Everything by Nathaniel M

Rotational kinetic energy, sustained and contained within a single object or system without need for constant travel, can be either a boon or a nuisance depending on the application.

This project focuses on the transfer of kinetic energy between rotational and linear, and on the devices used to do this: primarily, the piston.

Simple harmonic motion, in a nutshell, is when a moving system will return to its original state over and over in a repeating pattern, and will continue to do so stably unless the outside forces acting upon the system change.

In the case of pistons, rotational kinetic energy (RKE) is a form of simple harmonic motion (SHM). Pistons are useful because they translate RKE into linear kinetic energy (KE) without needing to change the overall position of the system, because the resulting KE is also SHM.

One major application of this technology lies in making pumps, in which RKE- easy to produce electrically or by hand, and easily scalable between power and time- is converted into KE to move fluids. However, another large application is the combustion engine, in which rapid, unsustained instances of force generate small amounts of KE, that are converted into stable, sustained RKE. Pistons are capable of converting both ways.

They are not, however, perfectly efficient. The rod (which connects the crankshaft to the piston head) never exerts a force on the piston head (or has a force exerted on it) in the same direction as the piston head moves (since the only times they are aligned, the net force is 0). Exactly how large this angle can grow depends on how large the crankshaft is in proportion to the rod, but in general, less than half of the crankshaft's RKE will become KE of the piston head. This is why it is difficult to construct a stable combustion engine with fewer than four pistons: twice as much RKE is required as KE for the same rate of displacement, and when force is only being exerted periodically on the piston heads in one direction, they exert half as much force as they would if bi-directional.

Another quirk of pistons is that though the motion on both ends may be SHM, it is not as "clean" as it would appear. Assuming the motion of the crankshaft remains constant, the piston head's motion will spike during near rotations (when the head is fully extended) and slow when it is contracted, and vice versa if the motion of the piston head forms a perfect sine wave with respect to time.

The derivatives of true sine waves (aka velocity and acceleration) are the same function translated to the side by Pi/2 * T. This graph makes it clearer and clearer as derivatives are taken that something more complex is going on.

Part of this is easy to explain. When the piston head is fully compressed or fully extended, the crankshaft will be either at 0 or Pi radians to its resting position (fully compressed). However, when the wheel is at Pi/2 or 3Pi/2 radians- the halfway point in terms of t (assuming constant rotation speed)- there is a significant offset in the location of the rod-crank joint perpendicular to the line the piston head moves along. As a result, the rod is the hypotenuse of a triangle formed between the rod-crank joint, the rod-piston joint, and the center of rotation. The leg of this triangle determines how high the piston is extended, and as such, it is extended significantly less than half way when the wheel is at its halfway marks. Using the pythagorean theorem, the total piston head offset below its halfway point as a function of the rod and crank lengths would be |rod| - sqrt(|rod|^2 - |crank|^2).

For energy transfers, however, the calculations are MUCH messier. Determining the overall force exerted and energy transferred over an interval of time would likely require a dot product with the result of an integrated other dot product, and they don't add up prettily. Fortunately, a single piston can smoothly translate RKE to KE at a relatively even rate regardless, and an array of four or more mistons can do the same for KE to RKE, so this underlying complexity does not diminish the system's usefulness.

Reflection questions!

Following this project and its explanations, I have a clearer idea about the complexities involved in how smooth the transition of energy is. I had assumed going in that everything would work out relatively nicely; by breaking the rotation of the point where the crankshaft joins the rod into horizontal sine and vertical cosine functions, one could be eliminated and the other would match the side-to-side motion of the piston head. This is clearly not the case, however.

The included graph of position, velocity and acceleration with respect to time was the most helpful resource I found for furthering my understanding.

I'm more interested now in exactly how the energy conversions and calculations would work out- in both this system, and in systems involving moving parts in general. While a large amount of Work = Force dot Direction would be involved, I'm certain by now there's more to it than that.

Sources: images are unhosted, so right click -> view image will give their original sources. Information is from class lectures, Wikipedia, epi-eng.com (information specific to combustion engines), and codecogs.com (various math resources).