Differential Quadrature Element Method Research Articles

Active control of thermally induced vibrations of temperature-dependent FGM circular plate with piezoelectric sensor/actuator layers

This study investigates the novel application of active control methods in mitigating thermally induced vibrations of functionally graded material (FGM) circular plates subjected to thermal shock. The top surface of the FGM circular plate is exposed to rapid surface heating, resulting in thermally induced vibrations. To counteract these deflections, two piezoelectric layers, one functioning as a sensor and the other as an actuator, are strategically integrated into the plate opposite sides. This layered configuration effectively distances the direct rapid surface heating from piezoelectric layers. All thermos-electro-mechanical properties of the used materials are considered temperature-dependent. The research employs the Generalized differential quadrature element (GDQE) method and the Crank-Nicolson time marching procedure to solve the nonlinear Fourier-type heat conduction equations for the layered structure. After obtaining the transient temperature profile, the study calculates thermal moments, forces, and pyroelectric terms. These thermally induced effects are then integrated into the motion and electrostatic equations, which inherently exhibit nonlinearity due to the von Kármán large deformations theory. The plate displacements are approximated using the first-order shear deformation theory (FSDT) along the plate thickness, while a quadrature distribution is considered for approximating the electric potential within the piezoelectric layers. The highly coupled electro-mechanical equations are solved using the GDQ method and the Newmark approach, with the Newton-Raphson iterative scheme utilized to address nonlinearity. A significant contribution of this research lies in the exploration of three distinct active control strategies: proportional control, derivative control, and proportional-derivative control. The study thoroughly analyzes the impact of each control type on the plate response, considering various boundary conditions.

Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton’s principle, the nonlinear governing equation of an axially functionally graded (AFG) pipe is established. The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam. Via the direct multi-scale method, the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method (DQEM). The influence of Young’s modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions. In general, the pipe in the supercritical region is more sensitive to the pulsating excitation. The nonlinearity changes from hard to soft, and the non-trivial equilibrium configuration introduces more frequency components to the vibration. Besides, the increasing Young’s modulus gradient improves the critical pulsating flow velocity of the parametric resonance, and further enhances the stability of the system. In addition, when the temperature increases along the axial direction, reducing the gradient parameter can enhance the response asymmetry. This work further complements the theoretical analysis of pipes conveying pulsating fluid.

Nonlinear transient heat transfer analysis of multilayered thermal protection systems by incremental differential quadrature element method

As a first attempt, the incremental differential quadrature element method (IDQEM) is used for the one-dimensional nonlinear transient heat transfer analysis of multilayered thermal protection systems (TPSs) under time-dependent aerodynamic heating. All thermophysical properties are considered to be temperature-dependent (TD). Based on the IDQEM, the multilayered TPS is divided into several spatial sub-domains along the layer interfaces, and the entire heating process is also divided into several temporal sub-domains. For each temporal sub-domain, the governing equations are discretized by the differential quadrature method and then solved by the Newton-Raphson method. The initial condition of each temporal sub-domain is defined by the temperature results at the end of the previous sub-domain. The temperature profile of the TPS during the entire heating process can be obtained by repeating the calculation procedure from the first temporal sub-domain to the last one. Convergence and accuracy of the method are confirmed. Numerical examples are carried out to show the effects of TD thermophysical properties, thermal boundary, and layer thickness on the temperature profile of a three-layer TPS. The results show that the temperature dependency of thermophysical properties is worthy of consideration in predicting the temperature profile of the TPSs.

Free vibration of functionally graded graphene platelets reinforced composite porous L-shaped folded plate

This paper investigates the free vibration of a functionally graded graphene platelets reinforced composite (FG-GPLRC) porous L-shaped folded plate. According to the folded position, the L-shaped folded plate is divided into three rectangular plates. Based on Mindlin-Reissner plate theory and Hamilton’s principle, the governing equations for free vibration of each subelement are deduced. Utilizing the generalized differential quadrature element (GDQE) method, the governing equations for each subelement are discretized and numerically solved to obtain the natural frequency. The study focuses on discussing the influence of several factors, including porosity distribution, porosity coefficient, plate size, crank angle, boundary conditions, GPL weight fraction, and GPL pattern on the natural frequency. The findings reveal that the incorporation of GPL enhances the mechanical performance of the porous folded plate compared to a pure porous plate. Dispersion of more GPL near the upper and lower layers enhances the stiffness of the FG-GPLRC porous folded plate, leading to higher natural frequency. Additionally, variations in the crank angle have a significant impact on the natural frequency.

An exact closed-form explicit solution of free transverse vibration for non-uniform multi-cracked beam

Non-uniformity and damage are the two primary subjects in studying the vibrations of the beam-type elements. An exact closed-form explicit solution for the transverse displacement of a non-uniform multi-cracked beam with any type of boundary conditions is introduced. The generalized functions and the distributional derivative concepts are adopted. Four fundamental functions are introduced. These functions make the boundary conditions' process and compute the frequency equation more convenient. By introducing the non-dimensional parameters, the non-dimensional motion equation of the damaged beam with an arbitrary count of cracks is derived, and its exact closed-form explicit solution is obtained based on the four introduced fundamental functions. The standard method of computing these functions is presented, and the closed-form of these functions is determined for eight cases like uniform and conical beams. The closed-form of the frequency equation and mode shapes of the non-uniform multi-cracked beam are derived for several boundary conditions. The influence of the count of cracks, their location and intensity, and the boundary conditions on the natural frequency and mode shape are assessed by running a numerical study. The first and second frequencies of a conical beam are computed to verify the obtained results by applying this newly presented closed-form solution and the Differential Quadrature Element Method. A good agreement is evident when the obtained results are compared.

Flexural wave manipulation in perforated metamaterial plates with acoustic black holes interconnected by piezoelectric studs

Aiming at achieving lower and broader bandgaps (BGs) and stronger collimation effect, this research proposes a novel perforated metamaterial plate with acoustic black holes (ABHs) interconnected by piezoelectric studs for flexural wave manipulation. Based on the differential quadrature element method together with the first-order shear deformation plate theory, the governing equations of the metamaterial plate are derived. The proposed model is validated by comparing with finite element simulation results. Finally, the wave propagation characteristics including the BGs, equi-frequency contours (EFCs), and group velocities are obtained. The results reveal that the introduction of the stud embedded with piezoelectric patches brings wider complete BGs and stronger collimation effect at lower frequencies. The boundary frequencies and bandwidth of BGs and the preferential direction of wave energy flow are tunable by adjusting geometry parameters of the unit cell. The balance between ABH and piezoelectric effects should be kept to enhance the collimation by generating the caustic lobes. Utilizing the BG and self-collimation mechanism, flexural wave propagation can be manipulated by creating point or line defects in the metamaterial plates. This research provides guidance on the design of metamaterial structure with ABHs and piezoelectric material to achieve broadband vibration absorption and directional wave propagation.

The effect of auxeticity on the vibration of conical sandwich shells with ring support under various boundary conditions

Auxetic materials, materials demonstrating negative Poisson's ratio, have revolutionized the use of materials in industries, as they demonstrate superb acoustic response, fracture resistance, and energy absorption. For the first time, this study embraces the free vibration of conical shells consisting of an auxetic core with and without ring support under various boundary conditions. First, the material characteristics of the auxetic core are calculated by means of a micromechanical approach. Afterwards, the kinematic motion equations of the conical shell are derived utilizing the first-order shear deformation theory. Finally, the governing equations are solved using the powerful generalized differential quadrature element method (GDQEM). The primary goal of this paper is to study the role of implementing an auxetic core as well as ring support in determining the vibrational behavior of the structure. The results of the study showed that the honeycomb interior angle and the presence of ring support can significantly affect the natural frequency of the structure. Lower frequencies can be reached as the interior angle increases. The importance of ring position is found to be highly dependent on the longitudinal mode shapes of vibration. The impact of ring position on natural frequencies is affected by the semi-vertex angle of the cone, and a shift in frequency peaks can be observed by increasing the semi-vertex angle.

Static bifurcation and nonlinear vibration of pipes conveying fluid in thermal environment

In practical engineering, changes of environment temperature can significantly alter natural characteristics of a pipe system. In current work, the exact non-trivial equilibrium configuration of the pipe conveying fluid in a thermal environment is investigated for the first time, including the effect of temperature on natural frequencies and bending vibrations. Firstly, the partial-differential-integral governing equation for bending vibrating pipes subjected to changing temperature is established based on the Euler-Bernoulli beam theory, by means of the generalized Hamilton's principle. Exact solutions for non-trivial equilibrium configurations and critical flow velocities are obtained analytically. Stability of these configurations is discussed and reveals that just the first one is stable. The nonlinear resonance properties of the pipe conveying fluid in a thermal environment are analyzed by the harmonic balance method (HBM) on the basis of Galerkin truncation. The analytical results are verified by the differential quadrature element method (DQEM). It is found that an increasing temperature leads to a larger non-trivial equilibrium configuration and a lower critical flow velocity, with a consequent decrease in the stability of the system. High temperature and high flow velocity reduce the resonant frequency in the subcritical region. In the supercritical region, they increase the resonant frequency. Unlike pipes in the subcritical region, the response in the supercritical region has more resonant peaks and exhibits soft characteristics due to the quadratic nonlinearity. Besides, the asymmetry of the response caused by the zero drift is more pronounced at low temperatures or flow velocities. This study provides theoretical guidance for the vibration design of pipes conveying fluids in a thermal environment.

Generalized differential quadrature element solution, swarm, and GA optimization technique to obtain the optimum frequency of the laminated rotary nanostructure

Optimization of the fiber angle in the laminated composite material to achieve the highest strength or stability has great importance in structural design. In the present study, we aim to utilize genetic algorithm (GA) and particle swarm optimization (PSO) together to find the best angle of composite layers in a multilayer cylindrical shell structure. To this aim, a detailed formulation of the composite cylindrical structure is presented using nonlocal strain gradient theory (NSGT), zigzag theory, and Hamilton's principle for a shear deformable shell displacement field. The modal equations of motion are further solved using the general differential quadrature element method (GDQEM). Having the route of obtaining the resonance frequency of the cylindrical composite structure, GA-PSO is engaged to obtain the optimal angle of the laminate composite to achieve the highest frequency. It is revealed that angle θ = 37.5° provides the highest frequency for a single-layer cylindrical shell. Using this angle, stacking sequences for 3, 5, and 7 layers’ composite are optimized to have the highest frequency. In the end, a detailed parametric study is presented using optimum condition shell composite. The results show that after 30 composite layers, changing the number of layers seems to be not effective on the resonance frequency of the cylinder for both simply supported and clamped boundary conditions.

Time-dependent dynamic response of cylindrical 2D-FG spinning microbeam based on different high-order beam theories applying the GDQEM and Newmark-beta techniques

The nonlocal influence on the cantilever constructions varies significantly. Some researchers feel that nonlocal cantilever structures predict stiffness hardening features; others argue that nonlocal cantilever models cause dynamic softening. In this regard, the vibration behavior of a spinning cylindrical cantilever imperfect functionally graded beam structure is analyzed using several mathematical models based on the nonlocal strain gradient theory. More specifically, higher-order shear deformation theories are investigated in the current study. In addition, the effect of boundary condition modeling is presented. Boundary conditions could be modeled with nonlocal effects or as a classical boundary condition. The governing equations are solved using a technique called generalized differential quadrature element (GDQE) method. The findings of a frequency analysis and a dynamic deflection study, both of which are impacted by a variety of diverse factors, are presented and analyzed. Specifically, the effects of boundary condition model selection are investigated in detail. The findings proved that nonlocality in boundary conditions caused the softer behavior of the cylindrical beam structure by decreasing the natural frequency. On the other hand, using different beam models and boundary conditions could result in a complete difference in obtained results.

Weak-form differential quadrature element analysis of plate on a tensionless and frictional foundation using a higher-order kinematics

Plate-foundation systems, as a structure common in engineering practices, have received extensive scientific research. A novel technical scheme is proposed to improve the computational efficiency in predicting the nonlinear responses of a plate on the foundation. The core work of this technical scheme lies in two aspects. The first aspect is to create a novel plate-foundation system, where a higher order kinematical assumption common in Carrera Unified Formulation is used to model the plate, and a plate-foundation interaction model taking into account the tensionless and frictional characteristics is created. The adjustable kinematical parameters of the higher-order kinematical assumption endow the plate with flexible accuracy. The second aspect is to derive a two-dimensional (2D) differential quadrature (DQ) formula for 2D interpolation based on the one-dimensional (1D) DQ formula, and thus a weak-form differential quadrature element method (WDQEM) for nonlinear quasi-static analysis of the plate-foundation system is proposed by using the principle of virtual work and the 2D interpolation derived. The proposed WDQEM supports flexible nodal configuration in the element, making it degree-of-freedom-flexible and suitable to carry out both p- and h-version strategies in the numerical simulation. Several numerical examples are given to (i) verify the proposed technical scheme and the corresponding computer codes; (ii) to investigate the numerical performance of the WDQEM through comparison between the classical plate model and the 3D solid model, and (iii) to study some physical behaviors related to the tensionless and frictional characteristics of the plate-foundation system. The example results show that (i) the WDQEM converges much faster than the conventional finite element method due to the higher order approximation; (ii) the higher order kinematics, due to its capability of adjusting the accuracy of kinematical assumptions, can effectively reduce the computational scale; (iii) the smoothness problem in the post process of stresses analysis is greatly alleviated, (iv) the tensionless and frictional features of the plate-foundation system should be considered when evaluating both the vertical and horizontal bearing capacities of the plate.

On the wave propagation in a high-speed rotating nanosystems via an optimization strategy and computer simulation

The current article attempted to investigate the vibration of laminated high-speed rotating cylindrical nanoshells using nonlocal stress/strain gradient theory (NSGT). Hybrid optimization, with low computational cost as well as high accuracy, is presented to find a solution for optimization problems. In this regard, genetic algorithm (GA), generalized differential quadrature element method, as well as particle swarm optimization are incorporated to enhance the frequency of the laminated high-speed rotating cylindrical nanoshells via obtaining fiber angle of layers and optimum frequency. The boundary conditions and the equation of motions associated with laminated high-speed rotating cylindrical nanoshells, are extracted by employing Hamilton’s principle in addition to first-order shear deformation theory. The particle swarm optimization method, as an operator of the GA, was utilized to enhance genetic algorithms. Also, the applicability, accuracy, and convergence of the presented method are exhibited. Also, it is demonstrated that the converged restates can be attained provided that more than sixteen iterations are employed.

Modified size-dependent theory for investigation of dynamic stability and critical voltage of piezoelectric curved system

Nowadays, using piezoelectric materials for electrical purposes has attracted much attention among researchers. So, in the current report, dynamic stability and critical voltage analysis of an nanoplate resting on an elastic foundation. The current nanostructure has been made of piezoelectric material. Modified couple stress theory (MCST) and Nonlocal theory (NT) are mixed (modified nonlocal couple stress theory) with one nonlocal and one length scale parameter for modeling the current nanostructure has been presented. Higher-order shear deformation plate theory (HSDPT), and Hamilton’s principle have been used for obtaining the boundary and governing equations of the piezoelectric nanoplate. A two-dimensional generalized differential quadrature element method (2D-GDQEM) has been presented for solving the displacement domains of the current nanostructure. By comparing the outputs of the current report and those of published articles, there is good agreement between the results. Finally, the results show that elastic foundation coefficients (Winkler and Pasternak type), boundary conditions, applied voltage, size-dependent parameters (length scale and nonlocal ones), and geometry properties have important role in the critical voltage and dynamic stability performance of the piezoelectric nanostructure. The outputs of the current report can be used for future researchers of NEMS by incorporating modified nonlocal couple stress theory (MNCST).

A size-dependent differential quadrature element model for vibration analysis of FG CNT reinforced composite microrods based on the higher order Love-Bishop rod model and the nonlocal strain gradient theory

In the current paper, a size-dependent differential quadrature element model using the higher-order Love-Bishop rod theory in conjunction with the nonlocal strain gradient theory for axial vibration analysis in the embedded functionally graded (FG) carbon nanotube (CNT) reinforced composite microrods with varying cross-section is presented for the first time. The CNT agglomeration effect is considered applying the two-parameter Eshelby-Mori-Tanaka homogenization scheme. The differential equation form of nonlocal strain gradient theory is employed to take into account the size-dependent effects. Based on the Hamilton's principle, the governing differential equation of the model is derived and solved using differential quadrature element method (DQEM). The higher-order variationally consistent boundary conditions are obtained using the weighted residual approach. The numerical results indicate that the present model is simple and it can be successfully employed for analysis of the small-sized structures. Moreover, the effects of small-scale parameters, the CNT distribution's patterns, the CNT agglomeration parameters, the non-uniformity of the cross section and the stiffness of the surrounding medium on the frequency are studied and discussed in details.

On the free vibrations of FG-GPLRC folded plates using GDQE procedure

The present study theoretically investigates the performance of the generalized differential quadrature element (GDQE) method to obtain the free vibration response of folded plates. The considered folded plate is assumed as a laminated functionally graded composite augmented with graphene platelets (GPLs). This kind of composite is identified as the FG-GPLRC structure. Each lamina is strengthened with randomly oriented and uniformly distributed GPLs. The GPL weight fraction of each layer changes based on several functionally graded (FG) models. The effective Young modulus of each ply is calculated utilizing a second-order correlation homogenization technique called Halpin–Tsai. Based on the GDQE method, the folded plate is split into two plate elements. The motion equations of each element are derived based on the first-order shear deformation theory and utilizing Hamilton’s principle. Equations of motion for each element are solved by employing the conventional GDQ method. After that, the suitable continuity conditions are applied to the common boundaries of the two elements. In this research, the influences of plate dimensions, crank angle, FG patterns, boundary conditions, and GPL weight fraction on the natural frequencies and corresponding mode shapes are examined.

Out-of-Plane Free Vibration Analysis of Continuous Curved Girders with Combined Linetypes Using Differential Quadrature Element Method

In this study, the out-of-plane free vibration equations of circular curved and clothoid transition curved girders with various boundary conditions were derived. The continuous curved girder with combined linetypes was discretized into separate curved girder elements. Based on the differential quadrature element method (DQEM), the Chebyshev–Gauss–Labatto unequal mesh division was applied to discretize the vibration equations of the continuous curved girder into those of the curved girder elements, along with the boundary conditions, internal geometric compatibility conditions, and force balance conditions. The out-of-plane natural frequency equations were derived considering the boundary conditions by the substitution method. The free vibration was solved for continuous curved girders with combined linetypes of circular and clothoid transition curves. The effectiveness of the DQEM was verified, and the effect of the number of grid points on the accuracy of the solution was evaluated. In addition, the effects of the bending/torsional stiffness ratio, warping coefficient, and boundary conditions on the natural frequencies of continuous curved girders with combined linetypes were investigated. The results showed that the DQEM can be efficiently used to solve the free vibration of multispan continuous curved girders with various linetype combinations. The frequencies of the continuous curved girder with combined linetypes decrease with the weakening of boundary constraints, but increase with increasing warping coefficient and bending/torsional stiffness ratio.

Static, stability and dynamic characteristics of asymmetric bi-directional functionally graded sandwich tapered elastic arches in thermo-mechanical environments

This research paper investigates the in-plane static, stability and dynamic characteristics of tapered Timoshenko bi-directional functionally graded (BDFG) sandwich curved elastic arches (symmetric and asymmetric) in thermo-mechanical environments using the differential quadrature element method (DQEM). Physical properties such as Young's modulus, the mass density, Poisson's ratio and the thermal expansion coefficient are defined to vary both with respect to temperature and position while shear deformation, axial deformation, and rotary inertia effects are incorporated in the analysis. Different types of arches namely circular, parabolic, catenary, elliptic and sinusoidal models are modelled by having non-uniform cross-section and the BDFG properties. General functions are used to vary the physical properties through the thickness and length of the curved. Incorporating the influence of having a thermal environment for simplified models, the current methodology for studying such structures is verified by comparing the results with previously published literature and finite element software simulations. The effect of the temperature rise, physical properties variation and geometrical parameters (such as the ratio of thickness to length, the opening angle and non-uniform cross-section) on the static deformation, buckling load and free vibration characteristics of BDFG arches with metal/ceramics in the inner surface and ceramics/metal in the outer surface of the arch is investigated. In addition, the mechanical behaviour of sandwich arches for various combinations such as homogeneous or FG for outer skins and the inner core is presented under thermo-mechanical loadings.

Tuned deep neural network for computer simulation of chaotic performance of the electromechanical annular system coupled with a piezoelectric patch

In this paper, an attempt is made to extend a nonlinear two-dimensional model for large oscillation and chaotic characteristics of the annular plate coupled with a piezoelectric patch subject to external excitation. For improving the mechanical properties, the core of the electrically structure is reinforced with graphene nanoplatelets. Von-Karman nonlinear theory and Hamilton’s principle are taken into consideration for the exact derivation of the general nonlinear governing equations and boundary conditions of the axisymmetric annular plate coupled with a piezoelectric patch. For developing a precise solution approach, the generalized differential quadrature element method, and Multiple scales technique, are eventually used. The results demonstrate that radius ratio, applied voltage, thickness to radius ratio, and harmonic load have an important role in the nonlinear large oscillation, and chaotic motion of the graphene platelets reinforced composite (GPLRC) annular plate coupled with a piezoelectric patch. The golden and fundamental result of the current research is that, as the thickness of the piezoelectric patch increases, the system's nonlinear frequency decreases, and the area of instability in responses and maximum amplitude or the peak point of the backbone curve of the smart annular plate increases.

A particle swarm optimization and coupled generalized differential quadrature element methods with genetic algorithm for stability analysis of the laminated microsystems

A particle swarm optimization and coupled generalized differential quadrature element methods with genetic algorithm for stability analysis of the laminated microsystems

Application of generalized differential quadrature element method to free vibration of FG-GPLRC T-shaped plates

Application of the generalized differential quadrature element (GDQE) method is illustrated for the free vibration analysis of T-shape plates. Plate is considered as a laminated composite which every layer is reinforced with the graphene platelets (GPLs). Random orientation and uniform distribution are assumed in the combination of reinforcing particles with the matrix. Also, variation of the GPL weight fraction from one layer to another is based on four models of functionally graded (FG). The Halpin-Tsai micromechanical rule is employed to obtain the effective elasticity modulus of media. This rule considers the size and geometry of applied GPLs. First order shear deformation theory with Hamilton’s principle are implemented to derive the equations of motion. It is worth noting that the equations of motion are extracted for each divided element by means of the finite element part of GDQE. Next, the GDQ tool is used to locate the Chebyshev-Gauss-Lobatto grid points on the elements and converting the related motion differential equations into a system of algebraic equations. After satisfying the compatibility conditions between stuck elements and the external edges’ conditions, the governing eigenvalue problem is solved. Some comparison studies are carried out to demonstrate the validity and efficiency of applied method. Henceforth, several novel results are shown to understand the free vibration behaviour of T-shaped FG-GPLRC plates. Furthermore, the effects of GPL weight fraction, functionally graded patterns, boundary conditions, and geometric parameters are presented on the frequency response and corresponding mode shapes.