4.2.1 Introduction

Nonlinear dynamical systems exhibiting impacts are often encountered in practice, for instance in the models of hammering in engines [Kunert and Pfeiffer (1989b)], [Pfeiffer and Prestl (1994)], rotor-casing dynamical systems [Li and Paidoussis (1994)], collisions of solids, ships moored at dockside, hand-held percussion machine [Babitsky (1998a)], [Babitsky (1999)], rigid blocks [Iyengar and Roy (1996)], control of joint s loose [Panet et. al. (1995)], [Paoli (1993)], [Paoli et. al. (1992)]. Many applications are given in [Babitsky (1998a)], [Brogliato (1996)]. Such mechanical systems have been mainly investigated in the single degree of freedom or the two degree of freedom case, in order to study the behaviour of the response: periodic responses [Babitsky (1998a)], [Babitsky and Krupenin (2001)], [Van De Vorst et. al. (1996)] including sticking motions [Budd and Dux (1994)], [Toulemonde and Gontier (1998)] in the undamped case and chaotic responses [Shaw and Shaw (1989)] have been found, bifurcations, transitions and global behaviours have been examined [Budd and Dux (1996)], [Peterka (1999a)], [Peterka and Vacik (1992)], [Whiston (1987)], [Yim and Lin (1991)] including grazing bifurcations [Foale and Bishop (1994)], [Frederiksson and Nordmark (1997)], [Nordmark (1991)] and nonsmooth phenomena [Lamba and Budd (1994)], [Lenci and Rega (1999)], [Whiston (1992)]. Some authors have studied more complicated models including both impacts and friction [Cone and Zadoks (1995)], [Peterka (1999b)], [Pfeiffer (1999)], or clearance variations [Budd et. al. (1995)]. Theoretical and experimental analysis of elastic systems undergoing vibro-impact oscillations were carried out in [Emaci et. al.