352 lines
10 KiB
Markdown
352 lines
10 KiB
Markdown
# Gleichungen Förderer aufwärts
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**violett** ist F wie _Förderer_
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**hellblau** ist S wie _Strecke_
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**AS** ist _Ausschleus_- Element
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**ES** ist _Einschleus_- Element
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## Gegeben
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$H_0$, $H_1$, $L_1$, $\alpha_F$, $\alpha_S$, $L_{ES}$, $H_{ES}$, $L_{AS}$, $H_{AS}$
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## Gesucht
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$L_F$, $L_S$, $H_F$, $H_S$
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---
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## Grundgleichungen ($H_1 > H_0$, Förderrichtung: von $H_0$ nach $H_1$)
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**(1) Horizontal:**
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$$L_1 = L_{ES} + L_F + L_S + L_{AS}$$
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**(2) Vertikal:**
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$$H_1 - H_0 = H_F - H_S - H_{ES} - H_{AS}$$
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**(3) Neigung F:**
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$$\tan(\alpha_F) = \frac{H_F}{L_F} \quad \Rightarrow \quad H_F = L_F \cdot \tan(\alpha_F)$$
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**(4) Neigung S:**
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$$\tan(\alpha_S) = \frac{H_S}{L_S} \quad \Rightarrow \quad H_S = L_S \cdot \tan(\alpha_S)$$
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---
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## Lösung (Einsetzen von (3),(4) in (1),(2))
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**(I)**
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$$L_F + L_S = L_1 - L_{ES} - L_{AS}$$
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**(II)**
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$$L_F \cdot \tan(\alpha_F) - L_S \cdot \tan(\alpha_S) = (H_1 - H_0) + H_{ES} + H_{AS}$$
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---
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## Ergebnis
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$$L_F = \frac{(H_1 - H_0 + H_{ES} + H_{AS}) + (L_1 - L_{ES} - L_{AS}) \cdot \tan(\alpha_S)}{\tan(\alpha_F) + \tan(\alpha_S)}$$
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$$L_S = (L_1 - L_{ES} - L_{AS}) - L_F$$
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$$H_F = L_F \cdot \tan(\alpha_F)$$
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$$H_S = L_S \cdot \tan(\alpha_S)$$
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# Gleichungen Förderer abwärts
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## Gegeben
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$H_0$, $H_1$, $L_1$, $\alpha_F$, $\alpha_S$, $L_{ES}$, $H_{ES}$, $L_{AS}$, $H_{AS}$
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## Gesucht
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$L_F$, $L_S$, $H_F$, $H_S$
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---
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## Grundgleichungen ($H_0 > H_1$, Förderrichtung: von $H_0$ nach $H_1$)
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**(1) Horizontal:**
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$$L_1 = L_{ES} + L_F + L_S + L_{AS}$$
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**(2) Vertikal:**
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$$H_0 - H_1 = H_{ES} + H_F + H_S + H_{AS}$$
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**(3) Neigung F:**
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$$\tan(\alpha_F) = \frac{H_F}{L_F} \quad \Rightarrow \quad H_F = L_F \cdot \tan(\alpha_F)$$
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**(4) Neigung S:**
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$$\tan(\alpha_S) = \frac{H_S}{L_S} \quad \Rightarrow \quad H_S = L_S \cdot \tan(\alpha_S)$$
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---
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## Lösung (Einsetzen von (3),(4) in (1),(2))
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**(I)**
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$$L_F + L_S = L_1 - L_{ES} - L_{AS}$$
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**(II)**
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$$L_F \cdot \tan(\alpha_F) + L_S \cdot \tan(\alpha_S) = (H_0 - H_1) - H_{ES} - H_{AS}$$
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---
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## Ergebnis
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$$L_F = \frac{(H_0 - H_1 - H_{ES} - H_{AS}) - (L_1 - L_{ES} - L_{AS}) \cdot \tan(\alpha_S)}{\tan(\alpha_F) - \tan(\alpha_S)}$$
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$$L_S = (L_1 - L_{ES} - L_{AS}) - L_F$$
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$$H_F = L_F \cdot \tan(\alpha_F)$$
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$$H_S = L_S \cdot \tan(\alpha_S)$$
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# Zusammenfassung
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Wobei $\alpha_S$ normalerweise immer bei 3° liegt und $\alpha_F$ von 3,6,9,12,15.. 51° läuft.
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```python
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#!/usr/bin/env python3
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"""
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Förderer-Berechnung in 2D – beide Fälle
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====================================================
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Aufwärts (H1 > H0):
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ΔH = H1 - H0
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L_F = (ΔH + H_ES + H_AS + L_rest·tan(α_S)) / (tan(α_F) + tan(α_S))
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Probe: H_F - H_S - H_ES - H_AS = ΔH
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Abwärts (H0 > H1):
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ΔH = H0 - H1
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L_F = (ΔH - H_ES - H_AS - L_rest·tan(α_S)) / (tan(α_F) - tan(α_S))
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Probe: H_ES + H_F + H_S + H_AS = ΔH
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"""
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import math
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# ============================================================
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# KONSTANTEN [alle in Meter / Grad]
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# ============================================================
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H0 = 2.0
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H1 = 5.0
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L1 = 8.0
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L_ES = 1.0
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H_ES = 0.3
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L_AS = 1.0
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H_AS = 0.3
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ALPHA_S = 3.0
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ALPHA_F_LIST = [3, 6, 9, 12, 15, 18, 24, 27, 33, 39, 45, 51]
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def berechne(h0, h1, l1, l_es, h_es, l_as, h_as, alpha_f_deg, alpha_s_deg):
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"""
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Berechnet L_F, L_S, H_F, H_S.
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Unterscheidet automatisch aufwärts/abwärts anhand h0 vs h1.
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"""
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alpha_f = math.radians(alpha_f_deg)
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alpha_s = math.radians(alpha_s_deg)
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tan_f = math.tan(alpha_f)
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tan_s = math.tan(alpha_s)
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l_rest = l1 - l_es - l_as
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if h1 >= h0:
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# === AUFWÄRTS ===
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# (2): H1 - H0 = H_F - H_S - H_ES - H_AS
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delta_h = h1 - h0
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nenner = tan_f + tan_s
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if abs(nenner) < 1e-12:
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return None
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zaehler = (delta_h + h_es + h_as) + l_rest * tan_s
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fall = "aufwärts"
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else:
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# === ABWÄRTS ===
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# (2): H0 - H1 = H_ES + H_F + H_S + H_AS
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delta_h = h0 - h1
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nenner = tan_f - tan_s
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if abs(nenner) < 1e-12:
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return None
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zaehler = (delta_h - h_es - h_as) - l_rest * tan_s
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fall = "abwärts"
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l_f = zaehler / nenner
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l_s = l_rest - l_f
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h_f = l_f * tan_f
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h_s = l_s * tan_s
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# Gegenprobe
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if fall == "aufwärts":
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probe = h_f - h_s - h_es - h_as # soll = H1-H0
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else:
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probe = h_es + h_f + h_s + h_as # soll = H0-H1
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return {
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"L_F": l_f, "L_S": l_s,
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"H_F": h_f, "H_S": h_s,
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"fall": fall, "delta_h": delta_h, "probe": probe,
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}
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def validierung(erg):
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probleme = []
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for key in ("L_F", "L_S", "H_F", "H_S"):
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if erg[key] < 0:
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probleme.append(f"{key} < 0")
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# L_F und L_S dürfen auch nicht > L_rest sein (implizit durch L_S < 0)
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return (len(probleme) == 0, probleme)
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def md_tabelle(fall_name, h0_val, h1_val):
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"""Erzeugt Markdown-Tabelle. Fall wird aus h0/h1 abgeleitet."""
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l_rest = L1 - L_ES - L_AS
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delta_h = abs(h1_val - h0_val)
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richtung = "aufwärts" if h1_val >= h0_val else "abwärts"
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lines = []
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lines.append(f"### {fall_name}")
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lines.append(f"H₀ = {h0_val:.3f} m, H₁ = {h1_val:.3f} m → **{richtung}**, "
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f"ΔH = {delta_h:.4f} m, L_rest = {l_rest:.3f} m\n")
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if richtung == "aufwärts":
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lines.append("Formel: L_F = (ΔH + H_ES + H_AS + L_rest·tan α_S) / "
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"(tan α_F **+** tan α_S)")
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lines.append("")
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lines.append("Probe: H_F − H_S − H_ES − H_AS = ΔH\n")
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else:
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lines.append("Formel: L_F = (ΔH − H_ES − H_AS − L_rest·tan α_S) / "
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"(tan α_F **−** tan α_S)")
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lines.append("")
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lines.append("Probe: H_ES + H_F + H_S + H_AS = ΔH\n")
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lines.append("| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Probe | Status |")
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lines.append("|--------:|--------:|--------:|--------:|--------:|------:|--------|")
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for alpha_f_deg in ALPHA_F_LIST:
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erg = berechne(h0_val, h1_val, L1, L_ES, H_ES, L_AS, H_AS,
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alpha_f_deg, ALPHA_S)
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if erg is None:
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lines.append(f"| {alpha_f_deg} | — | — | — | — | — | ⚠ Nenner = 0 |")
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continue
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gueltig, probleme = validierung(erg)
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status = "✓ gültig" if gueltig else f"✗ {', '.join(probleme)}"
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lines.append(
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f"| {alpha_f_deg} "
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f"| {erg['L_F']:.4f} "
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f"| {erg['L_S']:.4f} "
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f"| {erg['H_F']:.4f} "
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f"| {erg['H_S']:.4f} "
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f"| {erg['probe']:.4f} "
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f"| {status} |"
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)
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lines.append("")
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return "\n".join(lines)
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def main():
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md = []
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md.append("# Förderer-Berechnung (2D-Modell)\n")
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md.append("## Gegebene Werte\n")
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md.append("| Parameter | Wert |")
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md.append("|-----------|-----:|")
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md.append(f"| H₀ | {H0:.3f} m |")
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md.append(f"| H₁ | {H1:.3f} m |")
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md.append(f"| L₁ | {L1:.3f} m |")
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md.append(f"| L_ES | {L_ES:.3f} m |")
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md.append(f"| H_ES | {H_ES:.3f} m |")
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md.append(f"| L_AS | {L_AS:.3f} m |")
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md.append(f"| H_AS | {H_AS:.3f} m |")
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md.append(f"| α_S | {ALPHA_S:.1f}° |")
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md.append("")
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md.append("## Ergebnisse\n")
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# Fall 1: Aufwärts – H0=2 → H1=5
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md.append(md_tabelle("Fall 1: Aufwärts", h0_val=H0, h1_val=H1))
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# Fall 2: Abwärts – H0=5 → H1=2 (vertauscht!)
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md.append(md_tabelle("Fall 2: Abwärts", h0_val=H1, h1_val=H0))
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result = "\n".join(md)
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print(result)
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if __name__ == "__main__":
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main()
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```
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# Förderer-Berechnung (2D-Modell)
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## Gegebene Werte
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| Parameter | Wert |
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|-----------|-----:|
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| H₀ | 2.000 m |
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| H₁ | 5.000 m |
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| L₁ | 8.000 m |
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| L_ES | 1.000 m |
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| H_ES | 0.300 m |
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| L_AS | 1.000 m |
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| H_AS | 0.300 m |
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| α_S | 3.0° |
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## Ergebnisse
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### Fall 1: Aufwärts
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H₀ = 2.000 m, H₁ = 5.000 m → **aufwärts**, ΔH = 3.0000 m, L_rest = 6.000 m
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Formel: L_F = (ΔH + H_ES + H_AS + L_rest·tan α_S) / (tan α_F **+** tan α_S)
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Probe: H_F − H_S − H_ES − H_AS = ΔH
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| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Probe | Status |
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|--------:|--------:|--------:|--------:|--------:|------:|--------|
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| 3 | 37.3460 | -31.3460 | 1.9572 | -1.6428 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 6 | 24.8517 | -18.8517 | 2.6120 | -0.9880 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 9 | 18.5702 | -12.5702 | 2.9412 | -0.6588 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 12 | 14.7735 | -8.7735 | 3.1402 | -0.4598 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 15 | 12.2190 | -6.2190 | 3.2741 | -0.3259 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 18 | 10.3741 | -4.3741 | 3.3708 | -0.2292 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 24 | 7.8661 | -1.8661 | 3.5022 | -0.0978 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 27 | 6.9660 | -0.9660 | 3.5494 | -0.0506 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 33 | 5.5776 | 0.4224 | 3.6221 | 0.0221 | 3.0000 | ✓ gültig |
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| 39 | 4.5401 | 1.4599 | 3.6765 | 0.0765 | 3.0000 | ✓ gültig |
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| 45 | 3.7195 | 2.2805 | 3.7195 | 0.1195 | 3.0000 | ✓ gültig |
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| 51 | 3.0408 | 2.9592 | 3.7551 | 0.1551 | 3.0000 | ✓ gültig |
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### Fall 2: Abwärts
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H₀ = 5.000 m, H₁ = 2.000 m → **abwärts**, ΔH = 3.0000 m, L_rest = 6.000 m
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Formel: L_F = (ΔH − H_ES − H_AS − L_rest·tan α_S) / (tan α_F **−** tan α_S)
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Probe: H_ES + H_F + H_S + H_AS = ΔH
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| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Probe | Status |
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|--------:|--------:|--------:|--------:|--------:|------:|--------|
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| 3 | — | — | — | — | — | ⚠ Nenner = 0 |
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| 6 | 39.5767 | -33.5767 | 4.1597 | -1.7597 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 9 | 19.6794 | -13.6794 | 3.1169 | -0.7169 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 12 | 13.0226 | -7.0226 | 2.7680 | -0.3680 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 15 | 9.6759 | -3.6759 | 2.5926 | -0.1926 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 18 | 7.6531 | -1.6531 | 2.4866 | -0.0866 | 3.0000 | ✗ L_S < 0, H_S < 0 |
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| 24 | 5.3092 | 0.6908 | 2.3638 | 0.0362 | 3.0000 | ✓ gültig |
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| 27 | 4.5624 | 1.4376 | 2.3247 | 0.0753 | 3.0000 | ✓ gültig |
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| 33 | 3.4934 | 2.5066 | 2.2686 | 0.1314 | 3.0000 | ✓ gültig |
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| 39 | 2.7537 | 3.2463 | 2.2299 | 0.1701 | 3.0000 | ✓ gültig |
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| 45 | 2.2009 | 3.7991 | 2.2009 | 0.1991 | 3.0000 | ✓ gültig |
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| 51 | 1.7637 | 4.2363 | 2.1780 | 0.2220 | 3.0000 | ✓ gültig |
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