Mechanism Design & Analysis • Grashof Condition • Transmission Angle • Coupler Curves — Simulate • Explore • Practice • Quiz
The four-bar linkage is the simplest and most widely used closed-loop mechanism in mechanical engineering. It consists of four rigid links connected by four revolute (pin) joints, forming a closed kinematic chain. One link is fixed (the ground or frame), the input link (crank) is driven at a known angular velocity, the output link (follower or rocker) moves in response, and the connecting link (coupler) transmits motion between the crank and follower. This mechanism is the building block of countless real-world machines, from windshield wipers and sewing machines to aircraft landing gear and automotive suspensions.
Understanding the four-bar linkage is essential for any student of kinematics of machines or theory of machines. The mechanism can produce a wide variety of output motions depending on the relative lengths of its links. By changing link proportions, engineers can design mechanisms that convert continuous rotation into oscillation (crank-rocker), produce continuous rotation at the output (double-crank or drag link), or create oscillation at both the input and output (double-rocker). The parallelogram linkage, a special case, maintains parallel orientation between opposite links and is used in drafting machines, pantographs, and locomotive wheel couplings.
The behaviour of a four-bar linkage is governed by Grashof's law, which states that for at least one link to make a full revolution, the sum of the shortest and longest link lengths must be less than or equal to the sum of the remaining two link lengths: s + l ≤ p + q. If this condition is satisfied, the linkage is classified as a Grashof mechanism. When the shortest link is the crank (input), the mechanism is a crank-rocker. When the shortest link is the ground (frame), both the crank and follower can rotate fully, producing a double-crank (drag link). When the shortest link is the coupler, the result is another type of double-crank. If the Grashof condition is not satisfied, no link can complete a full revolution, and the mechanism is classified as a non-Grashof double-rocker (also called a triple-rocker in some texts).
The transmission angle (μ) is the acute angle between the coupler and the follower link. It is a critical measure of force transmission quality in a four-bar mechanism. When the transmission angle approaches 90°, force is transmitted most efficiently. As the transmission angle drops below 40°, mechanical advantage deteriorates significantly, and the mechanism may experience difficulty in transmitting motion. Designers typically aim to keep the minimum transmission angle above 40° throughout the full cycle of operation. This simulator highlights the transmission angle in real time and warns when it falls into the poor-transmission zone.
When a point on the coupler link is traced as the crank rotates through a full revolution, the resulting path is called a coupler curve. Four-bar coupler curves can produce an astonishing variety of shapes — circles, ellipses, figure-eights, and complex higher-order curves. Engineers use coupler curves for path generation, designing mechanisms where a specific point must follow a desired trajectory. The Atlas of Coupler Curves, cataloguing thousands of these shapes, remains an important reference in mechanism design. In this simulator, enabling the coupler curve toggle traces the midpoint of the coupler in real time, building up the complete curve over one revolution.
The kinematic analysis of a four-bar linkage begins with the loop-closure equation, which expresses the constraint that the four links form a closed polygon. Given the crank angle θ2, the coupler angle θ3 and follower angle θ4 can be found analytically using Freudenstein's equation or direct geometric methods. These analytical solutions avoid iterative procedures and are ideal for real-time simulation. Velocity analysis follows by differentiating the position equations, yielding the angular velocities of the coupler and follower as functions of the crank speed.
In Simulate mode, adjust the four link lengths (Ground, Crank, Coupler, Follower) using the sliders or select a preset configuration. The animated mechanism shows the crank rotating continuously, with the coupler and follower responding in real time. Toggle the coupler curve to see the beautiful path traced by the coupler midpoint. The readout cards display mechanism type, Grashof condition, crank angle, transmission angle, coupler angle, and angular velocity. Switch to Explore to study 12 key concepts across Linkage Basics, Kinematics, and Design. Practice mode generates problems on Grashof analysis, transmission angles, and link classification. Quiz tests your knowledge with 5 randomised multiple-choice questions from a pool of 15.