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Gleichungen und Bilder zum Variofoerderer dazu

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Michael Stangl
2026-03-24 16:56:56 +01:00
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# Gleichungen Förderer aufwärts
![Varioförderer aufwärts](bilder/Variofoerderer_aufwaerts.svg)
**violett** ist F wie _Förderer_
**hellblau** ist S wie _Strecke_
**AS** ist _Ausschleus_- Element
**ES** ist _Einschleus_- Element
## Gegeben
$H_0$, $H_1$, $L_1$, $\alpha_F$, $\alpha_S$, $L_{ES}$, $H_{ES}$, $L_{AS}$, $H_{AS}$
## Gesucht
$L_F$, $L_S$, $H_F$, $H_S$
---
## Grundgleichungen ($H_1 > H_0$, Förderrichtung: von $H_0$ nach $H_1$)
**(1) Horizontal:**
$$L_1 = L_{ES} + L_F + L_S + L_{AS}$$
**(2) Vertikal:**
$$H_1 - H_0 = H_F - H_S - H_{ES} - H_{AS}$$
**(3) Neigung F:**
$$\tan(\alpha_F) = \frac{H_F}{L_F} \quad \Rightarrow \quad H_F = L_F \cdot \tan(\alpha_F)$$
**(4) Neigung S:**
$$\tan(\alpha_S) = \frac{H_S}{L_S} \quad \Rightarrow \quad H_S = L_S \cdot \tan(\alpha_S)$$
---
## Lösung (Einsetzen von (3),(4) in (1),(2))
**(I)**
$$L_F + L_S = L_1 - L_{ES} - L_{AS}$$
**(II)**
$$L_F \cdot \tan(\alpha_F) - L_S \cdot \tan(\alpha_S) = (H_1 - H_0) + H_{ES} + H_{AS}$$
---
## Ergebnis
$$L_F = \frac{(H_1 - H_0 + H_{ES} + H_{AS}) + (L_1 - L_{ES} - L_{AS}) \cdot \tan(\alpha_S)}{\tan(\alpha_F) + \tan(\alpha_S)}$$
$$L_S = (L_1 - L_{ES} - L_{AS}) - L_F$$
$$H_F = L_F \cdot \tan(\alpha_F)$$
$$H_S = L_S \cdot \tan(\alpha_S)$$
# Gleichungen Förderer abwärts
![Varioförderer abwärts](bilder/Variofoerderer_abwaerts.svg)
## Gegeben
$H_0$, $H_1$, $L_1$, $\alpha_F$, $\alpha_S$, $L_{ES}$, $H_{ES}$, $L_{AS}$, $H_{AS}$
## Gesucht
$L_F$, $L_S$, $H_F$, $H_S$
---
## Grundgleichungen ($H_0 > H_1$, Förderrichtung: von $H_0$ nach $H_1$)
**(1) Horizontal:**
$$L_1 = L_{ES} + L_F + L_S + L_{AS}$$
**(2) Vertikal:**
$$H_0 - H_1 = H_{ES} + H_F + H_S + H_{AS}$$
**(3) Neigung F:**
$$\tan(\alpha_F) = \frac{H_F}{L_F} \quad \Rightarrow \quad H_F = L_F \cdot \tan(\alpha_F)$$
**(4) Neigung S:**
$$\tan(\alpha_S) = \frac{H_S}{L_S} \quad \Rightarrow \quad H_S = L_S \cdot \tan(\alpha_S)$$
---
## Lösung (Einsetzen von (3),(4) in (1),(2))
**(I)**
$$L_F + L_S = L_1 - L_{ES} - L_{AS}$$
**(II)**
$$L_F \cdot \tan(\alpha_F) + L_S \cdot \tan(\alpha_S) = (H_0 - H_1) - H_{ES} - H_{AS}$$
---
## Ergebnis
$$L_F = \frac{(H_0 - H_1 - H_{ES} - H_{AS}) - (L_1 - L_{ES} - L_{AS}) \cdot \tan(\alpha_S)}{\tan(\alpha_F) - \tan(\alpha_S)}$$
$$L_S = (L_1 - L_{ES} - L_{AS}) - L_F$$
$$H_F = L_F \cdot \tan(\alpha_F)$$
$$H_S = L_S \cdot \tan(\alpha_S)$$
# Zusammenfassung
Wobei $\alpha_S$ normalerweise immer bei 3° liegt und $\alpha_F$ von 3,6,9,12,15.. 51° läuft.
```python
#!/usr/bin/env python3
"""
Förderer-Berechnung in 2D – Markdown-Ausgabe
=============================================
"""
import math
# ============================================================
# KONSTANTEN [alle in Meter / Grad]
# ============================================================
H0 = 2.0
H1 = 5.0
L1 = 8.0
L_ES = 1.0
H_ES = 0.3
L_AS = 1.0
H_AS = 0.3
ALPHA_S = 3.0
ALPHA_F_LIST = [3, 6, 9, 12, 15, 18, 24, 27, 33, 39, 45, 51]
def berechne(delta_h, l1, l_es, h_es, l_as, h_as, alpha_f_deg, alpha_s_deg):
alpha_f = math.radians(alpha_f_deg)
alpha_s = math.radians(alpha_s_deg)
tan_f = math.tan(alpha_f)
tan_s = math.tan(alpha_s)
l_rest = l1 - l_es - l_as
if abs(tan_f - tan_s) < 1e-12:
return None
l_f = (delta_h - h_es - h_as - l_rest * tan_s) / (tan_f - tan_s)
l_s = l_rest - l_f
h_f = l_f * tan_f
h_s = l_s * tan_s
return {"L_F": l_f, "L_S": l_s, "H_F": h_f, "H_S": h_s}
def validierung(erg):
probleme = []
for key in ("L_F", "L_S", "H_F", "H_S"):
if erg[key] < 0:
probleme.append(f"{key} < 0")
return (len(probleme) == 0, probleme)
def md_tabelle(fall_name, h_anfang, h_ende):
"""Erzeugt eine Markdown-Tabelle. ΔH = |H_Ende - H_Anfang| (immer positiv)."""
delta_h = abs(h_ende - h_anfang)
lines = []
lines.append(f"### {fall_name}")
lines.append(f"H_Anfang = {h_anfang:.3f} m, H_Ende = {h_ende:.3f} m, "
f"ΔH = {delta_h:+.4f} m\n")
lines.append("| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Status |")
lines.append("|--------:|--------:|--------:|--------:|--------:|--------|")
for alpha_f_deg in ALPHA_F_LIST:
erg = berechne(delta_h, L1, L_ES, H_ES, L_AS, H_AS, alpha_f_deg, ALPHA_S)
if erg is None:
lines.append(f"| {alpha_f_deg} | — | — | — | — | ⚠ α_F = α_S |")
continue
gueltig, probleme = validierung(erg)
status = "✓ gültig" if gueltig else f"✗ {', '.join(probleme)}"
lines.append(
f"| {alpha_f_deg} "
f"| {erg['L_F']:.4f} "
f"| {erg['L_S']:.4f} "
f"| {erg['H_F']:.4f} "
f"| {erg['H_S']:.4f} "
f"| {status} |"
)
lines.append("")
return "\n".join(lines)
def main():
md = []
md.append("# Förderer-Berechnung (2D-Modell)\n")
md.append("## Gegebene Werte\n")
md.append("| Parameter | Wert |")
md.append("|-----------|-----:|")
md.append(f"| H₀ | {H0:.3f} m |")
md.append(f"| H₁ | {H1:.3f} m |")
md.append(f"| L₁ | {L1:.3f} m |")
md.append(f"| L_ES | {L_ES:.3f} m |")
md.append(f"| H_ES | {H_ES:.3f} m |")
md.append(f"| L_AS | {L_AS:.3f} m |")
md.append(f"| H_AS | {H_AS:.3f} m |")
md.append(f"| α_S | {ALPHA_S:.1f}° |")
md.append("")
md.append("## Ergebnisse\n")
# Fall 1: aufwärts – von H0 nach H1
md.append(md_tabelle("Aufwärts (H₀ → H₁)", h_anfang=H0, h_ende=H1))
# Fall 2: abwärts – von H1 nach H0, ΔH ebenfalls +3.0
md.append(md_tabelle("Abwärts (H₁ → H₀)", h_anfang=H1, h_ende=H0))
result = "\n".join(md)
with open("/mnt/user-data/outputs/foerderer_ergebnisse.md", "w", encoding="utf-8") as f:
f.write(result)
print(result)
if __name__ == "__main__":
main()
```
# Förderer-Berechnung
## Gegebene Werte
| Parameter | Wert |
|-----------|-----:|
| H₀ | 2.000 m |
| H₁ | 5.000 m |
| L₁ | 8.000 m |
| L_ES | 1.000 m |
| H_ES | 0.300 m |
| L_AS | 1.000 m |
| H_AS | 0.300 m |
| α_S | 3.0° |
## Ergebnisse
### Aufwärts (H₀ → H₁)
H_Anfang = 2.000 m, H_Ende = 5.000 m, ΔH = +3.0000 m
| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Status |
|--------:|--------:|--------:|--------:|--------:|--------|
| 3 | — | — | — | — | ⚠ α_F = α_S |
| 6 | 39.5767 | -33.5767 | 4.1597 | -1.7597 | ✗ L_S < 0, H_S < 0 |
| 9 | 19.6794 | -13.6794 | 3.1169 | -0.7169 | ✗ L_S < 0, H_S < 0 |
| 12 | 13.0226 | -7.0226 | 2.7680 | -0.3680 | ✗ L_S < 0, H_S < 0 |
| 15 | 9.6759 | -3.6759 | 2.5926 | -0.1926 | ✗ L_S < 0, H_S < 0 |
| 18 | 7.6531 | -1.6531 | 2.4866 | -0.0866 | ✗ L_S < 0, H_S < 0 |
| 24 | 5.3092 | 0.6908 | 2.3638 | 0.0362 | ✓ gültig |
| 27 | 4.5624 | 1.4376 | 2.3247 | 0.0753 | ✓ gültig |
| 33 | 3.4934 | 2.5066 | 2.2686 | 0.1314 | ✓ gültig |
| 39 | 2.7537 | 3.2463 | 2.2299 | 0.1701 | ✓ gültig |
| 45 | 2.2009 | 3.7991 | 2.2009 | 0.1991 | ✓ gültig |
| 51 | 1.7637 | 4.2363 | 2.1780 | 0.2220 | ✓ gültig |
### Abwärts (H₁ → H₀)
H_Anfang = 5.000 m, H_Ende = 2.000 m, ΔH = +3.0000 m
| α_F [°] | L_F [m] | L_S [m] | H_F [m] | H_S [m] | Status |
|--------:|--------:|--------:|--------:|--------:|--------|
| 3 | — | — | — | — | ⚠ α_F = α_S |
| 6 | 39.5767 | -33.5767 | 4.1597 | -1.7597 | ✗ L_S < 0, H_S < 0 |
| 9 | 19.6794 | -13.6794 | 3.1169 | -0.7169 | ✗ L_S < 0, H_S < 0 |
| 12 | 13.0226 | -7.0226 | 2.7680 | -0.3680 | ✗ L_S < 0, H_S < 0 |
| 15 | 9.6759 | -3.6759 | 2.5926 | -0.1926 | ✗ L_S < 0, H_S < 0 |
| 18 | 7.6531 | -1.6531 | 2.4866 | -0.0866 | ✗ L_S < 0, H_S < 0 |
| 24 | 5.3092 | 0.6908 | 2.3638 | 0.0362 | ✓ gültig |
| 27 | 4.5624 | 1.4376 | 2.3247 | 0.0753 | ✓ gültig |
| 33 | 3.4934 | 2.5066 | 2.2686 | 0.1314 | ✓ gültig |
| 39 | 2.7537 | 3.2463 | 2.2299 | 0.1701 | ✓ gültig |
| 45 | 2.2009 | 3.7991 | 2.2009 | 0.1991 | ✓ gültig |
| 51 | 1.7637 | 4.2363 | 2.1780 | 0.2220 | ✓ gültig |