Constrained Dynamics

// jWilliamDunn forked from:

// github.com/matthias-research/pages/blob/master/tenMinutePhysics/05-manyBeads.html

// see also: graphics.pixar.com/pbm2001/pdf/notesf.pdf

// this new version uses p5.js drawing and an independent loop for the physics simulation.

// it could be improved by removing the vector math code and using p5.js vectors instead

​

var // adjustables

    numBeads=5,

    friction=0.0003,

    restitution=0.9, //en.wikipedia.org/wiki/Coefficient_of_restitution

    g=30,            //gravity

    delay=30,        //delay in ms for physics loop

    steps=2;         //steps in physics calculations

​

let // fixed

    simMinWidth=2.0,

    cScale, simWidth, simHeight;

​

var mouseClicked=()=>setupScene(),

    keyPressed=()=>setupScene();

​

function setup() {

  createCanvas(windowWidth, windowHeight);

  cScale = Math.min(width, height) / simMinWidth;

  simWidth = canvas.width / cScale;

  simHeight = canvas.height / cScale;

  background(100);

  frameRate(60);

  setupScene();

  setInterval(simulate,delay); // loop for physics

}

​

function draw() {

  background(0);

  noFill();

  stroke(255);

  strokeWeight(2); // ring

  circle(cX(physicsScene.wireCenter), cY(physicsScene.wireCenter), 2*cScale*physicsScene.wireRadius);

​

  noStroke();

  fill(180,0,0); // beads

  for (var i = 0; i < physicsScene.beads.length; i++) {

    var bead = physicsScene.beads[i];

    circle(cX(bead.pos),cY(bead.pos), 2*cScale*bead.radius);

  }

}

​

function cX(pos) {

  return pos.x * cScale;

}

​

function cY(pos) {

  return height - pos.y * cScale;

}

​

// vector math -------------------------------------------------------

​

class Vector2 {

  constructor(x = 0.0, y = 0.0) {

    this.x = x; 

    this.y = y;

  }

​

  set(v) {

    this.x = v.x; this.y = v.y;

  }

​

  clone() {

    return new Vector2(this.x, this.y);

  }

​

  add(v, s = 1.0) {

    this.x += v.x * s;

    this.y += v.y * s;

    return this;

  }

​

  addVectors(a, b) {

    this.x = a.x + b.x;

    this.y = a.y + b.y;

    return this;

  }

​

  subtract(v, s = 1.0) {

    this.x -= v.x * s;

    this.y -= v.y * s;

    return this;

  }

​

  subtractVectors(a, b) {

    this.x = a.x - b.x;

    this.y = a.y - b.y;

    return this;      

  }

​

  length() {

    return Math.sqrt(this.x * this.x + this.y * this.y);

  }

​

  scale(s) {

    this.x *= s;

    this.y *= s;

    return this;

  }

​

  dot(v) {

    return this.x * v.x + this.y * v.y;

  }

​

  perp() {

    return new Vector2(-this.y, this.x);

  }

}

​

// scene -------------------------------------------------------

​

class Bead {

  constructor(radius, mass, pos) {

    this.radius = radius;

    this.mass = mass;

    this.pos = pos.clone();

    this.prevPos = pos.clone();

    this.vel = new Vector2();

  }

  startStep(dt, gravity) {

    this.vel.add(gravity, dt);

    this.prevPos.set(this.pos);

    this.pos.add(this.vel, dt);

  }

  keepOnWire(center, radius) {

    var dir = new Vector2();

    dir.subtractVectors(this.pos, center);

    var len = dir.length();

    if (len == 0.0)

      return;

    dir.scale(1.0 / len);

    var lambda = physicsScene.wireRadius - len;

    this.pos.add(dir, lambda);

    return lambda;

  }

  endStep(dt) {

    this.vel.subtractVectors(this.pos, this.prevPos);

    this.vel.scale((1.0-friction) / dt);

  }

​

}

​

var physicsScene = {

  gravity : new Vector2(0.0, -g),

  dt : 1.0 / 60.0,

  numSteps : steps,

  wireCenter : new Vector2(),

  wireRadius : 0.0,

  beads : [],

};

​

// -----------------------------------------------------

​

function setupScene() {

  physicsScene.beads = [];

​

  physicsScene.wireCenter.x = simWidth / 2.0;

  physicsScene.wireCenter.y = simHeight / 2.0;

  physicsScene.wireRadius = simMinWidth * 0.4;

​

  // numBeads is global

  var mass = 1.0;

​

  var r = 0.1;

  var angle = 0.0;

  for (i = 0; i < numBeads; i++) {

    mass = Math.PI * r * r;     

    var pos = new Vector2(

      physicsScene.wireCenter.x + physicsScene.wireRadius * Math.cos(angle), 

      physicsScene.wireCenter.y + physicsScene.wireRadius * Math.sin(angle));

​

    physicsScene.beads.push(new Bead(r, mass, pos));

    angle += Math.PI / numBeads +0.1;

    r = 0.05 + Math.random() * 0.1;

  }

}

​

// --- collision handling -------------------------------------------------------

​

function handleBeadBeadCollision(bead1, bead2) {

  //restitution is global

  var dir = new Vector2();

  dir.subtractVectors(bead2.pos, bead1.pos);

  var d = dir.length();

  if (d == 0.0 || d > bead1.radius + bead2.radius)

    return;

​

  dir.scale(1.0 / d);

​

  var corr = (bead1.radius + bead2.radius - d) / 2.0;

  bead1.pos.add(dir, -corr);

  bead2.pos.add(dir, corr);

​

  var v1 = bead1.vel.dot(dir);

  var v2 = bead2.vel.dot(dir);

​

  var m1 = bead1.mass;

  var m2 = bead2.mass;

​

  var newV1 = (m1 * v1 + m2 * v2 - m2 * (v1 - v2) * restitution) / (m1 + m2);

  var newV2 = (m1 * v1 + m2 * v2 - m1 * (v2 - v1) * restitution) / (m1 + m2);

​

  bead1.vel.add(dir, newV1 - v1);

  bead2.vel.add(dir, newV2 - v2);

}

​

// ------------------------------------------------

​

function simulate() {

  var step, i, sdt = physicsScene.dt / physicsScene.numSteps;

​

  for (step = 0; step < physicsScene.numSteps; step++) {

    for (i = 0; i < physicsScene.beads.length; i++)

      physicsScene.beads[i].startStep(sdt, physicsScene.gravity);

​

    for (i = 0; i < physicsScene.beads.length; i++) {

      physicsScene.beads[i].keepOnWire(

        physicsScene.wireCenter, physicsScene.wireRadius);

    }

​

    for (i = 0; i < physicsScene.beads.length; i++)

      physicsScene.beads[i].endStep(sdt);

​

    for (i = 0; i < physicsScene.beads.length; i++) {

      for (var j = 0; j < i; j++) {

        handleBeadBeadCollision(

          physicsScene.beads[i], physicsScene.beads[j]);

      }

    }

  }

}

​

/*

Copyright 2021 Matthias Müller - Ten Minute Physics

​

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Software is furnished to do so, subject to the following conditions:

​

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in all copies or substantial portions of the Software.

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OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS 

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*/