English translation: equilibrium of the Varignon Frame
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10:41 Nov 2, 2007
German to English translations [PRO] Bus/Financial - Mathematics & Statistics / Logistics
German term or phrase:Zugkraftausgleich auf dem Varignonschen Apparat
Die Transportkosten je Periode verlaufen proportional zu der mit den Transportmengen je Periode gewichteten zurückzulegenden Entfernung.
Das Optimierungskriterium lautet: XY
Damit ergibt sich folgender Ansatz zur Bestimmung des Standortes minimaler Transportkosten: XY
Das Optimum dieser Gleichung lässt sich nun durch Bildung der 1. und 2. Ableitung bestimmen.
Bekannte Lösungsverfahren sind hierbei Iterationsmethoden oder der mechanische Zugkraftausgleich auf dem Varignonschen Apparat.
This logistics text swings off into mathematical equations for a few pages, including this section.
balancing of tensile forces using a Varignon device/appratus
Explanation: Apparently Varignon was a relatlvely well known scientist at the time of Newton. He developed several theories related to mathematical systems and mechanical systems. IMO his name is practically unknown today outside of a few small academic circles.
Perhaps this text refers to the apparatus described in the following reference -- a balance scale to which weights and/or spring scales can be attached at various points on its arms.
In the above picture, Dl represents the distance the weight is from the fulcrum. (In the picture, Dl=4 since the weight is on a peg 4 units from the fulcrum.) Similarly, Dr represents the distance the spring balance is from the fulcrum. Fw is the force exerted by the weight, and Fs is the force exerted by the spring balance (due to the student's pulling on it). The force the students use on the spring balance should be enough to balance the weight (on the moment arm).
By experimenting with the locations of both the weight and the spring balance, students can discover Varignon's theorem, one of the most useful principles of mechanics. This theorem states that the moment of a force about any point is equal to the sum of the moments of the components of the force about the same point. Another way of saying this is that the sum of the moments (products of force and distance) about the fulcrum equals zero. A physics class could go deeper into the meaning of this theorem. (See the below explanation.)
Varignon was a friend of Newton, Leibniz, and the Bernoulli family. Varignon's principal contributions were to graphic statics and mechanics. Except for l'Hôpital, Varignon was the earliest and strongest French advocate of differential calculus. He recognized the importance of a test for the convergence of series, but analytical difficulties prevented his success. Nevertheless, he simplified the proofs of many propositions in mechanics, adapted Leibniz's calculus to the inertial mechanics of Newton's Principia, and treated mechanics in terms of the composition of forces in Projet d'une nouvelle mécanique in 1687. Among Varignon's other works was a 1699 publication concerning the application of differential calculus to fluid flow and to water clocks. In 1690 he created a mechanical explanation of gravitation. In 1702 he applied calculus to spring-driven clocks.
Ken Cox Local time: 11:38 Native speaker of: English PRO pts in category: 40