{
"cells": [
{
"cell_type": "markdown",
"id": "3a301b4f",
"metadata": {},
"source": [
"# Damped Harmonic Motion"
]
},
{
"cell_type": "markdown",
"id": "c36c2f24",
"metadata": {},
"source": [
"This notebook examines a model of a suspension mechanism, using the equation for damped harmonic motion, written in standard engineering notation."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "de025030",
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "9caddf19",
"metadata": {},
"outputs": [],
"source": [
"using SymPy"
]
},
{
"cell_type": "markdown",
"id": "3263b7e9",
"metadata": {},
"source": [
"### 1. defining the ordinary differential equation"
]
},
{
"cell_type": "markdown",
"id": "bd1f53f9",
"metadata": {},
"source": [
"The differential equation for damped harmonic motion is\n",
"\n",
"$$\n",
" \\frac{d^2}{d~\\!t^2} y(t) + 2 \\zeta \\omega \\frac{d}{d~\\!t} y(t) + \\omega^2 y(t) = 0.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "a9272bcf",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(t, omega, zeta)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t, omega, zeta = Sym(\"t, omega, zeta\")"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e1fe5f3f",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y$"
],
"text/plain": [
"y"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y = SymFunction(\"y\")"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "818e60ca",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - \\omega^{2} y{\\left(t \\right)} - 2 \\omega \\zeta \\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d 2 d \n",
"---(y(t)) = - omega *y(t) - 2*omega*zeta*--(y(t))\n",
" 2 dt \n",
"dt "
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ode = Eq(diff(y(t),t,2), - 2*zeta*omega*diff(y(t),t) - omega^2*y(t))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "809ae66e",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = C_{1} e^{\\omega t \\left(- \\zeta + \\sqrt{\\zeta^{2} - 1}\\right)} + C_{2} e^{- \\omega t \\left(\\zeta + \\sqrt{\\zeta^{2} - 1}\\right)}$"
],
"text/plain": [
" / ___________\\ / ________\n",
" | / 2 | | / 2 \n",
" omega*t*\\-zeta + \\/ zeta - 1 / -omega*t*\\zeta + \\/ zeta -\n",
"y(t) = C1*e + C2*e \n",
"\n",
"___\\\n",
" |\n",
" 1 /\n",
" "
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol = dsolve(ode)"
]
},
{
"cell_type": "markdown",
"id": "575e6d42",
"metadata": {},
"source": [
"Let us provide initial conditions:\n",
"\n",
"* initial displacement is at 1: y(0) = 1\n",
"\n",
"* initial velocity is zero: y'(0) = 0."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "f528f37b",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
"d \n",
"--(y(t))\n",
"dt "
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dy = diff(y(t),t)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "1df0516f",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\left. \\frac{d}{d t} y{\\left(t \\right)} \\right|_{\\substack{ t=0 }}$"
],
"text/plain": [
"/d \\| \n",
"|--(y(t))|| \n",
"\\dt /|t=0"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dy0 = subs(dy,t=>0)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "3f13de68",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{equation*}\\begin{cases}\\left. \\frac{d}{d t} y{\\left(t \\right)} \\right|_{\\substack{ t=0 }} & \\text{=>} &0\\\\y{\\left(0 \\right)} & \\text{=>} &1\\\\\\end{cases}\\end{equation*}"
],
"text/plain": [
"Dict{Sym{PyCall.PyObject}, Int64} with 2 entries:\n",
" Subs(Derivative(y(t), t), t, 0) => 0\n",
" y(0) => 1"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"initconds = Dict(y(0)=>1, dy0=>0)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "704f3b47",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = \\left(- \\frac{\\zeta}{2 \\sqrt{\\zeta^{2} - 1}} + \\frac{1}{2}\\right) e^{- \\omega t \\left(\\zeta + \\sqrt{\\zeta^{2} - 1}\\right)} + \\left(\\frac{\\zeta}{2 \\sqrt{\\zeta^{2} - 1}} + \\frac{1}{2}\\right) e^{\\omega t \\left(- \\zeta + \\sqrt{\\zeta^{2} - 1}\\right)}$"
],
"text/plain": [
" / ___________\\ \n",
" | / 2 | \n",
" / zeta 1\\ -omega*t*\\zeta + \\/ zeta - 1 / / zet\n",
"y(t) = |- ---------------- + -|*e + |---------\n",
" | ___________ 2| | ____\n",
" | / 2 | | / \n",
" \\ 2*\\/ zeta - 1 / \\2*\\/ zet\n",
"\n",
" / ___________\\\n",
" | / 2 |\n",
"a 1\\ omega*t*\\-zeta + \\/ zeta - 1 /\n",
"------- + -|*e \n",
"_______ 2| \n",
" 2 | \n",
"a - 1 / "
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol0 = dsolve(ode, ics=initconds)"
]
},
{
"cell_type": "markdown",
"id": "8fba7529",
"metadata": {},
"source": [
"Consider the general form of the solution, we observe that $\\zeta^2 - 1$ is a discriminant."
]
},
{
"cell_type": "markdown",
"id": "caf3854d",
"metadata": {},
"source": [
"### 2. The First Case: zeta < 1"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "8ab75d8b",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - y{\\left(t \\right)} - 1.0 \\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"---(y(t)) = -y(t) - --(y(t))\n",
" 2 dt \n",
"dt "
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ode1 = subs(ode, zeta=>1/2, omega=>1)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "76f50ff6",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = \\left(\\frac{\\sqrt{3} \\sin{\\left(\\frac{\\sqrt{3} t}{2} \\right)}}{3} + \\cos{\\left(\\frac{\\sqrt{3} t}{2} \\right)}\\right) e^{- \\frac{t}{2}}$"
],
"text/plain": [
" / / ___ \\ \\ \n",
" | ___ |\\/ 3 *t| | -t \n",
" |\\/ 3 *sin|-------| / ___ \\| ---\n",
" | \\ 2 / |\\/ 3 *t|| 2 \n",
"y(t) = |------------------ + cos|-------||*e \n",
" \\ 3 \\ 2 // "
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol1 = dsolve(ode1, ics=initconds)"
]
},
{
"cell_type": "markdown",
"id": "1d53003a",
"metadata": {},
"source": [
"To plot the solution, make a function of the right hand side."
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "590a60c6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"sol1fun (generic function with 1 method)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol1fun(x) = subs(sol1.rhs(), t=>x)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "c6fa7940",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(0:0.01:10, sol1fun, xlabel=\"t\", label=\"underdamped\")"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "29ebaee2",
"metadata": {},
"outputs": [],
"source": [
"# savefig(\"figdampedunder.png\")"
]
},
{
"cell_type": "markdown",
"id": "008a7413",
"metadata": {},
"source": [
"### 3. The Second Case: zeta = 1"
]
},
{
"cell_type": "markdown",
"id": "73e8e66c",
"metadata": {},
"source": [
"When $\\zeta = 1$, the characteristic equation has one double root."
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "e8af693f",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - y{\\left(t \\right)} - 2 \\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"---(y(t)) = -y(t) - 2*--(y(t))\n",
" 2 dt \n",
"dt "
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ode2 = subs(ode, zeta=>1, omega=>1)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "794bb383",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = \\left(t + 1\\right) e^{- t}$"
],
"text/plain": [
" -t\n",
"y(t) = (t + 1)*e "
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol2 = dsolve(ode2, ics=initconds)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "cf82378e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"sol2fun (generic function with 1 method)"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol2fun(x) = subs(sol2.rhs(), t=>x)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "6a9ba11c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(0:0.01:10, sol2fun, xlabel=\"t\", label=\"critically damped\")"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "73da6f6c",
"metadata": {},
"outputs": [],
"source": [
"# savefig(\"figdampedcritical.png\")"
]
},
{
"cell_type": "markdown",
"id": "92531238",
"metadata": {},
"source": [
"### 4. The Third Case: zeta > 1"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "1cc5cd7b",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - y{\\left(t \\right)} - 3.0 \\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"---(y(t)) = -y(t) - 3.0*--(y(t))\n",
" 2 dt \n",
"dt "
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ode3 = subs(ode, zeta=>3/2, omega=>1)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "5144b307",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = - 0.170820393249937 e^{- 2.61803398874989 t} + 1.17082039324994 e^{- 0.381966011250105 t}$"
],
"text/plain": [
" -2.61803398874989*t -0.3819660\n",
"y(t) = - 0.170820393249937*e + 1.17082039324994*e \n",
"\n",
"11250105*t\n",
" "
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol3 = dsolve(ode3, ics=initconds)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "394e66c9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"sol3fun (generic function with 1 method)"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol3fun(x) = subs(sol3.rhs(), t=>x)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "cb8a1d1f",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(0:0.01:10, sol3fun, xlabel=\"t\", label=\"overdamped\")"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "6e8a7f70",
"metadata": {},
"outputs": [],
"source": [
"# savefig(\"figdampedover.png\")"
]
},
{
"cell_type": "markdown",
"id": "bbe7c2dd",
"metadata": {},
"source": [
"### the role of omega"
]
},
{
"cell_type": "markdown",
"id": "d8e525d0",
"metadata": {},
"source": [
"Let us set the value of $\\omega$ to two and look for $\\zeta = 1$."
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "24fddbbe",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - 4 y{\\left(t \\right)} - 4 \\frac{d}{d t} y{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"---(y(t)) = -4*y(t) - 4*--(y(t))\n",
" 2 dt \n",
"dt "
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ode4 = subs(ode, omega=>2, zeta=>1)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "3cb21620",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$y{\\left(t \\right)} = \\left(2 t + 1\\right) e^{- 2 t}$"
],
"text/plain": [
" -2*t\n",
"y(t) = (2*t + 1)*e "
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol4 = dsolve(ode4, ics=initconds)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "f3ee2cf9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"sol4fun (generic function with 1 method)"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol4fun(x) = subs(sol4.rhs(), t=>x)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "7947955a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(0:0.01:10, sol4fun, xlabel=\"t\", label=\"omega = 2\")"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "ca4a7f10",
"metadata": {},
"outputs": [],
"source": [
"# savefig(\"figdampedomega2.png\")"
]
},
{
"cell_type": "markdown",
"id": "8bd8478b",
"metadata": {},
"source": [
"The value of $\\omega$ determines how fast the deviation disappears."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.12",
"language": "julia",
"name": "julia-1.12"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.12.4"
}
},
"nbformat": 4,
"nbformat_minor": 5
}