Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law

Abstract

The dynamics of a N-degree-of-freedom autonomous oscillator undergoing an energy-preserving impact law on one of its masses is investigated in the light of nonlinear modal analysis. The impacted rigid foundation provides a natural Poincaré section of the investigated system from which is formulated a smooth First Return Map well-defined away from the grazing trajectory. In order to focus on the impact-induced nonlinearity, the oscillator is assumed linear. Continuous one-parameter families of T-periodic orbits featuring one impact per period and lying on two-dimensional invariant manifolds in the state-space are shown to exist. The geometry of these piecewise-smooth manifolds is such that a linear "flat" portion (on which contact is not activated) is continuously attached to a purely nonlinear portion (on which contact is activated once per period) exhibiting a velocity discontinuity through a grazing orbit. These features explain the newly introduced terminology "Nonsmooth modal analysis". The stability of the periodic orbits lying on the invariant manifolds is also explored by calculating the eigenvalues of the linearized First Return Map. Internal resonances and multiple impacts per period are not addressed in this work. However, the pre-stressed case is succinctly described and extensions to multiple oscillators as well as self-contact are discussed.

Nonsmooth Modal Analysis of Piecewise-Linear Impact Oscillators

“…As in [20], the spectral radius of A.x 0 / is observed to be at least 1. If > 1 for some x 0 , then the periodic solutions induced by x 0 is unstable.…”

“…Periodic motions with multiple impacts per period have sometimes been mentioned for forced systems [17,22,25,33] but never systematically explored. Recently, a spring-mass system subject to a purely elastic impact law with one impact per period has been thoroughly investigated [20]. The present work significantly extends the results of the latter-thorough stability analysis set aside-to more complex geometries, non-diagonal mass matrices (such as in Finite-Element models) and multiple impacts per period, by providing a general analytic and minimal expression of periodic solutions for such systems for any number of k 2 N impacts per period.…”

Periodic Motions of Coupled Impact Oscillators

“…satisfy the inequality constraints corresponding to non-penetration of the obstacles. This has been achieved in a number of works in the case of breathers [17,18,37] and for nonsmooth modes close to grazing linear normal modes [27]. In the work [19], the analysis of [31] has been extended to several impacting particles, but the verification of the inequality constraints is still an open problem in that case.…”

“…In addition, several analytical approaches have been used to obtain time-periodic solutions formally for different types of piecewiselinear dynamical systems with rigid impacts. One can mention Fourier and Green function methods [4,5,8,17,18,19,23,24,25,31,37], modal decomposition [27,38] and sawtooth time transformations [32]. Most of the results obtained for discrete systems concern impacts localized on a single particle, and different types of waves have been constructed.…”

“…Most of the results obtained for discrete systems concern impacts localized on a single particle, and different types of waves have been constructed. In [27,32,38], nonsmooth normal modes have been obtained for general classes of conservative multiple degrees-of-freedom systems (the analysis of [32] is performed for a single or two impacting particles). Spatially-localized oscillations (breathers) with a single impacting node have been also studied for different classes of infinite or finite systems.…”