cam-follower

name	cam-follower
description	Cam and follower design — cam profiles (SHM, cycloidal, polynomial), follower motion, pressure angle, contact stress, follower jump, spring design, cam materials.
metadata	{"priority":5,"promptSignals":{"phrases":["cam","follower","cam design","cam profile","pressure angle","cycloidal","cam follower"],"minScore":4}}

Cam & Follower — Complete Skill

Cam Types

Disk (plate) cam: flat disk with profiled edge; most common Cylindrical cam: groove on cylinder; follower moves axially Face cam: groove on end face Translating / oscillating follower

Follower Types

Knife-edge: simple, high stress → theoretical only Roller follower: reduces contact stress, most practical (rolls, not slides) Flat-face follower: no pressure angle issue, higher friction, requires convex cam Spherical-face: compromise between roller and flat

Motion Profiles (Displacement Programs)

Simple Harmonic Motion (SHM)

Lift h, angle β (rise angle): y = h/2 × (1 - cos(πθ/β)) v = πh/(2β) × sin(πθ/β) a = π²h/(2β²) × cos(πθ/β)

Max acceleration: a_max = π²h/(2β²) [at θ=0 and θ=β] Disadvantage: infinite jerk at θ=0 and θ=β (discontinuous acceleration) → shock

Cycloidal Motion (no jerk)

y = h[θ/β - (1/2π)sin(2πθ/β)] v = h/β[1 - cos(2πθ/β)] a = 2πh/β² × sin(2πθ/β)

Max v: v_max = 2h/β (at β/2) Max a: a_max = 2πh/β² (at β/4 and 3β/4) Advantage: zero jerk everywhere → smooth operation; preferred for high speed

Constant Acceleration (Parabolic)

Rise h in angle β: First half (θ < β/2): y = 2h(θ/β)², a = 4h/β² (const.) Second half: y = h - 2h(1-θ/β)², a = -4h/β² (const.)

Min max acceleration of all symmetric profiles → useful for low-speed high-load cams Disadvantage: infinite jerk at β/2 transition

3-4-5 Polynomial (Cycloidal-like)

y/h = 10(θ/β)³ - 15(θ/β)⁴ + 6(θ/β)⁵ Smooth: y(0)=0, y(β)=h, y'(0)=0, y'(β)=0, y''(0)=0, y''(β)=0 Max v = 1.875h/β; max a = 5.774h/β²

Pressure Angle (Roller Follower, Translating)

φ = arctan[(dy/dθ - e) / (R_b + y)]

Where:

• e = eccentricity of follower (0 for inline)
• R_b = base circle radius
• φ_max ≤ 30° (recommended)

Design rule: increase R_b to reduce pressure angle R_b,min = max y' / tan(φ_max) - y_at_that_point [approximately] Offset (eccentricity) e: can reduce φ on one side (rise), increases on other (return)

Cam Geometry (Disk Cam, Translating Inline Roller)

Pitch curve radius: ρ_p = R_b + y(θ) Cam profile radius (to center of roller): r_profile = ρ_p Cam surface radius (for manufacturing): r_cam = ρ_p - r_roller (external cam)

Undercutting condition: Minimum radius of curvature of pitch curve ρ_min ≥ r_roller If ρ_min < r_roller → cam profile is undercut (physically impossible to machine)

ρ_pitch = [(ρ_p' + dρ_p/dθ)² + ρ_p²]^(3/2) / [(ρ_p')² - ρ_p×ρ_p'']... (polar curvature formula)

Contact Stress (Hertz, Roller on Cam)

For cylindrical roller (length L) on cam: q = 2F/(πL) × (1/ρ_contact)... (Hertzian line contact — see contact-stress skill) σ_c_max = 0.798 × √(F×E*/(L×ρ_eff))

ρ_eff = r_roller × R_cam / (r_roller + R_cam) [+ for external, - for internal] Limit: σ_c < 0.77×√(E×S_y) for infinite life

Follower Spring Design

Prevent follower jump (loss of contact): At max negative acceleration: required spring force F_spring ≥ m_follower × |a_max| + gravity (if applicable) k × (x₀ + y) ≥ m × a_max k = spring rate, x₀ = preload, y = follower lift

F_spring,min = m × a_max at max negative a (deceleration phase = return motion) Preload: F₀ = k × x₀ ≥ m × a_max - k × y_min_at_max_a

Critical speed of follower spring: ω_n,spring = √(k/(m_spring/3 + m_follower)) Operating cam speed < ω_n,spring/3 to avoid spring surge

Cam Materials and Hardness

Application	Cam Material	Hardness	Follower
Low speed	Cast iron (class 30-40)	200-250 HB	Cast iron
Medium speed	1045 steel, flame-hardened	48-55 HRC	52100 steel roller
High speed	8620 steel, case-hardened	58-62 HRC (case)	Hardened steel or ceramic
Automotive	4150 steel or chilled cast iron	55-65 HRC	Roller or flat tappet

Lubrication: critical for wear life; hydrodynamic possible for well-designed flat-face; roller followers have rolling contact (lower friction)

Cam Dynamics at Speed

Effective inertia force: F_inertia = m × a = m × (d²y/dt²) = m × ω² × d²y/dθ² (θ = ωt, so d²y/dt² = ω² × d²y/dθ²)

Maximum spring force at max acceleration: F_spring,max = F₀ + k × h [at full lift position] F_spring,min = F₀ [at base circle, if follower at rest]

Output

Provide: motion program (SHM, cycloidal, polynomial), y(θ), v_max, a_max, pressure angle φ_max [°], base circle radius R_b [mm], spring preload and rate [N/mm], cam surface hardness requirement.