On the dual side of operational dynamics: A formulation, applications, and implications of the impedance model

Abstract

This paper studies the operational dynamics of a structure in steady state under a given source. It is based on the duality of operational structural dynamics, which states that the structure in operation can be viewed either as a force-excited structure on the primal side or as a motion-excited structure on the dual side. The primal side is well known after Newtonian mechanics while the dual side is not as well. We formulate the primal and dual sides as partial differential equations and solve them analytically to produce the mobility model (Y model) and the impedance model (Z model), respectively. The Y model is described in terms of natural frequencies and normal modes while the Z model is in terms of anti-natural frequencies and anti-normal modes. Thus, the primal side shows the resonance phenomena while the dual side shows the anti-resonance phenomena. As both phenomena are common in practice, it is natural that we require both sides (models) to have a complete knowledge of the actual operational dynamics. Numerical applications and physical interpretations are further made to justify the formulation as well as to show the merits and implications of the Z model.