").append(c.replace(bN,"")).find(g):c)),d&&i.each(d,[c,b,a])}});return this},serialize:function(){return f.param(this.serializeArray())},serializeArray:function(){return this.map(function(){return this.elements?f.makeArray(this.elements):this}).filter(function(){return this.name&&!this.disabled&&(this.checked||bO.test(this.nodeName)||bI.test(this.type))}).map(function(a,b){var c=f(this).val();return c==null?null:f.isArray(c)?f.map(c,function(a,c){return{name:b.name,value:a.replace(bF,"\r\n")}}):{name:b.name,value:c.replace(bF,"\r\n")}}).get()}}),f.each("ajaxStart ajaxStop ajaxComplete ajaxError ajaxSuccess ajaxSend".split(" "),function(a,b){f.fn[b]=function(a){return this.on(b,a)}}),f.each(["get","post"],function(a,c){f[c]=function(a,d,e,g){f.isFunction(d)&&(g=g||e,e=d,d=b);return f.ajax({type:c,url:a,data:d,success:e,dataType:g})}}),f.extend({getScript:function(a,c){return f.get(a,b,c,"script")},getJSON:function(a,b,c){return f.get(a,b,c,"json")},ajaxSetup:function(a,b){b?b_(a,f.ajaxSettings):(b=a,a=f.ajaxSettings),b_(a,b);return a},ajaxSettings:{url:bV,isLocal:bJ.test(bW[1]),global:!0,type:"GET",contentType:"application/x-www-form-urlencoded",processData:!0,async:!0,accepts:{xml:"application/xml, text/xml",html:"text/html",text:"text/plain",json:"application/json, text/javascript","*":bX},contents:{xml:/xml/,html:/html/,json:/json/},responseFields:{xml:"responseXML",text:"responseText"},converters:{"* text":a.String,"text html":!0,"text json":f.parseJSON,"text xml":f.parseXML},flatOptions:{context:!0,url:!0}},ajaxPrefilter:bZ(bT),ajaxTransport:bZ(bU),ajax:function(a,c){function w(a,c,l,m){if(s!==2){s=2,q&&clearTimeout(q),p=b,n=m||"",v.readyState=a>0?4:0;var o,r,u,w=c,x=l?cb(d,v,l):b,y,z;if(a>=200&&a<300||a===304){if(d.ifModified){if(y=v.getResponseHeader("Last-Modified"))f.lastModified[k]=y;if(z=v.getResponseHeader("Etag"))f.etag[k]=z}if(a===304)w="notmodified",o=!0;else try{r=cc(d,x),w="success",o=!0}catch(A){w="parsererror",u=A}}else{u=w;if(!w||a)w="error",a<0&&(a=0)}v.status=a,v.statusText=""+(c||w),o?h.resolveWith(e,[r,w,v]):h.rejectWith(e,[v,w,u]),v.statusCode(j),j=b,t&&g.trigger("ajax"+(o?"Success":"Error"),[v,d,o?r:u]),i.fireWith(e,[v,w]),t&&(g.trigger("ajaxComplete",[v,d]),--f.active||f.event.trigger("ajaxStop"))}}typeof a=="object"&&(c=a,a=b),c=c||{};var d=f.ajaxSetup({},c),e=d.context||d,g=e!==d&&(e.nodeType||e instanceof f)?f(e):f.event,h=f.Deferred(),i=f.Callbacks("once memory"),j=d.statusCode||{},k,l={},m={},n,o,p,q,r,s=0,t,u,v={readyState:0,setRequestHeader:function(a,b){if(!s){var c=a.toLowerCase();a=m[c]=m[c]||a,l[a]=b}return this},getAllResponseHeaders:function(){return s===2?n:null},getResponseHeader:function(a){var c;if(s===2){if(!o){o={};while(c=bH.exec(n))o[c[1].toLowerCase()]=c[2]}c=o[a.toLowerCase()]}return c===b?null:c},overrideMimeType:function(a){s||(d.mimeType=a);return this},abort:function(a){a=a||"abort",p&&p.abort(a),w(0,a);return this}};h.promise(v),v.success=v.done,v.error=v.fail,v.complete=i.add,v.statusCode=function(a){if(a){var b;if(s<2)for(b in a)j[b]=[j[b],a[b]];else b=a[v.status],v.then(b,b)}return this},d.url=((a||d.url)+"").replace(bG,"").replace(bL,bW[1]+"//"),d.dataTypes=f.trim(d.dataType||"*").toLowerCase().split(bP),d.crossDomain==null&&(r=bR.exec(d.url.toLowerCase()),d.crossDomain=!(!r||r[1]==bW[1]&&r[2]==bW[2]&&(r[3]||(r[1]==="http:"?80:443))==(bW[3]||(bW[1]==="http:"?80:443)))),d.data&&d.processData&&typeof d.data!="string"&&(d.data=f.param(d.data,d.traditional)),b$(bT,d,c,v);if(s===2)return!1;t=d.global,d.type=d.type.toUpperCase(),d.hasContent=!bK.test(d.type),t&&f.active++===0&&f.event.trigger("ajaxStart");if(!d.hasContent){d.data&&(d.url+=(bM.test(d.url)?"&":"?")+d.data,delete d.data),k=d.url;if(d.cache===!1){var x=f.now(),y=d.url.replace(bQ,"$1_="+x);d.url=y+(y===d.url?(bM.test(d.url)?"&":"?")+"_="+x:"")}}(d.data&&d.hasContent&&d.contentType!==!1||c.contentType)&&v.setRequestHeader("Content-Type",d.contentType),d.ifModified&&(k=k||d.url,f.lastModified[k]&&v.setRequestHeader("If-Modified-Since",f.lastModified[k]),f.etag[k]&&v.setRequestHeader("If-None-Match",f.etag[k])),v.setRequestHeader("Accept",d.dataTypes[0]&&d.accepts[d.dataTypes[0]]?d.accepts[d.dataTypes[0]]+(d.dataTypes[0]!=="*"?", "+bX+"; q=0.01":""):d.accepts["*"]);for(u in d.headers)v.setRequestHeader(u,d.headers[u]);if(d.beforeSend&&(d.beforeSend.call(e,v,d)===!1||s===2)){v.abort();return!1}for(u in{success:1,error:1,complete:1})v[u](d[u]);p=b$(bU,d,c,v);if(!p)w(-1,"No Transport");else{v.readyState=1,t&&g.trigger("ajaxSend",[v,d]),d.async&&d.timeout>0&&(q=setTimeout(function(){v.abort("timeout")},d.timeout));try{s=1,p.send(l,w)}catch(z){if(s<2)w(-1,z);else throw z}}return v},param:function(a,c){var d=[],e=function(a,b){b=f.isFunction(b)?b():b,d[d.length]=encodeURIComponent(a)+"="+encodeURIComponent(b)};c===b&&(c=f.ajaxSettings.traditional);if(f.isArray(a)||a.jquery&&!f.isPlainObject(a))f.each(a,function(){e(this.name,this.value)});else for(var g in a)ca(g,a[g],c,e);return d.join("&").replace(bD,"+")}}),f.extend({active:0,lastModified:{},etag:{}});var cd=f.now(),ce=/(\=)\?(&|$)|\?\?/i;f.ajaxSetup({jsonp:"callback",jsonpCallback:function(){return f.expando+"_"+cd++}}),f.ajaxPrefilter("json jsonp",function(b,c,d){var e=b.contentType==="application/x-www-form-urlencoded"&&typeof b.data=="string";if(b.dataTypes[0]==="jsonp"||b.jsonp!==!1&&(ce.test(b.url)||e&&ce.test(b.data))){var g,h=b.jsonpCallback=f.isFunction(b.jsonpCallback)?b.jsonpCallback():b.jsonpCallback,i=a[h],j=b.url,k=b.data,l="$1"+h+"$2";b.jsonp!==!1&&(j=j.replace(ce,l),b.url===j&&(e&&(k=k.replace(ce,l)),b.data===k&&(j+=(/\?/.test(j)?"&":"?")+b.jsonp+"="+h))),b.url=j,b.data=k,a[h]=function(a){g=[a]},d.always(function(){a[h]=i,g&&f.isFunction(i)&&a[h](g[0])}),b.converters["script json"]=function(){g||f.error(h+" was not called");return g[0]},b.dataTypes[0]="json";return"script"}}),f.ajaxSetup({accepts:{script:"text/javascript, application/javascript, application/ecmascript, application/x-ecmascript"},contents:{script:/javascript|ecmascript/},converters:{"text script":function(a){f.globalEval(a);return a}}}),f.ajaxPrefilter("script",function(a){a.cache===b&&(a.cache=!1),a.crossDomain&&(a.type="GET",a.global=!1)}),f.ajaxTransport("script",function(a){if(a.crossDomain){var d,e=c.head||c.getElementsByTagName("head")[0]||c.documentElement;return{send:function(f,g){d=c.createElement("script"),d.async="async",a.scriptCharset&&(d.charset=a.scriptCharset),d.src=a.url,d.onload=d.onreadystatechange=function(a,c){if(c||!d.readyState||/loaded|complete/.test(d.readyState))d.onload=d.onreadystatechange=null,e&&d.parentNode&&e.removeChild(d),d=b,c||g(200,"success")},e.insertBefore(d,e.firstChild)},abort:function(){d&&d.onload(0,1)}}}});var cf=a.ActiveXObject?function(){for(var a in ch)ch[a](0,1)}:!1,cg=0,ch;f.ajaxSettings.xhr=a.ActiveXObject?function(){return!this.isLocal&&ci()||cj()}:ci,function(a){f.extend(f.support,{ajax:!!a,cors:!!a&&"withCredentials"in a})}(f.ajaxSettings.xhr()),f.support.ajax&&f.ajaxTransport(function(c){if(!c.crossDomain||f.support.cors){var d;return{send:function(e,g){var h=c.xhr(),i,j;c.username?h.open(c.type,c.url,c.async,c.username,c.password):h.open(c.type,c.url,c.async);if(c.xhrFields)for(j in c.xhrFields)h[j]=c.xhrFields[j];c.mimeType&&h.overrideMimeType&&h.overrideMimeType(c.mimeType),!c.crossDomain&&!e["X-Requested-With"]&&(e["X-Requested-With"]="XMLHttpRequest");try{for(j in e)h.setRequestHeader(j,e[j])}catch(k){}h.send(c.hasContent&&c.data||null),d=function(a,e){var j,k,l,m,n;try{if(d&&(e||h.readyState===4)){d=b,i&&(h.onreadystatechange=f.noop,cf&&delete ch[i]);if(e)h.readyState!==4&&h.abort();else{j=h.status,l=h.getAllResponseHeaders(),m={},n=h.responseXML,n&&n.documentElement&&(m.xml=n),m.text=h.responseText;try{k=h.statusText}catch(o){k=""}!j&&c.isLocal&&!c.crossDomain?j=m.text?200:404:j===1223&&(j=204)}}}catch(p){e||g(-1,p)}m&&g(j,k,m,l)},!c.async||h.readyState===4?d():(i=++cg,cf&&(ch||(ch={},f(a).unload(cf)),ch[i]=d),h.onreadystatechange=d)},abort:function(){d&&d(0,1)}}}});var ck={},cl,cm,cn=/^(?:toggle|show|hide)$/,co=/^([+\-]=)?([\d+.\-]+)([a-z%]*)$/i,cp,cq=[["height","marginTop","marginBottom","paddingTop","paddingBottom"],["width","marginLeft","marginRight","paddingLeft","paddingRight"],["opacity"]],cr;f.fn.extend({show:function(a,b,c){var d,e;if(a||a===0)return this.animate(cu("show",3),a,b,c);for(var g=0,h=this.length;g
=i.duration+this.startTime){this.now=this.end,this.pos=this.state=1,this.update(),i.animatedProperties[this.prop]=!0;for(b in i.animatedProperties)i.animatedProperties[b]!==!0&&(g=!1);if(g){i.overflow!=null&&!f.support.shrinkWrapBlocks&&f.each(["","X","Y"],function(a,b){h.style["overflow"+b]=i.overflow[a]}),i.hide&&f(h).hide();if(i.hide||i.show)for(b in i.animatedProperties)f.style(h,b,i.orig[b]),f.removeData(h,"fxshow"+b,!0),f.removeData(h,"toggle"+b,!0);d=i.complete,d&&(i.complete=!1,d.call(h))}return!1}i.duration==Infinity?this.now=e:(c=e-this.startTime,this.state=c/i.duration,this.pos=f.easing[i.animatedProperties[this.prop]](this.state,c,0,1,i.duration),this.now=this.start+(this.end-this.start)*this.pos),this.update();return!0}},f.extend(f.fx,{tick:function(){var a,b=f.timers,c=0;for(;c-1,k={},l={},m,n;j?(l=e.position(),m=l.top,n=l.left):(m=parseFloat(h)||0,n=parseFloat(i)||0),f.isFunction(b)&&(b=b.call(a,c,g)),b.top!=null&&(k.top=b.top-g.top+m),b.left!=null&&(k.left=b.left-g.left+n),"using"in b?b.using.call(a,k):e.css(k)}},f.fn.extend({position:function(){if(!this[0])return null;var a=this[0],b=this.offsetParent(),c=this.offset(),d=cx.test(b[0].nodeName)?{top:0,left:0}:b.offset();c.top-=parseFloat(f.css(a,"marginTop"))||0,c.left-=parseFloat(f.css(a,"marginLeft"))||0,d.top+=parseFloat(f.css(b[0],"borderTopWidth"))||0,d.left+=parseFloat(f.css(b[0],"borderLeftWidth"))||0;return{top:c.top-d.top,left:c.left-d.left}},offsetParent:function(){return this.map(function(){var a=this.offsetParent||c.body;while(a&&!cx.test(a.nodeName)&&f.css(a,"position")==="static")a=a.offsetParent;return a})}}),f.each(["Left","Top"],function(a,c){var d="scroll"+c;f.fn[d]=function(c){var e,g;if(c===b){e=this[0];if(!e)return null;g=cy(e);return g?"pageXOffset"in g?g[a?"pageYOffset":"pageXOffset"]:f.support.boxModel&&g.document.documentElement[d]||g.document.body[d]:e[d]}return this.each(function(){g=cy(this),g?g.scrollTo(a?f(g).scrollLeft():c,a?c:f(g).scrollTop()):this[d]=c})}}),f.each(["Height","Width"],function(a,c){var d=c.toLowerCase();f.fn["inner"+c]=function(){var a=this[0];return a?a.style?parseFloat(f.css(a,d,"padding")):this[d]():null},f.fn["outer"+c]=function(a){var b=this[0];return b?b.style?parseFloat(f.css(b,d,a?"margin":"border")):this[d]():null},f.fn[d]=function(a){var e=this[0];if(!e)return a==null?null:this;if(f.isFunction(a))return this.each(function(b){var c=f(this);c[d](a.call(this,b,c[d]()))});if(f.isWindow(e)){var g=e.document.documentElement["client"+c],h=e.document.body;return e.document.compatMode==="CSS1Compat"&&g||h&&h["client"+c]||g}if(e.nodeType===9)return Math.max(e.documentElement["client"+c],e.body["scroll"+c],e.documentElement["scroll"+c],e.body["offset"+c],e.documentElement["offset"+c]);if(a===b){var i=f.css(e,d),j=parseFloat(i);return f.isNumeric(j)?j:i}return this.css(d,typeof a=="string"?a:a+"px")}}),a.jQuery=a.$=f,typeof define=="function"&&define.amd&&define.amd.jQuery&&define("jquery",[],function(){return f})})(window);
\ No newline at end of file
diff --git a/experiment/simulation/js/jquery-ui-1.8.13.custom.min.js b/experiment/simulation/js/jquery-ui-1.8.13.custom.min.js
new file mode 100644
index 0000000..46d9452
--- /dev/null
+++ b/experiment/simulation/js/jquery-ui-1.8.13.custom.min.js
@@ -0,0 +1,653 @@
+/*!
+ * jQuery UI 1.8.13
+ *
+ * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about)
+ * Dual licensed under the MIT or GPL Version 2 licenses.
+ * http://jquery.org/license
+ *
+ * http://docs.jquery.com/UI
+ */
+(function(c,j){function k(a,b){var d=a.nodeName.toLowerCase();if("area"===d){b=a.parentNode;d=b.name;if(!a.href||!d||b.nodeName.toLowerCase()!=="map")return false;a=c("img[usemap=#"+d+"]")[0];return!!a&&l(a)}return(/input|select|textarea|button|object/.test(d)?!a.disabled:"a"==d?a.href||b:b)&&l(a)}function l(a){return!c(a).parents().andSelf().filter(function(){return c.curCSS(this,"visibility")==="hidden"||c.expr.filters.hidden(this)}).length}c.ui=c.ui||{};if(!c.ui.version){c.extend(c.ui,{version:"1.8.13",
+keyCode:{ALT:18,BACKSPACE:8,CAPS_LOCK:20,COMMA:188,COMMAND:91,COMMAND_LEFT:91,COMMAND_RIGHT:93,CONTROL:17,DELETE:46,DOWN:40,END:35,ENTER:13,ESCAPE:27,HOME:36,INSERT:45,LEFT:37,MENU:93,NUMPAD_ADD:107,NUMPAD_DECIMAL:110,NUMPAD_DIVIDE:111,NUMPAD_ENTER:108,NUMPAD_MULTIPLY:106,NUMPAD_SUBTRACT:109,PAGE_DOWN:34,PAGE_UP:33,PERIOD:190,RIGHT:39,SHIFT:16,SPACE:32,TAB:9,UP:38,WINDOWS:91}});c.fn.extend({_focus:c.fn.focus,focus:function(a,b){return typeof a==="number"?this.each(function(){var d=this;setTimeout(function(){c(d).focus();
+b&&b.call(d)},a)}):this._focus.apply(this,arguments)},scrollParent:function(){var a;a=c.browser.msie&&/(static|relative)/.test(this.css("position"))||/absolute/.test(this.css("position"))?this.parents().filter(function(){return/(relative|absolute|fixed)/.test(c.curCSS(this,"position",1))&&/(auto|scroll)/.test(c.curCSS(this,"overflow",1)+c.curCSS(this,"overflow-y",1)+c.curCSS(this,"overflow-x",1))}).eq(0):this.parents().filter(function(){return/(auto|scroll)/.test(c.curCSS(this,"overflow",1)+c.curCSS(this,
+"overflow-y",1)+c.curCSS(this,"overflow-x",1))}).eq(0);return/fixed/.test(this.css("position"))||!a.length?c(document):a},zIndex:function(a){if(a!==j)return this.css("zIndex",a);if(this.length){a=c(this[0]);for(var b;a.length&&a[0]!==document;){b=a.css("position");if(b==="absolute"||b==="relative"||b==="fixed"){b=parseInt(a.css("zIndex"),10);if(!isNaN(b)&&b!==0)return b}a=a.parent()}}return 0},disableSelection:function(){return this.bind((c.support.selectstart?"selectstart":"mousedown")+".ui-disableSelection",
+function(a){a.preventDefault()})},enableSelection:function(){return this.unbind(".ui-disableSelection")}});c.each(["Width","Height"],function(a,b){function d(f,g,m,n){c.each(e,function(){g-=parseFloat(c.curCSS(f,"padding"+this,true))||0;if(m)g-=parseFloat(c.curCSS(f,"border"+this+"Width",true))||0;if(n)g-=parseFloat(c.curCSS(f,"margin"+this,true))||0});return g}var e=b==="Width"?["Left","Right"]:["Top","Bottom"],h=b.toLowerCase(),i={innerWidth:c.fn.innerWidth,innerHeight:c.fn.innerHeight,outerWidth:c.fn.outerWidth,
+outerHeight:c.fn.outerHeight};c.fn["inner"+b]=function(f){if(f===j)return i["inner"+b].call(this);return this.each(function(){c(this).css(h,d(this,f)+"px")})};c.fn["outer"+b]=function(f,g){if(typeof f!=="number")return i["outer"+b].call(this,f);return this.each(function(){c(this).css(h,d(this,f,true,g)+"px")})}});c.extend(c.expr[":"],{data:function(a,b,d){return!!c.data(a,d[3])},focusable:function(a){return k(a,!isNaN(c.attr(a,"tabindex")))},tabbable:function(a){var b=c.attr(a,"tabindex"),d=isNaN(b);
+return(d||b>=0)&&k(a,!d)}});c(function(){var a=document.body,b=a.appendChild(b=document.createElement("div"));c.extend(b.style,{minHeight:"100px",height:"auto",padding:0,borderWidth:0});c.support.minHeight=b.offsetHeight===100;c.support.selectstart="onselectstart"in b;a.removeChild(b).style.display="none"});c.extend(c.ui,{plugin:{add:function(a,b,d){a=c.ui[a].prototype;for(var e in d){a.plugins[e]=a.plugins[e]||[];a.plugins[e].push([b,d[e]])}},call:function(a,b,d){if((b=a.plugins[b])&&a.element[0].parentNode)for(var e=
+0;e0)return true;a[b]=1;d=a[b]>0;a[b]=0;return d},isOverAxis:function(a,b,d){return a>b&&a=9)&&!a.button)return this._mouseUp(a);if(this._mouseStarted){this._mouseDrag(a);return a.preventDefault()}if(this._mouseDistanceMet(a)&&this._mouseDelayMet(a))(this._mouseStarted=this._mouseStart(this._mouseDownEvent,a)!==false)?this._mouseDrag(a):this._mouseUp(a);return!this._mouseStarted},_mouseUp:function(a){b(document).unbind("mousemove."+this.widgetName,this._mouseMoveDelegate).unbind("mouseup."+this.widgetName,this._mouseUpDelegate);if(this._mouseStarted){this._mouseStarted=
+false;a.target==this._mouseDownEvent.target&&b.data(a.target,this.widgetName+".preventClickEvent",true);this._mouseStop(a)}return false},_mouseDistanceMet:function(a){return Math.max(Math.abs(this._mouseDownEvent.pageX-a.pageX),Math.abs(this._mouseDownEvent.pageY-a.pageY))>=this.options.distance},_mouseDelayMet:function(){return this.mouseDelayMet},_mouseStart:function(){},_mouseDrag:function(){},_mouseStop:function(){},_mouseCapture:function(){return true}})})(jQuery);
+;/*
+ * jQuery UI Position 1.8.13
+ *
+ * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about)
+ * Dual licensed under the MIT or GPL Version 2 licenses.
+ * http://jquery.org/license
+ *
+ * http://docs.jquery.com/UI/Position
+ */
+(function(c){c.ui=c.ui||{};var n=/left|center|right/,o=/top|center|bottom/,t=c.fn.position,u=c.fn.offset;c.fn.position=function(b){if(!b||!b.of)return t.apply(this,arguments);b=c.extend({},b);var a=c(b.of),d=a[0],g=(b.collision||"flip").split(" "),e=b.offset?b.offset.split(" "):[0,0],h,k,j;if(d.nodeType===9){h=a.width();k=a.height();j={top:0,left:0}}else if(d.setTimeout){h=a.width();k=a.height();j={top:a.scrollTop(),left:a.scrollLeft()}}else if(d.preventDefault){b.at="left top";h=k=0;j={top:b.of.pageY,
+left:b.of.pageX}}else{h=a.outerWidth();k=a.outerHeight();j=a.offset()}c.each(["my","at"],function(){var f=(b[this]||"").split(" ");if(f.length===1)f=n.test(f[0])?f.concat(["center"]):o.test(f[0])?["center"].concat(f):["center","center"];f[0]=n.test(f[0])?f[0]:"center";f[1]=o.test(f[1])?f[1]:"center";b[this]=f});if(g.length===1)g[1]=g[0];e[0]=parseInt(e[0],10)||0;if(e.length===1)e[1]=e[0];e[1]=parseInt(e[1],10)||0;if(b.at[0]==="right")j.left+=h;else if(b.at[0]==="center")j.left+=h/2;if(b.at[1]==="bottom")j.top+=
+k;else if(b.at[1]==="center")j.top+=k/2;j.left+=e[0];j.top+=e[1];return this.each(function(){var f=c(this),l=f.outerWidth(),m=f.outerHeight(),p=parseInt(c.curCSS(this,"marginLeft",true))||0,q=parseInt(c.curCSS(this,"marginTop",true))||0,v=l+p+(parseInt(c.curCSS(this,"marginRight",true))||0),w=m+q+(parseInt(c.curCSS(this,"marginBottom",true))||0),i=c.extend({},j),r;if(b.my[0]==="right")i.left-=l;else if(b.my[0]==="center")i.left-=l/2;if(b.my[1]==="bottom")i.top-=m;else if(b.my[1]==="center")i.top-=
+m/2;i.left=Math.round(i.left);i.top=Math.round(i.top);r={left:i.left-p,top:i.top-q};c.each(["left","top"],function(s,x){c.ui.position[g[s]]&&c.ui.position[g[s]][x](i,{targetWidth:h,targetHeight:k,elemWidth:l,elemHeight:m,collisionPosition:r,collisionWidth:v,collisionHeight:w,offset:e,my:b.my,at:b.at})});c.fn.bgiframe&&f.bgiframe();f.offset(c.extend(i,{using:b.using}))})};c.ui.position={fit:{left:function(b,a){var d=c(window);d=a.collisionPosition.left+a.collisionWidth-d.width()-d.scrollLeft();b.left=
+d>0?b.left-d:Math.max(b.left-a.collisionPosition.left,b.left)},top:function(b,a){var d=c(window);d=a.collisionPosition.top+a.collisionHeight-d.height()-d.scrollTop();b.top=d>0?b.top-d:Math.max(b.top-a.collisionPosition.top,b.top)}},flip:{left:function(b,a){if(a.at[0]!=="center"){var d=c(window);d=a.collisionPosition.left+a.collisionWidth-d.width()-d.scrollLeft();var g=a.my[0]==="left"?-a.elemWidth:a.my[0]==="right"?a.elemWidth:0,e=a.at[0]==="left"?a.targetWidth:-a.targetWidth,h=-2*a.offset[0];b.left+=
+a.collisionPosition.left<0?g+e+h:d>0?g+e+h:0}},top:function(b,a){if(a.at[1]!=="center"){var d=c(window);d=a.collisionPosition.top+a.collisionHeight-d.height()-d.scrollTop();var g=a.my[1]==="top"?-a.elemHeight:a.my[1]==="bottom"?a.elemHeight:0,e=a.at[1]==="top"?a.targetHeight:-a.targetHeight,h=-2*a.offset[1];b.top+=a.collisionPosition.top<0?g+e+h:d>0?g+e+h:0}}}};if(!c.offset.setOffset){c.offset.setOffset=function(b,a){if(/static/.test(c.curCSS(b,"position")))b.style.position="relative";var d=c(b),
+g=d.offset(),e=parseInt(c.curCSS(b,"top",true),10)||0,h=parseInt(c.curCSS(b,"left",true),10)||0;g={top:a.top-g.top+e,left:a.left-g.left+h};"using"in a?a.using.call(b,g):d.css(g)};c.fn.offset=function(b){var a=this[0];if(!a||!a.ownerDocument)return null;if(b)return this.each(function(){c.offset.setOffset(this,b)});return u.call(this)}}})(jQuery);
+;/*
+ * jQuery UI Draggable 1.8.13
+ *
+ * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about)
+ * Dual licensed under the MIT or GPL Version 2 licenses.
+ * http://jquery.org/license
+ *
+ * http://docs.jquery.com/UI/Draggables
+ *
+ * Depends:
+ * jquery.ui.core.js
+ * jquery.ui.mouse.js
+ * jquery.ui.widget.js
+ */
+(function(d){d.widget("ui.draggable",d.ui.mouse,{widgetEventPrefix:"drag",options:{addClasses:true,appendTo:"parent",axis:false,connectToSortable:false,containment:false,cursor:"auto",cursorAt:false,grid:false,handle:false,helper:"original",iframeFix:false,opacity:false,refreshPositions:false,revert:false,revertDuration:500,scope:"default",scroll:true,scrollSensitivity:20,scrollSpeed:20,snap:false,snapMode:"both",snapTolerance:20,stack:false,zIndex:false},_create:function(){if(this.options.helper==
+"original"&&!/^(?:r|a|f)/.test(this.element.css("position")))this.element[0].style.position="relative";this.options.addClasses&&this.element.addClass("ui-draggable");this.options.disabled&&this.element.addClass("ui-draggable-disabled");this._mouseInit()},destroy:function(){if(this.element.data("draggable")){this.element.removeData("draggable").unbind(".draggable").removeClass("ui-draggable ui-draggable-dragging ui-draggable-disabled");this._mouseDestroy();return this}},_mouseCapture:function(a){var b=
+this.options;if(this.helper||b.disabled||d(a.target).is(".ui-resizable-handle"))return false;this.handle=this._getHandle(a);if(!this.handle)return false;d(b.iframeFix===true?"iframe":b.iframeFix).each(function(){d('').css({width:this.offsetWidth+"px",height:this.offsetHeight+"px",position:"absolute",opacity:"0.001",zIndex:1E3}).css(d(this).offset()).appendTo("body")});return true},_mouseStart:function(a){var b=this.options;this.helper=
+this._createHelper(a);this._cacheHelperProportions();if(d.ui.ddmanager)d.ui.ddmanager.current=this;this._cacheMargins();this.cssPosition=this.helper.css("position");this.scrollParent=this.helper.scrollParent();this.offset=this.positionAbs=this.element.offset();this.offset={top:this.offset.top-this.margins.top,left:this.offset.left-this.margins.left};d.extend(this.offset,{click:{left:a.pageX-this.offset.left,top:a.pageY-this.offset.top},parent:this._getParentOffset(),relative:this._getRelativeOffset()});
+this.originalPosition=this.position=this._generatePosition(a);this.originalPageX=a.pageX;this.originalPageY=a.pageY;b.cursorAt&&this._adjustOffsetFromHelper(b.cursorAt);b.containment&&this._setContainment();if(this._trigger("start",a)===false){this._clear();return false}this._cacheHelperProportions();d.ui.ddmanager&&!b.dropBehaviour&&d.ui.ddmanager.prepareOffsets(this,a);this.helper.addClass("ui-draggable-dragging");this._mouseDrag(a,true);return true},_mouseDrag:function(a,b){this.position=this._generatePosition(a);
+this.positionAbs=this._convertPositionTo("absolute");if(!b){b=this._uiHash();if(this._trigger("drag",a,b)===false){this._mouseUp({});return false}this.position=b.position}if(!this.options.axis||this.options.axis!="y")this.helper[0].style.left=this.position.left+"px";if(!this.options.axis||this.options.axis!="x")this.helper[0].style.top=this.position.top+"px";d.ui.ddmanager&&d.ui.ddmanager.drag(this,a);return false},_mouseStop:function(a){var b=false;if(d.ui.ddmanager&&!this.options.dropBehaviour)b=
+d.ui.ddmanager.drop(this,a);if(this.dropped){b=this.dropped;this.dropped=false}if((!this.element[0]||!this.element[0].parentNode)&&this.options.helper=="original")return false;if(this.options.revert=="invalid"&&!b||this.options.revert=="valid"&&b||this.options.revert===true||d.isFunction(this.options.revert)&&this.options.revert.call(this.element,b)){var c=this;d(this.helper).animate(this.originalPosition,parseInt(this.options.revertDuration,10),function(){c._trigger("stop",a)!==false&&c._clear()})}else this._trigger("stop",
+a)!==false&&this._clear();return false},_mouseUp:function(a){this.options.iframeFix===true&&d("div.ui-draggable-iframeFix").each(function(){this.parentNode.removeChild(this)});return d.ui.mouse.prototype._mouseUp.call(this,a)},cancel:function(){this.helper.is(".ui-draggable-dragging")?this._mouseUp({}):this._clear();return this},_getHandle:function(a){var b=!this.options.handle||!d(this.options.handle,this.element).length?true:false;d(this.options.handle,this.element).find("*").andSelf().each(function(){if(this==
+a.target)b=true});return b},_createHelper:function(a){var b=this.options;a=d.isFunction(b.helper)?d(b.helper.apply(this.element[0],[a])):b.helper=="clone"?this.element.clone().removeAttr("id"):this.element;a.parents("body").length||a.appendTo(b.appendTo=="parent"?this.element[0].parentNode:b.appendTo);a[0]!=this.element[0]&&!/(fixed|absolute)/.test(a.css("position"))&&a.css("position","absolute");return a},_adjustOffsetFromHelper:function(a){if(typeof a=="string")a=a.split(" ");if(d.isArray(a))a=
+{left:+a[0],top:+a[1]||0};if("left"in a)this.offset.click.left=a.left+this.margins.left;if("right"in a)this.offset.click.left=this.helperProportions.width-a.right+this.margins.left;if("top"in a)this.offset.click.top=a.top+this.margins.top;if("bottom"in a)this.offset.click.top=this.helperProportions.height-a.bottom+this.margins.top},_getParentOffset:function(){this.offsetParent=this.helper.offsetParent();var a=this.offsetParent.offset();if(this.cssPosition=="absolute"&&this.scrollParent[0]!=document&&
+d.ui.contains(this.scrollParent[0],this.offsetParent[0])){a.left+=this.scrollParent.scrollLeft();a.top+=this.scrollParent.scrollTop()}if(this.offsetParent[0]==document.body||this.offsetParent[0].tagName&&this.offsetParent[0].tagName.toLowerCase()=="html"&&d.browser.msie)a={top:0,left:0};return{top:a.top+(parseInt(this.offsetParent.css("borderTopWidth"),10)||0),left:a.left+(parseInt(this.offsetParent.css("borderLeftWidth"),10)||0)}},_getRelativeOffset:function(){if(this.cssPosition=="relative"){var a=
+this.element.position();return{top:a.top-(parseInt(this.helper.css("top"),10)||0)+this.scrollParent.scrollTop(),left:a.left-(parseInt(this.helper.css("left"),10)||0)+this.scrollParent.scrollLeft()}}else return{top:0,left:0}},_cacheMargins:function(){this.margins={left:parseInt(this.element.css("marginLeft"),10)||0,top:parseInt(this.element.css("marginTop"),10)||0,right:parseInt(this.element.css("marginRight"),10)||0,bottom:parseInt(this.element.css("marginBottom"),10)||0}},_cacheHelperProportions:function(){this.helperProportions=
+{width:this.helper.outerWidth(),height:this.helper.outerHeight()}},_setContainment:function(){var a=this.options;if(a.containment=="parent")a.containment=this.helper[0].parentNode;if(a.containment=="document"||a.containment=="window")this.containment=[(a.containment=="document"?0:d(window).scrollLeft())-this.offset.relative.left-this.offset.parent.left,(a.containment=="document"?0:d(window).scrollTop())-this.offset.relative.top-this.offset.parent.top,(a.containment=="document"?0:d(window).scrollLeft())+
+d(a.containment=="document"?document:window).width()-this.helperProportions.width-this.margins.left,(a.containment=="document"?0:d(window).scrollTop())+(d(a.containment=="document"?document:window).height()||document.body.parentNode.scrollHeight)-this.helperProportions.height-this.margins.top];if(!/^(document|window|parent)$/.test(a.containment)&&a.containment.constructor!=Array){a=d(a.containment);var b=a[0];if(b){a.offset();var c=d(b).css("overflow")!="hidden";this.containment=[(parseInt(d(b).css("borderLeftWidth"),
+10)||0)+(parseInt(d(b).css("paddingLeft"),10)||0),(parseInt(d(b).css("borderTopWidth"),10)||0)+(parseInt(d(b).css("paddingTop"),10)||0),(c?Math.max(b.scrollWidth,b.offsetWidth):b.offsetWidth)-(parseInt(d(b).css("borderLeftWidth"),10)||0)-(parseInt(d(b).css("paddingRight"),10)||0)-this.helperProportions.width-this.margins.left-this.margins.right,(c?Math.max(b.scrollHeight,b.offsetHeight):b.offsetHeight)-(parseInt(d(b).css("borderTopWidth"),10)||0)-(parseInt(d(b).css("paddingBottom"),10)||0)-this.helperProportions.height-
+this.margins.top-this.margins.bottom];this.relative_container=a}}else if(a.containment.constructor==Array)this.containment=a.containment},_convertPositionTo:function(a,b){if(!b)b=this.position;a=a=="absolute"?1:-1;var c=this.cssPosition=="absolute"&&!(this.scrollParent[0]!=document&&d.ui.contains(this.scrollParent[0],this.offsetParent[0]))?this.offsetParent:this.scrollParent,f=/(html|body)/i.test(c[0].tagName);return{top:b.top+this.offset.relative.top*a+this.offset.parent.top*a-(d.browser.safari&&
+d.browser.version<526&&this.cssPosition=="fixed"?0:(this.cssPosition=="fixed"?-this.scrollParent.scrollTop():f?0:c.scrollTop())*a),left:b.left+this.offset.relative.left*a+this.offset.parent.left*a-(d.browser.safari&&d.browser.version<526&&this.cssPosition=="fixed"?0:(this.cssPosition=="fixed"?-this.scrollParent.scrollLeft():f?0:c.scrollLeft())*a)}},_generatePosition:function(a){var b=this.options,c=this.cssPosition=="absolute"&&!(this.scrollParent[0]!=document&&d.ui.contains(this.scrollParent[0],
+this.offsetParent[0]))?this.offsetParent:this.scrollParent,f=/(html|body)/i.test(c[0].tagName),e=a.pageX,h=a.pageY;if(this.originalPosition){var g;if(this.containment){if(this.relative_container){g=this.relative_container.offset();g=[this.containment[0]+g.left,this.containment[1]+g.top,this.containment[2]+g.left,this.containment[3]+g.top]}else g=this.containment;if(a.pageX-this.offset.click.leftg[2])e=g[2]+this.offset.click.left;if(a.pageY-this.offset.click.top>g[3])h=g[3]+this.offset.click.top}if(b.grid){h=this.originalPageY+Math.round((h-this.originalPageY)/b.grid[1])*b.grid[1];h=g?!(h-this.offset.click.topg[3])?h:!(h-this.offset.click.topg[2])?e:!(e-this.offset.click.left<
+g[0])?e-b.grid[0]:e+b.grid[0]:e}}return{top:h-this.offset.click.top-this.offset.relative.top-this.offset.parent.top+(d.browser.safari&&d.browser.version<526&&this.cssPosition=="fixed"?0:this.cssPosition=="fixed"?-this.scrollParent.scrollTop():f?0:c.scrollTop()),left:e-this.offset.click.left-this.offset.relative.left-this.offset.parent.left+(d.browser.safari&&d.browser.version<526&&this.cssPosition=="fixed"?0:this.cssPosition=="fixed"?-this.scrollParent.scrollLeft():f?0:c.scrollLeft())}},_clear:function(){this.helper.removeClass("ui-draggable-dragging");
+this.helper[0]!=this.element[0]&&!this.cancelHelperRemoval&&this.helper.remove();this.helper=null;this.cancelHelperRemoval=false},_trigger:function(a,b,c){c=c||this._uiHash();d.ui.plugin.call(this,a,[b,c]);if(a=="drag")this.positionAbs=this._convertPositionTo("absolute");return d.Widget.prototype._trigger.call(this,a,b,c)},plugins:{},_uiHash:function(){return{helper:this.helper,position:this.position,originalPosition:this.originalPosition,offset:this.positionAbs}}});d.extend(d.ui.draggable,{version:"1.8.13"});
+d.ui.plugin.add("draggable","connectToSortable",{start:function(a,b){var c=d(this).data("draggable"),f=c.options,e=d.extend({},b,{item:c.element});c.sortables=[];d(f.connectToSortable).each(function(){var h=d.data(this,"sortable");if(h&&!h.options.disabled){c.sortables.push({instance:h,shouldRevert:h.options.revert});h.refreshPositions();h._trigger("activate",a,e)}})},stop:function(a,b){var c=d(this).data("draggable"),f=d.extend({},b,{item:c.element});d.each(c.sortables,function(){if(this.instance.isOver){this.instance.isOver=
+0;c.cancelHelperRemoval=true;this.instance.cancelHelperRemoval=false;if(this.shouldRevert)this.instance.options.revert=true;this.instance._mouseStop(a);this.instance.options.helper=this.instance.options._helper;c.options.helper=="original"&&this.instance.currentItem.css({top:"auto",left:"auto"})}else{this.instance.cancelHelperRemoval=false;this.instance._trigger("deactivate",a,f)}})},drag:function(a,b){var c=d(this).data("draggable"),f=this;d.each(c.sortables,function(){this.instance.positionAbs=
+c.positionAbs;this.instance.helperProportions=c.helperProportions;this.instance.offset.click=c.offset.click;if(this.instance._intersectsWith(this.instance.containerCache)){if(!this.instance.isOver){this.instance.isOver=1;this.instance.currentItem=d(f).clone().removeAttr("id").appendTo(this.instance.element).data("sortable-item",true);this.instance.options._helper=this.instance.options.helper;this.instance.options.helper=function(){return b.helper[0]};a.target=this.instance.currentItem[0];this.instance._mouseCapture(a,
+true);this.instance._mouseStart(a,true,true);this.instance.offset.click.top=c.offset.click.top;this.instance.offset.click.left=c.offset.click.left;this.instance.offset.parent.left-=c.offset.parent.left-this.instance.offset.parent.left;this.instance.offset.parent.top-=c.offset.parent.top-this.instance.offset.parent.top;c._trigger("toSortable",a);c.dropped=this.instance.element;c.currentItem=c.element;this.instance.fromOutside=c}this.instance.currentItem&&this.instance._mouseDrag(a)}else if(this.instance.isOver){this.instance.isOver=
+0;this.instance.cancelHelperRemoval=true;this.instance.options.revert=false;this.instance._trigger("out",a,this.instance._uiHash(this.instance));this.instance._mouseStop(a,true);this.instance.options.helper=this.instance.options._helper;this.instance.currentItem.remove();this.instance.placeholder&&this.instance.placeholder.remove();c._trigger("fromSortable",a);c.dropped=false}})}});d.ui.plugin.add("draggable","cursor",{start:function(){var a=d("body"),b=d(this).data("draggable").options;if(a.css("cursor"))b._cursor=
+a.css("cursor");a.css("cursor",b.cursor)},stop:function(){var a=d(this).data("draggable").options;a._cursor&&d("body").css("cursor",a._cursor)}});d.ui.plugin.add("draggable","opacity",{start:function(a,b){a=d(b.helper);b=d(this).data("draggable").options;if(a.css("opacity"))b._opacity=a.css("opacity");a.css("opacity",b.opacity)},stop:function(a,b){a=d(this).data("draggable").options;a._opacity&&d(b.helper).css("opacity",a._opacity)}});d.ui.plugin.add("draggable","scroll",{start:function(){var a=d(this).data("draggable");
+if(a.scrollParent[0]!=document&&a.scrollParent[0].tagName!="HTML")a.overflowOffset=a.scrollParent.offset()},drag:function(a){var b=d(this).data("draggable"),c=b.options,f=false;if(b.scrollParent[0]!=document&&b.scrollParent[0].tagName!="HTML"){if(!c.axis||c.axis!="x")if(b.overflowOffset.top+b.scrollParent[0].offsetHeight-a.pageY=0;i--){var j=c.snapElements[i].left,l=j+c.snapElements[i].width,k=c.snapElements[i].top,m=k+c.snapElements[i].height;if(j-e=j&&f<=l||h>=j&&h<=l||fl)&&(e>=
+i&&e<=k||g>=i&&g<=k||ek);default:return false}};d.ui.ddmanager={current:null,droppables:{"default":[]},prepareOffsets:function(a,b){var c=d.ui.ddmanager.droppables[a.options.scope]||[],e=b?b.type:null,g=(a.currentItem||a.element).find(":data(droppable)").andSelf(),f=0;a:for(;f
+ 
+
+where, Xst is the static deflection of the spring which is equal to the ratio of amplitude of the excitation force, F0,and the stiffness of the spring, K.
+ω - Excitation frequency,
+ωn - Naturalfrequency of the system.
+At ω = 0.95ωn, we get X/Xst = 10.26. The plot of the amplitude of response, X/Xst , versus frequency ratio, ω/ωn, is shown in the figure above.
+Now, we attach an auxiliary SDOF system with mass M2 and stiffness K2 to the main system which has a mass M1 and stiffness K1. When these systems were independent (separate), they would have their own natural frequencies given by ωn1 = (K1/M1)^0.5 for the main system, and ωn2 = (K1/M2)^0.5 for the auxiliary system
+It is to be noted that on attaching the auxiliary system to main system, another system is formed which is now a two degree of freedom system. It will have its own two natural frequencies but quite different from ω1 and ω2 that we considered above.
+The response of each mass of the system, now a 2-DOF system, to harmonic force acting on M1 (Amplitude of vibration of each mass is given by), is given by
+
+
+
+μ is the mass ratio = M2/M1. ωn1 = (K1/M1)^0.5 and ωn2 = (K1/M2)^0.5, However, the natural frequencies of the 2-degree of freedom system formed after attaching the auxiliary system are given by different expressions in terms of M1, M2, K1, and K2. Plot of the dimensionless responses of each of the two masses is shown above when a harmonic force acts on M1 when excitation frequency is close to natural frequency of main system,ω ≠ω1 ,, but much away from natural frequency of auxiliary system,ω ≠ω2, Mass ratio μ=M2/M1= 0.15.
+
+
+
+In this experiment, while the auxiliary system is tuned, you would choose different values of mass ratio and observe the responses of the primary and secondary systems. The responses can also be observed even if the auxiliary system is not tuned i.e. when the natural frequency of auxiliary system is not equal to the excitation frequency or the natural frequency of the main system. This is done by choosing values of M2 and K2 such that ω ≠ ω2 as ω2 = (K2/M2)^0.5.
+
+
+**Vibratory systems around us**
+
+Here are some examples of physical systems where the vibrations are prominent and can be observed easily. In musical instruments the vibrations are intentional. The parts of musical instruments are designed so that they generate sounds that are pleasant to listen. In many cases the vibrations are unwanted and we try to minimize them.
+
+1. A chandelier hanging from ceiling oscillates to and fro following an initial disturbance; maybe due to a breeze of air.
+
+2. The oscillations of the chandelier at cathedral of Pisa, Italy, were studied by the famous scientist Galileo Galilee.
+
+3. A load attached at end of a wire-rope of a crane oscillates to and fro due to initial disturbance; maybe due to sudden stopping of carriage of the crane while revolving about the vertical axis.
+
+4. The pendulum used in clock of olden days used to oscillate to and fro once every second. i.e. it had a period of oscillation of one second.
+
+5. String of a guitar, when plucked and left to its own, vibrates and makes a musical sound. It comes to rest after a while; the vibrations die out. Similarly, the diaphragm of a table vibrates when hit and left to its own. It also comes to rest after some time.
+
+All these are examples of vibratory systems that are set into vibration following an initial disturbance. All these systems have three components: mass, due to which the system possesses inertia; elasticity, due to which potential energy can be stored; and components that dissipate energy causing the vibratory motion to be damped which bring them to rest after some time.
+
+Vibration or vibratory systems are classified in number of ways. Some of the classifications are given below:
+
+ **Free and forced vibration** - A free vibration occurs due to initial displacement or velocity, or both, applied to the system only initially. There is no external force acting on the system when the system is vibrating. A forced vibration occurs when the system vibrates in response to external force applied continuously. When the force applied is periodic, i.e. it repeats itself after a fixed interval of time, the forced vibration is called periodic. If the periodic force and hence the resulting vibration varies sinusoidally with respect to time, the vibration is called harmonic. If the force is not periodic, the forced vibration is called aperiodic or random.
+
+ **Damped and undamped vibration** -
+When the vibratory system has elements that offer resistance to motion, energy is continuously dissipated and the free vibrations of such systems come to halt after some time. This is called damped vibration and such systems are called damped systems. Forced vibration of a damped system continues as long as the force acts but some of the work done by the external force is lost in overcoming the resistance offered by the damping elements. Systems without damping elements are called undamped systems and their vibrations are called undamped vibrations. All systems in nature have some or the other damping element and their natural vibrations are damped. Hence they come to rest after some time following free vibrations. Nevertheless, we study the vibration of undamped systems because the concepts developed in studying them are useful in analyzing and understanding the phenomena occurring in vibration of damped as well as complicated systems. When the force of resistance offered by a damping element is proportional to velocity of mass of the system, it is termed as viscous damping and the damping element is called a viscous damper. If the force of resistance has a constant value, it is termed as Coulomb damping. Damping due to dry friction shows this kind of behavior. Coulomb damping can occur when the system has components rubbing over each other. There are other types of damping also which shall be discussed later.
+
+
+**Degrees of freedom** : The vibratory systems are classified as single-degree-of-freedom systems, Multi-degree-of-freedom-systems or continuous systems. The number of degrees of freedom corresponds to the number of independent co-ordinates required to completely describe the motion of the system. In fact, it is the sum of the possible ways each mass can move independently of other masses. The translation of a mass along the three axes, X, Y and Z, and the three rotations about each of these axes constitute possible ways of motion of a mass. Many times, many of these six motions of a mass are restricted and a mass can have one or two degrees of freedom, i.e. only translation or translation and rotation of a single mass about any one of the axes.
+
+**Linear and Non-linear Vibrations** : Vibration is said to be linear if the damping force is proportional to velocity, inertia force is proportional to mass, and restoring force is proportional to displacement. If any of this proportionality is not satisfied, the system is said to be non-linear.
+
+**Solving engineering problems** : Analytical methods are usually applied to models of actual systems. We carry out experiments on models if physical systems are not available for testing. While preparing such models, we exclude superfluous details of the system but include all essential and important features of the actual system. While doing so, we idealize and approximate important behaviour of the system without affecting much the accuracy in predicting the behaviour. The system model so developed provides ease of application of analytical and experimental techniques. Once a satisfactory model is developed, laws of Physics can be applied which give a set of mathematical equations relating the properties and variables of the system. Such a set of mathematical equations is called mathematical model of the system. Solving the set of equations (or a single mathematical equation) provides expression for the system variable in terms of location and time. We call this as 'solution' of the problem. As an illustration of the concepts described above, see the example given below.
+
+
+
+The first term is the inertia force which is equal to mass multiplied by acceleration and the second term is the spring force given by stiffness of the spring multiplied by its elongation or compression.
+
+The differential equation is a mathematical model of the system.
+
+**General Solution**
+
+The solution to the above differential equation is given by
+
+
+
+A and B are constants that depend the initial conditions, i.e. the displacement and velocity of the mass when we started measuring our time.
+
+These are known as initial conditions.
+
+
is the natural frequency in radians per second and is given by
+
+
+
+**Particular solution obtained from the initial conditions**
+
+Substituting the initial conditions in above expression, we can obtain the values of A and B. Thus if X0 and V0 are the initial displacement and velocity, respectively, given to the mass, the above expression will yield values of A and B as 
and
+
+
+
+Now the expression for x becomes
+
+
+
+And we can obtain the value of x at any time t from this expression.
+
+Thus we have obtained the expressions for natural frequency and position of the head at any given time 't' and the problem stated by the problem statement is solved.