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In this research, the equations governing the vibrations of a composite curved beam reinforced with graphene platelets have been solved using a quasi-3D model of the beam. The investigated composite beam is multi-layered, each layer can be reinforced by different amount of graphene platelets, which leads to a functionally-graded distribution of reinforcements. According to the quasi-three-dimensional theory of beams, the effect of thickness stretching and shear strains is included and does not require a shear correction factor. It should be noted that according to the geometry of the assumed beam, deformations and strains are considered small and the structural relations of the material are linear. The governing equations of the curved beam are derived based on Hamilton’s principle after calculating the kinetic and strain energies. The boundary conditions of the beam are assumed to be simply supported on both ends. Therefore, the displacement parameters of the problem have been explained with Navier’s solution method and with the use of Fourier expansions, closed form expressions are obtained as elements of mass and stiffness matrices. The elasticity modulus of the beam is rewritten using the Halpin–Tsai model and Poisson’s ratio and mass density using the rule of mixtures. Finally, the results of each section are presented in the form of tables. First, the results of this research were verified with the existing results, and then new results were introduced to investigate the effective parameters of the problem. These results include the investigation of the distribution of graphene platelets, the effect of the weight percentage of the filler and the geometric properties of the beam. The results of this research on the dynamic response of the beam show that the effects of increasing the volume percentage of graphene, even to a small extent, can be seen in increasing the natural frequency of the curved beam.

Ballasted railway tracks are impaired due to either normal or abnormal operations. One of the problems is the differential settlements along the track. Clearly, there is the need to maintain periodically the track substructures by means of ballast packing/tamping. Inappropriate conducts result in the nonlinear distributions of support stiffness. This study firstly demonstrates the effects of improper ballast packing/tamping on the free vibration behaviors of in situ railway concrete sleepers. The two-dimensional finite element modeling of an in situ concrete sleeper was employed in the parametric studies. This model takes into account the coupled flexural-and-shear deformations of concrete sleepers, elastic displacements of fastening system, and nonlinear dynamic interaction between the sleeper and ballast support. Dynamic interaction between sleepers and ballast was investigated based on the nonlinear distribution of ballast support stiffness underneath the sleeper. Effects of both symmetrical and asymmetrical stiffness distributions on dynamic behaviors of the local in situ concrete sleeper were also highlighted.

This paper deals with the determination of free vibration characteristics and instability conditions of flexible spinning cylinders partially filled with fluid. Using the linearized Navier–Stokes equations for the incompressible, inviscid flow, a 2D model is developed for fluid motion at each section of the cylinder. The forces exerted on the cylinder wall as a result of the fluid motion are calculated as functions of lateral acceleration of the cylinder axis in the Laplace domain. Applying the Hamilton principle, the governing equations of flexural motion of the cylinder are derived and then combined with the equations describing the fluid forces to obtain the coupled field equations of the structural motion and the liquid sloshing dynamics. Using the obtained model, the free vibration and the stability conditions of the cylinder as well as the effect of material viscoelasticity on the stability boundaries are examined. The results of the stability analysis are presented for a number of examples, and some conclusions are outlined.

In this paper, a procedure based on the transfer matrix method for obtaining the exact solution to the equations of free vibration of damaged frame structures, considering the effects of axial extension, shear deformation, rotatory inertia, and all compliance components arising due to the presence of a crack, is presented. The crack is modeled by a rotational and/or translational spring based on the concept of linear elastic fracture mechanics. Only the in-plane motion of planar structures is considered. The formulation is validated through some examples existing in the literature. Additionally, the mode shapes and natural frequencies of a frame with pitched roof are provided. The variation of natural frequencies with respect to the crack location is presented. It is concluded that considering the axial compliance, and axial-bending coupling due to the presence of a crack results in different dynamic characteristics, which should be considered for problems where high precision is required, such as for the crack identification problems.

This paper presents a unified method for analyzing the dynamic behavior of spinning beams under elastic constraints. Based on the Timoshenko beam theory, a dynamic model of a spinning beam with elastic constraints is established. The displacement and bending angle are represented by a modified Fourier series. Compared with the traditional Fourier series, the improved Fourier series eliminates the discontinuity of the derivative at the boundary by introducing auxiliary polynomials, making it more suitable for elastic constraints. The governing equations and boundary conditions are coupled together using the energy method to form a set of standard linear equations. By solving this equation, the modes of the spinning beam structure under elastic constraints can be concisely and quickly obtained. Finally, by comparing with other methods, it is proved that the method has good convergence and practicability. Then, the effects of spinning speed, boundary stiffness and slenderness ratio on the whirling characteristics are analyzed. The results show that the linear spring has a more pronounced effect on the whirl frequency than the torsion spring. Different boundary constraints produce different elastic intervals. Mode exchange was found with increasing spinning speed. In the case of elastic constraints, the mode exchange occurs at lower spinning speed. This method has a certain universal applicability to the dynamic analysis of spinning beams under elastic constraints, and the research results can provide theoretical reference for subsequent research.

In this work, a new high accuracy solution for vibrations of structures made of plate segments is presented. Each plate segment is deforming both in and out of its plane, and the edges common to two plate segments must have compatible displacements, rotations, shear forces, and moments. It is obtained by using carefully chosen series that solve the combined partial differential equations of motion, for in-plane and out-of-plane deformations for all possible combinations of edge conditions. The number of terms in the series is taken such that convergence is ensured to the number of digits as shown. Examples of the new solutions are given and compared with available solutions for the same cases.

A study of free vibration of Timoshenko beams and Mindlin plates is presented whereby the analysis is based on the radial basis function method. Various approaches for computing eigenvalues using radial basis functions are also discussed together with the numerical results for various thickness-to-length ratios.

In this paper, the cell-based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young’s Modulus and the corresponding stochastic results are represented by Karhunen-Loéve expansion and polynomial chaos expansion, respectively. Young’s Modulus of structure is considered to be random for stochastic static as well as free vibration problems. Mathematical expressions and the solution procedure are articulated in detail to evaluate the statistical characteristics of responses in terms of the static displacements and the free vibration frequencies. The feasibility and the effectiveness of the proposed SGCS–FEM method in terms of accuracy and lower demand on the mesh size in the solution domain over that of conventional FEM for stochastic problems are demonstrated by carefully chosen numerical examples. From the numerical study, it is inferred that the proposed framework yields accurate results.

The free vibration problem of one-layered and two-layered metallic plates is investigated in this work. The thermomechanical effect is evaluated using a fully coupled thermomechanical model. The free frequency values of fully coupled problems are compared to the values of the pure mechanical problems. In pure mechanical models, the displacement is the only primary variable of the problem, while in fully coupled thermomechanical models, the temperature is also considered as a primary variable and the effect of the thermomechanical stiffness is evaluated. The thermoelastic coupling usually provides higher frequencies with respect to the pure mechanical case because it acts like a thermal source, which is proportional to the strain rate, which leads to a bigger global stiffness of the structure. Both thermomechanical and mechanical models are developed in the framework of Carrera's Unified Formulation (CUF). CUF permits several refined two-dimensional theories to be obtained with orders of expansion in the thickness direction, from linear to fourth-order, for both displacements and temperature. Both equivalent single layer and layer-wise approaches are considered for the multilayered plates. The thermomechanical effect is investigated, in terms of frequencies, for thick and thin one-layered and two-layered plates, and for lower and higher modes. It has mainly been concluded that the thermomechanical coupling: (a) Is correctly determined if both the thermal and mechanical parts are correctly approximated; (b) Is small for each investigated case; (c) Influences the various vibration modes in different ways; and (d) Has a limited dependence on the considered case, but this dependence vanishes if a global coupling is considered. Only fully coupled thermomechanical models allow to analyze this type of problem. The effect of the thermomechanical coupling on higher-order modes can only be investigated using refined two-dimensional theories.

A 3D free vibration analysis of multilayered structures is proposed. An exact solution is developed for the differential equations of equilibrium written in general orthogonal curvilinear coordinates. The equations consider a geometry for shells without simplifications and allow the analysis of spherical shell panels, cylindrical shell panels, cylindrical closed shells and plates. The method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structures. Results are given for multilayered composite and sandwich plates and shells. A free vibration analysis is proposed for a number of vibration modes, thickness ratios, imposed wave numbers, geometries and multilayer configurations embedding isotropic and orthotropic composite materials. These results can also be used as reference solutions for plate and shell 2D models developed for the analysis of multilayered structures.

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

In the present work, in accordance with the generalized theory of thermoelasticity with two thermal relaxation times, the vibration of a thick finite nanobeam resonator has been considered. Both the general thermoelasticity and coupled thermoelasticity (CT) theories with only one relaxation time can be deduced from the present model as special cases. Under clamped conditions for beam, the effect of relaxation times in nanobeam resonator has been investigated. Based on the analytical relationships, the beam deflection, temperature change, frequency shift and thermoelastic damping were investigated and the numerical results were graphically obtained. According to the observed results there is a clear difference between the CT theory, Lord and Shulman’s (LS) theory and Green and Lindsay’s (GL) theory.

An analytical method is presented for determination of vibration characteristics of high speed Double-Segment Compound Rotating disks. More specifically, a systematic approach based on established solution for linear in-plane vibration of each segment satisfying the displacement and stresses compatibility is developed. Fixed and free boundary conditions for the compound spinning annular disks are considered, and natural frequencies and mode shapes of rotating the disks are computed. The medium for each segment is considered to be homogenous, isotropic, and elastic. The developed analytical solution was achieved by implementing two-dimensional plane stress theory. The modal displacements and stresses at both inner and outer boundaries are determined. The dimensionless natural frequencies for different modes, rotating speeds, and thickness ratios are computed. The effect of stiffness changes for each segment on the natural frequencies are also studied.

This paper deals with the large amplitude free vibrations of inclined cables in a three dimensional space. Presented herein is a model formulation which is not limited to cables having small sag-to-span ratios and takes into consideration the axial deformation effect. The nonlinear equations of motion are obtained by the principle of virtual work-energy. The spatial and temporal problems of a multi-degree-of-freedom cable model are then solved by the finite difference method. Numerical examples for the case of a specified tension of arbitrarily inclined sagged cables are given. The effects of cable inclination, cable sag and internally resonant condition on cable nonlinear dynamics are demonstrated.