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.

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.

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.

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.

Stability of SDOF Linear Viscoelastic System Under the Excitation of Narrow-Band Noise

2007 ◽  
Author(s):  
Qinghua Huang ◽  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponents ◽  
Eigenvalue Problems ◽  
Narrow Band ◽  
Compressive Load ◽  
Bounded Noise ◽  
Viscoelastic System ◽  
Narrow Band Noise ◽  
Band Noise ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a single degree-of-freedom (SDOF) viscoelastic system under the excitation of a narrow-band noise, which is described as a bounded noise, is studied in this paper. An example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The equation of motion is an integro-differential equation with parametric excitation. The method of stochastic averaging for integro-differential equations, both first-order and second-order, is applied and the eigenvalue problems governing the moment Lyapunov exponents are established. Numerical results from Monte Carlo simulation are compared with the approximate analytical results, and the variations of the moment Lyapunov exponents with the change of different parameters are discussed.

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.

Stability of SDOF Linear Viscoelastic System Under the Excitation of Wideband Noise

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Qinghua Huang ◽  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponents ◽  
Compressive Load ◽  
Viscoelastic System ◽  
Single Degree Of Freedom ◽  
Wideband Noise ◽  
Viscoelastic Column ◽  
Linear Viscoelastic ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a single degree-of-freedom viscoelastic system under the excitation of a wideband noise are studied in this paper. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The method of averaging, both first order and second order, is applied. The averaged Itô differential equation governing the pth norm is established and the pth moment Lyapunov exponent is then obtained. White noise and real noise are considered as models of wideband noises. The variations of the moment Lyapunov exponents with the change of different parameters are discussed.

Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System

2000 ◽  
Vol 68 (3) ◽  
pp. 453-461 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Perturbation ◽  
Pitchfork Bifurcation ◽  
Two Dimensional ◽  
Point Boundary ◽  
Moment Lyapunov Exponent ◽  
Linearized System ◽  
The Moment ◽  
Moment Lyapunov Exponents

The Lyapunov exponents and moment Lyapunov exponents of a near-nilpotent system under stochastic parametric excitation are studied. The system considered is the linearized system of a two-dimensional nonlinear system exhibiting a pitchfork bifurcation. The effect of stochastic perturbation in the vicinity of static pitchfork bifurcation is investigated. Approximate analytical results of Lyapunov exponent are obtained. The eigenvalue problem for the moment Lyapunov exponent is converted to a two-point boundary value problem, which is solved numerically by the method of relaxation.

Monte Carlo Simulation of Moment Lyapunov Exponents

2005 ◽  
Vol 72 (2) ◽  
pp. 269-275 ◽  
Author(s):  
Wei-Chau Xie
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Lyapunov Exponents ◽  
Stochastic Systems ◽  
Simulation Method ◽  
Monte Carlo Simulation Method ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Moment Stability ◽  
Moment Lyapunov Exponents

A Monte Carlo simulation method for determining the pth moment Lyapunov exponents of stochastic systems, which governs the pth moment stability, is developed. Numerical results of two-dimensional systems under bounded noise and real noise excitations are presented to illustrate the approach.

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