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The gif that shows what I want to do was done manually, on each frame before being made into an action. Basically, I rotate the hinge, and the wire is meant to follow the bottom of that part of the lamp. Everything is on one mesh, with weight painting and auto IK.

The reason for this question is that I want to do this without using moving the wire manually every frame. Most of the methods I've tried simply stretch out without dealing with the loop. Is there a way to do this that's better than just going frame by frame and adjusting the wire as the lamp rotates?

Lamp moving

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Instead of using bones for the wire you can use soft body physics.

  1. create a line of vertices and subdivide it (this is your rope/softbody)

  2. give it these four modifiers (order matters!)

enter image description here

  1. the hook modifier should be set to the upper vertex which is then pinned to the armature. Easiest way is select the upper vertex and press CTRL-H -> this hooks it automatically to an empty

  2. add a copy location constraint to that empty and the part of your armature where you want to pin it

enter image description here

  1. give the softbody a goal with a vertex group, which includes both vertices (the first and last, which should be pinned to armature and lamp)

Set goal stiffness to highest value.

result:

enter image description here

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