MEN430 · Theory of Machines
Four-Bar Linkage Simulator
Developed by Najib Metni, PhD
Link Lengths
Type any length in mm — the field isn't capped, and the slider range stretches to fit whatever you enter.
Ground Pivots
O₂ is fixed at the origin (0, 0).
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Time, θ₂ (and θ₃/θ₄ or d₄), each link's angular speed and acceleration, joint accelerations, the crank driving torque (from the Forces tab inputs), and any trace points' coordinates, speed and acceleration — swept across the full feasible range at the current Speed setting.
Mechanism
Grashof Condition
—
| Role | Link | Length |
|---|---|---|
| Shortest (S) | — | — |
| Longest (L) | — | — |
| Other (P) | — | — |
| Other (Q) | — | — |
Stop / Toggle Positions
Quick-Return Motion
Angle vs θ₂
Swept across the full feasible range of θ₂ — this is geometry, not the live animation.
Distance vs θ₂
Same sweep — the prismatic joint's position, which doesn't share the angle graph's 0-360° axis.
Velocity Analysis
Live values at the current θ₂, driven by the crank's current angular velocity ω₂.
Angular Velocity vs θ₂
Swept at a constant ω₂ (the Speed setting) across the full feasible range of θ₂ — this is geometry, not the live animation.
Linear Velocity vs θ₂
Same sweep — speed of joint A (constant, dashed), joint B (solid), and any enabled trace points.
Peak Speeds
Acceleration Analysis
Live values at the current θ₂, for a constant crank speed (α₂ = 0). The coupler and rocker angular accelerations come from differentiating the velocity loop-closure equations a second time, and joint / trace-point accelerations from the rigid-body relation AP = Aref + α×r − ω²r. To study a non-zero crank acceleration, use the “Specified Position” section below.
Specified Position (with crank acceleration α₂)
Give the crank an angular acceleration α₂ at one chosen position θ₂. The result uses the current crank speed ω₂ (counterclockwise) and also feeds the Forces tab. It is marked on the graphs below (a triangle on a plot's edge means the value lies off that plot's scale).
Angular Acceleration vs θ₂
Swept at a constant ω₂ (the Speed setting) across the full feasible range of θ₂. Signed — positive is counterclockwise.
Linear Acceleration vs θ₂
Same sweep — magnitude of the acceleration of joint A (constant, dashed), joint B (solid), and any enabled trace points.
Peak Accelerations
Link Inertia Properties
Enter each link's mass and its moment of inertia about its own center of gravity. The center of gravity is given as (x, y) in each link's own coordinate system, whose origin is the link's first pin (crank: O₂, coupler: A, rocker: O₄), with x pointing along the link toward its far pin and y to its left (counter-clockwise side). When a link's shape is applied in Edit Shape, its center of gravity and moment of inertia are filled in from the shape (a uniform plate of the mass entered here; changing the mass rescales I until you type your own I). Every link is then solved as a free body (ΣF = m·aG, ΣMG = IG·α) to find all the pin forces and the driving torque T12 the ground must apply to the crank.
| Link | Mass (kg) | IG (kg·m²) | CG x-coord. (mm) | CG y-coord. (mm) |
|---|---|---|---|---|
| Crank a origin O₂ |
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| Coupler b origin A |
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| Rocker c origin O₄ |
In slider-crank mode the slider is treated as a point mass at joint B (only its mass matters); the ground pushes on it perpendicular to the slide direction.
External Force
The point is measured on the chosen link from its first pin (crank: O₂, coupler: A, rocker: O₄): x along the link toward its far pin, y to its left (+ = up when the link points right). The force keeps a constant magnitude and a fixed direction θ in the global frame (counterclockwise from +x) while the link moves, and is drawn on the mechanism.
Forces at the Current θ₂
Forces at the Specified Position
At θ₂ = — with the crank acceleration α₂ set in the Acceleration tab (currently —).
Flywheel
A flywheel on the crank shaft smooths the torque the motor has to supply. Give the coefficient of fluctuation k = (ωmax − ωmin) / ωavg you allow and the flywheel's moment of inertia Is (for a solid cylinder, ½·m·r²); ωavg is the Speed setting. The crank then speeds up and slows down by that much over each revolution, so the motor must supply T = T12 + Is·α2, which is drawn on the torque plot below. With Is equal to the size needed for k, the torque is (almost) flat at its mean.
Driving Torque T₁₂ vs θ₂
Torque the ground must apply to the crank to sustain a constant ω₂ (the Speed setting), swept across the full feasible range of θ₂. Positive is counterclockwise.
Torque in N·m (mm lengths are converted to metres internally).