The dynamics of bells as impacting system

The dynamics of impacting systems is the topic of this research study. As a familiar example of a simple impacting device, which can be used to demonstrate well-known features, the motion of a bell is analysed. Based on the abstract modelling as a double pendulum with state-dependent impacts given by the strikes of the clapper onto the bell, various forms are derived. In particular, it is shown that the size of the 'coupling' between clapper and bell determines the form of the oscillations showing such phenomena as falling or flying clapper, chatter, and the spectacular case of non-ringing which occurred in the case of the famous emperor's bell in the Cathedral of Cologne. Besides the specific results concerning the motion of the bell, the example can serve as a simple illustration of complicated dynamics in the presence of impacts.