One of my classes last quarter at UC Davis was EME171, which is modeling systems through differential equations. It requires using a computer to numerically integrate the equations over time, but it allows us to solve for the state of several systems that don't lend themselves well to the more traditional by-hand methods.
Normally, when an engineering student is given a mechanical system and told to wrote state equations for it, they start drawing force diagrams for each component and use equal and opposite forces to translate force throughout the entire system. This is fine for the little toy systems we used in ENG102 (Dynamics), but tends to break down when you're dealing with extraordinarily complex systems, or even those which have conversions between energy domains (translational, rotational, electrical, hydraulic, etc.), and the student inevitably forgets one of the "equal and opposite" forces, and then everything goes to hell very quickly (the damn bridge, she fall down).
This is where the thesis of EME171 comes in: Bond graphs.
A bond graph is a graphical representation of the modeled system, which is useful because it tracks the both the flow and the force through the system, not just the force like the typical drawing of a system. At first this is a little unintuitive, to be doing an analysis on a system using both the velocity and force on elements, after having spent so much time only considering velocity, or both current and voltage, instead of just voltage. This is done by using nine different elements to represent all energy flow, storage, and conversion.
Bond graph elements:
Newton's third law of motion for mechanical systems, and equivalent laws for other systems (eg electrical).
Ohm's law is the very familiar description of the relationship between the effort (voltage) and flow (current) across a resistor. It turns out that this same equation, E=R * F, holds for all of the other domains as well. The resistor is typically connected to a 0 or a 1 junction, which are used to tie the rest of the system together.
That's probably enough for a single post, but in the following posts I'll show how to use these elements to convert a simple system into a bond graph, and then use this bond graph to find the causality of power flow to learn more about the system.