Nonlinear Dynamics 34: 249–268, 2003. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. Regular Nonlinear Dynamics and Bifurcations of an Impacting System under General Periodic Excitation STEFANO LENCI ⋆ and GIUSEPPE REGA Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma ‘La Sapienza’, via A. Gramsci 53, I-00197 Roma, Italy; E-mail: lenci@univpm.it, giuseppe.rega@uniroma1.it (Received: 5 May 2001; accepted: 26 July 2002) Abstract. A class of periodic motions of an inverted pendulum with rigid lateral constraints is analyzed under the hypothesis that the system is forced by an arbitrary periodic excitation. The piecewise linear nature of the problem permits to obtain analytical results. The periodic solutions are determined as fixed points of the strobo- scopic Poincaré map, and it is shown that the stability is lost through classical saddle-node or period-doubling bifurcations. It is shown that the existence paths can be determined, both geometrically and analytically, on the basis of a function which can easily be derived from the excitation f(t). The main qualitative properties of these paths are discussed, and attention is paid to the detection of the bifurcations determining the stability of the solutions. None of the obtained results depend on the specific properties of the excitation, and all can be employed in the analysis of various cases with both symmetric and unsymmetric excitations. Some illustrative examples are reported at the end of the paper. Keywords: Periodic solutions, bifurcations, impacting system, general periodic excitation. 1. Introduction This paper deals with the analysis of the inverted pendulum with rigid lateral barriers and subjected to periodic external excitation (Figure 1). The study of this system has a double motivation. On the one hand, it arises in some practical application [1–3], such as rings, machine tools, rigid block dynamics [4–6], moored vessels [7, Section 15.2], rolling railway wheelset [8, 9] and so on. On the other hand, it represents a prototype of more involved systems as well as of other impacting oscillators [10–13], for which, thanks to the hypothesis of piecewise linearity, the dynamical complex behavior can be detected analytically. The dynamics of inverted pendulum have been repeatedly analyzed in the literature. Shaw and Rand [14] studied analytically the existence and the stability of two classes of periodic solutions and discussed their relations with the route to chaos established by the Melnikov’s method. This work was accompanied by the extensive experimental investigation of Moore and Shaw [15] aimed at checking some of the theoretical results. Successively, Shaw [16] suggested to modify the shape of the excitation to avoid homo- clinic intersection and, consequently, to control the chaotic dynamics of the system. He looked for the solution in a restricted class of excitations and found a two term optimal excitation (Section 7.1). The optimization problem proposed by Shaw was analytically solved by Lenci and Rega [17], which found that the optimal periodic solution is constituted by two equal and opposite impulses plus a bounded part, and by Lenci [18] who obtained the optimal solution ⋆ Now at Istituto di Scienza e Tecnica delle Costruzioni, Universit` a Politecnica delle Marche, via Brecce Bianche, I-60131 Ancona, Italy.