Missing frequencies in previous exact solutions of free vibrations of simply supported rectangular plates

Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories

“…For a square plate of length/thickness equal to 5, frequencies and through-the-thickness distributions of different stress components computed with a fifth-order plate theory were found to match very well with the analytical solution of Srinivas et al [14]. Batra and Aimmanee [15] have pointed out that both analyses and several others employing similar expansions for displacement fields missed some of the in-plane pure distortional modes that correspond to null transverse displacements. Soldatos and Hadjigergiou [16] had previously given some of these missing frequencies but did not give the corresponding mode shapes.…”

“…A comparison of results listed in Tables 1 and 3 reveals that edge conditions significantly influence not only the natural frequency but also the mode shape. As stated by Batra and Aimmanee [15] in-plane pure distortional modes are inadmissible in a clamped plate. With an increase in the aspect ratio from 4 to 20, the first nondimensional natural frequency decreases from 0.7854 to 0.0589 for a simply supported plate and from 1.4261 to 0.1154 for a clamped plate.…”

“…However, numerical techniques such as the RayleighRitz method employed by Liew et al [17] and the FEM used by Batra et al [21] to solve the 3-dimensional elasticity equations, and the compatible higher order plate theory of Batra and Vidoli [8] in conjunction with the meshless local Petrov-Galerkin method used by Table 2 Comparison of displacements and transverse stresses computed from the present plate theory with the exact solution of Srinivas et al [14] for a simply supported rectangular plate; L x = L y , L x =h ¼ 10 3 , mode = first antisymmetric (m = 1, n = 1) Nondimensional z-coordinate (2z/h) w(z)/w(0) u(z)/u(0) Qian et al [18] predict these frequencies and the in-plane modes of vibration. Batra and Aimmanee [15] have discussed in detail edge conditions and material symmetries which admit these pure distortional modes of vibration.…”

Free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers

“…All of the previous exact or analytical solutions, except that of Ding et al (2000), have missed the so-called special modes due to nil x-axis (m x = 0) or y-axis (m y = 0) wave numbers. The latter were briefly studied recently by Batra and Aimmanee (2003).…”

“…They have also analyzed the influence of the shear piezoceramic core thickness ratio, the sandwich elastic faces lamination scheme, and the piezoceramic lamina thickness position for elastic (EL), non-electroded (NE) and SC/OC electroded laminae (Benjeddou and Deü , 2003). However, the latter two references have, unfortunately, missed the special (m x = 0 or m y = 0) modes mentioned in (Ding et al, 2000;Batra and Aimmanee, 2003).…”