Bicycle mechanics

Summary: Bicycle geometry impacts performance, aerodynamics, efficiency, and stability.

The first bicycles of the early nineteenth century were simple designs of wooden frames and metal hoops for wheels. Though these early bicycles were propelled by feet pushing along the ground, soon pedals were added to the front axle allowing the rider to drive the front wheel for locomotion. It was not until the late 1880s when the first chain-driven bicycle was introduced, thereby separating the axles from the primary point of locomotion and overcoming problems with handling, steering, and weight distribution. This explosive decade of development also saw the first pneumatic tires, gearing, and coaster brakes, the latter allowing the rider to brake by pedaling backwards. Another series of innovations a century later was spurred by an explosion in frame design and fabrication techniques including the use of better materials such as aluminum, titanium, and, eventually, carbon fiber.

Bicycles serve as the primary means of transportation in several cultures, especially in southeast Asia. European communities are also known for embracing the bicycle as a legitimate form of transportation.

Mechanics

Bicycles have two in-line wheels and are driven by pedaling. The wheels each spin on axles rotating on bearing surfaces and most commonly support the rims via tension spokes. Pneumatic tires are secured to the outer surface of the rims to provide the primary contact with the ground. The centrally located bottom bracket is the rotating connection point of the pedals. Power is transferred to the rear wheel via a chain. Brakes are usually found on both wheels; most bicycles’ brakes squeeze braking pads on the rim surface to create friction and slow the wheel and, as a result, the bicycle. Many newer mountain bicycles use disc brakes for increased stopping power. The rider sits on a saddle atop the bicycle and leans forward on handle bars, which provide support and the ability to steer. Many bicycles, especially mountain bicycles, have shock absorbers built into the front fork to provide cushioning over rough terrain. Some bicycles also feature rear suspension, which allows the rear triangle of the frame to rotate and further absorb the impacts of uneven terrain.

Gears (chain rings on the bottom bracket, a cassette on the rear axle) allow the rider to alter the ratio of pedal rotation to wheel rotation in order to go faster or slower. The gear ratio is determined by the diameter of the chain ring divided by the diameter of the rear cog. Since the number of teeth is proportional to diameter, tooth count is more typically used. For example, a 39-tooth chain ring used with a 15-tooth cog produces a gear ratio of

that is, one revolution of the pedals produces 2.6 revolutions of the rear wheel. A standard 700C wheel (70 centimeters in diameter) will travel 0.7 π=2.2 meters (7.2 feet) along the road with each revolution. Thus, a single rotation of the pedals produces

Speed and distance traveled can then be calculated based upon the rider’s revolutions per minute.

Types of Bicycles

Reflecting their wide versatility, bicycles come in a multitude of different styles. One of the most common is the road bicycle, which is distinguished by thin tires; a drop-style handlebar; and a stiff, light frame. Road bicycles are designed for fast travel over smoother road surfaces. The other most common bicycle is the mountain bicycle, which features wide, knobby tires designed for increased traction in the dirt; flat handlebars for a more upright position; and a wide range of gears, including very low gears for steep climbing. Most mountain bicycles have a front suspension fork and many feature a rear suspension as well.

Cyclocross bikes are closely related to road bikes but have slightly wider tires and lower gears for racing on cyclocross race courses or for exploring gravel roads. Comfort bicycles, commuters, and hybrids are usually compromises between the stiffness of a road bicycle and the comfort of a mountain bicycle; these bicycles’ lower prices are often aimed at entry-level riders who are seeking practicality over high performance. Bicycle motocross (BMX) bicycles are single speed (no gears) with smaller, wider tires designed for racing on BMX courses. There are additional niche bicycles for special purposes such as time trialing, track racing, snow riding, and touring. Though most people cannot imagine a bicycle having anything but circular wheels, since that shape travels smoothly on flat roads, mathematicians have modeled as well as built wheels with other shapes, such as squares, three-leaf clovers, star-like shapes, and triangles. They found that a square-wheeled bike will travel smoothly on a road made of inverted catenaries, and each of the other types has at least one solution as well. A differential equation can be used to generally solve the problem of noncircular wheels.

Racing and Performance

Bicycle racing is a popular sport with a surprisingly active history. Near the end of the nineteenth century, bicycle racing was one of the most popular sports, drawing huge crowds of spectators across Europe and the United States. Today, bicycle racing is popular worldwide but has a stronger European following. Why certain cyclists are more successful than others can be analyzed in part using mathematics. Average riding speed, efficiency, and power are all calculated metrics useful for assessing performance. Seven-time Tour de France winner Lance Armstrong has been studied and modeled extensively throughout his career. American cyclist Greg LeMond overcame a 58-second deficit and won the 1989 Tour de France by 8 seconds over French favorite Laurent Fignon, which is generally attributed by most to the innovative aerodynamic handlebars he used in the last stage. Companies now routinely use mathematical modeling for cycling equipment, as well as to test aerodynamics and other essential properties, and teams use optimization strategies to construct bicycles within the sport’s guidelines, since seconds can make the difference between victory and second place.

For the average rider as well as for professionals, the geometry of a bicycle plays a large role in its overall performance and stability. For example, the distance between the axles and the angle the front fork makes with respect to the ground are both important, according to bicycle makers. Some mathematicians have explored stability issues. In study released in 2007, researchers investigated and dynamically modeled 25 parameters believed to be important, with the goal of being able to construct bicycles targeted toward riders’ specific needs.

Bibliography

Burke, Ed. High-Tech Cycling. 2nd ed. Champaign, IL: Human Kinetics, 2003.

Hall, Leon, and Stan Wagon. “Roads and Wheels.” Mathematics Magazine 65, no. 5 (1992).

Herlihy, David. Bicycle: The History. New Haven, CT: Yale University Press. 2006.

McGann, Bill. The Story of the Tour de France. Indianapolis, IN: Dog Ear Publishing, 2006.

Peveler, Will. The Complete Book of Road Cycling and Racing. New York: McGraw-Hill, 2008.