Multiple Scales via Galerkin Projections: Approximate Asymptotics for Strongly Nonlinear Oscillations

The method of multiple scales and the related method of averaging are commonly used tostudy slowly modulated oscillations. If the system of interest is a slightlyperturbed harmonic oscillator, then these techniques can be applied easily. If the unperturbed system is strongly nonlinear (though possiblyconservative), then these methods can run into difficulties due to the impossibilityof carrying out required analytical operations in closed form.In this paper, we abandon the requirement of closed form analyticaltreatment at all stages. Instead, Galerkin projections are used toobtain approximate realizations of the method of multiple scales. Thispaper adapts recent work using similar ideas for approximaterealizations of the method of averaging. A key contribution of thepresent work is in the systematic identification and removal of secularterms in the general nonlinear case, a procedure that is more difficultthan for the perturbed harmonic oscillator case, and that is unnecessaryfor averaging.A strength of the present work is that the heuristics (Galerkin)and asymptotics (multiple scales) are kept distinct,leaving room for systematic refinement of the formerwithout compromising the asymptotic features of the latter.     

The response of nonlinear systems to modulated high-frequency input

We study the response of a single-degree-of-freedom system with cubic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The only restriction on the amplitude modulation is that it contain frequencies much lower than the carrier frequency of the excitation. We apply the theory to different types of amplitude modulation and find that resonant excitation of the system may occur under some conditions.     

Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics

Analytical and experimental investigations are performed in order to characterize the dynamic behaviour of a non-smooth rotational oscillator, which exhibits multiple discontinuity boundaries in the phase space. The physical system consists of a rotating body subjected to an elastic restoring force and in contact with one or two rough discs rotating with constant driving velocities. The presence of multiple discontinuity boundaries caused by frictional contacts leads to non-standard bifurcations that are studied by means of a simple mechanical model.A test set-up has then been built to investigate the correctness of modelling of the friction force and the validity of the proposed model for technical applications: the experimental measurements qualitatively and quantitatively capture the basic scenarios anticipated by the model while a strong robustness of the phenomena pointed out by the theoretical analyses has been revealed in the experiments.     

Nonlinear Oscillations of a Nonresonant Cable under In-Plane Excitation with a Longitudinal Control

The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.     

Experimental investigations on the dynamic behavior of a 2-DOF airfoil in the transitional Re number regime based on digital-image correlation measurements

The present paper investigates the fluid–structure interaction (FSI) of a wing with two degrees of freedom (DOF), i.e., pitch and heave, in the transitional Reynolds number regime. This 2-DOF setup marks a classic configuration in aeroelasticity to demonstrate flutter stability of wings. In the past, mainly analytic approaches have been developed to investigate this challenging problem under simplifying assumptions such as potential flow. Although the classical theory offers satisfying results for certain cases, modern numerical simulations based on fully coupled approaches, which are more generally applicable and powerful, are still rarely found. Thus, the aim of this paper is to provide appropriate experimental reference data for well-defined configurations under clear operating conditions. In a follow-up contribution these will be used to demonstrate the capability of modern simulation techniques to capture instantaneous physical phenomena such as flutter. The measurements in a wind tunnel are carried out based on digital-image correlation (DIC). The investigated setup consists of a straight wing using a symmetric NACA 0012 airfoil. For the experiments the model is mounted into a frame by means of bending and torsional springs imitating the elastic behavior of the wing. Three different configurations of the wing possessing a fixed elastic axis are considered. For this purpose, the center of gravity is shifted along the chord line of the airfoil influencing the flutter stability of the setup. Still air free-oscillation tests are used to determine characteristic properties of the unloaded system (e.g. mass moment of inertia and damping ratios) for one (pitch or heave) and two degrees (pitch and heave) of freedom. The investigations on the coupled 2-DOF system in the wind tunnel are performed in an overall chord Reynolds number range of $9.66\times 10^3\le \text{Re}\le 8.77\times 10^4$. The effect of the fluid-load induced damping is studied for the three configurations. Furthermore, the cases of limit-cycle oscillation (LCO) as well as diverging flutter motion of the wing are characterized in detail. In addition to the DIC measurements, hot-film measurements of the wake flow for the rigid and the oscillating airfoil are presented in order to distinguish effects originating from the flow and the structure.     

On the fundamental sloshing frequency in the ice-fishing problemSur la fréquence de ballottement principale dans le problème de la glace avec trous pour pêcher

The sloshing problem is considered in a half-space covered by a rigid dock with apertures. The dependence of the fundamental sloshing frequency on the shape of the free surface region is studied. It is proved that the inequality holds between the fundamental eigenvalues corresponding to two different regions if some conditions are fulfilled. These conditions are verified for particular classes of regions of a fixed area in order to demonstrate that the disk yields the maximum of the fundamental eigenvalue for each of these classes. On the other hand, examples of regions are constructed for which the fundamental eigenfrequency is larger than that for the circular aperture of the same area and even as large as one wishes. To cite this article: V. Kozlov, N. Kuznetsov, C. R. Mecanique 330 (2002) 723–728.     

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.     

Frequency lock-in phenomenon for oscillating airfoils in buffeting flows

Navier-Stokes based computer simulations are conducted to determine the aerodynamic flow field response that is observed for a NACA0012 airfoil that undergoes prescribed harmonic oscillation in transonic buffeting flows, and also in pre-buffet flow conditions. Shock buffet is the term for the self-sustained shock oscillations that are observed for certain combinations of Mach number and steady mean flow angle of attack even in the absence of structural motion. The shock buffet frequencies are typically on the order of the elastic structural frequencies, and therefore may be a contributor to transonic aeroelastic response phenomena, including limit-cycle oscillations. Numerical simulations indicate that the pre-shock-buffet flow natural frequency increases with mean angle of attack, while the flow damping decreases and approaches zero at the onset of buffet. Airfoil harmonic heave motions are prescribed to study the interaction between the flow fields induced by the shock buffet and airfoil motion, respectively. At pre-shock-buffet conditions the flow response is predominantly at the airfoil motion frequency, with some smaller response at multiplies of this frequency. At shock buffet conditions, a key effect of prescribed airfoil motions on the buffeting flow is to create the possibility of a lock-in phenomenon, in which the shock buffet frequency is synchronized to the prescribed airfoil motion frequency for certain combinations of airfoil motion frequencies and amplitudes. Aerodynamic gain-phase models for the lock-in region, as well as for the pre-shock-buffet conditions are suggested, and also a possible relationship between the lock-in mechanism and limit-cycle oscillation is discussed.     

蒙卡燃耗中的氙振荡现象分析及计算研究

根据氙振荡产生的机理,按照燃耗步长来划分：在短步长条件下会产生物理氙振荡,在长步长条件下会产生数值氙振荡。研究发现,当燃耗步长增大到19 h,物理氙振荡的振幅最小,随着燃耗步长继续增大,表现为数值氙振荡,当燃耗步长27 d,数值氙振荡振幅最小。采用平衡氙方法强制使中子通量与氙浓度平衡,在堆芯物理计算程序(RMC)的输运模块中运用平衡氙方法对碘氙的浓度进行不断的迭代更新,在保证收敛的情况下抑制了氙振荡现象的发生。