M029 · Fixed-axis transmission

Gear Train & Idler

Turn the input, change tooth counts and insert an idler. Follow real involute teeth, signed shaft speeds and the torque traded for speed.

Input speed $\text{Ready}$

Idler speed $\text{Ready}$

Output speed $\text{Ready}$

Output / input speed $\text{Ready}$

Output external torque $\text{Ready}$

Net port power $\text{Ready}$

TEETH / SPEED / POWER

Change the teeth. Follow the motion.

Input teeth

N_1

24.00\,

Idler teeth

N_i

36.00\,

Output teeth

N_o

48.00\,

Insert idler

b_i

1.00\,

Input speed

n_{\mathrm{in}}

30.00\,

Applied input 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.

An idler reverses the output relative to a direct pair but its own tooth count cancels from the end-to-end ratio. Its center moves when its size changes. The stationary frame supplies the net reaction torque.

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.

Ideal rigid, lossless gearing at prescribed steady speeds. Involute flanks; schematic root joins, shafts and supports. No tooth stress, inertia or friction.

Start with the essentials

Focus question: Does a larger idler make the output slower?
One-sentence intuition: The idler ratio appears once in each direction and cancels from the final ratio.

Core mathematical model

A single mesh

N_a n_a+N_b n_b=0,\quad r_j=\frac{mN_j}{2}

Pitch-circle surface velocities agree at the reference contact. Gear axes are fixed.

With and without an idler

\frac{n_o}{n_1}=\begin{cases}-N_1/N_o&\text{direct}\\+N_1/N_o&\text{one idler}\end{cases}

Changing the idler size changes its own speed and the layout, but not the output speed.

Steady external torques

T_o=-\frac{T_1}{n_o/n_1},\quad T_{\mathrm{frame}}=-T_1-T_o

The applied output torque opposes the power supplied by the input. The held frame absorbs the net reaction. The ratio stays defined at zero speed.

Power balance

P_j=T_j\frac{2\pi n_j}{60},\quad P_1+P_o=0

Zero frame speed gives zero frame power even with a nonzero reaction torque.

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
Add one gear
Compare Direct pair with Insert an idler and scrub time.
What to observe: The output reverses direction, while the magnitude of the end-to-end ratio stays the same.
02
Resize only the idler
Change the middle tooth count while keeping the end gears fixed.
What to observe: The middle shaft moves and its speed changes; output speed and torque remain fixed.
03
Trade speed for torque
Increase the output tooth count and inspect the signed power row.
What to observe: Output speed falls, compatible load torque grows, and ideal input/output power cancel.