Generalized Differential Quadrature Method Research Articles

Non-linear free and forced vibration of bi-directional functionally graded truncated conical tube based on the nonlocal gradient strain theory

This paper deals with the free and forced, linear and nonlinear vibration of functionally graded (along with thickness) nanotube, considering the non-uniform cross-section that made a two-dimensional functionally graded (2D-FG) structure. The bi-dimensional nanostructure-dependent governing equation is modeled based on the classic beam theory (Euler-Bernoulli), and they are derived by Hamilton’s principle based on nonlocal gradient strain theory considering the von-Kármán nonlinear strain and the external harmonic load. To solve the general equations and calculate the results, the generalized differential quadrature method (GDQM) was used coupled with the numerical iteration method for the nonlinear response. The results are discussed for the different simply-supported and clamped boundary conditions. The 2D-FG nanotube’s behavior is analyzed for different nonlinear amplitudes, nonlocal strain gradient parameters and nonlocal parameters, different rates of cross-sections, and different material distributions.

Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution

The linear and nonlinear forced vibration response of axially functionally graded (AFG) cylindrical truncated conical and imperfect microbeam subjected to the dynamic harmonically load carried out in the presented research. Based on a couple of modified couple stress theory, the Euler-Bernoulli beam theory and von-Kármán theory, the linear and nonlinear governing equations and related boundary conditions for dynamic response of micro-size tubes are derived employing the Hamilton principle. We considered the uniform and nonuniform functions for the cross-section, in which the convex, linear and exponential functions are the nonuniform sections, and the porosity is regarded as an imperfection. The generalized differential quadrature method (GDQM) is used to prepare the initial conditions for homotopy perturbation (HP) techniques as the semi-analytical approach to calculate the linear and nonlinear results of dynamic responses. The obtained linear and nonlinear results of the free and forced vibration response show the negative and positive effects of some parameters such as the porosity parameter, the nonlinear amplitude, the small-scale parameter, AFG parameter, and different cross-section impact on the dynamic deflection and natural frequency of micro-scale tube and beams with both clamped and simply-supported boundary conditions.

On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Due to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

Free vibration and dynamic transient response of functionally graded composite beams reinforced with graphene nanoplatelets (GPLs) resting on elastic foundation in thermal environment

In this article, free vibration and dynamic transient response of a multilayer polymer nanocomposite beam resting on elastic foundation reinforced by graphene platelets (GPLs) non-uniformly distributed through the thickness direction in thermal environment is investigated. Theoretical formulations are derived based on Hamilton’s principle, Timoshenko beam theory relationships. The effective Young’s modulus of the GPL/polymer composite is estimated by Halpin–Tsai micromechanics model to account for the effects of GPL geometry and dimensions. The vibration frequencies of the beam are obtained numerically by generalized differential quadrature method (GDQM). The influences of the distribution pattern, weight fraction, the number of layers, thermal environment, slenderness ratio of the beam, boundary conditions, and types of elastic foundations on the free vibration behavior and transient response are presented. The results show that adding a very small amount of GPLs into polymer matrix as reinforcements together with the elastic foundation, significantly increases the natural frequencies of the beam. Placing more GPLs near the top and bottom surfaces of the beam are the most effective ways to strengthen the beam stiffness and increase the natural frequencies. Besides, with the increasing of foundation’s stiffness, the displacement amplitude of the functionally graded composite beam reinforced by GPLs subjected to impulsive load decreases.

Vibration of FG-CNT and FG-GNP sandwich composite coupled Conical-Cylindrical-Conical shell

In this paper, free vibration analysis of sandwich composite joined conical-cylindrical-conical shells is implemented using First-order Shear Deformation Theory (FSDT). The sandwich structure is made of three layers including two face sheets and one core. These layers are reinforced with functionally graded carbon nanotubes (FG-CNTs) and functionally graded graphene nanoplatelets (FG-GNPs). Eight patterns are considered for distribution of CNTs and GNPs throughout the thickness of layers. To obtain the equivalent mechanical properties of each layer, the well-known rule of mixture is utilized. The coefficients of material matrix is calculated using the Equivalent Single Layer (ESL) theory. Moreover, the Donnell's approach is employed for obtaining the equations of motion associated to the conical and cylindrical shells. The principle of Hamilton is used for achieve the governing system of partial differential equation (PDE) of joined shells. Afterwards, this system of PDE is solved by using an efficient semi-analytical solution method, namely Generalized Differential Quadrature Method (GDQM). Different cases of boundary conditions are investigated in this study. By comparing the obtained results with the reference solutions, the correctness and efficiency of the proposed formulation are proved. Furthermore, several new, complicated and applicable examples are considered to study the effect of different parameters of geometric, material and boundary conditions on the natural frequency of the joined shells as well.

RETRACTED ARTICLE: Training an innovative physics-learned deep neural network by the Adam optimization method to simulate the responses of the hybrid multilayer system

This article presents the high-exactitude analysis of hygro-thermo-mechanical vibration of the three-phase multi-scale hybrid composite angle-ply laminated rectangular plate (MHCALRP) for different couples of boundary conditions, within the context of the refined higher-order shear deformation theory with fifteen variables (RHOSDT15). Defining the system's kinematics according to RHOSDT15 is the essential step toward achieving sufficient accuracy in the elucidation of vibrational behavior of the moderately thick plates. The numerical solution is acquired by implementing the generalized differential quadrature method (GDQM). The accuracy of the employed approach is evaluated by comparing the results with the published studies. As a practical conclusion, it is revealed that the orientation angle of the macrofibers can effectively compensate for the lack of nano-reinforcement and the absence of extensional load to overcome the negative impacts of hygro-thermal environment in an affordable cost. The influence of extensional load, nano-/macro-scale reinforcements on the vibrational response of the MHCALRP is completely elucidated. Additionally, a comparative study is performed based on deep neural network (DNN) and its learning parameters are calculated according to adaptive Adam optimization method. Importance of this comparative study is related to providing a non-model-based system by means of a cutting-edge artificial intelligence method to predict vibrational response of the structure.

On The Free Vibration of Doubly Clamped Single-Walled Coiled Carbon Nanotubes: A Novel Size Dependent Continuum Model

In this paper, the size dependent vibration behavior of doubly clamped single-walled coiled carbon nanotubes (CCNTs) is investigated using nonlocal helical beam model. This model is based on Washizu’s beam theory so that all displacement components of CCNT in the equations of motion are defined at the centroidal principal axis and transverse shear deformations are considered. After deriving the nonlocal free vibration equations, they are solved by the generalized differential quadrature method (GDQM). Then, the natural frequencies and corresponding mode shapes are determined for the clamped-clamped boundary conditions (BCs). After that, a parametric study on the effect of different parameters, including the helix cylinder to the tube diameters ratio , the number of pitches, the helix pitch angle, and the nonlocal parameter on the natural frequencies is conducted. It is worth noting that the results of the proposed method would be useful in the practical applications of CCNTs such as using in nanoelectromechanical systems.

On the dynamics and resonance frequency of an electro-elastic rotary non-classical piezoelectric microdisk

In this research, frequency and resonance area information of a size-dependent electro-elastic rotary non-classical piezoelectric microdisk via modified couple stress theory (MCST) is presented. The computational formulation of the electrically microdisk, non-classical governing equations of electrically annular microdisk are derived by adding the symmetric rotation gradient and higher-order stress tensors to the strain energy. The current non-classical approach is capable for capturing the size-dependency in the electrically annular microdisk by using single material length scale factor. Finally, the non-classical governing equations are solved using generalized differential quadrature method (GDQM). Afterward, a parametric study is carried out to investigate the effects of applied ampere, length scale factor, applied voltage, and radius ratio on the resonance area and frequency responses of the electrically microdisk by considering MCST. The meaningful of resonance area in the current research is that, all designers should have attention to the lower values of natural frequency for obtaining the accurate results. Afterward, a parametric study is done to present the impacts of the radius ratio, length scale parameter, circumferential and radial mode number, geometry of piezoelectric material, applied voltage, and boundary conditions on the frequency responses of the piezoelectric microdisk by considering MCST. The results show that, when the foundation is considered, the effect of the radius ratio on the critical voltage of a piezoelectric microdisk reduces.

Nonlinear mechanics of a slender beam composited by three-directional functionally graded materials

In view of some advanced facilities such as aerospace and marine structures are often suffered by multi-directional loads in engineering, especially for slender structures undergoing large deformation and catastrophic failure, we use the three-directional (3D) functionally graded materials (FGMs) to fabricate the slender beams to resist 3D loads and study the nonlinear mechanics to provide an insight reference for possible applications. Assume the materials properties as continuously varying with an arbitrary function in three directions, the governing equation is derived based on Hamilton’s principle and geometric nonlinearity. The generalized differential quadrature method (GDQM) combined with an iterative technique is employed to calculate the nonlinear critical buckling load for predicting the dynamics. The Galerkin technique with the homotopy analysis method is utilized for determining the closed-form solutions to the nonlinear frequencies. Some numerical examples are presented to explore the effects of the physical parameters such as 3D FG indexes on the nonlinear mechanical behaviors in detail. It is found that the three directional FG indexes can tune the mechanical performance of the beam, such as improving on the load-bearing capacity and enhancing in the flexibility of dynamic design, which is tremendously different from nonlinear behaviors of 2D and 1D FGMs beams.

On instabilities and thermal post-buckling of the electrically annular system coupled with shape-memory alloy fibers

In this article, a mathematical derivation is made to develop a nonlinear static model for the thermal post-buckling characteristics of an annular system reinforced with graphene nanoplatelets (GPLs) and coupled with the piezoelectric actuator (PA) and shape memory alloy (SMA) under lateral loading. The matrix material is reinforced with GPLs at the nanoscale. The displacement-strain of postbuckling analysis of the functionally graded-graphene nanoplatelets reinforced composite (FG-GPLRC) annular system via higher-order shear deformation theory (HSDT) and using Von Karman nonlinearity is obtained. The governing equations and boundary conditions are formulated via the minimum total potential energy principle, and they are solved with the generalized differential quadrature method (GDQM). The direct iterative approach is presented for solving the set of equations that include highly nonlinear parameters. Finally, the results show that SMA fiber, radius ratio of outer to the inner, geometrical parameter of GPLs, applied voltage, piezoelectric thickness, and large amplitude play an essential impact on the thermal postbuckling response of the annular sandwich system. Another important consequence is that the highest thermal postbuckling to thermal buckling ratio is for the GPL-O pattern for all ranges of the large amplitude while considering the GPL-X pattern leads to the lowest thermal postbuckling to thermal buckling ratio for the smart annular system. This study’s more general conclusion is that designing the FG-GPLRC annular system should be more attention to the nonlinearity parameter.

Dynamic simulation of the ultra-fast-rotating sandwich cantilever disk via finite element and semi-numerical methods

In the presented research, vibrational, and amplitude behaviors of a sandwich spinning disk made of two laminated layers and graphene nanoplatelets reinforced composite (GPLRC) core has been reported. The Coriolis and centrifugal impacts have been taken into account due to its rotational feature. The stresses and strains have been obtained through the high-order shear deformable theory (HSDT). The structure’s boundary conditions (BCs) are determined using laminated rotating disk’s governing equations employing energy methods and ultimately have been solved via a computational approach called generalized differential quadrature method (GDQM). The rotational disk’s vibrations with different BCs have been explained using the curves drawn by MATLAB programming. Moreover, the hinged BCs have been considered to edges $$\theta = 3\pi /2$$ , and $$\theta = \pi /2$$ . Furthermore, cantilever (clamped–free) BCs, respectively, are taken into account in R = Ri, and R0. In addition to computational approach, a 3-D finite element (FE) simulation has been conducted via ABAQUS software employing the FE package to model the laminated cantilevered disk’s response. The outcomes determined by a FE simulation demonstrate a decent agreement with the semi-numerical approach’s results. Thereby the results reveal, disk’s angle of ply, number of layers, length scale, angular velocity, and nonlocal elements, and geometrical features have a significant influence on the vibration and amplitude characteristics of a sandwich spinning Clamped-Free disk. As a practical outcome in pertained industries, If the structure is manufactured of an even layers’ number, the system’s frequency response would be much better, specifically in a small radius ratio amount.

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

A numerical scheme based on backward differentiation formula (BDF) and generalized differential quadrature method (GDQM) has been developed. The proposed scheme has been employed to investigate three cases of Burgers' equation, one‐dimensional, two‐dimensional, and two‐dimensional coupled models. The results showed effectiveness accuracy in absolute error and error normsL2andL∞compared to other methods. The results were evaluated at different values of Reynold's number,Re, and viscosity,ν.

Computer simulation via a couple of homotopy perturbation methods and the generalized differential quadrature method for nonlinear vibration of functionally graded non-uniform micro-tube

In this paper, to improve the vibrational response of microstructures, the impact of the nonlinear modal analysis of axially functionally graded (AFG) truncated conical micro-scale tube including the thermal loading for the different type of cross sections such as uniform section, linear tapered section, convex section, the exponential section are studied that are applicable for various application, for example, the micro-thermal fins, macro-/micro-fluid-flow diffuser, fluid-flow nozzle, fluid-flow throat, micro-sensor, etc. The nonlinear equations are obtained applying Hamilton’s principles based on the modified couple stress to determine the size effect and Euler–Bernoulli beam theory considering the von-Karman’s nonlinear strain. The material combination varies along the tube’s length, denouncing the AFG tube made by metal and ceramic phases. The nonlinear equations are solved by applying a couple of homotopy perturbation methods (HPM) to calculating the nonlinear results and the generalized differential quadrature method (GDQM) to providing the initial conditions. The linear and nonlinear results presented the effect of various cross sections and other parameters on the micro-tube frequency that are valuable to design and manufacture the micro-electro-mechanical systems (MEMS).

Enhancing vibration performance of a spinning smart nanocomposite reinforced microstructure conveying fluid flow

In this article, critical angular velocity, critical velocity of fluid flow and vibration control analysis of a rotating multi-hybrid nanocomposite reinforced (MHCR) cylindrical microshell covered with a piezoelectric layer as sensor and actuator (PLSA) are presented. The current non-classical model is capable of capturing the size dependency in the microshells using only one material length scale parameter; moreover, the mathematical formulation of microshells based on the classical model can be recovered from the present model by neglecting the material length scale parameter. This structure is under conveying viscous fluid, and the related force is calculated by the modified formulation of Navier–Stokes. In addition, the current structure rotates around its axial direction. The Coriolis and centrifugal effects due to the rotation are considered. For semi-numerical method, the strains and stresses can be determined through 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 output of sensors. The boundary conditions are derived through governing equations of the MHCR cylindrical microshell using energy method known as Hamilton’s principle and finally are solved using generalized differential quadrature method (GDQM). The outcomes show that the angular velocity, velocity of fluid flow, external force, PD controller, external voltage and MHC’s weight fraction have a considerable impact on the amplitude, and vibration behavior of a spinning MHCR cylindrical shell conveying fluid flow.

Nonlinear transient thermoelastic response of FGM plate under sudden cryogenic cooling

Liquified natural gas (LNG) is increasingly used as fuel in marine industry. For transportation and storage purposes it is kept at −163 °C under atmospheric pressure. When LNG gets in contact with a plate, pipe or a tank, it causes very significant thermal load on the structure which has, until that point, been kept at environmental temperature. The current study presents the nonlinear transient thermoelastic response of circular/annular plates which experience sudden cryogenic thermal load. This is the first time that thermoelastic analysis has been presented for any structure under such condition. The material is composed of stainless steel (SUS304) and low-carbon steel (AISI1020) and is functionally graded through the thickness. The novel material configuration proposed in this study has been used for the first time in cryogenic and low-temperature applications. Material properties are evaluated based on available experimental data and assumed to be temperature dependent. The nonlinear governing equations and boundary conditions are obtained with the von Kármán assumption and the first order shear deformation theory. The governing equations are discretized and solved with the Generalized Differential Quadrature Method (GDQM) and Newton-Raphson iterative method. The temporal evolution of the temperature field is obtained by solving Fourier-type heat conduction equation using GDQM and Crank-Nicolson scheme. The results have been validated with axisymmetric FEM models using solid elements. A detailed parametric study has been carried out to investigate the effect of temperature dependency, material distribution, thermal/mechanical boundary conditions, geometrical nonlinearity, and plate size on the thermoelastic response. The results show that transient thermoelastic response of plates under sudden cryogenic cooling can be very different than steady state response. Significant deflections occur in a fraction of a second because of the sudden thermal load and decrease over time. The highest stresses appear predominantly in the initial phase, but can also appear later depending on the plate geometry and boundary conditions. For certain thermal boundary condition, the response of a plate under low-temperature thermal load can qualitatively differ from high-temperature thermal load.

Nonlinear frequency and chaotic motion of the honeycomb higher-order disk with graphene nanoplatelets face sheets subjected to harmonic excitation via two-dimensional analysis

Honeycomb structures are one type of structure that has the geometry of a honeycomb to allow the minimization of the amount of used material to reach minimal material cost and minimal weight. For this issue, in the current analysis, an attempt is made to develop the nonlinear mathematical model for the chaotic and large-amplitude motions of a sandwich disk with graphene nanoplatelet face sheets honeycomb core and subjected to external excitation. Using Hamilton’s principle, higher-order shear deformation theory (HSDT), and the Von-Karman nonlinear theory, the nonlinear governing equation is derived. The generalized differential quadrature method (GDQM) and perturbation approach (PA) are eventually used to develop a precise solution approach. This article’s fundamental and golden results are that for clamped–clamped boundary conditions, the softening of the system in zone 2 is less than zone 1, whereas the instable responses in zone 2 are more than zone 1. Also, for the sandwich disk with clamped–clamped boundary conditions, the system’s dynamic behavior tends to be more harmonious and less chaotic as the fiber’s angle increases. Therefore, increasing the fiber angle reduces the system’s chaos with clamped edges, which is very important for future works.

Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method

The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered.

Dynamic interaction between bi-directional functionally graded materials and magneto-electro-elastic fields: A nano-structure analysis

In the paper, a novel model of bi-directional (2D) functionally graded materials (FGMs) nanobeams resting on the Pasternak foundation under the magneto-electro-elastic (MEE) fields based on the Timoshenko beam theory is presented to investigate the dynamic interaction behavior. Taking the interaction between the 2D FGMs and MEE fields as well as the nonlocal elastic theory into account, the governing equations of the nanobeams resting on the Pasternak foundation are established by made use of the Hamilton’s principle. The generalized differential quadrature method (GDQM) is put forward to discretize the dynamic model of the nanobeams and carried out the calculations of the natural frequencies and associated mode shapes. We can find that the asymmetric modes in the MEE nanobeams are dependent on the 2D FGMs, which is significantly different from the former research. Numerical examples reveal the influences of the foundation coefficients, length-thickness ratio, magneto-electro-mechanical loadings and material distributions on the dynamic characteristics in details. It is notable that the analytical results are beneficial for designing accurately the smart nanostructures constructed from multi-ferroic composites.

Amplitude motion and frequency simulation of a composite viscoelastic microsystem within modified couple stress elasticity

In this research, amplitude motion and frequency simulation of a thick annular microsystem with graphene nanoplatelets (GPL) reinforcement in the framework of the modified couple stress theory (MCST) is undertaken. For obtaining the effective Poisson ratio, and mass density, the role of mixture is employed. As well as this, the Halpin–Tsai micromechanics model is presented for modeling the effective Young module of the current composite microstructure. The mathematical formulations of the current microstructure, size-dependent governing equations and boundary conditions are obtained by considering the MCST’s terms such as higher-order stress tensors, and symmetric rotation gradient into strain energy of the size-dependent GPL reinforced composite (GPLRC) microstructure. Finally, the generalized differential quadrature method (GDQM) is employed to obtain eigenvalue and eigenvectors of the GPLRC microsystem. Afterward, a parametric study is conducted to present the impacts of the radios ratio, length scale parameter, viscoelastic parameter, radial and circumferential mode number, and GPL’s geometry, on the amplitude motion, and frequency characteristics of the GPLRC annular microsystem. The results demonstrate that in the higher value of the radius ratio and time-dependent parameter, we can ignore the influence of length scale parameter on the amplitude and frequency of the annular microsystem. The useful suggestion of this study is that as the thickness of the GPLs increases, the impact of length scale parameter on the frequency of the annular microsystem decreases.

Thermal buckling analysis of FG graphene nanoplatelets reinforced porous nanocomposite MCST-based annular/circular microplates

Thermal buckling analysis of annular/circular microplates, which are made from functionally graded Graphene nanoplatelets (GNPs) reinforced porous nanocomposite is presented in this work. The microstructure is located on Pasternak elastic foundation, and its properties vary through the thickness direction according to the specific functions. Under the Gaussian random field scheme for closed-cell cellular solids and employing Halpin-Tsai and extended rule of mixture micromechanics models, effective properties of the structure are determined. Also, the modified couple stress theory is used to predict the results in the micro dimension. The virtual displacement principle and generalized differential quadrature method (GDQM) are chosen to derive and solve the governing equations, respectively. Previously published work is used to validate the results in a simpler state, and effects of the most important parameters on the critical buckling temperature are investigated. It is found that increasing the GNPs' weight fraction up to one percent leads the results to reduce about 7∼15 percent while increasing the porosity coefficient up to 0.7 improves thermal buckling behavior about 20∼50 percent contrary to expectations. The results of this study can be the benchmark for future studies. Since metal foams are widely used in different areas of technology and industries such as aerospace engineering, reinforcing them with nano-fillers makes them more applicable; therefore, outcomes of this study help to design and create the engineering structures with improved properties.