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I am trying to figure out how to compute torques in a network of rotating objects. I assume that each object has a given moment of inertia, and some objects are accelerated by external means. The objects are connected mechanically by zero-mass rigid connectors. Due to the rigid connections, all objects are forced to rotate at the same speed at any time.

For example, I could have three objects: A, B and C. All have a moment of inertia of 1kgm^2. A is connected to B, B is connected to C, and C is connected to A.

If we now accelerate A with 15W of power, then due to the equal moments of inertia, each object will gain kinetic energy at a rate of 5W. This will result in torques experienced by the connectors between the objects. I assume that I could compute the torques based on the power transmitted through a connection. In this simple example, due to the symmetry, there should be 5W transmitted from A to B, and 5W from A to C, while B to C transmits no power.

While in this simple example this is easy to figure out, I am interested in how to compute these transmitted powers in the general case, with many more objects.

If there was no cycle, then it would be easy. Any object that is connected to only one other object would receive the power P_1 it accelerates at. The next object would receive the power P_2 itself accelerates at, plus P_1, so P_1 + P_2. And so on. But with a cycle, there is no starting point from which I can compute everything else like this.

How can this problem be solved, given that there are cycles in the network?

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    $\begingroup$ torque isn't "transmitted" between objects in the same solid body, the solid body is treated as 1 object. You put 15W into the body. Unless you are trying to calculate stresses on the axle (or whatever) it doesn't matter. Torque on any point of a solid body applies to it all. $\endgroup$
    – Tiger Guy
    Feb 1 at 13:55
  • $\begingroup$ Follow the energy $\endgroup$
    – Abel
    Feb 1 at 13:58
  • $\begingroup$ @TigerGuy Imagine each object being a flywheel, and those are connected via axles. If one flywheel is accelerated, it transmits power through the axles to the neighboring flywheels. This transmission of power induces a torque inside the axle, right? $\endgroup$
    – Sibbo
    Feb 1 at 14:11
  • $\begingroup$ How about you present us with a problem and how you have tried to solve it instead of broad statements about indeterminate objects? Certainly stress during acceleration is solvable, start at A and the forces that would apply from it to the rest of the system. This would be the stress on any solid axle from inertia of the item being accelerated. $\endgroup$
    – Tiger Guy
    Feb 1 at 14:46

1 Answer 1

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This isn't really an answer, sorry in advance.

Not sure the exact language to use, but it seems to me that a cyclic network of rigid connections would be is overconstrained in some sense, and so I don't think we could know how much energy travels in each possible path.

What would be necessary to bring the network of flow in to a balance?

  • Would it be sufficient if we allow some elasticity (&damping?) ?
  • Or would it be necessary to require that the links be equally stiff?

If we can establish a balance principle, then we can use "0 flow across the balance point" to resolve the cyclic network.

Example below: 25W going into 5 objects, each of which absorbs 5W into its inertia. By symmetry, there would be no flow of energy at the opposite side of the circle in the diagram.

a

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  • $\begingroup$ Thanks. I agree with your assessment, every attempt in solving this have lead me to missing constraints so far. So likely, elasticity is needed. I'll make another question for that once I worked out a proper problem formulation. $\endgroup$
    – Sibbo
    Feb 6 at 13:52

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