The vectorial form of the Wave Propagation Method (VWM), regarding the dispersion of harmonic plain (elasto-dynamic) waves within certain wave-guides, is developed for the vibration analysis of circular cylindrical shells. To obtain this goal, all plain waves are divided into positive-negative going wave vectors along with the shell axis. Based on the Flügge thin shell theory, the shell continuity as well as boundary conditions are well satisfied by introducing the propagation and reflection matrices. Furthermore, all elements of the reflection matrix are derived for certain classical supports. As an example, for demonstrating the feasibility of VWM in the shell vibration analysis, a circular cylindrical shell with two ended flexible support is adopted. The natural frequencies of the systemaswell asmode shapes are obtained using VWM. The aquired results are compared with those of the previous works and found in excellent agreement. It is also found that VWM could mathematically provide a reduced dimensional matrix (dominant matrix) to calculate the natural frequencies of the system. Accordingly, the proposed method can provide high computational efficiency and remarkable accuracy, simultaneously.

Analyses of free vibration and thermal buckling of nanobeams using nonlocal shear deformation beam theories under various boundary conditions are precisely illustrated. The present beam is restricted by vertically distributed identical springs at the top and bottom surfaces of the beam. The equations of motion are derived using the dynamic version of Hamilton's principle. The governing equations are solved analytically when the edges of the beam are simply supported, clamped or free. Thermal buckling solution is formulated for two types of temperature change through the thickness of the beam: Uniform and linear temperature rise. To validate the accuracy of the results of the present analysis, the results are compared, as possible, with solutions found in the literature. Furthermore, the influences of nonlocal coefficient, stiffness of Winkler springs and span-to-thickness ratio on the frequencies and thermal buckling of the embedded nanobeams are examined.

This paper presents a mathematical model and a computational approach for the thermal post-buckling and free vibration in the vicinity of the buckled equilibrium position of microbeams based on the modified couple stress Euler-Bernoulli beam theory and geometrically accurate relation. The governing equations for the whole analysis are established with a intrinsic material length scale parameter to capture the size effect. These equations, in conjunction with the corresponding boundary conditions, are decomposed into two two-point boundary value problems, which are solved using the shooting method. For static deformation, the geometric nonlinearity is involved and the size dependent postbuckling configuration is obtained as a function of temperature rise. For dynamic one, the small amplitude free vibration about the prebuckled position is developed following an assumed harmonic time mode, and the length scale and temperature rise dependent fundamental natural frequency is presented. Numerical computations are executed to illustrate the size dependency in the thermal postbuckling behaviors and fundamental frequency of the vibration around the buckled configuration of microbeams.

In this paper, free vibration analysis of functionally graded rectangular plate is investigated based on the first order shear deformation theory and the effect of in-plane displacements on the natural frequencies of such plate is studied. The governing equations of motion are obtained, which are five coupled partial differential equations, without any simplification. Some mathematical manipulation leads us to decouple the equations. The decoupled equations are solved by the Levy's method for various boundary conditions. As the results show, in some boundary conditions the in-plane displacements cause a drastic change of frequencies. In other words, neglecting the in-plane displacement, which is assumed in some papers, is not proper for these boundary conditions. However, in the other boundary conditions, the natural frequencies are not significantly affected by the in-plane displacements. The results for various boundary conditions are discussed in detail and some interpretations for these differences are provided. Besides to the comparisons, the accurate natural frequencies of the plate for six different boundary conditions with several aspect ratios, thickness-length ratios and power law indices are presented. The natural frequencies of Mindlin functionally graded rectangular plates with considering the in-plane displacements are reported for the first time and can be used as benchmark.

In this paper, an analytical solution is developed for free vibration analysis of conicalfiber metal shells. In order to find constitutive relations, the assumptions of thinshells are used and the governing equations are based on Love’s theory. The Galerkin methodis employed to solve the governing equations in which beam functions are used toapproximate the mode shapes. Using beam functions enables us to assess the effects ofdifferent boundary conditions on the frequency response of the shells. Numericalcomparisons of the present and previously published results confirm the accuracy of thecurrent approach. Additionally, the influences of geometrical parameters and embeddingaluminum plies in different layers of the structure on natural frequency of the conicalshells with various boundary conditions are investigated. It can be observed that the morethe aluminum plies are used, the greater natural frequency of the structure will bereached. Except the clamped-free boundary conditions, the results also indicate that ifthe aluminum plies are embedded in the top and bottom layers of the laminate, naturalfrequency reaches its maximum value.

Fiber metal laminates (FMLs) are high-performance hybrid structures based on alternatingstacked arrangements of fiber-reinforced plastic (FRP) plies and metal alloy layers. Inthis paper, effect of some geometrical and material parameters on free vibrations of FMLplates was studied. The first-order shear deformation theory (FSDT) as well as the Fourierseries method was used to analytically solve the governing equations of the compositeplate. The accuracy of the used method was verified by comparing the Rayleigh-Ritzanalytical method and the ABAQUS finite element software (numerical) method. The resultsindicated that some of the important parameters like sequence of metal layers, aspectratio (a/b) of plate and orientation of compositefibers were important factors affecting free vibration of the FMLs.

Analytical solution is obtained for the free vibration of cross-ply laminated beams with multiple distributed extension piezoelectric actuators. The piezoelectric actuators are bonded at local position on the beam surface. The beam structure can contain one pair or two pairs or n pairs of piezoelectric actuators and it can be symmetric or unsymmetric about its mid-plane. The equations of motion and associated boundary conditions are derived for the beam model using Hamilton's principle. The state-space approach is used to find accurate natural frequencies and mode shapes for arbitrary combinations of boary conditions. The exact analytical solutions obtained are illustrated numerically in a number of figures revealing the influences of varying some parameters for the symmetric and unsymmetric cross-ply laminated beam for different type of piezoelectric actuators cases. The first order shear deformation beam theory (FOBT) is used to present the effect of actuators position and length on the nondimensional frequencies when one pair and two pairs of piezoelectric actuators are bonded at a local position on the beam surface.

In this paper, the effect of delamination on free vibration of a simply supported rectangular homogeneous plate with through-width delamination was investigated by the finite strip method. A constrained model was used and a finite strip with bending and in-plane stiffness was derived for the free vibration analysis. The effects of delamination length, delamination location in the thickness-wise and spanwise directions, and aspect ratio of the plate on the natural frequencies of the plate were presented. Results show that the delamination has considerable effect on the natural frequencies of the plate. The aspect ratio of the plate is also having significant effect on the natural frequency of the plate, especially on the mode 2 frequency of the plate.

This paper is intended to present a preliminary framework for dynamic interaction analysis of a maglev (magnetically levitated) vehicle running on a two-span guideway using a comprehensive iterative approach. A maglev vehicle with electrodynamic suspension (EDS) system is simplified as a two degrees-of-freedom (2-DOF) maglev oscillator tuned by a PID (Proportional-Integral-Derivative) controller. The guideway is modeled as a two-span continuous beam with uniform section. Two sets of equations of motion are written, with the first set for the guideway and the second set for the maglev oscillator traveling on the guideway through a motion-dependent magnetic force. To achieve the stable levitation gap for a maglev vehicle moving on a flexible guideway, Ziegler-Nicholas (Z-N) tuning rules are used to determine the tuning parameters of the PID controller. Numerical simulations demonstrate that the levitation gap affects the dynamic response of the maglev vehicle while little influence on the guideway response since the inertial force of the moving maglev vehicle is much lower than its static load.