As shown in the figure below, a screw is driven by a screw driver onto a suspended rod. The

rod is firmly held at one end. Six strain gages $($marked as A$,$ B$,$ C$,$ D$,$ E and F$)$ are used to

determine the driving torque on the screw and the bending torque on the suspended rod. All

material properties and geometric dimensions of screw, rod, and driving locations are

assumed to be given. The thermal effect can be ignored.

$(1)$ Express the relationship of the measurement from each strain gage to the principal strains of torsion $($on the screw$)$ and bending $($on the rod$)$;

$(2)$ Suggest the combinations of strain gage connections to the full$-$arm Whetstone Bridge that can be used to determine the torsional torque and bending torque.$$ Check the corresponding bridge constants to validate the suggested combination. You may also use $$dummy$$ stain$-$gages in your combination design.

$(3)$ Derive a formula to calculate the calibration number of torsional torque in your design.