Set the ground, crank, coupler, and rocker link lengths and watch the mechanism move through its range of motion, tracing the coupler-point path curve. Automatically classifies the linkage per the Grashof criterion.
This simulator solves the standard planar four-bar mechanism position problem in real time, animating the crank, coupler, and rocker links and tracing the path swept by a chosen point on the coupler link. It also applies the Grashof criterion to classify whether the mechanism produces continuous rotation or oscillation.
A four-bar linkage has four links: the fixed ground link (O₂O₄), the crank (O₂A, the driven input link), the coupler (AB, a floating link connected to nothing but the crank and rocker), and the rocker (BO₄, the output link). The vector loop equation O₂A + AB = O₂O₄ + O₄B must close for any valid crank angle θ₂. Given the crank angle, point A is fixed by simple trigonometry; point B must then lie on a circle of radius (coupler length) centered at A AND a circle of radius (rocker length) centered at the fixed pivot O₄ — a circle-circle intersection problem with up to two solutions, corresponding to the "open" and "crossed" assembly configurations of the same link lengths.
A coupler point P — a point rigidly attached to the coupler link but not on the line AB — traces a distinctive, often figure-eight-like or kidney-shaped path called a coupler curve as the mechanism moves through its cycle. Coupler curves are the basis for classic mechanisms like the Klann and Theo Jansen walking-leg linkages, Watt's straight-line mechanism, and pick-and-place robot arms.
Grashof's law states that for a four-bar linkage with shortest link length s, longest length l, and the other two lengths p and q, at least one link can rotate a full 360° relative to the others if and only if s + l ≤ p + q. When this condition holds, the mechanism is classified further by which link is shortest: if the ground link is shortest, both links adjacent to ground (crank and rocker) rotate fully — a double-crank or drag-link mechanism. If the crank (or rocker) is shortest, that link rotates fully while the opposite one only oscillates — the classic crank-rocker. If the coupler is shortest, it rotates fully relative to both neighbors while crank and rocker only oscillate — a double-rocker. When s + l > p + q, the linkage is non-Grashof: no link can complete a full rotation, and all three moving links only oscillate back and forth (triple-rocker) between limiting "toggle" positions.
Enter the four link lengths, and the simulator continuously drives the crank angle, solving for the coupler and rocker positions at each step and choosing the solution branch that keeps the motion continuous from frame to frame. When the current link combination is non-Grashof (or a double-rocker/reversed crank-rocker driven from the crank side), the crank itself cannot complete a full rotation — the simulator detects when no valid assembled position exists and reverses direction automatically, producing the correct rocking motion. Adjust the "Coupler Point Position" (how far along the coupler, from A toward B) and "Coupler Point Offset" (perpendicular distance from the coupler line, as a fraction of coupler length) sliders to explore how dramatically the traced coupler curve shape changes with the point's location on the coupler.
The transmission angle μ is the angle at the rocker's moving joint (B) between the coupler and rocker links. It indicates how efficiently force/torque transmits through the mechanism — when μ is close to 90°, transmission is efficient; when μ approaches 0° or 180° ("toggle" positions), the mechanism transmits very little useful torque and can bind or require excessive force. Good four-bar designs typically keep the transmission angle between about 40° and 140° throughout the full range of motion.
This happens whenever the Grashof condition s + l ≤ p + q is violated, or when it holds but the shortest link is the coupler or you are driving from the non-shortest adjacent link. In those cases, at some crank angles the required circle-circle intersection for the coupler-rocker joint simply does not exist (the coupler and rocker links cannot both reach the geometrically required point), so the mechanism physically cannot pass through those crank angles — it must reverse and oscillate within the feasible range instead.
Coupler curves — the path traced by a point on the floating coupler link — can approximate straight lines, closed loops, or complex figure-eight paths depending on link proportions and the point's location. Historical straight-line mechanisms (Watt's linkage, Chebyshev linkage) exploit specific coupler-curve shapes to guide a point along an approximately straight path using only rotating joints — useful in vehicle suspensions and steam engine valve gear before linear bearings were practical. Walking-robot leg mechanisms (like the Theo Jansen linkage, an eight-bar variant) use carefully tuned coupler curves to produce a natural stepping motion from a single rotating crank.
A slider-crank mechanism (used in piston engines and compressors) replaces the rocker link and one ground pivot with a sliding joint constrained to move along a straight line — effectively a four-bar linkage with the rocker link length taken to infinity. The same vector-loop position-analysis approach applies, but the constraint equation for the slider (perpendicular distance from the line of sliding) replaces the circle equation used for the rocker's fixed-length pivot arm.
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