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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Auslegungsrechnung: Lokomotive für Hängeförderbahn\n",
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"\n",
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"Diese Rechnung untersucht, wie schwer eine Lokomotive sein muss, um über Reibschluss eine Zugkraft von 100 N aufzubringen. Zusätzlich wird das nötige Antriebsmoment pro Rolle berechnet. Im Anschluss kann aus der Fahrgeschwindigkeit die Drehzahl der Rollen sowie die benötigte Antriebsleistung ermittelt werden. Erste Annahmen für Batteriekapazität werden getroffen.\n",
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"\n",
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"## Annahmen\n",
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"- Zugkraftbedarf (Anfahren): **100 N**\n",
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"- Anfahrbeschleuniugung: **0.1-0.2 m/s^2**\n",
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"- Zugkraftbedarf (Konstantfahrt): **10 N**\n",
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"\n",
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"- Reibwert Gummi auf Aluminium: variabel, realistisch: **0.3–0.6**\n",
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"- Rollendurchmesser: z. B. **80 mm**\n",
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"- Anzahl angetriebener Rollen: z. B. **2**\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fe11045c",
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"metadata": {},
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"source": [
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"## Mathematische Grundlagen\n",
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"\n",
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"Die Lokomotive bewegt sich auf einem Aluminiumprofil mit zwei 45° geneigten Flanken. Die Zugkraft wird über Reibung zwischen den angetriebenen Gummirädern und dem Profil erzeugt. Die folgenden Gleichungen beschreiben die physikalische Auslegung.\n",
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"\n",
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"---\n",
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"\n",
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"### 1. Gewicht der Lokomotive\n",
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"\n",
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"Aus der Gleichgewichtsbedingung bei Reibung auf 2× 45°-Flanken:\n",
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"\n",
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"$$\n",
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"F_{\\text{N,gesamt}} = 2 \\cdot \\left( \\frac{G}{2} \\cdot \\cos(45^\\circ) \\right) = \\frac{G}{\\sqrt{2}}\n",
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"$$\n",
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"\n",
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"Maximale Reibkraft:\n",
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"\n",
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"$$\n",
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"F_{\\text{Reib}} = \\mu \\cdot F_{\\text{N,gesamt}} = \\mu \\cdot \\frac{G}{\\sqrt{2}}\n",
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"$$\n",
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"\n",
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"Gleichsetzen mit Zugkraft:\n",
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"\n",
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"$$\n",
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"F_{\\text{Zug}} = \\mu \\cdot \\frac{G}{\\sqrt{2}} \\Rightarrow G = \\frac{F_{\\text{Zug}} \\cdot \\sqrt{2}}{\\mu}\n",
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"$$\n",
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"\n",
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"Masse der Lokomotive:\n",
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"\n",
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"$$\n",
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"m = \\frac{G}{g} = \\frac{F_{\\text{Zug}} \\cdot \\sqrt{2}}{\\mu \\cdot g}\n",
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"$$\n",
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"\n",
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"---\n",
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"\n",
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"### 2. Drehmoment an den Antriebsrollen\n",
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"\n",
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"Bei gleichmäßiger Verteilung auf $n_{\\text{Rollen}}$ Rollen mit Durchmesser $d$:\n",
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"\n",
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"$$\n",
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"F_{\\text{Rolle}} = \\frac{F_{\\text{Zug}}}{n_{\\text{Rollen}}}\n",
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"$$\n",
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"\n",
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"$$\n",
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"M = F_{\\text{Rolle}} \\cdot \\frac{d}{2}\n",
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"$$\n",
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"\n",
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"Gesamtmoment:\n",
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"\n",
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"$$\n",
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"M_{\\text{gesamt}} = n_{\\text{Rollen}} \\cdot M = F_{\\text{Zug}} \\cdot \\frac{d}{2}\n",
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"$$\n",
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"\n",
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"---\n",
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"\n",
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"### 3. Drehzahl der Rollen\n",
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"\n",
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"Bei Zielgeschwindigkeit $v$ und Rollendurchmesser $d$:\n",
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"\n",
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"$$\n",
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"n = \\frac{v}{\\pi \\cdot d} \\cdot 60\n",
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"$$\n",
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"\n",
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"---\n",
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"\n",
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"### 4. Leistung am Antrieb\n",
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"\n",
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"Die mechanische Leistung berechnet sich zu:\n",
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"\n",
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"$$\n",
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"P = M_{\\text{gesamt}} \\cdot \\omega = M_{\\text{gesamt}} \\cdot \\left( \\frac{2\\pi \\cdot n}{60} \\right)\n",
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"$$\n",
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"\n",
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"---\n",
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"\n",
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"### 5. Batteriekapazität (theoretisch)\n",
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"\n",
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"Bei Betriebsdauer $t$ (in Stunden) und Spannung $U$:\n",
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"\n",
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"$$\n",
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"Q_{\\text{Wh}} = P \\cdot t\n",
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"$$\n",
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"\n",
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"$$\n",
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"Q_{\\text{Ah}} = \\frac{Q_{\\text{Wh}}}{U}\n",
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"$$\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 189,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"\n",
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"# Konstanten\n",
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"g = 9.81 # Erdbeschleunigung in m/s²\n",
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"\n",
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"# Eingabeparameter\n",
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"f_zug_anfaren = 100 # Zugkraftbedarf max. in N\n",
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"f_zug_konstant = 10 # Zugkraftbedarf konst. in N\n",
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"fahrgeschwindigkeit_ms = 1.3 # Fahrgeschwindigkeit in m/s\n",
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"mu_min = 0.3 # minimaler Reibwert\n",
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"mu_max = 0.7 # maximaler Reibwert\n",
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"rollendurchmesser_mm = 50 # Rollendurchmesser in mm\n",
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"anzahl_rollen = 2 # Anzahl angetriebener Rollen\n",
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"\n",
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"\n",
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"# Zusätzliche Parameter für Plots\n",
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"betriebszeit_min = 1 # Minimale Betriebszeit in Stunden für Batterieplot\n",
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"betriebszeit_max = 8 # Maximale Betriebszeit in Stunden für Batterieplot\n",
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"betriebsspannung_v = 24 # Angenommene Betriebsspannung in Volt\n",
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"\n",
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"def berechne_lokgewicht(f_zug, mu_eff):\n",
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" \"\"\"Berechnet das notwendige Lokgewicht für eine gegebene effektive Reibung\"\"\"\n",
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" G = (f_zug * np.sqrt(2)) / mu_eff # Gewichtskraft in N, sqrt(2) aufgrund der 45° geneigten Lauffläche der Rollen\n",
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" m = G / g # Masse in kg\n",
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" return m\n",
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"\n",
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"def berechne_drehmoment(f_zug, rollendurchmesser_mm, anzahl_rollen):\n",
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" \"\"\"Berechnet das Drehmoment pro Rolle in Nm\"\"\"\n",
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" r = rollendurchmesser_mm / 2000 # Radius in Meter (Durchmesser / 2 / 1000)\n",
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" f_pro_rolle = f_zug / anzahl_rollen\n",
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" M_pro_rolle = f_pro_rolle * r\n",
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" return M_pro_rolle\n",
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"\n",
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"def berechne_drehzahl(fahrgeschwindigkeit_ms, rollendurchmesser_mm):\n",
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" \"\"\"Berechnet die Drehzahl der Rollen in Umdrehungen pro Minute (RPM).\"\"\"\n",
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" rollenradius_m = (rollendurchmesser_mm / 2) / 1000 # Radius in Metern\n",
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" drehzahl_rad_s = fahrgeschwindigkeit_ms / rollenradius_m\n",
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" drehzahl_rpm = drehzahl_rad_s * (60 / (2 * np.pi))\n",
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" return drehzahl_rpm\n",
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"\n",
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"\n",
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"def berechne_leistung(drehmoment_pro_rolle, drehzahl_rpm, anzahl_rollen):\n",
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" \"\"\"Berechnet die benötigte Antriebsleistung.\"\"\"\n",
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" drehzahl_rad_s = drehzahl_rpm * (2 * np.pi / 60)\n",
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" gesamt_drehmoment = drehmoment_pro_rolle * anzahl_rollen\n",
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" leistung_watt = gesamt_drehmoment * drehzahl_rad_s\n",
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" return leistung_watt\n",
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"\n",
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"def berechne_batteriekapazitaet(leistung_watt, betriebszeit_h, betriebsspannung_v, sicherheitsfaktor=1.2):\n",
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" \"\"\"Berechnet die benötigte Batteriekapazität in Amperestunden (Ah).\"\"\"\n",
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" energie_wh = leistung_watt * betriebszeit_h\n",
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" kapazitaet_ah = (energie_wh / betriebsspannung_v) * sicherheitsfaktor\n",
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" return kapazitaet_ah\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Plot: Lokgewicht über Reibwert\n",
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"Dieser Plot zeigt, wie stark das notwendige Lokgewicht vom angenommenen Reibwert abhängt. Reibwerte im oberen Bereich sollten angestrebt werden."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 190,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
|
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"text/plain": [
|
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"<Figure size 500x300 with 1 Axes>"
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"# Beispielwerte\n",
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"f_zug_anfaren = 100 # N\n",
|
||||
"mu_werte = np.linspace(mu_min, mu_max, 50)\n",
|
||||
"lokgewichte = [berechne_lokgewicht(f_zug_anfaren, mu) for mu in mu_werte]\n",
|
||||
"\n",
|
||||
"# Plot\n",
|
||||
"plt.figure(figsize=(5, 3))\n",
|
||||
"plt.plot(mu_werte, lokgewichte, label='Benötigtes Lokgewicht', color='blue')\n",
|
||||
"plt.xlabel(\"Effektiver Reibwert (μ)\")\n",
|
||||
"plt.ylabel(\"Erforderliches Lokgewicht (kg)\")\n",
|
||||
"plt.title(\"Einfluss des Reibwerts auf das Lokgewicht für 100 N Zugkraft\")\n",
|
||||
"plt.grid(True)\n",
|
||||
"plt.legend()\n",
|
||||
"plt.tight_layout()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Beispielrechnung mit festen Parametern"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 191,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"=== Typischer Mischbetrieb (95 % Konstantfahrt, 5 % Anfahren) ===\n",
|
||||
"Mittlere Antriebsleistung: 18.85 W\n",
|
||||
"Benötigtes Drehmoment im Anfahren: 1.25 Nm\n",
|
||||
"Benötigtes Drehmoment in Konstanntfahrt: 0.12 Nm\n",
|
||||
"Energieverbrauch pro Meter: 0.004 Wh/m\n",
|
||||
"Benötigte Masse (aus Spitzenlast): 28.83 kg\n",
|
||||
"Drehzahl der Rollen: 496.6 RPM\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Einheitlicher Reibwert\n",
|
||||
"mu = 0.5\n",
|
||||
"\n",
|
||||
"# Beschleunigungsbedarf\n",
|
||||
"a = 0.1 # m/s^2\n",
|
||||
"\n",
|
||||
"# Anteil der Betriebsmodi\n",
|
||||
"anteil_anfahren = 0.05\n",
|
||||
"anteil_konstant = 0.95\n",
|
||||
"\n",
|
||||
"# Berechnungen\n",
|
||||
"gewicht_anfahren = berechne_lokgewicht(f_zug_anfaren, mu)\n",
|
||||
"\n",
|
||||
"# Wir verwenden dieselbe Drehzahl für beide Fälle\n",
|
||||
"drehzahl_rpm = berechne_drehzahl(fahrgeschwindigkeit_ms, rollendurchmesser_mm)\n",
|
||||
"\n",
|
||||
"# Drehmomente und Leistungen\n",
|
||||
"drehmoment_anfahren = berechne_drehmoment(f_zug_anfaren, rollendurchmesser_mm, anzahl_rollen)\n",
|
||||
"leistung_anfahren = berechne_leistung(drehmoment_anfahren, drehzahl_rpm, anzahl_rollen)\n",
|
||||
"\n",
|
||||
"drehmoment_konstant = berechne_drehmoment(f_zug_konstant, rollendurchmesser_mm, anzahl_rollen)\n",
|
||||
"leistung_konstant = berechne_leistung(drehmoment_konstant, drehzahl_rpm, anzahl_rollen)\n",
|
||||
"\n",
|
||||
"# Gemittelte Leistung\n",
|
||||
"leistung_mittel = anteil_anfahren * leistung_anfahren + anteil_konstant * leistung_konstant\n",
|
||||
"\n",
|
||||
"# Energieverbrauch pro Meter (Wh/m)\n",
|
||||
"energieverbrauch_wh_pro_m = leistung_mittel / fahrgeschwindigkeit_ms / 3600 # (W / m/s) * (1/3600) = Wh/m\n",
|
||||
"\n",
|
||||
"# Batteriebedarf für typische Betriebszeiten\n",
|
||||
"betriebszeiten = np.linspace(betriebszeit_min, betriebszeit_max, 100)\n",
|
||||
"kapazitaet_ah_mittel = [berechne_batteriekapazitaet(leistung_mittel, t, betriebsspannung_v) for t in betriebszeiten]\n",
|
||||
"\n",
|
||||
"# Ausgabe\n",
|
||||
"print(\"=== Typischer Mischbetrieb (95 % Konstantfahrt, 5 % Anfahren) ===\")\n",
|
||||
"print(f\"Mittlere Antriebsleistung: {leistung_mittel:.2f} W\")\n",
|
||||
"print(f\"Benötigtes Drehmoment im Anfahren: {drehmoment_anfahren:.2f} Nm\")\n",
|
||||
"print(f\"Benötigtes Drehmoment in Konstanntfahrt: {drehmoment_konstant:.2f} Nm\")\n",
|
||||
"print(f\"Energieverbrauch pro Meter: {energieverbrauch_wh_pro_m:.3f} Wh/m\")\n",
|
||||
"print(f\"Benötigte Masse (aus Spitzenlast): {gewicht_anfahren:.2f} kg\")\n",
|
||||
"print(f\"Drehzahl der Rollen: {drehzahl_rpm:.1f} RPM\")\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 192,
|
||||
"id": "d894f650",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1000x500 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"# Parameter\n",
|
||||
"m = gewicht_anfahren # Masse in kg\n",
|
||||
"a = 0.1 # Beschleunigung in m/s²\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# Abgeleitete Größen\n",
|
||||
"t_beschleunigung = fahrgeschwindigkeit_ms / a # Zeit bis Zielgeschwindigkeit erreicht ist\n",
|
||||
"F_beschleunigung = m * a # Trägheitskraft in N\n",
|
||||
"\n",
|
||||
"# Zeitvektor\n",
|
||||
"t = np.linspace(0, t_beschleunigung + 2, 1000)\n",
|
||||
"\n",
|
||||
"# Geschwindigkeit über die Zeit\n",
|
||||
"v = np.where(t <= t_beschleunigung, a * t, fahrgeschwindigkeit_ms)\n",
|
||||
"\n",
|
||||
"# Momentane Leistung\n",
|
||||
"P_momentan = np.where(\n",
|
||||
" t <= t_beschleunigung,\n",
|
||||
" (F_beschleunigung + f_zug_konstant) * v,\n",
|
||||
" f_zug_konstant * fahrgeschwindigkeit_ms\n",
|
||||
")\n",
|
||||
"\n",
|
||||
"# Plot\n",
|
||||
"plt.figure(figsize=(10, 5))\n",
|
||||
"plt.plot(t, P_momentan, label=\"Momentanleistung [W]\", color=\"crimson\")\n",
|
||||
"plt.axvline(t_beschleunigung, linestyle=\"--\", color=\"gray\", label=\"Ende Beschleunigung\")\n",
|
||||
"plt.axhline(f_zug_konstant * fahrgeschwindigkeit_ms, linestyle=\":\", color=\"green\", label=\"Leistung bei Konstantfahrt\")\n",
|
||||
"plt.xlabel(\"Zeit [s]\")\n",
|
||||
"plt.ylabel(\"Leistung [W]\")\n",
|
||||
"plt.title(\"Momentanleistung während Anfahren und Konstantfahrt\")\n",
|
||||
"plt.grid(True)\n",
|
||||
"plt.legend()\n",
|
||||
"plt.tight_layout()\n",
|
||||
"plt.show()\n",
|
||||
"\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": ".venv",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.13.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Reference in new issue
Block a user