Four-Bar Linkage Calculator

Enter four link lengths and an input angle to find a closed four-bar assembly, its output and coupler angles, and the acute transmission angle. The animation shows how the fixed-length links move and where they cannot connect. Use these geometry results to explore a motion path before doing force, clearance and strength analysis.

What is a Four-Bar Linkage?

A four-bar linkage is a mechanical system made of 4 rigid links connected end-to-end in a closed loop by 4 pivot joints. You drive one link (the input), and the geometry of the other links determines how the output link moves.

Simple Explanation

Imagine four bars pinned into a closed loop, with one bar fixed to a base. Moving one adjacent bar forces the other two to follow while all four lengths stay constant. Depending on those lengths and which bar is fixed, a moving link may rotate fully or only rock through part of a turn.

📐 Browse all 1000+ Interactive Calculators

Four-Bar Linkage Mechanism Diagram

Four-Bar Linkage Calculator

How to Use This Calculator

1. Enter the ground link length (L₁), input link length (L₂), coupler link length (L₃), and output link length (L₄) in your chosen units — keep all 4 consistent.
2. Enter the input angle (θ₂) in degrees — this is the angle of your input crank relative to the ground link.
3. Check that your link lengths satisfy basic assemblability: the sum of any 3 links must exceed the fourth.
4. Click Calculate to see your result.

<> Embed this calculator → Found a mistake? Contact us here

Four-Bar Linkage Interactive Calculator

Play animationReturn to start

Cycle position

0°

Ground link L1

Input link L2

Coupler link L3

Output link L4

Input crank angle

Output angle

Coupler angle

Transmission angle

Coupler point B

B = (L2 cos θ2, L2 sin θ2). C is the selected continuous intersection of the circle about B of radius L3 and the circle about D of radius L4. All four links keep their entered lengths.

<> Embed this animation calculator → Found a mistake? Contact us here

Mathematical Equations

Put ground pivot A at (0,0) and D at (L₁,0). Input pivot B is (L₂ cos θ₂, L₂ sin θ₂). The vector loop is:

L₂e^iθ₂ + L₃e^iθ₃ = L₁ + L₄e^iθ₄

Its real and imaginary parts are L₂ cos θ₂ + L₃ cos θ₃ = L₁ + L₄ cos θ₄ and L₂ sin θ₂ + L₃ sin θ₃ = L₄ sin θ₄.

Find the moving pivot by circle intersection

Pivot C must lie on a circle of radius L₃ about B and a circle of radius L₄ about D. Let d = |D−B| and e = (D−B)/d. A closed assembly exists when |L₃−L₄| ≤ d ≤ L₃+L₄, with coincident-center cases requiring separate treatment.

Set a = (L₃²−L₄²+d²)/(2d), h = √(L₃²−a²), and P = B+ae. The two possible intersections are C = P ± h(−e_y, e_x). The calculator uses one intersection branch. At tangency, both intersections coincide.

Find θ₃ = atan2(C_y−B_y, C_x−B_x) and θ₄ = atan2(C_y, C_x−L₁). The displayed transmission angle is the smaller angle between the coupler and output-link lines, in the range 0°–90°.

Simple Example

With L₁ = 4, L₂ = 2, L₃ = 3, L₄ = 3.5 and θ₂ = 45°, the displayed branch gives θ₄ ≈ 96.67°, θ₃ ≈ 43.42° and an acute transmission angle of 53.25°. Pivot B is approximately (1.414,1.414), and C is (3.593,3.476), in the chosen length units.

The shortest and longest lengths sum to 6, less than the other two lengths’ sum of 6.5. With the shortest link as the input adjacent to ground, this is a crank-rocker arrangement.

Four-Bar Linkage Geometry

Which link can rotate?

For sorted link lengths s ≤ p ≤ q ≤ l, compare s+l with p+q. In the strict Grashof case, s+l < p+q, fixing the shortest link gives a double-crank; fixing a link adjacent to the shortest gives a crank-rocker; fixing the link opposite the shortest gives a double-rocker. Equality is a change-point case that needs special attention at collinear configurations.

For s+l > p+q, an assemblable linkage has no full relative revolution and both ground-adjacent links rock. The length comparison does not replace checking whether the links close at the particular angle you enter.

See Carnegie Mellon University’s planar-linkage reference for mechanism classifications and Grashof’s criterion.

Transmission angle

This calculator reports the acute angle between coupler and output lines. Near 0°, they are nearly collinear and the coupler force has little moment arm about the output pivot. An angle near 90° provides a larger moment arm for the same force and output-link length. Complete mechanical advantage also depends on the input geometry; this angle alone does not give actuator force or efficiency.

Worked geometry: an 80 mm input crank

Use ground L₁ = 400 mm, input L₂ = 80 mm, coupler L₃ = 300 mm, output L₄ = 250 mm and θ₂ = 45°. Here 80+400 < 250+300. Ground is adjacent to the shortest link, so the arrangement is crank-rocker.

The displayed branch gives output angle 113.20°, coupler angle 35.27° and acute transmission angle 77.93°. B is (56.57,56.57) mm and C is (301.51,229.78) mm. B is the input-to-coupler joint; it is not automatically a wiper tip or an arbitrary coupler point.

Tracing a point on the coupler

A general point needs its coordinates relative to the coupler. If those coordinates are (u,v) measured from B, its ground-frame position is B + (u cos θ₃−v sin θ₃, u sin θ₃+v cos θ₃). This calculator displays B; it does not have controls for an additional coupler point.

Driving and checking a physical linkage

An actuator can drive a crank through suitable pivoted brackets. Its required force, stroke and pin-to-pin lengths depend on bracket geometry and load through the whole motion. Replacing one fixed-length bar with a telescoping actuator changes the mechanism and requires a different model.

Check the intended assembly branch, available travel, part collisions, joint clearances, loads and structural strength. Velocity and acceleration analysis require input motion versus time; the animation’s playback speed is for inspection. These position results do not select a motor or predict wear.

Frequently Asked Questions

📐 Browse all 1000+ Interactive Calculators →

Questions about this calculator or found an error? Message our engineering team.