-
Notifications
You must be signed in to change notification settings - Fork 7
Expand file tree
/
Copy pathspring_damper.html
More file actions
613 lines (519 loc) · 18.5 KB
/
spring_damper.html
File metadata and controls
613 lines (519 loc) · 18.5 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
<!doctype html>
<html class="no-js" lang="en">
<head>
<meta charset="utf-8">
<style>
body {font-family: Helvetica, sans-serif;}
table {background-color:#CCDDEE;text-align:left}
</style>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
extensions: ["tex2jax.js"],
jax: ["input/TeX", "output/HTML-CSS"],
tex2jax: {
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
processEscapes: true
},
"HTML-CSS": { fonts: ["TeX"] }
});
</script>
<script type="text/javascript" aync src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js"></script>
<script src="https://cdn.plot.ly/plotly-2.5.1.min.js"></script>
<title>Mass-Spring-Damper Model</title>
</head>
<body>
<main>
<h1 style="text-align:center">Mass-Spring-Damper Model</h1>
<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
<col width="700">
<col width="400">
<tr>
<td>
<canvas id="simCanvas" width="700" height="600" style="border:2px solid #000000;border-radius: 20px;background-color:#FFFFFF">Your browser does not support the HTML5 canvas tag.</canvas>
</td>
<td>
<div id="plotOutput" style="width: 600px; height: 600px;border:2px solid #000000;border-radius: 0px;background-color:#EEEEEE"></div>
</td>
</tr>
<tr>
<td>
<table>
<col width="200" style="padding-right:10px">
<col width="100">
<tr>
<td><label>Current time</label></td>
<td><span id="time">0.00</span> s</td>
</tr>
<tr>
<td><label>Time per sim. step</label></td>
<td><span id="timePerStep">0.00</span> ms</td>
</tr>
<tr>
<td><label for="timeStepSizeInput">Time step size</label></td>
<td><input onchange="gui.sim.timeStepSize=parseFloat(value)" id="timeStepSizeInput" type="number" value="0.01" step="0.001"></td>
</tr>
<tr>
<td><label for="restLengthInput">Rest length</label></td>
<td><input id="restLengthInput" type="number" value="4.0" step="0.1"></td>
</tr>
<tr>
<td><label for="stiffnessInput">Stiffness</label></td>
<td><input onchange="gui.sim.stiffness=parseFloat(value)" id="stiffnessInput" type="number" value="5.0" step="1.0"></td>
</tr>
<tr>
<td><label for="dampingInput">Damping</label></td>
<td><input onchange="gui.sim.damping=parseFloat(value)" id="dampingInput" type="number" value="0.0" step="0.01"></td>
</tr>
<tr>
<td><label for="massInput">Mass</label></td>
<td><input onchange="gui.sim.mass=parseFloat(value)" id="massInput" type="number" value="0.5" step="0.01"></td>
</tr>
<tr>
<td><label for="timeIntegrationInput">Time integration method</label></td>
<td><select onchange="gui.sim.timeIntegrationMethod=value" id="timeIntegrationInput">
<option>Explicit Euler</option>
<option selected="selected">Symplectic Euler</option>
<option>Runge-Kutta 2</option>
<option>Implicit Euler</option>
</select>
</tr>
<tr>
<td></td>
<td><button onclick="gui.restart()" type="button" id="restart">Restart</button></td>
</tr>
<tr>
<td></td>
<td><button onclick="gui.doPause()" type="button" id="Pause">Pause</button></td>
</tr>
</table>
</td>
</tr>
<tr><td>
<h2>Mass spring algorithm:</h2>
This example shows a mass spring system that consists of particles linked by damped springs:
<ol>
<li>compute spring forces</li>
<li>time integration to get new particle positions and velocities</li>
</ol>
Note that the simulation is only conditionally stable since a conditionally stable explicit time integration method is used.
<h3>1. Compute spring forces</h3>
<p>In general a spring force can be obtained for any holonomic constraint.
In this example we use distance constraints
$$C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2}) = \| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d,$$
where $d$ is the rest length between particles $\mathbf{x}_{i_1}$ and $\mathbf{x}_{i_2}$.</p>
<h4>Potential energy</h4>
<p>For a scalar constraint we can define a potential energy as:
$$E(\mathbf x) = \frac k 2 C(\mathbf x)^2,$$
where $k$ is the stiffness of the spring.
<h4>General spring force</h4>
The spring force for a particle $j$ is then determined by the negative gradient of the potential energy function:
$$\mathbf f_j = - \frac{\partial E(\mathbf x)}{\partial \mathbf x_j} = -k \frac{\partial C(\mathbf x)}{\partial \mathbf x_j} C(\mathbf x).$$
<h4>General damping force</h4>
The corresponding damping force is obtained by using the time derivative of the constraint function:
$$\mathbf f^D_j = -\mu \frac{\partial C(\mathbf x)}{\partial \mathbf x_j} \dot{C}(\mathbf x).$$
<h4>Constraint gradients:</h4>
<p>To compute the spring and damping forces, the constraint gradients are required which are computed as: </p>
$$\begin{align*}
\frac{\partial C_i}{\partial \mathbf x_{i_1}} &= \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\frac{\partial C_i}{\partial \mathbf x_{i_2}} &= - \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}
\end{align*}$$
<h4>Spring force</h4>
So finally we get the following spring forces for a distance constraint $C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2})$:
$$\begin{align*}
\mathbf f_{i_1} = -k (\| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\mathbf f_{i_2} = +k (\| \mathbf x_{i_1} -\mathbf x_{i_2} \|-d) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}.
\end{align*}$$
<h4>Damping force</h4>
The damping forces for a distance constraint $C_i(\mathbf{x}_{i_1}, \mathbf{x}_{i_2})$ are determined as:
$$\begin{align*}
\mathbf f^D_{i_1} = -k \left ((\mathbf v_{i_1}- \mathbf v_{i_2}) \cdot \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \right ) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \\
\mathbf f^D_{i_2} = +k \left ((\mathbf v_{i_1}- \mathbf v_{i_2}) \cdot \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|} \right ) \frac{\mathbf x_{i_1} -\mathbf x_{i_2}}{\| \mathbf x_{i_1} -\mathbf x_{i_2} \|}.
\end{align*}$$
<h3>2. Time integration</h3>
Finally, the particles are advected by numerical time integration. In our case we use a symplectic Euler method:
$$\begin{align*}
\mathbf v(t + \Delta t) &= \mathbf v(t) + \frac{\Delta t}{m} \left (\mathbf f(t) + \mathbf f^D + \mathbf f^{\text{ext}} \right ) \\
\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t + \Delta t),
\end{align*}$$
where $\mathbf f^{\text{ext}}$ are the external forces.
</td></tr>
</table>
</main>
<script id="simulation_code" type="text/javascript">
class Simulation
{
constructor()
{
this.x = 6.0;
this.y = 0.0;
this.vx = 0.0;
this.vy = 0.0;
this.stiffness = 5.0;
this.restLength = 4.0;
this.damping = 0.05;
this.mass = 0.5;
this.timeStepSize = 0.01;
this.time = 0;
this.timeIntegrationMethod = "Symplectic Euler";
this.plotValues = {};
}
computeSpringForce(x, y, vx, vy)
{
let xl = Math.sqrt(x*x + y*y);
// compute displacement: C = ||x|| - restLength
let C = xl - this.restLength;
// determine gradient
let gradC_x = 0.0;
let gradC_y = 0.0;
if (xl > 0.0001)
{
gradC_x = x/xl;
gradC_y = y/xl;
}
// compute spring force
let Fs_x = -this.stiffness * C * gradC_x;
let Fs_y = -this.stiffness * C * gradC_y;
// compute damping force
let dot = vx * gradC_x + vy * gradC_y; // v*gradC
let Fd_x = -this.damping * dot * gradC_x;
let Fd_y = -this.damping * dot * gradC_y;
return [Fs_x + Fd_x, Fs_y + Fd_y];
}
explicitEuler()
{
// compute spring force
let force = this.computeSpringForce(this.x, this.y, this.vx, this.vy);
// explicit Euler step
this.x += this.timeStepSize * this.vx;
this.y += this.timeStepSize * this.vy;
this.vx += this.timeStepSize/this.mass * force[0];
this.vy += this.timeStepSize/this.mass * force[1];
}
symplecticEuler()
{
// compute spring force
let force = this.computeSpringForce(this.x, this.y, this.vx, this.vy);
// symplectic Euler step
this.vx += this.timeStepSize/this.mass * force[0];
this.vy += this.timeStepSize/this.mass * force[1];
this.x += this.timeStepSize * this.vx;
this.y += this.timeStepSize * this.vy;
}
rungeKutta2()
{
// compute spring force
let force1 = this.computeSpringForce(this.x, this.y, this.vx, this.vy);
// Runge-Kutta 2 step
let k1_x = this.timeStepSize * this.vx;
let k1_y = this.timeStepSize * this.vy;
let k1_vx = this.timeStepSize/this.mass * force1[0];
let k1_vy = this.timeStepSize/this.mass * force1[1];
// compute spring force
var force2 = this.computeSpringForce(this.x + 0.5*k1_x, this.y + 0.5*k1_y, this.vx + 0.5*k1_vx, this.vy + 0.5*k1_vy);
let k2_x = this.timeStepSize * (this.vx + 0.5*k1_vx);
let k2_y = this.timeStepSize * (this.vy + 0.5*k1_vy);
let k2_vx = this.timeStepSize/this.mass * force2[0];
let k2_vy = this.timeStepSize/this.mass * force2[1];
this.x += k2_x;
this.y += k2_y;
this.vx += k2_vx;
this.vy += k2_vy;
}
implicitEuler()
{
// compute spring force
let f0 = this.computeSpringForce(this.x, this.y, this.vx, this.vy);
// compute df/dx
let df_dx = this.compute_dF_dx(this.x, this.y);
// determine matrix:
// A = I - h^2 M^-1 df/dx
// Note that we set df/dv to zero. This means that damping is only handled explicitly.
let h = this.timeStepSize;
let h2 = h*h;
// M^-1 * df/dx
let Minv_df_dx = [[df_dx[0][0]/this.mass, df_dx[0][1]/this.mass], [df_dx[1][0]/this.mass, df_dx[1][1]/this.mass]];
let A = [[1.0 - h2*Minv_df_dx[0][0], -h2*Minv_df_dx[0][1]],
[-h2*Minv_df_dx[1][0], 1.0-h2*Minv_df_dx[1][1]]];
// determine right hand side of linear system
// b = h Minv (f_0 + h df/dx v_0)
let df_dx_v = this.mult(df_dx, [this.vx, this.vy]);
let b = [h/this.mass*(f0[0] + h*df_dx_v[0]), h/this.mass*(f0[1] + h*df_dx_v[1])];
// solve linear system by inverting A
let deltaV = this.mult(this.inverse(A), b);
// implicit Euler step
this.vx += deltaV[0];
this.vy += deltaV[1];
this.x += h * this.vx;
this.y += h * this.vy;
}
// matrix vector multiplication
mult(A, x)
{
return [A[0][0] * x[0] + A[0][1]*x[1], A[1][0] * x[0] + A[1][1]*x[1]];
}
// inverse of a matrix A
inverse(A)
{
let det = A[0][0]*A[1][1] - A[0][1]*A[1][0];
if (Math.abs(det) > 1.0e-6)
return [[A[1][1]/det, -A[0][1]/det], [-A[1][0]/det, A[0][0]/det]];
else
return A;
}
compute_dF_dx(x, y)
{
let xl = Math.sqrt(x*x + y*y);
// determine normalized x
let hatX = [0,0];
if (xl > 0.0001)
hatX = [x/xl, y/xl];
// determine x*x^T
let xxT = [[hatX[0]*hatX[0], hatX[0]*hatX[1]], [hatX[1]*hatX[0], hatX[1]*hatX[1]]];
// dhatX_dx = (I- hatX*hatX^T) / ||x||
let hatX_dx = [[(1.0 - xxT[0][0]) / xl, -xxT[0][1] / xl], [-xxT[1][0] / xl, (1.0 - xxT[1][1]) / xl]];
// df_dx = -k (dC/dx*dC/x^T + d^2C/d^2x C)
let CCT = xxT;
let C = xl - this.restLength;
return [[-this.stiffness * (CCT[0][0] + hatX_dx[0][0] * C), -this.stiffness * (CCT[0][1] + hatX_dx[0][1] * C)],
[-this.stiffness * (CCT[1][0] + hatX_dx[1][0] * C), -this.stiffness * (CCT[1][1] + hatX_dx[1][1] * C)]];
}
plotValue(name, val)
{
if (this.plotValues[name] == undefined)
this.plotValues[name] = [val];
else
this.plotValues[name].push(val);
}
// simulation step
simulationStep()
{
let dt = this.timeStepSize;
// time integration step
if (this.timeIntegrationMethod == "Explicit Euler")
this.explicitEuler();
else if (this.timeIntegrationMethod == "Symplectic Euler")
this.symplecticEuler();
else if (this.timeIntegrationMethod == "Runge-Kutta 2")
this.rungeKutta2();
else if (this.timeIntegrationMethod == "Implicit Euler")
this.implicitEuler();
// plot energies
let kE = 0.5*this.mass*(this.vx*this.vx + this.vy*this.vy);
let xl = Math.sqrt(this.x*this.x + this.y*this.y);
let pE = 0.5*this.stiffness * (xl - this.restLength)*(xl - this.restLength);
this.plotValue('Kinetic energy', kE);
this.plotValue('Potential energy', pE);
this.plotValue('Total energy', (kE+pE));
// update simulation time
this.time = this.time + dt;
}
}
class GUI
{
constructor()
{
this.canvas = document.getElementById("simCanvas");
this.c = this.canvas.getContext("2d");
this.requestID = -1;
this.timeSum = 0.0;
this.counter = 0;
this.pause = false;
this.origin = { x : this.canvas.width / 2 -200, y : this.canvas.height/2};
this.zoom = 80;
this.particleRadius = 0.025;
this.selectedParticle = -1;
// register mouse event listeners (zoom/selection)
this.canvas.addEventListener("mousedown", this.mouseDown.bind(this), false);
this.canvas.addEventListener("mousemove", this.mouseMove.bind(this), false);
this.canvas.addEventListener("mouseup", this.mouseUp.bind(this), false);
this.canvas.addEventListener("wheel", this.wheel.bind(this), false);
}
// set simulation parameters from GUI and start mainLoop
restart()
{
window.cancelAnimationFrame(this.requestID);
delete this.sim;
this.sim = new Simulation();
this.timeSum = 0.0;
this.counter = 0;
this.sim.stiffness = parseFloat(document.getElementById('stiffnessInput').value);
this.sim.damping = parseFloat(document.getElementById('dampingInput').value);
this.sim.timeStepSize = parseFloat(document.getElementById('timeStepSizeInput').value);
this.sim.mass = parseFloat(document.getElementById('massInput').value);
this.sim.restLength = parseFloat(document.getElementById('restLengthInput').value);
this.sim.timeIntegrationMethod = document.getElementById('timeIntegrationInput').value;
this.mainLoop();
}
drawCoordinateSystem()
{
// draw x-axis
this.c.strokeStyle = "#FF0000";
this.c.beginPath();
this.c.moveTo(this.origin.x, this.origin.y);
this.c.lineTo(this.origin.x+1*this.zoom, this.origin.y);
this.c.stroke();
// draw y-axis
this.c.strokeStyle = "#00FF00";
this.c.beginPath();
this.c.moveTo(this.origin.x, this.origin.y);
this.c.lineTo(this.origin.x, this.origin.y-1*this.zoom);
this.c.stroke();
}
draw()
{
this.c.clearRect(0, 0, this.canvas.width, this.canvas.height);
this.drawCoordinateSystem();
// draw springs as lines
this.c.strokeStyle = "#666666";
let num_steps = 50;
let dx = this.sim.x / num_steps;
let x = 0.0;
let y = 0.0;
for (let i = 0; i < num_steps; i++)
{
this.c.beginPath();
this.c.moveTo(this.origin.x + x*this.zoom, this.origin.y - y*this.zoom);
if (i == 0)
y = -0.25;
else if (i == num_steps-1)
y = 0.0;
else if (i % 2 == 0)
y -= 0.5;
else
y += 0.5;
this.c.lineTo(this.origin.x + (x+dx)*this.zoom, this.origin.y - y*this.zoom);
this.c.stroke();
x+=dx;
}
// draw line at rest length
this.c.strokeStyle = "#00AA11";
this.c.beginPath();
this.c.moveTo(this.origin.x + this.sim.restLength*this.zoom, this.origin.y+0.5*this.zoom);
this.c.lineTo(this.origin.x + this.sim.restLength*this.zoom, this.origin.y-0.5*this.zoom);
this.c.stroke();
// draw particles as circles
this.c.fillStyle = "#BB0000";
this.c.beginPath();
this.c.arc(this.origin.x, this.origin.y, 0.1 * this.zoom, 0, Math.PI*2, true);
this.c.closePath();
this.c.fill();
this.c.fillStyle = "#0000FF";
let px = this.origin.x + this.sim.x * this.zoom;
let py = this.origin.y - this.sim.y * this.zoom;
this.c.beginPath();
this.c.arc(px, py, 0.1 * this.zoom, 0, Math.PI*2, true);
this.c.closePath();
this.c.fill();
}
mainLoop()
{
// perform multiple sim steps per render step
for (let i=0; i < 8; i++)
{
let t0 = performance.now();
this.sim.simulationStep();
let t1 = performance.now();
this.timeSum += t1 - t0;
this.counter += 1;
if (this.counter % 50 == 0)
{
this.timeSum /= this.counter;
document.getElementById("timePerStep").innerHTML = this.timeSum.toFixed(2);
this.timeSum = 0.0;
this.counter = 0;
}
document.getElementById("time").innerHTML = this.sim.time.toFixed(2);
}
this.draw();
this.updatePlot();
if (!this.pause)
this.requestID = window.requestAnimationFrame(this.mainLoop.bind(this));
}
updatePlot()
{
let data = [];
for(let key in this.sim.plotValues)
{
let value = this.sim.plotValues[key];
while(this.sim.plotValues[key].length > 1000)
this.sim.plotValues[key].shift();
let trace = {
y: value,
type: 'scatter',
name: key
};
data.push(trace);
}
let layout = {
title: 'Energy',
yaxis: {
autorange: true,
range: [0, 3000],
type: 'linear'
}
}
Plotly.newPlot('plotOutput', data, layout);
}
doPause()
{
this.pause = !this.pause;
if (!this.pause)
this.mainLoop();
}
mouseDown(event)
{
// left mouse button down
if (event.which == 1)
{
let mousePos = this.getMousePos(this.canvas, event);
for (let i = 0; i < this.sim.particles.length; i++)
{
let p = this.sim.particles[i];
let px = this.origin.x + p.x * this.zoom;
let py = this.origin.y - p.y * this.zoom;
let dx = px - mousePos.x
let dy = py - mousePos.y
let dist2 = Math.sqrt(dx * dx + dy * dy)
if (dist2 < 10)
{
this.selectedParticle = i;
break;
}
}
}
}
getMousePos(canvas, event)
{
let rect = canvas.getBoundingClientRect();
return {
x: event.clientX - rect.left,
y: event.clientY - rect.top
};
}
mouseMove(event)
{
if (this.selectedParticle != -1)
{
let mousePos = this.getMousePos(this.canvas, event);
this.sim.particles[this.selectedParticle].x = (mousePos.x - this.origin.x) / this.zoom;
this.sim.particles[this.selectedParticle].y = -(mousePos.y - this.origin.y) / this.zoom;
}
}
mouseUp(event)
{
this.selectedParticle = -1;
}
wheel(event)
{
event.preventDefault();
this.zoom += event.deltaY * -0.05;
if (this.zoom < 1)
this.zoom = 1;
}
}
gui = new GUI();
gui.restart();
</script>
</body>
</html>