Maybe flexible steel rods/wires would work better anyway?
Just cut the right length of metal wire, bend round one end (tip) of the wire with a pair of pliers, thread on the 2 caps, springs and washers in the right order, and then bend round the other end...
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PS. It may also help to make a small kink in the middle of each wire to accommodate the other two wires, like this:
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In fact, you could probably make the wire with all the bends first, and then just thread on the different bits.
And placing the bends in precise position would ensure exactly the right tension, much better than fiddling around with screws and cores (or even knotting strings). Not to mention a lot cheaper and faster.
Also, springs may not even be necessary as the kink may act a bit like a spring, especially if it's more U-shaped.
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The only thing left is to prevent the cut ends of the wire scraping on the washers:
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And any excess friction would warm the steel rod and make it longer, so the tension would be self-correcting!!!
But, just in case it's still not springy enough:
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