From 8af1ff9b7bf4e8ce46af2b1c9316c48b81de64a1 Mon Sep 17 00:00:00 2001 From: mistangl Date: Tue, 28 May 2024 13:33:02 +0200 Subject: [PATCH] Berechnung aktualisiert --- Doku/Berechnung-foerderer-Längen.md | 31 +++---- Doku/Berechnung-foerderer-Längen_extended.md | 84 ++++++------------- Doku/bilder/Berechnung-Foerderer.png | Bin 0 -> 108084 bytes Doku/bilder/ChatGPT-Berechnung-extendend.png | Bin 0 -> 157060 bytes 4 files changed, 37 insertions(+), 78 deletions(-) create mode 100644 Doku/bilder/Berechnung-Foerderer.png create mode 100644 Doku/bilder/ChatGPT-Berechnung-extendend.png diff --git a/Doku/Berechnung-foerderer-Längen.md b/Doku/Berechnung-foerderer-Längen.md index 74d74d9..63db365 100644 --- a/Doku/Berechnung-foerderer-Längen.md +++ b/Doku/Berechnung-foerderer-Längen.md @@ -1,40 +1,33 @@ ![Alt text](bilder/SkizzeFördererStrecke.svg "Skizze von Förder und Strecke") Bekannt: - - $$ L_1, H_0, H_1, \alpha_F, \alpha_S, $$ Gesucht: - - $$ -L_F, L_S, \Delta H_F, \Delta H_S, (\rightarrow H_{Max}) +L_F, L_S, \Delta H_F, \Delta H_S, H_{Max}, x_{Verhältnis} $$ Beziehungen: - - - $$ - \frac{\Delta_{H_F}}{L_{F}} = tan(\alpha_F) [1] + \frac{\Delta{H_F}}{L_{F}} = tan(\alpha_F) [1] $$ $$ - \Delta_{H_F} = tan(\alpha_F) * L_{F} [1.2] + \Delta{H_F} = tan(\alpha_F) * L_{F} [1.2] $$ $$ - \frac{\Delta_{H_S}}{L_{S}} = tan(\alpha_S) [2] + \frac{\Delta{H_S}}{L_{S}} = tan(\alpha_S) [2] $$ $$ - \Delta_{H_S} = tan(\alpha_S) * L_{S} [2.2] + \Delta{H_S} = tan(\alpha_S) * L_{S} [2.2] $$ $$ - \Delta_{H_F} = H_1 - H_0 + \Delta_{H_S} [3] + \Delta{H_F} = H_1 - H_0 + \Delta{H_S} [3] $$ $$ L_1 = L_F + L_S [4] @@ -46,14 +39,14 @@ $$ [5] in [4]: L_S = L_F * (x_{Verhältnis} - 1) [6] $$ $$ -[2] in [3]: \Delta_{H_F} = H_1 - H_0 + tan(\alpha_S) * L_{S} [7] +[2] in [3]: \Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * L_{S} [7] $$ $$ -[6] in [7]: \Delta_{H_F} = H_1 - H_0 + tan(\alpha_S) * L_F * (x_{Verhältnis} - 1) [8] +[6] in [7]: \Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * L_F * (x_{Verhältnis} - 1) [8] $$ $$ -[5] in [8]: \Delta_{H_F} = H_1 - H_0 + tan(\alpha_S) * \frac{(x_{Verhältnis} - 1)}{x_{Verhältnis}} * L_1 [9] +[5] in [8]: \Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * \frac{(x_{Verhältnis} - 1)}{x_{Verhältnis}} * L_1 [9] $$ $$ [1] in [8]: tan(\alpha_F) = x_{Verhältnis}\frac{H_1 - H_0}{L_1} + tan(\alpha_S) * (x_{Verhältnis} - 1) [10] @@ -73,11 +66,9 @@ $$ --> auflösen nach x: - - Dann errechnet sich leicht - L_F aus Beziehung [5] -- \Delta_{H_F} aus [1] +- \Delta{H_F} aus [1] - und L_S aus [4] Bei der Annahme, dass $L_F$ immer ein "Teil", also kleiner als $L_1$ sein soll, erschliesst sich, dass nur x (grösser) gleich 1 gewünschte Ergebnisse liefert. Daraus ergibt sich, dass die folgende geometrischen Beziehung die gelten muß: @@ -94,5 +85,7 @@ $$ {tan(\alpha_F)} \geq {\frac{H_1 - H_0}{L_1}} $$ +Als Bild: +![Alt text](bilder/Berechnung-Foerderer.png) diff --git a/Doku/Berechnung-foerderer-Längen_extended.md b/Doku/Berechnung-foerderer-Längen_extended.md index 5ee3320..663a31d 100644 --- a/Doku/Berechnung-foerderer-Längen_extended.md +++ b/Doku/Berechnung-foerderer-Längen_extended.md @@ -10,7 +10,7 @@ Gesucht: $$ -L_F, L_S, \Delta H_F, \Delta H_S, (\rightarrow H_{Max}) +L_F, L_S, \Delta H_F, \Delta H_S, H_{Max}, x_{Verhältnis} $$ Beziehungen: @@ -21,13 +21,13 @@ $$ \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] + \Delta{H_F} = tan(\alpha_F) \cdot (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] + \Delta{H_S} = tan(\alpha_S) \cdot (L_{S}-L_{SEA}) + H_{SEA} [2.2] $$ $$ H_0 + \Delta{H_F} = H_1 + \Delta{H_S} [4] @@ -39,39 +39,47 @@ $$ L_1 = L_F + L_S [5] $$ $$ - L_1 = L_F * x_{Verhältnis} [6] + L_1 = L_F \cdot x_{Verhältnis} [6] $$ $L_1$ aus [6] in [5]: $$ - L_S = L_F * (x_{Verhältnis} - 1) [7] + L_S = L_F \cdot (x_{Verhältnis} - 1) [7] $$ $\Delta{H_S}$ aus [2.2] in [4.2]: $$ - \Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * (L_{S}-L_{SEA}) + H_{SEA}[9] + \Delta{H_F} = H_1 - H_0 + tan(\alpha_S) \cdot (L_{S}-L_{SEA}) + H_{SEA}[9] $$ $L_S$ aus [7] in [9]: $$ -\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * [L_F * (x_{Verhältnis} - 1) - L_{SEA})] + H_{FEA}[10] +\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) \cdot [L_F \cdot (x_{Verhältnis} - 1) - L_{SEA})] + H_{FEA}[10] $$ $L_F$ aus [6] in [10]: $$ -\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) * \Bigl[\frac{L_1}{x_{Verhältnis}} * (x_{Verhältnis} - 1) - L_{SEA})\Bigr] + H_{FEA}[10] +\Delta{H_F} = H_1 - H_0 + tan(\alpha_S) \cdot \Bigl[\frac{L_1}{x_{Verhältnis}} \cdot (x_{Verhältnis} - 1) - L_{SEA})\Bigr] + H_{FEA}[11] $$ -auflösen nach x: +$\Delta H_F$ aus [1.2] in [11]: $$ -\Delta{H_F} - H_{FEA} + H_0 - H_1 + tan(\alpha_S) * L_{SEA} = tan(\alpha_S) * L_F * (x_{Verhältnis} - 1) +tan(\alpha_F) \cdot (L_{F} - L_{FEA}) + H_{FEA} = H_1 - H_0 + tan(\alpha_S) \cdot \Bigl[\frac{L_1}{x_{Verhältnis}} \cdot (x_{Verhältnis} - 1) - L_{SEA})\Bigr] + H_{FEA}[12] +$$ +dann [12] auflösen nach x: +$$ +tan(\alpha_F) \cdot (L_{F} - L_{FEA}) - H_1 + H_0 + tan(\alpha_S) \cdot L_{SEA} = tan(\alpha_S) \cdot L_1 \cdot \Bigl[1 - \frac{1}{x_{Verhältnis}}\Bigr] $$ $$ -tan(\alpha_F) = x_{Verhältnis}*\Bigl[\frac{H_1 - H_0}{L_1} + tan(\alpha_S)\Bigr] - tan(\alpha_S) - \\ -tan(\alpha_F) + tan(\alpha_S) = x_{Verhältnis}*\Bigl[\frac{H_1 - H_0}{L_1} + tan(\alpha_S)\Bigr] +tan(\alpha_F) \cdot (L_{F} - L_{FEA}) - H_1 + H_0 + tan(\alpha_S) \cdot L_{SEA} - L_1 \cdot tan(\alpha_S) = - tan(\alpha_S) \cdot L_1 \cdot \Bigl[\frac{1}{x_{Verhältnis}}\Bigr] $$ $$ -\frac{tan(\alpha_F) + tan(\alpha_S)}{\frac{H_1 - H_0}{L_1} + tan(\alpha_S)} = x_{Verhältnis} +\frac{tan(\alpha_F) \cdot (L_{F} - L_{FEA}) - H_1 + H_0 + tan(\alpha_S) \cdot L_{SEA} - L_1 \cdot tan(\alpha_S)}{- tan(\alpha_S) \cdot L_1 }= \frac{1}{x_{Verhältnis}} +$$ +$$ +x_{Verhältnis}= \frac{- tan(\alpha_S) \cdot L_1 }{tan(\alpha_F) \cdot (L_{F} - L_{FEA}) - H_1 + H_0 + tan(\alpha_S) \cdot L_{SEA} - L_1 \cdot tan(\alpha_S)} +$$ +$$ +x_{Verhältnis}= \frac{tan(\alpha_S) \cdot L_1 }{-tan(\alpha_F) \cdot (L_{F} + L_{FEA}) + H_1 - H_0 - tan(\alpha_S) \cdot L_{SEA} + L_1 \cdot tan(\alpha_S)} $$ - + + $$ \frac{tan(\alpha_F) + tan(\alpha_S)}{\frac{H_1 - H_0}{L_1} + tan(\alpha_S)} \geq 1 $$ @@ -89,46 +97,4 @@ $$ $$ {tan(\alpha_F)} \geq {\frac{H_1 - H_0}{L_1}} $$ - -/// - -C-GPT - -# Bekannte Werte: -L1, H0, H1, alpha_F, alpha_S, H_SEA, L_SEA, H_FEA, L_FEA - -# Gesucht: -# L_F, L_S, ΔH_F, ΔH_S, (→ H_Max) - -# Beziehungen: -# [1] (ΔH_F - H_FEA) / (L_F - L_FEA) = tan(alpha_F) -# [1.2] ΔH_F = tan(alpha_F) * (L_F - L_FEA) + H_FEA -# [2] (ΔH_S - H_SEA) / (L_S - L_SEA) = tan(alpha_S) -# [2.2] ΔH_S = tan(alpha_S) * (L_S - L_SEA) + H_SEA -# [4] H_0 + ΔH_F = H_1 + ΔH_S -# [4.2] ΔH_F = H_1 - H_0 + ΔH_S -# [5] L_1 = L_F + L_S -# [6] L_1 = L_F * x_Verhältnis - -# Umstellen der Gleichungen für ΔH_F und ΔH_S: -ΔH_F = tan(alpha_F) * (L_F - L_FEA) + H_FEA -ΔH_S = tan(alpha_S) * (L_S - L_SEA) + H_SEA - -# Einsetzen von L_S = L_1 - L_F: -tan(alpha_F) * (L_F - L_FEA) + H_FEA = H_1 - H_0 + (tan(alpha_S) * ((L_1 - L_F) - L_SEA) + H_SEA) - -# Auflösen nach L_F: -tan(alpha_F) * L_F - tan(alpha_F) * L_FEA + H_FEA = H_1 - H_0 + tan(alpha_S) * L_1 - tan(alpha_S) * L_F - tan(alpha_S) * L_SEA + H_SEA - -# Gruppieren der Terme: -tan(alpha_F) * L_F + tan(alpha_S) * L_F = tan(alpha_S) * L_1 + H_1 - H_0 - tan(alpha_F) * L_FEA - H_FEA + tan(alpha_S) * L_SEA - H_SEA - -# L_F isolieren: -L_F = (tan(alpha_S) * L_1 + H_1 - H_0 - tan(alpha_F) * L_FEA - H_FEA + tan(alpha_S) * L_SEA - H_SEA) / (tan(alpha_F) + tan(alpha_S)) - -# Berechnung von x_Verhältnis: -x_Verhältnis = L_F / L_1 -x_Verhältnis = (tan(alpha_S) + (H_1 - H_0 - tan(alpha_F) * L_FEA - H_FEA + tan(alpha_S) * L_SEA - H_SEA) / L_1) / (tan(alpha_F) + tan(alpha_S)) - -# Vereinfachte Formel für x_Verhältnis: -x_Verhältnis = (tan(alpha_S) + ((H_1 - H_0) - tan(alpha_F) * L_FEA - H_FEA + tan(alpha_S) * L_SEA - H_SEA) / L_1) / (tan(alpha_F) + tan(alpha_S)) +--> diff --git a/Doku/bilder/Berechnung-Foerderer.png b/Doku/bilder/Berechnung-Foerderer.png new file mode 100644 index 0000000000000000000000000000000000000000..d8876ab7272caf8cd62106544aad27ffd02dd340 GIT binary patch literal 108084 zcmbSz1yogCyY;aEkrD(1X%VEmOAp=B4T5xchsr@pO1h;5q`Q&s?(Xhx{)K+;ci%hS zd++$h&ybDmz1Q0NtiATL<}>Gf@O>vGjEsnj2!TM5MMVT;A&~pa5Xd72g!^F6_jeWq z5Xe245kLPsQGR|ROKS^#BU3#HpJ^I2=yp9k19Ykx8j`s!*Q$PT4ebXMNc zR%DT1Fj|O8HV&hErmo0tVWqf&DAAtIB;kiQme_`DW|x;^rIX|z zZ*j^8DsKX?=r2W`i?W*oKPH*YEmUT=yVNhE&>y6Ve_0Z7r%Y0Zo9SJP^{d~bq0isw z$*~C>geQ_NQmwp)g7Rq}Q7_M_x~k5>{B~x&YZv~Tm37f~id|0dSRPx7sMtUtv@h?z 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