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Jigar, what you're effectively trying to do is violate the conservation of energy. Your proof is not clear.
First of all, if you have a variable radius, the center plate will be forced upward or downward, depending on which way you rotate. It seems you have ignored this. You would need to include this effect in your proof.
Secondly, it seems you want the center plate to apply a downward force on the pulleys through friction, yet want to lubricate it to prevent frictional losses. You can't do both. If you improve the clarity of your proof, we might be able to help show how the conservation of energy would prevent this idea.