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Electromechanical Dynamics (Part 1).0099

Electromechanical Dynamics (Part 1).0099 - Lumped-Parameter...

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Lumped-Parameter Electromechanics energy in (3.1.22) and (3.1.23): i W(aw , z) a , X) (3.1.22) f" = w ) (3.1.23) ax We now differentiate (3.1.22) with respect to x and (3.1.23) with respect to L. Then, because aja. 8xM a2w, a2wa the reciprocity relation results: ai(, X) _ af(, x) (3.1.58) ax N The process used in obtaining the reciprocity condition (3,1.58) shows that the condition is necessary for the system to be conservative. This same condition can also be shown to be sufficient to ensure that the system is conservative. The proof requires a straightforward but involved integration and is not carried out here primarily because it is a standard inclusion in some thermodynamics texts.* The reciprocity condition of (3.1.58) can be generalized to describe a conservative system with any number
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