A scheme for the non-smooth modal analysis of two-dimensional objects in unilateral contact is demonstrated for the case of a rectangular plate in frictionless unilateral contact with a rigid foundation. The unilateral contact conditions are described via the Signorini complementarity conditions. Here, application of nonsmooth modal analysis requires finding periodic solutions to the Signorini problem. To treat the Signorini boundary conditions, the nodal boundary method in the framework of the finite element method is used. The nodal boundary method allows to approximate the solution to the Signorini problem via a system of nonsmooth ordinary differential equations in the internal displacements. The resulting system exhibits periodic solutions without non-physical chattering. These periodic solutions are found via the harmonic balance method and sequential continuation is used for detection of nonsmooth modes.