Spectral Analysis of a System of Pendula Hanging from a Viscoelastic String and of its Synchronization Patterns

We propose and study a model for the mechanical system constituted by a chain of $n\geqslant 1$ identical pendula hanging from a viscoelastic string with fixed extrema. The novelty of our approach is to describe the string as a continuous system, specifically, as a one-dimensional viscoelastic Kelvin – Voigt string. The resulting system is a hybrid nonlinear system of coupled PDEs and ODEs. We linearize the system around the attractive equilibrium with pendula and string pointing downwards. The (infinite-dimensional) linearization decouples into a ``vertical'' and a ``horizontal'' subsystems. The former is a viscoelastic version of the well known Rayleigh loaded string, and its point spectrum is known. We thus consider the latter, which describes, at the linear level, the horizontal oscillations of string and pendula. We obtain closed form expressions for the eigenvalue equations and for the eigenfunctions for any value of $n$. Next, we study the point spectrum with a combination of analytical and numerical techniques, adopting a continuation approach from the limiting cases of massless pendula, which involves the well known spectrum of the Kelvin – Voigt string. Finally, we focus on the identification, particularly when $n=2$ and as a function of the parameters, of the eigenvalues closest to the imaginary axis, whose eigenfunction(s) dominate the asymptotic dynamics of the (horizontal) linearized systems and can explain the appearance of synchronization patterns in the chain of pendula.