Elevators
Elevators are vital mechanisms for transporting people and cargo vertically, and mathematics plays a crucial role in their design and function. Key mathematical concepts involved include dynamics, vibrations, and probability models, which help quantify aspects such as maximum speed and waiting time. For instance, hydraulic elevators operate based on Pascal's Law, allowing increased pressure in a confined fluid to generate force, while roped elevators utilize the conservation of energy through a counterweight system to facilitate movement.
In modern structures with multiple elevators, sophisticated algorithms optimize dispatching to minimize wait times and increase efficiency, responding to real-time passenger demands. The idea of a space elevator—a proposed structure designed for transporting objects from Earth's surface to geostationary orbit—hinges on complex engineering and mathematical principles. Challenges such as cable material strength, environmental hazards, and passenger safety are at the forefront of ongoing research, including initiatives by organizations like NASA. Overall, the intersection of elevators and mathematics not only enhances everyday transportation but also inspires futuristic engineering concepts.
Elevators and mathematics
Summary: Mathematics is used to quantify aspects such as the maximum speed and distance range of elevators as well as model vibration and optimize traffic flow.
An elevator is a mechanism for vertical transport of persons or cargo. Mathematics is used to quantify aspects such as the maximum speed and distance range of elevators, determined by their purpose, such as lifting passengers, cars, or aircraft. Applied mathematical models focus on the dynamics and vibrations within different types of elevator mechanisms, such as hydraulic or rope systems. Mathematicians also investigate questions related to aspects such as waiting time, using probability models. Systems of multiple elevators are modeled as high-dimensional spaces using dynamical systems. The number of passengers in an elevator system constantly changes, making an optimal policy for what is referred to as an “elevator group control” mathematically interesting. At the end of the nineteenth century, scientist Konstantin Tsiolkovsky conceived of a space elevator. He was self-taught and worked as a mathematics teacher.
Hydraulic Elevators
The main concept related to why hydraulic elevators work is Pascal’s Law, stating that when the pressure increases anywhere in a confined fluid, it equally increases everywhere. This, together with the fact that pressure (P) is equal to force (F) per unit area (A), can be exploited for an advantage of force. The elevator car stands on top of a piston ending in a wide shaft filled with oil, connected to a narrow shaft with oil. When a pump increases pressure in the narrow shaft, by applying a relatively small force, the equal pressure applies to the floor of the cabin, producing higher force because of the larger area: P1= P2, and
Hydraulic elevators are only used in relatively low buildings since the piston has to be as tall as the building to extend to the top floor but fully fit under the building when the elevator is on the ground floor. Digging as deep as a skyscraper is high to install an elevator is impractical. These elevators are mostly used for heavy loads in places such as car mechanic shops.
Roped Elevators
A mathematically interesting concept related to roped elevators is the conservation of energy. A roped elevator consists of two ends of a steel cable going around a pulley attached at the top, called a “sheave.” The elevator car is attached to one end of the cable, and the counterweight, which weighs about the same, is attached to the other end.
When the elevator car is at the bottom of the shaft, the counterweight is at the top, and its potential energy converts to force, helping move the elevator car up. When the elevator car is higher than the counterweight, their roles are reversed. This way, it takes very little additional force to make the sheave rotate and the elevator car move up and down.
Logistics
In modern buildings with multiple elevators, computer programs determine how to dispatch elevators to minimize wait time and to save energy. For example, a sensor may detect that an elevator is near capacity and will not stop it for any additional passengers. An elevator going down may not open its doors for people who want to go up, avoiding carrying them back and forth. More sophisticated elevator software can take into account typical traffic patterns, directing elevators to the busiest floors.
Space Elevator
A space elevator is a structure for escaping the gravity well of a planet, transporting objects between the surface and a geostationary orbit. This proposed structure would consist of a large satellite counterweight in orbit and a cable connecting it to the ground. The inertia of the counterweight rotating around the planet will balance the gravitational pull on the cable, keeping the cable taut. The National Aeronautics and Space Administration (NASA) is working on several efforts related to construction of a space elevator, including an annual engineering competition. The technological problems include avoiding meteorites and dangerous atmospheric weather systems, developing materials strong enough for the cable, designing the counterweight, protecting passengers from radiation, and powering the elevator cars. In 2008, Japan announced plans to build a space elevator in the immediate future. Space elevators have frequently appeared in science fiction since the early twentieth century.
Bibliography
Bangash, M. Y. H., and T. Bangash. Lifts, Elevators, Escalators and Moving Walkways/Travelators. Leiden, The Netherlands: Taylor and Francis, 2007.
Van Pelt, Michel. Space Tethers and Space Elevators. New York: Copernicus Books, 2009.
Wuffle, A. “The Pure Theory of Elevators.” Mathematics Magazine 55, no. 1 (January 1982).