Редактирование: Колебания груза на пружине

'''''Задача:''''' С помощью языка программирования JavaScript смоделировать колебания груза на пружине.
[[Файл:Kol.jpg|thumb|Система блоков с грузом (A — грузик, a - расстояние до пружины, l - расстояние до грузика, c - жесткость пружины.)]]

== Решение ==

{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Vorobev/Kolebani9/Graphics.html |width=900 |height=450 |border=0 }}

Программа: [[Медиа: Kolebani9.rar|скачать]]

<div class="mw-collapsible mw-collapsed">
'''Текст программы на языке JavaScript:''' <div class="mw-collapsible-content">
Файл '''"5.js"'''
<syntaxhighlight lang="javascript" line start="1" enclose="div">
function main()
{
	var step = 0;
	var rr = new THREE.WebGLRenderer();
    rr.setSize(window.innerWidth, window.innerHeight);
	rr.setClearColor(0xFFFFFFF,1);
    document.body.appendChild(rr.domElement);
    camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 1, 500);
    camera.position.set(0, 0, 100);
    camera.lookAt(new THREE.Vector3(0, 0, 0));
    var scene = new THREE.Scene();
	
	var material = new THREE.LineBasicMaterial({color: 0x000000});
	var material1 = new THREE.LineBasicMaterial({color: 0xbbbbbb});
	
	var radius = 5;
	var segments = 32;

	var circleGeometry = new THREE.CircleGeometry(radius, segments);				
	var circle = new THREE.Mesh(circleGeometry, material1);
	
	var radius1 = 0.5;
	var segments1 = 32;

	var circleGeometry = new THREE.CircleGeometry(radius1, segments1);				
	var circle1 = new THREE.Mesh(circleGeometry, material);
	
	var geometry = new THREE.Geometry();
    geometry.vertices.push(new THREE.Vector3(-40, 20, 0));
    geometry.vertices.push(new THREE.Vector3(0, 20, 0));
	
	var line = new THREE.Line(geometry, material);
	line.geometry.verticesNeedUpdate = true;
	geometry.dynamic = true;
	
 	var geometry1 = new THREE.Geometry();
    geometry1.vertices.push(new THREE.Vector3(-28, 28, 0));
    geometry1.vertices.push(new THREE.Vector3(-28, 27, 0));
	
	var line1 = new THREE.Line(geometry1, material);
	line1.geometry.verticesNeedUpdate = true;
	geometry1.dynamic = true; 
	
	var geometry2 = new THREE.Geometry();
    geometry2.vertices.push(new THREE.Vector3(-30, 22, 0));
    geometry2.vertices.push(new THREE.Vector3(-26, 21, 0));
	
	var line2 = new THREE.Line(geometry2, material);
	line2.geometry.verticesNeedUpdate = true;
	geometry2.dynamic = true;
	
	var geometry3 = new THREE.Geometry();
    geometry3.vertices.push(new THREE.Vector3(-26, 23, 0));
    geometry3.vertices.push(new THREE.Vector3(-30, 22, 0));
	
	var line3 = new THREE.Line(geometry3, material);
	line3.geometry.verticesNeedUpdate = true;
	geometry3.dynamic = true;
	
	var geometry4 = new THREE.Geometry();
    geometry4.vertices.push(new THREE.Vector3(-30, 24, 0));
    geometry4.vertices.push(new THREE.Vector3(-26, 23, 0));
	
	var line4 = new THREE.Line(geometry4, material);
	line4.geometry.verticesNeedUpdate = true;
	geometry4.dynamic = true;
	
	var geometry5 = new THREE.Geometry();
    geometry5.vertices.push(new THREE.Vector3(-26, 25, 0));
    geometry5.vertices.push(new THREE.Vector3(-30, 24, 0));
	
	var line5 = new THREE.Line(geometry5, material);
	line5.geometry.verticesNeedUpdate = true;
	geometry5.dynamic = true;
	
	var geometry6 = new THREE.Geometry();
    geometry6.vertices.push(new THREE.Vector3(-30, 26, 0));
    geometry6.vertices.push(new THREE.Vector3(-26, 25, 0));
	
	var line6 = new THREE.Line(geometry6, material);
	line6.geometry.verticesNeedUpdate = true;
	geometry6.dynamic = true;
	
	var geometry7 = new THREE.Geometry();
    geometry7.vertices.push(new THREE.Vector3(-26, 27, 0));
    geometry7.vertices.push(new THREE.Vector3(-30, 26, 0));
	
	var line7 = new THREE.Line(geometry7, material);
	line7.geometry.verticesNeedUpdate = true;
	geometry7.dynamic = true;
	
	var geometry8 = new THREE.Geometry();
    geometry8.vertices.push(new THREE.Vector3(-28, 28, 0));
    geometry8.vertices.push(new THREE.Vector3(-26, 27, 0));
	
	var line8 = new THREE.Line(geometry8, material);
	line8.geometry.verticesNeedUpdate = true;
	geometry8.dynamic = true;
	
	var geometry9 = new THREE.Geometry();
    geometry9.vertices.push(new THREE.Vector3(-30, 28, 0));
    geometry9.vertices.push(new THREE.Vector3(-26, 28, 0));
	
	var line9 = new THREE.Line(geometry9, material);
	
	var geometry10 = new THREE.Geometry();
    geometry10.vertices.push(new THREE.Vector3(-30, 28, 0));
    geometry10.vertices.push(new THREE.Vector3(-29, 30, 0));
	
	var line10 = new THREE.Line(geometry10, material);

	var geometry11 = new THREE.Geometry();
    geometry11.vertices.push(new THREE.Vector3(-29, 28, 0));
    geometry11.vertices.push(new THREE.Vector3(-28, 30, 0));
	
	var line11 = new THREE.Line(geometry11, material);
	
	var geometry12 = new THREE.Geometry();
    geometry12.vertices.push(new THREE.Vector3(-28, 28, 0));
    geometry12.vertices.push(new THREE.Vector3(-27, 30, 0));
	
	var line12 = new THREE.Line(geometry12, material);
	
	var geometry13 = new THREE.Geometry();
    geometry13.vertices.push(new THREE.Vector3(-27, 28, 0));
    geometry13.vertices.push(new THREE.Vector3(-26, 30, 0));
	
	var line13 = new THREE.Line(geometry13, material);
	
	var geometry14 = new THREE.Geometry();
    geometry14.vertices.push(new THREE.Vector3(-40, 20.25, 0));
    geometry14.vertices.push(new THREE.Vector3(-42, 21, 0));
	
	var line14 = new THREE.Line(geometry14, material);
	
	var geometry15 = new THREE.Geometry();
    geometry15.vertices.push(new THREE.Vector3(-40, 19.75, 0));
    geometry15.vertices.push(new THREE.Vector3(-42, 19, 0));
	
	var line15 = new THREE.Line(geometry15, material);
	
	var geometry16 = new THREE.Geometry();
    geometry16.vertices.push(new THREE.Vector3(-42, 19, 0));
    geometry16.vertices.push(new THREE.Vector3(-42, 21, 0));
	
	var line16 = new THREE.Line(geometry16, material);
	
	var geometry17 = new THREE.Geometry();
    geometry17.vertices.push(new THREE.Vector3(-42, 21, 0));
    geometry17.vertices.push(new THREE.Vector3(-43, 22, 0));
	
	var line17 = new THREE.Line(geometry17, material);
	
	var geometry18 = new THREE.Geometry();
    geometry18.vertices.push(new THREE.Vector3(-42, 20, 0));
    geometry18.vertices.push(new THREE.Vector3(-43, 21, 0));
	
	var line18 = new THREE.Line(geometry18, material);
	
	var geometry19 = new THREE.Geometry();
    geometry19.vertices.push(new THREE.Vector3(-42, 19, 0));
    geometry19.vertices.push(new THREE.Vector3(-43, 20, 0));
	
	var line19 = new THREE.Line(geometry19, material);
	
	scene.add(line);
 	scene.add(line1);
	scene.add(line2);
	scene.add(line3);
	scene.add(line4);
	scene.add(line5);
	scene.add(line6);
	scene.add(line7);
	scene.add(line8);
	scene.add(line9);
	scene.add(line10);
	scene.add(line11);
	scene.add(line12);
	scene.add(line13);
	scene.add(line14);
	scene.add(line15);
	scene.add(line16);
	scene.add(line17);
	scene.add(line18);
	scene.add(line19);
	scene.add(circle);
	scene.add(circle1);
    rr.render(scene, camera);
	
	var controls = new function()
		{
			this.jestkost = 0.8;
			this.mass = 0.2;
			this.g = 0.5;
			this.a = 4;
			this.l = 16;
		}
		
	var gui = new dat.GUI();
	gui.add(controls, 'jestkost', 0.8, Math.PI);
	gui.add(controls, 'mass', 0.001, 0.24);
	gui.add(controls, 'g', 0.1, 0.98);
	
	window.addEventListener('resize', onWindowResize, false);
	
	renderer();
		
	function onWindowResize()
	{
		camera.aspect = window.innerWidth/window.innerHeight;
		camera.updateProjectionMatrix();
		rr.setSize(window.innerWidth, window.innerHeight);
		renderer();
	}
	
	function renderer()
	{
		step+=0.1;
		
		var koef = controls.mass*controls.g*controls.l*controls.l/(controls.jestkost*controls.a*controls.a);
		var arg = Math.sqrt(controls.jestkost*controls.a*controls.a/(controls.mass*controls.l*controls.l));
		var koef1 = (line.geometry.vertices[1].y-20)*(line.geometry.vertices[1].y-20);
		var koef2 = controls.a/controls.l;
		
		line.geometry.vertices[1].y=-koef*Math.cos(arg*step)-koef+20;
		line.geometry.vertices[1].x=Math.sqrt(controls.l*controls.l-koef1);
		line.geometry.verticesNeedUpdate = true;
		line1.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+110)*koef2;;
		line1.geometry.verticesNeedUpdate = true;
		line2.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+80)*koef2;
		line2.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+86)*koef2;
		line2.geometry.verticesNeedUpdate = true;
		line3.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+86)*koef2;
		line3.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+90)*koef2;
		line3.geometry.verticesNeedUpdate = true;
		line4.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+90)*koef2;
		line4.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+94)*koef2;
		line4.geometry.verticesNeedUpdate = true;
		line5.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+94)*koef2;
		line5.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+98)*koef2;
		line5.geometry.verticesNeedUpdate = true;
		line6.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+98)*koef2;
		line6.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+102)*koef2;
		line6.geometry.verticesNeedUpdate = true;
		line7.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+102)*koef2;
		line7.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+106)*koef2;
		line7.geometry.verticesNeedUpdate = true;
		line8.geometry.vertices[1].y=(-koef*Math.cos(arg*step)-koef+106)*koef2;
		line8.geometry.vertices[0].y=(-koef*Math.cos(arg*step)-koef+110)*koef2;
		line8.geometry.verticesNeedUpdate = true;
		circle.position.x=Math.sqrt(controls.l*controls.l-koef1);
		circle.position.y=-koef*Math.cos(arg*step)-koef+20;;
		circle1.position.x=-40;
		circle1.position.y=20; 
		requestAnimationFrame(renderer);
		rr.render(scene, camera);
		
		document.getElementById("td1").innerHTML = line.geometry.vertices[1].x;
		document.getElementById("td2").innerHTML = line.geometry.vertices[1].y;
	}
}

</syntaxhighlight>
</div>

== Используемые библиотеки ==

* gui.js
* jquery.min.js
* orbit_controls.js
* script.js
* stats.js
* three.min.js

== Возможности программы ==

* изменение жесткости пружины
* изменение ускорения свободного падения

== Решение частного случая ==

'''''Условия задачи:''''' 
Стержень <math>OA</math> длины <math>l</math>, на конце которой помещен груз массы <math>m</math>, может поворачиваться вокруг оси <math>O</math>. На расстоянии <math>a</math> от оси <math>O</math> к стержню прикреплена пружина с коэффициентом жесткости <math>c</math>. Определить собственную частоту колебаний груза, если стержень <math>OA</math> в положении равновесия занимает горизонтальное положение. Массой стержня пренебречь.

'''''Решение:'''''
<math>y_c</math> - изменение координаты пружины по y;
<math>y</math> - изменение координаты грузика по y;

<math>T = \frac{m{\dot y}^2}{2}</math>;
<math>y_c = \frac{ya}{l}</math>;

<math>\Pi = -mg + \frac{c}{2}\frac{y^2a^2}{l^2}</math>;

<math> \frac{\partial \Pi}{\partial y} = -mg + \frac{ca^2y}{l^2}</math>;

<math>\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_i}\right) - \frac{\partial T}{\partial q_i} = \frac{\partial \Pi}{\partial y}</math>;

<math>\dot y = v</math>;

<math>m\dot v + \frac{ca^2y}{l^2} - mg = 0</math>;

<math>y = -\left(\frac{mgl^2}{ca^2}\right)Math.cos(Math.sqrt(\frac{ca^2}{ml^2})t + \frac{mgl^2}{ca^2}+y_0</math>;

== См. также ==

* [[Справка:Содержание|Справочная информация]]
* [[Проектная деятельность по информатике]]


[[Category: Студенческие проекты]]