Natural frequencies of rectangular orthotropic plates with a pair of parallel edges simply supported

Solutions of the exact characteristic equations for the title problem derived earlier by an extension of Bolotin's asymptotic method are considered. These solutions, which correspond to flexural modes with frequency factor, R, greater than unity, are expressed in convenient forms for all combinations of clamped, simply supported and free conditions at the remaining pair of parallel edges. As in the case of uniform beams, the eigenvalues in the CC case are found to be equal to those of elastic modes in the FF case provided that the Kirchoff's shear condition at a free edge is replaced by the condition $\text{?M}{}_{\mathrm{n}}\text{?n = 0}$. The flexural modes with frequency factor less than unity are also investigated in detail by introducing a suitable modification in the procedure. When Poisson's ratios are not zero, it is shown that the frequency factor corresponding to the first symmetric mode in the free-free case is less than unity for all values of side ratio and rigidity ratios. In the case of one edge clamped and the other free it is found that modes with frequency factor less than unity exist for certain dimensions of the plate—a fact hitherto unrecognized in the literature.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Natural frequencies of rectangular plates with free edges

This note presents vibration analysis of isotropic rectangular plates with free edges by the Rayleigh-Ritz method with B-spline functions. To show the accuracy of the present method, the results are compared with existing results based on other numerical methods and found to be in good agreement. Accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions. The effects of Poisson's ratio on natural frequencies of square plates with free edges are also investigated.     

Transverse vibrations of circular plates with stepped thickness

The study of the dynamic behaviour of circular plates with stepped thickness is of interest in view of their use in the construction of high frequency transducers. A simple analytical approach which allows for the prediction of their natural frequencies is proposed in the present Note.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

Natural frequencies of orthotropic, monoclinic and hexagonal plates by a meshless method

The collocation method with multiquadrics basis functions and a first-order shear deformation theory are used to find natural flexural frequencies of a square plate with various material symmetries and subjected to different boundary conditions. Computed results are found to agree well with the literature values obtained by the solution of the three-dimensional elasticity equations using the finite element method.     

Effect of thermal gradient on frequencies of an orthotropic rectangular plate whose thickness varies in two directions

A simple model is presented for the use of research workers in space technology, mechanical sciences and nuclear energy where certain components of the structures have to operate under elevated temperatures. The effect of a constant thermal gradient on the free vibrations of an orthotropic rectangular plate whose thickness varies linearly in two directions is considered. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz techniques with a two-term deflection function. The upper bound on the frequency corresponding to the first mode of vibrations is obtained for various boundary conditions, and for various values of the length to width ratio, two paper constants and the temperature gradient.     

Natural frequencies of twisted rotating plates

A detailed comparison is presented of the predicted eigenfrequencies of twisted rotating plates as obtained by using two different shape functions. Primarily, rotating twisted plates of two different aspect ratios and two different thickness ratios are considered. The effects of rotation are included by using a “stress smoothing” technique when calculating the augmented stiffness matrix. In addition, the effects of Coriolis acceleration, contributions from membrane behaviour, setting angle and sweep angle are considered. The effects of geometric non-linearity are briefly discussed. Finally, results of a brief study of cambered plates are presented.