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We study, via discrete element method simulations, the apparent mass (m, the ratio of a driving force to a resulting acceleration) and loss factor ( $$\eta$$ , the ratio of dissipated to stored energy) of granular dampers attached to a vertically driven, single degree of freedom mechanical system. Granular dampers (or particle dampers) consist in receptacles that contain macroscopic particles which dissipate energy, when they are subjected to vibration, thanks to the inelastic collisions and friction between them. Although many studies focus on $$\eta$$ , less work has been devoted to m. The apparent mass of granular dampers is an important characteristic since the grains, which are free to move or collide inside their receptacle, act as a non-constant and time-dependent mass which alters the mass of the main vibrating system in a non-trivial way. In particular, it has been recently demonstrated (Masmoudi et al. in Granul Matter 18:71, 2016) that m non-linearly depends on the driving acceleration $$\varGamma$$ according to a power law, $$m\propto \varGamma ^k$$ . Experiments using three-dimensional (3D) packings of particles suggest $$k=-2$$ . However, simulations with one-dimensional (1D) columns of particles on a vibrating plate and theoretical predictions based on the inelastic bouncing ball model (IBBM) suggest that $$k=-1$$ . These findings opened questions as to whether the apparent mass, which relies on how linear momentum is transferred from the damper to the primary system, depends on the dimensionality of the packing or on lateral interactions between walls and grains. Interestingly, $$\eta$$ was shown to follow a universal curve, $$\eta \propto \varGamma ^{-1}$$ , whatever the dimensionality and the constraints in the motion of the grains. In this work, we consider granular dampers without a lid under different confinement conditions in the motion of the particles (1D, quasi-1D, quasi-2D and full 3D). We find that the mechanical response of the granular damper (m and $$\eta$$ ) is not sensitive to the lateral confinement or dimensionality. However, we have observed two distinct regimes, depending on whether the driving frequency is above or below the resonant frequency of the primary system. In the inertial regime, $$\eta$$ decays according to the IBBM for all dimensions, $$\eta \propto \varGamma ^{-1}$$ , while m falls as $$\varGamma ^{-2}$$ for all dimensions, in agreement with Masmoudi’s experiments. However, the power law for m is valid only for moderate acceleration, before becoming negative at high accelerations. In the quasi-static regime, both m and $$\eta$$ display a complex behavior as functions of the excitation amplitude, but the mean trend is consistent with the IBBM predictions, i.e., $$m\propto \varGamma ^{-1}$$ and $$\eta \propto \varGamma ^{-1}$$ .

We consider in this chapter the dynamics of rigid multibody systems subjected to frictionless unilateral constraints. Starting from the mechanical description of the problem, we derive its formulation as a second-order Measure Differential Inclusion and we introduce the corresponding mathematical framework, namely functions of Bounded Variation and Stieltjes measures. Then, the main difficulties in the study of vibro-impact problems are described and an overview of the state of the art about existence results and relevant numerical methods (penalty approach, time-stepping schemes at the position or velocity level) is proposed. Throughout this chapter, the bouncing ball model problem is considered to highlight the key points of the mathematical analysis without too many technicalities.

The bouncing of an inelastic ball on a vibrating plate is a popular model used in various fields, from granular gases to nanometer-sized mechanical contacts. For random plate motion, so far, the model has been studied using Poincar{\'e} maps in which the excitation by the plate at successive bounces is assumed to be a discrete Markovian (memoryless) process. Here, we investigate numerically the behaviour of the model for continuous random excitations with tunable correlation time. We show that the system dynamics are controlled by the ratio of the Markovian mean flight time of the ball and the mean time between successive peaks in the motion of the exciting plate. When this ratio, which depends on the bandwidth of the excitation signal, exceeds a certain value, the Markovian approach is appropriate; below, memory of preceding excitations arises, leading to a significant decrease of the jump duration; at the smallest values of the ratio, chattering occurs. Overall, our results open the way for uses of the model in the low excitation regime, which is still poorly understood., Comment: Final published version, 5 pages, 4 figures

International audience; The problem of interval observer design is studied for a class of linear hybrid systems. Several observers are designed oriented on different conditions of positivity and stability for estimation error dynamics. Efficiency of the proposed approach is demonstrated by computer experiments for academic and bouncing ball systems.

International audience; We present an experimental study on the statistical properties of the injected power needed to maintain an inelastic ball bouncing constantly on a randomly accelerating piston in the presence of gravity. We compute the injected power at each collision of the ball with the moving piston by measuring the velocity of the piston and the force exerted on the piston by the ball. The probability density function of the injected power has its most probable value close to zero and displays two asymmetric exponential tails, depending on the restitution coefficient, the piston acceleration, and its frequency content. This distribution can be deduced from a simple model assuming quasi-Gaussian statistics for the force and velocity of the piston.

International audience; The contribution deals with timestepping schemes for nonsmooth dynamical systems. Traditionally, these schemes are locally of integration order one, both in non-impulsive and impulsive periods. This is inefficient for applications with infinitely many events but large non-impulsive phases like circuit breakers, valve trains or slider-crank mechanisms. To improve the behaviour during non-impulsive episodes, we start activities twofold. First, we include the classic schemes in time discontinuous Galerkin methods. Second, we split non-impulsive and impulsive force propagation. The correct mathematical setting is established with mollifier functions, Clenshaw-Curtis quadrature rules and an appropriate impact representation. The result is a Petrov-Galerkin distributional differential inclusion. It defines two Runge-Kutta collocation families and enables higher integration order during non-impulsive transition phases. As the framework contains the classic Moreau-Jean timestepping schemes for constant ansatz and test functions on velocity level, it can be considered as a consistent enhancement. An experimental convergence analysis with the bouncing ball example illustrates the capabilities.

We compare the statistical properties of eigenvalue sequences for a gamma=1 Bunimovich stadium billiard. The eigenvalues have been obtained by two ways: one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator (real system) and the other set is calculated numerically (ideal system). The influence of the mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system are shown. We also discuss the influence of a family of marginally stable orbits, the bouncing ball orbits, in two microwave stadium billiards with different geometrical dimensions., Comment: RevTex, 8 pages, 8 figures (postscript), to be published in Phys. Rev. E