Gear Train and Four-Bar Linkage Lab (2D)

What is modelled

The front view of a small bench machine: a motor drives a two-stage spur-gear reducer, and the slow output shaft turns the crank of a four-bar linkage whose rocker swings a load. Everything on screen is drawn from the same equations used in the readouts, so the machine and the numbers cannot disagree. Gears use a module of 0.1 length units per tooth of pitch diameter.

Equations

• Gear train: each external mesh reverses the direction and gives ωout/ωin = -Zdriver/Zdriven. Two stages: i = (Z2/Z1)(Z4/Z3) input turns per output turn. Teeth stay in mesh when θb = α + π - π/Zb - (θa - α) Za/Zb, where α is the direction of the line of centres. The centre distance is the sum of the pitch radii r = m Z / 2.
• Torque and power: motor torque T = P/ω; through the train Tcrank = T i ηn and ωcrank = ω/i with n = 2 meshes and efficiency η per mesh (typical spur gears 0.97 to 0.99).
• Four-bar position: the coupler point C is the intersection of the circle of radius b around the crank pin B and the circle of radius c around the rocker pivot O₄. It exists only if |b - c| ≤ |BO4| ≤ b + c.
• Velocity ratio from instant centres: ω4/ω2 = a sin(θ2 - θ3) / (c sin(θ4 - θ3)). For an ideal (frictionless) linkage power in equals power out, so Trocker = Tcrank/|ω4/ω2|.
• Transmission angle: cos μ = (b2 + c2 - D2)/(2 b c) where D is the distance from B to O₄. Designers keep μ between about 40 and 140 degrees; near 0 or 180 the rocker has a huge mechanical disadvantage.
• Grashof rule: with shortest link s, longest l and the other two p, q, at least one link can turn fully if s + l ≤ p + q. The crank turns fully when the shortest link is the crank or the ground link. Otherwise the crank cannot turn through every angle: the simulated motor reverses at the limit, as a real reversing drive would.

Checks run in the page

• Gear meshing: tooth and gap alignment on the line of centres for 2000 random angle and tooth-count pairs.
• Train ratio from the tooth counts against the turns measured through the meshing formulas.
• Four-bar closure (link lengths constant) and the velocity-analysis ratio against a finite difference.
• The Grashof rule against a brute-force 360 degree reach test for 300 random link sets.

Honest simplifications

Rigid links, no clearances, no pivot friction (only a constant efficiency per gear mesh), no inertia or dynamics (the motion is purely kinematic, slowed down by the slow-motion factor), straight trapezoid teeth instead of involute profiles, no shaft or bearing loads and no stress analysis. Torques are shown for a steady motor power; a real motor has a torque-speed curve.

What to try

• Choose "Double-rocker (crank jams)" and watch the motor reverse at the limit; the Grashof readout explains why.
• Shorten the ground link until the rocker also turns fully (drag link).
• "Poor transmission angle": see the transmission-angle graph dip under 40 degrees and the rocker torque spike.
• Swap the gear teeth: 12-48-12-48 turns the crank 16 times slower and multiplies its torque by about 15 (98 percent per mesh).