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

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.

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

10.1115/1.1445143 ◽

2002 ◽

Vol 69 (3) ◽

pp. 346-357 ◽

Cited By ~ 8

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 ◽

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.

Numerical Determination of the Moment Lyapunov Exponents

Applied Mechanics and Biomedical Technology ◽

10.1115/imece2003-55486 ◽

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 ◽

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

10.1115/1.2775496 ◽

2008 ◽

Vol 75 (2) ◽

Cited By ~ 23

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 ◽

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.

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

10.1115/1.2999427 ◽

2009 ◽

Vol 76 (4) ◽

Cited By ~ 5

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 ◽

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.

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

10.1115/imece2000-1752 ◽

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 ◽

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.

The Response of a Rectangular Thin Plate Subjected to In-Plane Narrow Band Noise Excitation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.764-765.379 ◽

2015 ◽

Vol 764-765 ◽

pp. 379-382

Author(s):

Gen Ge ◽

Jing Xian Wei ◽

Wen Di Zhang

Keyword(s):

Floquet Theory ◽

Narrow Band ◽

Multiple Scale ◽

Order Moment ◽

Narrow Band Noise ◽

Band Noise ◽

Steady State Responses ◽

The Moment ◽

State Responses ◽

Moment Stability

One stochastic dynamical model of a thin rectangular plate subject to in-plate parametrical narrow band noise excitation is proposed based on elastic theory and Galerkin’s approach. At first the model is simplified applying the multiple scale method and the averaged equation in Ito form is obtained. Secondly, the stochastic moment stability of the steady state responses are analyzed by Floquet theory and the moment method. Finally, the second order moment of the system is obtained, which can be used to estimate the variance of the responses. The numerical results agree with the theatrical analysis.

Numerical Determination of Moment Lyapunov Exponents of Two-Dimensional Systems

10.1115/1.2041663 ◽

2005 ◽

Vol 73 (1) ◽

pp. 120-127 ◽

Cited By ~ 1

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 ◽

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.

Response of a Shape Memory Alloy Beam Model under Narrow Band Noise Excitation

Mathematical Problems in Engineering ◽

10.1155/2014/985467 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Gen Ge

Keyword(s):

Shape Memory Alloy ◽

Shape Memory ◽

Narrow Band ◽

Multiple Scale ◽

Nonlinear Damping ◽

Beam Model ◽

Narrow Band Noise ◽

Band Noise ◽

The Stability ◽

The Moment

To describe the hysteretic nonlinear characteristic of the strain-stress relation of shape memory alloy (SMA), a Van-der-Pol hysteretic cycle is applied to simulate the hysteretic loops. Then, the model of a simply supported SMA beam subject to transverse narrow band noise excitation with nonlinear damping was proposed. The deterministic and the stochastic responses are studied, respectively, applying the multiple scale method. The stability of the steady state responses is analyzed by Floquet theory and the moment method. The numerical simulation results quite agree with the theoretical analysis.