Authoritative reference, formulas, and guidance
This gear ratio calculator helps mechanical engineers, technicians, makers, cyclists, and students quickly determine the overall ratio of single or multi‑stage gear trains, estimate output speed and torque, and understand direction changes and efficiency losses.
Data Source and Methodology
Authoritative Data Source: Shigley’s Mechanical Engineering Design, 11th Edition (2019), McGraw‑Hill. See Chapter on Gears: Kinematics and Power Transmission. Publisher link.
Secondary Reference: Machinery’s Handbook, 31st Edition (2020), Industrial Press — sections on gear kinematics and proportions. Publisher link.
All calculations strictly follow the formulas and data provided by these sources.
The Formula Explained
For a single stage with driver gear 1 and driven gear 2:
Inline teeth/diameter equivalence:
$$ i = \frac{\omega_1}{\omega_2} = \frac{n_1}{n_2} = \frac{z_2}{z_1} = \frac{D_{p2}}{D_{p1}} $$
Multi‑stage (k = 1…m) overall ratio:
$$ i_{\text{total}} = \prod_{k=1}^{m} i_k $$
Output speed from input speed:
$$ n_{\text{out}} = \frac{n_{\text{in}}}{i_{\text{total}}} $$
Overall efficiency (product of stage efficiencies):
$$ \eta_{\text{total}} = \prod_{k=1}^{m} \eta_k $$
Output torque (ideal torque scaled by efficiency):
$$ T_{\text{out}} = T_{\text{in}} \cdot i_{\text{total}} \cdot \eta_{\text{total}} $$
Direction of rotation for external meshes:
$$ \text{Direction factor} = (-1)^{m} \quad \Rightarrow \quad \text{reversed if m is odd} $$
Glossary of Variables
- z1, z2 — Tooth count of driver and driven gears respectively.
- Dp1, Dp2 — Pitch diameters of driver and driven gears (consistent units).
- i — Gear ratio of a stage, equal to driven/driver (z2/z1 or Dp2/Dp1). i > 1 is reduction; i < 1 is overdrive.
- i_total — Product of all stage ratios.
- n_in, n_out — Input and output rotational speed (RPM).
- T_in, T_out — Input and output torque.
- η_k, η_total — Efficiency of stage k and product efficiency (0–1).
- Direction — Whether output rotation matches or reverses the input (external meshes flip each time).
How It Works: A Step‑by‑Step Example
Scenario: Two external gear stages. Stage 1: driver 20 teeth, driven 60 teeth. Stage 2: driver 18 teeth, driven 54 teeth. Each stage at 98% efficiency. Input speed 1500 rpm, input torque 2.0 N·m.
- Stage ratios: i1 = 60/20 = 3.0; i2 = 54/18 = 3.0.
- Overall ratio: i_total = 3.0 × 3.0 = 9.0 (reduction).
- Direction: 2 meshes → even → output direction same as input.
- Overall efficiency: η_total = 0.98 × 0.98 = 0.9604.
- Output speed: n_out = 1500 / 9 = 166.67 rpm.
- Output torque: T_out = 2.0 × 9 × 0.9604 ≈ 17.29 N·m.
Frequently Asked Questions (FAQ)
Is the “ratio” displayed as x:1 or 1:x?
This tool reports the magnitude as x:1 where x = driven/driver. Values above 1 indicate reduction; below 1 indicate overdrive.
Do idler gears change the overall ratio?
No. Idlers reverse direction but do not change the magnitude of the ratio.
Can I mix teeth and diameter inputs?
Yes. Each stage can be defined by teeth or pitch diameters. Just ensure both driver and driven for that stage use the same method.
What efficiency should I use?
For quality spur/helical gears, 97–99% per mesh is typical under good lubrication. Chains/belts may be lower. Consult vendor data.
Does the calculator include backlash or slip?
No. Backlash does not affect the nominal ratio. Slip is not modeled for gears; belt/chain slip is not considered unless you lower efficiency accordingly.
How precise are the results?
They are as precise as your inputs. The kinematic relations are exact; efficiency is an approximation based on your assumption.
What unit should I use for pitch diameter?
Any consistent length unit (mm, in). The ratio uses a ratio of diameters, so units cancel.