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A simple, efficient spring animation library for smooth, natural motion.

Harmonica OpenGL Demo

It even works well on the command line.

Harmonica TUI Demo


Harmonica is framework-agnostic and works well in 2D and 3D contexts. Simply call NewSpring with your settings to initialize and Update on each frame to animate.

import ""

// A thing we want to animate.
sprite := struct{
    x, xVelocity float64
    y, yVelocity float64

// Where we want to animate it.
const targetX = 50.0
const targetY = 100.0

// Initialize a spring with framerate, angular frequency, and damping values.
spring := harmonica.NewSpring(harmonica.FPS(60), 6.0, 0.5)

// Animate!
for {
    sprite.x, sprite.xVelocity = spring.Update(sprite.x, sprite.xVelocity, targetX)
    sprite.y, sprite.yVelocity = spring.Update(sprite.y, sprite.yVelocity, targetY)

For details, see the examples and the docs.


NewSpring takes three values:

  • Time Delta: the time step to operate on. Game engines typically provide a way to determine the time delta, however if that's not available you can simply set the framerate with the included FPS(int) utility function. Make the framerate you set here matches your actual framerate.
  • Angular Velocity: this translates roughly to the speed. Higher values are faster.
  • Damping Ratio: the springiness of the animation, generally between 0 and 1, though it can go higher. Lower values are springier. For details, see below.

Damping Ratios

The damping ratio affects the motion in one of three different ways depending on how it's set.


A spring is under-damped when its damping ratio is less than 1. An under-damped spring reaches equilibrium the fastest, but overshoots and will continue to oscillate as its amplitude decays over time.

Critical Damping

A spring is critically-damped the damping ratio is exactly 1. A critically damped spring will reach equilibrium as fast as possible without oscillating.


A spring is over-damped the damping ratio is greater than 1. An over-damped spring will never oscillate, but reaches equilibrium at a slower rate than a critically damped spring.


This library is a fairly straightforward port of Ryan Juckett’s excellent damped simple harmonic oscillator originally written in C++ in 2008 and published in 2012. Ryan’s writeup on the subject is fantastic.



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