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We present a formulation for the size-affected vibration study of multi-cracked non-uniform Timoshenko beams based on the well-posed stress-driven nonlocal elastic theory with discontinuities. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. The presence of cracks divides the beam into segments connected by translational and rotational springs, and compatibility conditions are established to address the geometric discontinuities introduced by these cracks. The stress-driven constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at the two ends of the entire structure and multi-sets of constitutive continuity conditions at the junctions of the sub-structures. To solve the equations of motion, the constraint conditions and the integrals involved, we employ the differential quadrature method (DQM) alongside an interpolation quadrature formula, which allows us to efficiently compute the frequencies of the cracked beams across various boundary types. After validating our approach against results in the existing literature, we present numerical studies that examine the effects of the nonlocal parameter, the slope of the beam's thickness variation, crack location, severity, number, and the stiffness of the springs on the vibrational behavior of the beams. [ABSTRACT FROM AUTHOR]

This study presents a methodology for analyzing wave propagation in tensegrity lattices within a non-dimensional framework. Two strategies are employed to predict pass bands and bandgaps. The first examines wave dispersion through a single representative unit cell, using dispersion curves to identify bandgaps and pass bands. The second models the structure as a finite system, using modal analysis to compute the steady-state response to harmonic loads and plot the frequency response function. Two unit cells are analyzed: a two-dimensional D-bar unit and a three-dimensional tensegrity prism. The study investigates axial and in-plane bending wave propagation in a D-bar chain and axial wave propagation in a prism chain. After non-dimensionalizing the equations of motion, key parameters such as ratios of stiffness per unit length, mass per unit length, and prestress are identified. Its is shown that varying these parameters shifts the bandgap locations and widths. Prestress has minimal effect in the D-bar case, while a slight shift in the first bandgap is observed in the prism. The predictions from unit cell and finite structure analyses show good agreement. • Derived the non-dimensional EOMs of tensegrity structures. • Studied axial linear wave propagation in a 1D tensegrity D-bar/3D prism lattice. • Studied in-plane bending linear wave propagation in a 1D tensegrity D-bar lattice. • Validated unit cell results with finite structure results using the modal analysis approach. • Performed a parametric study on the identified non-dimensional parameters. [ABSTRACT FROM AUTHOR]

The mechanical behaviours of bio-inspired helicoidal symmetric laminated composite (BIHLC) beams are investigated via the Ritz method. By exploiting the variational formulation, equations of motion along with element stiffness, geometrical stiffness, and mass matrices are derived. The study conducts a thorough examination, covering bending, buckling stability, and free vibration analyses of BIHLC beams with various lamination schemes. The developed model is verified against existing literature on conventional composite laminated and BIHLC beams. The study also examines the mechanical response of BIHLCs, considering boundary conditions, lamination schemes, orthotropy ratios, and aspect ratios. Notably, deflections, critical buckling loads, and fundamental frequencies demonstrate variations dependent on the specific lamination scheme, boundary condition, and aspect ratio. Novel findings, presented for the first time, offer valuable insights for future studies in this area. [ABSTRACT FROM AUTHOR]

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young's modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed. [ABSTRACT FROM AUTHOR]

Based on the theory of peridynamic differential operator (PDDO), a novel three-dimensional (3D) peridynamic (PD) model for isotropic and orthotropic material is proposed to investigate the elastic mechanical behaviors and stress distribution of Ceramic Matrix Composites (CMC) microstructure. The non-local expressions of strain and stress tensors are derived by employing PDDO and the classical constitutive equations. The bond forces in interface region crossing two different constituent materials are obtained by converting the partial differential terms into the non-local forms. The weak form of PD equations of motion is derived by using the principle of virtual work and PDDO. The validity of this approach is verified by predicting the elastic response and stress distributions of isotropic and orthotropic materials under uniaxial tension and pure shear deformation. Finally, simulations are conducted on a CMC microstructure with fiber and matrix under periodic boundary conditions. It is demonstrated that the current approach can effectively build 3D CMC microstructure with uniform or non-uniform discretization, and accurately predict the stress fields and effective elastic properties. [ABSTRACT FROM AUTHOR]

Vibration control plays a vital role in engineering structures, especially for low-frequency range, because designing structures with small dimensions makes it difficult to control large-wavelength vibrations. The present study seeks another road to realize low-frequency vibration control from the view of the real vibration responses. Similar to mode shapes, deflection modes are the main distributions of vibration amplitudes on structures in low frequency. Therefore, they are also essential vibration characteristics of structures. This paper proposes an effective passive vibration control method under both non-resonant and resonant excitations based on deflection mode theory and optimal algorithm. In fact, it is just like an axially functionally graded design for structures. The equations of motion for non-uniform beam structures are formulated by Hamilton's principle. Firstly, a desired deflection mode is artificially designed, and the beam structure is discretely divided into subunits. Then, by adjusting the thickness or elastic modulus of each subunit to make the deflection mode of the beam coincide with that of the desired one. This process is completed by the genetic algorithm (GA). After that, by experimental and simulation analyses, the deflection mode of the optimally designed beam structures coincides with that of the desired one. Therefore, the present vibration control method is verified to be correct and effective. [ABSTRACT FROM AUTHOR]

This work presents geometrically nonlinear transient analyses of various rotating blades. The structures are discretized through refined beams or multi-dimensional finite element models, which are generated using the Carrera Unified Formulation (CUF). The CUF offers a procedure to develop low- and high-fidelity one-dimensional, two-dimensional, and three-dimensional finite element models hierarchically and automatically. Various beam models were developed using different kinematics models based on Taylor or Lagrange expansion functions. Multi-dimensional models were obtained by merging beam and solid elements, exploiting the unique feature of Lagrange polynomials to have only pure displacements as unknowns. This property allows beam and solid elements to be coupled at the node level without requiring complicated mathematical formulations. By utilizing the Finite Element Method in conjunction with the CUF, the governing equations are written by including all rotation effects, namely the Coriolis term, spin-softening, and geometrical stiffening. In a total Lagrangian scenario, the Hilbert-Hughes-Taylor-α method and the iterative Newton-Raphson scheme are employed to solve the equations of motion. The proposed methodology has been applied to evaluate different blade configurations, comparing the solution obtained using linear, linearized, and nonlinear approaches. The results have been verified and validated by comparing them with existing solutions present in the literature. [ABSTRACT FROM AUTHOR]

The study focuses on the free vibration analysis of beams composed of functionally graded porous materials and characterized by a variable cross-section along their length. A broad range of beams is examined encompassing various tapered configurations, porosity profiles, and porosity content. The equations of motion are derived using Hamilton's principle within the framework of Timoshenko beam theory. These equations are solved semi-analytically using the differential transform method, which has been adapted to incorporate various boundary conditions such as clamped–clamped, clamped–free, clamped–pinned, and pinned–pinned constraints within a general formulation of the beam eigenvalue problem. To validate the proposed solution technique, computed natural frequencies are compared with existing literature results for tapered inhomogeneous beams and uniform porous beams. Notably, new results are obtained for tapered porous beams. In this regard, a comprehensive parametric study explores the influence of various factors on the natural frequencies and mode shapes of functionally graded porous beams with variable cross-sections. These factors include the type of porosity profiles, a range of porosity parameters, cross-section taper ratios, and specific boundary conditions. The findings deepen our understanding of the modal characteristics of functionally graded porous beams, providing valuable guidance for engineering design and structural optimization in relevant applications. Additionally, they may serve as benchmarks for other researchers. [ABSTRACT FROM AUTHOR]

• Nonlinear (NL) vibration of shear-deformable multilayer nonhomogeneous plates is studied. • NL basic equations of nonhomogenous multilayer plates within SDT are established. • NL-PDEs are reduced to NL-ODE with second and third order nonlinearity. • Solution of NL-ODE is performed by the modified Poincaré −Lindstedt method. • New amplitude dependent expressions for NL frequency of multilayer plates is obtained. In this study, one of the first attempts is made to solve the nonlinear (NL) vibration problem of shear-deformable multilayer plates consisting of nonhomogeneous orthotropic layers (NHOLs) using the Poincaré-Lindstedt method. First, the shear deformation theory (SDT) for homogeneous plates is extended to multilayer plates composed of NHOLs. In the framework of von-Karman type nonlinear theory, the basic relations of the plates in question are established and then NL equations of motion based on four functions such as rotation angles, Airy stress and deflection functions are derived. Then, NL-partial differential equations (NL-PDEs) are reduced to NL-ordinary differential equations (NL-ODE) and the solution of NL-ODE is performed for the first time by the modified Poincaré −Lindstedt method, yielding new amplitude dependent expressions for NL frequency, and for the ratio of NL frequency to linear (NL/L) frequency for multilayer plates consisting of NHOLs. Finally, detailed parametric studies are carried out to gain insight into the effects of various factors such as shear strains, non-homogeneity, number and array of layers on the NL frequencies under different rectangular plate characteristics. [ABSTRACT FROM AUTHOR]

• New sensitivity-improving approach for mode-localized MEMS mass sensors is presented. • This improvement is based on incorporating piezoelectric thin films pattern. • Nonlinear Ritz-based ROM of the design is obtained through the Hamilton principle. • System's nonlinear dynamics is investigated employing the harmonic balance method. • This approach improves the mode-localized sensors sensitivity more than two times. This investigation focuses on developing a new sensitivity-improving approach for high-order mode-localized mass micro-sensors by utilizing the capabilities of piezoelectric materials. To this end, an electrostatically coupled micro-beam as the building block of MEMS mass sensors is considered. The present design includes the incorporation of a patterned arrangement of piezoelectric thin films placed on the lower electrode of the system. The nonlinear reduced equations of motion for the introduced tunable system are derived by employing the Hamilton principle in conjunction with the Euler-Bernoulli beam theory and the Ritz discretization procedure. These equations are subsequently solved using the harmonic balance method. The present findings are validated by those available in the literature for the case of static excitation. In addition, the eigenvalue loci of the proposed system have been compared and verified by those obtained through three-dimensional finite element simulations carried out in COMSOL Multiphysics commercial software. Taking the shift of the amplitude ratio as the measure demonstrating the sensitivity of the proposed design, it is observed that incorporating piezoelectric excitation can significantly enhance the efficiency of these systems more than two times in comparison to conventional mode-localized mass micro-sensors without piezoelectric layers. [ABSTRACT FROM AUTHOR]

• Predicting dynamic responses of MEE composite plates resting on visco-Pasternak foundation in hygro-thermal environment subjected to blast load is developed. • The variation of electric and magnetic potentials is determined according to electromagnetic boundary conditions and the Maxwell equation. • Effects of electric and magnetic loads, geometrical and foundation parameters on dynamic responses of MEE composite plates are investigated as new numerical results. • Elastic foundation, viscoelasticity, temperature, and moisture have been demonstrated to contribute considerably to the reduction of structural vibration. This is the first endeavor to present a finite element approach to predict free oscillation and transient characteristics of magneto-electro-elastic (MEE) composite rectangular and elliptical plates resting on the visco-Pasternak medium in a hygro-thermal environment subjected to blast load. The variation of electric and magnetic potentials along the thickness direction of the MEE plate is determined via the Maxwell equation and magneto-electric boundary condition. The governing equations of motion for the MEE plate are obtained using refined higher-order shear plate theory and Hamilton's principle. Stiffness matrices, mass matrices, damping matrices, and force vectors of the plate are derived using a four-node quadrilateral element with eight degrees of freedom per node approximated using C1-order non-conforming Hermite and Lagrange functions. In addition, the Navier close-form solutions for rectangular plates with simply supported boundary conditions are used as a useful comparison tool for the numerical solutions. Dynamic response results of plates were obtained using Newmark's direct integration written by Matlab's programming. The accuracy of the method is verified through numerical comparison with confidence statements. The results show that the applied magnetic and electric potential highly inspires the transient responses. These results can be used in vibration control studies of structures and structures subjected to explosive or low-velocity impact loads. [ABSTRACT FROM AUTHOR]

Viscoelastic constrained layer damping is a simple and efficient method for reducing noise, vibration, and fatigue in metal structures. In this work, a spectral collocation method utilizing a layer-wise plate theory was established to inspect the vibration characteristics of a sandwich plate with a viscoelastic core. The displacements of each layer satisfied the Mindlin plate theory. The core's transverse shear stress remained constant. The three layers' transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces' continuity of the displacements. The equations of motion were obtained by utilizing Hamilton's principle. The viscoelastic core's material properties were considered as frequency-dependent. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the proposed method. The modified Oberst beam method was applied by fitting the impulse response function to obtain the adhesive's frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the present theoretical method agreed well with the measured results. [ABSTRACT FROM AUTHOR]

In the current investigation, the small and large amplitude free vibration characteristics of long cylindrical panels made of FGMs is investigated using the higher order shear deformation shell theory and Donnell kinematic assumptions. The interaction of the panel with a nonlinear hardening/softening foundation in also included into the formulation. The properties of the panel are assumed to be graded across the thickness of the panel and assumed to be temperature dependent. Using the neutral surface concept, the three coupled motion equations of the shell are established and solved for the case of long panels which are immovable simply supported on straight edges. The solution method is based on the two step perturbation technique which results in a closed form expression for the frequencies of the panel as a function of the mid-point deflection. Numerical studies are first validated for the case of long FGM plates. Afterwards novel numerical results are given to show the effect of geometrical parameters, power law index, foundation stiffnesses and temperature elevation. As shown, temperature elevation results in the reduction in natural frequencies and enhancement of the frequency ratio of the panel. Also temperature dependency reduces the natural frequencies and frequency ratios of the panel. [ABSTRACT FROM AUTHOR]

The free vibration of different kinds of stepped coupled doubly-curved shell structures with elastically constrained edges are investigated by adopting the Jacobi-Ritz method for the first time. Coupled structures are comprised of substructures, where paraboloidal, hyperbolical, elliptical, and cylindrical stepped shells are typical ones. The Flügge's thin shell theory is utilized to construct the analytical model, together with the multi-segment partitioning strategy. For each shell segment, despite of various boundary conditions, the displacement components along the meridional directions are expressed by Jacobi polynomials and those along the circumferential directions are represented by Fourier series. Then the unknown coefficients of the displacements are obtained by introducing the Rayleigh-Ritz method. The solutions proposed here for coupled structures have two main advantages: first, there is no need to vary the displacement or the motion equations; and secondly, the efficiency of modeling can be notably enhanced. By comparison with (Finite Element method) FEM and others' results, the reliability of current method can be validated. At last, the free vibrations of different kinds of coupled structures containing stepped shell are analyzed by presenting several numerical examples, the results of which may be served as reference data. [ABSTRACT FROM AUTHOR]

In this paper, a general approach is provided for the free vibration analysis of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells with arbitrary boundary conditions. General formulations are derived based on the first-order shear deformation theory, the Donnell-type kinematic assumptions, and the artificial spring technique. Coriolis and centrifugal effects due to rotation are taken into account in the shell model. By employing Chebyshev polynomials as admissible functions, the Rayleigh-Ritz method is employed to derive the equations of motion for rotating FG-CNTRC cylindrical shells. The approach proposed is validated by comparing the present results with those reported in literature. The traveling wave motions of rotating FG-CNTRC shells are investigated. The effects of geometric parameters, volume fraction of carbon nanotubes, and boundary conditions on shell vibrations are also evaluated. [ABSTRACT FROM AUTHOR]

This study is devoted to illustrate the mechanical buckling and free vibration analyses of double-porous functionally graded (FG) nanoplates embedded in an elastic foundation. A new quasi-3D refined plate theory is presented to model the displacement field. This theory contains only five unknown functions and considers the shear strain as well as thickness stretching. Based on the modified Mooney-type exponential relation, a new exponential law is presented to govern the materials variation and porosities distribution through the thickness of the nanoplates. The two porous nanoplates are bonded together by a set of parallel elastic springs and surrounded by Pasternak medium. The nonlocal strain gradient theory containing the nonlocal parameter and gradient coefficient is utilized to study the size-dependent effects. Based on Hamilton's principle, the equations of motion are drawn including the material parameters, elastic foundation reaction and biaxial compressive forces. An analytical approach for simply-supported and clamped bilayer porous FG nanoplates is implemented. The obtained results are compared with those available in the literature. Additional numerical calculations are introduced to show the influences of the material length scale parameters, inhomogeneity parameter, porosity factor and other parameters on the critical buckling and frequencies of the double-porous FG nanoplates. [ABSTRACT FROM AUTHOR]

The subject of the paper is a rectangular plate with symmetrically varying mechanical properties in the thickness direction. The nonlinear hypothesis of deformation of the straight line normal to the plate neutral surface is assumed. The field of displacements of the plate is formulated assuming this hypothesis. Based on the Hamilton's principle three differential equations of motion are obtained. The system of equations is analytically solved. The critical loads and fundamental natural frequencies for exemplary plates are derived. Moreover, the FEM model of the plate in the ABAQUS system is developed, and analogical calculations for exemplary plates are carried out. The calculation results of these two methods are compared. [ABSTRACT FROM AUTHOR]

In this study, nonlocal strain gradient theory (NSGT) is applied to examine the dynamic instability of embedded viscoelastic graphene sheet under periodic axial load including thermal effects. The foundation is simulated by visco-Pasternak model containing springs, dampers and a shear layer. The motion equations are derived according to the four-variable refined shear deformation plate theory and via Hamilton's principle. The equations are converted into a linear system of Mathieu-Hill equations by means of Navier's method. Afterwards, Bolotin's approach is utilized to determine the principle unstable region of graphene sheet. The influences of nonlocal parameter, structural damping coefficient, length scale parameter, static load factor, temperature variation, foundation type as well as aspect ratio on the dynamic stability of graphene sheet are investigated. Based on the numerical results, it is indicated that with enlarging the nonlocal parameter, static load factor and temperature change, the excitation frequency decreases and so, instability region shifts to left side while the effect of length scale parameter is on the contrary. Additionally, it is indicated that when the length scale parameter enhances, the effects of temperature and foundation on the instability region of graphene sheet reduce. [ABSTRACT FROM AUTHOR]

Abstract In this paper, some methods to determine the absolute frequencies of traveling waves in a rotating cross-ply laminated cylindrical shell with elastic supports are investigated. Based on the Sanders' shell theory and by taking into account the Hamilton principle, the governing equations of motion are derived in the rotating coordinate system, which considers the effects of initial hoop tension, the centrifugal and the Coriolis forces due to the rotation as well. The constraint equations of elastic supports are modelled by using artificial distributed elastic springs in the possible directions. By substitution of mode shape profile functions into equations and using the differential quadrature method, the eigenvalue equations of the rotaing shell are derived in both rotating and fixed systems. To make more comparison, the eigenvalue equation of synchronous critical speeds is also derived. Convergence and comparison of the proposed method is investigated through comparing its results with available literature. The comparison results shows that the direction of the corresponding traveling waves and the graphical determination of critical speeds are determined by a more convenient criteria with easier physical interpretation in the fixed system, specially by using the direct method which is more efficient in computation than the converting method. [ABSTRACT FROM AUTHOR]

In this study, a comprehensive study on the vibration behaviour of functionally graded thick microplates with material imperfections is presented for free and forced vibrations within a quasi-3D model and the modified couple stress (MCS) theory. Axial, transverse, rotation, and stretching motions are considered for modelling the thick microplate in the framework of the modified power-law scheme and quasi-3D and MCS theories. The microplate deformation is assumed to be infinitesimal and is modelled using the linear strain–displacement relationships. Using a virtual work method, the governing motion equations are derived. For fourfold coupled (axial-transverse-rotation-stretching) characteristics, the partial differential equations components are discretised via trigonometric expressions and the corresponding natural frequencies and time histories (time-dependent deflections) are determined numerically. The methodology and model are initially validated through a comparative analysis between the natural frequencies of the macrolevel simplified structure and those obtained using finite element software. Additionally, the simulation is compared, for validation purposes, with other simplified versions from the literature. Once model's validity is confirmed, an investigation into the quasi-3D free and forced vibrations of the structure is conducted for the new model. The results demonstrate the considerable effect of thickness stretching and material imperfection on the vibration characteristics of the poroelastic thick microplates. [ABSTRACT FROM AUTHOR]

In this paper, a locally resonant sandwich metamaterial beam with inertial amplification mechanisms is presented for flexural vibration attenuation. This sandwich metamaterial is proposed as a lightweight structure with a low-frequency stopband. An analytical model was developed to derive the equations of motion for the resonator core and skin plate using Hamilton's principle. Analytical solutions for the stopband boundary frequencies were derived from the equations of motion. The analytical solutions demonstrated that inertial amplification resonators can exhibit lower resonance frequencies than linear spring-mass resonators with the same weight and spring constant. It was also shown that the proposed sandwich beam exhibited a lower frequency stopband in exchange for the stopband width. Finite element method analyses and experiments were performed to investigate the frequency response of the sandwich beam. The flexural vibration of the sandwich beam was significantly attenuated within a certain frequency range. The frequency range with low transmissibility agreed well with the stopband predicted by the analytical solutions. [ABSTRACT FROM AUTHOR]

This paper investigates the free vibration analysis of a microplate, interacting with a stationary fluid. For the fluid part, the potential flow assumptions are considered, and it is supposed to be incompressible, irrotational and inviscid. Different positions for the fluid relative to the structure are studied: a structure in contact with a fluid with a free surface, a submerged microplate and a floating structure. For the structure, the mechanical properties of the microplate such as density, Young modulus and shear modulus are assumed to be a function of structure thickness. Besides, the modified strain gradient theory on the basis of higher-order plate theory is adopted to capture the size effects of the microplate. Using Bernoulli's relation and fluid velocity potential function, the fluid pressure affecting the microplate is calculated, and then Hamilton's principle is employed to derive coupled fluid–solid equations of motion. After validation of the presented formulations, a comprehensive parametric study is conducted to exhibit the response of the system under variations of some parameters such as structure mechanical properties, fluid parameters and size-effect parameters. • Studied the dynamic response of a microplate under the effects of fluid medium. • Studied the influences of small-scale parameters using modified strain gradient theory. • Made comparisons between modified strain gradient theory, modified couple stress theory with classical approach. • Considered the impacts of transverse shear stresses and rotary inertias using a general formulation, which can reduce to various shear deformable plate theories. • Modelled different positions for the fluid relative to the structure. [ABSTRACT FROM AUTHOR]

Abstract A numerical and experimental investigation of the natural frequencies and frequency response of a cantilevered three-layer sandwich beam carrying a tip-mass is presented. The tip-mass is such that its center of mass is offset from the point of attachment. The higher-order sandwich panel theory (HSAPT) and the Hamilton's principle are employed in deriving the system governing equations of motion and boundary conditions, which are solved using the generalized differential quadrature (GDQ) method. The obtained results are compared with those obtained from the use of commercial finite element software ANSYS, and they are both experimentally validated. The results are generally in good agreement. Parametric studies are conducted to examine the influence of some design variables on system vibrational behavior. [ABSTRACT FROM AUTHOR]

Abstract The present paper is concerned with active vibration control of functionally graded piezoelectric material (FGPM) plate using the piezoelectric material component as actuator. Using the classical laminated plate theory, the equation of motion for the FGPM plate is deduced based on Hamilton's principle and Rayleigh-Ritz method. A velocity feedback control method is used to obtain an effective active damping in the vibration control. The influences of distribution type, volume fraction index and total volume fraction of piezoelectric material on the vibration control of FGPM plate are investigated. The calculation results show that the vibration control result is strongly affected by the distribution of piezoelectric materials in FGPM plate. It is also found that the total volume fraction, especially the volume fraction index play an important role in the vibration control. Furthermore, under the non-uniform electric field the effect of external voltage position on active vibration control is studied. The results show that one can obtain an excellent control effect by optimizing the structure of FGPM plate and the position of external control voltage. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new nonlinear model of nanobeams made of bi-directional (2D) functionally graded material (FGM) is presented. The material properties are assumed to obey an exponential gradation along both the thickness and length directions. The equation of motion is derived based on Euler-Bernoulli beam theory, von Kármán geometric nonlinearity and non-local elasticity theory. The nonlinear bending, buckling and vibration of the nanobeams with size effect is investigated. Results show that the 2D FGM introduces an additional stiffness term in governing equation, so it cannot be solved by classical analytical methods. Therefore, the differential quadrature method (DQM) is used to solve the nonlinear problem. The size-dependent nonlinear critical load and frequencies are calculated. The effect of different material parameters, boundary conditions and size scale are discussed in details. [ABSTRACT FROM AUTHOR]

The vibrations of axially functionally graded (AFG) beams is examined incorporating shear deformation and an imperfection. A coupled nonlinear model is developed in the Hamiltonian framework for an AFG imperfect beam. Different sources of displacement nonlinearities are involved in this problem, namely, the geometrical nonlinearities due to the large deformations, and the nonlinearities due to the presence of a geometrical imperfection along the length of the beam. Expressions for the virtual work of external excitation and damping as well as formulas for the motion and elastic energies of the system are inserted into Hamilton’s energy/work principle. The equations are truncated employing Galerkin’s technique while incorporating symmetric as well as asymmetric modes. It is shown that the nonuniform material properties along with the tapered geometry of the beam as well as the presence of geometric imperfections results in the contribution of both asymmetric and symmetric vibration modes. Hence, a well-optimised scheme on the basis of a continuation method is developed to solve the high DOF nonlinear problem. Numerical simulations are performed to investigate the influences of the gradient index, the imperfection and the taper values. [ABSTRACT FROM AUTHOR]

This paper proposes novel Ritz functions for the size-dependent analysis of micro laminated composite beams with arbitrary lay-ups. Displacement field is based on a higher-order deformation beam theory and size effect is captured by the modified couple stress theory. Lagrange’s equations are used to obtain the governing equations of motion. The present beam model, which can recover the classical one by neglecting the material length scale parameter, is used to predict the size-dependent responses of micro composite beams. The results indicate that the present study is efficient for bending, vibration and buckling problems of micro composite beams. Some new results are given to serve as benchmarks for future studies. [ABSTRACT FROM AUTHOR]

This paper presents the free vibration behavior of two directional functionally graded beams subjected to various sets of boundary conditions which are simply supported (SS), clamped-simply supported (CS), clamped-clamped (CC) and clamped-free (CF) by employing a third order shear deformation theory. The material properties of the beam vary exponentially in both directions. In order to investigate the free vibration response, the equations of motion are derived by means of Lagrange equations. The axial, transverse deflections and rotation of the cross sections are expressed in polynomial forms including auxiliary functions which are used to satisfy the boundary conditions. The verification and convergence studies are performed by using computed results from a previous study which is based on the Timoshenko beam formulation. The results for extensive studies are provided to understand the influences of the different gradient indexes, various aspect ratios and boundary conditions on the free vibration responses of the two directional functionally graded beams. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamic properties of three-layered composite sandwich open circular cylindrical shells (CSOCCS). First, the equations of motion that govern the free vibrations of CSOCCS are derived by applying Hamilton’s principle based on the first-order shear deformation shell theory. Owing to considering the effect of rotary inertias and shear deformation, thin-to-moderately thick shells can be analyzed. Next, these equations are solved by means of the closed-form Navier method. The calculated results are compared with the findings of previous studies and those obtained by the finite element method, and a good agreement is observed. The variation of modal loss factor and frequency with system parameters is evaluated and presented graphically. It is the first time to study the dynamic properties of composite open circular cylindrical shells with constrained viscoelastic core. [ABSTRACT FROM AUTHOR]

The priority of this paper is to explore the computational characteristics of Harmonic Differential Quadrature (HDQ) method for free flexural vibration analysis of functionally graded piezoelectric material (FGPM) beam. The modified Timoshenko beam theory is used in this study where electric potential is assumed to have sinusoidal variation across the depth. The equations of motion and boundary conditions are derived by employing Hamilton‘s principle. The material properties are assumed to have a power law or sigmoid law variation across the depth. The available equations of motion are then solved using the Harmonic Differential Quadrature (HDQ) method to obtain the natural frequencies of the FGPM beam. The efficacy of the present method is validated by comparing the results with the previous published work. It is observed that the results obtained by HDQ method are at the same level of accuracy than those of previous analysis using GDQ method. [ABSTRACT FROM AUTHOR]

The full dynamic response of FG-CNT reinforced composite plates with arbitrary geometry subjected to impact loading is considered in the present paper. The CNT reinforcement distribution is considered either uniform or functionally graded along the plate thickness and the equivalent mechanical properties of reinforced composite plates are estimated according to the extended rule of mixtures. The derived governing equations are based on high-order shear deformation theory, using Hamilton’s principle. An integration scheme appropriate for calculating double integral with variable limits is developed based on the Simpson’s rule to evaluate the components of Hamilton’s equation. A two-dimensional Ritz formulation appropriate for general boundary conditions is incorporated in the nonlinear Hertzian contact law to establish the equations of motion. A well-known fourth-order Runge-Kutta method is employed to solve the resulting equations in time domain. The validation of the proposed model is accomplished by comparing its results and those published in the literature and good agreement is achieved. A comprehensive sensitivity analysis is conducted to study the effect of various involved parameters such as CNT volume fraction and its distribution profile along the thickness, boundary conditions, temperature rising and in-plane loading on impact characteristics of both circular and triangular plates. [ABSTRACT FROM AUTHOR]

In this paper, a simplified way of deriving the equation of motion is presented for the general laminated composite plates under time-dependent compression. The considered plates with simply supported boundary conditions are subjected to the in-plane harmonic loads. By employing a novel semi-inverse method (SIM) based on least square (LS) fitting, the equation of motion is derived from the static equilibrium path. Dynamic responses of glass fiber reinforced polymer (GFRP) laminated plates with different layer arrangements are analyzed by such equations and the dynamic buckling loads are obtained based on Budiansky–Hutchinson and Volmir criteria. Static equilibrium paths are obtained based on commercial finite element method (FEM), i.e., ANSYS software. Again, the results of SIM are compared with those of FEM in ANSYS software and the verifications of dynamic responses are proved considering full types of material couplings. The presented results show that the proposed simplified way of the derivation of the equation of motion allows for a very low time consumption analysis. [ABSTRACT FROM AUTHOR]

An accurate, efficient and reliable four-noded quadrilateral plate element is developed for the free vibration analysis of laminated composite plates. The element formulation is based on a refined third-order shear deformation plate theory (TSDT) and the quasi-conforming element technique. The equations of motion and the variational consistent mass matrix are derived through Hamilton's principle. The resulting composite plate element is free from shear locking. Furthermore, both its element stiffness and mass matrices are computed explicitly. The convergence and accuracy of the present element are evaluated by the standard static and dynamic examples of isotropic plates. The accuracy of the computed natural frequencies of laminated composite plates is verified by the examples with various lamination schemes, elastic modulus ratios, aspect ratios and boundary conditions and by the comparison with other results. The effect of the associated boundary conditions in different TSDTs is also evaluated through numerical results. The influence of the higher-order and coupling mass matrices on the frequencies of higher-mode vibration is conducted too. The numerical results clearly demonstrate that the present plate element is not only very efficient but also capable of yielding accurate natural frequencies of both the fundamental and the higher-mode vibrations of laminated composite plates. [ABSTRACT FROM AUTHOR]

In the present work, anti-plane wave propagation with oblique incidence in a magneto-electro-elastic multi-laminated composite periodic structure with perfect contact is studied. A general asymptotic homogenization method is applied to analyze dispersion phenomena and dynamic process. On assuming the solution having a single-frequency dependency, the higher-order terms for the displacement, electric potential and magnetic potential in asymptotic expansions are studied. Higher order local problems up to a finite order and a generalization of the local problem of infinite order are derived through rigorous calculations satisfying the necessary and sufficient condition for the existence of 1-periodic solutions. The effective equation of motion also known as "Good" Boussinesq equation has been achieved directly which is one of the highlight of the present work. Moreover, closed-form expression of dispersion equation and closed-form solutions of first and second local problems have been obtained. The analytical results have been validated for particular elastic case. Graphical illustrations have been done for tri-laminated magneto-electro-elastic composite periodic structure and the effect of size of unit cell, angle of incidence of wave, volume fraction of the composite and magneto-electric properties on the dispersion curve and also slowness curves have been examined. The numerical results have been validated with previous work and compared with results obtained by other approximation method. The present dynamic homogenization method has been proven to provide a more dispersive system and better approximation. [ABSTRACT FROM AUTHOR]

• Primary resonance of laminated nanocomposite plates is studied considering nonlinearity. • Nonlinear basic equations are derived within first order shear deformation theory (FSDT). • The governing equations are solved using Galerkin and multi-scale methods. • Nonlinear forced frequency depending on the amplitude is found at primary resonance. • Backbone curve and nonlinear to linear frequency ratio within FSDT are determined. In this study, the primary resonance of laminated plates consisting of heterogeneous nanocomposite layers is investigated comparatively within first order shear deformation theory (FSDT) and classical lamination plate theory (CLPT). One of the features of the work is the generalization of the FSDT for homogeneous anisotropic laminated plates to laminated functionally graded anisotropic nanocomposite plates. The mechanical properties of the matrix reinforced with carbon nanotube (CNT), and fundamental relations of laminated moderately thick plates consisting of heterogeneous anisotropic nanocomposite layers are modeled theoretically within FSDT using von Kármán type nonlinear theory. It then derives equations of motion in the form of nonlinear partial differential equations (NL-PDEs) within FSDT. NL-PDEs equations are transformed into nonlinear ordinary differential equations (NL-ODEs) by Galerkin method and are solved by multi-scale method. The nonlinear forced vibration frequency as the function of amplitude at primary resonance within both theories are obtained for the first time. In addition, backbone curve and nonlinear frequency/linear frequency ratio are found. The reliability and accuracy of the proposed formulation is verified by comparing with the results in the literature. Finally, the effects of external excitation, non-linearity, and variation of CNT patterns on the forced vibration frequencies are examined. [ABSTRACT FROM AUTHOR]

This paper investigates out-of-plane dynamic instability for functionally graded porous (FGP) graphene platelets reinforced composites (GPLRC) circular shallow arches subjected to a radial periodic concentrated load in a thermal environment for the first time. Hamilton principle is employed to derive the differential equations of motion of the arches associated with out-of-plane displacements. The dynamic instability regions (DIR) with period 2 T are determined by using Bolotin's method and verified by finite element software ANSYS. The effects of uniform temperature difference,out-of-plane boundary condition, pore distribution mode, the weight fraction of graphene platelets (GPLs), GPLs dimension coefficients, and the location of the concentrated periodic load on the DIRs are comprehensively analyzed. The results show that higher temperature difference can weaken the dynamic stability of the arch. The load position has no influence on the boundary excitation frequency at the lowest point of the DIR but plays a negative impact on the area of the DIRs and a positive influence on the lowest value of the load. The parametric resonance of the arch is more likely to occur if the concentrated periodic load is located near the arch crown. [ABSTRACT FROM AUTHOR]

[Display omitted] This paper investigates the dynamic characteristics of non-uniform multi-span functionally graded (FG) 3D graphene foams (3D-GFs) reinforced beams with elastic boundary conditions. It is assumed that the foams are distributed according to four porosity distributions along the thickness direction and the beams sections varies in height and width along the length with the power-law distribution. Based on Timoshenko beam theory and Hamilton's principle, the equations of motion for multi-span beams are derived and the discrete singular convolution (DSC) method is adopted to solve them numerically. The present analysis is rigorously validated through direct comparisons against the results by those available in open literatures. A comprehensive parametric study is conducted to study the effects of porosity distribution, slenderness ratio, taper ratio and different combination types of multi-span beams with non-uniform sections. This study provides a new conception of multi-span beam composition, providing designers with choices of beam forms under vibration response. [ABSTRACT FROM AUTHOR]

Dielectric Elastomer (DE) devices have gained widespread attention in recent years for their ability to generate large, reversible strains in response to electric actuation. However, some DE materials exhibit a viscoelastic nature that detracts from their electromechanical performance in applications requiring precise time-dependent actuation. This study presents an investigation into the feasibility of using multilayered DE composites, comprising commercially available acrylic-based VHB 4905 and natural rubber membranes, to overcome the viscoelastic effects. A comparison of the electromechanical performance of bi-layer and tri-layer DE composites with that of a mono-layer VHB 4910 membrane is carried out through numerical and experimental methods. We present a theoretical model of the DE composite-based bulging actuator, using an energy-based approach and the assumption of homogeneous deformations. The model predicts the time-dependent deflection of the inflated/deflated multilayered DE composites, taking into account the effect of the intervening air column. The predictive capabilities of the model are validated through experiments under both DC and AC operating conditions, and the electromechanical performance of the DE composites is examined for various actuation responses such as resonance, creep, and hysteresis. The bi-layer DE composite is found to be superior in terms of actuation levels, reduced hysteresis, and reduced creep response. The findings of this study hold promise for the development of DE materials with improved actuation capabilities while maintaining dielectric breakdown strength. • A theoretical model is introduced to analyze the dynamic electromechanical behavior of multi-layered DE composites. • The dynamic behavior of DE-based composites is studied using the Euler–Lagrange equation of motion. • The influence of the intervening air column on the time-dependent deflection of inflated/deflated DE composites is considered. • To mitigate viscoelastic effects, detailed experimental investigations of multi-layered DE composites are conducted. • The electromechanical performance of the developed multi-layered DE composites is compared with that of a mono-layer VHB 4910 membrane. [ABSTRACT FROM AUTHOR]

The coupled vibration of flexible rods in contact with rigid cylinders has been observed to generate disruptive noise and significantly decrease the service life. Although the coupling effect of cylindrical rotation with transverse vibration or axial motion of slender rods has been well studied, the full coupling of the three motions remains inadequately understood. To address this issue, we propose an improved discrete differential geometry model with a tri-linear friction method to deliver accurate reproduction of resonant frequencies and vibration responses. Our model features several advances: (i) Viscous forces associated with stretching and bending strain rates are incorporated into the motion equation of the viscoelastic rod, replacing the assumption of quasi-static axial motion; (ii) A tri-linear friction method is introduced to obtain the relative motion between the slender rod and cylinders on the contact zone, thus eliminating errors stemming from pre-prescribed contact regions; (iii) The model fully couples the axial and transverse motion over the entire rod as well as the rotation of cylinders, surpassing the limitation of the coupled rotational-transverse model which only establishes equations of transverse motion on the non-contact zone of the rod. The numerical results indicate that the axial and transverse vibrations of the upper and lower span of the slender rod and the cylindrical rotational vibration are mutually excited, with their resonant frequencies having overlapping sites, which validates the proposed coupling model. Moreover, we are able to capture the phenomenon of contact surface sliding during the coupled vibration, confirm that the sliding limits the vibration amplitude, and determine the nonlinear relationship between the excitation and vibration response. This work provides a theoretical basis for further optimization of composite transmission structures, particularly in terms of suppressing vibration and improving efficiency of power transmission. [ABSTRACT FROM AUTHOR]

A numerical approach is adopted for the multiscale analysis of vibrations of single-walled carbon nanotubes (SWCNTs). The SWCNT is modeled by a hyperelastic membrane whose kinematics is described using the higher-order Cauchy-Born rule. The constitutive model is formulated exclusively in terms of the interatomic potential, so, it inherits the atomistic information and involves no other phenomenological input. The variational differential quadrature (VDQ) method is employed in which the continuum model is discretized using DQ, and a weak form of equation of motion is obtained via a variational approach. VDQ is computationally advantageous since it has a fast rate of convergence and can reproduce the results of molecular dynamics simulations. Detailed investigations into frequencies and mode shapes of SWCNTs with different geometrical parameters, boundary conditions and chiralities are carried out. It is found that short nanotubes display a coupling between the axial/torsional and bending modes. Also, as the tube diameter or length increases, mode transitions are made at several critical points. If the edge supports are more flexible and tube length is longer, the critical diameters are larger. Eventually, the vibration characteristics of axially strained nanotubes are analyzed, and it is concluded that SWCNTs with smaller radii have higher strain sensitivity. [ABSTRACT FROM AUTHOR]

This paper develops a nonlocal theory of elasticity for the thermoelastic vibration of nanoplates. A mixed variational formula based upon Hamilton’s principle is extended to deal with nanoplates. The present nonlocal first-order shear deformation theory does not need any shear correction factors. Equations of motion and boundary conditions are obtained simultaneously through the mixed variational formula. Governing equations of a mixed nonlocal smart nanoplate are obtained and their solutions are given. The accuracy of the present results is investigated through many comparison examples in the literature. The influences of different factors such as nonlocal parameter, aspect ratio, side-to-thickness ratio, and mode numbers on the natural vibration characteristics of nanoplates are studied. [ABSTRACT FROM AUTHOR]

A novel analytical model that accurately represents the free vibration response of intact/damaged composite plates accounting for the bending-torsional coupling is developed. Unlike the existing models for the bending-torsion vibrations of composite plates, in the present model, the equations of motion are derived in an uncoupled form. It is then easier to derive analytical solutions for composite plates under coupled bending-torsion vibrations. In addition, for the first time, analytical solutions for the free bending and torsional vibrations of cantilever and clamped-clamped laminated composite plates with discontinuities including circular holes, rectangular holes, notches, and internal and edge cracks are easily obtained. Shifts in the natural frequencies due to discontinuities are modeled where the reductions in the plate stiffnesses are related to the amount of the energy release rate due to a defect formation. In order to verify the accuracy of the proposed model and the derived solutions, the proposed model is compared to the available conventional models. Moreover, experimental measurements of the free bending and torsional vibrations of carbon-fiber-laminated composite plates with circular holes is performed. A set of finite element analyses are carried out and compared to the experimental and the proposed model results. Then, a parametric study is presented in order to investigate the effects of the discontinuity size and location on the bending and torsional natural frequencies of laminated composite plates with various forms of discontinuities. The performed analyses demonstrate the accuracy of the proposed models to represent the free vibrations of laminated composite plates with discontinuities. [ABSTRACT FROM AUTHOR]

In this work, the dynamics of a truss core sandwich beam with NES (nonlinear energy sink) device are investigated. Based on the principle of nonlinear TET (targeted energy transfer), an NES device is placed in the interior of a sandwich beam to suppress vibration of the beam. The governing equation of motion for the sandwich beam with NES device attached is derived. The vibration responses of the system are analyzed under impulse and harmonic loads. Feasibility of the vibration suppression scheme is demonstrated by investigating the relation between the mass of the NES device and suppressing effect as well as the relative motion between the sandwich beam and the NES device. In addition, the influences of the NES parameters on the suppressing effect are studied. It is worth noting that the vibration suppression is enhanced with increasing the damping of NES when the system is excited by harmonic loads and with decreasing the damping when the system is excited by impulse loads. The influence of the nonlinear characteristic of spring on the vibration suppression depends on the selection of the other parameters of the NES. [ABSTRACT FROM AUTHOR]

This study deals with the response of an FGM annular plate under lateral thermal shock load. The equations of motion are obtained using the first order shear deformation plate theory. The governing equations are solved using the Laplace transformation and Galerkin finite element method. Finally, numerical inversion of the Laplace transform is carried out to obtain the results in real time domain. It is shown that coupling coefficient has a damping effect on the radial force resultant and deflection. [ABSTRACT FROM AUTHOR]

This article develops a novel finite element formulation based on nonlocal theory to analyze the vibration of viscoelastic orthotropic nanoplates resting on the variable viscoelastic foundation (VEF). The mechanical properties of nanoplate are assumed to be viscoelastic orthotropic according to Kelvin's model. The variable VEF consists of two layers: a shear layer with constant stiffness, the other layer is described as a system composed of alternating damping and springs, in which the spring stiffness and the damping coefficient vary x -axis only. Based on Hamilton's principle, a refined higher-order shear deformation plate theory (HSDT) and nonlocal theory are used to establish motion equations of the nanoplates. A four-node quadrilateral finite element is created by combining two interpolation functions including Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement. The obtained results in our work are verified through reliable publications. A series of factors influencing the vibration of orthotropic nanoplates resting on the variable VEF is discovered and discussed. [ABSTRACT FROM AUTHOR]

In this study, a nonlocal strain gradient isogeometric model, which includes the higher-order shear deformation theory, nonlocal strain gradient theory and isogeometric analysis method, for free vibration analysis of functionally graded (FG) nanoplates made of magneto-electro-elastic (MEE) materials is the first time presented. The mechanism of reduction and increase in the stiffness of MEE-FG nanoplates are shown in the proposed model under the influence of two scale parameters. For simplicity of calculation, the material properties of the MEE-FG nanoplates are described by the power-law scheme. A weak form of governing equations of motion of the MEE-FG nanoplates is obtained by applying the principle of extended virtual displacement. The natural frequency of the MEE-FG nanoplates is evaluated through the change of power index, geometrical parameter, nonlocal parameter, strain gradient parameter, electric voltage and magnetic potential. The results obtained through nonlocal strain gradient theory (NSGT) are lower than those predicted by classical theory when the nonlocal parameter is larger than or equal to the strain gradient parameter. On the contrary, natural frequencies obtained by NSGT are higher than those estimated by classical theory as the nonlocal parameter is smaller than the strain gradient parameter. [ABSTRACT FROM AUTHOR]

• A novel dynamic model on nonlinear vibrations of blade are investigated. • Blade is simplified to a FGGP reinforced rotating pretwisted composite cantilever plate. • Dynamic model expresses changes of non-uniform axial and contact forces through tip clearance and rotating pretwisted plate-casing rubbing. • Influences of graphene distribution patterns and axial excitation on frequencies are investigated. • Nonlinear vibrations under 1:1 internal resonance at critical rotating speeds are studied. A novel dynamic model of the functionally graded graphene platelet (FGGP) reinforced rotating pretwisted composite blade under the aerodynamic force and blade-casing rubbing is established for the first time. The blade is simplified to a FGGP reinforced rotating pretwisted composite cantilever plate. The dynamic model of the axial excitation produced by considering the aerodynamic force in the tip clearance includes two trigonometric functions with different frequencies. The subsonic airflow is considered as a transverse excitation which is derived by utilizing the vortex lattice method (VLM). The dynamic model expresses the changes of the non-uniform axial and contact forces through the tip clearance and rotating pretwisted plate-casing rubbing when the shaft is eccentric. Based on Rayleigh-Ritz method, the linear frequencies and mode shapes are obtained for the FGGP reinforced rotating pretwisted plate. The influences of the graphene distribution pattern, rotating pretwisted plate-casing rubbing, axial excitation in the tip clearance and geometric parameter on the frequencies are investigated. Using the obtained mode shapes, von-Karman type nonlinear geometric assumptions and Lagrange equation, the differential governing equations of motion are derived for the FGGP reinforced rotating pretwisted plate. The nonlinear vibrations under the 1:1 internal resonance at two critical rotating speeds is studied by using Runge-Kutta method. The amplitude-frequency and force-frequency response curves under the low and high critical rotating speeds are investigated by using numerical calculations. The obtained results demonstrate the influence of the rotating speed, frequency ratios and incoming flow speeds on the nonlinear vibrations of the FGGP reinforced rotating pretwisted plate. The dynamic model of the rubbing-impact for the rotating blade provides a theoretical basis for the blade-casing rubbing analysis. At the same time, this study is also used as a theoretical guidance to reduce the damage of the blade-casing rubbing and blade design. [ABSTRACT FROM AUTHOR]

Curvilinear reinforcement fibres allow to tailor the elastic properties of composite laminates in space, leading to Variable Stiffness Composite Laminates (VSCL). In this work, the aeroelastic stability of VSCL cylindrical shells under a supersonic airflow along the axial direction is analysed in order to verify if curvilinear fibres can be used to increase the flutter speed. A linear mathematical model for thin circular cylindrical VSCL shells is developed for that purpose. The aerodynamic pressure caused by the supersonic airflow is modelled recurring to the linearised piston theory. The p -version of the finite element method is used to transform the differential equations of motion from partial to ordinary. Using the latter, eigenvalue problems are defined to obtain the natural frequencies and mode shapes of vibration, and the critical flutter conditions. Various types of curvilinear fibre paths are explored, to find paths that result in higher values of critical flutter free stream pressure. The values are compared with the ones of conventional, constant stiffness, composite laminated shells. [ABSTRACT FROM AUTHOR]

Functionally graded sandwich structures are an advanced class of sandwich structures with material properties varying smoothly in one or more spatial directions. In this work, the effect of the homogenization scheme and partial Pasternak foundation support on nonlinear vibration of bidirectional functionally graded sandwich (BFGSW) beams is studied. The core of the sandwich beams is homogeneous, while the two face layers are made from a three-phase functionally graded material (FGM). Two homogenization schemes, the Voigt model and Maxwell formula, are employed to estimate the effective material properties of the face layers. Based on the refined third-order shear deformation theory, the discrete nonlinear equation of motion is derived with the aid of a two-node beam element, and then solved by a direct iterative procedure. Nonlinear frequencies of the sandwich beams with different boundary conditions are evaluated for various initial vibration amplitudes and foundation parameters. The obtained result reveals that the difference between the nonlinear frequency ratios obtained from the Voigt model and the Maxwell formula is more significant for the beam associated with a higher initial vibration amplitude. The influence of the material distribution and the foundation support on the nonlinear vibration of the sandwich beams is studied in detail and highlighted. • Nonlinear vibration of bidirectional functionally graded sandwich beams is studied. • Influence of homogenization scheme and partial foundation support is considered. • Discrete equation of motion is derived using 2-node beam element and solved by iterative method. • Effect of material gradation, foundation support on nonlinear vibration is investigated. [ABSTRACT FROM AUTHOR]

Featured by the two material length parameters in the nonlocal strain gradient theory, it is still unknown that what are the boundary conditions of nonlocal strain gradient beams, since the equations of motion and boundary conditions of these beam models appear in the same form as those of the classical ones. Based on the weighted residual approaches, this paper provides the boundary value problems of Euler–Bernoulli beams within the framework of the nonlocal strain gradient theory in conjunction with the von Kármán nonlinear geometric relation. The closed-form solutions for bending and buckling loads of nonlocal strain gradient beams are obtained. Numerical results show that the higher-order boundary conditions have no effect on the static bending deflection of beams for the cases studied. However, the higher-order boundary conditions and the material length parameters have a significant effect on the buckling loads. Finally, when the two material length parameters are the same, the buckling loads can not always reduce to the classical solutions, the findings of which violate our expectations. The results provided in this work are expected to be helpful for the applications of this theory to the analysis of engineering structures. [ABSTRACT FROM AUTHOR]