Gaussian White-noise Force Research Articles

Effect of stochastic forcing on the dynamic behavior of a self-sustained oscillator coupled to a non-linear energy sink

In this paper the influence of stochasticity (i.e. a Gaussian white noise forcing) on the dynamic behavior of a self-sustained oscillator coupled to a non-linear energy sink is investigated. To this end, the standard stochastic averaging is used to compute the slow flow dynamics of the system. Preliminary results show that the reasoning which allows to predict the system behavior in the deterministic case can be contradicted in presence of stochasticity. Then, by means of the Monte Carlo method, the stochastic averaging procedure is validated. Finally, two quantities are introduced to highlight more precisely the special features of the stochastic system behavior compared to that of the deterministic system. These are the probability of being in a harmless regime and the First-Passage Time to reach a harmful regime which are computed and investigated combining again the Monte Carlo approach with numerical integrations of the slow flow dynamics. The results obtained show afresh that the stochastic forcing can modify significantly the dynamic behavior of the corresponding deterministic system. Indeed, when they are computed on the latter, the two quantities aforementioned have a discontinuity at the mitigation limit (i.e. the value of the bifurcation parameter under consideration below which the NES acts and above which it no longer acts) revealing an abrupt change of behavior of the coupled system. The paper shows that this typical characteristic of the deterministic system is lost in the presence of stochasticity, the stochastic system becoming smooth at the mitigation limit.

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole $\mathbb{R}^d$, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the domain truncation, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in one- and two-dimensional space, confirming our theoretical results.

Three-dimensional instability of a flow past a sphere: Mach evolution of the regular and Hopf bifurcations

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

A new flutter speed analysis method using stochastic approach

In this study a new analytical method for analyzing flutter speed of aeroelastic systems is presented. In this method, apart from the aerodynamic lift and moment deduced from the aerodynamic theories, one Gaussian white noise force was also added to the lift force. Then the spectral density of response was calculated with respect to the frequency response of the system as well as the spectral density of the excitation. The variance of the response was determined with respect to the airspeed. The flutter speed was obtained by investigating the variation of the response-variance against the airspeed. The maximum of the variance of response represented the flutter speed. To verify this new approach, two case studies were considered. The first case study was a 2-D airfoil under unsteady subsonic flow and the second one was a slender wing in an unsteady incompressible flow with Goland wing properties. The results of the proposed method showed a good agreement with the results of the conventional flutter analysis. One of the major advantages of this approach over the other numerical methods such as the p–k method was that no convergence of reduced frequency was needed so the computational cost was reduced dramatically.

Uniqueness of the generators of the 2D Euler and Navier–Stokes flows

A uniqueness result is proven for the infinitesimal generator associated with the 2D Euler flow with periodic boundary conditions in the space L 2 ( μ ) with respect to the natural Gibbs measure μ given by the enstrophy. This result remains true for the generator of the stochastic process associated with a 2D Navier–Stokes equation perturbed by a space–time Gaussian white noise force. The corresponding Liouville operator N defined on the space C b , cyl 1 of smooth cylinder bounded functions has a unique skew-adjoint m -dissipative extension in the class of closed operators in L 2 ( μ ) × V ′ where V = D ( N ¯ ) .

Transitions across a barrier induced by deterministic forcings.

The response of a bistable dynamical system to a deterministic forcing is studied with emphasis on the kinetics of the passage across the barrier separating the two states, and compared to classical Kramers' theory describing the response to a Gaussian white noise forcing. The existence of nontrivial thresholds for the occurrence of transitions is established. Analytic results complemented by numerical simulations are derived for the characteristics of these transitions for periodic and chaotic forcings. The probabilistic properties of the response are finally addressed and some connections are established with the universal stable distributions of probability theory.

Instanton calculus in shell models of turbulence

It has been shown recently that the intermittency of the Gledzer-Ohkitani-Yamada (GOY) shell model of turbulence has to be related to singular structures whose dynamics in the inertial range includes interactions with a background of fluctuations. In this paper we propose a statistical theory of these objects by modeling the incoherent background as a Gaussian white-noise forcing of small strength Gamma. A general scheme is developed for constructing instantons in spatially discrete dynamical systems and the Cramer function governing the probability distribution of effective singularities of exponent z is computed up to first order in a semiclassical expansion in powers of Gamma. The resulting predictions are compared with the statistics of coherent structures deduced from full simulations of the GOY model at very high Reynolds numbers.

Optimal weight absorber designs for vibrating structures exposed to random excitations

AbstractOptimal mass ratios that minimize the response of a laminated beam with an attached absorber are tabulated for various values of beam damping. The beam is treated as an equivalent one degree of freedom (1DOF) main system vibrating in the fundamental mode. The beam is subjected to Gaussian white noise force and Gaussian white noise base frame acceleration. Optimal absorber frequency ratios and absorber damping ratios have been tabulated by others; the results for the classical 1DOF main system with attached absorber suggest that the optimized non‐dimensional response decreases monotonically as the mass ratio increases. However, to generalize this monotonic relation may lead to inappropriate conclusions. If we define a constraint such that an increase in absorber mass leads to a proportional decrease in available beam construction material, i.e. effectively the combined mass of the beam and absorber is minimized, then variations in the mass ratio will affect the beam's parameters such as mass, stiffness and damping. Since some of these parameters are used for non‐dimensionalising the response, inspection of non‐dimensional responses may in some cases lead to inappropriate conclusions. This paper shows the optimal mass ratios for minimizing the response of a structure exposed to earthquake or fluid flow type random excitations.

Power Spectra of Systems Described by a Nonlinear Langevin Equation Driven by Random and Periodic Forces

Simulation was performed both by a digital computer and by an analog circuit connected to a micro-computer, and stationary power spectra were measured. When only Gaussian white noise force was applied additively to an equation of inactive type, was observed the Lorentzian power spectrum, whose density decreased with increasing nonlinearity in low frequency region. An equation of active type also exhibits the Lorentzian power spectrum if random force was week. When random and periodic forces were applied simultaneously, the continuous part of the power spectrum was also Lorentzian, but the low frequency density was reduced considerably. The experimentally obtained results were consistent with those obtained theoretically by using the statistical linearization approximation developed by Budger, Lindenberg and Shuler.