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.

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.

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.

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.

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.

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.

scholarly journals Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Di Liu ◽  
Yanru Wu ◽  
Xiufeng Xie
Keyword(s):  
Lyapunov Exponent ◽  
White Noise ◽  
Lyapunov Exponents ◽  
Stochastic Stability ◽  
Time History ◽  
Kernel Functions ◽  
Direct Monte Carlo Simulation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment

Nonviscously damped structural system has been raised in many engineering fields, in which the damping forces depend on the past time history of velocities via convolution integrals over some kernel functions. This paper investigates stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents. Using the coordinate transformation, the coupled Itô stochastic differential equations of the norm of the response and angles process are obtained. Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue problem, and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system. Lyapunov exponent also can be obtained according to the relationship between moment Lyapunov exponent and Lyapunov exponent. Finally, the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed in detail. These results are validated by the direct Monte Carlo simulation technique.

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.

scholarly journals Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Stability ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Real Noise ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

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