Cubic root-locus format for structures with a local viscous damping device

Abstract

The tuning and efficiency of a local viscous damper on a flexible structure is described via an approximate representation of the characteristic equation in the form of a cubic polynomial. Increasing the viscosity to the limit of locking of the device typically leads to an increased modal resonance frequency, but may lead to a decrease. A cubic approximation is developed covering both cases in which the effect of the non-resonant modes is represented by an equivalent stiffness or inertia term, respectively. The magnitude of the stiffness or inertia parameter is determined by the change in resonance frequency following from locking of the device, and thus the cubic root-locus curve is determined by only two parameters: the original undamped resonance frequency, and the frequency after locking. The length parameter along the curve is given by a suitably scaled non-dimensional damping parameter. It is demonstrated that when introducing the geometric mean of the unlocked and locked frequencies the complex root locus curve for increasing damping parameter consists of inverse points with respect to a circle with the geometric mean frequency as radius, and furthermore the inverse points correspond to reciprocal values of the normalized damping parameter. The explicit results of optimal damper tuning and the resulting structural damping enable direct design based solely on the three parameters, that can typically be extracted from a standard finite element analysis of the structure. Two examples demonstrate the accuracy of the cubic format, and clarify the reason for decrease of the modal frequency by damping for certain cases.