Mechanismentechnik - Aufgabe 11.11
Roberts'sche Ersatzgetriebe
Demetrius Lorenz 1
1 Fachhochschule DortmundJun 2020
Keywords: Kinematik, Mechanismentechnik, Ersatzgetriebe, Samuel Roberts, Konstruktion, microJam
Abstract
tbd...
Aufgabenstellung
Ermitteln Sie zum gegebenen Gelenkviereck die fehlenden Gelenkpunkte der Ersatzgetriebe in der zugehörigen Stellung und fügen Sie die Ersatzgetriebe der Skizze hinzu.
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Abb. 1: Skizze der Aufgabenstellung
1 Satz von Roberts
Satz von Roberts/Tschebyschew Jede Koppelkurve eines Viergelenkgetriebes lässt sich durch zwei weitere Gelenkvierecke erzeugen.
(vgl. Gössner 2017, S.215)
In anderen Worten; es existieren zwei weitere Mechanismen, die die gleiche Koppelkurve eines Viergelenkgetriebes abbilden. Der Nutzen ist darin begründet, dass der Bauraum in der Praxis naturgemäß begrenzt ist. Sollte dieser Fall auftretten, können zwei Ersatzgetriebe nach Roberts mit der identischen Koppelkurve zur Bauraumuntersuchung hinzugezogen werden.
Die Ersatzgetriebe nach Roberts können sowohl zeichnerisch als auch rechnerisch ermittelt werden. Die zeichnerische Methode eignet sich auch beim Einsatz moderner CAD-Systeme.
1.1 Zeichnerische Vorgehensweise
Schritt 1 „Übertragen der Kurbel-, Koppel und Schwingenlänge a , b , c a,b,c a , b , c (vgl. Gössner 2017, S.215)
Schritt 2 „Errichten des Koppeldreiecks über der Koppellänge b.“ (vgl. Gössner 2017, S.215)
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Abb. 2: Vollständige Strecklage
Schritt 3 „Antragen an den freien Enden von a a a b b b f f f g g g (vgl. Gössner 2017, S.215)
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Abb. 3: Antragen der parallelen Geraden zu f und g
Schritt 4 „Ziehen einer parallelen Geraden zur Grundlinie durch die Spitze des Koppeldreiecks.“ (vgl. Gössner 2017, S.215)
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Abb. 4: Antragen der parallelen Geraden zur Grundlinie
Schritt 5 „Verlängern der Seiten f f f g g g (vgl. Gössner 2017, S.215)
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Abb. 5: Verlängern der Seiten f und g
Schritt 6 „Rückübertragung der gefundenen Gliedlängen der Ersatzgelenkvierecke an das Ursprungsviergelenk.“ (vgl. Gössner 2017, S.215)
Die Gliedlängen lassen sich am einfachsten mit einem Zirkel übertragen. Dafür wird der Zirkel anhand der konstruierten Ersatzvierecke (Schritt 1-5) eingestellt. Anschließend werden Hilfskreise in die Skizze der Aufgabenstellung eingezeichnet. Mit Hilfe der Schnittpunkte werden die parallelen Seitenlängen übertragen.
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Abb. 6: Rückübertragung der Gliedlängen
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Abb. 7: Rückübertragung der Gliedlängen
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Abb. 8: Rückübertragung der Gliedlängen
Damit sind die Ersatzgetriebe nach Roberts zeichnerisch bestimmt.
2 Modell
Zum Schluss soll nachgewiesen werden, dass die Koppelkurven der Ersatzgetriebe tatsächen der Koppelkurve des originalen Viergelenks entsprechen.
2.1 Original Getriebe
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Modell 1: kinematisches Modell
2.2 Ersatzgetriebe Orange
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{ "id":"a","p1":"A0","p2":"A","len":{ "type":"const" }},
{ "id":"b","p1":"C0","p2":"B","len":{ "type":"const" } },
{ "id":"c","p1":"B","p2":"K","len":{ "type":"const" } },
{ "id":"d","p1":"K","p2":"A","len":{ "type":"const" } },
{ "id":"e","p1":"A","p2":"B","len":{ "type":"const" },"ori":{ "type":"drive","input":true,"Dt":3,"Dw":6.283185307179586 } }
],
"shapes": [
{ "type":"fix","p":"C0" },
{ "type":"fix","p":"A0" }
],
"views": [
{ "show":"pos","of":"K","as":"trace","mode":"dynamic","id":"view1","Dt":3,"stroke":"rgba(255,0,0,1)","fill":"darkorange" },
{ "show":"w","of":"a","as":"info","x":10,"y":75,"Dt":1.9,"id":"view1" }
]
}
Modell 2: kinematisches Modell
2.3 Ersatzgetriebe Grün
{
"gravity":true,
"nodes": [
{ "id":"B0","x":260,"y":20,"base":true },
{ "id":"C0","x":300,"y":140,"base":true },
{ "id":"A","x":180,"y":20,"idloc":"ne" },
{ "id":"B","x":300,"y":100},
{ "id":"K","x":20,"y":260}
],
"constraints": [
{ "id":"a","p1":"B0","p2":"A","len":{ "type":"const" } },
{ "id":"b","p1":"C0","p2":"B","len":{ "type":"const" },"ori":{ "type":"drive","input":true,"Dt":3,"Dw":6.283185307179586 } },
{ "id":"c","p1":"B","p2":"K","len":{ "type":"const" } },
{ "id":"d","p1":"K","p2":"A","len":{ "type":"const" } },
{ "id":"e","p1":"A","p2":"B","len":{ "type":"const" } }
],
"shapes": [
{ "type":"fix","p":"C0" },
{ "type":"fix","p":"B0" }
],
"views": [
{ "show":"pos","of":"K","as":"trace","mode":"dynamic","id":"view1","Dt":3,"stroke":"rgba(255,0,0,1)","fill":"yellowgreen" },
{ "show":"w","of":"a","as":"info","x":10,"y":75,"Dt":1.9,"id":"view1" }
]
}
Modell 3: kinematisches Modell
References
Gössner, S., 2017. Mechanismentechnik: Vektorielle Analyse ebener Mechanismen. Berlin: Logos