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Continuous rails are often modeled as infinite beams. In this paper, closed-form solution is newly derived by the residue theorem and Green’s function for the dynamic response of an infinite beam resting on viscoelastic foundation under a harmonic moving load. The most suitable span length will be determined for the analogic finite beam (which can be more conveniently used in practice) to best represent the deflection of the infinite beam for practical reasons. Starting from the equations of motion, closed-form solution is firstly derived for the infinite beam. Then, it is verified by the finite element method (FEM) using semi-infinite elements to simulate the infinite boundary conditions. The deflection curves of the infinite beam and analogic finite beam are compared for various parameters. It is concluded that: (1) a larger span length should be adopted for the finite beam under higher load speeds or with foundations of softer stiffness or larger damping; typically, for load speeds $<90$m/s (324km/h); (2) typically, a span length of 50 m can be adopted for the analogic beam for UIC rails resting on foundations with stiffness $k>3.503\times 10^7{\mathrm{\text{N/m}}}^2$ and damping coefficient $c<1.7325\times 10^6{\mathrm{\text{Ns/m}}}^2$; (3) the peak deflection of the infinite and analogic finite beams occurs after the moving load for foundations with larger damping; and (4) the shape and amplitude of the deflection curve vary as the load frequency and amplitude varies, but the difference between the two beams is insignificant for moving loads with high frequencies.

In this research, an approach combining the precise time integration method (PTIM) and mode decomposition method is proposed to compute the response of beam structures resting on viscoelastic foundation. The PTIM has high precision, high efficiency, but it still suffers from the problem of large-size matrices when directly computing structure. And this problem can be overcome by mode superposition. Thus, the present paper integrates the PTIM and mode decomposition method together which holds the explicit recurrence form of precise algorithm. Comparing with the other numerical methods, it is found that the presented method is much more precisely and time-saving. The effect of the speed of moving load, the foundation stiffness and the length of the beam on the response of beam have also been studied. These numerical computation results show that the present method is effective and feasible.

In the present contribution, the nonlinear dynamics of beams under axial time-dependent excitation is studied. The beams, which rest on a linear viscoelastic foundation, are assumed as axially graded both in terms of geometrical and material properties. The transversal motion equation is derived under two main hypotheses, namely the beams satisfy the requirements of Euler–Bernoulli theory and undergo moderately large deflections (MLDs).

Presented herein is a novel computational approach using the moving element method (MEM) for simulating the dynamic response of Mindlin plate resting on a viscoelastic foundation and subjected to moving loads. The governing equations and the element mass, damping and stiffness matrices are formulated in a convected coordinate system in which the origin is attached to the point of the moving applied load. Thus, the method simply treats moving loads as ‘stationary’ at the nodes of the plate to avoid updating the locations of moving loads due to the change of the contact points on the plate. To verify the accuracy of the proposed computational approach, static and free vibration analyses of plates are investigated first. Next, the dynamic response of plate resting on a viscoelastic foundation subjected to a moving load is examined. A parametric study is performed to determine the effects of the load’s velocity, foundation damping and foundation stiffness on the dynamic response of a plate. Finally, the comparisons of the dynamic response of plates resting on viscoelastic foundation and subjected to moving vehicles with three models of load (single-wheel, single-axle and tandem-axle) are discussed.

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

Carbon nanotubes (CNTs) have emerged as efficient tools in drug delivery systems; therefore, it is essential to refer to the importance of the magnetic field, in addition to the fluid flow on the dynamic behavior of CNTs. Additionally, in such medical applications, the actual working environment of nanotube often contains temperature changes, and CNTs are surrounded by soft tissues with viscoelastic mechanical properties. In this study, the vibrational behavior of CNTs conveying magnetic-fluid flow and resting on a viscoelastic foundation is investigated under various temperature variations. To incorporate the influence of slip velocity at the nanoscale, a correction factor is employed on the basis of the Beskok–Karniadakis model. The nanotube is modeled by the Euler–Bernoulli beam theory, and governing equations of motion are derived by implementing Hamilton’s principle based on Eringen’s nonlocal elasticity theory. Results indicate that by applying a magnetic field with an intensity of 30T, the dimensionless critical flow velocity increases from 4.345 to 12.603. Also, the critical flow velocity shows an increase from 4.345 to 5.854 in the presence of a viscoelastic foundation. Furthermore, a temperature variation equal to 20K reduces the critical flow velocity dramatically from 4.345 to 1.802 at low temperatures, while an increase from 4.345 to 5.443 is observed at high temperatures. Consequently, while the magnetic field and the viscoelastic foundation affect the system stability, the temperature variation may improve or deteriorate the stability. Therefore, to plan for a medical application, the inclusion of temperature variation is required.

This article elaborates on the dispersion of waves in piezoelectric sandwich nanoplates resting on a viscoelastic foundation. The nanoplate comprises a functionally graded (FG) graphene-reinforced composite core layer with two piezoelectric surface layers. By combining the Halpin–Tsai model and related mixture rules, the properties of the composite material have been obtained. The Euler–Lagrange equation is obtained using the third-order shear deformation theory (TSDT) and Hamilton’s principle. Subsequently, based on the nonlocal strain gradient theory (NSGT), the equation of motion is presented. Finally, the effects of scale parameters, hygrothermal conditions, graphene distribution, and viscoelastic foundation on the propagation characteristics are numerically studied. The results reveal that the scale effect is more evident when the wave number is larger. Furthermore, critical damping increases with a rise in the wavenumber and Winkler modulus.

Studying the dynamics of beams subjected to a moving mass is important due to their wide applications, including railways, machining processes, and microelectromechanical systems (MEMS). Various numerical and analytical approaches have been used for modeling such structures. In this analytical study, we have used a combination of the Optimal homotopy analysis method (Optimal HAM) and enriched multiple scales (MS) to analytically study the dynamics of a simply supported Euler–Bernoulli beam traversed by a moving mass and resting on a viscoelastic foundation. The viscoelastic foundation contributes to the modeling by adding a linear and nonlinear term to the formulation. Further, we have considered a fifth-order nonlinear term to account for the bending vibration of the flexible beam. Using the Galerkin method, we have formed the corresponding ordinary differential equation (ODE). Then, we used the enriched MS Optimal HAM to calculate the dynamic response of the beam. After validating our method by comparing our results with the dynamic results of the beam obtained from finite element analysis (FEA), we investigated the effects of the determining parameters on the beam dynamic response. The effects of the foundation nonlinear and linear terms, the moving load weight, and its velocity have been investigated by studying the variation of the normalized beam lateral deflection versus the normalized moving mass instantaneous position in each case. We showed that the difference between linear and nonlinear modeling results is pronounced, and it becomes more pronounced for faster and heavier moving loads.

This paper modeled forced and free vibrations of an axially functionally graded (AFG) porous Rayleigh nanotube containing fluid surrounded by a viscoelastic foundation and acted by a traveling load. The material characteristics of the nanotube are graded longitudinally according to a power-law rule. Also, different porosity models in the longitudinal direction of the system are taken into account. The nonlocal strain gradient theory (NSGT) is applied to reflect the size-dependency of the model. The governing dynamic equation of the system incorporating the scale effects is derived by considering the effects of slip boundary conditions for the internal fluid. The vibration response of the system is obtained analytically. Several comparison studies verify this study’s results. Various dynamic phenomena, such as cancellation and maximum free response, are discussed in different operating conditions of the system. Besides, a detailed parametric investigation is conducted to identify the effects of influential parameters, including material gradient, porosity coefficient, fluid velocity, scale parameter ratio, and substrate characteristics, on the dynamic behavior of the system. These research results could be valuable in designing and optimizing advanced nanosensors and targeted drug delivery systems.

This paper aims to analyze the wave propagation in functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams placed on a viscoelastic foundation utilizing an improved first-order shear deformation theory (FSDT). The material properties are derived from the mixture rule. Four carbon nanotube distribution patterns are considered in the analysis. The extended Hamilton’s Principle is utilized to derive the governing wave equations for the CNTRC beam. A comparison between the present theory results and those in the literature is conducted for validation. The wave dispersion investigation is mainly based on the phase and group velocities. The results illustrate the wave propagation responses for the different CNT configurations. In addition, the influence of the CNTs volume fraction, foundation stiffness parameters, and damping coefficient on the wave characteristics is examined.

Smart control and dynamic investigation of a graphene nanoplatelets reinforced composite (GPLRC) cylindrical shell surrounded by a piezoelectric layer as actuator and sensor based on a numerical solution method called generalized differential quadrature method (GDQM) are presented for the first time. The strains and stresses can be determined via the first-order shear deformable theory (FSDT). For accessing to various mass densities, thermal expansion as well as Poisson ratio, the rule of mixture is applied, although a modified Halpin–Tsai theory is used for obtaining the module of elasticity. The external voltage is applied to the sensor layer, while a proportional-derivative (PD) controller has been utilized for controlling the output of sensor. GPLRCs boundary conditions are derived through governing equations of the cylindrical shell using an energy method known as Hamilton’s principle. The outcomes show that the PD controller, viscoelastic foundation, slenderness factor ($L$/$R)$, external voltage and graphene nanoplatelets (GPLs) weight fraction have a considerable impact on the amplitude, and vibration behavior of a GPLRC cylindrical shell. As an applicable result in related industries, the parameter and consideration of the PD controller have a positive effect on the static and dynamic behaviors of the structure.

In this research, an approach combining the precise time integration method (PTIM) and mode decomposition method is proposed to compute the response of beam structures resting on viscoelastic foundation. The PTIM has high precision, high efficiency, but it still suffers from the problem of large-size matrices when directly computing structure. And this problem can be overcome by mode superposition. Thus, the present paper integrates the PTIM and mode decomposition method together which holds the explicit recurrence form of precise algorithm. Comparing with the other numerical methods, it is found that the presented method is much more precisely and time-saving. The effect of the speed of moving load, the foundation stiffness and the length of the beam on the response of beam have also been studied. These numerical computation results show that the present method is effective and feasible.