Dynamics of pendula hanging from a string

This thesis aims to explain the phenomenon of the synchronization in mechanical~--~discrete and continuous~--~systems as an outcome of damping contributions affecting the normal modes of oscillation. We present and prove a result for finite-dimensional linear mechanical systems asserting the existence~--~under suitable conditions~--~of an attractive invariant undamped subspace in the phase space, on which the dynamics consists of small oscillations. Thereafter, we construct a hybrid model composed of a homogeneous flexible and elastic string, with fixed extremities, on which are suspended first one, and then two, pendula. We include a dissipative contribution due to internal damping within the string, modelling it as visco-elastic friction. The study, making use of the Lagrangian formalism, provides a linear analysis based on the determination of the undamped and damped normal modes of oscillation. Hence, we investigate beats and synchronization by means of a numerical study as the parameters that characterise the system change.