Revistas

The objective of this paper is to propose a time integration scheme for nonsmooth mechanical systems involving one-sided contact, impact and Coulomb friction, that respects the principles of discrete-time energy balance with positive dissipation. To obtain energetic consistency in the continuous time model when an impact occurs, we work with an impact law with friction inspired by the work of M. Frémond (Frémond, 1995, 2001, 2002, 2017) which ensures that dissipation is positive, i.e. that the Clausius–Duhem inequality is satisfied for the impulses and the velocity jumps. On this basis, we propose a time integration method based on the Moreau–Jean scheme (Jean and Moreau, 1987; Moreau, 1988) with a discrete version of the Frémond impact law, and show that this method has correct dissipation properties.

Acary, Vincent

Arbitrary order expansions for the automatic reduction and solutions of nonlinear vibratory systems have been developed successfully within the realm of the direct parametrisation of invariant manifolds. Whereas the method has been used with high-order expansions and large dimensional systems, this article proposes to look at the same problem from the opposite view angle. By using low-dimensional systems, symbolic computations, analytical developments and numerical verifications, this contribution analyzes the reduced dynamics appearing in cases where a single master mode is involved, reviewing typical scenarios in nonlinear vibrations: primary resonance, sub- and superharmonic resonances and parametric excitation. To achieve this task, the normal form style is preferentially used. A symbolic open-source package is also provided to generalize the presented results to other styles, higher orders, and different scenarios. It is shown how the low-order terms allow recovering the classical solutions given by perturbation methods, and how the automated expansions allow one to generalize the analysis to arbitrary orders. When analytical solutions are not tractable anymore, numerical solutions are employed to underline how converged solutions are at hand when the validity limit of the expansions is not reached. All the results presented in this paper can thus be used to better understand the nonlinear dynamical solutions occurring in nonlinear vibrations, as well as from a system identification perspective, since the normal form is the simplest dynamical system displaying a given resonance scenario.

de Figueiredo Stabile, André