2e:null 2f:["$","title",null,{"children":"1. Movement of a Single Spring and other ODE | CD | Observable"}] 34:[] 32:{"id":"93501d53feebf40c","slug":"1-movement-of-a-single-spring-and-other-ode","title":"1. Movement of a Single Spring and other ODE","thumbnail":"b593109057d554f2984ac938a420c7e60b55d83e6b383a59d8ac87be8bcc376b","access_level":"public","create_time":"$D2020-09-20T15:22:17.062Z","update_time":"$D2020-09-22T13:08:41.726Z","view_time":null,"first_public_time":"$D2020-09-21T17:47:41.394Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"30f043e2993c044e","login":"cdham","name":"CD","avatar_url":"https://avatars-next.observableusercontent.com/80589672f9b361e3704b1f4f64afd254600384efc7c7a0ba464bdcbad36ed312","tier":"starter_2024"},"version":381,"publish_version":381,"role":null,"authored":false,"authors":[{"type":"individual","id":"0280e7acabd34fb0","login":"cdham","name":"CD","avatar_url":"https://avatars-next.observableusercontent.com/80589672f9b361e3704b1f4f64afd254600384efc7c7a0ba464bdcbad36ed312","tier":null}],"members":[{"type":"individual","id":"0280e7acabd34fb0","login":"cdham","name":"CD","avatar_url":"https://avatars-next.observableusercontent.com/80589672f9b361e3704b1f4f64afd254600384efc7c7a0ba464bdcbad36ed312","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"1. Movement of a Single Spring and other ODE","cells":[{"id":0,"value":"md`# 1. Movement of a Single Spring and other ODE\n## Introduction\nApplication of Runge Kutta with \n\n - x = position of the block\n - v = x' = velocity of the block\n - m = mass of the block\n - R = rest length of the spring\n - k = spring stiffness\n - b = damping constant (friction)\n\n## Implementation\nAssigning parameter: `\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"params = ({ beta: 0.5, k: 5, x0: 2, v0: -3, m: 5, B: 25 });","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":45,"value":"jxg_plot = {\n return html`

`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":47,"value":"draw_first = draw_pic(params)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"function draw_pic(params) {\n let N = 1000;\n let a = 0;\n let b = params.B;\n let sol = rk4(\n F(params.beta, params.k, params.m),\n [params.x0, params.v0],\n 0,\n a,\n b,\n N\n ).map(xy => xy[0]);\n sol.N = N;\n sol.a = a;\n sol.b = b;\n\n let brd = JXG.JSXGraph.initBoard('jxg_plot', {\n boundingbox: [-b / 20, 4, b, -4],\n showCopyright: false,\n showNavigation: false,\n axis: true,\n // Weird way to hide the ticks per\n // https://stackoverflow.com/questions/52068445/\n defaultAxes: {\n y: { ticks: { visible: true, majorHeight: 5 }},\n x: { ticks: { visible: true, majorHeight: 5 } }\n }\n });\n let curve = brd.create(\n 'curve',\n [sol.map((x, i) => ((sol.b - sol.a) * i) / sol.N), sol],\n {\n strokeColor: 'black',\n strokeWidth: 2\n }\n );\n curve.highlight = function() {};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"function F(b, k, m) {\n return ([x, y], t) => [\n y,\n - k / m * x - b / m * y\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":80,"value":"md`# 2. Other Examples\n## Chemical\nApplication of Runge Kutta for concentration of salt x in a home made soap maker. This and all following examples are from [here](http://nm.mathforcollege.com/topics/runge_kutta_4th_method.html) \n\n### Itteration in 2, 4, or 8 steps\nwritten to one array.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":189,"value":"{\n var yn = [[0, 50]]\n var steps = 2\n var dt = 3 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 37.5 - 3.5 *y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":214,"value":"{\n var yn = [[0, 50]]\n var steps = 4\n var dt = 3 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 37.5 - 3.5 *y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":218,"value":"{\n var yn = [[0, 50]]\n var steps = 8\n var dt = 3 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 37.5 - 3.5 *y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":282,"value":"md`Result requires 7 or 8 steps in order to be sufficiently close to the exact solution which is 10.7.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":222,"value":"md`## Civil Engineering\nApplication of Runge Kutta for concentration of bacteria \n\n### Itteration in 2, 4, or 8 steps\nwritten to one array.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"{\n var yn = [[0, 10000000]]\n var steps = 2\n var dt = 7 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], - 0.06 *y[1],], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":230,"value":"{\n var yn = [[0, 10000000]]\n var steps = 4\n var dt = 7 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], - 0.06 *y[1],], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":233,"value":"{\n var yn = [[0, 10000000]]\n var steps = 8\n var dt = 7 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], - 0.06 *y[1],], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":279,"value":"md`Result is sufficiently close to the exact solution after 2 steps.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":245,"value":"md`## Industrial Engineering\nApplication of Runge Kutta for calcualtion of speed of a motor \n\n### Itteration in 2,4, and 8 steps\nwritten to one array.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":248,"value":"{\n var yn = [[0, 0]]\n var steps = 2\n var dt = 0.8 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 1000-3*y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":269,"value":"{\n var yn = [[0, 0]]\n var steps = 4\n var dt = 0.8 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 1000-3*y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":270,"value":"{\n var yn = [[0, 0]]\n var steps = 8\n var dt = 0.8 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0], 1000-3*y[1]], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":264,"value":"md`Result is sufficiently close to the exact solution with 303rad/s after 2 steps.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":285,"value":"md`## Mechanial Engineering\nApplication of Runge Kutta for calcualtion of contraction of a steel shaft. \n\n### Itteration in 2, 4, and 8 steps\nwritten to one array.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":288,"value":"{\n var yn = [[0, 27]]\n var steps = 2\n var dt = 86400 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0],\n -5.33e-6*(-3.69e-6*y[1]**4\n +2.33e-5*y[1]**3\n +1.35e-3*y[1]**2\n +5.42e-2*y[1]\n +5.588)*(y[1]+33)], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":364,"value":"{\n var yn = [[0, 27]]\n var steps = 4\n var dt = 86400 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0],\n -5.33e-6*(-3.69e-6*y[1]**4\n +2.33e-5*y[1]**3\n +1.35e-3*y[1]**2\n +5.42e-2*y[1]\n +5.588)*(y[1]+33)], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":367,"value":"{\n var yn = [[0, 27]]\n var steps = 8\n var dt = 86400 / steps\n for (var i = 0; i < steps; i++) {\n yn.push(odeRK4([], yn[i], (_, y) => [y[0],\n -5.33e-6*(-3.69e-6*y[1]**4\n +2.33e-5*y[1]**3\n +1.35e-3*y[1]**2\n +5.42e-2*y[1]\n +5.588)*(y[1]+33)], dt))\n }\n return yn\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":370,"value":"md`Result is sufficiently close to the exact solution after 4 steps only.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":25,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"import {odeRK4} from '@rreusser/integration'\n","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"JXG = require('jsxgraph')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":208,"value":"import { rk4 } from '@mcmcclur/runge-kutta-for-systems-of-odes'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"d3 = require('d3-selection@2', 'd3-scale@3', 'd3-drag@2', 'd3-format@2')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 33:"$32:body"