I’m rather hoping there’s something better, but various CAD formats support specifying assemblies of objects, and joints between those objects that can represent properties like that. Often this comes with at least some level of simulation, or if not simulation imposed constraints like in the FreeCAD assembly workbench, allowing you to move connected parts in the assembly but only through the range permitted by the “joint”. I quote that because that includes things like meshed gears, linear slides, ball joints, all kinds of things like that some of which I would not call joints as such.

It's an intriguing idea, but the scope of any such formal definition would essentially be the entire scope of physics, materials science, thermodynamics, etc. For much more bounded problems (like that very fun website you linked) I think something like that would be more attaintable, but still challenging.

Take the example of the differential gearing shown. I doubt there exists any functional differential/mass produced assembly that looks exactly like the example presented. The concept of differential gearing may be able to be broken down into more symbolic representation of forces and motion, but at some point it becomes simplified to the point of impracticality.

The movements from your first URL are actually from a 100-year old book, and its chapter on mechanical terminology (which has been oddly stripped for the web page) might be a good start:

https://www.thalia.at/shop/home/artikeldetails/A1032836899

You know, I find myself thinking the same thing sometimes. And the closest I can think of would be some mapping of force transmitted through linkages on to current transmitted through wires and from there its the same formalism as electronic circuits. Though the usefulness of the abstracted form really depends on what you're trying to compute.

In robotics the URDF format can be used to specify and simulate rigid-body systems. You can also check various physics engine input file formats like MuJoCo (MJCF format). (It's just some variant of XML).

If you want soft-body dynamics, you can have decompose your objects into particles of different types which interact, (examples are liquidfun or powder toy). (It's just a list of particles and a particle-particle interaction matrix) (More or less base on Smoothed Particle Hydrodynamics).

If you want something more serious, you'll have to look at finite-element based multi-physics tools, like Ansys or Comsol, which will allow you to specify the various domains and solve the differential equations on each domains.

If you want to go very small scale you have Molecular Dynamics tools with standards like GROMACS.

If you are more into the topological optimization, you can use neural networks to represent the quantities which are of interest to you and Physically Inspired Neural Networks (PINN), or neural ODE, or "sim-to-real" tools, in which case you'll probably be using pytorch.

Or text if you use WorldSim like LLM tools and only need an abstract representation. Agents playing with CAD software tools will probably standardize some convenient way to solve engineering tasks.

The Stem Boom is another somewhat mind bending mechanical idea. You have a roll that when opened, creates a boom. This can be very small when stored and very long when unfurled. Or two rolls that together form a boom.

https://ntrs.nasa.gov/api/citations/20170003919/downloads/20...

another good video is this channel, which explains exactly how a telescoping boom works in somewhat detail (using a standing desk as an example: https://www.youtube.com/watch?v=UVTDMO7rlhs