Nonlinear behaviour of the instability of steady natural convection in a vertical air layer

In this paper we deal with the flow of natural convection between two vertical planes with horizontal temperature gradient. We show that periodic steady flow occur when Rayleigh number increases. Critical values are obtained numerically and a nonlinear analysis is presented according to Center manifold method.     

Non smooth dynamics of mechanical systems with history term

Models involving maximal monotone term and history (delay) term are considered in a mathematical and numerical point of view.     

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.     

Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity

The concept of the vibratory energy transfer between a linear master DOF and a nonsmooth nonlinear energy sink (NES) in the presence of gravity forces is studied. Different invariant manifolds of the system at different time scales are revealed and necessary conditions for leading the behavior of the system to strongly modulated response (SMR) are enlightened.     

Persoz’s gephyroidal model described by a maximal monotone differential inclusion

Persoz’s gephyroidal model, which consists of elementary rheological models (dry friction element and linear spring), can be covered by the existence and uniqueness theory for maximal monotone operators. Moreover, classical results of numerical analysis allow one to use a numerical implicit Euler scheme, with convergence order of the scheme equal to one. Some numerical simulations are presented.     

Targeted energy transfer with parallel nonlinear energy sinks. Part I: Design theory and numerical results

In this paper, we study a Targeted Energy Transfer (TET) problem between a p degrees-of-freedom (dof) linear master structure and several coupled parallel slave Nonlinear Energy Sink (NES) systems. In detail, each lth dof l=1,2,…,p contains n _l parallel NES; so the linear structure has (n ₁+n ₂+⋅⋅⋅+n _l +⋅⋅⋅+n _p ) NES. We are interested to study analytically the TET phenomenon during the first mode of the compound system. To this end, complexification, averaging, and multiple scales methods are used.     

A stochastic differential equation from friction mechanics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented. To cite this article: F. Bernardin et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems

We present new analytical approach to the problem of energy pumping in strongly non- homogeneous nonlinear two-degree-of-freedom (2DOF) systems with single anchor spring under condition of initial impact. Energy pumping is a passive, almost irreversible transfer of mechanical energy from the main substructure of the system to the light auxiliary attachment. The mechanism of energy pumping in the system under consideration is a resonance capture. The approach is based on application of Laplace transformation to the principal asymptotic approximation of the equations of motion in complex form and using the power expansion of the solution in terms of time. Obtained temporal dependence of the system energetic characteristics gives a tool for estimation of energy pumping efficiency. In particular, we show that the system without an anchor spring in attachment is more efficient than the system with such a spring. Numerical simulations confirm the analytical results.     

Targeted energy transfer with parallel nonlinear energy sinks,part II: theory and experiments

In this paper governing equations of general multi degrees of freedom (dof) systems with trees of parallel Nonlinear Energy Sink (NES) devices at each dof are derived. Then these equations are summarized for a 4 dof structure with two parallel NES at the 4^th dof in order to control the first mode of the system. A prototype four storey structure with two parallel NES at the top floor is studied experimentally. The NES of the mentioned system is designed by endowing the suggested method in the Part I of this paper. A couple of experimental tests are carried out on the structure at the DGCB laboratory of the ENTPE, France. The aim is to control the first mode of the compound nonlinear system by demonstrating the efficiency of the parallel NES systems on the intended task.