M029 · Moving-axis transmission

Planetary Gears & Two Inputs

Hold the ring, sun or carrier, then unlock two inputs. See three planets spin and orbit while the same tooth constraint balances every speed and power port.

Sun speed $\text{Ready}$

Ring speed $\text{Ready}$

Carrier speed $\text{Ready}$

Planet spin in fixed frame $\text{Ready}$

Carrier external torque $\text{Ready}$

Net port power $\text{Ready}$

TEETH / SPEED / POWER

Change the teeth. Follow the motion.

Sun teeth

N_s

24.00\,

Planet teeth

N_p

18.00\,

Held member selection

b_m

0.00\,

Driven member speed

n_{\mathrm{in}}

30.00\,

Second input: ring speed

n_r

0.00\,

Applied drive torque

T_{\mathrm{in}}

1.00\,

All shafts point toward the viewer in front view; counterclockwise is positive. Tooth module is two millimetres, reference pressure angle twenty degrees. All geometry uses one uniform millimetre scale. Playback runs one model second per real second and starts paused. White dots mark material points.

Torque means external torque applied to the gear set, not delivered load torque. Positive power enters the set; negative power leaves it. A held member can carry reaction torque while its power is zero. Torques are quasistatic compatible loads; speeds do not respond dynamically to them.

The ring tooth count is derived from sun and planet counts. Choices preserve three-planet assembly phasing and clearance. Two-input mode prescribes sun and ring speeds independently; the carrier is their tooth-weighted average. This is a simple spur planetary differential, not an automotive bevel differential.

The twenty-degree involute flanks are sampled into a mesh. Tooth roots use schematic radial joins instead of a generated cutter fillet; shafts, bearings, face width and carrier arms are illustrative. No contact-force integration, elastic deflection, friction, backlash, impact, stress, thermal or manufacturing certification is implied. Pitch-contact markers are kinematic references, not the actual moving tooth contact.

One ideal simple planetary set with three equally spaced planets. Prescribed steady speeds and external port torques; no bevel differential, clutch dynamics, backlash or efficiency model.

Start with the essentials

Focus question: Why does changing the held member change the ratio?
One-sentence intuition: Subtract carrier rotation first; in that frame every planet axis is fixed.

Core mathematical model

Assembly geometry

N_r=N_s+2N_p,\quad r_j=\frac{mN_j}{2}

Equal module and concentric sun/ring fix the planet-center distance. The module here is two millimetres.

One speed constraint

N_s(n_s-n_c)+N_r(n_r-n_c)=0

The relation is nonsingular even when all three members turn together.

Absolute planet spin

n_p=n_c-\frac{N_s}{N_p}(n_s-n_c)

The white material-point marker uses fixed-frame spin, not just spin relative to its carrier.

Compatible external torques

(T_s,T_r,T_c)=q\,(N_s,N_r,-N_s-N_r)

The multiplier is fixed by the chosen drive torque. The held port supplies its reaction.

Signed power at every port

P_j=T_j\frac{2\pi n_j}{60},\quad P_s+P_r+P_c=0

Power is in watts for torque in newton metres and speed in revolutions per minute. Positive enters the set.

Common difficulties

A slower shaft can carry more torque

Typical misconceptionSpeed reduction destroys power.

Better mental modelIn this ideal steady model, torque increases as speed decreases. Real losses and transient inertia require additional physics.

Reference contact is not a material point

Typical misconceptionThe pitch circles describe all actual tooth contact.

Better mental modelInvolute tooth contact travels along the line of action and generally includes sliding. The marked pitch point only verifies the transmission kinematics.

Run the experiment

01
Hold the ring
Play the default set and compare gold sun, blue planets and coral carrier.
What to observe: The carrier turns more slowly in the same direction. The stationary ring carries torque with zero power.
02
Change the held member
Select Carrier held, then Sun held. Use the front view to read tooth motion.
What to observe: With the carrier held the ring reverses. With the sun held the ring drives the carrier.
03
Use two inputs
Choose Both inputs together, then Opposing inputs. Change the second input speed.
What to observe: Equal inputs rotate as a rigid assembly. Opposing inputs can stop the carrier while the teeth continue to move.