## Abstract

We study the qualitative dynamics of a simple mass-spring system involving non regularized unilateral contact and Coulomb friction and submitted to an oscillating external force. The period-amplitude plane of the excitation appears to be essentially divided into two ranges of sliding solutions. At each point of the lower range there exist infinitely many equilibrium points and all the trajectories go to equilibrium in finite time. In the upper range, there no longer exist equilibria. Different kinds of periodic solutions are shown to exist in different zones and the transitions between these zones are explicitly computed. The upper boundary of this range, where the mass looses contact, is also computed and special attention is paid to the dependence of this upper boundary with respect to the period of the excitation.

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## Notes

Let us compare problems (21) and (25). It appears that proving that (21) has a solution everywhere in zone \(\Upomega_5\) amounts to proving that problem (25) has a solution such that

*u*_{4}is constant everywhere in a nonzero measure subset of the \(\{T,\varepsilon\}\) plane, which could have been missed by a direct study of problem (25), and which is a result interesting in itself. Moreover distinguishing Zones \(\Upomega_5\) and \(\Upomega_6\) seems easier for an intuitive introduction to the partition of the plane. Nevertheless, the distinction between these two zones would not be necessary if we were dealing only with the transition to the loss of contact.In fact in this range the complete calculations are obviously also carried out using Maple, but here from explicit formula.

A difficulty has already been encountered when studying the transition from Zone \(\Upomega_5\) to Zone \(\Upomega_6: \) since we don’t know a lot about the qualitative behaviour, another guess could be that the loss of contact arises only through non periodic solutions so that the line \(\rbrack T_{\alpha} + T_{\beta}, + \infty \lbrack \times \{\frac{4\mu \mathcal{A}}{K_t+3\mu W}\}\) would be the boundary for the loss of periodicity, instead of a transition between different types of periodic solutions. The answer is given if problem (27) has a solution.

It is interesting to observe that, while giving the transition between Zone \(\Upomega_1\) and Zone \(\Upomega_2\) only amounts to plotting the graph of an explicit function, calculating all the other curves of Fig. 18 requires an implicit computation that can be very time consuming.

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Léger, A., Pratt, E. Qualitative analysis of a forced nonsmooth oscillator with contact and friction.
*Ann. Solid Struct. Mech.* **2**, 1–17 (2011). https://doi.org/10.1007/s12356-011-0015-7

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DOI: https://doi.org/10.1007/s12356-011-0015-7

### Keywords

- Coulomb friction
- Mass-spring systems
- Nonsmooth dynamics
- Stability
- Unilateral contact