---
title: "Four-Bar Linkage Interactive Lab | Sandbox Physics"
description: "One crank. A whole family of paths. Can these links turn all the way around?"
canonical_url: "https://sandboxphysics.com/labs/four-bar-linkage/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/four-bar-linkage/"
last_modified: "2026-10-19"
---

Mechanism design · Closed loops

# Four-Bar Linkage

Turn a three-dimensional four-bar mechanism. Change link lengths, flip its assembly and move a tracer on the coupler. Find which designs rotate freely and which reach a geometric limit.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Rigid planar links and ideal pins. Exact position geometry, analytic velocity and acceleration, and quasistatic signed load transfer. Prescribed input speed; no inertia, friction, collisions or structural strength model.

Physics tutorial

## A closed loop designs the motion

**Background** Rigid links impose geometric constraints before any force calculation.

**Why it matters** Watch the same solved joint positions drive the 3D apparatus, tracer and quantitative views.

Start with the essentials

- **Focus question**: Which link lengths allow this input to complete a turn?
- **One-sentence intuition**: Assembly is the intersection of two circles. Where circles separate, no rigid assembly exists; where they touch, the velocity constraint becomes singular.

### Core mathematical model

#### Vector loop closure

$$
a\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}+b\begin{bmatrix}\cos\beta\\\sin\beta\end{bmatrix}=\begin{bmatrix}d\\0\end{bmatrix}+c\begin{bmatrix}\cos\phi\\\sin\phi\end{bmatrix}
$$

The fixed pivots lie on the horizontal axis. The two oriented circle intersections define the assembly branches.

#### Continuous input rotation

$$
d+a<b+c,\qquad |d-a|>|b-c|
$$

These strict inequalities keep this chosen input away from circle tangency for an entire turn. Equality is a change or limiting configuration and is treated separately.

#### Signed virtual power

$$
\tau_{\rm in}=-\tau_o\phi'(\theta),\qquad\tau_{\rm in}\dot\theta+\tau_o\dot\phi=0
$$

The output torque is an applied external torque, positive counterclockwise. This relation requires a regular closure and negligible link inertia.

### Common difficulties

#### Zero is not undefined

**Typical misconception** Every stationary output is a singular solution.

**Better mental model** At an ordinary output reversal the output velocity is zero and the solution can remain regular. At a closure singularity the differential equations cannot uniquely determine the motion.

#### Prescribed motion

**Typical misconception** The animation proves this motor can drive the real mechanism.

**Better mental model** Input speed is prescribed. Link inertia, friction, compliance and collision dynamics are omitted; power balance is quasistatic.

### Run the experiment

1. 01

   #### Turn it

   Play, then pause and drag the gold crank pin in the plan view.

   **What to observe:** The 3D links and all plots share the same input angle.

2. 02

   #### Change a length

   Compare the two assemblies, then select Geometric limit.

   **What to observe:** Only feasible positions are drawn. A warning replaces missing links instead of stretching them.

3. 03

   #### Inspect the limiting pose

   Select At the toggle and inspect the undefined derivatives. Then export the sweep.

   **What to observe:** Missing and singular values remain distinct from valid zero values.

Physical conclusion

Geometry sets the motion and its limits. Quasistatic power transfer does not establish dynamic performance.

Theory & apparatus: [Carnegie Mellon · Planar Linkages](https://www.cs.cmu.edu/~rapidproto/mechanisms/chpt5.html) · [Universidad de Almería · Four-bar mechanisms](https://ingmec.ual.es/tmm/L01_03a_four_bars.html)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)