Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

Nonlinear free vibration analysis of SSMFG cylindrical shells resting on nonlinear viscoelastic foundation in thermal environment

In this paper, a semi-analytical method for the free vibration behavior of spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells under the thermal environment is investigated. The distribution of linear and uniform temperature along the direction of thickness is assumed. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The cylindrical shell has three layers consist of ceramic, FGM, and metal in two cases. In the first model i.e. Ceramic-FGM-Metal (CFM), the exterior layer of the cylindrical shell is rich ceramic while the interior layer is rich metal and the functionally graded material is located between these layers and the material distribution is in reverse order in the second model i.e. Metal-FGM-Ceramic (MFC). The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von Kármán equations and the smeared stiffeners technique, the problem of nonlinear vibration has been solved. In order to find the nonlinear vibration responses, the fourth order Runge–Kutta method is utilized. The results show that the different angles of stiffeners and nonlinear elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells. Also, the results illustrate that the vibration amplitude and the natural frequency for CFM and MFC shells with the first longitudinal and third transversal modes (m = 1, n = 3) with the stiffeners angle θ = 30°, β = 60° and θ = β = 30° is less than and more than others, respectively.     

On the dynamics of Rayleigh beams resting on fractional-order viscoelastic Pasternak foundations subjected to moving loads

The standard averaging method is used to provide an analytical explanation on the effects of spacing loads, load velocity, order of the fractional viscoelastic property of shear layer material on the amplitude of the beam. The geometric nonlinearity is taken into account in the model. The analysis shows that, when the moving loads are uniformly distributed upon all the length of the structure, it vibrates the least possible. Moreover, as the order of the derivative increases, the resonant amplitude of the beam vibration decreases. In other hand, by means of Melnikov technique, a necessary condition for onset of horseshoes chaos resulting from heteroclinic bifurcation is derived analytically. We point out the critical weight of moving loads and order of the fractional derivative above which the system becomes unstable.     

Homoclinic bifurcations and chaotic dynamics of non-planar waves in axially moving beam subjected to thermal load

The homoclinic bifurcations and nonplanar chaotic waves in axially moving beam (AMB) under thermal excitation are investigated. By the multiple scale technique, the equivalent nonlinear system is derived to explore qualitatively the dynamical characteristics of AMB system for the case of primary resonance. Using Melnikov approach as well as geometric analysis, the criterion for homoclinic chaos and complex nonplanar motions for AMB system is discussed. The theoretical predictions are tested by the numerical approach. For the design and application of the AMB, some inspiration and guidance are provided by the results from theory and simulation.     

Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation

Dynamic instability of cantilever carbon nanotubes conveying fluid embedded in viscoelastic foundation under a partially distributed tangential force is investigated based on nonlocal elasticity theory and Euler–Bernouli beam theory. The present study has incorporated the effects of nonlocal parameter, Knudsen number, surface effects and magnetic field. And two main parameters have also considered, namely partially distributed tangential force and foundation. It is assumed that viscoelastic foundation has modeled as Kelvin–Voigt, Maxwell and Standard linear solid types. The size-dependent governing equation of transverse vibration is derived using Hamilton’s variational principle and discretized by the Galerkin truncation method. A detailed parameter study is carried out, indicating the stability behavior of the nanotubes. In the light of numerical results, it is shown that variables considered in nondimensional equations have significant effects on natural frequencies and flutter velocities, especially for the foundation distribution length and model as well as the partially distributed tangential force.     

Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction

This paper presents an investigation on partially fluid-filled cylindrical shells made of functionally graded materials (FGM) surrounded by elastic foundations (Pasternak elastic foundation) in thermal environment. Material properties are assumed to be temperature dependent and radially variable in terms of volume fraction of ceramic and metal according to a simple power law distribution. The shells are reinforced by stiffeners attached to their inside and outside in which the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The formulations are derived based on smeared stiffeners technique and classical shell theory using higher-order shear deformation theory which accounts for shear flexibility through shell's thickness. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The governing equations of liquid motion are derived using a finite strip element formulation of incompressible inviscid potential flow. The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. A detailed numerical study is carried out to investigate the effects of power-law index of functional graded material, fluid depth, stiffeners, boundary conditions, temperature and geometry of the shell on the natural frequency of eccentrically stiffened functionally graded shell surrounded by Pasternak foundations.     

The critical speed of a moving load on a prestressed plate resting on a prestressed half-plane

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the dynamical response of a system consisting of a prestressed covering layer and a prestressed half-plane to a moving load applied to the free face of the covering layer is investigated. Two types (complete and incomplete) of contact conditions on the interface are considered. The subsonic state is considered, and numerical results for the critical speed of the moving load are presented. The influence of problem parameters on the critical speed is analyzed. In particular, it is established that the prestressing of the covering layer and half-plane increases the critical speed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 257–270, March–April, 2007.     

Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.     

Using spectral element method for analyzing continuous beams and bridges subjected to a moving load

Spectral element method in frequency domain is employed to analyze continuous beams and bridges subjected to a moving load. The formulation is developed for an Euler beam under a moving load with an arbitrary amplitude and velocity. It is shown that the procedure is simplified for a moving load with a constant amplitude and velocity. Static Green’s function is used as a modifying function to improve the moment and shear force results. It is further shown that while modifying function is used in conjunction with spectral element method, fewer elements will be required to achieve proper results. The numerical examples show the accuracy of the method.     

Flexoelectric and surface effects on the electromechanical behavior of graphene-based nanobeams

In this novel work, the electromechanical behavior of graphene-based nanocomposite (GNC) beams with flexoelectric and surface effects were investigated using size-dependent Euler-Bernoulli theory, linear piezoelectricity and Galerkin's weighted residual method along with modified strength of materials and finite element (FE) approaches. In addition, analytical and FE models were developed to study the static response of flexoelectric GNC nanobeams with various boundary conditions: cantilever, simply-supported and clamped-clamped. The developed models predict that the effective piezoelectric coefficients of GNC are responsible for the actuation capability of a graphene layer in the transverse direction due to the applied field in its axial direction and the predictions by both the models are found to be in good agreement. Results reveal that the flexoelectric and surface effects on the static response of GNC nanobeams are significant and should be taken into account. The electromechanical response of GNC nanobeams can be tailored to achieve the required coupled electromechanical characteristics of a vast range of NEMS using various boundary conditions and thickness of nanobeam as well as volume fraction of graphene. Our fundamental study sheds a light on the possibility of developing high-performance and lightweight graphene-based NEMS such as nanosensors, nanogenerators and nanoresonators using non-piezoelectric graphene.     

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

Forced vibration of the pre-stressed bi-layered plate-strip with finite length resting on a rigid foundation

Within the scope of the piecewise homogeneous body model utilizing Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies the time-harmonic dynamical stress field in the pre-stressed bi-layered plate-strip with finite length resting on the rigid foundation is investigated. The materials of the layers are assumed to be isotropic. The FEM modeling is developed for the solution to the corresponding boundary-value-contact problem. The numerical results regarding the influence of the finiteness of the layers’ length on the stress distribution on the interface planes are presented and discussed. In particular, it is shown that with increasing the plate length the results obtained for the considered case approach to the corresponding ones attained for the bi-layered plate with infinite length.     

An inverse vibration analysis of a tower subjected to wind drags on a shaking ground

A method is presented to estimate the strength of wind drags on an elevated tower and the magnitude of the vibration of the ground on which the tower stands. The governing equations for the motions of the tower are discretized using the finite-difference method. Based on these discretized governing equations, a linear inverse model is constructed to identify the external wind drags and the ground vibration. The optimized solution of the model is determined by the linear least-square error method, which requires no numerical iteration. The uniqueness of the solution can be identified by linear algebra theory. A numerical example is given to demonstrate the feasibility of the method. The results show that the original wind drags and ground vibration may be estimated from the measured deflection at several locations along the tower. The reasonable estimations are achievable even though there exist certain measurement errors. The loading conditions are checked at different locations and the deflection information at different location may be used. The procedure is easy and effective. It may be extended to many inverse applications that the discretized governing equations were derived.     

Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces

A study on the buckling and dynamic stability of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces is performed in this research. The static and dynamic governing equations of the nanobeam are established with Galerkin method and under Euler–Bernoulli hypothesis. The buckling, post-buckling and nonlinear dynamic stability character of the nanobeam is presented. The quasi-elastic method, Leibnitz’s rule, Runge–Kutta method and the incremental harmonic balanced method are employed for obtaining the buckling voltage, post-buckling characteristics and the boundaries of the principal instability region of the dynamic system. Effects of the electrostatic load, van der Waals force, creep quantity, inner damping, geometric nonlinearity and other factors on the post-buckling and the principal region of instability are investigated.     

Action of a variable concentrated load on a viscous half-space with allowance for surface tension

An attempt is made to estimate the effect of surface tension on the distribution of velocities and pressure and the shape of the free surface under the action of a variable concentrated load.Mekhanika Polimerov, Vol. 3, No. 2, pp. 273–278, 1967     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

Buckling and vibration analysis of nonlocal axially functionally graded nanobeams based on Hencky-bar chain model

Buckling and free vibration analyses of nonlocal axially functionally graded Euler nanobeams is the main objective of this paper. Due to its simplicity, the Eringen's differential constitutive model is adopted for describing the nonlocal size dependency of nanostructure beam. The nonlocal equilibrium equation is derived using the principle of the minimum potential energy principle, and discretized by using the link-spring model known in literature as Hencky bar-chain model. The general applicability of the proposed approach allows analyses of functional graded microbeams without any restriction on variability, boundary and loading conditions. A comparison with results available in the literature shows the reliability of the method.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.