Dynamics of Non-viscously Damped Distributed Parameter Systems

Abstract

Linear dynamics of Euler-Bernoulli beams with non-viscous non-local damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at difierent points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such general damping model results in a linear partial integro-difierential equation. Exact closed-form expressions of the natural frequencies and mode-shapes of the beam are derived. The analytical method is capable of handling complex boundary conditions. Numerical examples are provided to illustrate the new results. he flnite element method, coupled with model updating techniques, allow us to accurately model the mass and the stifiness properties of complex engineering structures and subsequently to analyze their dynamics. Determination of the dynamic response, which is crucial for the design of a structure, not only depends on the mass and stifiness properties but also heavily depends on the energy dissipation properties or ‘damping’. The capabilities of modern design tools, in terms of the modelling and analyzing of damping properties, are not as advanced as for the mass and stifiness properties. There are several reasons behind this: (a) by contrast with inertia and stifiness forces, it is not in general clear which state variables are relevant to determine the damping forces, (b) the spatial location of the damping sources are generally unclear - often the structural joints are more responsible for the energy dissipation than the (solid) material, (c) the functional form of the damping model is di‐cult to establish experimentally, and flnally, (d) even if one manages to address the previous issues, what parameters should be used in a chosen model is still very much an open problem. Because of these di‐culties modelling of damping from flrst principles is very di‐cult, if not impossible, for complex engineering structures. Over the years, and even now, structural dynamicists have bypassed these problems by using the viscous damping model. A vast literature, starting from Lord Rayleigh’s 1 classic monograph, ‘Theory of Sound’, is available on the viscous damping model. With a viscous damping model, it is assumed that the instantaneous generalized velocities are the only relevant state variables that determine damping. Viscous damping is by no means the only damping model