Search Results (36)

The present research analyzed the nonlinear vibration of a CNTRC embedded in a nonlinear Winkler–Pasternak foundation in the presence of an electromagnetic actuator and mechanical impact. A higher-order shear deformation beam theory was applied to various models, as well as Euler–Bernoulli, Timoshenko, Reddy, and other beams, using a unified NSGT. The governing equations were obtained based on the extended shear and normal strain component of the von Karman theory and a Hamilton principle. The system was discretized by means of the Galerkin–Bubnov procedure, and the OAFM was applied to solve a complex nonlinear problem. The buckling and bending problems were studied analytically by using the HPM, the Galerkin method in combination with the weighted residual method, and finally, by the optimization of results for a simply supported composite beam. These results were validated by comparing our results for the linear problem with those available in literature, and a good agreement was proved. The influence of some parameters was examined. The results obtained for the extended models of the Euler–Bernoulli and Timoshenko beams were almost the same for the linear cases, but the results of the nonlinear cases were substantially different in comparison with the results obtained for the linear cases. Full article

This study numerically investigated free vibration characteristics of functionally graded material (FGM) beams on Winkler–Pasternak three-parameter elastic foundations using the modified generalized differential quadrature (MGDQ) method. To compare the effects of different beam theories on the predicted frequency responses, an nth order generalized beam theory was employed to establish the governing equations of the system’s dynamic model within the Hamilton framework. As a pioneering effort, a MATLAB (version 2021a) computational program implementing the MGDQ method was developed to obtain the free vibration responses of foundation-supported FGM beams. Parametric analyses were conducted through numerical simulations to systematically examine the influences of various factors, including beam theories, damping coefficients, foundation stiffness parameters, boundary conditions, gradient indices, and span-to-thickness ratios, on the natural frequencies and damping ratios of FGM beams. The findings provide an essential theoretical foundation for dynamic characteristic analysis and functional design of foundation-supported FGM beam structures. Full article

To explore the nonlinear dynamics of a piezoelectric energy harvesting device, we consider the simultaneous parametric and external excitations. Based on Bernoulli–Euler beam theory, a new dynamic model is proposed taking into account the curvature of the beam, geometric and electro-mechanical coupling nonlinearities, and damping nonlinearity, with inextensible deformation. The system is discretized by using the Galerkin–Bubnov procedure and then is investigated by the optimal auxiliary functions method. Explicit analytical expressions of the approximate solutions are presented for a complex problem near the primary resonance. The main novelty of our approach relies on the presence of different auxiliary functions, the involvement of a few convergence-control parameters, the construction of the initial and first iteration, and much freedom in selecting the procedure for obtaining the optimal values of the convergence-control parameters. Our procedure proves to be very efficient, simple, easy to implement, and very accurate to solve a complicated nonlinear dynamical system. To study the stability of equilibrium points, the Routh–Hurwitz criterion is adopted. The Hopf and saddle node bifurcations are studied. Global stability is analyzed by the Lyapunov function, La Salle’s invariance principle, and Pontryagin’s principle with respect to the control variables. Full article

To create a discretized prediction model for the deformation of an adjacent pipeline, the pipeline structure is discretized, the differential equations governing the longitudinal deformation of the pipeline are inferred, and the displacement expressions and the solution methods of the virtual nodes of each unit are provided after discretization. This approach is based on the Pasternak foundation beam theory. It aims to address the issue of the difficulty in predicting the deformation of the adjacent pipeline caused by shield tunneling in a saturated soft ground layer in the Yangtze River Delta. The deformation pattern of the surrounding soil is determined and confirmed through additional numerical simulation, and the discretized prediction model is contrasted with the conventional Winkler foundation beam model and the Pasternak foundation beam model. The findings demonstrate that the discrete prediction model is simpler to solve and more accurately describes the deformation characteristics of the adjacent pipeline as well as the deformation distribution law. The calculated deformation characteristics primarily appear as the adjacent pipeline’s deformation due to the double tunnel boring exhibiting a “mono-peak shape” with a large middle and small ends, which is consistent with the actual situation. The two main factors influencing the pipeline deformation are the shield tunneling distance and pipeline spacing; the former has a negative correlation with the pipeline deformation, while the latter has a positive correlation. This work can offer a straightforward deformation prediction technique for shield tunneling in the Yangtze River Delta’s saturated soft ground next to existing pipelines. Full article

This paper explores the vibration behaviour of an elastically constrained graphene origami-enabled auxetic metamaterial beam subject to a harmonic external force. The effective mechanical properties of the metamaterial are approximated using a micromechanical model trained via a genetic algorithm provided in the literature. The three coupled equations of motion are solved numerically; a set of trigonometric functions is used to approximate the displacement components. The accuracy of the proposed model is confirmed by comparing it with the natural frequencies of a simplified non-metamaterial structure available in the literature. Following this validation, the investigation extends to investigate the forced vibration response of the metamaterial beam, examining the influence of the graphene origami distribution pattern and content, graphene folding degree, linear and shear layer stiffness, and geometrical parameters on the dynamic behaviour of the structure. The results generally highlight the considerable effect of the shear layer, modelled as a Pasternak foundation, on the vibration behaviour of the elastically constrained metamaterial beams. Full article

This work presents accurate values for the dynamic stiffness matrix coefficients of Levinson beams under axial loading embedded in a Winkler–Pasternak elastic foundation. Levinson’s theory accounts for greater shear deformation than the Euler–Bernoulli or Timoshenko theories. Using the dynamic stiffness approach, an explicit algebraic expression is derived from the homogeneous solution of the governing equations. The dynamic stiffness matrix links forces and displacements at the beam’s ends. The Wittrick–Williams algorithm solves the eigenvalue problem for the free vibration and buckling of uniform cross-section parts. Numerical results are validated against published data, and reliability is confirmed through consistency tests. Parametric studies explore the effects of aspect ratio, boundary conditions, elastic medium parameters, and axial force on beam vibration properties. The relative deviation for the fundamental frequency is almost $6.89\%$ for a cantilever beam embedded in the Pasternak foundation, $5.16\%$ for a fully clamped beam, and $4.79\%$ for a clamped–hinged beam. Therefore, Levinson beam theory can be used for calculations relevant to loads with short durations that generate transient responses, such as impulsive loads from high-speed railways, using the mode superposition method. Full article

The present study dealt with a comprehensive mathematical model to explore the nonlinear forced vibration of a magnetostrictive laminated beam. This system was subjected to mechanical impact, a nonlinear Winkler–Pasternak foundation, and an electromagnetic actuator considering the thickness effect. The expressions of the nonlinear differential equations were obtained for the pinned–pinned boundary conditions with the help of the Galerkin–Bubnov procedure and Hamiltonian approach. The nonlinear differential equations were studied using an original, explicit, and very efficient technique, namely the optimal auxiliary functions method (OAFM). It should be emphasized that our procedure assures a rapid convergence of the approximate analytical solutions after only one iteration, without the presence of a small parameter in the governing equations or boundary conditions. Detailed results are presented on the effects of some parameters, among them being analyzed were the damping, frequency, electromagnetic, and nonlinear elastic foundation coefficients. The local stability of the equilibrium points was performed by introducing two variable expansion method, the homotopy perturbation method, and then applying the Routh–Hurwitz criteria and eigenvalues of the Jacobian matrix. Full article

This paper presents numerical investigations into the free vibration properties of a sandwich composite plate with two fiber-reinforced plastic (FRP) face sheets and a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core made of functionally graded carbon nanotube-reinforced composite resting on Winkler/Pasternak elastic foundation. The material properties of the FG-CNTRC core are gradient change along the thickness direction with four distinct carbon nanotubes reinforcement distribution patterns. The Hamilton energy concept is used to develop the equations of motion, which are based on the high-order shear deformation theory (HSDT). The Navier method is then used to obtain the free vibration solutions. By contrasting the acquired results with those using finite elements and with the previous literature, the accuracy of the present approach is confirmed. Moreover, the effects of the modulus of elasticity, the carbon nanotube (CNT) volume fractions, the CNT distribution patterns, the gradient index p, the geometric parameters and the dimensionless natural frequencies’ elastic basis characteristics are examined. The results show that the FG-CNTRC sandwich composite plate has higher dimensionless frequencies than the functionally graded material (FGM) plate or sandwich plate. And the volume fraction of carbon nanotubes and other geometric factors significantly affect the dimensionless frequency of the sandwich composite plate. Full article

This study proposes an investigation into the nonlinear vibration of a simply supported, flexible, uniform microbeam associated with its curvature considering the mechanical impact, the electromagnetic actuation, the nonlinear Winkler–Pasternak foundation, and the longitudinal magnetic field. The governing differential equations and the boundary conditions are modeled within the framework of a Euler–Bernoulli beam considering an element of the length of the beam at rest and using the second-order approximation of the deflected beam and the Galerkin–Bubnov procedure. In this work, we present a novel characterization of the microbeam and a novel method to solve the nonlinear vibration of the microactuator. The resulting equation of this complex problem is studied using the Optimal Homotopy Asymptotic Method, employing some auxiliary functions derived from the terms that appear in the equation of motion. An explicit closed-form analytical solution is proposed, proving that our procedure is a powerful tool for solving a nonlinear problem without the presence of small or large parameters. The presence of some convergence-control parameters assures the rapid convergence of the solutions. These parameters are evaluated using some rigorous mathematical procedures. The present approach is very accurate and easy to implement, even for complicated nonlinear problems. The local stability near the primary resonance is studied. Full article

Based on the three-dimensional elasticity theory, the free vibration of functionally graded porous (FGP) sandwich rectangular plates is studied, and a unified solution for free vibration of the plates is proposed in this study. The arbitrary boundary conditions of FGP sandwich rectangular plates are simulated by using the Rayleigh–Ritz method combined with artificial spring theory. The calculation performances of the unified solution for FGP sandwich rectangular plates such as convergence speed and computational efficiency are compared extensively under different displacement functions. In addition, three kinds of elastic foundation (Winkler/Pasternak/Kerr foundations) and three porosity distributions are considered. Some benchmark results and accurate values for the free vibration of FGP sandwich rectangular plates resting on elastic foundations are given. Finally, the effects of diverse structural parameters, elastic foundations with different parameters, and boundary conditions on the free vibration of the FGP sandwich rectangular plates are analyzed. Full article

This research represents the first theoretical investigation about the vibration behavior of circular organic solar cells. Therefore, the vibration response of asymmetric circular organic solar cells that represent a perfect renewable energy source is demonstrated. For this purpose, the differential quadrature method (DQM) is employed. The organic solar cell is modeled as a laminated plate consisting of five layers of Al, P3HT:PCBM, PEDOT:PSS, ITO, and Glass. This cell is rested on a Winkler–Pasternak elastic foundation and assumed to be exposed to various types of hygrothermal loadings. There are three different kinds of temperature and moisture variations that are taken into account: uniform, linear, and nonlinear distribution throughout the cell’s thickness. The displacement field is presented based on a new inverse hyperbolic shear deformation theory considering only two unknowns. The motion equations including hygrothermal effect and plate–foundation interaction are established within the framework of Hamilton’s principle. The DQM is utilized to solve these equations. In order to ensure the accuracy of the proposed theory, the present results are compared with those reported by other higher-order theories. A comprehensive parametric illustration is conducted on the impacts of different parameters involving the geometrical configuration, elastic foundation parameters, temperature, and moisture concentration on the deduced eigenfrequency of the circular organic solar cells. Full article

Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz’s method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied. Full article

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated. Full article

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa). Full article

Given their considerable specific surface area and amorphous characteristics, nanoparticles exhibit excellent pozzolanic activity, and when undergoing a reaction with calcium hydroxide, this leads to the generation of a denser matrix by promoting the formation of a greater amount of C-S-H gel, thereby enhancing the strength and durability of the concrete and fortifying the overall structure. Indeed, the present study investigates a comparative study of the buckling and free vibration analyses of concrete beams reinforced with various types of nanoparticles. For its simplicity and accuracy, a higher-order shear deformation theory will be used to analytically model the reinforced concrete beam. Furthermore, the powerful Eshelby’s model is used to derive the equivalent nanocomposite properties. The soil medium is simulated with Pasternak elastic foundation, including a shear layer, and Winkler’s spring, interlinked with a Kerr foundation. The motion equations are derived using Hamilton’s principle. Moreover, based on Navier’s analytical methods, the closed-form solutions of simply supported beams have been obtained. Different parameters, such as type and volume percent of nanoparticles, geometrical parameters, choice of theory and soil medium, on the buckling and dynamic behavior of the beam, are exercised and shown. The major findings of this work indicate that the use of nanoparticles in concretes increases better mechanical resistance and amplifies the natural frequencies. In addition, the elastic foundation has a significant impact on the buckling and vibration performances of concrete beams. Full article