---
title: "Universal Joint Interactive Lab | Sandbox Physics"
description: "One turn each. Two pulses of speed. Why does a bent joint speed up twice each turn?"
canonical_url: "https://sandboxphysics.com/labs/universal-joint/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/universal-joint/"
last_modified: "2026-10-20"
---

Spatial mechanisms · Uneven rotation

# Universal Joint

Follow two intersecting shafts joined by a gold right-angle cross. Increase the bend, compare input and output phase markers, and measure the speed ripple hidden inside a complete revolution.

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

Single rigid Hooke–Cardan joint with ideal intersecting axes and prescribed input speed. Exact kinematics and massless quasistatic load transfer; no inertia, friction, bearing forces, flexibility or interference rating.

Physics tutorial

## A complete turn can hide uneven speed

**Background** Couplings carry rotation between shafts whose axes do not coincide.

**Why it matters** Watch the connecting geometry before reading the speed curve.

Start with the essentials

- **Focus question**: Why does a bent universal joint change output speed twice each turn?
- **One-sentence intuition**: The right-angle cross constrains the two fork orientations, so equal angle increments cannot pass through unchanged.

### Core mathematical model

#### Right-angle cross constraint

$$
\tan\phi=\cos\beta\tan\theta,\qquad\phi=\operatorname{unwrap}\!\left[\operatorname{atan2}(\cos\beta\sin\theta,\cos\theta)\right]
$$

Quadrant-safe unwrapping preserves a full output turn; the tangent relation alone loses quadrant information.

#### Instantaneous speed ratio

$$
R=\frac{\dot\phi}{\omega}=\frac{\cos\beta}{1-\sin^2\beta\sin^2\theta},\quad R_{\min}=\cos\beta,\quad R_{\max}=\frac1{\cos\beta}
$$

The extrema occur a quarter input turn apart and the pattern repeats each half-turn.

#### Output acceleration

$$
\ddot\phi=\omega^2\frac{\cos\beta\sin^2\beta\sin2\theta}{(1-\sin^2\beta\sin^2\theta)^2}
$$

Input angular speed is fixed; output angular acceleration can still be nonzero.

#### Massless quasistatic virtual power

$$
\tau_{\rm in}=-\tau_o\frac{\mathrm d\phi}{\mathrm d\theta},\qquad\tau_{\rm in}\omega+\tau_o\dot\phi=0
$$

No frictional loss or inertial power is included.

### Common difficulties

#### Equal turns versus equal speed

**Typical misconception** One output turn per input turn guarantees constant speed.

**Better mental model** A universal joint has a varying instantaneous ratio; an ideal parallel Oldham coupling has a constant ratio.

#### Kinematics versus dynamics

**Typical misconception** A smooth animation establishes acceptable bearing loads and wear.

**Better mental model** Input motion is prescribed, and the connecting parts are massless and frictionless. Dynamic inertia, real clearances and contact forces require a different model.

### Run the experiment

1. 01

   #### Establish the aligned limit

   Select aligned shafts and play one turn.

   **What to observe:** Both shaft phase markers agree and the speed ratio stays unity.

2. 02

   #### Increase the bend

   Compare the slowest and fastest output presets, then inspect the cross from different views.

   **What to observe:** The output speed oscillates twice while the orthogonality residual remains near numerical precision.

3. 03

   #### Inspect the transfer of power

   Reverse the signed load and export the cycle.

   **What to observe:** Torque and power change sign; the prescribed motion does not.

Physical conclusion

A geometric velocity relation does not determine contact forces or real drive performance.

Theory & apparatus: [Carnegie Mellon · Universal Joint](https://www.cs.cmu.edu/~rapidproto/mechanisms/chpt8.html) · [Ruland · Oldham Coupling](https://www.ruland.com/oldham-coupling-install)

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)