Нелинейные колебания груза с вынуждающей силой

thumb|Nonlinear Oscillations of cargo with the driving force|500px

Annotation to the project

In this project we study the nonlinear oscillations of the load acting on it with a periodic force

Statement of the problem

Let suspended on a non-linear spring load mass m experiences the external force F, the change in law F = sin (t)

• write a program in JavaScript, simulating the behavior of the load when setting various parameters of the system..
  

General information about the subject

If the oscillatory system is affected by periodically changing external force, then the system makes fluctuations which nature to some extent repeats nature of change of this force. Such fluctuations are called compelled.

F0 is called the amplitude of the force which is also the highest value of force.

Thanks to the work carried out by an external force, increasing the maximum values, which reach the potential energy of the spring and the kinetic energy of the load. Loss to overcome the resistance forces will increase. Finally, the moment will come when the work of the external force will exactly offset the energy losses in the system. Further increase of fluctuations in system will stop, and fluctuations with some constant amplitude will be established. || Характер амплитуды

equation of motion: [math]m\ddot x = -kx -{k_1}x^3 + {F_0}sin(t) - B \dot x[/math]

Visualization in JavaScript

Download program: SpringNoLine.rar

'

Файл "Spring.js" <syntaxhighlight lang="javascript" enclose="div">

   window.addEventListener("load", Main_Spring, true);
   function Main_Spring() {
   var canvas = spring_canvas;
   canvas.onselectstart = function () {return false;};     // ban selection canvas
   var ctx = canvas.getContext("2d");                      // drawing on ctx
   var w = canvas.width;                                   // width of the window in the calculated coordinates
   var h = canvas.height;                                  // the height of the window in the calculated coordinates
   var Pi = 3.1415926;    	      	                    // "Pi"
   var m0 = 1;    		      	                    // weight scale
   var T0 = 1;    		      	                    // time scale(period of fluctuations of initial system)
   var t = 0;
   var k0 = 2 * Pi / T0;           	                    // frequency scale
   var C0 = m0 * k0 * k0;          	                    // hardness scale
   var B0 = 2 * m0 * k0;  	      	                    // viscosity scale
   var omega = 10;
   
   // *** physical parameters ***
   var F = 80;
   var m = 1 * m0;                 	                    // mass
   var C = 1 * C0;                 	                    // rigidity
   var C1 = 1 * C0;                 	                    // rigidity1
   var B = .1 * B0;                 	                    // viscosity
   
   slider_m.value = (m / m0).toFixed(1); number_m.value = (m / m0).toFixed(1);
   slider_C.value = (C / C0).toFixed(1); number_C.value = (C / C0).toFixed(1);
   slider_C1.value = (C / C0).toFixed(1); number_C1.value = (C / C0).toFixed(1);
   slider_B.value = (B / B0).toFixed(1); number_B.value = (B / B0).toFixed(1);
   slider_F.value = (F / 40).toFixed(1); number_F.value = (F / 40).toFixed(1);

   // *** computing parameters***

   var fps = 300;		      	            // frames per second 
   var spf = 100;		      	            // steps per frame   
   var dt  = 0.05 * T0 / fps;    	            // integration step
   var steps = 0;                                  // number of integration steps

   function setM(new_m) {m = new_m * m0;}
   function setC(new_C) {C = new_C * C0;}
   function setC1(new_C1) {C1 = new_C1 * C0 * 0.067;}
   function setB(new_B) {B = new_B * B0;}
   function setF(new_F) {F = new_F * 40;}

   slider_m.oninput = function() {number_m.value = slider_m.value;       setM(slider_m.value);};
   number_m.oninput = function() {slider_m.value = number_m.value;       setM(number_m.value);};
   slider_C.oninput = function() {number_C.value = slider_C.value;       setC(slider_C.value);};
   number_C.oninput = function() {slider_C.value = number_C.value;       setC(number_C.value);};
   slider_C1.oninput = function() {number_C1.value = slider_C1.value;       setC1(slider_C1.value);};
   number_C1.oninput = function() {slider_C1.value = number_C1.value;       setC1(number_C1.value);};
   slider_B.oninput = function() {number_B.value = slider_B.value;       setB(slider_B.value);};
   number_B.oninput = function() {slider_B.value = number_B.value;       setB(number_B.value);};
   slider_F.oninput = function() {number_F.value = slider_F.value;       setF(slider_F.value);};
   number_F.oninput = function() {slider_F.value = number_F.value;       setF(number_F.value);};

   var count = true;                   // whether to carry out calculation of system
   var v = 0;				// body speed

   var rw = canvas.width / 30;    	
   var rh = canvas.height / 1.5;
   var x0 = 15 * rw - rw / 2;     	
   var y0 = rh / 1.33 - rh / 2;

   // spring parameters
   var coil = 10;        // number of turns
   var startX = 0;       // fixing of a spring

   //  create a rectangle=
   var rect = {
       x: x0,  width: rw,
       y: y0,	height: rh,
       fill: "rgba(0, 0, 255, 1)"    	// color
   };

   // capture a rectangle with the mouse
   var mx_;                                    // position buffer of a mouse
   document.onmousedown = function(e) {        // function when pressing a key of a mouse
       var m = mouseCoords(e);                 // receive settlement coordinates of the cursor of a mouse

       var x = rect.x;
       var xw = rect.x + rect.width;
       var y = rect.y;
       var yh = rect.y + rect.height;
       if (x <= m.x && xw >= m.x   && y <= m.y && yh >= m.y) {
           if (e.which == 1) {                         // Left mouse button is pressed
               rect.xPlus = rect.x - m.x;              // shift of the cursor on an axis x
               rect.yPlus = rect.y - m.y;              // shift of the cursor on an axis y
               mx_ = m.x;
               count = false;
               document.onmousemove = mouseMove;       // until a key is pressed - movement function works
           }
       }
   };

   document.onmouseup = function(e) {          // function when you release the mouse button
       document.onmousemove = null;              // when the key is released - there is no function of movement
       count = true;
   };

   function mouseMove(e) {                     // function when you move the mouse (It works only while holding LMB)
       var m = mouseCoords(e);                 // We get estimated coordinates of the mouse cursor
       rect.x = m.x + rect.xPlus;

// v = 6.0 * (m.x - mx_) / dt / fps; // inertia preservation

       v = 0;
       mx_ = m.x;
   }

   function mouseCoords(e) {                   // function returns the calculated coordinates of the mouse cursor
       var m = [];
       var rect = canvas.getBoundingClientRect();
       m.x = (e.clientX - rect.left);
       m.y = (e.clientY - rect.top);
       return m;
   }

   // graph
   var vGraph = new TM_graph(                  // define the graph
       "#vGraph",                              // html #vGraph
       250,                                    // how many steps on the "x" axis shows
       -1, 1, 0.2);                            // min. value of axis Y, max. value of axis Y, step on axis Y

   function control() {
       calculate();
       draw();
       requestAnimationFrame(control);
   }
   control();

// setInterval(control, 1000 / fps); // Starting system

   function calculate() {
       if (!count) return;
       for (var s=1; s<=spf; s++) {
           var f = -B*v - C * (rect.x - x0) - C1*Math.pow(rect.x - x0,3)+2*F*Math.sin(t);

v += f / m * dt; //console.log(f);

           rect.x += v * dt;

t+= dt;

           steps++;
           if (steps % 80 == 0) vGraph.graphIter(steps, (rect.x-x0)/canvas.width*2);   // подать данные на график
       }

   }

   function draw() {
       ctx.clearRect(0, 0, w, h);

draw_spring(startX, rect.x, h/2, 10, 50);

       ctx.fillStyle = "#0000ff";
       ctx.fillRect(rect.x, rect.y, rect.width, rect.height);
   }

function draw_spring(x_start, x_end, y, n, h) { ctx.lineWidth = 2;

       ctx.strokeStyle = "#7394cb";

var L = x_end - x_start; for (var i = 0; i < n; i++) { var x_st = x_start + L / n * i; var x_end = x_start + L / n * (i + 1); var l = x_end - x_st; ctx.beginPath(); ctx.bezierCurveTo(x_st, y, x_st + l / 4, y + h, x_st + l / 2, y); ctx.bezierCurveTo(x_st + l / 2, y, x_st + 3 * l / 4, y - h, x_st + l, y); ctx.stroke(); } }

}