Lagrangians for Electromechanical Systems

William Manhães, Rubens Sampaio, Roberta Lima, Peter Hagedorn, Jean-François Deü


Electromechanical systems are very common. The importance of constructing the dynamical equations of motors coupled with mechanical systems suggests a new strategy. In the literature, often, the derivation of the dynamical equations is wrong. One thinks that the standard derivations of the dynamical equations of purely mechanical systems can be mimicked to electromechanical systems. Unfortunately, it cannot. The main reason is that in electromechanical systems one deals with the presence of electromagnetic fields, continuous entities. This fields store electrical and mechanical energies. In purely mechanical systems the conservative mechanical energy is stored as elastic or gravitational energy, and the nonconservative terms enter the equation as nonconservative forces. This cannot be done in electromagnetic systems. This paper shows the right way to derive the dynamical equations applying the results for two systems. Both systems are formed by a motor, a coupling mechanism, and a mechanical subsystem. In one case the coupling is the scotch yoke mechanism and the mechanical part is a cart. In the other the coupling mechanism is a slider crank mechanism and the mechanical system is a pendulum. To explain clearly the ideas, the dynamical equations are derived in a different way, putting in evidence the common errors. The examples were taken from the recent literature, but the same mistake is found in several places.

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