APPLICATION OF THE VIBRATION ANALYSIS OF LINEAR SYSTEMS WITH TIME-PERIODIC COEFFICIENTS TO THE DYNAMICS OF A ROLLING TYRE

In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

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Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4026989 ◽

2015 ◽

Vol 10 (2) ◽

Cited By ~ 4

Author(s):

Anwar Sadath ◽

C. P. Vyasarayani

Keyword(s):

Differential Equations ◽

Delay Differential Equations ◽

Approximate Model ◽

Periodic Coefficients ◽

Galerkin Approximations ◽

Approximate System ◽

The Stability ◽

Delay Differential ◽

Time Periodic ◽

Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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Vibration Analysis of Non-Classically Damped Linear Systems

Journal of Vibration and Acoustics ◽

10.1115/1.1760563 ◽

2004 ◽

Vol 126 (3) ◽

pp. 456-458 ◽

Cited By ~ 1

Author(s):

Z. S. Liu, ◽

D. T. Song, and ◽

C. Huang ◽

D. J. Wang ◽

S. H. Chen

Keyword(s):

Linear Systems ◽

Physical Meaning ◽

Vibration Analysis ◽

Original System ◽

New Method ◽

Order System ◽

Damping Term ◽

Clear Physical Meaning ◽

Damped System

This Technical Brief presents a new method for vibration analysis of a non-classically damped system. The basic idea is to introduce a transformation, which bears clear physical meaning, so that the original non-classical damped system is transformed into a new 2nd-order system that does not have the damping term. The transformed system not only provides an alternative of calculating response, but also reveals more clearly vibration behaviors of the original system.

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Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.3124093 ◽

2009 ◽

Vol 4 (3) ◽

Cited By ~ 1

Author(s):

Yandong Zhang ◽

S. C. Sinha

Keyword(s):

Normal Form ◽

Design Problem ◽

Controller Design ◽

Design Method ◽

Observer Design ◽

Periodic Systems ◽

Normal Form Theory ◽

Periodic Coefficients ◽

System States ◽

Time Periodic

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

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Reducibility of linear systems of difference equations with almost periodic coefficients

Ukrainian Mathematical Journal ◽

10.1007/bf01061357 ◽

1993 ◽

Vol 45 (12) ◽

pp. 1869-1877

Author(s):

Yu. A. Mitropol'skii ◽

D. I. Martynyuk ◽

V. I. Tynnyi

Keyword(s):

Linear Systems ◽

Difference Equations ◽

Almost Periodic ◽

Periodic Coefficients

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