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.

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.

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.

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.

scholarly journals Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems

2021 ◽  
Vol 11 (8) ◽  
pp. 3430
Author(s):  
Erik Cuevas ◽  
Héctor Becerra ◽  
Héctor Escobar ◽  
Alberto Luque-Chang ◽  
Marco Pérez ◽  
...  
Keyword(s):  
Convergence Rates ◽  
Optimization Problems ◽  
Search Space ◽  
Second Order ◽  
Order System ◽  
Search Patterns ◽  
Multiple Behaviors ◽  
Complex Optimization ◽  
Second Order Systems ◽  
Order Algorithm

Recently, several new metaheuristic schemes have been introduced in the literature. Although all these approaches consider very different phenomena as metaphors, the search patterns used to explore the search space are very similar. On the other hand, second-order systems are models that present different temporal behaviors depending on the value of their parameters. Such temporal behaviors can be conceived as search patterns with multiple behaviors and simple configurations. In this paper, a set of new search patterns are introduced to explore the search space efficiently. They emulate the response of a second-order system. The proposed set of search patterns have been integrated as a complete search strategy, called Second-Order Algorithm (SOA), to obtain the global solution of complex optimization problems. To analyze the performance of the proposed scheme, it has been compared in a set of representative optimization problems, including multimodal, unimodal, and hybrid benchmark formulations. Numerical results demonstrate that the proposed SOA method exhibits remarkable performance in terms of accuracy and high convergence rates.

Two-sided form, differentiation and second-order observation in Escher’s artworks and Calvino’s stories

Kybernetes ◽  
2019 ◽  
Vol 48 (5) ◽  
pp. 1060-1077
Author(s):  
Laura Appignanesi
Keyword(s):  
Twentieth Century ◽  
Systems Theory ◽  
Second Order ◽  
Order System ◽  
Literary Works ◽  
Content Type ◽  
Theoretical Concepts ◽  
Art And Literature ◽  
Second Order Systems ◽  
Late Works

Purpose The purpose of this paper is to find a leading idea of the mid-twentieth century, demonstrating the pervasive nature of some concepts belonging to second-order systems theory. To achieve this objective, the paper looks at the art and literature of this era, to identify the principles developed by Luhmann in his late works. In particular, Escher’s drawings, Calvino’s stories and Luhmann’s concepts seem to express, in different ways, the same functioning mechanism of the complex social system. Design/methodology/approach With reference to theoretical approach and methodology, this paper carries out an interdisciplinary demonstration by alternative modes of logos and mythos. Some of the pillars of general systems theory are examined through the logical articulation of concepts developed by Spencer-Brown, von Foerster, and first of all through the late works of Luhmann, as well as through the analysis of Escher’s artworks and Calvino’s literary works. This paper interprets these artistic and literary works using cybernetic principles and systemic concepts, in particular, “two-sided forms,” “system–environment differentiation” and “second-order observation.” Findings In general, the main finding is the similarity of fascination with paradoxes and forms, with post-ontological reasoning, in twentieth century. The result of the cross-reading of Escher, Calvino and Luhmann reveals the presence of what Simmel called the “hidden king”: a philosophical paradigm of an era. In mid-1900s, this leading idea seems to express itself in the discoveries of biology and cybernetics, such as in Luhmann’s theory, art and literature. Escher’s drawings, Calvino’s stories and the concepts of Luhmann are projections of second-order system theory, in its constructivist value. Originality/value The originality of this paper lies mainly in the demonstration of theoretical concepts through the alternative modes of logos and mythos. These reflections can provide a new perspective to investigate social sciences from a cultural angle. This particular approach allows a deep awareness of the theory. The concrete value is to provide a better understanding to manage complexity.

Periodic Behavior of a Nonlinear Third Order Vibrating System

2002 ◽  
Author(s):  
Gholamreza Nakhaie Jazar ◽  
Mohammad H. Alimi ◽  
Mohammad Mahinfalah ◽  
Ali Khazaei
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Ad Hoc ◽  
Order Differential Equation ◽  
Second Order ◽  
Order System ◽  
Third Order ◽  
Modeling Process ◽  
Third Order Differential Equation ◽  
Second Order Systems

In modeling of dynamical systems, differential equations, either ordinary or partial, are a common outcome of the modeling process. The basic problem becomes the existence of solution of these deferential equations. In the early days of the solution of deferential equations at the beginning of the eighteenth century the methods for determining the existence of nontrivial solution were so limited and developed very much on an ad hoc basis. Most of the efforts on dynamical system are related to the second order systems, derived by applying Newton equation of motion to dynamical systems. But, behavior of some dynamical systems is governed by equations falling down in the general nonlinear third order differential equation x″′+f(t,x,x′,x″)=0, sometimes as a result of combination of a first and a second order system. It is shown in this paper that these equations could have nontrivial solutions, if x, x′, x″, and f(t,x,x′,x″) are bounded. Furthermore, it is shown that the third order differential equation has a τ-periodic solution if f(t,x,x′,x″) is an even function with respect to x′. For this purpose, the concept of Green’s function and the Schauder’s fixed-point theorem has been used.

scholarly journals Filtering Method Based on Symmetrical Second Order Systems

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 813
Author(s):  
Cristian Toma
Keyword(s):  
Time Moment ◽  
Sampling Procedure ◽  
Second Order ◽  
Sampling Time ◽  
Half Period ◽  
Order System ◽  
Filtering Method ◽  
Oscillating Systems ◽  
Sampling Structure ◽  
Second Order Systems

This study presents a filtering and sampling structure based on symmetrical second order systems working on half-period. It is shown that undamped second order oscillating systems working on half-period could provide: (i) a large attenuation coefficient for an alternating signal (due to the filtering second order system), and (ii) a robust sampling procedure (the slope of the generated output being zero at the sampling time moment). Unlike previous studies on the same topics, these results are achieved without the use of an additional integrator.

scholarly journals Extracting second-order system matrices from state space system

2006 ◽  
Vol 220 (8) ◽  
pp. 1147-1149 ◽  
Author(s):  
P R Houlston
Keyword(s):  
Technical Note ◽  
Second Order ◽  
Order System ◽  
System Modelling ◽  
Direct Control ◽  
Space System ◽  
First Order ◽  
The Reformation ◽  
State Space System ◽  
Second Order Systems

This technical note concerns the reformation of a second-order system from an arbitrary first-order system. At present, the majority of control literature is concerned with controlling systems within the first-order linearization of a system. The author is part of a growing community looking to expand the direct control of second-order systems and the benefits associated in doing so. However, there are potential stages of system modelling that may result in it being necessary to form the first-order form of the system, such as model reduction. This may have the effect of destroying the second-order notion of the system. The purpose of this note is to regain the structure of the second-order system and thus enable the benefits of direct second-order control to be realized. Although the problem itself has been previously resolved, the author proposes the virtue of a simpler method.

scholarly journals Second-Order Systems With Singular Mass Matrix and an Extension of Guyan Reduction

1995 ◽  
Author(s):  
Sanjay P. Bhat ◽  
Dennis S. Bernstein
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Mass Matrix ◽  
Regular System ◽  
Second Order ◽  
Order System ◽  
Coupled Subsystems ◽  
Singular Mass Matrix ◽  
Guyan Reduction ◽  
Second Order Systems

Abstract The set of consistent initial conditions for a second-order system with singular mass matrix is obtained. In general, such a system can be decomposed (i.e., partitioned) into three coupled subsystems of which the first is algebraic, the second is a regular system of first-order differential equations, and the third is a regular system of second-order differential equations. Under specialized conditions, these subsystems are decoupled. This result provides an extension of Guyan reduction to include viscous damping.

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