mechanism-design

name	mechanism-design
description	Mechanism design — four-bar linkage, slider-crank, Grashof condition, velocity/acceleration analysis (graphical & analytical), instant center, kinematic synthesis, degrees of freedom.
metadata	{"priority":7,"promptSignals":{"phrases":["mechanism","linkage","four-bar","slider-crank","kinematic","Grashof","instant center","mechanism design","coupler"],"minScore":4}}

Mechanism Design — Complete Skill

Degrees of Freedom (Kutzbach / Grübler)

M = 3(L-1) - 2J₁ - J₂

M = mobility (DOF of mechanism) L = number of links (including ground) J₁ = full joints (1 DOF constraint: revolute, prismatic) J₂ = half joints (2 DOF constraint: roll-slide)

Examples: Four-bar linkage: L=4, J₁=4, J₂=0 → M = 3(4-1) - 2(4) - 0 = 1 ✓ (1 DOF) Slider-crank: L=4, J₁=4 → M = 1 ✓ 5-bar linkage: L=5, J₁=5 → M = 2 (2 DOF — requires 2 inputs)

For over-constrained (M<0): structure (rigid) For under-constrained (M>1): requires multiple actuators

Four-Bar Linkage

Link Notation

• Link 1 (d): ground/frame
• Link 2 (a): crank (driver)
• Link 3 (b): coupler
• Link 4 (c): follower/rocker
• Angles: θ₁=0 (ground), θ₂ (crank), θ₃ (coupler), θ₄ (follower)

Grashof Condition

Let s = shortest link, l = longest link, p, q = other two links s + l ≤ p + q: Grashof (at least one link rotates fully) s + l > p + q: non-Grashof (all links oscillate only)

Grashof types (which link is fixed):

• Ground = shortest s: double-crank (both cranks rotate fully)
• Ground adjacent to shortest: crank-rocker (crank rotates, rocker oscillates)
• Ground opposite to shortest: double-rocker (both oscillate, coupler rotates)

Loop Closure Equation

Vector loop: r₂ + r₃ - r₄ - r₁ = 0 In components: a·cos(θ₂) + b·cos(θ₃) = d + c·cos(θ₄) a·sin(θ₂) + b·sin(θ₃) = c·sin(θ₄)

Solve for θ₃, θ₄ (given θ₂): Define K₁ = d/a, K₂ = d/c, K₃ = (a²-b²+c²+d²)/(2ac), K₄ = d/b, K₅ = (c²-d²-a²-b²)/(2ab)

A·cos(θ₄) + B·sin(θ₄) + C = 0 (quadratic in half-angle tan(θ₄/2)) A = cosθ₂ - K₁ - K₂·cosθ₂ + K₃ B = -2sinθ₂ C = K₁ - (K₂+1)cosθ₂ + K₃ θ₄ = 2·arctan[(-B ± √(B²-4AC))/(2A)] → two solutions (open/crossed)

Velocity Analysis

Relative Velocity Method

V_A = V_O₂ + ω₂ × r_AO₂ (vector cross product) V_B = V_A + V_BA (relative velocity) V_B = ω₄ × r_BO₄ (constraint: B on link 4)

Scalar solution (graphical velocity polygon): Know: magnitude and direction of V_A; direction of V_B (⊥ to link 4) Draw velocity polygon → find V_B magnitude and ω₃, ω₄

Analytical velocity: ω₃ = (a·ω₂·sin(θ₄-θ₂)) / (b·sin(θ₃-θ₄)) ω₄ = (a·ω₂·sin(θ₂-θ₃)) / (c·sin(θ₄-θ₃))

V_coupler = √(V_Ax² + V_Ay²) at coupler point P V_Px = V_Ax - ω₃(y_P - y_A), V_Py = V_Ay + ω₃(x_P - x_A)

Acceleration Analysis

A_A = A_O₂ + α₂ × r_AO₂ - ω₂² × r_AO₂ (centripetal + tangential) A_B = A_A + A_BA [relative accel] = A_O₄ [link 4 constraint]

A_BA = α₃ × r_BA - ω₃² × r_BA (normal + tangential components) A_B = α₄ × r_BO₄ - ω₄² × r_BO₄

Analytical accelerations: α₃, α₄ from differentiating velocity equations (complex but systematic)

Instant Centers

Kennedy's Theorem

For n links: n(n-1)/2 instant centers exist Primary (obvious): joint locations Secondary: found by Kennedy's theorem — 3 instant centers of any 3 links are collinear

Instant center I₁₃ (links 1 and 3): Line through I₁₂ and I₂₃; also line through I₁₄ and I₄₃ → intersection = I₁₃

Velocity using instant centers: ω₃ = ω₂ × (I₁₂I₁₃)/(I₁₂I₁₃) ... from lever proportion: V = ω₂ × r(link₂ from I₁₂) = ω₃ × r(link₃ from I₁₃)

Slider-Crank Mechanism

Position (Crank r, connecting rod l, offset e=0)

x_B = r·cos(θ) + √(l² - r²sin²(θ)) Approximate (for l >> r): x_B ≈ r·cosθ + l - r²/(4l)·sin²θ (harmonic approximation)

Velocity

ẋ_B = -r·ω·sinθ·(1 + r·cosθ/√(l²-r²sin²θ))

Approximate: ẋ_B ≈ -rω(sinθ + r/(2l)·sin2θ) [terms: fundamental + second harmonic]

Acceleration

ẍ_B ≈ -rω²(cosθ + r/l·cos2θ) Primary: rω²cosθ, Secondary: rω²(r/l)cos2θ

Balancing: counter-rotating balance masses cancel primary forces; secondary harder (Lanchester balancer)

Three-Position Synthesis (Graphical)

Synthesize four-bar to guide coupler through 3 prescribed positions:

1. Choose 3 positions of coupler point P: P₁, P₂, P₃
2. Construct displacement poles W₁₂, W₁₃ (intersection of perpendicular bisectors of P₁P₂, P₁P₃)
3. Choose pivot O₂ → find O₄ from reflection about W₁₂ and W₁₃ lines
4. Verify Grashof condition, transmission angle

Transmission angle γ: angle between coupler and follower (b and c) γ_min > 45° for good force transmission (45° = limit; 0° = toggle = bad) Toggle position: crank and coupler aligned → mechanism locks; use as design checkpoint

Special Mechanisms

Scotch-Yoke

Crank + slider in slot → perfect sinusoidal output x = r·cos(ωt) [pure harmonic motion] Used in: oscillating pumps, testing machines

Oldham Coupling

Three-plate coupling: transmits rotation between parallel offset shafts Middle plate slides in orthogonal slots; constant velocity, even with offset

Geneva Mechanism

Intermittent rotation: 4-slot Geneva = 90° rotation per step, 270° dwell Input crank with driving pin engages slots Used in: film projectors, indexing tables, clocks

Hooke (Universal) Joint

Velocity ratio: ω₂/ω₁ = cosβ/(1-sin²β·cos²θ₁) (not constant!) Non-uniform velocity → 2× rotation/revolution vibration Double Hooke joint (equal angles, aligned): cancels non-uniformity → constant velocity ratio

Output

Provide: DOF (mobility M), Grashof type, θ₃ and θ₄ for given θ₂, ω₃ and ω₄ [rad/s], coupler point velocity and acceleration, transmission angle γ_min [°], instant center I₁₃ location.