ohio State Engineer 23, no. 5 (1940).

Peterson, Ivars. “Ivars Peterson’s MathTrek: Laying Track.” http://www.maa.org/mathland/mathtrek‗01‗08‗07.html.