Initially, heavy loads were moved by putting several straight, round logs underneath the object and providing muscle power to shove the object over them. The logs would roll naturally as the object moved, thus carrying the load. This required several logs and a repeated transfer of the rearmost log to the front as the load was propelled forward.
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This didn’t work well on hills or rough ground, and it typically took a lot of muscle, which meant a lot of people and/or animals. At some point, the wheel and the axle were invented, but muscle power still provided the force. Big, strong animals typically were used for this, and as wheel and axle technology improved, so did steering and suspension. The steam engine brought a new source of power, and the train was born, but it needed tracks on which to run.
As for the personal vehicle, Nicholas Otto typically is credited with the invention of the gasoline internal combustion engine. When he got it right, he built a motorcycle around it. Otto’s engine used reciprocating piston motion to turn a crank that provided torque to do work. Pulleys, belts, drive and driven pulleys, axles, brakes and methods of steering continually improved over time, and in the natural course of engineering, drive shaft technology has evolved repeatedly. Some of the early vehicles used a torque tube, which was a pipe-like affair that not only provided protection for the spinning drive shaft, but actually transmitted the pushing power of the drive axle to the vehicle frame.
The engine and transaxle that provide the driving power is sitting on rubber mounts connected to the frame, but the wheels and the final drive have, for decades, been unsprung weight. What that means is that the wheels and the final drive assembly (the rear axle) are riding directly on the ground and have to follow the bumps and whatnot while the body and frame ride smoothly above the springs. Because the axle is mounted to the body with hinge-like rods and beams that push the vehicle as the wheels do their work, the torque tube has pretty much disappeared and the propeller shaft (driveshaft) spins in the open air.
The driveshaft is relatively stationary on the transmission end as it transmits power to the final drive. But during the course of driving, it has to continually provide torque to the drive axle while absorbing angle changes related to torque, load and terrain. The changes aren’t that steep in most cases, but it goes without saying that the propeller shaft not only has to continue providing torque during those angle changes
But the real-world dynamics of suspension geometry demands that the shaft gets longer and shorter as the axle travels up and down. So there is a flexible joint on each end of the shaft, which typically is connected to the splined output shaft on the transmission with a long sleeve that is smooth on the outside (for sealing) and splined to match the tranny output on the inside. These splines are heavily lubricated so they’ll move freely to lengthen and shorten the shaft, necessarily accommodating geometry changes that come with axle movement.