Modal response of a beam with a sensor–actuator pair for the implementation of velocity feedback control

A theoretical analysis is presented of the flexural vibration of a beam with a control system which implements direct velocity feedback using either an ideal collocated force actuator or a closely located piezoelectric patch actuator. The aim of this study is to describe the vibration of the beam as the control gain is raised. Both control systems generate active damping which reduces the vibration level at resonance frequencies. However, it is shown that when the gain passes an optimal value then the vibration of the beam is rearranged into a new set of lightly damped resonance frequencies, since the control systems impose new boundary conditions at the control position on the beam, in which the velocity is driven to zero in both cases but different spatial derivatives of the velocity are driven to zero in the case of the force actuator and the piezoelectric patch actuator. The new ‘‘natural frequencies’’ and ‘‘natural modes’’ of the beam constrained by the two feedback control systems with large control gains are derived analytically. The new resonance frequencies and mode shapes seen in the simulations are consistent with the natural frequencies and natural modes of the constrained beams derived analytically.