Presets
Playback
Gear train
Tooth counts of the four gears and the efficiency of each mesh.
Motor and load
Supply voltage, hanging mass, winch drum radius and drag.
Time
Slow motion makes the fast gears visible.
Layers
Model checks
The science
What this lab models
A small brushed DC motor drives a two-stage compound spur-gear train: pinion A drives gear B, pinion C on the same shaft drives gear D, and gear D turns a winch drum that lifts a weight. The motor, the mesh efficiency, the tooth counts, the drum radius and the load are all adjustable, and the whole run is integrated in time.
Gear ratio, speed and torque
Two meshing gears have the same pitch-line speed, so w1 r1 = w2 r2 and, with the pitch radius r = m z / 2, the speed ratio is the inverse of the tooth ratio. For the compound train the overall ratio is i = (z2 / z1)(z4 / z3). Power is conserved up to losses, so the torque is multiplied by the same factor: T_out = i eta T_in, where eta is the product of the mesh efficiencies. The tangential tooth force at a mesh is F = T / r_pitch.
The DC motor
Armature current I = (V - Ke w) / R, torque T = Kt I and back-EMF Ke w give a straight torque-speed line from the stall torque Kt V / R down to the no-load speed V / Ke. Mechanical power T w peaks at half of the no-load speed with P_max = V^2 Kt / (4 R Ke). This is why a gear train matters: it moves the load to a speed where the motor is efficient.
Dynamics
Reflected to the motor shaft the whole machine has the inertia J_eq = J_motor + I_A + (I_B + I_C)/i1^2 + (I_D + J_load)/i^2, and J_eq dw/dt = T_motor - T_load / (i eta^k) - b w. The gear efficiency enters as 1/eta when the motor drives the load and as eta when the load drives the motor (lowering the weight), so spur gears are not self-locking. At steady state the speed is w = (Kt V / R - T_load / (i eta^2)) / (Kt Ke / R + b).
Energy ledger
The electrical energy V I dt is split into copper loss I^2 R, motor friction, gear loss, drag at the drum, potential energy of the weight and kinetic energy. The check below the controls verifies that the books close.
Numerical checks on this page
(1) The simulated steady lifting speed matches the closed-form expression above. (2) The involute tooth outlines of both meshes are rotated through a full tooth pitch with the kinematic mesh law, and no tooth cuts into the other gear (this also exposes the real interference of a small pinion with a large gear). (3) The energy ledger closes and the motor power curve peaks at the textbook value.
Honest simplifications
The teeth are ideal 20 degree involutes without backlash, undercut, deflection or lubrication; the mesh efficiency is a constant you choose. The motor model is linear (no brush drop, temperature or saturation) and the drum rope is rigid and massless. The weight stops at the top and bottom limits of the 28 cm travel, where the motor stalls.
What to try
- Lift 1.5 kg at 12 V: watch the weight rise and compare the rpm of A, B and D; the output is 16 times slower than the motor.
- High ratio versus Low ratio: a bigger ratio lifts heavier loads but slower.
- Overload: the motor stalls: the current rises to V / R and the motor is no longer turning.
- Compare Lossy gears with Ideal gears in graph 4: the efficiency curve shifts down by roughly eta squared.
- Lower the weight, motor shorted: the weight drives the motor as a generator and its own back-EMF brakes the descent.