Iterative Methods for Eigenvalues of Viscoelastic Systems

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Sondipon Adhikari ◽  
Blanca Pascual
Keyword(s):  
Past History ◽  
Iterative Algorithms ◽  
Kernel Functions ◽  
Degree Of Freedom ◽  
Internal Variables ◽  
State Space Approach ◽  
Convolution Integrals ◽  
Complex Eigenvalues ◽  
History Of ◽  
Space Approach

This paper proposes a new iterative approach for the calculation of eigenvalues of single and multiple degree-of-freedom viscoelastic systems. The Biot model of viscoelasticity is assumed. With this model, the viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. The method proposed in this paper is aimed to address this issue. In total, five iterative algorithms for the real and complex eigenvalues of single and multiple degree-of-freedom systems have been proposed. The results are obtained in terms of explicit closed-form expressions. This enables one to approximately calculate the eigenvalues of complex viscoelastic systems using the eigenvalues of the underlying elastic systems. Representative numerical examples are given to verify the accuracy of the derived expressions.

Analysis of Asymmetric Nonviscously Damped Linear Dynamic Systems

2003 ◽  
Vol 70 (6) ◽  
pp. 885-893 ◽  
Author(s):  
S. Adhikari ◽  
N. Wagner
Keyword(s):  
State Space ◽  
Past History ◽  
Kernel Functions ◽  
Extended State ◽  
State Space Approach ◽  
Linear Dynamic Systems ◽  
Convolution Integrals ◽  
History Of ◽  
Damped Systems ◽  
Space Approach

Multiple-degree-of-freedom linear asymmetric nonviscously damped systems are considered. It is assumed that the nonviscous damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. An extended state-space approach involving a single asymmetric matrix is proposed. The nature of the eigensolutions in the extended state space has been explored. Some useful results relating the modal matrix in the extended state space and the modal matrix in the original space has been derived. Numerical examples are provided to illustrate the results.

A Reduced Second-Order Approach for Linear Viscoelastic Oscillators

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Sondipon Adhikari
Keyword(s):  
Past History ◽  
Kernel Functions ◽  
Second Order ◽  
Order System ◽  
Closed Form Expression ◽  
Internal Variables ◽  
Dynamical Response ◽  
Convolution Integrals ◽  
History Of ◽  
Second Order Systems

This paper proposes a new approach for the reduction in the model-order of linear multiple-degree-of-freedom viscoelastic systems via equivalent second-order systems. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here, an approximate reduced second-order approach is proposed for this type of problems. The proposed approximation utilizes the idea of generalized proportional damping and expressions of approximate eigenvalues of the system. A closed-form expression of the equivalent second-order system has been derived. The new expression is obtained by elementary operations involving the mass, stiffness, and the kernel function matrix only. This enables one to approximately calculate the dynamical response of complex viscoelastic systems using the standard tools for conventional second-order systems. Representative numerical examples are given to verify the accuracy of the derived expressions.

Modal Analysis of Nonviscously Damped Beams

2006 ◽  
Vol 74 (5) ◽  
pp. 1026-1030 ◽  
Author(s):  
S. Adhikari ◽  
M. I. Friswell ◽  
Y. Lei
Keyword(s):  
Natural Frequencies ◽  
Past History ◽  
Kernel Functions ◽  
Mode Shapes ◽  
Damping Force ◽  
The Past ◽  
Convolution Integrals ◽  
Special Cases ◽  
History Of ◽  
Damping Model

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal 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 different 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 a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Lancaster’s Method of Damping Identification Revisited

2002 ◽  
Vol 124 (4) ◽  
pp. 617-627 ◽  
Author(s):  
Sondipon Adhikari
Keyword(s):  
Transfer Functions ◽  
Past History ◽  
A Priori ◽  
Kernel Functions ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Damping Matrix ◽  
Convolution Integrals ◽  
History Of ◽  
Damped Systems

Identification of damping is an active area of research in structural dynamics. In one of the earliest works, Lancaster [1] proposed a method to identify the viscous damping matrix from measured natural frequencies and mode shapes. His method requires the modes to be normalized in a particular way, which in turn a priori needs the very same viscous damping matrix. A method, based on the poles and residues of the measured transfer functions, has been proposed to overcome this basic difficulty associated with Lancaster’s method. This approach is then extended to a class of nonviscously damped systems where the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions. Suitable numerical examples are given to illustrate the modified Lancaster’s method developed here.

Exact determination of critical damping in multiple exponential kernel-based viscoelastic single-degree-of-freedom systems

2019 ◽  
Vol 24 (12) ◽  
pp. 3843-3861 ◽  
Author(s):  
Mario Lázaro
Keyword(s):  
Past History ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Critical Curves ◽  
History Of ◽  
Definition Of ◽  
Analytical Expressions ◽  
Critical Damping

In this paper, exact closed forms of critical damping manifolds for multiple-kernel-based nonviscous single-degree-of-freedom oscillators are derived. The dissipative forces are assumed to depend on the past history of the velocity response via hereditary exponential kernels. The damping model depends on several parameters, considered variables in the context of this paper. Those parameter combinations which establish thresholds between induced overdamped and underdamped motion are called critical damping manifolds. If such manifolds are represented on a coordinate plane of two damping parameters, then they are named critical curves, so that overdamped regions are bounded by them. Analytical expressions of critical curves are deduced in parametric form, considering certain local nondimensional parameters based on the Laplace variable in the frequency domain. The definition of the new parameter (called the critical parameter) is supported by several theoretical results. The proposed expressions are validated through numerical examples showing perfect fitting of the determined critical curves and overdamped regions.

Filtering Out Expected Dividends and Expected Returns

2012 ◽  
Vol 02 (03) ◽  
pp. 1250012 ◽  
Author(s):  
Oleg Rytchkov
Keyword(s):  
Structural Breaks ◽  
Return Predictability ◽  
Expected Returns ◽  
Ratio Test ◽  
Price Ratio ◽  
Stock Return Predictability ◽  
Suggested Approach ◽  
State Space Approach ◽  
History Of ◽  
Space Approach

This paper applies a state space approach to the analysis of stock return predictability. It acknowledges that expected returns and expected dividends are unobservable and uses the Kalman filter to extract them from the observed history of realized dividends and returns. The suggested approach explicitly takes into account the time variation in expected dividend growth rates and exploits the present value relation. The obtained predictors for future returns are robust to structural breaks in the means of expected dividends and returns and more efficient than the dividend–price ratio. The likelihood ratio test reliably rejects the hypothesis of constant expected returns.

Nonviscous Modes of Nonproportionally Damped Viscoelastic Systems

2015 ◽  
Vol 82 (12) ◽  
Author(s):  
Mario Lázaro
Keyword(s):  
Eigenvalue Problem ◽  
State Space ◽  
Nonlinear Eigenvalue Problem ◽  
Time History ◽  
Kernel Functions ◽  
The State ◽  
Physical Models ◽  
Eigenvalues And Eigenvectors ◽  
State Space Approach ◽  
Space Approach

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.

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