This study investigates the generalized stiffness of laterally functionally graded materials (LFGMs) and applies these findings to dynamic beam elements. The generalized stiffnesses of LFGM, coupled with material and cross-sectional properties such as flexural and axial rigidity, mass per unit length, and mass-moment of inertia, are explicitly formulated. In the context of LFGM, material properties depend on an asymmetrical power law function with respect to cross-sectional depth. An example of the generalized numerical stiffness of a circular cross-section is provided for various material properties. To illustrate the application of generalized stiffness to dynamic beam elements, free vibration of LFGM beams with rotary inertia is considered. The dimensionless differential equation governing the free vibration of such beams is derived and numerically solved to obtain natural frequencies and corresponding mode shapes. Numerical results demonstrate a good consistency with the finite element method.

This paper deals with free vibrations of the axially functionally graded (AFG) horseshoe arch. The modulus of elasticity and the mass density of AFG material of arch are chosen as a univariate quadratic function. The differential equations with the boundary conditions that govern the free vibration of such arch are derived and numerically solved to calculate natural frequencies and mode shapes. Natural frequencies of this study agree well with those of the finite element ADINA. Parametric studies of the geometrical and mechanical properties of the arch on frequencies and mode shapes are performed and extensively discussed.

In this paper, free vibrations of the tapered shear deformable column are studied. The column is clamped at the bottom and is free or hinged or clamped at the top. The column has a tapered cross-section, square and circular, under the condition of constant volume. The axial compressive load is acted to the top. The differential equations of such column were derived, in which the effects of the rotary inertia and shear deformation were included. For computing natural frequencies with mode shapes, the differential equations were solved numerically. The effects of column parameters of the frequencies and mode shapes were performed, and its results were extensively discussed.

In the present world stiffeners are widely used to fulfill the requirement of high stiffness. Stiffened plates are used in most of the mechanical structures. In this paper free vibration analysis of eccentric and concentric stiffened isotropic plate with central stiffener and double stiffener has been studied and effect of various parameters such as boundary conditions, aspect ratio on non-dimensional frequency parameter of plate are investigated. Comparison of non-dimensional frequency parameter for eccentric and concentric isotropic stiffened plate with effect of number of stiffener at different mode shapes, aspect ratios and boundary conditions is studied using ANSYS parametric design language (APDL) code. A ten nodded element (SOLID 187) from ANSYS element library is used for the discretization of proposed stiffened isotropic plate. Convergence study of developed model with respect to the number of modes has been done and results are compared from the related available published literature. Non dimensional frequencies are higher for fully clamped boundary condition for eccentric and concentric stiffened isotropic plate in comparison to other boundary condition. It is also seen that non dimensional frequencies are increases in a moderate value for eccentric and concentric isotopic plate with double stiffener in comparison to the central stiffener.

This paper presents the free vibration analysis of stiffened isotropic plate by means of finite element method. Stiffeners are used in plates to increase the strength and stiffness. The effect of position of stiffeners on isotropic plate has been studied which involve the possible combination of clamped and free edge condition. The model has been discretized using a 20-node solid element (SOLID186) from the ANSYS element library. The natural frequencies are calculated using Block-Lanczos algorithm. The comparisons of stiffened plate with the available results are found to be in good uniformity. The effect of different boundary conditions, stiffeners location, thickness ratio, stiffener thickness to plate thickness and aspect ratio on the vibration analysis of stiffened plates has been studied.

In the present paper, the differential transformation method is employed to develop a semi-analytical solution for free transverse vibration of single-walled carbon nanotube (SWCNT) with arbitrary boundary conditions. The small scale effect is taken into consideration via Eringen’s nonlocal elasticity theory while the transverse shear deformation effects and rotary inertia are taken into account in presented Timoshenko beam theory. Through variational formulation and the Hamilton & apos; s principle the governing differential equations and the boundary conditions are derived and then solved by a semi-analytical method called differential transformation method (DTM) for various frequency modes of beams and different edge conditions. Comparisons made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the presented approach. The effects of several parameters such as transverse shear deformation effects, slenderness ratios, boundary conditions and small scale on vibration characteristics of SWCNT are examined. The present study illustrates that the vibration characteristics of an SWCNT are strongly dependent on the small scale parameters.

Cutouts are commonly used as access port for mechanical and electrical structures. Most of the structures generally work under severe dynamic loading and different constrained conditions during their service life. This may lead to vibration of the structure. Therefore, it is necessary to predict the vibration responses of laminated composite plates with cutouts precisely with less computational cost and good accuracy of these complex structures. A suitable finite element model is proposed and developed based on first order shear deformation theory using ANSYS parametric design language (APDL) code. The model has been discretized using an appropriate eight nodded element (SHELL 281) from the ANSYS element library. The free vibrations are computed using Block-Lanczos algorithm. The convergence study has been done of the developed model and compared with those available published literature. Effects of different geometric parameters (aspect ratio, thickness ratio, boundary conditions, number of layers, angle of lamina geometry of cutout, cutout side to plate side ratio and distance between cutouts) and material properties on the free vibration responses are discussed in detail. The frequency increases with increase in the number of layers, modulus ratio of plate and angle of lamina. The frequency decreases with increase in aspect ratio, thickness ratio, size of cutout and distance between cutouts. The boundary conditions of the plate play an important role in the free vibrations of the plate with cutouts. The Non-dimensional frequencies are higher for fully clamped boundary condition in comparison to other boundary conditions.

In this paper, an analytical method is proposed for calculation of natural frequencies of a delaminated composite beam from both free and constrained mode frequencies. In previous studies, the frequencies of a delaminated composite beam were computed with assumption of occurring open or close delamination during the vibration. According to this assumption, two separated modes, i.e., “free mode” and “constrained mode”, are occurred in vibration of the delaminated beam. In fact, a delamination may breathe (open and close) during the vibration and the assumptions of the free or constrained mode models are not completely correct in the whole of the vibration period. For this reason, a new formulation is proposed for calculation of natural frequencies based on the breathing of delamination. The obtained results are compared with various theoretical and experimental results available in the literature. Thus, the effects of location and size of delamination can be investigated on the natural frequencies of delaminated beams.

Finite element method is used to study the free vibration analysis of functionally graded skew plates. The material properties of the skew plates are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. The first order shear deformation theory is used to incorporate the effects of transverse shear deformation and rotary inertia. Convergence study with respect to the number of nodes has been carried out and the results are compared with those from past investigations available in the literature. Two types of functionally graded skew plates - Al/ZrO2 and Al/Al2O3 are considered in this study and the effects of the volume fraction, different external boundary conditions and thickness ratio on the natural frequencies are studied in detail.

This paper deals with exact solution for free vibration analysis of simply supported rectangular plates on elastic foundation. The solution is on the basis of three dimensional elasticity theory. The foundation is described by the Pasternak (two-parameter) model. First, the Navier equations of motion are replaced by three decoupled equations in terms of displacement components. Then, these equations are solved in a semi-inverse method. The obtained displacement field satisfies all the boundary conditions of the problem in a point wise manner. The solution is in the form of a double Fourier sine series. Then free-vibration characteristics of rectangular plates resting on elastic foundations for different thickness/span ratios and foundation parameters are studied. The numerical results are compared with the available results in the literature. Important parameters on the accuracy of plate theories and free-vibration characteristics of rectangular plates resting on elastic foundations are discussed.