In this paper, we study a mathematical model for in-plane vibrations of planar networks of beams, incorporating the interaction of longitudinal and transversal displacements at the nodes. We generalize the model proposed in [13] for the special case of a hexagonal network of identical beams, constructed for the description of vibrations of honeycombs. We construct a self-adjoint operator adapted to reformulate the concrete dynamic problem into an abstract wave equation, which is equivalent to an abstract principle of stationary action (see [3]). This allows us to prove the convergence of an expansion of the solution in terms of eigenfunctions of this operator. In view of the explicit study of these eigenfunctions in special geometric configurations, we formulate an associated characteristic equation for the general case.