Vibrations of initially stressed cylinders of variable thickness

Natural frequencies and buckling loads for cylindrical shells having linearly varying thickness are obtained by using a segmentation technique. The present results for free vibration of a cylinder compare very well with those obtained previously. The effect of the thickness variation on the frequencies of a cylindrical shell is studied. Frequencies are also calculated for a cylinder of variable thickness under axial compression and a relationship between the frequency and axial compression is obtained for a particular wave number.     

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.     

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.     

Vibrations of rectangular plates with elastically restrained edges

The Rayleigh-Ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges. By treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions, a relatively small number of functions is required; consequently only a modest quantity of computation is necessary. The good accuracy of the method is demonstrated by solving test problems. The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes in addition to the fundamental. The usefulness and accuracy of existing simplified approaches to the problem are assessed. The effect of in-plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also.     

Large amplitude vibration of an initially stressed moderately thick plate

Non-linear equations of motion for a transversely isotropic moderately thick plate in a general state of non-uniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The large amplitude flexural vibration of a simply supported rectangular moderately thick plate subjected to initial stress is investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are used to solve the vibrations problem by the Galerkin method. The effects of various parameters on the non-linear vibration frequencies are studied.     

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 orthotropic rectangular plates with stepped thickness

It is shown how one can derive an approximate formula for estimating a natural frequency of an orthotropic rectangular plate with stepped thickness by using several natural frequencies of the corresponding isotropic plate reduced from the orthotropic one. To justify the method, an orthotropic two-part rectangular plate with simply supported sides is discussed, and an approximate formula is proposed for estimating the fundamental natural frequency.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Random vibration of an initially stressed thick plate on an elastic foundation

The mean-square bending moment of a thick rectangular plate excited by a uniform distribution of stationary random forces that are uncorrelated in space is calculated. The plate has in-plane compressive or tensile stresses. In addition, the plate is mounted on an elastic foundation. Numerical results are given for plates with uniform initial stress when the temporal correlation function of the excitation possesses an exponential decay. In general it can be said that the position on the plate where the mean-square moment takes on a maximum value depends upon the relative values of the initial stress, the stiffness of the foundation and the aspect ratio of the plate. The mean-square response amplitude of the plate on a foundation never exceeds that of the plate without a foundation, regardless of the intensity of the initial stress or the geometrical configuration of the plate.     

Vibrations of initially imperfect circular plates including the shear and rotatory inertia effects

An analysis of the free flexural vibrations of elastic circular plates with initial imperfections is presented. The analysis includes the effects of transverse shear and rotatory inertia. The vibration amplitudes are assumed to be large, and two non-linear differential equations are obtained for free vibration of the plate and solved numerically. The period of the plate has been calculated as a function of the initial amplitude for four typical supporting conditions.     

Vibrations of non-uniform rectangular plates: A spline technique method of solution

The fourth order differential equation governing the transverse motion of an elastic rectangular plate of variable thickness has been solved, by using the quintic spline interpolation technique. An algorithm for computing the solution of this differential equation is presented, for the case of equal intervals. Frequencies, mode shapes and moments for doubly symmetric, antisymmetric-symmetric and second symmetric-symmetric modes of vibration are presented for various cases of boundary conditions.     

Vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions

Results are presented for the fundamental frequency of a rectangular plate having a thickness which varies in a bilinear fashion in the x-direction. Translational and rotational flexibilities are taken into account at all edges. A simple algorithm, which allows one to evaluate the fundamental frequency of vibration, is derived by making use of the Ritz method and expressing the fundamental displacement function in terms of a polynomial co-ordinate function which satisfies approximately the natural boundary conditions.     

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

We use the first-order shear deformation theory and a meshless method based on radial basis functions in a pseudospectral framework for predicting the free vibration behavior of thick orthotropic, monoclinic and hexagonal plates. The shape parameter is obtained by an optimization procedure. The three translational and two rotational degrees of freedom of a point of the laminate are independently approximated. Through numerical experiments, the capability and efficiency of the radial basis functions—pseudospectral method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are examined.