Defective two adjacent single degree of freedom systems linked by spring-dashpot-inerter for vibration control

The paper deals with the analytical formulation of two adjacent single degree of freedom systems coupled by spring-dashpot-inerter elements adequately tuned in order to build a defective system. From modal analysis analytical expressions of eigenvalues, damping factor and connection parameters in order to build defective systems are given. It is demonstrated that is always possible to realize a defective system when a connection with spring-dashpot-inerter elements is utilized, irrespectively of the assumed structural parameters. Given a couple of oscillators, it is possible to build infinite defective systems by conveniently varying one parameter of the connection, chosen as independent variable. Among the feasible choices of parameters, a design criterion for the connection is proposed maximizing the modal damping factor and closed-form expressions of the design parameters are carried out. Maps of the natural frequency, damping factor and the damped frequency of the defective system with maximum modal damping factor are furnished. A procedure to obtain the eigenvectors forming a complete base to compute the system response analytically is given. For these systems, the response to free vibration with initial conditions and to base harmonic motion is investigated. The main characteristics of the response are discussed and considerations about the dynamics and effectiveness of the connection for control purposes are outlined.