Screw jack performance: What is the maximum theoretical efficiency of a screw jack expressed in terms of the angle of friction phi (for a square-thread jack in raising mode)?

Introduction / Context:

Screw jacks convert torque into axial lifting force via a power screw. Efficiency depends on thread lead angle and friction. There exists a condition that maximizes efficiency for a given angle of friction, leading to a compact expression for the maximum value achievable (ignoring collar friction and other losses).

Given Data / Assumptions:

• Square-thread jack, raising a load slowly (quasi-static).
• Angle of friction = phi, with mu = tan phi.
• Collar friction and other parasitic losses neglected for the formula.

Concept / Approach:

For raising, efficiency eta can be written in terms of thread lead angle alpha and friction as eta = tan alpha / (tan alpha + mu * sec alpha). Maximizing eta with respect to alpha for a given phi yields a closed-form maximum depending only on phi.

Step-by-Step Solution:

Verification / Alternative check:

Check limits: as phi → 0 (no friction), sin phi → 0, so eta_max → 1 (100%). As phi increases, eta_max decreases monotonically, which matches physical intuition.

Why Other Options Are Wrong:

• tan- and cos-based forms listed do not emerge from maximizing the screw jack efficiency expression.
• cos^2 phi does not match known limiting behavior as phi → 0.
• 1 - tan phi can be negative for moderate phi and is dimensionally inconsistent as an efficiency formula.

Common Pitfalls:

• Confusing the general efficiency expression (depends on alpha) with the maximum efficiency (depends only on phi).

Final Answer:

eta_max = (1 - sin phi) / (1 + sin phi)