'Geometric' Viscous Damping Model for Nearly Constant Beam Modal Damping

This paper considers the damped transverse vibration of flexural structures. Viscous damping models available to date, such as proportional damping, suffer from the deficiency that the resulting modal damping is strongly frequency dependent, which is a situation not representative of experiments with built-up structures. The focus model addresses a viscous geometric damping term in which an internal 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 boundary-value eigenvalue problem for free vibration can nevertheless be posed and solved. For small damping the method of weighted residuals provides an alternate approach to the development of approximate modal equations of motion and estimation of modal damping. 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 beams having any combination of hinged and guided boundary conditions this model yields constant modal damping that is independent of frequency as well as real mode shapes. For more general boundary conditions modal damping varies somewhat, approaching the expected constant value with increasing mode number; furthermore, the mode shapes are complex. This viscous damping model should prove useful to researchers and engineers who need a simple time-domain damping model that exhibits more realistic variation of damping with frequency than the alternatives.