Torsion Shafts

Overview

Engineering reference data for Torsion Shafts in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Polar Moment of Inertia Explained

The Polar Moment of Inertia ($J$) is a measure of a shaft's ability to resist torsion. It is defined with respect to an axis perpendicular to the cross-sectional area and is analogous to the Area Moment of Inertia ($I$), which characterizes a beam's ability to resist bending.

Note: "Polar Moment of Inertia" is also referred to as "Second Moment of Area," "Area Moment of Inertia," "Polar Moment of Area," or "Second Area Moment."

Shaft Diameter Formula

The required diameter of a solid shaft can be calculated from:

$D = 1.72 \left( \frac{T_{max}}{\tau_{max}} \right)^{1/3}$

where:

• $D$ = shaft diameter (m, in)
• $T_{max}$ = maximum twisting torque (Nm, lbf·ft)
• $\tau_{max}$ = maximum allowable shear stress (Pa, lbf/ft²)

Angular Deflection in Degrees

For practical calculations, the angular deflection in degrees:

Solid shaft:

$\alpha_{degrees} \approx \frac{584 \, L \, T}{G \, D^4}$

Hollow shaft:

$\alpha_{degrees} \approx \frac{584 \, L \, T}{G \, (D^4 - d^4)}$

Torsional Resisting Moments by Cross Section

References

• Original Source