{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Eigenschaften des Baumes"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# Höhe des Stammes: h in [m]\n",
"L = var('L')\n",
"\n",
"# Durchmesser der Krone: D in [m]\n",
"D = var('D')\n",
"\n",
"# Querschnittfläche der Krone: A in [m^2]\n",
"A = pi*(D/2)^2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Eigenschaften des Windes"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"# Dichte von Luft: rho in [kg/m^3]\n",
"rho = var('rho')\n",
"\n",
"# Geschwindigkeit: v_km_std in [km/h]\n",
"v_km_h = var('v_kmh')\n",
"\n",
"# Geschwindigkeit: v in [m/s]\n",
"v = v_km_h / 3.6"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### angegebene Werte"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"[v_kmh == 50, rho == 1, L == 5, D == 10]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"baeumchen = [v_km_h==50, rho==1, L==5, D==10]\n",
"show(baeumchen)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Kraft F und Drehmoment D des Windes"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"15.2"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"75.8"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Kraft in [kN]\n",
"Kraft_Baum = rho * A * v^2 / 1e3\n",
"show( Kraft_Baum.subs(baeumchen).numerical_approx(digits=3))\n",
"\n",
"# Drehmoment in [kN m]\n",
"Dreh_Baum = Kraft_Baum * L\n",
"show( Dreh_Baum.subs(baeumchen).numerical_approx(digits=3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Kraft F der Slackline (Theorie)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"# Masse des Probanden: M in [kg]\n",
"M = var('M')\n",
"\n",
"# Länge der Slackline: l in [m]\n",
"l = var('l')\n",
"\n",
"# Durchhang der Slackline: d in [m]\n",
"d = var('d')\n",
"\n",
"# Höhe des Aufhängepunktes: h in [m]\n",
"h = var('h')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### angegebene Formel und Werte (Vorsicht: die Einheiten stimmen hier nicht!!)\n",
"### Quelle: https://de.wikipedia.org/wiki/Slackline#Bruchlast"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"[M == 80, l == 10, d == 0.100000000000000, h == 2]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Kraft in kN\n",
"F_SL = M*l/(400*d)\n",
"\n",
"slackline = [M==80, l==10, d==0.1, h==2]\n",
"show(slackline)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Kraft F und Drehmoment D der Slackline"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"20.0"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"40.0"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Kraft in [kN]\n",
"show( F_SL.subs(slackline).numerical_approx(digits=3))\n",
"\n",
"# Drehmoment in [kN m]\n",
"D = F_SL * h\n",
"show( D.subs(slackline).numerical_approx(digits=3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# ... das sollte ungefähr klappen — kleinere Bäume aber besser nicht!"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 8.1",
"language": "",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.15rc1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}