Колебания в цилиндре — различия между версиями
Материал из Department of Theoretical and Applied Mechanics
| Строка 22: | Строка 22: | ||
<math> θ= \frac{1}{2}\ M\ R ^{2} </math> | <math> θ= \frac{1}{2}\ M\ R ^{2} </math> | ||
| − | <math> T= \frac{1}{4}\ M\ R ^{2} \ dot\varphi ^{2} + \frac{1}{2}\ m\ V ^{2} </math> | + | <math> T= \frac{1}{4}\ M\ R ^{2} \dot\varphi ^{2} + \frac{1}{2}\ m\ V ^{2} </math> |
<math> \Pi = m \ g\ x \sin(\varphi )\ </math> | <math> \Pi = m \ g\ x \sin(\varphi )\ </math> | ||
Версия 11:37, 22 декабря 2017
Постановка задачи
Определить движение однородного стержня массы М, скользящего внутри цилиндра радиуса R. Внутри однородного стержня движется точка массы m с постоянной скоростью v.
Решение
Решаем, пользуясь уравнением Лагранжа 2-ого рода:
, где
T - кинетическая энергия системы П - потенциальная энергия системы q - независимые обобщенные координаты
1) У данной системы одна степень свободы. В качестве обобщенной координаты возьмем угол отклонения стержня от заданного положения.
2) Выразим кинетическую и потенциальную энергию через эту координату и найдем производные, необходимые для решения уравнения Лагранжа:
3)Продифференцировав энергии по обобщенной координате и ее производной, получаем уравнение:
4)Решив это дифференциальное уравнение, с нулевыми начальными условиями и зафиксированным углом наклона стержня, получим уравнение движения:
Визуализация процесса
Для моделирования колебаний данного маятника используются cледующие библиотеки:
- three.js
- dat.gui.js
- stats.js
Текст программы на языке JavaScript:
Файл "Generalwork2.js"
1 (function () {
2
3
4 var stats = initStats();
5
6 // create a scene, that will hold all our elements such as objects, cameras and lights.
7 var scene = new THREE.Scene();
8
9 // create a camera, which defines where we're looking at.
10 var camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 0.1, 1000);
11
12 // create a render and set the size
13 var renderer = new THREE.WebGLRenderer();
14
15
16 renderer.setClearColor(new THREE.Color(0xEEEEEE, 1.0));
17 renderer.setSize(window.innerWidth, window.innerHeight);
18 renderer.shadowMapEnabled = true;
19
20 var OrbitControls= new THREE.OrbitControls(camera);
21 OrbitControls.autoRotate = false;
22 var clock = new THREE.Clock();
23
24
25 var planeGeometry = new THREE.PlaneGeometry(150,60);
26 var planeMaterial = new THREE.MeshBasicMaterial({color:0xE1EfE1});
27 var plane = new THREE.Mesh(planeGeometry,planeMaterial);
28 plane.receiveShadow = true;
29
30 // rotate and position the plane
31 plane.rotation.x=-0.5*Math.PI;
32 plane.position.x=10
33 plane.position.y=-4
34 plane.position.z=0
35
36 // add the plane to the scene
37 scene.add(plane);
38
39 var sphereGeometry = new THREE.SphereGeometry(2/5, 20, 20);
40 var sphereMaterial = new THREE.MeshLambertMaterial({color: 0xffdddd,transparent:false,opacity:0.2});
41 var sphere = new THREE.Mesh(sphereGeometry, sphereMaterial);
42
43 // position the sphere
44 sphere.position.x =0;
45 sphere.position.y =0;
46 sphere.position.z =0;
47 // add the sphere to the scene
48 scene.add(sphere);
49
50 //create a stick
51 var stickGeometry = new THREE.CylinderGeometry(1/2,1/2,7.7);
52 var stickMaterial = new THREE.MeshLambertMaterial({color: 0x7777ff,transparent:true,opacity:0.7});
53 var stick = new THREE.Mesh(stickGeometry, stickMaterial);
54
55 // stick.castShadow = true;
56
57 stick.position.x=0;
58 stick.position.y=0;
59 stick.position.z=0;
60
61 // add the cube to the scene
62 scene.add(stick);
63
64 // create a cube
65
66 var cylinderGeometry = new THREE.CylinderGeometry(4,4,10,30,10, true );
67 var cylinderMaterial = new THREE.MeshLambertMaterial({color: 0xb3a1aa});
68 cylinderMaterial.side = THREE.DoubleSide;
69 var cylinder = new THREE.Mesh(cylinderGeometry, cylinderMaterial);
70 cylinder.castShadow=true;
71 // cylinder.castShadow = true;
72
73 // position the cylinder
74 cylinder.rotation.x=-0.5*Math.PI;
75 cylinder.position.x=0;
76 cylinder.position.y=0;
77 cylinder.position.z=0;
78
79 // add the cube to the scene
80 scene.add(cylinder);
81
82 var group = new THREE.Object3D();
83 group.add(sphere);
84 group.add(stick);
85 scene.add(group);
86
87 var arrowHelper = new THREE.ArrowHelper(new THREE.Vector3( 0, 1.0, 0.0 ), group.position, 10, 0xacd1c5);
88 scene.add( arrowHelper );
89 var arrowHelper2 = new THREE.ArrowHelper(new THREE.Vector3( 1.0, 0, 0.0 ), group.position, 10, 0xacd1c5 );
90 scene.add( arrowHelper2 );
91 var arrowHelper3 = new THREE.ArrowHelper(new THREE.Vector3( 0, 0, 1.0 ), group.position, 10, 0xacd1c5 );
92 scene.add( arrowHelper3 );
93
94
95 // position and point the camera to the center of the scene
96 camera.position.x = 5;
97 camera.position.y = 10;
98 camera.position.z = 30;
99 camera.lookAt(scene.position);
100
101
102
103 // add spotlight for the shadows
104 var ambientLight=new THREE.AmbientLight(0x0c0c0c);
105 //scene.add(ambientLight);
106 var spotLight = new THREE.SpotLight( 0xEEEEEE );
107 spotLight.position.set( 40, 60, 20 );
108 spotLight.castShadow = true;
109 scene.add( spotLight );
110
111 var pointColor = "#65a54e";
112 var pointLight1 = new THREE.PointLight(pointColor);
113 pointLight1.distance = 20;
114 scene.add(pointLight1);
115
116 var sphereLight = new THREE.SphereGeometry(0.2);
117 var sphereLightMaterial = new THREE.MeshLambertMaterial({color:0x65a54e});
118 var sphereLightMesh = new THREE.Mesh(sphereLight, sphereLightMaterial);
119 sphereLightMesh.castShadow = true;
120
121 sphereLightMesh.position = new THREE.Vector3(6, 5, -10);
122 pointLight1.position = sphereLightMesh.position;
123 scene.add(sphereLightMesh);
124
125
126 var pointLight2 = new THREE.PointLight(0xff000d);
127 pointLight2.distance = 20;
128 scene.add(pointLight2);
129
130 var sphereLight2 = new THREE.SphereGeometry(0.2);
131 var sphereLight2Material = new THREE.MeshLambertMaterial({color:0xff000d});
132 var sphereLight2Mesh = new THREE.Mesh(sphereLight2, sphereLight2Material);
133 sphereLight2Mesh.castShadow = true;
134
135 sphereLight2Mesh.position = new THREE.Vector3(3, 8, 8);
136 pointLight2.position = sphereLight2Mesh.position;
137 scene.add(sphereLight2Mesh);
138
139 var pointLight3 = new THREE.PointLight(222672);
140 pointLight3.distance = 20;
141 scene.add(pointLight3);
142
143 var sphereLight3 = new THREE.SphereGeometry(0.2);
144 var sphereLight3Material = new THREE.MeshLambertMaterial({color:222672});
145 var sphereLight3Mesh = new THREE.Mesh(sphereLight3, sphereLight3Material);
146 sphereLight2Mesh.castShadow = true;
147
148 sphereLight3Mesh.position = new THREE.Vector3(-8, 8, 6);
149 pointLight3.position = sphereLight3Mesh.position;
150 scene.add(sphereLight3Mesh);
151
152 var pointLight4 = new THREE.PointLight(0x634ea5);
153 pointLight4.distance = 20;
154 scene.add(pointLight4);
155
156 var sphereLight4 = new THREE.SphereGeometry(0.2);
157 var sphereLight4Material = new THREE.MeshLambertMaterial({color:0x634ea5});
158 var sphereLight4Mesh = new THREE.Mesh(sphereLight4, sphereLight4Material);
159 sphereLight4Mesh.castShadow = true;
160
161 sphereLight4Mesh.position = new THREE.Vector4(6, 7, 4);
162 pointLight4.position = sphereLight4Mesh.position;
163 scene.add(sphereLight4Mesh);
164
165 var pointLight5 = new THREE.PointLight(0xff000d);
166 pointLight5.distance = 20;
167 scene.add(pointLight5);
168
169 var sphereLight5 = new THREE.SphereGeometry(0.2);
170 var sphereLight5Material = new THREE.MeshLambertMaterial({color:0xff000d});
171 var sphereLight5Mesh = new THREE.Mesh(sphereLight5, sphereLight5Material);
172 sphereLight5Mesh.castShadow = true;
173
174 sphereLight5Mesh.position = new THREE.Vector3(-9, 4, -2);
175 pointLight5.position = sphereLight5Mesh.position;
176 scene.add(sphereLight5Mesh);
177
178 var pointLight6 = new THREE.PointLight(0x4e9fa5);
179 pointLight6.distance = 20;
180 scene.add(pointLight6);
181
182 var sphereLight6 = new THREE.SphereGeometry(0.2);
183 var sphereLight6Material = new THREE.MeshLambertMaterial({color:0x4e9fa5});
184 var sphereLight6Mesh = new THREE.Mesh(sphereLight6, sphereLight6Material);
185 sphereLight6Mesh.castShadow = true;
186
187 sphereLight6Mesh.position = new THREE.Vector3(-4, 5, 12);
188 pointLight6.position = sphereLight6Mesh.position;
189 scene.add(sphereLight6Mesh);
190
191 var pointLight7 = new THREE.PointLight(0xa54e9f);
192 pointLight7.distance = 20;
193 scene.add(pointLight7);
194
195 var sphereLight7 = new THREE.SphereGeometry(0.2);
196 var sphereLight7Material = new THREE.MeshLambertMaterial({color:0xa54e9f});
197 var sphereLight7Mesh = new THREE.Mesh(sphereLight7, sphereLight7Material);
198 sphereLight7Mesh.castShadow = true;
199
200 sphereLight7Mesh.position = new THREE.Vector3(-1, 8, -7);
201 pointLight7.position = sphereLight7Mesh.position;
202 scene.add(sphereLight7Mesh);
203
204 // add the output of the renderer to the html element
205 $("#WebGL-output").append(renderer.domElement);
206
207 // call the render function
208 renderer.render(scene, camera);
209 var step = 1;
210 var stepg=0.8;
211 var d=1;
212 var g=1;
213 var oldContext = null;
214
215 // setup the control gui
216 var omega,phi,c,time,dt,m , M;
217 var controls = new function () {
218 // we need the first child, since it's a multimaterial
219 this.orbit=false
220 this.radiusTop = 4;
221 this.radiusBottom = 4;
222 this.height = 10;
223
224 this.segmentsX = 30;
225 this.segmentsY = 10;
226
227 this.openEnded = true;
228 this.opacity = stickMaterial.opacity;
229
230 this.pointColor = pointColor;
231 this.intensity = 0;
232 this.distance = 100;
233
234 this.visible = cylinderMaterial.visible;
235 this.side = "front";
236 this.c=1;
237
238 this.m=2
239 this.M=10
240
241
242 this.redraw=function()
243 {
244 dt=0.01;
245 time=0;
246 delta=0;
247 phi=Math.PI/3;
248 c=controls.c;
249 stick.rotation.z=phi;
250 sphere.position.x=-2.9
251 sphere.position.y=1.7
252 }
253
254 }
255
256
257
258
259 var gui = new dat.GUI();
260
261 gui.add(controls, 'opacity', 0, 1).onChange(function (e) {
262 stickMaterial.opacity = e
263 });
264
265 gui.add(controls, 'm', 0.1, 10);
266 gui.add(controls, 'M', 0, 10);
267 gui.add(controls, 'orbit', false).onChange(function (e) {
268 OrbitControls.autoRotate = e;
269 })
270 gui.add(controls, 'redraw');
271
272
273
274
275 controls.redraw();
276
277 render();
278 controls.redraw();
279
280
281 function render() {
282
283 sphere.position.x+= d*0.034;
284 sphere.position.y-= d*0.02;
285 if (sphere.position.x>3.1) {d=-1}
286 if (sphere.position.x<-3) {d=1}
287
288
289
290 time+=dt;
291 delta=(-d)*(Math.atan(0.04*time/4*Math.sqrt(controls.M/3*controls.m)))*dt;
292 phi+=delta;
293 group.rotation.z=phi;
294
295
296 var x = clock.getDelta();
297 OrbitControls.update(x);
298
299 stats.update();
300 requestAnimationFrame(render);
301 renderer.render(scene, camera);
302 }
303
304
305 function initStats() {
306
307 var stats = new Stats();
308 stats.setMode(0); // 0: fps, 1: ms
309 // Align top-left
310 stats.domElement.style.position = 'absolute';
311 stats.domElement.style.left = '0px';
312 stats.domElement.style.top = '0px';
313
314 $("#Stats-output").append(stats.domElement);
315
316 return stats;
317 }
318
319
320 });
321 window.onload = init;
