Mario Lázaro

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Abstract Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non-proportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in fixed point theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.

Nonviscous Modes of Nonproportionally Damped Viscoelastic Systems

Nonviscously damped vibrating systems are characterized by dissipative mechanisms depending on the time-history of the response velocity, introduced in the physical models using convolution integrals involving hereditary kernel functions. One of the most used damping viscoelastic models is the Biots model, whose hereditary functions are assumed to be exponential kernels. The free-motion equations of these types of nonviscous systems lead to a nonlinear eigenvalue problem enclosing certain number of the so-called nonviscous modes with nonoscillatory nature. Traditionally, the nonviscous modes (eigenvalues and eigenvectors) for nonproportional systems have been computed using the state-space approach, computationally expensive. In this paper, we address this problem developing a new method, computationally more efficient than that based on the state-space approach. It will be shown that real eigenvalues and eigenvectors of viscoelastically damped system can be obtained from a linear eigenvalue problem with the same size as the physical system. The numerical approach can even be enhanced to solve highly damped problems. The theoretical results are validated using a numerical example.

Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

The authors gratefully acknowledge the support of the Polytechnic University of Valencia under Program Nos. PAID 02-11-1828 and 05-10-2674.

Analysis of Nonviscous Oscillators Based on the Damping Model Perturbation

A novel numerical approach to compute the eigenvalues of linear viscoelastic oscillators is developed. The dissipative forces of these systems are characterized by convolution integrals with kernel functions, which in turn contain a set of damping parameters. The free-motion characteristic equation defines implicitly the eigenvalues as functions of such parameters. After choosing one of them as independent variable, the key idea of the current paper is to obtain a differential equation whose solution can be considered, under certain conditions, a good approximation. The method is validated with several numerical examples related to damping models based on exponential kernels, on fractional derivatives, and on the well-known viscous model. Taylor series expansions up to the second order are obtained and in addition analytical solutions for the viscous model are achieved. The numerical results are very close to the exact ones for light and medium levels of damping and also very good for high levels if the chosen parameter is close to initial values that are defined for every case.

Nonviscous Modes of Viscoelastically Damped Vibrating Systems

Nonviscously damped vibrating systems are characterized by dissipative mecha‐ nisms depending on the time history of the response velocity, introduced in the physical models using convolution integrals involving hereditary kernel functions. One of the most used damping viscoelastic models is Biot’s model, whose hereditary functions are assumed to be exponential kernels. The free-motion equations of these types of nonviscous systems lead to a nonlinear eigenvalue problem enclosing certain number of the so-called nonviscous modes with nonoscillatory nature. Traditionally, the nonviscous modes (eigenvalues and eigenvectors) for nonproportional systems have been computed using the state-space approach, computationally expensive. This number of real eigenvalues is directly related to the rank of the damping matrices associated with the exponential kernels. The state-space approach has traditionally been used up to now as the only method to compute the nonviscous modes for nonpropor‐ tionally damped systems. Motivated by this open problem, we propose in this chapter to describe the available numerical methods for classically damped systems and present the recent methods for nonclassically damped systems. It is shown that the problem of finding the nonviscous modes can be reduced to solve as a set of linear eigenvalue problems. The presented methods are compared through a numerical example.

The Polynomial Pivots as Initial Values for a New Root-Finding Iterative Method

A new iterative method for polynomial root-finding based on the development of two novel recursive functions is proposed. In addition, the concept of polynomial pivots associated with these functions is introduced. The pivots present the property of lying close to some of the roots under certain conditions; this closeness leads us to propose them as efficient starting points for the proposed iterative sequences. Conditions for local convergence are studied demonstrating that the new recursive sequences converge with linear velocity. Furthermore, an a priori checkable global convergence test inside pivots-centered balls is proposed. In order to accelerate the convergence from linear to quadratic velocity, new recursive functions together with their associated sequences are constructed. Both the recursive functions (linear) and the corrected (quadratic convergence) are validated with two nontrivial numerical examples. In them, the efficiency of the pivots as starting points, the quadratic convergence of the proposed functions, and the validity of the theoretical results are visualized.

Proposal of a Viscous Model for Nonviscously Damped Beams Based on Fractional Derivatives

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.