Dynamic Response of Structures with Frequency Dependent Damping Models

Abstract

The calculation of dynamic response of multiple-degree-of-freedom linear systems with frequency dependent damping is considered. It is assumed that the damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. Exact closed-form expressions for the dynamic response due to general forces and initial conditions are derived in terms of the eigensolutions of the system in the original space. Eigensolutions of the damped system in turn are obtained approximately as functions of the undamped eigensolutions. This enables one to approximately calculate the dynamic response of complex systems with frequency dependent damping by simple post-processing of the undamped eigensolutions. Suitable examples are given to illustrate the derived results. he characterization of energy dissipation or damping in complex vibrating structures such as aircrafts and helicopters is of fundamental importance. Noise and vibration are not only uncomfortable to the users of these complex dynamical systems, but also may lead to fatigue, fracture and even failure of such systems. Increasing use of composite structural materials, active control and health-monitoring systems in the industry requires the development of new reliable damping models and corresponding computational methods. This paper is aimed at developing computationally e‐cient and physically insightful approximate numerical methods for linear dynamical systems with frequency dependent damping. Frequency depended or non-viscous/viscoelastic damping can be incorporated in various ways. Here we use the Biot model 1 (also known as the exponential model) which allows one to incorporate wide range of functions in the frequency domain by means of summation of simple ‘pole residue forms’. Several authors have considered this model due to its simplicity and generality. Viscous damping is by far the most common model for modelling of vibration damping in linear systems. It used widely for their simplicity and mathematical convenience even though the behavior of real structural materials is, at best, poorly mimicked by simple viscous models. It is well recognized that in general a physically realistic model of damping will not be viscous. Damping models in which the dissipative forces depend on any quantity other than the instantaneous generalized velocities are non-viscous damping models. Mathematically, any causal model which makes the energy dissipation functional non-negative is a possible candidate for a non-viscous damping model. Possibly the most general way to model damping within the linear range is to use non-viscous damping models which depend on the past history of motion via convolution integrals over kernel functions. The equations of motion of a N-degree-of-freedom linear system with such damping can be expressed by