One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

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Chattering and Related Behaviour in Impact Oscillators

In the present paper two types of nonsmooth oscillators are investigated: an impact oscillator and a self-sustained friction oscillator. Both are nonsmooth one degree of freedom oscillators with harmonic external excitation. Here the different types of motion, bifurcation diagrams and Poincare maps are determined from experiments. These results will be compared with numerical results on the basis of the identified impact and friction models. The nonsmooth third-order systems show rich bifurcational behaviour which is analysed by numerical simulations but also using mapping approaches. Two different formalisms for the calculation of the Lyapunov exponents are applied. The latter one requires special considerations in the given case of nonsmooth systems. Furthermore, the embedding dimension is gained applying the method of false nearest neighbours. In the case of coexisting solutions further analysis is done by means of bifurcation and stability analysis and the cell-mapping approach.

Dynamics of oscillators with impact and friction

Extraction of impacting signals using blind deconvolution

Impacting systems are found in a great variety of mechanical constructions and they are intrinsically nonlinear. In this paper it is shown how near-grazing systems, i.e. systems in which the impacts take place at low speed, can be described by discrete mappings. The derivation of this mapping for a harmonic oscillator with a stop is dealt with in detail. It is found that the resulting mapping for rigid obstacles is somewhat different from those presented earlier in the literature. The derivations are extended to systems with a compliant obstacle. We find that the map for impacts with a compliant obstacle is very similar to the one describing collisions with a rigid obstacle. A notable difference is a change of scale of the bifurcation parameter. We illustrate our findings in the limit of large damping, where the mechanism of period adding can be analysed exactly. The relevance of our results to experiments on practical impact systems is indicated.

/pdf/mappings-of-grazing-impact-oscillators-3qijv47pll.pdf

Mappings of grazing-impact oscillators

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurca...

F. Dux

Papers

The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators