Equal modal damping design for a family of resonant vibration control formats

The principle of equal modal damping is used to give a unified presentation and calibration of resonant control of structures for different control formats, based on velocity, acceleration–position or position feedback. When introducing a resonant controller the original resonant mode splits into two, and if these are required to have the same modal damping ratio, the characteristic equation conforms to a two-parameter format. By selecting a suitable relative separation of the modal frequencies, the design problem defines a one-parameter family – determined, for example, in terms of the resulting modal damping ratio. While velocity feedback, and the associated acceleration–position formats, lead to near-equal resonant peak heights of the velocity in a frequency response diagram, position feedback leads to balanced acceleration peaks. It is demonstrated, how a simple additional time derivative term in the control coupling can change these properties into balanced position and velocity peaks, respectively. In particular this gives an improved control format based on measurement of structural displacement or deformation. In all cases the optimal calibration in terms of a root locus identification leads to a simple explicit pair of design formulae for controller frequency and damping ratio based on a simple two -degrees-of-freedom system. Unconditional stability is demonstrated for a general multi-degrees-of-freedom system with multiple controllers for the velocity and acceleration-velocity formats, while the position and extended position feedback format give a simple stability condition in terms of the gain factors and the structure flexibility matrix. The paper concludes with a simple design procedure based on the desired effective damping of a flexible structure with equal modal control in any of the discussed resonant formats.