Free vibration analysis of a thin rectangular elastic plate constrained at the corner points by a rigid frame

The free vibration of a thin rectangular elastic plate constrained at the corner points by a rigid frame is investigated in this paper. Different types of single-point connections are described as compatibility conditions between the vibration displacements of the plate and the frame, and with the geometric relations among different points on the rigid frame, the frame is transformed into constraint equations imposed on the elastic plate. The Lagrange multipliers are employed to enforce the constraint equations, and the natural frequencies and the mode shapes of the plate are obtained using the Rayleigh-Ritz method. Through a comparison with the results of the literature the finite element method software, the validity of the present method is confirmed. Finally, a comparison is carried out among models with various combinations of single-point connections at four corners, and the results indicate that the difference in the connections between the frame and the plate leads to obvious changes in the natural frequencies and the mode shapes, which confirms the value of the present method in the vibration analysis of such structures. Plates of different aspect ratios are also considered.