A Viscous "Geometric" Beam Damping Model that Results in Weak Frequency Dependence of Modal Damping

Abstract

This paper considers the damped transverse vibration of flexural structures. Viscous damping models available to date, e.g. proportional damping, suffer from the deficiency that the resulting modal damping is strongly frequency-dependent, a situation not representative of experiments with built-up structures. Strain-based viscous damping, which corresponds to the case of stiffness-proportional damping, yields modal damping that increases linearly with frequency. Motion-based viscous damping, which corresponds to the case of massproportional damping, yields modal damping that decreases linearly with frequency. The focus model addresses a viscous “geometric” damping term in which a resisting shear force is proportional to the time rate of change of the slope. Separation of variables does not lead directly to a solution of the governing partial differential equation, although a boundaryvalue eigenvalue problem for free vibration can nevertheless be posed and solved numerically. In a discretized finite element context, the resulting damping matrix resembles the geometric stiffness matrix used to account for the effects of membrane loads on lateral stiffness. For a beam having simply-supported boundary conditions, this model yields constant modal damping that is independent of frequency. For more general boundary conditions, modal damping varies somewhat, approaching the expected constant value with increasing mode number. For simply-supported boundary conditions, the corresponding mode shapes are real, even though the damping is non-proportional. Such a viscous damping model should prove useful to researchers and engineers who need a time-domain damping model that exhibits realistic frequency-independent damping.