Four-Bar Linkage (Position)

kinematics

Verified build 2 relations · 4 identities proven · 2 modeling steps · 6 parity samples

Pin four rigid links into a loop and you own the oldest machine element there is: the four-bar linkage. Bicycle suspension, engine hood hinges, locking pliers, oil-well pumpjacks, your own knee — all four-bars. One degree of freedom: turn the crank ( $\theta_2$ ) and every other angle is decided by geometry, no springs, no slip, no choice.

Except — two choices. The position problem reduces (via Freudenstein’s equation and the tan-half-angle trick in the derivation below) to a quadratic, and a quadratic has two roots. Both are real, both are honest equilibrium assemblies of the same four links at the same crank angle: the open circuit and the crossed circuit, mirror-ish twins that no continuous motion connects. You’d have to pull a pin to switch. This THING is the first in the catalog where a single input has two verified answers, and the widget grows a circuit selector to pick between them — each branch independently machine-proved against the loop equations.

Three warnings the widget will raise, all geometric facts rather than strength limits:

Assembly. If the coupler and rocker together can’t reach between the crank pin and the far ground pivot, there is no linkage at that crank angle — the discriminant goes negative and the page says so rather than inventing a number.
Grashof. Whether any link can fully rotate is decided by one inequality on the four lengths: $s + l \le p + q$ . Satisfy it (the default lengths do) and the crank cranks; violate it and you own a triple-rocker that jams partway around.
Transmission angle. Even an assembled linkage can be a bad linkage: when the coupler pushes the rocker nearly along the rocker’s own length, the force goes into the bearings instead of the motion. Watch the warning fire near the extremes of travel.

There is no material picker here, and that’s the lesson: kinematics is the geometry of motion. Steel, titanium, or 3D-printed nylon, the angles are identical — material enters only when forces and stiffness do (the torque-power and stress THINGs).

Try it

Governing relations

a\cos\theta_2 + b\cos\theta_3 - c\cos\theta_4 - d = 0

Assumes: rigid links, ideal revolute (pin) joints, no clearances; planar mechanism; ground pivots at distance d on the x-axis · Valid while: The links cannot assemble at this crank angle — the coupler and rocker together cannot reach between the crank pin and the fixed pivot. (Non-Grashof linkages have whole arcs of crank angle where this happens.) Transmission angle is outside 30°–150°. The coupler is pushing the rocker nearly along its own length; force transmission is poor, and a real linkage here binds, chatters, or stalls. Designers keep the transmission angle above ~40°. Non-Grashof linkage (shortest + longest exceeds the sum of the other two): no link can fully rotate. This is a triple-rocker — the crank will hit assembly limits within one revolution.

Source: Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 2 (Grashof condition), ch. 4 (vector-loop position analysis, the K-constant/Freudenstein closed form, open vs. crossed circuits, transmission angle).

a\sin\theta_2 + b\sin\theta_3 - c\sin\theta_4 = 0

Assumes: rigid links, ideal revolute (pin) joints, no clearances

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

a \cos{\left(\theta_{2} \right)} + b \cos{\left(\theta_{3} \right)} = c \cos{\left(\theta_{4} \right)} + d

1. Walk the linkage as vectors: crank then coupler must land on the same point as ground then rocker. This is the x-component of that loop. Rigid links and pin joints are the entire physics of the device — everything below is algebra. — vector loop, x-component modeling step

a \sin{\left(\theta_{2} \right)} + b \sin{\left(\theta_{3} \right)} = c \sin{\left(\theta_{4} \right)}

2. The y-component of the same loop (the ground link lies along x, so it contributes nothing here). Two equations, two unknown angles θ3 and θ4 — but they are tangled in sines and cosines, not linear in anything. — vector loop, y-component modeling step

b^{2} = \left(- a \sin{\left(\theta_{2} \right)} + c \sin{\left(\theta_{4} \right)}\right)^{2} + \left(- a \cos{\left(\theta_{2} \right)} + c \cos{\left(\theta_{4} \right)} + d\right)^{2}

3. Isolate the θ3 terms — b·cosθ3 from the first equation, b·sinθ3 from the second — then square and add. The identity cos² + sin² = 1 swallows θ3 whole, leaving one equation in the one unknown θ4. Eliminating a variable you don't want is the oldest trick in kinematics. — square and add to eliminate θ3

- \frac{d \cos{\left(\theta_{2} \right)}}{c} + \frac{d \cos{\left(\theta_{4} \right)}}{a} + \frac{a^{2} - b^{2} + c^{2} + d^{2}}{2 a c} = \cos{\left(\theta_{2} - \theta_{4} \right)}

4. Expand, divide through by 2ac, and gather the geometry into three constants K1 = d/a, K2 = d/c, K3 = (a² − b² + c² + d²)/2ac: Freudenstein's equation, the 1954 form that made linkage synthesis a pencil-and-paper subject. Given θ2, everything but θ4 is a number. — Freudenstein's equation

u^{2} \left(\cos{\left(\theta_{2} \right)} - \frac{d \cos{\left(\theta_{2} \right)}}{c} - \frac{d}{a} + \frac{a^{2} - b^{2} + c^{2} + d^{2}}{2 a c}\right) - 2 u \sin{\left(\theta_{2} \right)} - \left(1 + \frac{d}{c}\right) \cos{\left(\theta_{2} \right)} + \frac{d}{a} + \frac{a^{2} - b^{2} + c^{2} + d^{2}}{2 a c} = 0

5. The half-angle substitution u = tan(θ4/2) turns cosθ4 into (1−u²)/(1+u²) and sinθ4 into 2u/(1+u²), collapsing the trigonometry into an ordinary quadratic A·u² + B·u + C = 0. (Handed the raw loop equations, a computer algebra system's blind solver hangs; this 200-year-old substitution is the difference between intractable and trivial.) — Weierstrass (tan-half-angle) substitution

\theta_{4} = 2 \arctan{\left(u \right)}

6. Invert the substitution. A quadratic has TWO roots, and both are real wherever the linkage assembles: the open circuit and the crossed circuit — two genuinely different ways to pin the same four links together at the same crank angle. No continuous motion connects them; you must disassemble the linkage to switch. The widget's circuit selector chooses, and the build verifies each root against the loop equations independently. — two roots, two circuits

How it fails

A linkage’s failures are mostly kinematic — it stops doing the motion you wanted — long before anything breaks:

Binding at poor transmission angles. As the transmission angle leaves the comfortable 40°–140° band, the coupler force aims down the rocker’s length: friction multiplies, bearings load up sideways, and the mechanism stutters or locks. The widget’s warning band is the designer’s first check on any candidate linkage.
Dead points (toggle positions). When crank and coupler line up, the output momentarily has no mechanical advantage at all — fine if you coast through with inertia (engines do), fatal if you must start from rest there. Locking pliers exploit the toggle deliberately, parking just past it so working force drives the jaws tighter.
Circuit and branch defects in synthesized linkages. A linkage designed to hit three target positions may technically reach them — on different circuits. On paper the positions check out; the physical mechanism would need disassembly partway through its task. Catching circuit defects is a standard (and humbling) step in linkage synthesis.
Joint clearances and backlash. Every real pin has play. Clearances let the coupler rattle through a small parallelogram of poses at each crank angle; under load reversal the slop crosses over with a clack — felt as backlash, heard as wear accelerating.
Link flexibility. The rigid-link model fails quietly at speed: a long slender coupler bows under inertial load, the output lags and overshoots, and a mechanism that was geometrically perfect at 10 rpm misses its timing at 1000.
Wear, then fatigue. Pins and bushings wear first (growing the clearances above), and the links themselves — loaded and unloaded every revolution — eventually answer to fatigue, the subject the stress THINGs handle.

DC Motor (Permanent Magnet)
The machine that turns current into torque — and its whole personality is one straight line. At fixed voltage a PM DC motor trades speed for torque along T = T_stall(1 − ω/ω₀): two datasheet numbers pin every operating point, and the peak power hides at half the no-load speed.
- torque-power
- kinematics
Slider-Crank (Exact Kinematics and Gas Torque)
Crank, connecting rod, piston — the four-bar linkage with one pivot pushed to infinity, and the heart of every reciprocating engine and pump. Spin the crank at a fixed speed and the piston's position, velocity, and acceleration follow by pure geometry and two derivatives; push on the piston with a gas force and the connecting-rod obliquity turns it into a crank torque that swings from zero to a peak and back every revolution. The classic two-term r/l approximation rides alongside the exact form so you can watch it drift.
- kinematics
- torque-power
Flywheel (Solid Rotating Disk)
The machine that stores work as spin — and loads itself doing it. Centrifugal self-loading grows with ρω²R², so the energy a flywheel can hold per kilogram is capped not by its size but by one material index: strength over density.
- energy-storage
- stress
- mass-cost
Torsional Oscillator (Disk on a Shaft)
A disk on an elastic shaft is a torsion pendulum: twist it and let go and it rings at one natural frequency, ω_n = √(k_t/J_d). The pitch is set entirely by the shaft's stiffness and the disk's inertia — not by how hard you twist it — while the stress it survives is set by the amplitude.
- dynamics
- stress
- torque-power
Planetary (Epicyclic) Gearset
Three coaxial members — sun, ring, planet carrier — share one gear mesh law. With two degrees of freedom, it has no single "ratio": fix a different member and the same hardware becomes a different transmission.
- kinematics
- torque-power

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Symmetric Two-Bar Truss
- theta3 alpha
- theta4 alpha
Belt Drive (Flat Belt / Capstan)
- theta3 theta_w
- theta4 theta_w
Slider-Crank (Exact Kinematics and Gas Torque)
- theta3 theta
- theta4 theta
Torsional Oscillator (Disk on a Shaft)
- theta3 Theta
- theta4 Theta

Sources

Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 2 (Grashof condition), ch. 4 (vector-loop position analysis, the K-constant/Freudenstein closed form, open vs. crossed circuits, transmission angle).