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 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. 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.
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 nonproportional 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.
In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-histoiy of the response velocity, which is mathematically represented by convolution integrals involving hereditaiy functions. The widespread Biot damping model assumes that such functions are exponential kernels, which modify the eigenvalues' set so that as many real eigenvalues (named nonviscous eigenvalues) as kernels are added to the system. This paper is focused on the study of a mathematical characterization of the nonviscous eigenvalues. The theoretical results allow the bounding of a set belonging to the real negative numbers, called the nonviscous set, constructed as the union of closed intervals. Exact analytical solutions of the nonviscous set for one and two exponential kernels and approximated solutions for the general case ofN kernels are developed. In addition, the nonviscous set is used to build closedform expressions to compute the nonviscous eigenvalues. The results are validated with numerical examples covering single and multiple degree-of-freedom systems where the proposed method is compared with other existing one-step approaches available in the literature.
In structural dynamics, energy dissipative mechanisms with non-viscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are called critical damping surfaces (or manifolds, since we can have a lot of parameters). In this paper a general method to obtain critical damping surfaces for nonviscously damped systems is proposed. The approach is based on transforming the algebraic equations which defined implicitly the critical curves into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.
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