Automobiles - Math Matters - Maplesoft

Math Matters


The modern automobile is a complex collection of mechanical, chemical, electrical, hydraulic, and other types of physical systems, all of which have strong connections to mathematics.

During the 19th century, scientists developed ways to mathematically model energy, and through these models they were able to build the practical engines that led to the modern car.

Otto (4 stroke) Cycle:

Nikolaus Otto (German, 1832-1891), a scientist, developed a version of the internal combustion engine that was efficient enough for practical automobiles. Today, his Otto thermodynamic cycle is the scientific foundation for most gas-powered cars.

1 » 2 Intake stroke (isentropic compression)
2 » 3 Compression stroke (constant volume heating)
3 » 4 Power stroke (isentropic expansion)
4 » 1 Exhaust stroke (constant volume cooling)

The maximum efficiency for engines that use this cycle is given by:
where, is the compression ratio and is the heat capacity ratio.

Gottlieb Daimler
(German, 1834-1900)
transformed the Otto cycle idea to practical vehicles, including the world’s first motorcycle and the first generation of automobiles.
Rudolph Diesel
(German, 1858-1913)
invented the Diesel engine that applied a different thermodynamic cycle (and fuel) than the Otto cycle.
Lord Kelvin
(born William Thomson, Scottish, 1824-1907)
established the mathematical and scientific foundation for energy physics (Laws of Thermodynamics).