Gruebler's Equation (planar):
DOF = 3(n-1) - 2j1 - j2

Where:
n = number of links (including ground)
j1 = number of full joints (pin, slider)
j2 = number of half joints (cam, gear)

DOF = 1: Constrained mechanism
DOF = 0: Structure
DOF < 0: Over-constrained

Grashof criterion:
s + l <= p + q

Where:
s = shortest link
l = longest link
p, q = intermediate links

If satisfied: At least one link can rotate fully

Types:
- Crank-rocker: Shortest link is crank
- Double-crank: Shortest link is ground
- Double-rocker: No full rotation

Loop closure equation:
r2*e^(i*theta2) + r3*e^(i*theta3) - r4*e^(i*theta4) - r1 = 0

Solve for theta3, theta4 given theta2 (input)

Velocity:
omega3 = omega2 * r2 * sin(theta4-theta2) / (r3 * sin(theta4-theta3))

mu = angle between coupler and output link

Ideal: mu = 90 degrees
Acceptable: 40 < mu < 140 degrees
Poor: mu < 30 or mu > 150 degrees

Common profiles:

1. Parabolic (constant acceleration)
   s = (1/2) * a * t^2 for first half
   Good: Simple, smooth
   Bad: Infinite jerk at transition

2. Simple harmonic
   s = (h/2) * (1 - cos(pi*t/T))
   Good: Zero velocity at ends
   Bad: Finite acceleration at ends

3. Cycloidal
   s = h * (t/T - sin(2*pi*t/T)/(2*pi))
   Good: Zero acceleration at ends
   Bad: Higher peak acceleration

4. Modified trapezoid
   Combines constant acceleration with transitions
   Good: Low peak acceleration
   Bad: More complex

tan(alpha) = (dy/dtheta) / (rb + y)

Where:
alpha = pressure angle
dy/dtheta = slope of displacement curve
rb = base circle radius
y = follower displacement

Limit: alpha < 30 degrees (typically)

Simple gear train:
i = N2/N1 = omega1/omega2

Compound gear train:
i_total = product of individual ratios

Planetary gear train:
i = 1 + Nring/Nsun (sun fixed)
i = 1/(1 + Nsun/Nring) (ring fixed)

Module: m = d/N
Pitch: p = pi * m
Addendum: a = m
Dedendum: b = 1.25 * m
Center distance: C = m * (N1 + N2) / 2

Contact ratio:
CR = (Arc of action) / (Circular pitch)
Minimum CR > 1.2 recommended

Newton-Euler method:
Sum F = m * a_g (for each link)
Sum M_g = I_g * alpha (about mass center)

D'Alembert approach:
Add inertia forces: -m*a, -I*alpha
Solve as static equilibrium

Shaking force = -Sum(m_i * a_i)
Shaking moment = -Sum(I_i * alpha_i + r_i x m_i * a_i)

Balancing strategies:
1. Add counterweights
2. Optimize mass distribution
3. Use multiple cylinders (phase)

{
  "mechanism_type": "linkage|cam|gear|custom",
  "motion_requirements": {
    "input_motion": "rotation|translation",
    "output_motion": "rotation|translation",
    "motion_profile": "string or array",
    "speed": "number (RPM or m/s)"
  },
  "constraints": {
    "space_envelope": "object",
    "force_requirements": "number",
    "accuracy": "number"
  },
  "operating_conditions": {
    "load": "number",
    "speed_range": "array [min, max]",
    "duty_cycle": "string"
  }
}

{
  "mechanism_design": {
    "type": "string",
    "configuration": "object",
    "link_dimensions": "array"
  },
  "kinematic_results": {
    "position_analysis": "array or function",
    "velocity_analysis": "array or function",
    "acceleration_analysis": "array or function",
    "transmission_angle": "number"
  },
  "dynamic_results": {
    "forces": "array",
    "torques": "array",
    "shaking_forces": "object"
  },
  "performance_metrics": {
    "pressure_angle": "number (cams)",
    "contact_ratio": "number (gears)",
    "efficiency": "number"
  },
  "design_documentation": "reference"
}