Four-Bar Linkage

What a four-bar linkage is

A four-bar linkage is the simplest closed kinematic chain that does useful work. Four rigid links are joined end to end by four pin (revolute) joints into a loop, and then one of the links is bolted down as the fixed ground. The conventional names for the four links are:

• Ground (frame): the fixed link, link 1, that the other links pivot against.
• Crank (input): link 2, the driven link, often connected to a motor.
• Coupler (floating link): link 3, pinned to ground at neither end — it both translates and rotates.
• Rocker / follower (output): link 4, the link whose motion you actually want.

The reason this arrangement is so prized is its degree-of-freedom count. For a planar mechanism the Kutzbach-Gruebler criterion gives:

DOF = 3(n − 1) − 2·j₁ − j₂

  n  = number of links            = 4
  j₁ = number of 1-DOF joints (pins, sliders) = 4
  j₂ = number of 2-DOF joints     = 0

DOF = 3(4 − 1) − 2(4) − 0
    = 9 − 8
    = 1

One degree of freedom means one input fixes everything. Spin the crank to any angle θ₂ and the coupler angle θ₃ and rocker angle θ₄ are forced — there is exactly one assembly configuration (well, two, the "open" and "crossed" branches, but the mechanism stays on whichever branch it was assembled in). That is why you can drive a four-bar with a single, dumb motor and get perfectly repeatable, synchronized output without any feedback control.

The loop-closure equation

The whole kinematics of a four-bar follows from one vector statement: walk around the closed loop and you return to where you started. Treating each link as a complex vector r·e^iθ gives the loop-closure equation:

r₂·e^(iθ₂) + r₃·e^(iθ₃) − r₄·e^(iθ₄) − r₁ = 0

  r₁ = ground length     (fixed link)
  r₂ = crank length      (input,  angle θ₂ known)
  r₃ = coupler length
  r₄ = rocker length     (output, angle θ₄ unknown)

Splitting into real and imaginary parts gives two scalar equations in the two unknowns θ₃ and θ₄ for any commanded crank angle θ₂. Solving them (the closed form is Freudenstein's equation) yields the position; differentiating once gives the angular-velocity ratios and differentiating again gives accelerations. The velocity ratio between output and input is governed by the transmission angle — more on that below. Freudenstein's equation, written with the three ratios K₁ = r₁/r₂, K₂ = r₁/r₄, K₃ = (r₁² + r₂² − r₃² + r₄²)/(2·r₂·r₄), is the algebraic backbone of nearly all four-bar synthesis.

Grashof's law and linkage types

Whether a link can rotate a full 360° is decided by link lengths alone, through Grashof's law. Sort the four link lengths and label the shortest s, the longest l, and the other two p and q. Then:

Grashof (a link can fully rotate):     s + l ≤ p + q
Non-Grashof (no full rotation):        s + l > p + q
Change-point (special, indeterminate): s + l = p + q

Which link can rotate, and therefore what the mechanism is called, depends on which link is grounded relative to the shortest link s:

Type	Grashof?	Grounded link	Input motion	Output motion	Typical use
Crank-rocker	Yes (s+l < p+q)	Link adjacent to shortest	Crank fully rotates 360°	Rocker oscillates (arc)	Windshield wiper, sewing-machine needle bar
Double-crank (drag-link)	Yes	The shortest link itself	Crank fully rotates	Output fully rotates (uneven speed)	Quick-return planers, conveyor drives
Double-rocker	Yes	Link opposite the shortest	Input oscillates	Output oscillates; coupler fully rotates	Pantographs, some wipers
Triple-rocker (non-Grashof)	No (s+l > p+q)	Any link	Input oscillates only	Output oscillates only	Landing-gear retraction, suspension arms
Change-point (parallelogram, deltoid)	Equality (s+l = p+q)	Any	Rotates, but passes through collinear toggle	Indeterminate at change point	Locomotive coupling rods, drafting tables

The crank-rocker is the most common arrangement and the one shown in the visualization above: ground the link next to the shortest, drive the shortest link as the crank, and the opposite rocker sweeps back and forth. A pumpjack ("nodding donkey") on an oil well is a crank-rocker scaled up to several metres; the motor spins the crank at perhaps 6–20 strokes per minute and the walking beam rocks the polished rod up and down.

Coupler curves

The coupler link is pinned to ground at neither end, so it both rotates and translates. Pick any point rigidly attached to it — even one off the line between its two pins — and that point traces a closed path called a coupler curve as the crank turns once. These curves are algebraic curves of degree six (sextics), which is why they can be so varied: ovals, kidney shapes, figure-eights, teardrops, and curves with near-straight or near-stationary (dwell) segments.

Engineers harvest coupler curves to get motions that would otherwise need a cam or a computer:

• Straight-line motion. Watt's linkage (used in his steam-engine to guide the piston rod) produces an approximate straight line over part of the curve; the Chebyshev and Hoeken linkages do the same with specific length ratios such as 1 : 2.5 : 2.5 : 2.5. The Peaucellier-Lipkin linkage (eight bars, not four) produces an exact straight line, but the four-bar approximations are simpler and good enough for most machines.
• Walking gaits. Theo Jansen's strandbeests use an eleven-bar leg built from coupled four-bars; the foot's coupler curve is a flattened oval — roughly straight on the ground stroke (so the body doesn't bob) and lifted on the return stroke.
• Dwell mechanisms. A coupler point whose curve has a near-circular arc segment will "pause" while the crank keeps turning, advancing film one frame at a time in a movie projector, or indexing parts on an assembly line.

Transmission angle and mechanical advantage

Not every crank position transmits force well. The transmission angle μ is the angle between the coupler and the rocker at the joint they share. The component of the coupler force that actually drives the rocker is proportional to sin μ:

• At μ = 90°, the full coupler force turns the rocker — maximum mechanical advantage.
• As μ approaches 0° or 180°, sin μ → 0; almost all the force pushes straight through the pin into the bearing, doing no useful work. Bearing loads spike, friction climbs, and the mechanism can toggle (lock).

The rule of thumb for good design is to keep μ between about 40° and 140° throughout the entire revolution. Below ~40° the linkage feels sluggish and wears its pins quickly; designers either re-proportion the link lengths or accept the limitation. Toggle positions are not always bad: a toggle clamp or a vise-grip pliers deliberately operates at the toggle point, where infinite mechanical advantage holds the clamp shut with almost no holding force.

Four-bar versus other motion converters

	Four-bar linkage	Slider-crank	Cam-follower	Gear train
Joints	4 revolute pins	3 revolute + 1 prismatic (slider)	Higher pair (line contact)	Higher pairs (tooth contact)
Input → output	Rotation → oscillation / rotation	Rotation ↔ linear stroke	Rotation → any programmed lift	Rotation → rotation (ratio)
Output motion flexibility	High (coupler curves)	Fixed sinusoidal-ish stroke	Arbitrary (cut the cam profile)	Constant ratio only
Speed / load capacity	High; all low-friction pin joints	High	Limited by contact stress & wear	Very high
Cost / complexity	Low — four bars, four pins	Low	Higher — profile must be machined	Moderate to high
Typical use	Wipers, pumpjacks, suspensions	Piston engines, compressors	Valve trains, indexing	Transmissions, reducers

The slider-crank is really just a four-bar with one revolute joint replaced by a slider (the limiting case where the rocker length goes to infinity), which is why piston engines and four-bars share the same loop-closure mathematics. Where a cam can produce any motion profile but suffers contact-stress wear and is costly to machine, a four-bar gives a fixed but smooth, all-pin-joint motion at low cost and high reliability — so designers reach for the four-bar first and only resort to a cam when the required motion can't be approximated by a coupler curve.

Worked example: checking Grashof and the transmission angle

Suppose a designer proposes a four-bar with ground r₁ = 100 mm, crank r₂ = 30 mm, coupler r₃ = 90 mm, rocker r₄ = 80 mm. Is it a crank-rocker, and is the transmission angle acceptable?

Step 1 — Grashof test:
  shortest s = 30 (crank), longest l = 100 (ground)
  others    p = 90, q = 80
  s + l = 30 + 100 = 130
  p + q = 90 + 80  = 170
  130 ≤ 170  →  Grashof  ✓  (a link can fully rotate)

Step 2 — type:  shortest link (crank) is adjacent to ground,
  and ground is the link next to the shortest → CRANK-ROCKER ✓

Step 3 — extreme transmission angle (crank & ground collinear):
  At θ₂ = 0° the diagonal across the coupler-rocker triangle is
    d = r₁ − r₂ = 100 − 30 = 70 mm   (crank folded onto ground)
  Law of cosines at the rocker joint:
    cos μ = (r₃² + r₄² − d²) / (2·r₃·r₄)
          = (90² + 80² − 70²) / (2·90·80)
          = (8100 + 6400 − 4900) / 14400
          = 9600 / 14400 = 0.667
    μ = 48.2°            (acceptable, > 40°)

  At θ₂ = 180° (crank stretched along ground):
    d = r₁ + r₂ = 130 mm
    cos μ = (8100 + 6400 − 16900) / 14400 = −0.167
    μ = 99.6°            (excellent)

The minimum transmission angle over the cycle is about 48°, comfortably above the 40° floor, so this is a healthy crank-rocker. Had the ground been longer or the rocker shorter, μ could have dropped below 40° at one extreme and the linkage would feel notchy and load its pins heavily there.

Failure modes and trade-offs

• Toggle / lock-up. If the transmission angle reaches 0° or 180° away from a deliberate clamping toggle, the mechanism jams or the output direction becomes indeterminate. Fix by re-proportioning links so μ stays inside 40°–140°.
• Branch (assembly) defect. The position equations have two solutions — the open and crossed configurations. A linkage synthesized to hit a set of points may need to switch branches to reach them all, which is physically impossible without disassembly. Always verify a single branch reaches every design point.
• Pin-joint wear and backlash. Clearance in the four pins accumulates as positional error at the coupler point. High-cycle mechanisms (a wiper does > 10 million cycles in a car's life) need hardened pins, bushings, and grease, or the coupler curve drifts.
• Dynamic shaking forces. At speed the moving links throw inertial forces into the frame. A crank-rocker is harder to balance than a pure rotor; counterweights on the crank cancel only part of the shaking force, and the rocker's reciprocation leaves a residual that vibrates the mount.
• Order defect. A linkage can pass through the right positions but in the wrong sequence as the crank turns — fine for displaying poses, useless if timing matters. Synthesis must check that the precision points occur in order.
• Tolerance stack-up. Because output is so sensitive to the four link lengths, manufacturing tolerance on each bar compounds; precision four-bars (e.g. surgical or aerospace) call out length tolerances of a few hundredths of a millimetre.