84 lines
2.4 KiB
Markdown
84 lines
2.4 KiB
Markdown
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Bekannt:
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$$
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L_1, H_0, H_1, \alpha_F, \alpha_S,\\
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H_{SEA}, L_{SEA}, H_{FEA}, L_{FEA}
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$$
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Gesucht:
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$$
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L_F, L_S, \Delta H_F, \Delta H_S, H_{Max}, x_{Verhältnis}
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$$
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Beziehungen:
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$$
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\frac{\Delta{H_F} - H_{FEA}}{L_{F} - L_{FEA}} = tan(\alpha_F) [1]
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$$
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$$
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\Delta{H_F} = tan(\alpha_F) \cdot (L_{F} - L_{FEA}) + H_{FEA}[1.2]
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$$
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$$
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\frac{\Delta{H_S}-H_{SEA}}{L_{S}-L_{SEA}} = tan(\alpha_S) [2]
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$$
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$$
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\Delta{H_S} = tan(\alpha_S) \cdot (L_{S}-L_{SEA}) + H_{SEA} [2.2]
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$$
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$$
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H_0 + \Delta{H_F} = H_1 + \Delta{H_S} [4]
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$$
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$$
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\Delta{H_F} = H_1 - H_0 + \Delta{H_S} [4.2]
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$$
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$$
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L_1 = L_F + L_S [5]
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$$
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$$
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L_1 = L_F \cdot x_{Verhältnis} [6]
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$$
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$\Delta H_S$ aus [2.2] in [4.2]:
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$$
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\Delta H_F = H_1 - H_0 + (\tan(\alpha_S) \cdot (L_S - L_{SEA}) + H_{SEA}) [7]
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$$
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$\Delta H_F$ aus [1.2] in [7]:
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$$ \tan(\alpha_F) \cdot (L_F - L_{FEA}) + H_{FEA} = H_1 - H_0 + (\tan(\alpha_S) \cdot (L_S - L_{SEA}) + H_{SEA}) [8]$$
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$L_S$ aus [5] in [8]:
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$$ \tan(\alpha_F) \cdot (L_F - L_{FEA}) + H_{FEA} = H_1 - H_0 + \tan(\alpha_S) \cdot (L_1 - L_F - L_{SEA}) + H_{SEA} [9]$$
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### auflösen der Gleichung nach $L_F$.
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1. **Erweitere die Gleichung:**
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$$ \tan(\alpha_F) \cdot L_F - \tan(\alpha_F) \cdot L_{FEA} + H_{FEA} = H_1 - H_0 + \tan(\alpha_S) \cdot L_1 - \tan(\alpha_S) \cdot L_F - \tan(\alpha_S) \cdot L_{SEA} + H_{SEA} $$
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2. **Sortiere die Gleichung nach $L_F$:**
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$$ \tan(\alpha_F) \cdot L_F + \tan(\alpha_S) \cdot L_F = H_1 - H_0 + \tan(\alpha_S) \cdot L_1 - \tan(\alpha_S) \cdot L_{SEA} + H_{SEA} + \tan(\alpha_F) \cdot L_{FEA} - H_{FEA} $$
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$$ L_F (\tan(\alpha_F) + \tan(\alpha_S)) = H_1 - H_0 + \tan(\alpha_S) \cdot L_1 - \tan(\alpha_S) \cdot L_{SEA} + H_{SEA} + \tan(\alpha_F) \cdot L_{FEA} - H_{FEA} $$
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3. **Löse nach $L_F$ auf:**
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$$ L_F = \frac{H_1 - H_0 + \tan(\alpha_S) \cdot L_1 - \tan(\alpha_S) \cdot L_{SEA} + H_{SEA} + \tan(\alpha_F) \cdot L_{FEA} - H_{FEA}}{\tan(\alpha_F) + \tan(\alpha_S)} $$
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### Bestimmung der restlichen Größen:
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1. **$L_S$:**
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$$ L_S = L_1 - L_F $$
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2. **$\Delta H_F$:**
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$$ \Delta H_F = \tan(\alpha_F) \cdot (L_F - L_{FEA}) + H_{FEA} $$
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3. **$\Delta H_S$:**
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$$ \Delta H_S = \tan(\alpha_S) \cdot (L_S - L_{SEA}) + H_{SEA} $$
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4. **$H_{Max}$:**
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$$ H_{Max} = H_0 + \Delta H_F $$
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5. **$x_{Verhältnis}$:**
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$$ x_{Verhältnis} = \frac{L_1}{L_F} $$
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