A subsystems characterization of the zero modes for flexible mechanical structures

Transmission zero frequencies, zero input directions and zero state directions constitute the zero modes structure of a linear system. In this paper, these quantities are analyzed and characterized from a physical and structural perspective, for the class of multi-actuated flexible mechanical systems described in second order matrix differential form. The zero modes are shown to be closely related to the eigen-structure of energetically isolated subsystems of the original system, defined by the positions of sensors and actuators over the structure. Relations between the zero and pole frequencies are provided in form of structural bounds that define regions of the complex plane where the zeros must lie, and an efficient algorithm for structural zeros computation is proposed. Also, expressions for the sensitivity of the zero frequencies to structural variations are provided, and examples involving a lumped masses system and a beam in transverse vibration are presented to illustrate the application of the proposed theory to finite-dimensional and continuous system models.