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Non-Circular wheels, upon initial impression, should not work. On the flat roads that modern humans use, circular wheels are by far the most smooth and fast wheels. When looking deeper, it becomes obvious why this is the case. Flat roads are easy to walk on, and require little adaptation to implement circular wheels. However, it is important to recognize that non-circular wheels are certainly usable on modern day vehicles, they simply require a road that is specially designed for that shape.
History and Background
The ancient Egyptians furthered these experiments by introducing a roller system, demonstrating an early understanding of the mechanics of load movement. These rollers were capable of moving large amounts of weight, as shown with the Great Pyramids, whose stones were likely moved by such tools. As history progressed, various cultures continued to experiment with non-circular wheel designs.These innovations were often driven by the practical need to traverse challenging terrains and transport heavy loads. (Solved, 2014)
Fast forward to the modern era, and the legacy of non-circular wheels endures in specialized applications. The historical experimentation with square, polygonal, or cylindrical wheel shapes has influenced contemporary off-road vehicle design. The adaptability of non-circular wheels to diverse terrains, inspired by ancient engineering principles, is evident in off-road trucks, military vehicles, and all-terrain vehicles. From ancient experiments in Mesopotamia and Egypt to the modern applications in off-road vehicles, the evolution of non-circular wheels reflects a persistent human quest for efficient transportation solutions across varied landscapes. (How Off-Road Tires Work, n.d.)
Explanation of Mathematics
Roads for non-circular wheels have an intricate relationship between the wheel's geometry, mechanics, and the surface it traverses. The mathematical process to design such roads involves an in-depth understanding of these factors. For the non circular wheel, one must first choose a non-circular shape. There are a myriad of shapes that can be chosen, as long as their equation can be expressed in parametric equations. These parametric equations define how the wheel moves and interacts with the road. For example, if the wheel is square, the equations might describe how the x and y coordinates change as the wheel rotates. (Roads and Wheels, n.d.) (Hall & Wagon, 2006)
It is also important to understand the contact points between the non-circular wheel and the road surface. Mathematically modeling how the shape of the wheel influences these contact points is what allows us to understand the shape of the road that can handle these wheels. The mathematical reasoning behind these shapes depends on the type of shape. The non-circular wheel is made up of two curves A and B, which can form a wheel-road couple if B can roll without slipping on A. This is shown through a fixed point on the plane of A, which serves as the wheel hub. If A has a linear trajectory in the fixed plane, then B can roll without slipping on A. It is basically the movement of a plane over a fixed plane, the base of which is A, the rolling curve of which is B, and a roulette of which is linear. (Roads and Wheels, n.d.) (Hall & Wagon, 2006)
Essentially, the motion of a non-circular wheel on a paired road is made up of two parts: the round part and the flat part. For most modern people, the flat portion is the paved roads that are driven on. The round part is the wheel itself, a complete circle. However, if the wheel was to be square, how would that change the road? The most important factor is that the wheel has four flat surfaces, meaning that the road will need to be round. To find that road a hub will need to be designated, which is the point at which the wheel rotates. For almost every wheel ever, this hub is at the center of the wheel in order to simplify the process. Should a wheel ever have the hub not on the center of the wheel, the resulting road will become increasingly unpredictable, to the point that a road may not be possible. The contact point, the point on the wheel that is contacting the ground, will always be on a straight line between the ground and the hub. If the wheel is spun in a full rotation, tracing that point that is hanging towards the ground, the paired road can be found. There will be no slipping, and the car should glide as evenly on this new road as a round wheel on a modern road.
Significance and Application
The significance of non-circular wheels lies in their ability to navigate challenging terrains and overcome obstacles that circular wheels might struggle with. These specialized roads have historical roots, dating back to ancient civilizations, where the ingenuity of non-circular wheel designs was crucial for transportation across varied landscapes. (Dempster, 2009)
In the modern era, the applications of roads for non-circular wheels extend across diverse fields. One of the notable applications is in off-road vehicles. Non-circular wheels, shaped to optimize traction and stability, are employed in off-road trucks, all-terrain vehicles, and military vehicles. These specialized wheels enhance mobility in rugged environments, providing a crucial advantage in scenarios where conventional circular wheels may falter. (How Off-Road Tires Work, n.d.)
Moreover, the principles derived from the historical development of non-circular wheels find contemporary application in space exploration. Lunar rovers, designed for planetary exploration, often incorporate wheels with irregular shapes to navigate the uneven and rocky surfaces of celestial bodies. The adaptability of non-circular wheels becomes particularly valuable in extraterrestrial environments where traditional road infrastructure is nonexistent. (Smith, 2017)
In robotics, the concept of non-circular wheels has been embraced for navigating complex terrains. Robots designed for search and rescue missions, exploration of disaster-stricken areas, or even planetary exploration draw inspiration from ancient wheel designs. The ability to maneuver through challenging landscapes is critical in scenarios where circular wheels might limit mobility. (Knowledge | Types of Robot Wheels, n.d.)
Works Cited
Dempster, T. (2009, June 17). Chinese man reinvents the wheel. Reuters. Retrieved October 31, 2023, from https://www.reuters.com/article/us-china-bicycle/chinese-man-reinvents-the-wheel-idUSTRE55G1GB20090617
Hall, L., & Wagon, S. (2006, November 27). Roads and Wheels. MST.edu. Retrieved October 31, 2023, from https://web.mst.edu/~lmhall/Personal/RoadsWheels/RoadsWheels.pdf
How Off-Road Tires Work. (n.d.). How Off-Road Tires Work When Street Driving. Retrieved October 31, 2023, from https://www.utires.com/articles/how-off-road-tires-work/
Knowledge | Types of Robot Wheels. (n.d.). Robot Platform. Retrieved October 31, 2023, from http://www.robotplatform.com/knowledge/Classification_of_Robots/Types_of_robot_wheels.html
Roads and wheels. (n.d.). MATHCURVE.COM. Retrieved October 31, 2023, from https://mathcurve.com/courbes2d.gb/engrenage/engrenage2.shtml
Smith, N. (2017, October 26). Reinventing the Wheel. NASA. Retrieved October 31, 2023, from https://www3.nasa.gov/specials/wheels/
Solved! How Ancient Egyptians Moved Massive Pyramid Stones. (2014, May 1). NBC News. Retrieved October 31, 2023, from https://www.nbcnews.com/science/science-news/solved-how-ancient-egyptians-moved-massive-pyramid-stones-n95171
Woolley, L. (n.d.). sledge (?); chariot (?). British Museum. Retrieved October 31, 2023, from https://www.britishmuseum.org/collection/object/W_1928-1010-2