Understanding Spur Gear Design and Tooth Strength
What This Gearbox Calculator Solves
A single-stage spur gearbox transmits power between two parallel shafts through a pair of gears whose teeth are straight and parallel to the axis. This gearbox calculator takes the basic machine-design inputs — tooth counts, module, face width, input torque and speed — and returns the gear ratio, output torque and RPM, pitch diameters, center distance, tangential tooth load, and an estimate of the Lewis bending stress at the tooth root. It is intended as a fast first-pass sizing tool for the mechanical design of a spur gear pair.
Governing Equations
The first design decision is the gear ratio, fixed entirely by the tooth counts:
i = Z2 / Z1 = d2 / d1
Once the ratio is set, output speed and torque follow from the input conditions and the mesh efficiency:
T2 = T1 · i · η
The metric size parameter of a gear is the module, defined as pitch diameter over tooth count:
m = d / Z
Two meshing gears must share the same module, otherwise the teeth cannot engage. The center distance between shafts is simply a = (d1 + d2) / 2, which locks the housing geometry.
The tangential force transmitted at the pitch circle is:
Ft = 2 · T1 / d1
The classical Lewis equation treats each tooth as a cantilever beam loaded at the tip by this tangential force. The resulting root bending stress is:
σ = Ft / (b · m · Y)
where b is the face width, m is the module, and Y is the Lewis form factor, which depends on tooth count and pressure angle.
Calculation Method
The solver follows a straightforward machine-design sequence:
- Compute the gear ratio from the tooth counts.
- Compute pitch diameters from the module and tooth counts.
- Compute output speed as input speed divided by the ratio.
- Compute output torque as input torque times ratio times efficiency.
- Compute tangential load at the pinion pitch circle.
- Compute the Lewis bending stress using the appropriate form factor.
- Compare the resulting stress to the material allowable and derive a safety factor.
Input Parameters
Tooth counts (Z1, Z2) are integers that fix the ratio. In a reduction stage, the smaller pinion is the faster, more heavily cycled gear and is therefore the life-limiting element.
Module (m) is the tooth size in millimetres. Increasing module while keeping tooth counts fixed scales the whole gear up and directly raises bending strength.
Face width (b) is the axial length of the tooth. Wider teeth reduce bending stress approximately linearly but increase weight, misalignment sensitivity and bearing loads.
Input torque (T1) and speed define the power going through the mesh. A common input mistake is mixing units (Nm vs. Nmm, rpm vs. rad/s) or forgetting to apply the mesh efficiency.
Engineering Interpretation
The Lewis bending stress is a static root stress. It should be compared to the material's allowable bending stress, with a safety factor of at least 1.2–1.5 for steady industrial service and higher for shock or reversing loads. In parallel with the bending check, a full gear design must verify surface (Hertz) contact stress, which is often the limiting failure mode for through-hardened steels, and must consider dynamic factors from pitch-line velocity, lubrication regime and fatigue life.
A calculated stress below the allowable does not by itself prove that a gearbox is safe or code-compliant; it is one input to a broader design review.
Units and Conversion
The calculator works in consistent metric units: millimetres for length, newtons for force, newton-metres for torque and megapascals for stress. Useful conversions include 1 Nm = 1000 Nmm, 1 MPa = 1 N/mm2, and 1 kW ≈ 9550 Nm·rpm. Because the Lewis formula divides force by (b · m), a unit mismatch between torque (Nm) and diameter (mm) is the most common source of large errors.
Assumptions and Limitations
- Single-stage, parallel-shaft spur gear pair only.
- Rigid, perfectly aligned shafts and ideal conjugate tooth action.
- Load is shared equally across the face width (no misalignment or crowning).
- Steady-state torque; no dynamic amplification or shock factor is applied.
- Lewis form factor is taken as a function of tooth count; it does not include stress concentration, size or surface factors.
- No surface durability (Hertz contact), scuffing, wear or fatigue-life check is performed.
- Helical, bevel, worm and planetary gearboxes are outside the scope of this workspace.
Practical Engineering Use
This gearbox design workspace is useful for quick sizing of reduction or step-up stages in machine tools, conveyors, actuators, robotics joints and small power-transmission drives. Typical first-pass questions it answers are: what module and face width are needed to keep the pinion tooth root below the allowable bending stress, what output torque and speed result from a given ratio, and how does the center distance change if the module is adjusted. Detailed gear rating to AGMA 2001 or ISO 6336, bearing selection and housing stiffness analysis are left to dedicated design procedures.