From 67f13f1d8087b640bf3cb47802cc51d506af65c0 Mon Sep 17 00:00:00 2001
From: mistangl <m.stangl@schoenenberger.de>
Date: Mon, 27 May 2024 17:08:45 +0200
Subject: [PATCH] =?UTF-8?q?F=C3=B6rderer=20und=20Gef=C3=A4llestrecke=20um?=
 =?UTF-8?q?=20Ein=20und=20Auslauf=20erg=C3=A4nzt?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 Doku/Berechnung-foerderer-Längen_extended.md  |  91 ++++
 .../SkizzeFördererStrecke_mitEinAuslauf.svg   | 466 ++++++++++++++++++
 2 files changed, 557 insertions(+)
 create mode 100644 Doku/Berechnung-foerderer-Längen_extended.md
 create mode 100644 Doku/bilder/SkizzeFördererStrecke_mitEinAuslauf.svg

diff --git a/Doku/Berechnung-foerderer-Längen_extended.md b/Doku/Berechnung-foerderer-Längen_extended.md
new file mode 100644
index 0000000..128e54d
--- /dev/null
+++ b/Doku/Berechnung-foerderer-Längen_extended.md
@@ -0,0 +1,91 @@
+![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)-->
+$$
+\frac{tan(\alpha_F) + tan(\alpha_S)}{\frac{H_1 - H_0}{L_1} + tan(\alpha_S)} \geq 1
+$$
+$$
+{tan(\alpha_F) + tan(\alpha_S)} \geq {\frac{H_1 - H_0}{L_1} + tan(\alpha_S)}
+$$
+$$
+{tan(\alpha_F)} \geq {\frac{H_1 - H_0}{L_1}}
+$$
diff --git a/Doku/bilder/SkizzeFördererStrecke_mitEinAuslauf.svg b/Doku/bilder/SkizzeFördererStrecke_mitEinAuslauf.svg
new file mode 100644
index 0000000..37a634d
--- /dev/null
+++ b/Doku/bilder/SkizzeFördererStrecke_mitEinAuslauf.svg
@@ -0,0 +1,466 @@
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