Цепочка частиц с вращательными степенями свободы — различия между версиями
Материал из Department of Theoretical and Applied Mechanics
| Строка 4: | Строка 4: | ||
Рассматривается совокупность твердых тел, образующих цепочки. Центры масс фиксированы. Взаимодействия осуществляются посредством балок Бернулли-Эйлера, соединяющих тела. | Рассматривается совокупность твердых тел, образующих цепочки. Центры масс фиксированы. Взаимодействия осуществляются посредством балок Бернулли-Эйлера, соединяющих тела. | ||
| + | Дифференциальное уравнение изгиба балки: | ||
| + | ::<math> | ||
| + | EJ\ddddot{\bf y} = 0 | ||
| + | </math>, | ||
| + | где E - модуль юнга, J - полярный момент инерции. | ||
| + | Момент взаимодействия: | ||
| + | ::<math> | ||
| + | EJ\ddot{\bf y} = M | ||
| + | </math> | ||
| + | Уравнение движения: | ||
| + | ::<math> | ||
| + | Q\ddot{\bf ϕ}_{k} = -2EJ/l({\bf ϕ}_{k-1}-2{\bf ϕ}_{k}+{\bf ϕ}_{k})-12EJ/l({\bf ϕ}_{k}) | ||
| + | </math> | ||
== Реализации цепочки == | == Реализации цепочки == | ||
Версия 20:54, 22 мая 2016
Виртуальная лаборатория > Цепочка частиц с вращательными степенями свободыКраткое описание
Рассматривается совокупность твердых тел, образующих цепочки. Центры масс фиксированы. Взаимодействия осуществляются посредством балок Бернулли-Эйлера, соединяющих тела. Дифференциальное уравнение изгиба балки:
- ,
где E - модуль юнга, J - полярный момент инерции. Момент взаимодействия:
Уравнение движения:
Реализации цепочки
1 window.addEventListener("load", MainSystem, true);
2
3 function MainSystem(){
4 var context_s = canvasSystem.getContext('2d');
5 var context_g = canvasGraph.getContext('2d');
6 var context_g_1 = canvasGraph_1.getContext('2d');
7 var context_g_2 = canvasGraph_2.getContext('2d');
8
9 const Pi = 3.1415926;
10 const m0 = 1;
11 const T0 = 1;
12 const l0 = 1;
13 const E0 = 1;
14
15 //Width of canvas - width of browser
16 const distance_between_canvases = 5; //5px
17 canvasSystem.width = document.body.clientWidth;
18 canvasGraph.width = document.body.clientWidth / 2 - distance_between_canvases;
19 canvasGraph_1.width = document.body.clientWidth / 2 - distance_between_canvases;
20 canvasGraph_2.width = document.body.clientWidth;
21
22 /* -- Used constans -- */
23 var Db = 0.1 * l0; // Diameter of beam
24 const l = 30 * l0; //Length of beam
25 const a = 60 * l0; //Length of object
26 var Db2 = Db * Db;
27 var J = Pi * Db2 * Db2 / 64; //Polar moment of inertia
28 const E = 10000000 * E0; //Youngs modulus
29 var C = E * J / l;
30 var N = parseFloat(number_of_objects.value) + 1; //number_of_objects.value is number of objects
31 const m = 0.01 * m0; //Mass of object
32 const Q = m * a * a / 12; //Moment of inertia
33 const w_c = Math.sqrt(2 * C / Q); //Self frequency
34
35 const fps = 50; // frames per second
36 var spf = calcul_speed.value; // steps per frame
37 const frequency = 1000 / fps; //frequency of call function - 1000 milliseconds/ fps
38 const dt = 0.05 * T0 / fps; //Step of integration
39
40 var scale = canvasSystem.width / N; //Scale of graph of system
41 var scale1 = canvasGraph_2.width / (N + 2); //Scale of graph of angels
42
43 //For wave
44 const n = 1; //Number of full-wave
45 var k_ = 2 * Pi / (l * (N - 2) * n); //Spatial frequencyw
46 var w_ = Math.sqrt((-2 * C / Q * l * l) * k_ * k_ + (12 * C / Q));
47
48 /* -- Used variables -- */
49 var K0 = 0; var P0 = 0; var E_p0 = 0; var L0 = 0; //Energies at i-step
50 var K1 = 0; var P1 = 0; var E_p1 = 0; var L1 = 0; // Energies at (i+1)-step
51 var E_m = 0; //Maximum of Energy at the first moment
52 var t = 0; //Time
53
54 var U = []; //Exact solution for wave
55 var shaft = []; //Objects
56
57 var pause = false;
58 const stretch_graphics = 3;
59 var help = stretch_graphics * canvasGraph.width; //Scale of graph of energies
60 var firstCalculation = true;
61 /* -- */
62
63 //Restart the programm with new parameters
64 restart.onclick = function(){
65 N = parseFloat(number_of_objects.value) + 1;
66 scale = canvasSystem.width / N;
67 scale1 = canvasGraph_2.width / (N + 2);
68 spf = calcul_speed.value;
69 J = Pi * Db2 * Db2 / 64;
70 C = E * J / l;
71
72 context_s.clearRect(0, 0, canvasSystem.width, canvasSystem.height);
73 context_g.clearRect(0, 0, canvasGraph.width, canvasGraph.height);
74 context_g_1.clearRect(0, 0, canvasGraph_1.width, canvasGraph_1.height);
75 context_g_2.clearRect(0, 0, canvasGraph_2.width, canvasGraph_2.height);
76
77 shaft = [];
78 addSystem(shaft);
79
80 firstCalculation = true;
81 t = 0;
82 P1 = 0;
83 K1 = 0;
84 E_m = 0;
85 }
86
87 //Pause
88 pause_button.onclick = function(){
89 pause = !pause;
90 if(pause == false)
91 pause_button.value = "Pause";
92 else
93 pause_button.value = "Run";
94 }
95
96 //Calculate all parameters of system
97 function control(){
98 if(!pause){
99 /* -- Find the maximum of energy -- */
100 if(firstCalculation){
101 for (var i = 1; i < N; i++){
102 E_m += Q * shaft[i].w * shaft[i].w / 2;
103 }
104
105 for (var i = 1; i < N; i++){
106 E_m += C / 2 * (12 * shaft[i].fi * shaft[i].fi - ((shaft[i-1].fi - shaft[i].fi) * (shaft[i-1].fi - shaft[i].fi) +
107 (shaft[i].fi - shaft[i+1].fi) * (shaft[i].fi - shaft[i+1].fi)));
108 }
109
110 L0 = E_m;
111 E_p0 = E_m / 2;
112 firstCalculation = false;
113 }
114 /* -- */
115
116 physics();
117 draw();
118
119 if(t*help > canvasGraph.width){
120 t = 0;
121 context_g.clearRect(0, 0, canvasGraph.width, canvasGraph.height);
122 context_g_1.clearRect(0, 0, canvasGraph_1.width, canvasGraph_1.height);
123 }
124
125 draw_Graph_energy(t*help, (t + dt)*help);
126 draw_Graph_angels();
127
128 //exact_solution_for_wave(t*help);
129
130 P0 = P1;
131 K0 = K1;
132 L0 = L1;
133 E_p0 = E_p1;
134 E_p1 = 0;
135 L1 = 0;
136 P1 = 0;
137 K1 = 0;
138 t += dt;
139 }
140 }
141
142 //Physics - calculate the positions of objects
143 function physics(){
144 for (var s = 1; s <= spf; s++){
145 //Periodic initial conditions
146 shaft[0].fi = shaft[N-1].fi;
147 shaft[N].fi = shaft[1].fi;
148
149 for (var i = 1; i < N; i++){
150 shaft[i].M = - 2 * C * (shaft[i-1].fi + 2 * shaft[i].fi) - 2 * C * (2 * shaft[i].fi + shaft[i+1].fi);
151 }
152
153 for (var i = 1; i < N; i++){
154 shaft[i].w += shaft[i].M / Q * dt;
155 shaft[i].fi += shaft[i].w * dt;
156 }
157
158 for (var i = 1; i < N; i++){
159 shaft[i].M = 0;
160 }
161 }
162 }
163
164 //Draw the graph of system
165 function draw(){
166 context_s.clearRect(0, 0, canvasSystem.width, canvasSystem.height);
167
168 for (var i = 1; i < N; i++){
169 context_s.beginPath();
170 context_s.moveTo(shaft[i].x - (a/2) * Math.sin(shaft[i].fi), shaft[i].y - (a/2) * Math.cos(shaft[i].fi));
171 context_s.lineTo(shaft[i].x + (a/2) * Math.sin(shaft[i].fi), shaft[i].y + (a/2) * Math.cos(shaft[i].fi));
172 context_s.closePath();
173 context_s.stroke();
174 }
175 }
176
177 //Draw the graph of angels
178 function draw_Graph_angels(){
179 context_g_2.clearRect(0, 0, canvasGraph_2.width, canvasGraph_2.height);
180
181 for(var i = 0; i < N; i++){
182 context_g_2.beginPath();
183 context_g_2.moveTo(scale1 * (i+1), -shaft[i].fi / (Pi/2) * canvasGraph_2.height / 2 + canvasGraph_2.height / 2);
184 context_g_2.lineTo(scale1 * (i+2), -shaft[i+1].fi / (Pi/2) * canvasGraph_2.height / 2 + canvasGraph_2.height / 2);
185 context_g_2.closePath();
186 context_g_2.stroke();
187 }
188 }
189
190 //Draw the graphics of energies
191 function draw_Graph_energy(x0, x1){
192 //Potential
193 context_g_1.beginPath();
194 context_g_1.strokeStyle = "#FF0000";
195 context_g_1.moveTo(x0, -P0 / E_m * canvasGraph_1.height + canvasGraph_1.height);
196
197 for (var i = 1; i < N; i++){
198 P1 += C / 2 * (12 * shaft[i].fi * shaft[i].fi - ((shaft[i-1].fi - shaft[i].fi) * (shaft[i-1].fi - shaft[i].fi) +
199 (shaft[i].fi - shaft[i+1].fi) * (shaft[i].fi - shaft[i+1].fi)));
200 }
201
202 context_g_1.lineTo(x1, -P1 / E_m * canvasGraph_1.height + canvasGraph_1.height);
203 context_g_1.closePath();
204 context_g_1.stroke();
205
206 //Kinetical
207 context_g_1.beginPath();
208 context_g_1.strokeStyle = "#000000";
209 context_g_1.moveTo(x0, -K0 / E_m * canvasGraph_1.height + canvasGraph_1.height);
210
211 for (var i = 1; i < N; i++){
212 K1 += Q * shaft[i].w * shaft[i].w / 2;
213 }
214
215 context_g_1.lineTo(x1, -K1 / E_m * canvasGraph_1.height + canvasGraph_1.height);
216 context_g_1.closePath();
217 context_g_1.stroke();
218
219 //Full energy
220 context_g_1.beginPath();
221 context_g_1.strokeStyle = "blue";
222 context_g_1.moveTo(x0, -E_p0 / E_m * canvasGraph.height + canvasGraph.height);
223
224 E_p1 = (K1 + P1) / 2;
225
226 context_g_1.lineTo(x1, -E_p1 / E_m * canvasGraph.height + canvasGraph.height);
227 context_g_1.closePath();
228 context_g_1.stroke();
229
230 //Lagrangian
231 context_g.beginPath();
232 context_g.strokeStyle = "orange";
233 context_g.moveTo(x0, -L0 / E_m * canvasGraph.height / 2 + canvasGraph.height / 2);
234
235 L1 = K1 - P1;
236
237 context_g.lineTo(x1, -L1 / E_m * canvasGraph.height / 2 + canvasGraph.height / 2);
238 context_g.closePath();
239 context_g.stroke();
240 }
241
242 //Add the system of objects
243 function addSystem(shaft){
244 for (var i = 0; i < N + 1; i++){
245 var shaft_new = [];
246
247 shaft_new.x = scale * i;
248 shaft_new.y = canvasSystem.height / 2;
249 shaft_new.fi = 0;
250 shaft_new.w = 0;
251 shaft_new.M = 0;
252 shaft[shaft.length] = shaft_new;
253 }
254
255 /* --Initial conditions-- */
256 //Random velocities
257 if(all_.checked){
258 var average_w = 0; //Average velocity
259
260 for (var i = 0; i < N; i++){
261 shaft[i].w = Math.random() * w_c;
262 average_w += shaft[i].w;
263 }
264
265 average_w /= N;
266
267 for (var i = 0; i < N; i++){
268 shaft[i].w -= average_w;
269 }
270 }
271
272 // N/10 - Central part of objects by sin
273 if(part.checked){
274 for (var i = Math.floor(-Math.floor(N / 10) / 2); i < Math.floor(Math.floor(N / 10) / 2); i++){
275 shaft[Math.floor(N / 2) + i + 1].fi =
276 Math.sin(2 * Pi * (Math.floor(Math.floor(N / 10) / 2) - i) * (Math.floor(Math.floor(N / 10) / 2) + i) / N/2);
277 }
278 }
279
280 //Central object
281 if(one.checked){
282 shaft[Math.floor(N / 2)].w = w_c;
283 }
284
285 //Wave
286 if(wave.checked){
287 for (var i = 1; i < N; i++){
288 shaft[i].fi = Math.sin(k_ * (l * i));
289 shaft[i].w = -w_ * Math.cos(k_ * (l * i));
290 }
291 }
292 }
293
294 //Exact solution for wave
295 function exact_solution_for_wave(t) {
296 for (var i = 1; i < N; i++){
297 U[i] = Math.sin(k_ * (l * i) - w_ * t / 200);
298 }
299 //context_g_2.clearRect(0, 0, canvasGraph_2.width, canvasGraph_2.height);
300 for(var i = 0; i < N; i++){
301 context_g_2.beginPath();
302 context_g_2.moveTo(scale1 * (i+1), -U[i] / (Pi/2) * canvasGraph_2.height / 2 + canvasGraph_2.height / 2);
303 context_g_2.lineTo(scale1 * (i+2), -U[i+1] / (Pi/2) * canvasGraph_2.height / 2 + canvasGraph_2.height / 2);
304 context_g_2.closePath();
305 context_g_2.stroke();
306 }
307 }
308
309 addSystem(shaft); //Adding our system of objects
310
311 setInterval(control, frequency);
312 }
1 <!DOCTYPE html>
2 <html>
3 <body>
4 <canvas id="canvasSystem" width="1200" height="300" style="border:1px solid #000000;"></canvas><br><br>
5
6 Number of objects: <input type="number" id="number_of_objects" value="500" step=1 style="width: 5em">,
7 Calculation speed: <input type="range" id="calcul_speed" value="100" step=0.01 min=10 max=300><br>
8
9 Initial conditions:<br>
10 <input type="radio" checked="checked" name="initial_conditions" id="all_"/>Random velocities<br>
11 <input type="radio" name="initial_conditions" id="part"/>Central part of objects by sin<br>
12 <input type="radio" name="initial_conditions" id="one"/>One object<br>
13 <input type="radio" name="initial_conditions" id="wave"/>Wave<br>
14 <input type="button" id="restart" value="Restart">
15 <input type="button" id="pause_button" value="Pause"><br><br>
16
17 Graphics:<br>
18 <canvas id="canvasGraph" width="600" height="300" style="border:1px solid #000000;"></canvas>
19 <canvas id="canvasGraph_1" width="600" height="300" style="border:1px solid #000000;"></canvas><br>
20 L<hr align="left" width="50" size="3" color="orange" />
21 E / 2<hr align="left" width="50" size="3" color="blue" />
22 K<hr align="left" width="50" size="3" color="#000000" />
23 P<hr align="left" width="50" size="3" color="#FF0000" /><br>
24
25 Angles:<br>
26 <canvas id="canvasGraph_2" width="1200" height="300" style="border:1px solid #000000;"></canvas><br>
27
28 <script src="simulation.js"></script>
29 </body>
30 </html>
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