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.

Stochastic Dynamics of Impact Oscillators

2004 ◽  
Vol 72 (6) ◽  
pp. 862-870 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Jun H. Park
Keyword(s):  
Model Reduction ◽  
Stochastic Dynamics ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Exit Times ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
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.

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.

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.

Chattering and related behaviour in impact oscillators

1994 ◽  
Vol 347 (1683) ◽  
pp. 365-389 ◽  
Keyword(s):  
Initial Data ◽  
Systematic Study ◽  
Infinite Number ◽  
Periodic Motion ◽  
Finite Time ◽  
Impact Oscillator ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
Single Degree ◽  
Ball Bouncing

One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

Co-dimension Two Bifurcations in the Presence of Noise

1991 ◽  
Vol 58 (1) ◽  
pp. 259-265 ◽  
Author(s):  
N. Sri Namachchivaya
Keyword(s):  
Normal Form ◽  
Diffusion Process ◽  
Stochastic Averaging ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Exit Times ◽  
Double Zero ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stochastic Bifurcations

Some results pertaining to co-dimension two stochastic bifurcations are presented. The normal form associated with non-semi-simple double-zero eigenvalues is considered. The method of stochastic averaging applicable for singularly perturbed stochastic differential equations is used to further reduce the problem to a one dimensional diffusion process. Probability density, most probable values, stability conditions in probability, and mean exit times are obtained for the reduced system.

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.

UNIFIED APPROACH FOR NOISY NONLINEAR MATHIEU-TYPE SYSTEMS

2001 ◽  
Vol 01 (03) ◽  
pp. 405-450 ◽  
Author(s):  
N. SRI NAMACHCHIVAYA ◽  
RICHARD B. SOWERS
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Unified Approach ◽  
Van Der Pol ◽  
Periodic Hamiltonian System ◽  
Time Periodic ◽  
Stochastic Bifurcations

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. As a prototype, we consider the noisy Duffing–van der Pol–Mathieu equation and achieve a reduction by developing rigorous methods to replace, in a limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a lower-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.

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