Stability and performance limits for the active vibration isolation on a 2 DOF system using a reactive force actuator and velocity feedback

In this paper a theoretical study of active vibration isolation on a two degree of freedom system is presented. The system consists of two lumped masses connected by a coupling spring. Either mass is also attached to a firm reference base by a mounting spring. The vibrations of the lower mass are excited by a point force and transmitted to the upper mass via the coupling spring. A reactive control force actuator is used between the two masses in parallel with the coupling spring. Either mass is equipped with an absolute velocity sensor. The two sensors and the actuator are used to implement velocity feedback control loops to actively isolate the upper mass form the vibrations of the lower mass over a broad range of frequencies. The primary concern of the study is to determine what type of velocity feedback is suitable with respect to the five system parameters (the three spring stiffnesses and the two masses). It is shown analytically that there are two groups of passive systems; supercritical and subcritical. Supercritical systems are characterised by the ratio of the upper mass to the lower mass being larger than the ratio of the upper mounting spring stiffness to the lower mounting spring stiffness. With supercritical systems, feeding back the absolute upper mass velocity to the reactive force actuator results in an unconditionally stable feedback loop. The vibration isolation objective can be fully achieved without an overshot at higher frequencies. In contrast, if the ratio of the upper mass to the lower mass is smaller than the ratio of the upper mounting spring stiffness to the lower mounting spring stiffness, the system is subcritical. With subcritical systems the upper mass velocity feedback is conditionally stable and the vibrationisolation objective can not be accomplished in a broad frequency band. For subcritical systems a blended velocity feedback is proposed. The error velocity signal is obtained by subtracting weighted outputs of the two sensors. It is shown that the stability and the performance of the active vibration isolation greatly depend upon the velocity weighting factor used. It is shown that the use of an appropriate velocity weighting factor for subcritical systems can significantly improve the vibration isolation performance with reference to the passive isolation.