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Roll up a chain of parts

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Hi, I have been struggling on a assembly with mechanism on/off for months, and still no solution found, so maybe you could help

It is a tube where a chain has to be rolled up on (diameter gets bigger, like a spiral). Almost like a bike chain with multiple hinges, on a roller blind, my chain is just wider

 

 

I draw the spiral (converted to a spline), which is motion of the chain. The chain-parts (red and green)  are assembled together at the hinge-point, and this point is also attached to the spiral/spline.

First drag will go okay, with no problem if you do it slowly, but at some point, one or more hinge-points will jump to another radius inside the spiral. I tried to start the assembly at the straight line and inside the spiral, with same result,

 

 

I would like to hear your ideas, how to do this?


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