On vertex conditions for elastic systems

• Matching conditions corresponding to the system of three planar beams touching each other are derived.
• It is shown that the vertex conditions depend explicitly on the angles between the beams in the equilibrium position.
• All known examples of vertex conditions appear as special cases of the derived conditions.

In this paper vertex conditions for the differential operator of fourth derivative on the simplest metric graph – the Y-graph, – are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values of functions and their derivatives at the central vertex. It is shown that such conditions corresponding to the free movement of beams depend on the angles between the beams in the equilibrium position.