Difference between revisions of "Modeling of a shock absorber"
From Department of Theoretical and Applied Mechanics
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[[File:Amor.jpg|350px|left| Automobile shock absorber]] | [[File:Amor.jpg|350px|left| Automobile shock absorber]] | ||
| − | The equations of the | + | The equations of motion have the form (the velocity vector is directed upwards): |
::<math> | ::<math> | ||
| − | m \ddot{y} = -mg - c \Delta y - F_{com}^{reb} \dot{y} \\ | + | m \ddot{y} = -mg - c \Delta y - F_{com}^{reb} \dot{y}, \\ |
| − | F_{com} = \frac{\pi^{2} * E *\pi * d_{rod}^{4}}{64 * L^{2}}\\ | + | F_{com} = \frac{\pi^{2} * E *\pi * d_{rod}^{4}}{64 * L^{2}},\\ |
| − | F_{ | + | F_{ten} = \frac{d_{pist}^{2} * \pi * p} {4},\\ |
</math> | </math> | ||
| − | where | + | where <math>E</math> is the modulus of elasticity (Young's modulus) <math>E= 2.05 \cdot 10^11</math> Pa; |
| − | + | <math>F_{com}</math> is the force directed opposite to the velocity vector (acts during the ''com''pression of a piston); | |
| − | + | <math>F_{ten}</math> is the force directed along the velocity vector (acts on the piston during the ''ten''sion). | |
| − | + | You can drag and drop the yellow square to load the damper. | |
| − | The | + | The time dependence of 'y' coordinate is plotted in real time: |
{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Bogdanov/shock_absorbers/index.html |width=1000 |height=750 |border=0 }} | {{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Bogdanov/shock_absorbers/index.html |width=1000 |height=750 |border=0 }} | ||
Revision as of 18:32, 26 May 2016
Virtual laboratory > Modeling of shock absorberThis demonstration stand shows the work of a shock absorber
A shock absorber (in reality, a shock "damper") is a mechanical or hydraulic device designed to absorb and damp shock impulses. It does this by converting the kinetic energy of the shock into another form of energy (typically heat) which is then dissipated.
The equations of motion have the form (the velocity vector is directed upwards):
where is the modulus of elasticity (Young's modulus) Pa;
is the force directed opposite to the velocity vector (acts during the compression of a piston);
is the force directed along the velocity vector (acts on the piston during the tension).
You can drag and drop the yellow square to load the damper.
The time dependence of 'y' coordinate is plotted in real time:
Download Shock_absorbers.zip.
The program text on JavaScript (developer Bogdanov Dmitriy, the code based on the program byTsvetkov Denis):
File "Spring.js"
1 window.addEventListener("load", Main_Spring, true);
2 function Main_Spring() {
3
4 var canvas = spring_canvas;
5 canvas.onselectstart = function () {return false;}; // the prohibition of canvas selection
6 var ctx = canvas.getContext("2d"); // the drawing takes place on ctx
7 var w = canvas.width; // the width of the window in the calculated coordinates
8 var h = canvas.height; // the height of the window in the calculated coordinates
9
10 var Pi = Math.PI; // the 'Pi' number
11 var g = 9.81;
12 var m0 = 1; // the scale of weight
13 var T0 = 1; // the scale of time (period of oscillation the initial system)
14
15 var k0 = 2 * Pi / T0; // the scale of frequency
16 var C0 = 1; // the scale of stiffness
17 var D0 = 0.001; // the diameter
18 var p0 = 1; // the pressure
19 var E0 = 1e-6;
20 var L = 300;
21 // *** Setting the physical parameters ***
22 var E = 2.05e11 * E0; // the elastic modulus
23 var m = 3 * m0; // the mass
24 var C = 15 * C0; // the stiffness
25 var Dp = 43.1 * D0; // the piston diameter
26 var Dsh = 17.3 * D0; // the rod diameter
27 var p = 4 * p0; // the pressure
28 slider_m.value = (m / m0).toFixed(1); number_m.value = (m / m0).toFixed(1);
29 slider_C.value = (C / C0).toFixed(1); number_C.value = (C / C0).toFixed(1);
30 slider_Dp.value = (Dp / D0).toFixed(1); number_Dp.value = (Dp / D0).toFixed(1);
31 slider_Dsh.value = (Dsh / D0).toFixed(1); number_Dsh.value = (Dsh / D0).toFixed(1);
32 slider_p.value = (p / p0).toFixed(1); number_p.value = (p / p0).toFixed(1);
33 slider_L.value = (L).toFixed(1); number_L.value = (L).toFixed(1);
34
35 // *** Setting the parameters of computing ***
36
37 var fps = 100; // frames per second (the display quality)
38 var spf = 50; // steps per frame - the number of integration steps between frames (increase the calculation speed)
39 var dt = 0.05 * T0 / fps; // the integration step (the quality of calculation)
40 var steps = 0; // the number of integration steps
41
42 function setM(new_m) {m = new_m * m0;}
43 function setC(new_C) {C = new_C * C0;}
44 function setDp(new_Dp) {Dp = new_Dp * D0;}
45 function setP(new_p) {p = new_p * p0;}
46 function setDsh(new_Dsh) {Dsh = new_Dsh * D0;}
47 function setL(new_L) {L = new_L;}
48
49 slider_m.oninput = function() {number_m.value = slider_m.value; setM(slider_m.value);};
50 number_m.oninput = function() {slider_m.value = number_m.value; setM(number_m.value);};
51 slider_C.oninput = function() {number_C.value = slider_C.value; setC(slider_C.value);};
52 number_C.oninput = function() {slider_C.value = number_C.value; setC(number_C.value);};
53 slider_p.oninput = function() {number_p.value = slider_p.value; setP(slider_p.value);};
54 number_p.oninput = function() {slider_p.value = number_p.value; setP(number_p.value);};
55 slider_Dp.oninput = function() {number_Dp.value = slider_Dp.value; setDp(slider_Dp.value);};
56 number_Dp.oninput = function() {slider_Dp.value = number_Dp.value; setDp(number_Dp.value);};
57 slider_Dsh.oninput = function() {number_Dsh.value = slider_Dsh.value; setDsh(slider_Dsh.value);};
58 number_Dsh.oninput = function() {slider_Dsh.value = number_Dsh.value; setDsh(number_Dsh.value);};
59 slider_L.oninput = function() {number_L.value = slider_L.value; setL(slider_L.value);};
60 number_L.oninput = function() {slider_L.value = number_L.value; setL(number_L.value);};
61
62 var count = true; // the verification of the system calculation
63 var v = 0; // the body speed
64
65 var rw = canvas.width / 10; var rh = canvas.height;
66 var x0 = canvas.width / 2 - 25; var y0 = rh/2-25;
67
68 // the spring settings
69 var coil = 11; // the number of coils
70 var startY = h; // the spring fixation
71
72 // the creation of a rectangle-sinker
73 var rect = {
74 x: x0, width: 50,
75 y: y0, height: 50,
76 fill: "rgba(112, 155, 255, 1)" // colour
77 };
78
79 // the capture a rectangle by the mouse
80 var my_; // the buffer of mouse position (to calculate the speed of the ball releasing)
81 document.onmousedown = function(e) { // the function of mouse click
82 if (Dp <= Dsh)
83 {
84 alert("The diameter of the piston must be larger than the diameter of the rod");
85 slider_Dsh.value = (Dp / D0 - 1).toFixed(1);
86 number_Dsh.value = (Dp / D0 - 1).toFixed(1);
87 setDsh(slider_Dsh.value);
88 setDsh(number_Dsh.value);
89 }
90 else {
91 var m = mouseCoords(e); // getting calculated coordinates of the mouse cursor
92 var x = rect.x;
93 var xw = rect.x + rect.width;
94 var y = rect.y;
95 var yh = rect.y + rect.height;
96 if (x <= m.x && xw >= m.x && y <= m.y && yh >= m.y) {
97 if (e.which == 1) { // left mouse button is pressed
98 rect.xPlus = rect.x - m.x; // cursor displacement with respect to the sinker on the X axis
99 rect.yPlus = rect.y - m.y; // cursor displacement with respect to the sinker on the Y axis
100 my_ = m.y;
101 count = false;
102 document.onmousemove = mouseMove; // while the mouse button is pressed the function of movement works
103 }
104 }
105 }
106 };
107
108 document.onmouseup = function(e) { // the working function when you release the mouse button
109 document.onmousemove = null; // when the key is released displacement function does not work
110 count = true;
111 };
112
113 function mouseMove(e) { // the function works only while holding LMB
114 var m = mouseCoords(e); // getting calculated coordinates of the mouse cursor
115 rect.y = m.y + rect.yPlus;
116 // v = 6.0 * (m.x - mx_) / dt / fps; // inertia conservation
117 v = 0;
118 my_ = m.y;
119 }
120
121 function mouseCoords(e) { // the function returns the calculated coordinates of the mouse cursor
122 var m = [];
123 var rect = canvas.getBoundingClientRect();
124 m.x = (e.clientX - rect.left);
125 m.y = (e.clientY - rect.top);
126 return m;
127 }
128
129 // the graph
130 var vGraph = new New_graph( // the determination of the graph
131 "#vGraph", // at html-page #vGraph
132 250, // the number of steps by X axis
133 -1, 1, 0.2); // the minimum value of Y axis, the maximum value of Y axis, step along the Y axis
134
135 function control() {
136 calculate();
137 draw();
138 requestAnimationFrame(control);
139 }
140 control();
141 // setInterval(control, 1000 / fps); // starting system
142
143
144 function calculate() {
145 if (!count) return;
146 for (var s=1; s<=spf; s++) {
147 var Fo = Math.pow(Dp/D0, 2) * Pi * p / 4;
148 var Fs = Math.pow(Pi, 3) * E * Math.pow(Dsh/D0, 4) / (64 * L * L * 100);
149 document.getElementById('Fo').innerHTML = '<b>F отб. =</b>'+Fo.toFixed(2)+' H';
150 document.getElementById('Fs').innerHTML = '<b>F сж. =</b>'+Fs.toFixed(2)+' H';
151 var f;
152 if (v >= 0){
153 f = -C * (rect.y - y0) - Fs * D0 * v;
154 };
155 if (v < 0){
156 f = -C * (rect.y - y0) - Fo * D0 * v;
157 };
158 v += f / m * dt;
159 rect.y += v * dt;
160 steps++;
161 if (steps % 80 == 0) vGraph.graphIter(steps, -(rect.y-y0)/canvas.height*2); // submitting the data on graph
162 }
163
164 }
165 function draw() {
166 ctx.clearRect(0, 0, w, h);
167 ctx.fillStyle = "#4B4747";
168 ctx.fillRect(rect.x-22 , 480, 100, 20 );
169 ctx.fillRect(rect.x + 18, rect.y + 40, 14, 900 )
170 ctx.strokeStyle = "#000";
171 ctx.beginPath();
172 for (var i = 0; i <= coil; i++ ) {
173 var x;
174 var y;
175 if (i != coil + 1) {
176 y = startY + ((rect.y - startY))/coil*i;
177 x = canvas.width/2 + ((i%2==0)?-1:1)*15 + (rect.x - x0)/coil*i;
178 } else {
179 y = startY + ((rect.y - startY))/coil*(i+1);
180 x = canvas.width/2 + ((i%2==0)?1:-1)*15 + (rect.x - x0)/coil*(i+1);
181 }
182 if (i==0) x = x0+25;
183 ctx.lineTo(x, y+5);
184 }
185 ctx.stroke();
186 ctx.fillStyle = "#FFFC06";
187 ctx.fillRect(rect.x, rect.y, rect.width, rect.height);
188 }
189 }
190 function New_graph(htmlElement, yArrayLen, minY, maxY, stepY){
191 this.htmlElement = htmlElement;
192 this.yArrayLen = yArrayLen;
193 this.minY = minY;
194 this.maxY = maxY;
195 this.stepY = stepY;
196 this.vArray = [];
197 }
198 New_graph.prototype.graphIter = function(x, y){
199 this.vArray.push([x, y]); // adding value to the end of the array
200 if (this.vArray.length > this.yArrayLen) this.vArray.shift(); // if the array has more 'yArrayLen' values, we remove the first one
201 var htmlElement1 = this.htmlElement;
202 var vArray1 = this.vArray;
203 var minY1 = this.minY;
204 var maxY1 = this.maxY;
205 var stepY1 = this.stepY;
206 $(function() {
207 var options = {
208 yaxis: {
209 min: minY1,
210 max: maxY1,
211 tickSize: stepY1
212 }
213 };
214 $.plot(htmlElement1, [vArray1], options); // drawing graph at the element "vGraph"
215 });
216 };
217
218 New_graph.prototype.graph = function(data){
219 this.vArray = data;
220 var htmlElement1 = this.htmlElement;
221 var vArray1 = this.vArray;
222 var minY1 = this.minY;
223 var maxY1 = this.maxY;
224 var stepY1 = this.stepY;
225 $(function() {
226 var options = {
227 yaxis: {
228 min: minY1,
229 max: maxY1,
230 tickSize: stepY1
231 }
232 };
233 $.plot(htmlElement1, [vArray1], options); // drawing graph at the element "vGraph"
234 });
235 };
236
237
238
239 function New(){}
240 New.prototype.addSlider = function(htmlSliderElement, htmlValueElement, minVal, maxVal, stepVal, startVal, setFunc){
241 $(function() {
242 $( htmlSliderElement ).slider({ // slider on div - element "slider_m"
243 value:startVal, min: minVal, max: maxVal, step: stepVal,
244 slide: function( event, ui ) { // it works during movement of the slider
245 $( htmlValueElement ).text( ui.value.toFixed(2) ); // it assigns a value to the text field "value_m"
246 setFunc(ui.value);
247 }
248 });
249 });
250 };
251 New.prototype.addInputSlider = function(htmlSliderElement, htmlValueElement, minVal, maxVal, stepVal, startVal, setFunc, pressFunc){
252 window[pressFunc] = function(event){
253 var regExpPattern = /[0-9]+[.]?[0-9]+/;
254 var inputVal = document.getElementById(htmlValueElement.substr(1)).value;
255 if (regExpPattern.test(inputVal.toString()) && inputVal != 0){setFunc(inputVal);}
256 };
257
258 $(function() {
259 $( htmlSliderElement ).slider({
260 value:startVal, min: minVal, max: maxVal, step: stepVal,
261 slide: function( event, ui ) {
262 $( htmlValueElement ).val( ui.value.toFixed(2) );
263 setFunc(ui.value);
264 }
265 });
266 $( htmlValueElement ).val($( htmlSliderElement ).slider( "value" ).toFixed(2) );
267 });
268 };
File "index.html"
1 <!DOCTYPE html>
2 <html>
3 <head>
4 <meta charset="UTF-8" />
5 <title>Амортизатор</title>
6 <script src="Spring.js"></script>
7 <script src="jquery.min.js"></script>
8 <script src="jquery.flot.js"></script>
9 </head>
10 <body>
11 <table>
12 <tr>
13 <td><canvas id="spring_canvas" width="300" height="500" style="border:1px solid #000000;"></canvas></td>
14 <td><div id="vGraph" style="width:600px; height:300px; clear:both;"></div></td>
15 </tr>
16 <tr>
17 <td><input type="range" id="slider_m" min="0.1" max="10" step=0.1 style="width: 120px;" />
18 M = <input type="number" id="number_m" min="0.1" max="10" step=0.1 style="width: 40px;" /> kg<br>
19 <input type="range" id="slider_C" min="0" max="50" step=0.1 style="width: 120px;" />
20 C = <input type="number" id="number_C" min="0" max="50" step=0.1 style="width: 40px;" /> *10^5 N/m<br>
21 <input type="range" id="slider_Dp" min="0" max="100" step=0.1 style="width: 120px;" />
22 dpist = <input type="number" id="number_Dp" min="0" max="100" step=0.1 style="width: 40px;" /> mm<br>
23 <input type="range" id="slider_Dsh" min="0" max="100" step=0.1 style="width: 120px;" />
24 drod = <input type="number" id="number_Dsh" min="0" max="98" step=0.1 style="width: 40px;" /> mm<br>
25 <input type="range" id="slider_L" min="50" max="500" step=1 style="width: 120px;" />
26 L = <input type="number" id="number_L" min="50" max="500" step=1 style="width: 40px;" /> mm<br>
27 <input type="range" id="slider_p" min="0" max="10" step=0.1 style="width: 120px;" />
28 p = <input type="number" id="number_p" min="0" max="10" step=0.1 style="width: 40px;" /> MPa<br><br></td>
29 <td>
30 <div id = "Fo"> </div>
31 <div id = "Fs"> </div>
32 <div id = "info"> where: m - mass, C - coefficient of elasticity <br>dpist - piston diameter, drod - rod diameter<br>L - rod length, p - pressure in the shock absorber<br>Freb - force of rebound stroke<br>Fcom - force of the compression stroke
33 </td>
34 </tr>
35 </body>
36 </html>
