◈Overview
The Spur Gear Set Design Engine takes your mechanical requirements —
center distance, speed ratio, load, and material — and automatically searches thousands
of gear combinations to find the best standard pitch value and tooth count that satisfies
all constraints simultaneously.
The engine evaluates candidates using the Lewis beam strength equation
and the Barth velocity factor, scores them against your load band,
and returns the single best result. It supports all three major pitch systems
(Diametral Pitch, Module, and Circular Pitch) and can run all three at once in
Auto mode.
When the required center distance does not fall exactly on a standard pitch circle,
the engine automatically calculates the required profile shift coefficients
and adjusts the capacity estimate accordingly.
ℹ
Results are engineering estimates based on the Lewis beam equation. They are suitable
for preliminary design and selection. Final designs for critical applications must be
verified by a qualified mechanical engineer using full AGMA or ISO standards.
▶Quick Start
- Select a Pitch System or leave it on Auto to let the engine choose the best system for your application.
- Enter the Center Distance — the distance between the two shaft centerlines. Use inches for DP/CP systems, or millimeters if Module is
selected.
- Set the Input Speed and Output Speed in RPM. The ratio is derived automatically.
- Enter a Service Factor appropriate for your application (see the reference table in the calculator or the Service Factor section
below).
- Choose a Material from the dropdown. The allowable bending stress is filled in automatically.
- Optionally enter an Output Torque or Horsepower. If left blank, the engine returns maximum safe capacity instead.
- Results appear instantly as you type. Review the Safety Margin, Face Width, Tooth Counts, and
Profile Shift cards.
- Toggle Measurement Over Wires/Pins if you need inspection dimensions.
⊙Pitch System
Diametral Pitch (DP)
Inch Standard · USA
Number of teeth per inch of pitch diameter. Higher DP = finer, smaller teeth.
The most common system in North American manufacturing.
Module (m)
Metric Standard · Global
Pitch diameter in mm divided by tooth count. Higher module = coarser, larger teeth.
Enter center distance and space limit in millimeters when Module is selected.
Circular Pitch (p)
Inch Legacy · Arc Length
Arc length along the pitch circle between adjacent teeth (inches).
Legacy US system still used in some large-gear applications.
Relationship: p = π / DP.
Standard pitch values evaluated:
Diametral Pitch — 21 values
(catalog-only = non-AGMA preferred)
2 2.5 3 4 5 6 8 10 12 14 (catalog only) 16 18 (catalog only) 20 24 28 (catalog only) 32 36 (catalog only) 40 (catalog only) 48 56 (catalog only) 64
Module — 12 values
m1
m1.25
m1.5
m2
m2.5
m3
m4
m5
m6
m8
m10
m12
Circular Pitch — 14 values
1-1/2"
1-1/4"
1"
3/4"
5/8"
1/2"
3/8"
5/16"
1/4"
3/16"
1/8"
1/10"
1/16"
1/20"
✓
Auto mode runs all three pitch systems in parallel and picks the single
best candidate across all systems. Use it when you have no system preference — it guarantees
the optimal pitch for your constraints. The winning system is identified in the results.
↔Center Distance & Space Limit
Center Distance
The distance between the centerlines of the two gear shafts. This is the primary
constraint — the engine scores candidates with a 12× penalty for
center distance error versus only 1.5× for ratio error. Candidates must be within
8% of this value to be considered.
Enter in inches for DP and Circular Pitch systems.
Enter in millimeters when Module is selected (the label updates automatically).
If you switch pitch systems, the field converts automatically.
⚠ When profile shift is applied, the actual gear geometry
operates at exactly your input center distance — the pitch circles are expanded or
contracted to meet precisely.
Space Limit / Max OD (optional)
The maximum allowable outside diameter of either gear. Any candidate whose pinion
or gear outside diameter exceeds this value is rejected outright.
Leave blank if you have no envelope constraint.
The outside diameter is calculated as (N + 2) / DP for standard gears,
or (N + 2 + 2x) / DP when profile shift is applied.
∠Pressure Angle
The pressure angle is the angle between the tooth profile at the pitch point and
a line tangent to the pitch circle. It affects tooth strength, undercutting risk,
and separating force between shafts.
14.5°
Legacy standard. Shallower, weaker tooth form. Generates lower separating
forces. Higher risk of undercutting at low tooth counts (undercutting threshold: 32 teeth).
Rarely used in new designs.
20° (default — recommended)
Current AGMA standard. Stronger tooth form than 14.5°.
Good balance of strength, noise, and separating force. Undercutting threshold: 17 teeth.
The correct choice for virtually all new spur gear designs.
25°
High-strength tooth form. Generates the highest separating forces —
requires stiffer shaft and bearing systems. Lowest undercutting risk (threshold: 13 teeth).
Best for high-load, low-speed applications where bearing loads are manageable.
SFService Factor
The AGMA service factor (Ksf) is a multiplier applied to the required tangential
tooth load that accounts for shock loading, duty cycle, and application severity.
A service factor of 1.0 means the gear set is sized exactly for the rated load — no margin.
A factor of 1.5 means the gears must be capable of carrying 1.5× the nominal load.
Use the Reference Table button in the calculator to look up typical values.
The table below summarizes common applications:
| Load Character |
Typical Application |
1-Shift |
2-Shift |
Continuous |
| Uniform |
Generators, fans, light conveyors |
1.00 |
1.25 |
1.50 |
| Moderate shock |
Centrifugal pumps, mixers, bottling |
1.25 |
1.50 |
1.75 |
| Moderate shock |
Heavy conveyors |
1.50 |
1.75 |
2.00 |
| Heavy shock |
Crushers, punch presses, ball mills |
1.75 |
2.00 |
2.25 |
| Heavy shock |
Hoists, cranes, reciprocating compressors |
1.75 |
2.00 |
2.25 |
⚠
Values are AGMA 2001 guidance rounded to the nearest 0.25. Always verify with the
applicable AGMA standard for safety-critical applications.
⚡Torque, Speed & Power
Input Speed (rpm)
Rotational speed of the driving shaft (pinion). Used to
calculate pitch-line velocity and the Barth velocity factor. Required field.
Output Speed (rpm)
Target rotational speed of the driven shaft (gear). The engine
derives the required speed ratio from Input ÷ Output. The actual ratio may differ
slightly since tooth counts are whole numbers — any mismatch is reported in the
results with a warning if it exceeds 0.5%.
Output Torque (ft-lb or in-lb)
The required torque at the output (gear) shaft. The engine
uses this to calculate minimum face width via the Lewis equation. Use the unit
selector to switch between ft-lb and in-lb — the field converts automatically.
Leave blank to run in Capacity Mode — the engine returns the
maximum torque the recommended gear set can safely carry.
If both Horsepower and Torque are entered, HP takes precedence
and the torque field is calculated automatically.
Horsepower (optional)
When HP is entered along with Output Speed, the engine converts
to output torque using: T = HP × 5252 / RPM × η × (1 − f), where η is
efficiency and f is friction loss fraction. The torque field is then locked and
displays the derived value.
Efficiency (%)
Mechanical efficiency of the gear mesh, typically 95–99% for
well-lubricated spur gears. Defaults to 95%. Used to back-calculate input torque from
output torque, and to convert HP to torque when HP is entered.
Friction Loss (%)
Additional friction losses beyond the basic mesh efficiency —
bearing friction, seal drag, churning losses. Combined with Efficiency to form an
effective power factor. Defaults to 0%. Most spur gear applications set this to 0
unless specific bearing loss data is available.
MMaterial & Allowable Stress
Select the gear blank material. The engine looks up the allowable bending stress
from the table below and uses it directly in the Lewis beam equation to determine
face width and load capacity. All values represent conservative allowable bending
stress for preliminary design — not ultimate strength.
Select Custom to enter your own allowable bending stress in psi if
your material is not listed or if you have heat-treat or case-hardening data that
differs from these defaults.
| Material |
Allowable Bending Stress |
ksi |
| 1018 CF Steel |
22,000 psi |
22.0 ksi |
| 1045 Steel |
28,000 psi |
28.0 ksi |
| 1144 HR Steel |
26,000 psi |
26.0 ksi |
| 4140 Steel |
35,000 psi |
35.0 ksi |
| 4140 HT Steel |
45,000 psi |
45.0 ksi |
| 4150 Steel |
36,000 psi |
36.0 ksi |
| 4150 HT Steel |
48,000 psi |
48.0 ksi |
| 8620 Steel |
32,000 psi |
32.0 ksi |
| 9310 Steel |
40,000 psi |
40.0 ksi |
| 303 Stainless Steel |
20,000 psi |
20.0 ksi |
| 304 Stainless Steel |
21,000 psi |
21.0 ksi |
| 316 Stainless Steel |
22,000 psi |
22.0 ksi |
| 15-5 PH Stainless |
42,000 psi |
42.0 ksi |
| 17-4 PH Stainless |
45,000 psi |
45.0 ksi |
| Class 40 Cast Iron |
14,000 psi |
14.0 ksi |
| Aluminum 6061 |
12,000 psi |
12.0 ksi |
| 2024 Aluminum |
16,000 psi |
16.0 ksi |
| Phosphor Bronze |
18,000 psi |
18.0 ksi |
| Nylon 6/6 |
6,500 psi |
6.5 ksi |
| Acetal/Delrin |
8,000 psi |
8.0 ksi |
| Phenolic Grade C |
7,000 psi |
7.0 ksi |
| Phenolic Grade L |
6,000 psi |
6.0 ksi |
⚠
These are conservative preliminary-design values. Case-hardened, nitrided,
or shot-peened gears may support significantly higher allowable stresses. Consult
AGMA 2101 or your material supplier for certified allowable stress numbers.
⌀Measurement Over Wires / Pins
Measurement Over Wires or Pins (MOW) is the standard shop-floor method for
verifying gear tooth thickness using a micrometer and precision pins seated
in the tooth spaces. It is more accurate than span measurement for gears with
odd tooth counts and gives a direct check of the effective tooth thickness
including any profile shift.
Toggle the Measurement Over Wires/Pins switch to show or hide
these dimensions in the results. The engine calculates the measurement for both
the pinion and the gear.
Standard Pin Diameter
Calculated as 1.728 / DP per Machinery's Handbook.
This is the wire or pin diameter that sits at approximately the pitch circle, giving
the most accurate tooth thickness measurement. The engine displays this value
automatically once a gear set is found.
Custom Pin Diameter (optional)
If you have a specific pin size available in your inspection
tooling, enter it here. The engine will compute MOW for your custom pin instead.
Both the standard and custom results are displayed side-by-side for comparison.
inv(φ₁) = t/d + inv(φ) + d_pin/d_b − π/N
Even tooth count: M = d_b / cos(φ₁) + d_pin
Odd tooth count: M = d_b · cos(90°/N) / cos(φ₁) + d_pin
ℹ
When profile shift is applied, the effective tooth thickness changes: the engine
adjusts t → t + 2x·tan(φ)/P and recalculates φ₁ for the shifted gear. The results
panel shows both the standard (unshifted) and profile-shift-corrected MOW values
side by side, with the difference ΔM highlighted.
▣Understanding the Result Cards
Pitch System
The pitch standard of the winning candidate. In Auto mode this
is the best result across all three systems. Colored cyan (DP), purple (Module),
or orange (CP).
Pitch Value
The specific pitch value selected — e.g. "8 DP", "m2.50",
or "p = 1/2"". AGMA preferred values receive a slight scoring bonus.
Pressure Angle
Shows both the standard pressure angle and the operating angle.
These differ when profile shift is applied — the operating angle increases as the
center distance opens up.
Teeth (Pinion / Gear)
Tooth counts for the pinion (driver) and gear (driven). The
engine tests ±12 offsets from the ideal tooth count and ±4 gear tooth offsets per
pitch value to find the best fit.
Face Width
Minimum face width to carry the load with the service factor
applied, calculated by the Lewis equation. Rounded up to the nearest 0.25 in (DP ≤ 24)
or 0.01 in (fine pitch, DP > 24). A 10% margin is added before rounding when torque
is specified. Capped at 10/DP.
Ratio
Actual gear ratio (gear teeth ÷ pinion teeth). If this differs
from your target by more than 0.5% the card turns amber with the deviation shown.
The actual output speed card reflects this ratio.
Center Distance
Actual pitch-circle center distance of the chosen gear set.
May differ slightly from input — any difference exceeding 0.5% triggers an amber
warning. Profile shift corrects the operating geometry to your exact input distance.
Outside Diameters
Standard OD = (N+2)/DP. Modified OD = (N+2+2x)/DP when profile
shift is applied. Use the modified OD for blanks when specifying profile-shifted gears.
Horsepower
Power based on output torque and output speed. If HP was entered,
this reflects the effective HP after efficiency and friction corrections.
Input / Output Torque
Output torque is either your required load or the maximum safe
capacity. Input torque is back-calculated through the gear ratio and efficiency factor.
Toggle the unit selector (ft-lb / in-lb) to convert.
Velocity Factor (Kv)
Barth velocity factor — a derating factor for dynamic load at
speed. Kv = 600 / (600 + V) where V is pitch-line velocity in ft/min.
Lower Kv = higher speed = lower allowable load.
Profile Shift
Shows x-sum and individual shift coefficients when the actual
center distance differs from your input. Green = negligible; amber = moderate;
red = excessive (may compromise tooth geometry).
xProfile Shift Explained
Standard spur gears are cut with the tool rack centered on the pitch circle. When
you need a center distance that does not fall exactly on a standard pitch combination,
the tooth profiles are shifted radially inward or outward — this is profile shift.
The total shift coefficient is x_sum = (C_required − C_actual) × DP.
Positive x_sum means the gears are moved apart (positive shift); negative means they
are moved closer (negative shift, also called undershift).
x_sum = (C_required − C_actual) × DP
x1 = x_sum × pinionFraction (up to 80% of x_sum)
x2 = x_sum − x1
inv(φ_op) = inv(φ) + 2 × x_sum × tan(φ) / (N1 + N2)
Shift quality guide (|x_sum|):
0 – 0.05 · Negligible
Standard gears will work as-is. No special drawing callout needed.
0.05 – 0.25 · Minor
Well within standard practice. Specify x1 and x2 on drawings.
0.25 – 0.50 · Moderate
Acceptable. Specify profile shift on drawings and verify tip clearance.
0.50 – 0.90 · Large
Consult a gear engineer. Check tip narrowing, contact ratio, and backlash.
Above 0.90 · Excessive
Tooth geometry may be compromised. Consider a different pitch value
or change the center distance requirement.
ℹ
When the pinion tooth count is below the undercutting threshold for the selected
pressure angle, the engine automatically redistributes more shift to the pinion
(anti-undercut redistribution). This is shown in the Profile Shift card with the
pinion fraction highlighted in amber.
✓Safety Margin Badge
The badge in the top-right of the results panel shows the safety margin — the ratio
of the gear set's calculated load capacity to the required torque, after the service
factor has been applied.
Green — Margin ≥ 1.0
The gear set can carry the required load with the service factor
applied. A margin of 1.0 means the gears are sized exactly for the load; 1.5 means
they have 50% additional capacity.
Red — Margin < 1.0
The gear set cannot carry the required load. Try a stronger
material, reduce the torque input, increase center distance, or lower the service factor.
Cyan — Capacity Mode
No torque was entered. The engine is showing the maximum safe
torque capacity of the recommended gear set rather than checking against a specific load.
✓
When profile shift is applied, the safety margin shown is the corrected
value — it accounts for the updated Lewis Y factor (applied individually to each gear
using x1 and x2), the operating pressure angle, and the pitch-line velocity at the
operating pitch circle. This is shown with the "(PS corrected)" notation.
ExWorked Example
The following example walks through a complete design from requirements to results.
Scenario: A centrifugal pump drive. The gearbox must fit within
existing housing with a 4.000 inch center distance. Input shaft turns at 1750 rpm;
output shaft must turn near 583 rpm (3:1 ratio). Pump requires 25 ft-lb output
torque. Material is 4140 Steel. Service factor for centrifugal pumps running
two-shift is 1.50.
Typical Results
Pitch Value8 DP
Teeth (Pinion/Gear)21 / 63
Actual Ratio3.000:1
Actual C.D.5.250 in
x_sum−10.0
Face Width~1.00 in
Safety Margin> 1.0 ✓
✓
Try entering these values in the calculator now to see the full result set — pitch
equivalents, profile shift details, operating pressure angle, outside diameters,
and input torque all update instantly.
AzGlossary
AGMA
American Gear Manufacturers Association. Sets gear design standards including tooth geometry, rating methods, and allowable
stress values used throughout this engine.
Base Circle
The circle from which the involute tooth profile is generated. Diameter d_b = d × cos(φ), where d is pitch diameter and φ is
pressure angle.
Center Distance
Distance between the centerlines of the two meshing gear shafts. For standard gears: C = (N₁ + N₂) / (2 × DP).
Diametral Pitch (DP)
Number of teeth per inch of pitch diameter. The primary pitch parameter in the US inch system. Higher DP = finer teeth.
Face Width
The length of the gear tooth measured along the shaft axis. Determines load capacity — wider face = more load. Limited to
8/DP – 10/DP for spur gears.
Involute
The tooth profile curve used for virtually all modern spur and helical gears. Generated by unwrapping a string from the base
circle. Ensures smooth meshing regardless of minor center distance variations.
Lewis Form Factor (Y)
A factor in the Lewis beam equation that accounts for tooth shape as a function of tooth count and pressure angle. Higher Y
= stronger tooth form.
Module (m)
Metric pitch parameter. m = pitch diameter (mm) / tooth count = 25.4 / DP. Higher module = coarser, larger teeth.
MOW (Measurement Over Wires/Pins)
A shop-floor inspection method for verifying gear tooth thickness using precision pins seated in tooth spaces and measured
with a micrometer.
Pinion
The smaller of the two meshing gears — always the driver in a speed-reduction arrangement. Has the lower tooth count.
Pitch Circle
The imaginary circle on which two gears effectively roll without sliding. For standard gears: d = N / DP.
Pitch-Line Velocity (V)
The tangential velocity at the pitch circle in ft/min. V = π × d_pinion × N_rpm / 12. High velocity requires dynamic
derating via the Barth velocity factor.
Pressure Angle (φ)
The angle between the tooth profile at the pitch point and a tangent to the pitch circle. Common values: 14.5°, 20°, 25°.
Affects tooth strength and separating force.
Profile Shift (x)
Radial displacement of the gear cutter from the standard position. Used to achieve non-standard center distances, prevent
undercutting, or equalize strength between pinion and gear.
Safety Margin
Ratio of gear set load capacity to required load, after applying the service factor. A margin of 1.0 or above means the
design is adequate.
Service Factor (SF)
A multiplier on the required load that accounts for shock, duty cycle, and application severity. Per AGMA 2001 guidelines.
Undercutting
Removal of material at the tooth root during cutting when the tooth count is too low. Weakens the tooth and can interfere
with meshing. Prevented by positive profile shift or increasing tooth count above the threshold.
Velocity Factor (Kv)
The Barth velocity factor — a derating multiplier for dynamic load at speed. Kv = 600 / (600 + V). Ranges from 1.0
(stationary) downward as speed increases.
Spur Gear Set Design Engine™ is a trademark of Rush Gears Inc.
Copyright © 2026 Rush Gears Inc. All rights reserved.
Results are for reference only. Final designs must be verified by a qualified engineer.