An Averaging Approach for Noisy Strongly Nonlinear Periodically Forced Systems

2002 ◽  
Author(s):  
Seunggil Choi ◽  
N. Sri Namachchivaya
Keyword(s):  
Stochastic Averaging ◽  
External Noise ◽  
Degree Of Freedom ◽  
Periodic Force ◽  
Single Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Single Degree ◽  
Periodically Driven ◽  
Statistical Measures ◽  
Stochastic Bifurcations

The purpose of this work is to develop a unified approach to study the dynamics of single degree of freedom systems excited by both periodic and random perturbations. The near resonant motion of such systems is not well understood. We will study this problem in depth with the aim of discovering a common geometric structure in the phase space, and to determine the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. Depending on the strength of the noise, reduced Markov process takes its values on a line or on graph with certain gluing conditions at the vertex of the graph. The reduced model will provide a framework for computing standard statistical measures of dynamics and stability, namely, mean exit times, probability density functions, and stochastic bifurcations. This work will also explain a counter-intuitive phenomena of stochastic resonance, in which a weak periodic force in a nonlinear system can be enhanced by the addition of external noise.

Unified Approach for Noisy Nonlinear Mathieu-Type Systems

2001 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Richard B. Sowers ◽  
H. J. Van Roessel
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Unified Approach ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Low Dimensional

Abstract The purpose of this work is to develop a unified approach to study the dynamics of a single degree of freedom system excited by both periodic and random perturbations. We consider the noisy Duffing-van der Pol-Mathieu equation as a prototypical single degree of freedom system and achieve a reduction by developing rigorous methods to replace, in some limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

Noisy Impact Oscillators

2004 ◽  
Author(s):  
Jun H. Park ◽  
N. Sri Namachchivaya
Keyword(s):  
Model Reduction ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Exit Times ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
Single Degree ◽  
Saddle Type ◽  
Dissipative Hamiltonian System ◽  
Stochastic Bifurcations

The purpose of this work is to develop an averaging approach to study the dynamics of a vibro-impact system excited by random perturbations. As a prototype, we consider a noisy single-degree-of-freedom equation with both positive and negative stiffness and achieve a model reduction; i.e., the development of rigorous methods to replace in some asymptotic regime, a complicated system by a simpler one. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative Hamiltonian system with either center type or saddle type fixed points. We achieve the model-reduction through stochastic averaging. Examination of the reduced Markov process on a graph yields mean exit times, probability density functions, and stochastic bifurcations.

Passive Targeted Energy Transfers and Strong Modal Interactions in a Thin Plate With Strongly Nonlinear End Attachments of Different Configurations

2007 ◽  
Author(s):  
F. Georgiades ◽  
A. F. Vakakis
Keyword(s):  
Wavelet Transforms ◽  
Degree Of Freedom ◽  
Vibration Energy ◽  
Modal Interactions ◽  
Single Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Single Degree ◽  
Energy Transfers ◽  
Parametric Studies ◽  
Underlying Mechanisms

In this paper we examine Targeted Energy Transfers (TETs) and nonlinear modal interactions occurring in a thin cantilever plate lying on an elastic foundation with strongly nonlinear lightweight attachments of different configurations. Under shock excitation of the plate we systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena between the plate and attachments of the following configurations: a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) NES; multiple SDOF attachments attached at different points of the plate; and a single multi-degree-of-freedom (MDOF) attachment with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and location of the attachments, in order to optimize TETs from the plate to the NES. We examine in detail the underlying mechanisms influencing TETs by means of Hilbert-Huang Transforms in combination with Wavelet Transforms. These transforms enable one to systematically study the strong modal interactions between the essentially nonlinear attachments and different plate modes. The efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed.

scholarly journals External Periodic Force Control of a Single-Degree-of-Freedom Vibroimpact System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jingyue Wang ◽  
Haotian Wang ◽  
Tie Wang
Keyword(s):  
Periodic Orbits ◽  
Force Control ◽  
Time History ◽  
Degree Of Freedom ◽  
Bifurcation And Chaos ◽  
Periodic Force ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Impact System

A single-degree-of-freedom mechanical model of vibro-impact system is established. Bifurcation and chaos in the system are revealed with the time history diagram, phase trajectory map, and Poincaré map. According to the bifurcation and chaos of the actual vibro-impact system, the paper puts forward external periodic force control strategy. The method of controlling chaos by external periodic force feedback controller is developed to guide chaotic motions towards regular motions. The stability of the control system is also analyzed especially by theory. By selecting appropriate feedback coefficients, the unstable periodic orbits of the original chaotic orbit can be stabilized to the stable periodic orbits. The effectiveness of this control method is verified by numerical simulation.

scholarly journals Higher order stochastic averaging method for mechanical systems having two degrees of freedom

2004 ◽  
Vol 26 (2) ◽  
pp. 103-110
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Degrees Of Freedom ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Degree Of Freedom ◽  
Approximation Solution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Two Degree Of Freedom

Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises.In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained.

Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
Daniil Iourtchenko ◽  
Eirik Mo ◽  
Arvid Naess
Keyword(s):  
Path Integration ◽  
Integration Method ◽  
First Passage Time ◽  
Degree Of Freedom ◽  
First Passage ◽  
Controlled System ◽  
Single Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Single Degree ◽  
Path Integration Method

This paper presents a first passage type reliability analysis of strongly nonlinear stochastic single-degree-of-freedom systems. Specifically, the systems considered are a dry friction system, a stiffness controlled system, an inertia controlled system, and a swing. These systems appear as a result of implementation of the quasioptimal bounded in magnitude control law. The path integration method is used to obtain the reliability function and the first passage time.

Sign in / Sign up

Export Citation Format

Share Document