I am trying to configure a force sensor (a piezoresistive force sensor) in a device to measure small forces (less than 0.5 mN). This is difficult due to noise of the system. One of the ways I have considered to address this problem is to mechanically amplify the force applied to the force sensor with a gearbox. The small force applied would be transferred to the gearbox with a gear on a drive bar, and the amplified force from the increased torque would be transferred to the sensor with another gear and drive bar (see schematic). However, I have no experience with gearboxes and would like some opinion on feasibility or if this system even makes sense. I am particularly worried that static friction forces in the gears would cause inaccuracies, especially because gearboxes are designed for a completely different purpose.
Any gearbox or lever linkage is going to have backlash to go with your other concerns. By the time you get to a system using this kind of mechanical amplifier to get the sensitivity you need along with the low noise, accuracy and precision you want, you're into making a what's essentially a precision watch... Can it be done? Sure. It's going to be time consuming and expensive though.
It will be faster, cheaper and more efficient to simply purchase a load cell designed to measure micro Newton level Forces. If you're a student, doing a project on a student budget with no corporate sponsor, you should call the company anyway, they might help you anyway.
If you really need to build the measurement device yourself, I suggest looking into a Torsion Balance. They're typically used to measure things like the gravitational attraction force between two objects, so the concept is in line with your measurements. You could also build a strain gage based sensor, perhaps some kind of canteliever beam.
You may actually be interested in using a lever instead of a gearbox, as its a simpler construction and has less friction. But produces same torque increase, as a gear is essentially a folded up lever. If you want to be really fancy you couple this with two pins but that is not nessesary.
Just be sure to calibrate the new system somehow after you have built it. On the other hand you can just invest in a strain gauge and connect it to a shaft with known dimensions, as this may be more reliable than relying on the accuracy of your rotating joint to transfer force form one place to another