OP

Quoted from DiabloRush:

The pic below is a flipper linkage. Plunger, plastic tab, and pinch bolt.* There's slop in all of it. Any moving object has 6 degrees of freedom in free space: 3 for translation and 3 for rotation (X,Y,Z, roll, pitch, yaw).** Some of these are constrained when hooked to other parts. If there's slop between those parts, each element can retain a limited range of motion in all 6 degrees, though some of them will be small, some very small. So, that part alone has 6 for the plunger, 6 more for the plastic tab, and 6 further with the pinch bolt linkage.
Now, a few of these are fairly well constrained, but not zero given the low tolerance in the connections. Add 6 more for the flipper shaft and bat. So, just adding those up, you've got 24 degrees. Finally, the coil stop is yet another component that needs to be considered. There's flex in the bracket and coil mount ( a coil will move slightly back and forther with respect to the coil stop. So you need to add another 6 degrees for the coil stop. I'm now up to ~30. There's other, complicating effects that I'm not considering. I haven't counted the coil sleeve as a separate element. That could be an error on my part. You assume rigid, homogenous and unchanging material properties in each element, for example. Non-rigid (flexing) parts can really complicate this and can add many more degrees of freedom. I assume constant material properities. Strength, density, elastic modulus. Those can vary, too, and add additional degrees (as components swell with temperature, for example). But, you get the idea....
Now, to be clear, some of these are very small ranges indeed, but not zero. If you want to mechanically describe the motion of this system, you need to know the position in space and rate of acceleration and rotation of each of those elements. Over 30 of them. My figure of 15 was a judgement that maybe half of these are so small as to be insignificant (for example, its probably ok to treat the pinch bolt, shaft, and bat as a single element). This is why I said "just eyeballing it". A proper analysis of which of these elements dominates the motion (often called a sensitivity analysis) could be done with an appropriate FEA (finite element analysis) code. This is done all the time in industry; we do it with the world's fastest computers where I work that take months (and sometimes a year or two) to complete for a single system.
Finally, just an off hand comment. This discussion explains why I'm such a lousy golfer. The golf swing has literally hundreds of degrees of freedom associated with it. Despite 50+ years of training, I've been unable to control the vast majority of those. I will never have glory on the PGA tour as a result. This is true for basically all athletic endeavors. Getting all those motions to be consistent is non-trivial. Peace.
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*it's probably more correct to treat this as 4 elements instead of 3. The bolt attaching the pinch bolt to the plastic tab is very sloppy in many of these. Hence, that bolt is yet another element. None-the-less, I think you get the idea here....
**you sometimes see X,Y, and Z referred to by the alternate terms surge, sway, and heave.