A general linear mathematical model of power flow analysis and control for integrated structure–control systems

Generalized integrated structure–control dynamical systems consisting of any number of active/passive controllers and three-dimensional rigid/flexible substructures are investigated. The developed mathematical model assessing the behaviour of these complex systems includes description of general boundary conditions, the interaction mechanisms between structures, power flows and control characteristics. Three active control strategies are examined. That is, multiple channel absolute/relative velocity feedback controllers, their hybrid combination and an existing passive control system to which the former control systems are attached in order to improve overall control efficiency. From the viewpoint of continuum mechanics, an analytical solution of this generalized structure–control system has been developed allowing predictions of the dynamic responses at any point on or in substructures of the coupled system. Absolute or relative dynamic response or receptance, transmissibility, mobility, transfer functions have been derived to evaluate complex dynamic interaction mechanisms through various transmission paths. The instantaneous and time-averaged power flow of energy input, transmission and dissipation or absorption within and between the source substructure, control subsystems and controlled substructure are presented. The general theory developed provides an integrated framework to solve various vibration isolation and control problems and provides a basis to develop a general algorithm that may allow the user to build arbitrarily complex linear control models using simple commands and inputs. The proposed approach is applied to a practical example to illustrate and validate the mathematical model as well as to assess control effectiveness and to provide important guidelines to assist vibration control designers.