---
title: "Oldham Coupling Interactive Lab | Sandbox Physics"
description: "The shafts turn evenly. The middle disc slides. Can offset shafts turn evenly while the center disc slides?"
canonical_url: "https://sandboxphysics.com/labs/oldham-coupling/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/oldham-coupling/"
last_modified: "2026-10-20"
---

Spatial mechanisms · Parallel offset

# Oldham Coupling

Connect parallel shafts that do not share a center. Inspect the perpendicular tongue-and-groove faces, follow the floating disc around its orbit, and compare constant output speed with changing sliding speed.

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

Ideal parallel shafts and frictionless perpendicular slides. The displayed offset and part dimensions are schematic, not manufacturer ratings. No angular misalignment, wear, contact force, inertia or backlash model.

Physics tutorial

## Equal rotation, unequal sliding

**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**: How can offset shafts turn evenly while the disc center moves?
- **One-sentence intuition**: Two perpendicular slides absorb the lateral offset without changing the angular phase.

### Core mathematical model

#### Two perpendicular sliding constraints

$$
\mathbf u=(\cos\theta,\sin\theta),\quad\mathbf v=(-\sin\theta,\cos\theta),\quad\mathbf C=e\cos\theta\,\mathbf u=(e,0)+e\sin\theta\,\mathbf v
$$

Coordinates are in the transverse upward–depth plane, with the input center at the origin.

#### Twice-around center orbit

$$
\mathbf C=\left(\frac e2(1+\cos2\theta),\frac e2\sin2\theta\right),\quad |\dot{\mathbf C}|=e\omega,\quad|\ddot{\mathbf C}|=2e\omega^2
$$

The center orbit has radius half the axis offset; it is not either shaft orbiting.

#### Constant transmission with changing sliding

$$
\phi=\theta,\quad\frac{\dot\phi}{\omega}=1,\quad\dot s_1=-e\omega\sin\theta,\quad\dot s_2=e\omega\cos\theta
$$

Travel is signed from the corresponding shaft center along its own groove direction.

#### 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

   #### Follow the floating center

   Increase the offset and compare the two slide-speed curves.

   **What to observe:** The center goes around twice per input turn. Sliding speeds are in quadrature while shaft speeds stay equal.

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)