static friction of gear pair

Theoretically, a well lubricated, aligned, and maintaned gear box should have theoretically zero static friction. If you are after estimating what the torque is to get it to move, to my experience, it is always easier and best to test the torque required to start it moving.

The reason is that apart from the individual elements, how you implement and setup the gear box can have a detrimental effect. I'll count some of the sources that can really throw off your calculations:

• poor alignment of the geometrical axis of the shaft with the axis that the center of the bearings form.
• tightening too much (or too little) a fastener on the system
• dimensional variations within the acceptable tolerances
• temperature setting
• condition of the gear box.

In the theoretical best case scenario, you'd have

• spur gears almost zero losses

• helical gears (due to the axial force component that is generated) you'd have higher losses (depending on the angle of the teeth)

• bevel gears: even higher losses

Bottom line is that you'd have a better chance with a design that uses spur gears. You should spend some time designing and finetuning the assembly, and then actually measure the torque required to get it to move.

A final note, is that if acceleration is important, then you should also take into account the moment of inertia of the rotating parts (that will have a significant impact on the starting torque).