Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation

1992 ◽  
Vol 59 (3) ◽  
pp. 664-673 ◽  
Author(s):  
S. T. Ariaratnam ◽  
Wei-Chau Xie
Keyword(s):  
Asymptotic Stability ◽  
Degrees Of Freedom ◽  
Elastic Beam ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
System Parameters ◽  
Asymptotic Expressions ◽  
Small Intensity ◽  
Central Cross Section ◽  
Torsional Instability

The almost-sure asymptotic stability of a class of coupled multi-degrees-of-freedom systems subjected to parametric excitation by an ergodic stochastic process of small intensity is studied. Explicit asymptotic expressions for the largest Lyapunov exponent for various values of the system parameters are obtained by using a combination of the method of stochastic averaging and a well-known procedure due to Khas’minskii, from which the asymptotic stability boundaries are determined. As an application, the example of the flexural-torsional instability of a thin elastic beam acted upon by a stochastically fluctuating load at the central cross-section of the beam is investigated.

scholarly journals Asymptotic Stability of Delay-Controlled Nonlinear Stochastic Systems with Actuator Failures

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
N. Zhou ◽  
R. H. Huan
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
Control Force ◽  
Nonlinear Stochastic Systems ◽  
Actuator Failures

The problem of asymptotic stability of delay-controlled nonlinear stochastic systems with actuator failures is investigated in this paper. Such a system is formulated as a continuous-discrete hybrid system based on the random switch model of failure-prone actuator. Time delay control force is converted into delay-free one by randomly periodic characteristic of the system. Using limit theorem and stochastic averaging, an approximate formula for the largest Lyapunov exponent of the original system is then derived, from which necessary and sufficient conditions for asymptotic stability are obtained. The validity and utility of the proposed procedure are demonstrated by using a stochastically driven nonlinear two-degree system with time delay feedback and actuator failure.

Stability of Rotating Pretwisted, Tapered Beams With Randomly Varying Speeds

1998 ◽  
Vol 120 (3) ◽  
pp. 784-790 ◽  
Author(s):  
T. H. Young ◽  
T. M. Lin
Keyword(s):  
Averaging Method ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bending Vibration ◽  
Approximation Solution ◽  
Rotating Speed ◽  
System Parameters ◽  
Sample Stability ◽  
Small Intensity

This paper presents a study of stability of coupled bending-bending vibration of pretwisted, tapered beams rotating at randomly varying speeds. The rotating speed is characterized as a wide-band, stationary random process with a zero mean and small intensity superimposed on a constant speed. This randomly varying speed may result in the existence of parametric random instability of the beam. The stochastic averaging method is used to derive Ito’s equation for an approximation solution, and expressions for stability boundaries of the system are obtained by the second-moment and sample stability criteria, respectively. It is observed that those system parameters which will raise the first natural frequency but not enhance the centrifugal force of the beam tend to stabilize the system.

Lyapunov Exponents and Stochastic Stability of Quasi-Integrable-Hamiltonian Systems

1999 ◽  
Vol 66 (1) ◽  
pp. 211-217 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Stability ◽  
Degrees Of Freedom ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Largest Lyapunov Exponent ◽  
Integrable Hamiltonian Systems ◽  
Averaged Equations ◽  
Bifurcation Phenomena

The averaged equations of integrable and nonresonant Hamiltonian systems of multi-degree-of-freedom subject to light damping and real noise excitations of small intensities are first derived. Then, the expression for the largest Lyapunov exponent of the square root of the Hamiltonian is formulated by generalizing the well-known procedure due to Khasminskii to the averaged equations, from which the stochastic stability and bifurcation phenomena of the original systems can be determined approximately. Linear and nonlinear stochastic systems of two degrees-of-freedom are investigated to illustrate the application of the proposed combination approach of the stochastic averaging method for quasi-integrable Hamiltonian systems and Khasminskii’s procedure.

Dynamics of Spinning Disc Parametrically Excited by Noise

1999 ◽  
Author(s):  
Narayanan Ramakrishnan ◽  
N. Sri Namachchivaya
Keyword(s):  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Averaging ◽  
Transverse Motion ◽  
Probability Density Distribution ◽  
Integral Of Motion ◽  
Parametrically Excited ◽  
One Dimensional ◽  
Small Intensity ◽  
Spinning Disc

Abstract The nonlinear dynamics of a circular spinning disc parametrically excited by noise of small intensity is investigated. The governing PDEs are reduced using a Galerkin reduction procedure to a two-DOF system of ODEs which, govern the transverse motion of the disc. The dynamics is simplified by exploiting the S1 invariance of the equations of motion of the reduced system and further, reduced by performing stochastic averaging. The resulting one-dimensional Markov diffusive process is studied in detail. The stationary probability density distribution is obtained by solving the Fokker-Planck equation along with the appropriate boundary conditions. The boundary behaviour is studied using an asymptotic approach. Some aspects of dynamical and phenomenological bifurcations of the stationary solution are also investigated. The scheme of things presented here can be applied in principle to a four-dimensional Hamiltonian system possessing one integral of motion in addition to the hamiltonian and having one fixed point.

scholarly journals Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on a Foundation having Fractional Order Viscoelastic Physical Properties

2021 ◽  
Author(s):  
Lionel Merveil Anague Tabejieu ◽  
Blaise Roméo Nana Nbendjo ◽  
Giovanni Filatrella
Keyword(s):  
Physical Properties ◽  
Fractional Order ◽  
Elastic Beam ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Moderate Increase ◽  
Moving Loads ◽  
Resonant Amplitude ◽  
Wind Turbulence ◽  
And Shifts

The present chapter investigates both the effects of moving loads and of stochastic wind on the steady-state vibration of a first mode Rayleigh elastic beam. The beam is assumed to lay on foundations (bearings) that are characterized by fractional-order viscoelastic material. The viscoelastic property of the foundation is modeled using the constitutive equation of Kelvin-Voigt type, which contain fractional derivatives of real order. Based to the stochastic averaging method, an analytical explanation on the effects of the viscoelastic physical properties and number of the bearings, additive and parametric wind turbulence on the beam oscillations is provided. In particular, it is found that as the number of bearings increase, the resonant amplitude of the beam decreases and shifts towards larger frequency values. The results also indicate that as the order of the fractional derivative increases, the amplitude response decreases. We are also demonstrated that a moderate increase of the additive and parametric wind turbulence contributes to decrease the chance for the beam to reach the resonance. The remarkable agreement between the analytical and numerical results is also presented in this chapter.

scholarly journals A Hidden Chaotic System with Multiple Attractors

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1341
Author(s):  
Xiefu Zhang ◽  
Zean Tian ◽  
Jian Li ◽  
Xianming Wu ◽  
Zhongwei Cui
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Numerical Methods ◽  
Chaotic System ◽  
Time Domain ◽  
Chaotic Attractor ◽  
Largest Lyapunov Exponent ◽  
Spectral Entropy ◽  
Multiple Attractors ◽  
System Parameters

This paper reports a hidden chaotic system without equilibrium point. The proposed system is studied by the software of MATLAB R2018 through several numerical methods, including Largest Lyapunov exponent, bifurcation diagram, phase diagram, Poincaré map, time-domain waveform, attractive basin and Spectral Entropy. Seven types of attractors are found through altering the system parameters and some interesting characteristics such as coexistence attractors, controllability of chaotic attractor, hyperchaotic behavior and transition behavior are observed. Particularly, the Spectral Entropy algorithm is used to analyze the system and based on the normalized values of Spectral Entropy, the state of the studied system can be identified. Furthermore, the system has been implemented physically to verify the realizability.

Asymptotic Stability of Controlled Nonlinear Stochastic Systems Considering the Dynamics of Sensors and Actuators

2021 ◽  
pp. 2150180
Author(s):  
Jiaojiao Sun ◽  
Zuguang Ying ◽  
Ronghua Huan ◽  
Weiqiu Zhu
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stable Region ◽  
High Dimensional ◽  
Nonlinear Stochastic System ◽  
Dimensional System ◽  
Sensors And Actuators ◽  
Controlled System

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. In this paper, asymptotic stability of trivial solutions of a controlled nonlinear stochastic system considering the dynamics of sensors and actuators is investigated. Considering the inherent and intentional nonlinearities and random loadings, the coupled dynamic equations of the controlled system with sensors and actuators are given, which are further formulated by a controlled, randomly excited, dissipated Hamiltonian system. The Hamiltonian of the controlled system is introduced, and, based on the stochastic averaging method, the original high-dimensional system is reduced to a one-dimensional averaged system. The analytical expression of Lyapunov exponent of the averaged system is derived, which gives the approximately necessary and sufficient condition of the asymptotic stability of trivial solutions of the original high-dimensional system. The validation of the proposed method is demonstrated by a four-degree-of-freedom controlled system under pure stochastically parametric excitations in detail. A comparative analysis, which is related to the stochastic asymptotic stability of the system with and without considering the dynamics of sensors and actuators, is carried out to investigate the effect of their dynamics on the motion of the controlled system. Results show that ignoring the dynamics of sensors and actuators will get a shrink stable region of the controlled system.

scholarly journals On the Role of Nonsynchronous Rotating Damping in Rotordynamics

2000 ◽  
Vol 6 (6) ◽  
pp. 467-475 ◽  
Author(s):  
Giancarlo Genta ◽  
Eugenio Brusa
Keyword(s):  
General Model ◽  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
System Parameters ◽  
Spin Speed ◽  
Rotating Systems ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Rotor Model

Nonsynchronous rotating damping, i.e. energy dissipations occurring in elements rotating at a speed different from the spin speed of a rotor, can have substantial effects on the dynamic behaviour and above all on the stability of rotating systems.The free whirling and unbalance response for systems with nonsynchronous damping are studied using Jeffcott rotor model. The system parameters affecting stability are identified and the threshold of instability is computed. A general model for a multi-degrees of freedom model for a general isotropic machine is then presented. The possibility of synthesizing nonsynchronous rotating and nonrotating damping using rotor- and stator-fixed active dampers is then discussed for the general case of rotors with many degrees of freedom.

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

2018 ◽  
Vol 140 (4) ◽  
Author(s):  
Y. J. Wu ◽  
H. Y. Wang
Keyword(s):  
Harmonic Function ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Digital Simulation ◽  
Stochastic Averaging ◽  
Harmonic Excitation ◽  
Van Der Pol Oscillator ◽  
Strongly Nonlinear ◽  
Set Up ◽  
White Noises

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

A Novel Parameter Optimization Method for the Driving System of High-Speed Parallel Robots

2018 ◽  
Vol 10 (4) ◽  
Author(s):  
Xin-Jun Liu ◽  
Gang Han ◽  
Fugui Xie ◽  
Qizhi Meng ◽  
Sai Zhang
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Dynamic Performance ◽  
Optimization Method ◽  
Parallel Robots ◽  
Parameters Optimization ◽  
Driving System ◽  
System Parameters ◽  
Driving Torque ◽  
Instantaneous Acceleration

Driving system parameters optimization, especially the optimal selection of specifications of motor and gearbox, is very important for improving high-speed parallel robots' performance. A very challenging issue is parallel robots' performance evaluation that should be able to illustrate robots' performance accurately and guide driving system parameters optimization effectively. However, this issue is complicated by parallel robots' anisotropic translational and rotational dynamic performance, and the multiparameters of motors and gearboxes. In this paper, by separating the influence of translational and rotational degrees-of-freedom (DOFs) on robots' performance, a new dynamic performance index is proposed to reflect the driving torque in instantaneous acceleration. Then, the influence of driving system's multiparameters on robots' driving torque in instantaneous acceleration and cycle time in continuous motion is investigated. Based on the investigation, an inertia matching index is further derived which is more suitable for minimizing the driving torque of parallel robots with translational and rotational DOFs. A comprehensive parameterized performance atlas is finally established. Based on this atlas, the performance of a high-speed parallel robot developed in this paper can be clearly evaluated, and the optimal combination of motors and gearboxes can be quickly selected to ensure low driving torque and high pick-and-place frequency.

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