Stochastic Bifurcations of a Nonlinear Acousto-Elastic System

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
W. Dheelibun Remigius ◽  
Sunetra Sarkar
Keyword(s):  
Hopf Bifurcation ◽  
Stochastic System ◽  
Elastic System ◽  
Spectral Projection ◽  
Stochastic Bifurcation ◽  
Deterministic System ◽  
Acoustic Oscillations ◽  
Field Equations ◽  
Coupled Mode ◽  
Coupled Field Equations

The nonlinear stochastic behavior of a nonconservative acousto-elastic system is in focus in the present work. The deterministic acousto-elastic system consists of a spinning disk in a compressible fluid filled enclosure. The nonlinear rotating plate dynamics is coupled with the linear acoustic oscillations of the surrounding fluid, and the coupled field equations are discretized and solved at various rotation speeds. The deterministic system reveals the presence of a supercritical Hopf bifurcation when a specific coupled mode undergoes a flutter instability at a particular rotation speed. The effect of randomness associated with the damping parameters are investigated and quantified on the coupled dynamics and the stochastic bifurcation behavior is studied. The quantification of the parametric randomness has been undertaken by means of a spectral projection based polynomial chaos expansion (PCE) technique. From the marginal probability density functions (PDFs), it is observed that the stochastic system exhibits stochastic phenomenological bifurcations (P-bifurcation). The study provides insights into the behavior of the stochastic system during its P-bifurcation with reference to the deterministic Hopf bifurcation.

Bifurcations and Uncertainty Quantifications in Nonlinear Acousto-Elastic Systems

2016 ◽  
Author(s):  
W. Dheelibun Remigius ◽  
Sunetra Sarkar
Keyword(s):  
Elastic System ◽  
Coupled System ◽  
Spectral Projection ◽  
Acoustic Oscillations ◽  
Field Equations ◽  
Acoustic Damping ◽  
Expansion Technique ◽  
Elastic Systems ◽  
Coupled Field Equations ◽  
Surrounding Fluid

The study of nonlinear aeroelastic instability mechanism of nonconservative acousto-elastic system is the focus here. The acousto-elastic system consists of a spinning disc in a compressible fluid filled enclosure. The nonlinear rotating plate is coupled with the linear acoustic oscillations of the surrounding fluid. Based on the acousto-elastic theory, the coupled field equations are discretized and solved for various rotation speeds in order to obtain the coupled system dynamics. The study shows that the coupled system undergoes a flutter instability at a particular rotation speed and the instability takes the form of supercritical Hopf bifurcation. Subsequently, the effect of randomness associated with the structural and the acoustic damping parameters are quantified on the nonlinear instability behaviour by means of a spectral projection based polynomial chaos expansion technique.

Bleustein-Gulyaev Waves in an Inhomogeneous Layered Piezoelectric Half-Space

2006 ◽  
Vol 306-308 ◽  
pp. 1223-1228
Author(s):  
Fei Peng ◽  
Hua Rui Liu
Keyword(s):  
Fourier Transform ◽  
Phase Velocity ◽  
Half Space ◽  
Electric Potential ◽  
Piezoelectric Layer ◽  
Field Equations ◽  
Transform Method ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
The Fourier Transform

The propagation of Bleustein-Gulyaev (BG) waves in an inhomogeneous layered piezoelectric half-space is investigated in this paper. Application of the Fourier transform method and by solving the electromechanically coupled field equations, solutions to the mechanical displacement and electric potential are obtained for the piezoelectric layer and substrate, respectively. The phase velocity equations for BG waves are obtained for the surface electrically shorted case. When the layer and the substrate are homogenous, the dispersion equations are in agreement with the corresponding results. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO3 except that they are polarized in opposite directions. Effects of the inhomogeneities induced by either the layer or substrate are discussed in detail.

scholarly journals A Coupled Thermodynamic Model for Transport Properties of Thin Films during Physical Aging

Polymers ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 387 ◽  
Author(s):  
Hongjiu Hu ◽  
Xiaoming Fan ◽  
Yaolong He
Keyword(s):  
Thin Films ◽  
Transport Properties ◽  
Free Volume ◽  
Physical Aging ◽  
Field Equations ◽  
Lattice Contraction ◽  
Strongly Coupled ◽  
Coupled Diffusion ◽  
Coupled Field Equations ◽  
Short Time

A coupled diffusion model based on continuum thermodynamics is developed to quantitatively describe the transport properties of glassy thin films during physical aging. The coupled field equations are then embodied and applied to simulate the transport behaviors of O2 and CO2 within aging polymeric membranes to validate the model and demonstrate the coupling phenomenon, respectively. It is found that due to the introduction of the concentration gradient, the proposed direct calculating method on permeability can produce relatively better consistency with the experimental results for various film thicknesses. In addition, by assuming that the free volume induced by lattice contraction is renewed upon CO2 exposure, the experimental permeability of O2 within Matrimid® thin film after short-time exposure to CO2 is well reproduced in this work. Remarkably, with the help of the validated straightforward permeability calculation method and free volume recovery mechanism, the permeability behavior of CO2 is also well elucidated, with the results implying that the transport process of CO2 and the variation of free volume are strongly coupled.

Exact soliton solutions for a class of coupled field equations

1993 ◽  
Vol 173 (1) ◽  
pp. 30-32 ◽  
Author(s):  
Xin-Yi Wang ◽  
Bing-Chang Xu ◽  
Philip L. Taylor
Keyword(s):  
Soliton Solutions ◽  
Field Equations ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
Exact Soliton Solutions

On the exact soliton solutions for a class of coupled field equations

1993 ◽  
Vol 182 (2-3) ◽  
pp. 300-301 ◽  
Author(s):  
Xiaowu Huang ◽  
Jiahua Han ◽  
Kaiyi Qian ◽  
Wei Qian
Keyword(s):  
Soliton Solutions ◽  
Field Equations ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
Exact Soliton Solutions

Dynamic Stability and Failure Probability Analysis of Dome Structures Under Stochastic Seismic Excitation

2014 ◽  
Vol 14 (05) ◽  
pp. 1440001 ◽  
Author(s):  
Jie Li ◽  
Jun Xu
Keyword(s):  
Dynamic Stability ◽  
Failure Probability ◽  
Stochastic System ◽  
Dynamic Instability ◽  
Seismic Excitation ◽  
Stochastic Dynamic ◽  
Deterministic System ◽  
Density Evolution ◽  
Stochastic Response ◽  
Strength Failure

The intrinsic relationship between deterministic system and stochastic system is profoundly revealed by the probability density evolution method (PDEM) with introduction of physical law into the stochastic system. On this basis, stochastic dynamic stability analysis of single-layer dome structures under stochastic seismic excitation is firstly studied via incorporating an energetic physical criterion for identification of dynamic instability of dome structures into PDEM, which yields to sample stability (stable reliability). However, dynamic instability is not identical to structural failure definitely, where strength failure can be experienced not only in the stable structure but also when the structure is out of dynamic stability. It is practically feasible to decouple the stochastic dynamic response of dome structures to be a stable one and an unstable one according to the generalized density evolution equation (GDEE). Consequently, the global failure probability can be investigated separately based on the corresponding independent stochastic response. For unstable failure probability assessment, the failure probability is the unstable probability if the dome's failure is attributed to instability, whereas inverse absorbing is firstly implemented to get rid of the stochastic response before instability and a complementary process is filled in the safe domain immediately to finally assess the probability of strength failure after dynamic instability.

TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION

1996 ◽  
Vol 06 (11) ◽  
pp. 1947-1975 ◽  
Author(s):  
LUDWIG ARNOLD ◽  
N. SRI NAMACHCHIVAYA ◽  
KLAUS R. SCHENK-HOPPÉ
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Lyapunov Exponents ◽  
Variational Equation ◽  
Van Der Pol Oscillator ◽  
Stochastic Bifurcation ◽  
Van Der Pol ◽  
Truncated Normal ◽  
Stochastic Hopf Bifurcation ◽  
Bifurcation Behavior

In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenological (based on the Fokker-Planck equation), and dynamical (based on Lyapunov exponents). The method of stochastic averaging of the nonlinear system yields a set of equations which, together with its variational equation, can be explicitly solved and hence its bifurcation behavior completely analyzed. We augment this analysis by asymptotic expansions of the Lyapunov exponents of the variational equation at zero. Finally, the stochastic normal form of the noisy Duffing-van der Pol oscillator is derived, and its bifurcation behavior is analyzed numerically. The result is that the (truncated) normal form retains the essential bifurcation characteristics of the full equation.

scholarly journals Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Stochastic System ◽  
Critical Parameter ◽  
Random Parameter ◽  
Order System ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Existence Conditions ◽  
Theoretical Results

The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.

scholarly journals Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jiying Ma ◽  
Qing Yi
Keyword(s):  
Stationary Distribution ◽  
Stochastic System ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Deterministic System ◽  
Stochastic Epidemic ◽  
Lyapunov Function Method ◽  
Deterministic Framework ◽  
Theoretical Results

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.

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