Abstract
Viscoelastic materials are widely used in structural dynamics for the control of the vibrations and energy dissipation. They are characterized by damping forces that depend on the history of the velocity response via hereditary functions involved in convolution integrals, leading to a frequency-dependent damping matrix. In this paper, one-dimensional beam structures with viscoelastic materials based on fractional derivatives are considered. In this work, the construction of a new equivalent viscous system with fictitious parameters but capable of reproducing the response of the viscoelastic original one with acceptable accuracy is proposed. This allows us to take advantage of the well-known available numerical tools for viscous systems and use them to find response of viscoelastic structures. The process requires the numerical computation of complex frequencies. The new fictitious viscous parameters are found to be matching the information provided by the frequency response functions. New mass, damping, and stiffness matrices are found, which in addition have the property of proportionality, so they become diagonal in the modal space. The theoretical results are contrasted with two different numerical examples.
Highlights
The relevance of the correct modeling of damping mechanisms in structural dynamics and in particular mechanical vibrations is well known
In the specific case of a viscoelastic system based on fractional calculus, Makris [31] proposed an equivalent viscous model giving an equivalent damping ratio and a new natural frequency for viscoelastic dampers in single-degreeof-freedom systems
[−mω2 + φ (ω) k] û = F (ω) and we introduce H(ω) as the frequency response function (FRF) of the system: û = H (ω) F (ω), H (ω)
Summary
The relevance of the correct modeling of damping mechanisms in structural dynamics and in particular mechanical vibrations is well known. The challenge is to find an equivalent viscous model, for the particular case of one-dimensional viscoelastic structures, defining an equivalent damping matrix Ce and a new mass matrix Me ≠ M and a new stiffness matrix Ke ≠ K, so that the response given by the equations. In the specific case of a viscoelastic system based on fractional calculus, Makris [31] proposed an equivalent viscous model giving an equivalent damping ratio and a new natural frequency for viscoelastic dampers in single-degreeof-freedom systems. The method allows us to find three system matrices forming a new equivalent viscous system (3), say mass, damping, and stiffness, capable of representing the response of the original viscoelastic model given by (1). The motivation of this paper is double: to build a proportional viscous model whose response can be a good approximation to the exact one, given by the viscoelastic model and extending the resulting model to systems with a relatively high viscoelasticity
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