Machine design / Sheet 06
Gearbox calculations
Compound, planetary and harmonic drives. Ratio multiplies torque and divides speed, but it divides reflected inertia by the square, and that is usually the reason the gearbox is there.
Scope: Spur, planetary, strain waveSections: 13Units: SI, N·m and mmRevision: A
74 formulas
Gear geometry
i = ωin/ωout = Nout/Nin = dout/din
Teeth ratio, diameter ratio and speed ratio are the same number.
Module
m = d / N
mm. Imperial equivalent is diametral pitch P = N/d
Pitch diameter
d = m N
Meshing gears must share the same module
Circular pitch
p = π m
Arc distance between adjacent teeth
Centre distance
C = m (N1 + N2) / 2
External mesh; subtract for internal
Base circle
db = d cos φ
φ = 20° standard pressure angle
Tooth proportions
a = m, b = 1.25 m
Addendum and dedendum, full depth teeth
Minimum teeth
Nmin = 2 / sin²φ ≈ 17
Below this the pinion undercuts, unless profile shifted
Contact ratio
mp = length of action / base pitch
Need above 1.2, prefer 1.4; below 1.0 the mesh drops out
Helical, normal module
mn = mt cos β
β is the helix angle, usually 15° to 30°
Helical, virtual teeth
Ne = N / cos³β
Use Ne for strength lookups
Worm ratio
i = Nwheel / Nstarts
Starts, not teeth. Very high ratio from one stage
Worm self-locking
λ < friction angle, about 5°
λ is the lead angle; self-locking costs efficiency
Forces, power and inertia
Power
P = T ω, ω = 2πN/60
Power is constant through an ideal gearbox
Output torque
Tout = Tin · i · η
Ratio multiplies torque, efficiency subtracts from it
Tangential force
Ft = 2 T / d
The force that actually does work
Radial force, spur
Fr = Ft tan φ
Separating force, loads the bearings
Axial thrust, helical
Fa = Ft tan β
The price of quiet running; needs a thrust bearing
Pitch line velocity
V = π d n / 60
Drives the dynamic factor and lubrication regime
Reflected inertia
Jref = Jload / i²
Divided by the SQUARE. This is why gearboxes exist in robotics
Inertia matching
Jref / Jmotor = 1 to 10
Target band for responsive servo control
Heat generated
Q = Pin (1 - η)
Often the real packaging constraint
Tooth strength
Lewis bending
σ = Ft / (b m Y)
b is face width, Y the Lewis form factor
Barth velocity factor
Kv = (6.1 + V) / 6.1
V in m/s; multiply the stress by this
AGMA bending
σ = Ft Ko Kv Ks (1 / b mt)(KH KB / YJ)
Ko overload, KH load distribution
AGMA contact
σc = ZE √(Ft Ko Kv Ks KH / (b dw1 ZI))
Hertzian pitting; usually governs for hardened gears
Face width rule
b = 9m to 14m
Too wide and load distribution degrades
Two failure modes
tooth root bending, flank pitting
Bending is sudden, pitting is progressive; check both
Bending fatigue at the tooth root is a fatigue problem, so everything on sheet 05 applies: the root fillet radius is a stress concentration, and shot peening the root is a standard fix.
Simple and compound trains
Simple train
i = Nlast / Nfirst
Idlers change direction only, never the ratio
Compound train
i = (N2 N4 N6…) / (N1 N3 N5…)
Product of driven over driver, stage by stage
Reverted train
N1 + N2 = N3 + N4
Condition for coaxial input and output, same module
Efficiency
ηtotal = η1 · η2 · η3
Three stages at 0.98 give 0.94 overall
Ratio per stage
up to 6:1, occasionally 7:1
Beyond that the pinion gets too small
Referred backlash
divided by the downstream ratio
The LAST stage dominates output backlash
Direction
external mesh reverses, internal does not
Count the external meshes to get the output sign
Planetary geometry
Tooth relation
NR = NS + 2 NP
Geometric necessity, not a design choice
Assembly condition
(NS + NR) / n = integer
n is the planet count, for equal spacing
Planet clearance
(NS + NP) sin(π/n) > NP + 2
Adjacent planets must not collide
Carrier radius
rC = m (NS + NP) / 2
Sun and planet centre distance
Load sharing
F per planet = Ftotal / n · Kγ
Kγ = 1.1 to 1.3, sharing is never perfect
Planetary ratios
| Held | Input | Output | Ratio | Direction |
|---|---|---|---|---|
| Ring | Sun | Carrier | 1 + NR/NS | same |
| Ring | Carrier | Sun | 1 / (1 + NR/NS) | same, overdrive |
| Sun | Ring | Carrier | 1 + NS/NR | same, 1.2 to 1.5 |
| Sun | Carrier | Ring | 1 / (1 + NS/NR) | same, overdrive |
| Carrier | Sun | Ring | -NR/NS | reversed |
| Any two locked | - | - | 1 | direct drive |
Willis equation
(ωS - ωC) / (ωR - ωC) = -NR/NS
The general relation; every row above falls out of it
Tabular method
lock and rotate +1, then arrest
Superposition fallback if you forget the table
Torque balance
TS + TR + TC = 0
External torques sum to zero in steady state
Practical ratio per stage
3:1 to 10:1
Ceiling is geometric, see the note below
Efficiency
0.97 to 0.98 per stage
Higher than compound spur for the same ratio
The 10:1 ceiling is geometry, not convention. i = 1 + NR/NS needs a tiny sun and a huge ring, while NR = NS + 2NP forces the planets to grow to match, and the sun undercuts below 17 teeth. That gap is exactly what harmonic drives fill.
Harmonic (strain wave) drive
Three parts
wave generator, flexspline, circular spline
Elliptical bearing inside a flexible cup inside a rigid ring
Tooth difference
NCS - NFS = 2
The ellipse creates two mesh regions, so two teeth
Circular spline held
i = -NFS / 2
WG in, flexspline out; output reverses
Flexspline held
i = +NCS / 2
WG in, circular spline out; same direction
General form
i = NFS / (NCS - NFS)
100:1 needs a 200 tooth flexspline
Typical ratios
30:1 to 320:1 in one stage
Nothing else comes close in one coaxial stage
Teeth in mesh
15 to 30 percent of all teeth
The source of the torque density and zero backlash
Torsional windup
θ = T / K
K is nonlinear, quoted as K1, K2, K3 by torque band
Average torque
Tav = [Σ(ni Ti³) / Σni]1/3
Cubic weighting, same form as bearing life
Life
L = L10 (Trated/Tav)³ (nrated/nin)
Wave generator bearing usually limits it
Four torque ratings
rated, repeated peak, momentary peak, ratcheting
Ratcheting is where flexspline teeth jump; unique to this drive
Efficiency
0.70 to 0.90
Falls sharply at low speed and low temperature
Architecture comparison
| Type | Ratio per stage | Efficiency | Backlash | Notes |
|---|---|---|---|---|
| Spur, compound | up to 6:1 | 0.98 | moderate | cheapest, offset shafts |
| Helical | up to 8:1 | 0.98 | moderate | quieter, adds axial thrust |
| Bevel | up to 5:1 | 0.97 | moderate | right angle drive |
| Worm | 5:1 to 100:1 | 0.40 to 0.90 | low | can self-lock, runs hot |
| Planetary | 3:1 to 10:1 | 0.97 | moderate to low | coaxial, torque dense |
| Cycloidal | 10:1 to 120:1 | 0.85 to 0.93 | very low | stiff, shock tolerant, heavy |
| Harmonic | 30:1 to 320:1 | 0.70 to 0.90 | essentially zero | light, expensive, winds up |
Selection logic
Need high ratio, coaxial, zero backlash
harmonic
Robot joints, gimbals, precision positioners
Need high ratio plus shock capacity
cycloidal
Stiffer than harmonic, tolerates impact
Need torque density and efficiency
planetary
The default for servo actuators up to 100:1 in stages
Cost is the constraint
compound spur or helical
Accept offset shafts and backlash
Need self-locking
worm
Accept the efficiency and heat penalty
Worked chain: sizing a drive
Step 1
Tout, nout, duty cycle
Peak and continuous separately, from the load
Step 2
i = nmotor / nout
Then Tmotor = Tout / (i η)
Step 3
Jref = Jload/i², compare to Jmotor
Aim for a ratio between 1 and 10
Step 4
check peak torque and ratcheting
Emergency stop and collision cases, not just running load
Step 5
Q = Pin(1 - η)
Thermal check; continuous duty often limits before torque
Step 6
backlash and torsional stiffness
Only if positioning accuracy matters; then add an output encoder
Step 7
life from the cubic average torque
Not from the peak, and not from the average of torques
Worked example: 100:1 robot joint
Single planetary attempt
1 + NR/NS = 100 → NR = 99 NS
With NS = 18 that needs NR = 1782. Impossible
Two planetary stages
10 × 10
Each stage near its geometric ceiling; long package
Three compound spur stages
4.64 × 4.64 × 4.64
η = 0.94, offset shafts, backlash from three meshes
Single harmonic
NFS = 200, NCS = 202
One coaxial stage, zero backlash, short axial length
What you give up
stiffness, efficiency, cost
Windup under load is invisible to a motor-side encoder
Numbers worth memorizing
| Quantity | Value | Comment |
|---|---|---|
| Pressure angle | 20° | 14.5° legacy, 25° for higher capacity |
| Minimum pinion teeth | 17 | use 18 for margin, or profile shift |
| Contact ratio target | 1.4 | 1.2 minimum |
| Helix angle | 15° to 30° | thrust rises with tan β |
| Face width | 9m to 14m | wider degrades load distribution |
| Spur mesh efficiency | 0.98 to 0.99 | per mesh, multiplies per stage |
| Planet count | 3 typical | 4 or 5 for higher torque density |
| Harmonic tooth difference | 2 | ratio = teeth / 2 |
| Inertia ratio target | 1:1 to 10:1 | reflected load to motor |
Concepts that decide the interview
- Ratio multiplies torque linearly but divides reflected inertia by the square. A 100:1 reduction makes the load inertia 10,000 times smaller at the motor, and that is what makes a joint controllable. It is the real answer to why direct drive is hard.
- Efficiency compounds while ratio does not, and the losses become heat. A 500 W input at 80 percent efficiency puts 100 W into the housing, which often constrains the package before torque does.
- Planetary torque density comes from load sharing across the planets, plus coaxial input and output. That combination is why nearly every robot joint and automatic transmission uses them.
- The single-stage planetary ceiling near 10:1 is geometric, forced by NR = NS + 2NP and the 17 tooth undercut limit. Harmonic drives exist to fill exactly that gap.
- Harmonic zero backlash comes from teeth preloaded into mesh by flexspline deflection, and 15 to 30 percent of teeth engaged at once. The cost is modest nonlinear torsional stiffness, so the joint winds up under load.
- Windup is invisible to a motor-side encoder, which is why precision robots add a second encoder on the output. Good detail to volunteer when asked about accuracy versus repeatability.
- Backlash referred to the output is divided by the downstream ratio, so the last stage dominates. Spend precision on the output stage, not the input one.
- Always size on both peak and continuous. Continuous sets thermal life, peak decides whether teeth survive an emergency stop, and harmonic drives add a separate ratcheting limit.
- Tooth root bending is a fatigue problem, so root fillet radius, surface finish and shot peening all apply exactly as they do on the fatigue sheet.
Sheet 06 of 07 / Rev AInvolute teeth, standard proportions unless noted