We propose and study a model for the mechanical system constituted bya chain of \(n\geqslant 1\) identical pendula hanging from a viscoelasticstring with fixed extrema. The novelty of our approach is todescribe the string as a continuous system, specifically, as aone-dimensional viscoelastic Kelvin – Voigt string. The resultingsystem is a hybrid nonlinear system of coupled PDEs and ODEs. Welinearize the system around the attractive equilibrium with pendulaand string pointing downwards. The (infinite-dimensional)linearization decouples into a “vertical” and a “horizontal”subsystems. The former is a viscoelastic version of the well knownRayleigh loaded string, and its point spectrum is known. We thusconsider the latter, which describes, at the linear level, thehorizontal oscillations of string and pendula. We obtain closed formexpressions for the eigenvalue equations and for the eigenfunctionsfor any value of \(n\) . Next, we study the point spectrum with acombination of analytical and numerical techniques, adopting acontinuation approach from the limiting cases of massless pendula, whichinvolves the well known spectrum of the Kelvin – Voigt string. Finally,we focus on the identification, particularly when \(n=2\) and as afunction of the parameters, of the eigenvalues closest to theimaginary axis, whose eigenfunction(s) dominate the asymptoticdynamics of the (horizontal) linearized systems and can explain theappearance of synchronization patterns in the chain of pendula.
