The eigenvalue problem and transverse response of a circular plate, that is free at the periphery and that slides freely along the axis of symmetry without bending rotation, are theoretically analyzed. The occurance of eigenvalues in the boundary conditions is accounted for with an extended operator definition in the equation of transverse motion. The stability of these plates under concentrated loads moving at uniform speed is analyzed for (i) harmonic transverse loading and (ii) loading proportional to transverse displacement and velocity. The harmonic loading case leads to a classical, critical‐speed analysis. The proportional loading case represents the excitation of the plate by transverse position guides. The number, orientation, and mechanical properties of the guides determine the transverse stability of the plate‐guide dynamic system.