Abstract
The equations of motion of an asymmetric Timoshenko shaft, that is having unequal principal moments of inertia, are derived within the framework of the Lagrangian formulation for continuous systems and fields. The Lagrangian density of the system is calculated in a moving frame, that is a rotating frame attached to the deformed shaft, and proves to depend on the four Lagrangian variables (fields) of the system and their first derivatives w.r.t. space and time.
On account of general results of the theory of continuous systems and fields, the four Lagrange's equations of motion are derived from the Lagrangian density and are successively reduced to the two usual equations in the displacements.
The procedure described in this work is compared with both a different Lagrangian formulation, based on the use of a floating frame, that is a rotating frame attached to the undeformed shaft, and the well-known Newtonian approach adopted by Dimentberg.
Sommario. Si applica la formulazione lagrangiana per i sistemi continui e i campi per ricavare le equazioni del moto di un albero di Timoshenko asimmetrico, la cui sezione presenta cioé momenti principali d'inerzia diversi. La densità di lagrangiana è calcolata in un sistema di riferimento rotante solidale all'albero deformato e risulta essere funzione delle quattro variabili lagrangiane (campi) del sistema e delle loro derivate prime rispetto allo spazio e al tempo.
In accordo con i risultati generali della teoria dei sistemi continui e dei campi, si ricavano, a partire dalla densità di lagrangiana, le quattro equazioni di Lagrange successivamente ridotte alle due classiche equazioni rispetto ai soli spostamenti.
Il procedimento proposto viene messo a confronto con una diversa formulazione lagrangiana, basata sull'uso di un sistema di riferimento rotante solidale all'albero indeformato, e con la ben nota formulazione newtoniana adottata da Dimentberg.
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Raffa, F.A., Vatta, F. Equations of Motion of an Asymmetric Timoshenko Shaft. Meccanica 36, 201–211 (2001). https://doi.org/10.1023/A:1013079613566
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DOI: https://doi.org/10.1023/A:1013079613566