---
title: "How Coupled Springs Move Together Interactive Lab | Sandbox Physics"
description: "Can one bead's swing grow and fade while the energy in every normal mode stays fixed? It can."
canonical_url: "https://sandboxphysics.com/labs/coupled-oscillators/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/coupled-oscillators/"
last_modified: "2026-10-27"
---

M065 · Normal modes

# Coupled Oscillators

**Can one bead's swing grow and fade while the energy in every normal mode stays fixed?** It can. Written in normal modes, the chain splits into independent oscillators, and each one keeps its own energy forever. A single bead, though, moves with a mix of modes. When two of those modes have nearby frequencies, they drift in and out of step, and the bead's swing grows and fades: a beat.

Release a chain of beads and springs with fixed ends, either in one sine-shaped mode or in two neighboring high modes; the beads move along the chain. Pick any bead and compare the forces from its two springs with its acceleration, follow how each mode adds to its displacement, and track the energy computed directly from bead positions and velocities. The three-dimensional apparatus, the event cursor and the plots all show the same exact solution.

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

The masses are equal and normalized, the springs are identical ideal Hooke springs, the ends are fixed and every bead starts at rest. The beads move along the chain; this is not a model of a string vibrating sideways. Each normal mode evolves analytically, with no time stepping, and the energy computed from bead positions and velocities is found separately from the mode energies. Beats use the two highest neighboring modes. The reported period is how long the absolute value of the beat envelope takes to repeat, which need not be when the whole chain returns to the same state. Damping, sideways motion, spring mass, coil mechanics and collisions are left out. The preset amplitudes keep the ideal springs within the stated normalized strain range. The cursor shows five decimal places, but that does not mean experimental accuracy.

Physics tutorial

## Coupled oscillators: find normal modes inside complex motion

**Background** Springs link each mass to its neighbors. Certain shapes of the whole chain keep their form as they move, with every mass oscillating at the same frequency.

**Why it matters** Molecular vibrations, phonons, vibrating structures and quantum fields all rely on the same idea: replace coupled coordinates with independent modes.

Start with the essentials

- **Focus question**: Why can a chain of interacting masses still be split into as many independent simple motions as there are masses?
- **One-sentence intuition**: In the right collective coordinates, a coupled system becomes a set of independent harmonic oscillators.

### Core mathematical model

#### Discrete spring chain

$$
m\ddot{x}_j=k(x_{j+1}-2x_j+x_{j-1})
$$

Mass $j$ feels the sum of restoring forces from its two neighbors.

#### Fixed-end normal frequencies

$$
\omega_p=2\sqrt{\frac{k}{m}}\sin\!\left(\frac{p\pi}{2(N+1)}\right)
$$

Each mode $p$ has its own frequency and fixed node pattern.

### Common difficulties

#### A mode is not one mass

**Typical misconception** Mode 2 means the motion of the second mass.

**Better mental model** One mode is a displacement pattern involving every mass.

#### Beats are not a new eigenfrequency

**Typical misconception** The slow rise and fall of the amplitude means a new low-frequency mode has appeared.

**Better mental model** The beat is an envelope created by adding two nearby frequencies.

### Run the experiment

1. 01

   #### Find the lowest mode

   Choose Mode 1, press First-mode quarter period and look at each bead.

   **What to observe:** All masses move broadly in the same direction.

2. 02

   #### Count nodes

   Compare Mode 2 with Mode 3.

   **What to observe:** Higher mode number adds internal nodes and raises frequency.

3. 03

   #### Add two modes together

   Choose Neighboring-mode beats.

   **What to observe:** A slow envelope wraps the fast oscillation, yet the energy in each mode stays constant.

Physical conclusion

Normal modes are the coordinate system that makes coupling simple.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an 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)