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The aim is to study the boundary controllability of a system modelling the vibrations of a network of $N$ Euler-Bernoulli beams serially connected by $(N-1)$ vibrating interior point masses. Using the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W. H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required for the observability inequality to hold. They are established using an adequate basis.