STUDY ON THE FLEXURAL VIBRATION OF RECTANGULAR THIN PLATES WITH FREE BOUNDARY CONDITIONS

An analytical method is presented for the flexural vibration of rectangular thin plates with free boundary conditions. Based on the apparent elasticity method, the flexural vibration of a rectangular thin plate is reduced to two one-dimensional flexural vibrations of slender rods. Then the two-dimensional flexural vibration of a rectangular thin plate with free boundary conditions is considered as the coupling of these two equivalent flexural vibrations with different equivalent elastic constants. It should be noted that these two equivalent flexural vibrations are different from the traditional one-dimensional flexural vibrations of slender rods. They are coupled to each other by the introduced mechanical coupling coefficient. The analytical solutions for the isotropic rectangular thin plate in flexural vibration are derived and the resonance frequency equation is obtained. The natural vibrational mode is analyzed and the frequency spectra are calculated. It is found that the normal modes and the natural frequencies of the rectangular thin plate in flexural vibration are abundant. Theoretical analyses show that one-dimensional flexural vibration of a slender rod based on the classical elementary flexural theory, as well as the stripe mode vibration of a rectangular thin plate is a limiting vibrational mode of rectangular thin plates. Experiments show that the measured resonance frequencies are in good agreeement with the calculated results, and the displacement nodal line pattern is also observed experimentally.