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In this paper, the bending, buckling, and free vibration of functionally graded porous (FGP) beams are studied based on two beam theories (with or without considering thickness stretching, respectively). The effect of thickness stretching is obtained by comparing the results of the two theories. Two symmetrical distributions and one asymmetrical distribution of pores are considered. Both Young’s modulus and mass density of the FGP beams are in gradient variation in the thickness direction. The governing equations are constructed using Hamilton’s principle. The analytical solutions are obtained by Navier’s method. The effects of slenderness ratios, pore distribution, porosity and thickness stretching on FGP beams have been investigated. The results show that the inhomogeneity of FGP beams in the thickness direction is positively correlated with the effect of thickness stretching.

The free vibration analysis of a perovskite solar cell (PSC) within thermal environments is studied through a quasi-3D plate model. This model is designed to account for the stretching effects and non-uniform shear strains throughout the thickness. The PSC film is represented in the model as a thin laminated plate, comprising five distinct plies: ITO, PEDOT:PSS, perovskite, PCBM, and Au. A graphene platelets reinforced composite (GPLRC) substrate, made of Poly (methyl methacrylate), is assumed to be located under the noted five layers. The GPLRC substrate is conceptualized in the model as a porous foundation with finite depth. The foundation stiffnesses are determined using a modified Vlasov model, and the equivalent Young modulus of the foundation is evaluated using a modified Halpin–Tsai model, introducing the porosity coefficient. It is also considered that the material properties of GPLRC substrates exhibit temperature dependency. For the PSC with all edges simply supported, Navier solution method is applied to obtain the frequencies and associated mode numbers. Based on the results presented in this study, an increase in the porosity and thickness of the substrate can negatively impact the frequencies of the structure. However, natural frequencies experience almost no change if the temperature rises.

This study aims to analyze the free vibration of an ogival arch in which two symmetrical arched shapes are discontinuously met at the mid-arc. Differential equations governing the free vibration of an ogival circular arch were derived, where the rotatory inertia and shear deformation effects are included. The governing equations were numerically solved to calculate natural frequencies and mode shapes. In numerical solution methods, the symmetric and anti-symmetric boundary conditions at the mid-arc were focused rather than the boundary conditions of supported end due to the discontinuity of the mid-arc. For the first time, the free vibration problem for discontinuous arches, such as ogival arches, is solved in this study. Calculation results of this study for natural frequencies are compared well with those of the finite element method. The effects of various arch parameters on natural frequencies were highlighted and discussed in detail.

In this paper, the free vibration of a sandwich plate with an anisogrid core and two face sheets reinforced with graphene platelets (GPLs) is investigated. A continuous approach is considered for the lattice core and the equivalent properties are calculated. Adopting the Halpin–Tsai micromechanics, the effective Young’s modulus of the nanocomposites/graphene platelets is extracted. Also, mass density and Poisson’s ratio are earned with the simple rule of mixtures. A quasi-3D theory is applied to model the kinematics of the sandwich plate with simply supported boundary conditions. Hamilton’s principle is implemented to obtain the equations of motion that are solved based on the Navier solution. The validity of the results of this study is confirmed by comparing the analytical results with those presented in other researches and also a finite element model. The effect of the parameters of the lattice core such as the width of ribs, the number of helical ribs in one direction, and the ratio of thickness of face sheets to core on the natural frequencies of the sandwich plate was investigated. Additionally, the impact of the pattern of graphene platelets and their weight fraction on the natural frequencies were investigated. The results show that by decreasing the ratio of the thickness of face sheets to the thickness of core and increasing the number of ribs and their width, the natural frequencies will decrease. Moreover, the patterns FG-V and FG-A have the highest and the lowest natural frequencies, respectively, among the other distribution of graphene platelets.

The free vibration analysis of joined conical–spherical panels and shells of revolution is the primary aim of this investigation. The equations of motion are derived by considering Donnell’s kinematic assumptions and the first-order shear deformation theory (FSDT), which considers the effects of shear deformation as well as rotational inertia. The equations of motion are discretized by using the two-dimensional generalized differential quadratic (2D-GDQ) method in the $(x,𝜃)$ and $(\phi ,𝜃)$ directions for conical and spherical panels and shells, respectively. Natural frequencies of joined conical–spherical shells and panels are determined for different boundary conditions. The accuracy of results has been compared with the results of the other researchers and those obtained using Abaqus and Ansys, which showed good agreement. Moreover, the effects of length, thickness, and the angle of the semi-vertex of the conical shell on the dynamic behavior of joined spherical–conical panels/shells are investigated.

The present research applies a 2D refined plate theory and isogeometric analysis (IGA) for free vibration analysis of functionally graded (FG) sandwich plates, whose governing equations are treated based on a unified formulation (UF), and nonuniform rational Lagrange (NURL)-based IGA technique. The constitutive model of FG materials is approximated via a Voigt’s rule of mixture based on an equivalent single-layer (ESL) theory. The present framework offers several advantages, including high precision of vibration response by employing higher-order plate theory and the capability of NURL basis functions to capture the exact form of plate geometries. Moreover, higher-order theories postulated by the UF are exempt from the Poisson locking phenomenon and do not require a shear correction factor. Additionally, by employing UF, the effect of thickness stretching on vibration response is considered. Furthermore, higher-order NURL basis functions effectively mitigate shear locking. A large numerical investigation shows the accuracy of results and investigates the effects of several key parameters, such as gradient index, thickness-to-length ratios, layer-to-thickness ratios, and boundary conditions, on the vibration response of FG sandwich plates.

Due to the challenges in mathematical solving, accurate analytical solutions for rectangular thin plates are mainly limited to these plates subject to simple Lévy-type boundary conditions (BCs). Conversely, plates with more complicated BCs are more frequently encountered in practical engineering applications. This study employs the finite integral transform method to analyze the free vibration behavior of isotropic rectangular thin plates with three edges rotationally-restrained and one edge free, which is difficult for other methods. Primarily, the method adopted eliminates the necessity for predetermining trial function for plate deflection, offering simplicity and generality for plate mechanical problems. Furthermore, the approach establishes one way to allow one solving linear algebraic equations instead of handling complex boundary value problems of higher-order vibration partial differential equation (PDE), resulting precise and rapidly converging analytical solutions. Finally, new accurate free vibration studies for plates with four non-Lévy-type BCs and three non-classical BCs are presented via altering the introduced rotating fixed coefficients. The results obtained are contrasted with the numerical simulations from ABAQUS software and existing literature works, confirming the accuracy of this method and providing benchmark results for further exploration of free vibration problems of rectangular plates.

This study aims to determine the natural frequencies of axially functionally graded porous material (FGPM) beams with non-uniform cross-sections under a variety of boundary conditions. Two types of pore distribution were used: even and uneven. Initially, an analytical method was used to determine the natural frequencies of FGPM beams with different cross-sections. The Fredholm integral approach was used to obtain the characteristic equations. Furthermore, the collected results are confirmed and compared to the existing literature. The artificial neural network (ANN) technique is then used to predict the fluctuations in the natural frequency of a clamped–simply supported axially FGPM beam with a non-uniform cross-section. The prediction of natural frequencies by ANN is based on a large dataset of over 1100 data points acquired from the analytical solution obtained in this study and data available in the literature. This research conducts a parametric analysis to evaluate the influence of numerous aspects, including as beam characteristics, material properties, geometric details, gradient parameters, and porosity distribution, on functionally graded (FG) beam vibration behavior. Finally, both ANN and computational analysis demonstrate that porosity distributions and non-uniform cross-sectional area have a substantial effect on the natural frequencies of axially FG beams.

In this paper, a comprehensive dynamical analysis of porous-reinforced carbon nanotube (CNT) nanocomposite beams on viscoelastic foundations is presented. Using a three-unknown shear beam theory, the two-nodded finite element model was formulated and adapted for both $C^1$ and $C^0$ continuities for displacement variables. This model captured the distinct characteristics and interactions of the porous medium, CNT reinforcements, and the intrinsic damping introduced by the viscoelastic foundation. The dynamical investigation incorporates both free and forced vibration analysis, and the Newmark method is employed for time-dependent analysis, ensuring accurate temporal behavior prediction. The study also emphasizes the impact of different CNT distribution patterns and porosity variations on the mechanical properties of the beams, particularly in terms of stiffness and damping characteristics. Further depth is added by analyzing the interactions between beams and both Winkler and Pasternak foundation models, with findings that highlight the importance of foundation parameters in the overall dynamic response. The results shed light on the critical role each component plays in the composite beam’s dynamic response, offering insights for potential innovative designs and applications in both advanced structural engineering and nanotechnology domains.

The weak form quadrature element method is applied to free vibration analysis of Mindlin sectorial plates. Sharp corner functions are introduced in the vicinity of the corner to eliminate the influence of stress singularity. The outer annular subdomain is analyzed according to weak form quadrature procedures. The continuity of displacement and rotation angles on the interface is enforced afterwards. For various combinations of boundary conditions and vertex angles, the first six nondimensional frequencies are computed by the present formulation. Comparative analysis against existing analytical and numerical results confirms the validity and efficiency of the proposed formulation for analyzing the free vibration behavior of Mindlin sectorial plates.

In this study, the effects of cracks and geometric imperfections on the free vibration characteristics of L-shaped functionally graded beams are comprehensively investigated. The governing equations are established based on the Timoshenko beam theory and Hamilton’s principle. The cracks are modeled using massless rotating springs, and the material properties vary along the beam’s thickness direction. The free vibration frequencies of the system are obtained by using the differential quadrature method. The results are verified by comparing them with the findings in the existing literature. Additionally, the influence of deformation type, geometric imperfection mode, crack location, and crack depth on the frequency of L-shaped functionally graded beams are analyzed.

A more accurate kinematic model is extended in this work for vibrational analysis of a nanocomposite plate reinforced with graphene origami subjected to mechanical and thermal loading. The stretching part of transverse deflection is included in the kinematic relations in addition to the shear and bending components to arrive at a general formulation. The Hamiltonian procedure is extended to derive the governing equations of motion. The parametric results are obtained using the analytical method for investigating the impact of graphene origami reinforcement such as foldability and content as well as geometric parameters on the natural frequency responses. The results of this work can be used for the design of optimized structures and systems with high stiffness and low density applicable in multi-field loading conditions. As a novel conclusion, one can conclude that accounting thickness stretching leads to a decrease in natural frequencies because of more deformability and consequently a decrease in structural stiffness of stretchable plate.

The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibration and the upper bound solution of the forced vibration of solid structures, by making use of the important softening effects of the node-based smoothed finite element method (NS-FEM). Through the gradient smoothing technique, the strain-displacement matrix is obtained in the smoothing domain based on the element mesh nodes. Subsequently, the stiffness matrix is computed in a manner consistent with the standard finite element method (FEM). Here, the practical Lanczos algorithm and the modal superposition technique are employed to obtain the frequencies, modes, and transient responses of a given homogeneous structure. For three-dimensional (3D) solid structures, the automatically generated four-node tetrahedron (T4) element meshes are utilized. The results obtained from the NS-FEM are compared with the standard FEM in terms of accuracy, convergence and computational efficiency.

This paper investigates the free vibration of multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on the nonlocal elasticity theory which allows the effects of small length scale and the more refined van der Waals (vdW) interaction formulas, the equation of motion is first derived and then solved analytically. The results reveal that the effects of the small length scale are significant for small aspect ratios and high radial vibration modes, and are instead insensitive to the number of layers of MWCNTs and weakly dependent on the wall thickness of MWNTs. This finding means that the effects of small length scale on complicated MWCNTs may be simplified to double-walled carbon nanotubes (DWCNTs) or even single-walled carbon nanotubes (SWCNTs).

The aim of this paper is to investigate the free and forced axial vibrations under the two various linear and harmonic axial concentrated forces in zigzag single-walled carbon nanotube (SWCNT). Two different boundary conditions, namely clamped–clamped and clamped-free, are established. Eringen’s nonlocal elasticity is employed to justify the nonlocal behavior of constitutive relations. The governing equation and the associated boundary condition are derived based on Hamilton’s principle. In order to solve the derived equation numerically, the assumed modes method is utilized. In the free axial vibration section, the first three natural frequencies are obtained for the various values of the nonlocal parameter. The results are in good agreement in comparison with another study. The fundamental natural frequencies with respect to the nonlocal parameter of the case study as a semiconducting nanotube with boron nitride nanotube (BNNT) as a semiconducting nanotube and SWCNT (5,5) as a metallic nanotube are compared. The effects of the nonlocal parameter, thickness and ratio of the excitation-to-natural frequencies overtime on dimensional and nondimensional axial displacements are studied.

Background: This paper is concerned with a nonlinear semi-continuum model for an ultrathin structure. The basic equations of the theoretical model for silicon micro/nanosheets are derived, and the geometric nonlinearity is introduced in the model. Methods: From two different approaches including the new strain energy and the new external potential energy, we establish the nonlinear semi-continuum theoretical model of silicon micro/nanosheets, respectively. A new dimensionless nonlinear semi-continuum parameter is defined. Based on the theoretical model, the characteristics of bending deformation and free vibration are revealed. Results: The relationships between bending deflection and atomic layers in thickness direction as well as the relaxation coefficient between atomic layers are analyzed. The resonance frequencies of free vibration and their relationship with atomic layers are calculated. By introducing the specific property parameters of silicon micro/nanomaterials, several numerical calculations have been carried out. Conclusion: The theoretical results are compared with other studies in the literature, such as nonlinear finite element method (FEM), experimental and classical results, to validate the semi-continuum model established in the present research. This work can provide new ideas for the mechanical analyses of micro/nanomaterials and structures, and the results could be foundations for the design and application of silicon micro/nanosheets.

This study aims at investigation of the resonance frequencies of carbon nanopeapods constructed by a single wall carbon nanotube and encapsulated buckyball molecules (C₆₀). A nanopeapod can be used as a nanoscale variable frequency beam resonator according to the number and positions of the encapsulated fullerenes. Using the molecular structural mechanics method the covalence bonds are simulated by equivalent beam elements and the van der Waal interactions between the buckyballs and nanotube are modeled as linear springs. Also, an equivalent beam model is proposed for the nanopeapod with sectional properties which are obtained by the molecular structural mechanics model. The beam-like modes of free vibrations are obtained using both models and the effect of position and number of buckyballs on the resonance frequencies are investigated.

In this paper, we study the vibration of curved nano-sandwich (CNS) with considering the influence of core shear based on the Eringen nonlocal theory. The equation of motion is derived and exact solution for the natural frequencies of CNS is presented. The proposed nonlocal model includes a material length scale parameter that can capture the size effect in CNS beam. The effects of important parameters, such as the thickness to length ratio, nonlocal parameter and mode number on the frequencies of CNS are investigated. The result of our research shows that as the opening angle increases, the amount of natural frequencies decrease. We have additionally validate, our results against previous research works which showed good agreement.

The coupled free vibration analysis of the thin-walled laminated composite I-beams with bisymmetric and monosymmetric cross sections considering shear effects is developed. The laminated composite beam takes into account the transverse shear and the restrained warping induced shear deformation based on the first-order shear deformation beam theory. The analytical technique is used to derive the constitutive equations and the equations of motion of the beam in a systematic manner considering all deformations and their mutual couplings. The explicit expressions for displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the dynamic stiffness matrix is determined using the force–displacement relationships. In addition, for comparison, a finite beam element with two-nodes and fourteen-degrees-of-freedom is presented to solve the equations of motion. The performance of the dynamic stiffness matrix developed by study is tested through the solutions of numerical examples and the obtained results are compared with results available in literature and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The vibrational behavior and the effect of shear deformation are investigated with respect to the modulus ratios and the fiber angle change.

Stiffened skew plates find application in various engineering fields. The free vibration characteristics of such plates have been studied by various methods. An orthogonally stiffened skew plate is a skew plate with stiffeners running orthogonal to two opposite edges. To the best knowledge of the present investigators, no previous work has been done for free vibration characteristics of skew plates of such stiffening geometry. The present work studies the free vibration of such plates. The pb-2 Rayleigh–Ritz method was employed due to its accuracy and computational efficiency. The conventional finite element method was also used as a comparative check. A convergence study was first performed for various boundary conditions. Then the vibration of orthogonally stiffened skew plates with different boundary conditions was studied. Close agreement was found between these two methods. The variations of natural frequencies with different parameters, including skew angle ϕ, edge ratio b/a, and height-thickness ratio f/h, were investigated for three types of boundary conditions.