Examples of self-energizing systems might include The Politics of Divisiveness, Running an Empire, Economic Bubbles and The Fractional Banking System. Under certain conditions, it also includes automotive brakes.
Apologies in advance
I apologize in advance. This will discussion will NOT be in a straight line. I will bend every effort to compensate by making the story understandable and even entertaining in places.
Through the 1960s and into the early 1970s most vehicles had drum brakes on all four corners. They did the job and were inexpensive.
The effects of Ralph Nadar's Unsafe at any Speed and the advent of the first oil crisis made improved performance and weight savings a higher priority. That, and the fact that the demands on the braking system kept increasing as vehicle weights went up, engine power went up, towing expectations were going up.
The automotive industry responded by putting disk brakes on the front two corners but left the two rear corners as drum brakes. As a general rule-of-thumb, +65% of the braking potential is in the front. The weight of the motor and transmission is actually forward of the front wheels on most vehicles so two-thirds of the static weight passes through the front wheels. That ratio increases under most dynamic braking as there is a "weight shift" forward as braking throws the mass of the vehicle forwards.
In addition to their superior dynamic braking characteristics (less fade), they are typically lighter in weight. That became more of an issue as front wheel drive vehicles almost universally adopted some form of McPherson Struts for front suspensions. Engineers wanted to minimize as much unsprung mass as possible in the front corners.
The continued pressure to reduce component mass led brake engineers to take another look at squeezing more performance out of drum brakes.
A traditional set of drum brakes consists of a "trailing" shoe and a "leading" shoe.
For the purposes of illustration: Imagine that you were given an 8' long 2X4 that had rubber tips. Would it easier to drag the 2X4 or push the 2X4? It should be obvious that it is easier to drag it.
Dragging the 2X4 results in the moment caused by the friction forces lifting up on the aft end of the 2X4 helping it skate across the ground.
Pushing the 2X4 results in the friction forces camming the rubber tipped end into the pavement which greatly increases it resistance to movement. The pushed 2X4 is, in effect, a self-energizing system.
The trailing shoe is dragging.
The leading shoe is being pushed.
Every person who was formally taught to change traditional disk brake shoes was coached to make sure to keep leading and trailing shoes straight. And most shoes had a much thicker application of friction material on the leading shoe.
The brake engineers must have looked at this and thought, it is a pity there is not some way to make both shoes leading shoes. If there was, it would be possible to downsize (and reduce the weight of) all of the components.
They found a way. The trailing shoe transmits force to the block by way of a pivot pin. The trailing shoe cannot "trail" if the pivot is replaced by a sloppy yoke or clevis. It must transmit the braking force through the leading shoe via compression. In effect, both shoes are being pushed, both shoes function as leading shoes.
My car has the super-duper drum brakes in back. They are quirky. After driving on wet pavement and parking for more than 20 minutes they become grabby.
I suspect the reason is because if you draw free-body-diagrams of the pushed 2X4 experiment and derive the braking force you might get a formula that looks like this
Braking force = Mu * (weight of 2X4)/(2*(1-Mu*h/l)) where Mu is the coefficient of friction, h is the height of the hand holding the 2X4 and l is the distance the rubber contacts the ground in front of the person pushing.
Of particular note is the term (1-Mu*h/l). This term gets smaller as Mu gets larger. There are two possible consequences.
"Brake Fade" becomes more pronounced as the pad-drum interface heats up during prolonged braking. Think about it in non-mathematical terms: The system performance is multiplied by friction....the system performance will degrade by that multiple when the friction force diminishes.
The math suggests another outcome when Mu*h/l approaches 1.0. We approach a divide-by-zero situation where the brakes become "grabby".
The larger lesson
Self-energizing systems pay for themselves by burying hyper-sensitivities in unexpected places.
There-in lies my fear and loathing of self-energizing systems. A simple, straightforward, predictable system was replaced by a more complex system with the appearance of a perpetual motion machine. The complex system has goofy, unpredictable performance characteristics. Some of those goofy quirks might result in violent, divide-by-zero events.
Self-energizing systems make it more difficult to create a robust system "A robust process is a process that can absorb all anticipatable variation in the inputs and still produces an acceptable output." because the self-energizing mechanism makes it impossible to identify all of the inputs. Who could have anticipated that exposure to water mist would, after a delay, drastically change braking performance? Tiny little perturbations in minor, unmonitored, unnoticed features in the system can result in the system screeching to a halt.
It is instructive that very, very few vehicles now have drum brakes, even on the back two wheels.