Grubler’s criterion (planar): F = 3(n−1) − 2j₁ − j₂

n = links, j₁ = lower pairs, j₂ = higher pairs. F = 1 → constrained mechanism.

Grashof’s law: s + l ≤ p + q (for four-bar with full rotation)

Slider-crank displacement: x = r(1−cosθ) + r²sin²θ/(2l)

Fundamentals

Speed ratio: N₁/N₂ = T₂/T₁ (meshing gears)

Compound train: Overall ratio = Product of driven teeth / Product of driving teeth

Module: m = D/T (mm). Circular pitch: p_c = πm

Centre distance: C = m(T₁ + T₂)/2 (external), C = m(T₂ − T₁)/2 (internal)

Epicyclic (Planetary)

Tooth relation: T_ring = T_sun + 2T_planet

Tabular method:

Fix arm, rotate sun by +x → All gear speeds as ratios of x

Add +y to all → Arm = y, Sun = x+y, Ring = y − x(T_S/T_R)

Apply boundary conditions to solve for x and y.

Free Undamped

ω_n = √(k/m) rad/s

f_n = (1/2π)√(k/m) Hz

T_n = 2π√(m/k) s

Static deflection method: ω_n = √(g/δ_st)

Damped Vibrations

Damping ratio: ξ = c/(2√(km)) = c/(2mω_n)

Critical damping: c_c = 2√(km) = 2mω_n

Damped frequency: ω_d = ω_n√(1 − ξ²)

Log decrement: δ = ln(x_n/x_n+1) = 2πξ/√(1−ξ²)

ξ < 1: underdamped. ξ = 1: critically damped. ξ > 1: overdamped.

Forced Vibrations

MF = 1/√[(1−r²)² + (2ξr)²], where r = ω/ω_n

Phase: φ = tan⁻¹[2ξr/(1−r²)]

At resonance (r=1): MF = 1/(2ξ)

Transmissibility: TR = √[1+(2ξr)²]/√[(1−r²)²+(2ξr)²]

Isolation when r > √2 (TR < 1).

Springs

Parallel: k_eq = k₁ + k₂

Series: 1/k_eq = 1/k₁ + 1/k₂

Energy fluctuation: ΔE = I × ω² × C_s

Where C_s = coefficient of fluctuation of speed = (ω_max−ω_min)/ω_avg

I = ΔE / (ω² × C_s) — required moment of inertia

For a solid disc: I = ½mr²

For a rim-type flywheel: I ≈ mr² (mass concentrated at rim)

Height of Watt governor: h = g/ω² = 895/N² (m, N in rpm)

Sensitivity: = 2(N₂−N₁)/(N₂+N₁)

Isochronous governor: maintains constant speed at all loads (range = 0)

Hunting: Governor oscillates continuously above and below the mean speed

Static Balancing (Single Plane)

Σmrω² = 0 → Σmr = 0 (since ω is common)

Vector sum of all mr products must close to zero.

Dynamic Balancing (Multiple Planes)

Σmr = 0 (force balance) AND Σmrl = 0 (moment balance)

Where l = axial distance from reference plane. Both vector equations must be satisfied.

Module: m = D/T = p_c/π

Pitch circle diameter: D = mT

Addendum: a = m (standard full-depth)

Dedendum: d = 1.25m

Tooth height: h = 2.25m

Minimum teeth (no interference): T_min = 2/sin²φ (for full-depth, meshing with rack)

For φ = 20°: T_min = 2/sin²20° = 2/0.1170 ≈ 17 teeth