The Routh-Hurwitz method for stability determination of linear differential-difference systems†

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear differential equations modelling wave propagation in spatially inhomogeneous media at caustic (turning) points. Here the formalism is adapted to determine a class of asymptotic solutions at caustic points for those equations modelling wave propagation in media with both spatial and temporal inhomogeneities. The analogous Schrodinger equation is also considered.

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Linear differential and difference systems: EGδ- and EGσ- eliminations

Programming and Computer Software ◽

10.1134/s0361768813020023 ◽

2013 ◽

Vol 39 (2) ◽

pp. 91-109 ◽

Cited By ~ 12

Author(s):

S. A. Abramov ◽

D. E. Khmelnov

Keyword(s):

Difference Systems ◽

Linear Differential

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5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500016553 ◽

1976 ◽

Vol 74 ◽

pp. 71-79

Author(s):

R. Datko

Keyword(s):

Asymptotic Stability ◽

Differential Equations ◽

Difference Equations ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Uniform Asymptotic Stability ◽

Linear Ordinary Differential Equations ◽

Linear Differential ◽

Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

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Small Vibrations Superimposed on Non-Linear Rigid Body Motion

Volume 1A: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4204 ◽

1997 ◽

Author(s):

A. L. Schwab ◽

J. P. Meijaard

Keyword(s):

Rigid Body ◽

Equations Of Motion ◽

Harmonic Balance Method ◽

Rigid Motion ◽

Rigid Body Motion ◽

Body Motion ◽

Connecting Rod ◽

Non Linear ◽

Linear Differential

Abstract In the case of small elastic deformations in a flexible multi-body system, the periodic motion of the system can be modelled as a superposition of a small linear vibration and a non-linear rigid body motion. For the small deformations this analysis results in a set of linear differential equations with periodic coefficients. These equations give more insight in the vibration phenomena and are computationally more efficient than a direct non-linear analysis by numeric integration. The realization of the method in a program for flexible multibody systems is discussed which requires, besides the determination of the periodic rigid motion, the determination of the linearized equations of motion. The periodic solutions for the linear equations are determined with a harmonic balance method, while transient solutions are obtained by averaging. The stability of the periodic solution is considered. The method is applied to a pendulum with a circular motion of its support point and a slider-crank mechanism with flexible connecting rod. A comparison is made with previous non-linear results.

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Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

Journal of Applied Mechanics ◽

10.1115/1.4049836 ◽

2021 ◽

Vol 88 (5) ◽

Author(s):

Fan Kong ◽

Pol D. Spanos

Keyword(s):

Harmonic Balance Method ◽

Linearization Method ◽

System Response ◽

Statistical Linearization ◽

Algebraic Equations ◽

Stochastic Response ◽

Single Degree Of Freedom ◽

Stochastic Component ◽

Linear Differential

Abstract A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

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