Time Varying Loads and White Noise on an SDOF System with Bouc Hysteresis Using Gaussian Closure

In a recent publication [H. Waubke and C. Kasess, Gaussian closure technique applied to the hysteretic Bouc model with nonzero mean white noise excitation, J. Sound Vibr. 382 (2016) 258–273], the response of a single-degree-of-freedom (SDOF) system under Gaussian white noise and a constant dead load is presented. The system has a hysteresis described by Bouc [R. Bouc, Forced vibration of mechanical systems with hysteresis, in Proc. Fourth Conference on Nonlinear Oscillation (Prague, 1967), p. 315]. New is the usage of a slowly time-varying deterministic load added to the Gaussian white noise process. The transient solution is calculated using the Gaussian closure technique together with an explicit time step procedure. All moments in the Gaussian closure technique are evaluated analytically. The results of the Gaussian closure technique are in good agreement with the results from the Monte-Carlo method.

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

On the Anisotropic Gaussian Velocity Closure for Inertial-Particle Laden Flows

The accurate simulation of disperse two-phase flows, where a discrete particulate condensed phase is transported by a carrier gas, is crucial for many applications; Eulerian approaches are well suited for high performance computations of such flows. However when the particles from the disperse phase have a significant inertia compared to the time scales of the flow, particle trajectory crossing (PTC) occurs i.e. the particle velocity distribution at a given location can become multi-valued. To properly account for such a phenomenon many Eulerian moment methods have been recently proposed in the literature. The resulting models hardly comply with a full set of desired criteria involving: 1- ability to reproduce the physics of PTC, at least for a given range of particle inertia, 2- well-posedness of the resulting set of PDEs on the chosen moments as well as guaranteed realizability, 3- capability of the model to be associated with a high order realizable numerical scheme for the accurate resolution of particle segregation in turbulent flows. The purpose of the present contribution is to introduce a multi-variate Anisotropic Gaussian closure for such particulate flows, in the spirit of the closure that has been suggested for out-of-equilibrium gas dynamics and which satisfies the three criteria. The novelty of the contribution is three-fold. First we derive the related moment system of conservation laws with source terms, and justify the use of such a model in the context of high Knudsen numbers, where collision operators play no role. We exhibit the main features and advantages in terms of mathematical structure and realizability. Then a second order accurate and realizable MUSCL/HLL scheme is proposed and validated. Finally the behavior of the method for the description of PTC is thoroughly investigated and its ability to account accurately for inertial particulate flow dynamics in typical configurations is assessed.