Resonance Induced by Bounded Noise

2007 ◽  
Author(s):  
Jinyu Zhu ◽  
W.-C. Xie ◽  
Ronald M. C. So
Keyword(s):  
Lyapunov Exponent ◽  
Degrees Of Freedom ◽  
Singular Perturbation Method ◽  
Bounded Noise ◽  
Stochastic Dynamical System ◽  
Moment Lyapunov Exponent ◽  
Band Characteristic ◽  
The Moment ◽  
Moment Lyapunov Exponents

Dynamic stability of a two degrees-of-freedom system under bounded noise excitation with a narrow band characteristic is studied through the determination of the moment Lyapunov exponent. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established using the theory of stochastic dynamical system. For weak noise excitations, a singular perturbation method is employed to obtain second-order expansions of the moment Lyapunov exponents. The case when the system is in combination additive resonance in the absence of noise is considered and the effect of noise on the parametric resonance is investigated.

Parametric Resonance of a Two Degrees-of-Freedom System Induced by Bounded Noise

2009 ◽  
Vol 76 (4) ◽  
Author(s):  
Jinyu Zhu ◽  
W.-C. Xie ◽  
Ronald M. C. So ◽  
X. Q. Wang
Keyword(s):  
Lyapunov Exponents ◽  
Parametric Resonance ◽  
Degrees Of Freedom ◽  
Singular Perturbation Method ◽  
Bounded Noise ◽  
Two Degrees Of Freedom ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Combined Resonance ◽  
Moment Lyapunov Exponents

The dynamic stability of a two degrees-of-freedom system under bounded noise excitation with a narrowband characteristic is studied through the determination of moment Lyapunov exponents. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. For weak noise excitations, a singular perturbation method is employed to obtain second-order expansions of the moment Lyapunov exponents and Lyapunov exponents, which are shown to be in good agreement with those obtained using Monte Carlo simulation. The different cases when the system is in subharmonic resonance, combination additive resonance, and combined resonance in the absence of noise, respectively, are considered. The effects of noise and frequency detuning on the parametric resonance are investigated.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

2002 ◽  
Vol 69 (3) ◽  
pp. 346-357 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Stochastic Fluctuation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

scholarly journals MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

2021 ◽  
Vol 19 (2) ◽  
pp. 209
Author(s):  
Goran Janevski ◽  
Predrag Kozić ◽  
Ratko Pavlović ◽  
Strain Posavljak
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Degrees Of Freedom ◽  
Simulation Method ◽  
Stochastic Dynamic ◽  
Regular Perturbation ◽  
Thin Walled ◽  
The Moment ◽  
Moment Stability ◽  
Moment Lyapunov Exponents

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments.  The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Weak Noise Expansion of Moment Lyapunov Exponents of a Two-Dimensional System Under Bounded Noise Excitation

2000 ◽  
Author(s):  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Dimensional System ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Weak Noise ◽  
Two Dimensional System ◽  
The Moment ◽  
Moment Lyapunov Exponents

Abstract The moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation are studied in this paper. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter.

Numerical Determination of Moment Lyapunov Exponents of Two-Dimensional Systems

2005 ◽  
Vol 73 (1) ◽  
pp. 120-127 ◽  
Author(s):  
Wei-Chau Xie ◽  
Ronald M. C. So
Keyword(s):  
Lyapunov Exponents ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Dimensional System ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Partial Differential ◽  
Stochastic Dynamical System ◽  
Moment Lyapunov Exponents

The pth moment Lyapunov exponent of an n-dimensional linear stochastic system is the principal eigenvalue of a second-order partial differential eigenvalue problem, which can be established using the theory of stochastic dynamical system. An analytical-numerical approach for the determination of the pth moment Lyapunov exponents, for all values of p, is presented. The approach is illustrated through a two-dimensional system under bounded noise or real noise parametric excitation. Series expansions of the eigenfunctions using orthogonal functions are employed to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

Simulation of Moment Lyapunov Exponents for Linear Homogeneous Stochastic Systems

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Wei-Chau Xie ◽  
Qinghua Huang
Keyword(s):  
Lyapunov Exponents ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Numerical Algorithms ◽  
Stochastic Dynamical Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

Moment Lyapunov exponents are important characteristic numbers for describing the dynamic stability of a stochastic system. When the pth moment Lyapunov exponent is negative, the pth moment of the solution of the stochastic system is stable. Monte Carlo simulation approaches complement approximate analytical methods in the determination of moment Lyapunov exponents and provides criteria on assessing the accuracy of approximate analytical results. For stochastic dynamical systems described by Itô stochastic differential equations, the solutions are diffusion processes and their variances may increase with time. Due to the large variances of the solutions and round-off errors, bias errors in the simulation of moment Lyapunov exponents are significant in improper numerical algorithms. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented in this paper.

Numerical Determination of the Moment Lyapunov Exponents

2003 ◽  
Author(s):  
Wei-Chau Xie ◽  
Ronald M. C. So
Keyword(s):  
Lyapunov Exponents ◽  
Eigenvalue Problems ◽  
Double Series ◽  
Numerical Approach ◽  
Dimensional System ◽  
Bounded Noise ◽  
Partial Differential ◽  
The Moment ◽  
Moment Lyapunov Exponents

Two numerical methods for the determination of the pth moment Lyapunov exponents of a two-dimensional system under bounded noise or real noise parametric excitation are presented. The first method is an analytical-numerical approach, in which the partial differential eigenvalue problems governing the moment Lyapunov exponents are established using the theory of stochastic dynamical systems. The eigenfunctions are expanded in double series to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The second method is a Monte Carlo simulation approach. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise

2001 ◽  
Vol 68 (6) ◽  
pp. 903-914 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Lyapunov Exponent ◽  
Asymptotic Approximation ◽  
Coupled Oscillators ◽  
Stability Index ◽  
Stationary Solutions ◽  
Real Noise ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment

In a recent paper an asymptotic approximation for the moment Lyapunov exponent, gp, of two coupled oscillators driven by a small intensity real noise was obtained. The utility of that result is limited by the fact that it was obtained under the assumption that the moment p is small, a limitation which precludes, for example, the determination of the stability index. In this paper that limitation is removed and an asymptotic approximation valid for arbitrary p is obtained. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation.

The moment Lyapunov exponent of a Timoshenko beam under bounded noise excitation

2010 ◽  
Vol 81 (4) ◽  
pp. 403-417 ◽  
Author(s):  
Goran Janevski ◽  
Predrag Kozić ◽  
Ratko Pavlović ◽  
Zoran Golubović
Keyword(s):  
Lyapunov Exponent ◽  
Timoshenko Beam ◽  
Bounded Noise ◽  
Moment Lyapunov Exponent ◽  
The Moment

Moment Lyapunov Exponent and Stochastic Stability of Binary Airfoil under Combined Harmonic and Non-Gaussian Colored Noise Excitations

2018 ◽  
Vol 17 (02) ◽  
pp. 1850010 ◽  
Author(s):  
D. L. Hu ◽  
X. B. Liu
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Colored Noise ◽  
Singular Perturbation Method ◽  
Random Forces ◽  
Gaussian Colored Noise ◽  
Moment Lyapunov Exponent ◽  
Path Integral Method ◽  
The Moment ◽  
Non Gaussian

Both periodic loading and random forces commonly co-exist in real engineering applications. However, the dynamic behavior, especially dynamic stability of systems under parametric periodic and random excitations has been reported little in the literature. In this study, the moment Lyapunov exponent and stochastic stability of binary airfoil under combined harmonic and non–Gaussian colored noise excitations are investigated. The noise is simplified to an Ornstein-Uhlenbeck process by applying the path-integral method. Via the singular perturbation method, the second-order expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulation. Finally, the effects of the noise and parametric resonance (such as subharmonic resonance and combination additive resonance) on the stochastic stability of the binary airfoil system are discussed.

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