EDIT: I changed the title of this question and made this edit based on a conversation with a friend. While I am dealing with mechanical cams the plain fact is that what I have is an oscillation in mechanical form and my friend says oscillations are essentially sine waves. He doesn't know how to answer this question but per his advice I have made this edit to focus this question on how to predict when multiple oscillations will synchronize. The rest of the question I am leaving alone as it still accurately describes the problem.
I am not a mathematician (minimal algebra from 35 years ago as an undergrad is the limit of my math studies) but I am dabbling in some ideas involving two-dimensional elliptical paths and I would like to browse any materials that might deal with relevant concepts but to do that I need to know the right words to look for. I realize this is just the tip of the iceberg but I would at least like to know which iceberg to stand on. :)
EDIT: Consider all orbits invariable with no gravitational issues. I have called this an analogy in the most accurate sense of that word. I chose to use this particular analogy because it would take pages to describe the system as it actually is and furthermore the design is still in evolution. What I am actually working on is a very complex mechanical cam system (this is a hobby of mine) kind of like the one illustrated above, only my "toy" is not a solar system but something far more complicated, yet at its heart it is still a fixed cyclic system with synchronization events between the cams.
I will try and describe what I am asking about by the analogous idea of a planetary system orbiting a central star (sun). We all know that every planet has its own orbit on some kind of elliptical path and that each orbit has a time period for one complete circuit of the sun. Assuming a common orbital plane (or a two-dimensional overhead perspective) at some point in time all of the planets will synchronize so that they form a straight line that passes through the center of each of them and through the sun. At some later point in time all the planets will again synchronize, but this time the line may be at a different tangent to the sun. Now add to this analogy several comets that also form part of the orbital family of the system but which have very different orbital patterns (but which can still be mapped from a two-dimensional overhead perspective).
WHAT I AM MOST INTERESTED IN IS: Identifying the related field/terminology for math that studies the journeys from one synch event to the next, in order to predict these synchronization events. I know such a system could be simulated in accelerated time and by sheer brute force these events could be predicted but I am hoping there are forms of math that will actually shortcut the process and find these predictions more efficiently than having to calculate every little step of every journey.
EDIT: Please do not change the tags without explaining your changes. These may not all be correct (osculating-circles is a guess) but since I am limited to only 5 tags I would like to keep the best choices possible. If you really think a tag needs to be changed please explain why in the comments.
EDIT: I am not asking for actual (references-works), just identifiers for the field/terminology related to this problem. I am pretty good at finding my own references once I know the right terminology to look for. Because of this I removed the refworks tag but that does not mean I will actually object to references being mentioned, it's just they are not necessary to answer the question.
EDIT: I do not think this question is about (conic-sections) per se, it is about the movement of items along repeating/cyclical paths that happen to be elliptical. I would not be surprised if math for circular pathways would work just as well but I am not certain of that.
Looking at your problem from a general perspective, if each individual orbit/cam/function is Periodic then take two of the orbits and find when they overlap using the ratios of their periods. This gives you another period in which they are in synchronization. Compare this to the period between these and a third object. The period in which they will all align is the least common multiple (LCM) of the periods. For instance if object A aligns with object B every 4 seconds and object A aligns with object C every 6 seconds then objects A,B, and C will align twelve seconds after their initial alignment because 12 is the LCM of 4 and 6.
If the objects have a more complex relationship between each other and are either not periodic or their period is dependent on other objects in the system then things get more complex. The principle of superposition still will likely apply. If you can figure out when two objects are in synchronization then the only time that other objects can be in synchronization is when those objects are in sync. Again you can do that part with LCM.
To use LCM in Excel for multiple numbers simply enter all your numbers in one column, and add the formula somewhere else "=LCM(A1:A10)" Where A1:A10 is the range of cells that contain your numbers.
If that doesn't work then there may be some Chaos involved as there is in the real solar system. Look up Chaos theory, period three implies chaos, and Nonlinear dynamics and Chaos.
Since your question is pretty general without giving specific math I just suggested some general strategies for solving it. Comment if you would like to see some of these ideas explored a little deeper.
Terms which might be helpful or might make good tags:
Periodic
Least Common Multiple
Wave interference
Moire Patterns