- \( \phi > 0 \) : counter clockwise
- \( \phi < 0 \) : clockwise

- Statically indeterminate: must consider shaft deformations.
- Multi-planar: equilibrium requires the existence of shear stresses on the faces formed by the two planes containing the axis of the shaft

- For circular shafts (hollow and solid): cross-sections remain plane and undistorted due to axisymmetric geometry
- For non-circular shafts: cross-sections are distorted when subject to torsion
- Linear and elastic deformation

The twist rate is given by.

Moving terms.

Arc length.

- is proportional to the angle of twist
- varies linearly with the distance from the axis of the shaft
- is
**maximum**at the surface

Geometry

Hooke's law.

From equilibrium.

Elastic torsion formula.

Power.

Solid shaft (radius and diameter).

Hollow shaft (inner and outer radius).

- The angle of twist of the shaft is proportional to the applied torque
- The angle of twist of the shaft is proportional to the length
- The angle of twist of the shaft decreases when the diameter of the shaft increases

Angle of twist.

Torsional stiffness.

Torsional flexibility.

Heads up - Extra!

** Torsion of thin-walled hallow shafts ** builds on this content.

In general, the maximum shear stress is given by

Common assuptions:

- Gears are perfectly rigid.
- The roation axis is perfectly fixed in space.
- Gear teeth are evenly spaced and perfectly shaped so there is no gap to create lost motion.
- Gear tooth faces are perfectly smooth so there is no slip.
- Mated gears twist through the same arc length.

To solve:

- Draw a FBD for each shaft.
- Use the torsion equilibrium equations of each shaft to find the recation forces and torques.
- Use the shear stress equation to find the stresses in the shaft.

Gear ratio.

Angle of twist

Extra!

** Power transmition ** can be used to find shear stress in gear shafts.

Maximum shear stress: