Stochastic Stability of Linear Gyroscopic Systems: Application to Pipes Conveying Fluid

2002 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Perturbation Approach ◽  
Almost Sure Stability ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Gyroscopic Systems ◽  
Two Degree Of Freedom ◽  
Pulsating Fluid ◽  
Moment Stability

In this paper we obtain asymptotic approximations for the moment Lyapunov exponent, g(p), and the Lyapunov exponent,λ, for a two-degree-of-freedom gyroscopic system close to a double zero resonance and subjected to small damping and noisy disturbances. Using a perturbation approach, we show analytically that the moment and the top Lyapunov exponent grow in proportion to ε1/3 when the damping and noise respectively are of O(ε) and O(ε). These results, pertaining to pth moment stability and almost-sure stability of the trivial solution, are applied to study the stochastic stability of a pipe conveying pulsating fluid.

Stochastic Stability of Coupled Oscillators in Resonance: A Perturbation Approach

2004 ◽  
Vol 71 (6) ◽  
pp. 759-768 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Lyapunov Exponent ◽  
Eigenvalue Problem ◽  
Coupled Oscillators ◽  
Orthogonal Expansion ◽  
Perturbation Approach ◽  
Almost Sure Stability ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
The Galerkin Method

A perturbation approach is used to obtain an approximation for the moment Lyapunov exponent of two coupled oscillators with commensurable frequencies driven by a small intensity real noise with dissipation. The generator for the eigenvalue problem associated with the moment Lyapunov exponent is derived without any restriction on the size of pth moment. An orthogonal expansion for the eigenvalue problem based on the Galerkin method is used to derive the stability results in terms of spectral densities. These results can be applied to study the moment and almost-sure stability of structural and mechanical systems subjected to stochastic excitation.

Maximal Lyapunov Exponent and Almost-Sure Stability for Coupled Two-Degree-of-Freedom Stochastic Systems

1994 ◽  
Vol 61 (2) ◽  
pp. 446-452 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel ◽  
S. Talwar
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Asymptotic Formulas ◽  
Perturbation Approach ◽  
Maximum Lyapunov Exponent ◽  
Almost Sure Stability ◽  
Asymptotic Growth Rate ◽  
Two Degree Of Freedom

In this paper, a perturbation approach is used to calculate the asymptotic growth rate of stochastically coupled two-degree-of-freedom systems. The noise is assumed to be white and of small intensity in order to calculate the explicit asymptotic formulas for the maximum Lyapunov exponent, The Lyapunov exponents and rotation number for each degree-of-freedom are obtained in the Appendix. The almost-sure stability or instability of the four-dimensional stochastic system depends on the sign of the maximum Lyapunov exponent. The results presented here match those presented by the first author and others using the method of stochastic averaging, where approximate Itoˆ equations in amplitudes and phase are obtained in the sense of weak convergence.

scholarly journals DYNAMIC BEHAVIOR OF TWO ELASTICALLY CONNECTED NANOBEAMS UNDER A WHITE NOISE PROCESS

2020 ◽  
Vol 18 (2) ◽  
pp. 219
Author(s):  
Ivan R. Pavlović ◽  
Ratko Pavlović ◽  
Goran Janevski ◽  
Nikola Despenić ◽  
Vladimir Pajković
Keyword(s):  
Lyapunov Exponent ◽  
White Noise ◽  
Axial Loading ◽  
Regular Perturbation ◽  
White Noise Process ◽  
Noise Process ◽  
Asymptotic Expressions ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Stability

This paper investigates the almost-sure and moment stability of a double nanobeam system under stochastic compressive axial loading. By means of the Lyapunov exponent and the moment Lyapunov exponent method the stochastic stability of the nano system is analyzed for different system parameters under an axial load modeled as a wideband white noise process. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises.

scholarly journals Stochastic Stability of a Fractional Viscoelastic Plate Driven by Non-Gaussian Colored Noise

2021 ◽  
Author(s):  
Dongliang Hu ◽  
Yong Huang
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Colored Noise ◽  
Viscoelastic Plate ◽  
Dynamic Equations ◽  
Stochastic Dynamic ◽  
Gaussian Colored Noise ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Non Gaussian

Abstract In this paper, the moment Lyapunov exponent and stochastic stability of a fractional viscoelastic plate driven by non-Gaussian colored noise is investigated. Firstly, the stochastic dynamic equations with two degrees of freedom are established by piston theory and Galerkin approximate method. The fractional Kelvin–Voigt constitutive relation is used to describe the material properties of the viscoelastic plate, which leads to that the fractional derivation term is introduced into the stochastic dynamic equations. And the noise is simplified into an Ornstein-Uhlenbeck process by utilizing the path-integral method. Then, via the singular perturbation method, the approximate expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulations. Finally, the effects of the noise, viscoelastic parameters and system parameters on the stochastic dynamics of the viscoelastic plate are discussed.

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.

Stochastic Stability of a Planner Gyropendulum System with Synchronous Motor Driven by Gaussian White Noises

2019 ◽  
Vol 19 (10) ◽  
pp. 1971006
Author(s):  
Shenghong Li ◽  
Zurun Xu
Keyword(s):  
Lyapunov Exponent ◽  
Synchronous Motor ◽  
Infinite Matrix ◽  
Almost Sure Stability ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
White Noises

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.

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.

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.

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