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Förderer und Gefällestrecke um Ein und Auslauf ergänzt

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Michael Stangl
2024-05-27 17:08:45 +02:00
parent de9d6f2379
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Doku/Berechnung-foerderer-Längen_extended.md
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![Alt text](bilder/SkizzeFördererStrecke_mitEinAuslauf.svg "Skizze von Förder und Strecke")
Bekannt:
<!--![Alt text](bilder/Formel_bekannt.png)-->
$$
L_1, H_0, H_1, \alpha_F, \alpha_S,\\
H_{SEA}, L_{SEA}, H_{FEA}, L_{FEA}
$$
Gesucht:
<!--![Alt text](bilder/Formel_gesucht.png)-->
$$
L_F, L_S, \Delta H_F, \Delta H_S, (\rightarrow H_{Max})
$$
Beziehungen:
<!--![Alt text](bilder/Formel_beziehungen_1_8.png)-->
<!--![Alt text](bilder/Formel_beziehungen_9_13.png)-->
$$
\frac{\Delta{H_F} - H_{FEA}}{L_{F} - L_{FEA}} = tan(\alpha_F) [1]
$$
$$
\Delta{H_F} = tan(\alpha_F) * (L_{F} - L_{FEA}) + H_{FEA}[1.2]
$$
$$
\frac{\Delta{H_S}-H_{SEA}}{L_{S}-L_{SEA}} = tan(\alpha_S) [2]
$$
$$
\Delta{H_S} = tan(\alpha_S) * (L_{S}-L_{SEA}) + H_{SEA} [2.2]
$$
$$
H_0 + \Delta{H_F} = H_1 + \Delta{H_S} [4]
$$
$$
\Delta{H_F} = H_1 - H_0 + \Delta{H_S} [4.2]
$$
$$
L_1 = L_F + L_S [5]
$$
$$
L_1 = L_F * x_{Verhältnis} [6]
$$
$L_1$ aus [6] in [5]:
$$
L_S = L_F * (x_{Verhältnis} - 1) [7]
$$
$\Delta{H_S}$ aus [2.2] in [4.2]:
$$
\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * (L_{S}-L_{SEA}) + H_{SEA}[9]
$$
$L_S$ aus [7] in [9]:
$$
\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * [L_F * (x_{Verhältnis} - 1) - L_{SEA})] + H_{FEA}[10]
$$
$L_F$ aus [6] in [10]:
$$
\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * \Bigl[\frac{L_1}{x_{Verhältnis}} * (x_{Verhältnis} - 1) - L_{SEA})\Bigr] + H_{FEA}[10]
$$
auflösen nach x:
$$
\Delta{H_F} - H_{FEA} + H_0 - H_1 + tan(\alpha_S) * L_{SEA} = tan(\alpha_S) * L_F * (x_{Verhältnis} - 1)
$$
$$
tan(\alpha_F) = x_{Verhältnis}*\Bigl[\frac{H_1 - H_0}{L_1} + tan(\alpha_S)\Bigr] - tan(\alpha_S)
\\
tan(\alpha_F) + tan(\alpha_S) = x_{Verhältnis}*\Bigl[\frac{H_1 - H_0}{L_1} + tan(\alpha_S)\Bigr]
$$
$$
\frac{tan(\alpha_F) + tan(\alpha_S)}{\frac{H_1 - H_0}{L_1} + tan(\alpha_S)} = x_{Verhältnis}
$$
<!--![Alt text](bilder/Formel_aufloesen_x.png)-->
Dann errechnet sich leicht
- L_F aus Beziehung [6]
- \Delta{H_F} aus [1]
- und L_S aus [5]
Bei der Annahme, dass $L_F$ immer ein "Teil", also kleiner als $L_1$ sein soll, erschliesst sich, dass nur x (grösser) gleich 1 gewünschte Ergebnisse liefert. Daraus ergibt sich, dass die folgende geometrischen Beziehung die gelten muß:
<!--![Alt text](bilder/Formel_verhaeltnis_ergebnis.png)-->
$$
\frac{tan(\alpha_F) + tan(\alpha_S)}{\frac{H_1 - H_0}{L_1} + tan(\alpha_S)} \geq 1
$$
$$
{tan(\alpha_F) + tan(\alpha_S)} \geq {\frac{H_1 - H_0}{L_1} + tan(\alpha_S)}
$$
$$
{tan(\alpha_F)} \geq {\frac{H_1 - H_0}{L_1}}
$$
Doku/bilder/SkizzeFördererStrecke_mitEinAuslauf.svg
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