Asymptotic Treatment of Non-Classically Damped Linear Systems

1995 ◽  
Vol 48 (11S) ◽  
pp. S111-S117
Author(s):  
W. Fang ◽  
J.-G. Tseng ◽  
J. A. Wickert
Keyword(s):  
Eigenvalue Problem ◽  
Dynamic System ◽  
Perturbation Method ◽  
Quadratic Forms ◽  
Vibration Modes ◽  
Discrete Dynamic System ◽  
Damped System ◽  
Base Solution ◽  
The Stability ◽  
Dissipative Terms

The presence of non-classical dissipation in a general discrete dynamic system is investigated through a perturbation method for the eigenvalues and vectors. Results accurate to second-order are obtained, with corrections to the base solution being expressed in terms of readily-calculated quadratic forms. Exact solutions, and the derived asymptotic ones, are compared with the predictions of the so-called method of approximate decoupling, in which certain non-classical dissipative terms are omitted from calculations in the eigenvalue problem. The perturbation method is discussed through its application in several examples, indicating circumstances in which a non-classically damped system can be well-approximated by an “equivalent” classically damped one. Somewhat surprisingly, the addition of non-classical damping does not necessarily increase the stability of all vibration modes, and the perturbation method is shown to be useful in identifying those critical modes.

Transient Optimization of an Electrically-Damped Cantilever-Supported Microactuator and the Pull-In Analysis

2000 ◽  
Author(s):  
Yijian Chen ◽  
Yashesh Shroff ◽  
William G. Oldham
Keyword(s):  
Optimal Control ◽  
Simple Model ◽  
Dynamic System ◽  
Perturbation Method ◽  
Performance Index ◽  
Transient Behavior ◽  
Linear Control ◽  
Control Parameters ◽  
Linear Control Theory ◽  
The Stability

Abstract Analytic modeling of the transient behavior of an electrically-damped cantilever-supported microactuator using the perturbation method and linear control theory is presented. Five control parameters are identified and the transient optimization of the dynamic system to reduce the overshoot and settling time is carried out. With the ITAE performance index minimized, the optimal control parameters are obtained and the resultant optimized transient behavior is shown. We apply the Routh-Hurwitz criterion to analyze the stability of the dynamic system and three inequality relations for a stable system are derived. The pull-in phenomenon for a short-cantilever actuator is investigated with this simple model.

scholarly journals A Modal Perturbation Method for Eigenvalue Problem of Non-Proportionally Damped System

2020 ◽  
Vol 10 (1) ◽  
pp. 341
Author(s):  
Danguang Pan ◽  
Xiangqiu Fu ◽  
Qingjun Chen ◽  
Pan Lu ◽  
Jinpeng Tan
Keyword(s):  
Perturbation Method ◽  
State Space ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Algebraic Equations ◽  
Engineering Structures ◽  
Damped System ◽  
Real Mode ◽  
Nonlinear Algebraic Equations ◽  
Practical Engineering

The non-proportionally damped system is very common in practical engineering structures. The dynamic equations for these systems, in which the damping matrices are coupled, are very time consuming to solve. In this paper, a modal perturbation method is proposed, which only requires the first few lower real mode shapes of a corresponding undamped system to obtain the complex mode shapes of non-proportionally damped system. In this method, an equivalent proportionally damped system is constructed by taking the real mode shapes of a corresponding undamped system and then transforming the characteristic equation of state space into a set of nonlinear algebraic equations by using the vibration modes of an equivalent proportionally damped system. Two numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method. The numerical results show that: (1) with the increase of vibration modes of the corresponding undamped system, the eigenvalues and eigenvectors monotonically converge to exact solutions; (2) the accuracy of the proposed method is significantly higher than the first-order perturbation method and proportional damping method. The calculation time of the proposed method is shorter than the state space method; (3) the method is particularly suitable for finding a few individual orders of frequency and mode of a system with highly non-proportional damping.

Stability and Natural Characteristics of a Supported Beam

2011 ◽  
Vol 338 ◽  
pp. 467-472 ◽  
Author(s):  
Ji Duo Jin ◽  
Xiao Dong Yang ◽  
Yu Fei Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Axial Force ◽  
Critical Values ◽  
Rotational Spring ◽  
Analytical Expression ◽  
The Stability ◽  
Spring Constants ◽  
Critical Axial Force ◽  
Natural Characteristics

The stability, natural characteristics and critical axial force of a supported beam are analyzed. The both ends of the beam are held by the pinned supports with rotational spring constraints. The eigenvalue problem of the beam with these boundary conditions is investigated firstly, and then, the stability of the beam is analyzed using the derived eigenfuntions. According to the analytical expression obtained, the effect of the spring constants on the critical values of the axial force is discussed.

Stability Considerations in Thermoelastic Contact

1980 ◽  
Vol 47 (4) ◽  
pp. 871-874 ◽  
Author(s):  
J. R. Barber ◽  
J. Dundurs ◽  
M. Comninou
Keyword(s):  
Steady State ◽  
Perturbation Method ◽  
Energy Function ◽  
First Principles ◽  
Dimensional Model ◽  
Contact Conditions ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
The Stability ◽  
Steady State Solutions

A simple one-dimensional model is described in which thermoelastic contact conditions give rise to nonuniqueness of solution. The stability of the various steady-state solutions discovered is investigated using a perturbation method. The results can be expressed in terms of the minimization of a certain energy function, but the authors have so far been unable to justify the use of such a function from first principles in view of the nonconservative nature of the system.

scholarly journals Dynamics of f(R) gravity models and asymmetry of time

2018 ◽  
Vol 27 (02) ◽  
pp. 1850002 ◽  
Author(s):  
Murli Manohar Verma ◽  
Bal Krishna Yadav
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Scale Factor ◽  
Gravity Models ◽  
Field Equations ◽  
Classical Level ◽  
The Matrix ◽  
The Stability ◽  
Effects Of Radiation ◽  
Linear And Quadratic Forms

We solve the field equations of modified gravity for [Formula: see text] model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase-space analysis of [Formula: see text] models, both with and without the effects of radiation. The stability of these points is studied against the perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analyzing the dynamics of the system during the radiation, matter and acceleration-dominated phases of the universe. Certain linear and quadratic forms of [Formula: see text] are determined from the geometrical and physical considerations and the behavior of the scale factor is found for those forms. Further, we also determine the Hubble parameter [Formula: see text], the Ricci scalar [Formula: see text] and the scale factor [Formula: see text] for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the [Formula: see text] gravity in the Einstein frame that may lead to an arrow of time at a classical level.

scholarly journals On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 712 ◽  
Author(s):  
Manuel De la Sen
Keyword(s):  
Dynamic System ◽  
Output Feedback ◽  
Iteration Step ◽  
Stability And Convergence ◽  
Feedback Controls ◽  
Time Aggregation ◽  
The Stability ◽  
The Given ◽  
Stability Results

This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

scholarly journals Instability Through Anisotropy in Spherical Stellar Systems

1987 ◽  
Vol 127 ◽  
pp. 515-516
Author(s):  
P.L. Palmer ◽  
J. Papaloizou
Keyword(s):  
Growth Rate ◽  
Eigenvalue Problem ◽  
Growth Rates ◽  
Stellar Systems ◽  
Matrix Eigenvalue Problem ◽  
Simple Method ◽  
Anisotropic Models ◽  
Poisson Equations ◽  
Degree Of Anisotropy ◽  
The Stability

We consider the linear stability of spherical stellar systems by solving the Vlasov and Poisson equations which yield a matrix eigenvalue problem to determine the growth rate. We consider this for purely growing modes in the limit of vanishing growth rate. We show that a large class of anisotropic models are unstable and derive growth rates for the particular example of generalized polytropic models. We present a simple method for testing the stability of general anisotropic models. Our anlysis shows that instability occurs even when the degree of anisotropy is very slight.

scholarly journals FAULT DETECTION IN DISCRETE DYNAMIC SYSTEMS WITH UNCERTAINTY BASED ON INTERVAL OPTIMIZATION

2011 ◽  
Vol 16 (4) ◽  
pp. 549-557 ◽  
Author(s):  
Wei Li ◽  
Xiaoli Tian
Keyword(s):  
Fault Detection ◽  
Dynamic System ◽  
Dynamic Systems ◽  
Optimization Model ◽  
Uncertain Parameters ◽  
Interval Optimization ◽  
Numerical Examples ◽  
Discrete Dynamic ◽  
Discrete Dynamic System ◽  
Discrete Dynamic Systems

The imprecision and the uncertainty of many systems can be expressed with interval models. This paper presents a method for fault detection in uncertain discrete dynamic systems. First, the discrete dynamic system with uncertain parameters is formulated as an interval optimization model. In this model, we also assume that there are some errors of observation values of the inputs/outputs. Then, M. Hladík's newly proposed algorithm is exploited for this model. Some numerical examples are also included to illustrate the method efficiency.

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