SME
Kinematics of Mechanisms · Problem type
Planetary train (Willis’s formula)
Use this for “A planetary set has a 24-tooth sun and a 72-tooth ring. The ring is held and the sun turns at 1200 rpm. How fast does the arm turn?”
Chart
Sun teeth Nₛ = 24
Ring teeth Nᵣ = 72
Planet teeth Nₚ = 24
This is an example. Type your own values to start a new problem.
Nᵣ
×
(
nᵣ (rpm)
−
nₐ (rpm)
)
=
−
Nₛ
×
(
nₛ (rpm)
−
nₐ (rpm)
)
Nₚ
Planet teeth
worked out (not typed)
Formulas
Nᵣ × (nᵣ − nₐ) = −Nₛ × (nₛ − nₐ)
72 × (0 − 300) = −24 × (1,200 − 300)
Nₚ
=
Nᵣ − Nₛ
2
24
=
72 − 24
2
Assumptions
•
One sun, its planets and the ring share a module, so Nᵣ = Nₛ + 2Nₚ.
•
Speeds are + for one way of turning and − for the other.
•
Seen from the arm, the train is fixed-axis: sun to ring is −Nₛ ÷ Nᵣ.
Step-by-step
WE KNOW
Nₛ = 24, Nᵣ = 72, nₛ = +1,200 rpm, nᵣ = 0 rpm
FIND
arm speed (nₐ), planet teeth (Nₚ)
Step 1 · Find arm speed (nₐ)
Use Nᵣ × (nᵣ − nₐ) = −Nₛ × (nₛ − nₐ)
Seen
from
the
arm,
the
ring
turns
−
Nₛ
Nᵣ
times
the
sun;
gather
the
arm’s
speed
on
one
side.
nₐ
=
Nᵣ × nᵣ + Nₛ × nₛ
Nᵣ + Nₛ
nₐ
=
72 × 0 + 24 × 1,200
72 + 24
nₐ
=
28,800
96
nₐ = +300 rpm
Step 2 · Find planet teeth (Nₚ)
Use Nₚ = (Nᵣ − Nₛ) ÷ 2
Across the ring: one sun and two planets fill it, so each planet takes half of what is left.
Nₚ
=
72 − 24
2
Nₚ
=
48
2
Nₚ = 24
Check
Put the numbers back in. Both sides should match.
72 × (0 − 300) = −24 × (1,200 − 300) ✓
24
=
72 − 24
2
✓
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