An analytical model based on lumped parameters for the dynamic analysis of church bells

In this paper a theoretical model able to simulate the dynamic behaviour of church bells is presented. An analytical frame based on Lagrange’s equations is applied to a lumped parameters model, which represents the real yoke–bell–clapper system. Two different approaches are employed to deal with the clapper to bell impact: the first one simply introduces a discontinuity in the equations of motion when impacts occur, while the second one maintains the integration of the equations of motion during the whole contact time, thus providing a description of the contact phenomenon, even though simplified. The proposed model is validated by comparing the numerical results obtained from several simulations on existing bell–clapper systems with the available experimental data in terms of impact acceleration and bell period. As a result, the model is seen to be a reliable predictive tool. Simulation results are also employed to develop some design and installation criteria that can be exploited to reduce the risk of damage for both the bell and the clapper.