Spectrum of a network of Euler–Bernoulli beams

A network of N flexible beams connected by n vibrating point masses is considered. The spectrum of the spatial operator involved in this evolution problem is studied. If λ2 is any real number outside a discrete set of values S and if λ is an eigenvalue, then it satisfies a characteristic equation which is given. The associated eigenvectors are also characterized. If λ2 lies in S and if the N beams are identical (same mechanical properties), another characteristic equation is available. It is not the case for different beams: no general result can be stated. Some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization. At last the asymptotic behaviour of the eigenvalues is investigated by proving the so-called Weyl's formula.