Acme threads are a commonly used thread profile in power screws for driving loads and transmitting power. However, designing a durable and efficient threaded system using this profile requires precise calculations to ensure it can withstand the intended loads.

This complete guide provides an in-depth explanation of Acme thread calculations, including their design, measurement, and tolerance considerations.

Table of Contents

## Acme Thread Calculations

Acme thread calculations refer to the mathematical calculations and considerations involved in the design and manufacturing of Acme threads. These calculations involve determining the pitch diameter, lead, and flank angles, as well as the tolerances required to ensure the thread can withstand the intended loads.

### Acme Thread Geometry

Before learning about the various calculations involved in the design of Acme threads, it is important to first understand their geometry.

Acme threads are a specially designed screw thread with a 29-degree thread angle and a flat crest and root, producing a trapezoidal tooth profile, as illustrated in the diagram below.

Acme threads are named after the company that first produced them, Acme Machinery Co. of Cleveland, Ohio. Developed in the 1890s as an improvement over square threads, which had straight-sided flanks and were challenging to manufacture, Acme threads have a trapezoidal shape that is easier to cut using single-point threading or die.

Despite being less efficient than square threads at power transmission, Acme threads are stronger and wear better due to their less acute flank angle. However, while Acme threads are among the strongest symmetric thread profiles, asymmetric buttress threads can bear greater loads for applications that only require a single direction.

The standard nominal diameters and pitches for Acme threads are shown in the table below:

### Depth of Acme Thread

The depth or height of an Acme thread varies depending on its type. There are three main types of Acme threads: General Purpose (GPA), Centralizing (CA), and Stub Acme.

GPA threads are the standard Acme thread profile used for common mechanical applications, CA threads are used in applications that require a self-centering feature, such as in lead screws and power transmission devices, and Stub Acme threads are often used in applications where space is limited, such as in small hand-operated tools.

Generally, the depth of GPA and CA threads is approximately half the pitch, while the depth of Stub Acme threads is shallower, depending on space limitations.

### Torque Calculations for Acme Threads

Force analysis on a power screw is normally done by unrolling one turn of a thread and treating it as an inclined plane. Consider a load, F, acting on a power screw, as shown in the diagram below:

Calculating the torque depends on whether the load is being raised or lowered. For a load being raised, the free-body diagram for a thread becomes:

The angle of incline, λ, can be obtained using the formula:

Where:

- λ = lead angle of the screw or angle of incline [degrees]
- L = lead [m]
- d
_{m}= pitch diameter [m]

Note that the pitch diameter can be approximated to be the average of the minor and major diameters of the thread.

From the free-body diagram above, the torque required to raise the load can be obtained using the formula:

Where:

- T
_{up}= torque required to lift the load [N-m] - F = force due to the load [N]
- ϕ= face angle of Acme thread [14.5⁰]
- f = coefficient of friction [unitless]

Otherwise, for a load being lowered, the free-body diagram becomes:

The torque required to lower the load then becomes:

Where:

- T
_{down}= torque required to lower the load [N-m]

### Virtual Coefficient of Friction

Initially, power screws were designed with square threads, and as a result, most established formulas for power screws are based on square threads. While these formulas can generally be applied to Acme threads, the two types differ in terms of frictional force. This is because, in Acme threads, the normal reaction between the screw and nut is higher than in square threads due to the axial component of this reaction needing to be equal in magnitude to the axial load, as depicted in the diagram below.

Hence, a virtual entity called the virtual coefficient of friction has been established to allow the power screw formulas for square threads to be compatible with Acme threads. That is, the frictional force on an Acme thread can be calculated using the formula:

Where:

- F
_{f}= frictional force on the thread [N] - F = axial load [N]
- R
_{N}= normal reaction force due to the axial load [N] - f
_{1}= virtual coefficient of friction [unitless]

For Acme threads, the virtual coefficient of friction may be substituted in place of the true coefficient of friction. Note that the coefficient of friction is affected by various factors such as the type of material, temperature, surface finish, lubrication, coatings, and wear.

## Stress Calculations

There are four main types of stresses that can occur in an Acme screw assembly: direct tensile or compressive stress due to axial load, shear stress due to torsion, shear stress due to axial load, and bearing pressure. When designing threaded components, it is important to ensure that the principal stresses resulting from the combination of these stresses are less than the permissible stresses on the material.

### Direct Stress Due to Axial Load

When the load is compressive and the unsupported length between the load and the nut is short, the direct stress can be computed by simply dividing the load by the cross-sectional area of the screw. This can be mathematically expressed as:

Where:

- σ
_{c}= direct compressive stress [Pa] - A
_{c}= cross-sectional area of the screw [m^{2}]

Otherwise, if the unsupported length is huge, then column theory must be considered and the stress can be calculated using the Rankine-Gordon formula, as follows:

Where:

- σ
_{y}= yield stress [Pa] - C = end-fixity coefficient [unitless]
- E = modulus of elasticity [Pa]
- L = length of the screw [m]
- k = least radius of gyration [m]

### Shear Stress Due to Torsion

The torsional shear stress is induced by the torque applied to the screw. It can be calculated using the formula:

Where:

- τ = torsional shear stress [Pa]
- T = torque applied to the screw [N-m]

### Shear Stress Due to Axial Load

Shear stress may also be induced by the axial load at the root of the screw threads as well as at the major diameter of the nut threads. At the screw threads, the shear stress can be calculated using the formula:

Where:

- τ
_{screw}= shear stress at the screw due to axial load [Pa] - n = number of engaged threads [unitless]
- d
_{i}= minor diameter of the screw [m] - t = thickness of the thread [m]

On the other hand, at the nut threads, the shear stress can be calculated using the formula:

Where:

- τ
_{nut}= shear stress at the nut due to axial load [Pa] - d
_{o}= major diameter of the screw [m]

The number of engaged threads can be calculated by dividing the height of the nut by the pitch of the thread.

### Bearing Pressure

The bearing pressure refers to the pressure exerted on the contact surfaces between the screw and the nut. It is calculated by dividing the load by the effective thread area, which is also the area of the contact surface of the thread, as shown in the following equation:

Where:

- P
_{b}= bearing pressure [Pa]