Power Screw Torque and Efficiency Equations

This engineering reference provides equations for calculating the torque required to raise and lower axial loads using power screws. It includes the effects of screw-thread friction, thrust-collar friction, thread geometry, lead angle, efficiency and self-locking or overhauling behaviour.

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These equations are applicable to conventional sliding-thread power screws such as square, Acme and similar thread forms. Ball screws and roller screws have different friction characteristics and should normally be selected using manufacturer data.

Power Screw Torque Calculations

A power screw converts rotary motion and torque into axial motion and force. Typical applications include screw jacks, presses, machine-tool lead screws and mechanical positioning systems.

The following derivation considers a screw carrying an axial load W. Although the original example considers a vertically acting load, the equations apply equally to an axial force acting along the screw axis.

Load notation: If W represents a suspended mass, it must first be converted to force. For example, a mass of 1000 kg corresponds to a gravitational force of approximately 9810 N when g = 9.81 m/s².

Nomenclature

Symbol	Description	SI Unit
θ	Thread half-angle measured in an axial plane	rad
θ_n	Thread half-angle in the plane normal to the helix	rad
η	Screw efficiency	–
d_m	Mean screw diameter	m
d_mc	Mean thrust-collar diameter	m
μ_c	Coefficient of friction at thrust collar	–
μ_s	Coefficient of friction at screw threads	–
F	Tangential force applied at the mean screw radius	N
F_n	Normal reaction force at screw thread	N
F_f	Thread friction force	N
l	Thread lead; axial advance per revolution	m
n	Number of thread starts where l = np	–
p	Thread pitch	m
α	Lead or helix angle at mean diameter	rad
r_ci	Thrust-collar inside radius	m
r_co	Thrust-collar outside radius	m
r_m	Mean screw radius = d_m/2	m
r_mc	Mean thrust-collar radius	m
W	Axial load carried by screw	N
T_R	Torque required to raise load	N·m
T_L	Torque required to lower load	N·m
T_o	Torque associated with overhauling/backdriving	N·m

Lead or Helix Angle

The lead of a screw is the axial distance travelled in one complete revolution. For a multiple-start thread:

l = n p

The lead angle at the mean screw diameter is:

α = \tan^{- 1} (\frac{l}{π d_{m}})

Therefore:

\tan α = \frac{l}{π d_{m}}

Torque Required to Raise a Load

Power screw showing axial load, operating torque, thread geometry and thrust collar

Consider a tangential force F applied at the mean screw radius r_m to raise the axial load W. In the arrangement shown, the nut rotates while the screw is prevented from rotating.

Power screw thread force diagram showing normal reaction and friction force

The normal reaction F_n acts on the thread surface. The corresponding friction force is:

F_{f} = μ_{s} F_{n}

The following diagram shows the horizontal and vertical components acting at a representative point at the mean screw radius.

Power screw force components resolved at the mean thread radius

Resolving the forces in the axial direction gives:

F_{n} \cos θ_{n} \cos α = W + F_{f} \sin α

Substituting F_f = μ_sF_n gives:

F_{n} = \frac{W}{\cos θ_{n} \cos α - μ_{s} \sin α}

The torque required at the screw thread is therefore:

T_{R} = r_{m} F_{n} (μ_{s} \cos α + \cos θ_{n} \sin α)

Thrust-Collar Friction

Where the axial load is reacted through a sliding thrust collar, an additional friction torque is required. Using the mean collar radius r_mc:

T_{c} = μ_{c} W r_{mc}

For the simple mean-radius approximation used on this page:

r_{mc} = \frac{r_{ci} + r_{co}}{2}

The total raising torque including collar friction is consequently:

T_{R} = r_{m} F_{n} (μ_{s} \cos α + \cos θ_{n} \sin α) + μ_{c} W r_{mc}

Substituting for the normal force and using r_m = d_m/2 gives:

T_{R} = \frac{W d_{m}}{2} \frac{μ_{s} \cos α + \cos θ_{n} \sin α}{\cos θ_{n} \cos α - μ_{s} \sin α} + \frac{W d_{mc} μ_{c}}{2}

Dividing the numerator and denominator of the thread term by cos α gives the useful form:

T_{R} = \frac{W d_{m}}{2} \frac{μ_{s} + \cos θ_{n} \tan α}{\cos θ_{n} - μ_{s} \tan α} + \frac{W d_{mc} μ_{c}}{2}

Equation B – Torque required to raise the load.

Normal Thread Angle

For a thread having a significant helix angle, the normal thread angle θ_n differs slightly from the thread angle measured in the axial plane.

\tan θ_{n} = \cos α \tan θ

Therefore:

θ_{n} = \tan^{- 1} (\cos α \tan θ)

For many conventional single-start, fine-lead power screws the lead angle is small and cos α is close to unity. In these cases, θ_n may be approximated by θ.

For multi-start screws and screws having relatively large leads, use the normal thread angle θ_n rather than assuming θ_n = θ.

Power screw raising torque equation using the thread angle approximation

Torque Required to Lower a Load

When the load is lowered, the direction of the thread friction force reverses. For a self-locking screw, an applied torque is required to lower the load.

Including thrust-collar friction, the lowering torque is:

T_{L} = \frac{W d_{m}}{2} \frac{μ_{s} - \cos θ_{n} \tan α}{\cos θ_{n} + μ_{s} \tan α} + \frac{W d_{mc} μ_{c}}{2}

Equations in Terms of Thread Lead

Since:

\tan α = \frac{l}{π d_{m}}

the raising and lowering equations can also be expressed directly in terms of the thread lead.

Power screw raising and lowering torque equations expressed in terms of thread lead

Low-Friction Thrust Bearings

Where the axial thrust is carried by a suitable ball or roller thrust bearing, the collar-friction torque may be sufficiently small for μ_c to be neglected for an approximate calculation. The equations then reduce to the thread-friction terms alone.

Power screw raising and lowering torque equations neglecting collar friction

T_{R} \approx \frac{W d_{m}}{2} \frac{μ_{s} + \cos θ_{n} \tan α}{\cos θ_{n} - μ_{s} \tan α}

T_{L} \approx \frac{W d_{m}}{2} \frac{μ_{s} - \cos θ_{n} \tan α}{\cos θ_{n} + μ_{s} \tan α}

Self-Locking and Overhauling

A power screw is self-locking when the axial load cannot backdrive the screw in the absence of an applied operating torque. If the lead angle is sufficiently large relative to friction, the load can instead drive the screw backwards. This condition is commonly described as overhauling or backdriving.

Neglecting thrust-collar friction, the boundary between self-locking and overhauling occurs when the lowering torque is zero:

\tan α = \frac{μ_{s}}{\cos θ_{n}}

Therefore, neglecting collar friction:

Condition	Relationship
Self-locking	tan α < μ_s / cos θ_n
Limiting condition	tan α = μ_s / cos θ_n
Overhauling / backdriving	tan α > μ_s / cos θ_n

Important: Self-locking should not automatically be treated as a safety brake. Actual behaviour depends on friction, lubrication, wear, vibration, thread geometry and the complete drive arrangement. Where uncontrolled movement could create a hazard, an appropriate braking or locking system should be provided.

Power Screw Efficiency

The forward or raising efficiency can be defined as the ideal torque required to raise the load in the absence of friction divided by the actual torque required.

For the thread alone, the ideal friction-free raising torque is:

T_{ideal} = \frac{W d_{m}}{2} \tan α

The corresponding raising efficiency is:

η = \frac{T_{ideal}}{T_{R}}

Power screw raising efficiency equation including friction

Where thrust-collar friction is negligible, substitution of the thread torque equation gives:

η = \frac{\tan α (\cos θ_{n} - μ_{s} \tan α)}{μ_{s} + \cos θ_{n} \tan α}

Backdriving / Lowering Efficiency

For a screw that can be backdriven, lowering efficiency may be expressed as the actual torque generated during backdriving divided by the corresponding friction-free torque.

Power screw backdriving efficiency equation including friction

If thrust-collar friction is negligible, the expression reduces to the thread-friction relationship shown below.

Simplified power screw backdriving efficiency equation

Power Screw Friction Coefficients

The calculated torque and efficiency are strongly dependent on the assumed coefficients of friction. RoyMech provides typical screw-thread and thrust-collar friction values at:

Power Screw Friction Factors

Actual friction depends on the material combination, lubrication, surface finish, operating speed, temperature, wear and other service conditions. Manufacturer or test data should be used where reliable torque prediction is important.

Related RoyMech References

External References and Further Reading

Module 6 – Power Screws – Detailed notes and power screw equations.
Roton – Power Screw Formulas – Power screw calculation equations.
Tribology-ABC Power Screw Calculator – External power screw calculation resource.

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