Mechanisms & Linkages

Degrees of Freedom (Gruebler)

Planar: F = 3(n−1) − 2·j₁ − j₂.

Links n (incl. ground)

Full joints j₁ (1-DOF)

Half joints j₂ (2-DOF)

Four-Bar Grashof

Order: ground, crank (input), coupler, rocker (output).

Ground

Crank

Coupler

Rocker

Spur Gear Ratio & Geometry

i = N₂/N₁; d = m·N; center = (d₁+d₂)/2.

Driver teeth N₁

Driven teeth N₂

Module m (mm)

Input speed (rpm)

Input torque (N·m)

Planetary (Epicyclic) Gear

(ω_ring−ω_carrier)/(ω_sun−ω_carrier) = −N_sun/N_ring.

Sun teeth N_s

Ring teeth N_r

Sun speed (rpm)

Carrier speed (rpm)

Slider-Crank

x = r·cosθ + l·√(1−(r/l)²sin²θ).

Crank radius r (mm)

Rod length l (mm)

Crank angle θ (°)

Crank speed (rpm)

Cam — Cycloidal Motion

s = L[θ/β − sin(2πθ/β)/2π]. Smooth, finite jerk.

Total lift L (mm)

Rise angle β (°)

Cam angle θ (°, 0–β)

Cam speed (rpm)

Geneva Drive

External Geneva: driven wheel indexes 360/n per crank turn.

Number of slots n

Crank radius R (mm) — for center distance

Formula reference

DOF: planar F=3(n−1)−2j₁−j₂; F=1 driven mechanism, F=0 structure, F<0 over-constrained.

Grashof: with shortest s, longest l, others p,q: s+l≤p+q ⇒ Grashof. Shortest = ground → double-crank; = a side link → crank-rocker; = coupler → double-rocker. s+l>p+q ⇒ triple-rocker.

Gears: ω₂/ω₁=N₁/N₂; pitch dia d=m·N; center distance (d₁+d₂)/2; torque scales with the ratio (ideal).

Planetary: (ω_r−ω_c)/(ω_s−ω_c)=−N_s/N_r; planet teeth (N_r−N_s)/2; ring-fixed sun→carrier ratio 1+N_r/N_s.

Slider-crank: x=r·cosθ+l·√(1−(r/l)²sin²θ); stroke 2r; velocity v=ω·dx/dθ.

Cam (cycloidal): s=L[θ/β−sin(2πθ/β)/2π]; peak velocity 2Lω/β; peak accel 2πLω²/β² (β in rad).

Geneva: index angle 360/n; crank motion during indexing 180°−360°/n; dwell 180°+360°/n; center distance R/sin(180°/n).