Differential Quadrature Method Research Articles

Effect of free vibration on mechanical characteristics of Bi-directional functionally graded beams under hygrothermal effect

ABSTRACT The effect of free vibration on mechanical characteristics of bi-directional functionally graded material (BDFGM) beams under the influence of hygrothermal effects involves the application of the harmonic differential quadrature method (HDQM). The material parameters are supposed to vary in both axial and thickness directions. The temperature distribution is employed along the length direction of the BDFGM beams while maintaining a constant moisture concentration. An exponential law with axial thermal variation under moisture effect is investigated to analyse vibration behaviour in the BDFGM beams using the HDQM. Hamilton’s principle is used to obtain governing equations. Three kinds of temperature distribution profile were considered i.e. uniform (UTR), linear (LTR) and nonlinear temperature rise (NLTR). Free vibration analysis is performed using the HDQM. The precision of the HDQM is validated by comparing it with prior studies. Various configurations of boundary conditions, including supported, clamped, and free, are considered. Parametric results are investigated to explore the effect of the exponent parameters, temperature profile and constant humidity content. Results indicate that temperature, constant moisture content, material properties, and different end conditions significantly influence natural frequency. The outcomes can be applied to designing two or more directional FGM structures for different environmental conditions.

Enhancing the stability and efficiency of the advanced concrete structures surrounded by an auxetic foundation

This study delves into the vibrational characteristics of concrete annular plates reinforced with graphene oxide powders (GOP), underpinned by an innovative auxetic-elastic foundation. By applying Hamilton’s principle, the governing dynamic equations are meticulously derived, incorporating the effects of internal forces and GOP geometrical properties. The Differential Quadrature Method (DQM) is utilized to solve these equations, providing a robust and efficient numerical approach for analyzing the system’s complex dynamic behavior. GOP reinforcement significantly enhances the mechanical properties of the concrete plates, yielding improved stiffness and superior damping capabilities. The auxetic-elastic foundation, known for its unique property of becoming thicker perpendicular to the applied force, plays a pivotal role in altering the vibrational response of the system. The study meticulously examines the influence of this foundation on the system’s vibrational behavior, highlighting its potential to enhance performance under varying operational conditions. A comprehensive analysis is conducted to understand the effects of different boundary conditions on the vibrational characteristics of the reinforced concrete annular plates. The results underscore the importance of integrating advanced materials like GOP and innovative foundation designs in optimizing structural components. This study provides valuable insights into the design and application of reinforced concrete structures, emphasizing the effectiveness of GOP reinforcement and auxetic-elastic foundations in enhancing vibrational performance. These findings offer a significant contribution to the field of structural engineering, paving the way for future innovations in the design of resilient and efficient structural systems.

On the measurement of the aerodynamic features of composite structures under subsonic airflow using mathematical modeling and machine learning method

This study investigates the aerodynamic features of functionally graded triply periodic minimal surface (FG-TPMS) structures under subsonic airflow using a combination of mathematical modeling and machine learning methods. The modified supersonic piston theory of Krumhaar is used to quantify the aerodynamic stiffness and damping matrices. Using characteristics and six distinct patterns of volume distribution of FG-TPMS materials in terms of their mechanical qualities, Hamilton’s principle, and parabolic shear deformation theory are used to extract the governing equations of the presented composite structure. After that, the differential quadrature method at the discrete places and Lagrange interpolated polynomials and Kronecker product are used to solve the governing equations in the numerical domain. This study highlights the potential of integrating advanced computational techniques with machine learning to address complex engineering problems. The findings provide valuable insights into the aerodynamic optimization of sandwich structures, paving the way for their more efficient and effective design in aerospace engineering. As an important outcome for related industries, the displacement field along with the Z direction increases by increasing the angle of airflow by approximately 9 % while this value for the X, and Y directions are 7 %, and 6 %, respectively. The proposed methodology offers a powerful tool for engineers and researchers seeking to optimize the performance of advanced composite materials under subsonic airflow conditions.

Transient Dynamic Response of Generally Shaped Arches under Interval Uncertainties

This paper endeavors to investigate the characteristics of the transient dynamic response of a generally shaped arch when influenced by uncertain parameters while being subjected to specific external excitation. The equations of motion of the generally shaped arches are derived by the differential quadrature (DQ) method, and the deterministic dynamic responses are calculated using the Newmark-β method. By employing the Chebyshev inclusive function, an interval method based on a non-intrusive polynomial surrogate model is developed, and the uncertain dynamic responses are reckoned by enabling numerical simulations. The results of the proposed interval method are compared with those obtained from the scanning method for validation. The effects of various shapes and rise span ratios on the dynamic responses are investigated through a parametric study. The results suggest that the degree of fluctuation in the uncertain dynamic behavior is influenced by the type of parameter. Additionally, the responses of each shaped arch decrease with the increase in the rise span ratios, and with the same rise span ratio, the deterministic responses and corresponding uncertain responses are also affected by the shape of the arch, and they are considered to be at a minimum when the arch shape is parabolic. This study will enhance understanding of the dynamic properties of arches with uncertainties and provide some basis for the assessment and health monitoring of arch structures.

Frequency Responses of a Graphene Oxide Reinforced Concrete Structure

This paper presents a comprehensive investigation on the vibrations of reinforced concrete structure by graphene oxide powders (GOPs) using a polynomial displacement field and the Generalized Differential Quadrature Method (GDQM). The study focuses on analyzing the dynamic behavior of the structure and assessing the effects of three different distribution patterns of GOPs on its vibrations. To accurately model the deformation of the pressure vessel, a polynomial displacement field is employed, taking into account the complex geometrical and material properties of the structure. The results highlight the significant influence of the distribution pattern of GOPs on the natural frequencies of the spherical concrete pressure vessel. The analysis reveals that variations in the weight fraction and arrangement of GOPs have a direct impact on the stiffness and dynamic characteristics of the structure. Specifically, increasing the weight fraction of GOPs generally leads to higher natural frequencies, indicating enhanced structural rigidity. Moreover, the polynomial displacement field and GDQM demonstrate their effectiveness in accurately predicting the vibrations of the reinforced pressure vessel. The combination of these numerical techniques enables efficient and reliable analysis of the dynamic response, allowing for optimization of the design and performance of spherical concrete pressure vessels.

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams.

This paper examines the accuracy and effectiveness of various beam theories in predicting the critical buckling loads and fundamental frequencies of functionally graded porous (FGP) beams whose material properties change continuously across the thickness. The beam theories considered are classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and the broken-line hypothesis-based shear deformation beam theory (BSDBT). Governing equations for those beam theories are formulated by using the Hamilton's principle and are then solved by means of the generalised differential quadrature method. Finite element simulation solutions are provided as reference results to assess the predictions of those beam theories. Comprehensive numerical results are presented to evaluate the influences of the porosity distribution and coefficient, slenderness ratio, and boundary condition on the difference between theoretical predictions and simulation results. It is found that the differences significantly increase as the porosity coefficient rises, and this effect becomes more noticeable for the rigid beam with a smaller slenderness ratio. Nonetheless, the results produced by the BSDBT are always the closest to simulation ones. The findings in this paper will contribute to the establishment of more refined theories for the mechanical analysis of FGP structures.

On nonlinear dynamic analysis of hyperbolic tangent FG-GPL-reinforced shallow spherical cap under supersonic flow

Shallow spherical caps are commonly encountered in the design of spacecraft components, such as fuel tanks or structural elements. Understanding the dynamic behavior can contribute to the identification of potential failure modes. The modulus of elasticity of the augmented composite upper layer is obtained using the model Halpin–Tsai, and this layer is augmented with graphene platelets (GPLs). Also, GPLs’ volume fraction from the top layer is considered for different cases of GPLs. The bottom layer is assumed functionally graded (FG) with two kinds, type A: FG layers and homogeneous core, and type B: FG core and homogeneous layers. To approximate the displacement field, the new hyperbolic tangent shear deformation theory (HTSDT) has been used which is no need for shear modification coefficients. By adopting Hamilton’s principle, the equations of motion are obtained and are solved by the discrete differential quadrature method (DQM) as well as Newmark's time marching scheme by implementing the famous Newton–Raphson iterative technique. When the FG index is infinite, it is observed that the point of maximum displacement decreases by 12.5%, and critical load increases by about 23% compared to the case where it is zero.

Investigation of free vibration of piezoelectric actuator reinforced with functionally graded Boron Nitride nanotube using timoshenko beam model and differential quadrature method

The vibrational characteristics and natural frequencies of Boron Nitride Nanotube (BNNT)-reinforced piezoelectric beams, based on the Timoshenko model, have been studied for the first time in this paper. Four types of functionally graded (FG) distribution patterns of BNNT reinforcements were considered for the FG-BNNT-reinforced piezoelectric beam. It is assumed that the material properties of FG-BNNT-reinforced beams are estimated using the Representative Volume Element (RVE) micromechanical model. The FG-BNNT-reinforced piezoelectric beam with different Boundary Conditions (BCs) is exposed to electrical voltage. Coupled partial differential equations of the system have been derived with the aid of Hamilton's principle and then converted into corresponding dimensionless equations using dimensionless variables. The differential quadrature method (DQM) has been used to obtain the natural frequencies and shape modes. According to the convergence analysis, twelve grid points were enough to obtain an accurate result for the analysis. To confirm the accuracy and reliability of the present study, the obtained results were validated with frequencies from the literature. Also, the effects of BNNT volume fraction and its reinforcement pattern, as well as applied voltage, on dimensionless natural frequencies have been investigated for different BCs. The results show that applying electric loadings can be utilized as well as controlling parameters to modify the resonance frequency of the composite actuator. More specifically, it is shown that applying voltages to a FG-BNNT-reinforced piezoelectric beam changes the natural frequency for each reinforcement pattern, BCs, and mode numbers. Such ability could be employed when designing reinforced piezoelectric composites for vibrational applications.

An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations

This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.

Magneto Axisymmetric Vibration of FG-GPLs Reinforced Annular Sandwich Plates with an FG Porous Core Using DQM and a New Shear Deformation Theory

Based on the differential quadrature procedure (DQP), the vibrational response of functionally graded (FG) sandwich annular plates enhanced with graphene platelets (GPLs) and with an FG porous core is illustrated in this paper. The current annular plate is assumed to deform axisymmetrically and expose to a radial magnetic field. The Lorentz magnetic body force is deduced via Maxwell’s relations. The effective physical properties of the upper and lower layers of the sandwich plate are obtained by employing the Halpin–Tsai model. Our technique depends on a new four-unknown shear deformation theory to depict the displacements. In addition, the motion equations are established via Hamilton’s principle. The motion equations are solved by employing the DQP. In order to study the convergence of the DQ method, the minimum number of grid points needed for a converged solution is ascertained. In addition, the current theory’s outcomes are compared with those of previous higher-order theories. The effects of the porosity distribution type, porosity factor, GPLs distribution pattern, GPLs weight fraction, inner-to-outer radius ratio, outer radius-to-thickness ratio, magnetic field parameters, core thickness, and elastic substrate parameters on the nondimensional vibration frequencies are discussed.

3D nonlocal solution with layerwise approach and DQM for multilayered functionally graded shallow nanoshells

A nonlocal layerwise 3D bending solution for multilayered functionally graded shallow nanoshells is presented. Eringen’s nonlocal elasticity theory is used for approaching the small length scale effect. Curvilinear orthogonal coordinate system is employed for writing the constitutive and equilibrium equations. The middle-plane of displacements are assumed as a Navier solution which are only valid for simply supported spherical, cylindrical panels and rectangular plates. The thickness domain is discretized by the well-known Chebyshev–Gauss–Lobatto, and its derivatives are calculated numerically by the Differential Quadrature method (DQM). Lagrange interpolation polynomials are utilized as the basis functions for DQM. The traction conditions at the top and the bottom of the shell panel and continuity conditions of the interlaminar adjacent layers for transverse stresses are considered, so this method presents layerwise capabilities. The results are compared with higher order theories proposed in the literature, and due to the 3D nature of the presented formulation, benchmark solutions are provided.

Introducing hybrid deep neural networks to predict the transient vibrations of the sandwich systems reinforced by multi-phase nanocomposites

In advanced materials engineering, understanding transient vibrations in sandwich annular plates reinforced by multi-phase nanocomposites is essential. To adequately represent the geometry of these systems, a generic cylindrical coordination system is often used in modeling. The heterogeneous laminated plate is homogenized into an analogous uniform medium to facilitate analysis. In order to represent precise stress and strain distributions, taking into account rotating inertia and transverse shear deformation, a higher-order theory based on a hyperbolic function is used. The governing equations of motion are derived using Hamilton’s principle in polar coordinates, guaranteeing compliance with energy laws. The 2D generalized differential quadrature (GDQ) method effectively discretizes the governing equations for numerical solutions while retaining a high degree of accuracy in recording transient reactions. Furthermore, by solving the differential equations in the frequency domain, the Laplace transform approach sheds light on the dynamic behavior of the system. Using the deep neural networks (DNNs)-fuzzy approach improves the prediction of these composite structures’ vibrational activity. This hybrid approach uses fuzzy logic to control uncertainty and DNN for pattern identification. The DNN-fuzzy approach reliably predicts the intricate vibrational properties of multi-phase sandwich systems reinforced with nanocomposite using datasets from mathematical modeling. This combined use of mathematical techniques, theoretical models, and predictive algorithms improves our knowledge of complex composite structures’ transient vibrations and makes them more useful in a range of engineering domains.

Nonlinear vibrations and critical angular velocity information of the high-speed rotating eccentric disk made of Gori-metamaterials: Introducing data-driven solution for solving the nonlinear problems

Eccentric discs in rotational motion are commonly utilized in various technical fields, including gas turbine engines, flywheels, gears, and brakes. So, improving its critical angular velocity and frequency characteristics is a challenging issue for engineers. So, in this work for the first time, nonlinear vibrations and buckling analysis of the high-speed rotating eccentric disks using mathematical simulation and data-driven solutions are presented. One of the suggestions for improving its mechanical properties is considering metamaterials in the construction of the eccentric disk. Metamaterial is a novel synthetic material that has distinctive physical and mechanical capabilities that are unattainable in natural materials due to its well-designed structure. The properties of the eccentric disks are controlled by the amount of graphene and the degree of folding of graphene origami (GOri) across the thickness of the eccentric disks. These properties, such as Poisson's ratio, vary depending on the position and can be estimated using micromechanical models assisted by genetic programming (GP). Using von-Karman nonlinearity, transformed differential quadrature method (TDQM), and Newton's method the nonlinear governing equations are obtained and solved, respectively. The results show that when the radius ratio of the rotating eccentric disk increases by 8%, the critical point is decreased from 420 HZ to 370 HZ, a reduction of around 12 %. Another suggestion for improving its mechanical properties is controlling the geometrics and physics of the presented structure according to the designer's purposes.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction:: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method:: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result:: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion:: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

Micromechanical analysis of fiber-reinforced ceramic matrix composites by a geometrically nonlinear hierarchical quadrature element model

The stress fields of fiber-reinforced ceramic matrix composites (CMCs) under longitudinal tension are predicted using a non-linear hierarchical quadrature element method (HQEM) in this work. The HQEM combines the differential quadrature method (DQM) with the hierarchical finite element method (HFEM), which is a typical p-version finite element method (FEM). High-order interpolation functions of the HQEM allow accurate evaluation of stress distribution. Fiber, matrix, and interfaces of the CMCs are modeled as linear elastic materials that may have large displacements. The nonlinear HQEM estimation of CMCs stress distributions are validated by comparing with analytical results obtained using classical BHE shear-lag model. Stress distribution in three characteristic stages in the damage evolution process, namely, interface perfectly bonded, interface debonding, and fiber failure are investigated. It is shown that when fiber failure happens, geometric non-linearity must be considered to avoid excessive stress concentration. Furthermore, the stress distribution within three-dimensional cylindrical CMCs cell is studied, which sheds light on the future exploration of three-dimensional CMCs analyses.

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Free vibration analysis of a sandwich conical shell with a magnetorheological elastomer core, enriched with carbone nanotubes and functionally graded porous face layers

In this paper, the free vibration behaviors of a sandwich conical shell with a magnetorheological elastomer (MRE) core enriched with carbon nanotubes (CNTs) and two functionally graded (FG) porous face layers are investigated. The mathematical modeling of the shell is performed via the first-order shear deformation theory (FSDT) incorporating the continuity conditions between the face layers and the core. The porosity parameters are adjusted to provide the same mass for all porosity dispersion patterns. The governing equations and boundary conditions are derived via Hamilton’s principle and are solved through a semi-analytical solution to attain the natural frequencies and the loss factors for various boundary conditions. First, appropriate triangular functions are utilized to provide an exact solution in the circumferential direction. Then, a numerical solution is presented in the meridional direction utilizing the differential quadrature method (DQM). The influences of various factors on the natural frequencies and loss factors are investigated including intensity of the applied magnetic field, mass fraction of the CNTs subjoined to the core, thickness of the CNT-reinforced MRE core, thickness of the FG porous face layers, porosity parameter and dispersion pattern of the pores in the FG porous face layers, and the boundary conditions. Numerical results show that although subjoining CNTs to the MRE core and applying magnetic field have weak effects on the natural frequencies of the shell, increases in the mass fraction of the CNTs and intensity of the applied magnetic field result in remarkable increases in the loss factors, and provide stronger vibration suppression.

Mathematical Models of Wound Healing and Tissue Regeneration: Insights with Numerical Solution

Understanding and forecasting the dynamics of wound healing processes heavily relies on mathematical modeling. Researchers can understand the complicated connections between cells, tissues, and biochemical components involved in wound healing by using mathematical equations and computational simulations. One such mathematical model that has seeks the interest of researchers in exploring wound healing phenomenon is Fisher’s equation. In the present work, the equation is solved using differential quadrature method with trigonometric tension B-spline (TTBs) basis function. The obtained results are compared with the results obtained from earlier studies and the results are presented in form of tables and figures. The implementation of the particle swarm optimization algorithm has removes the dilemma of unknown parameter involvement in the TTBs and hence resulted in a novel technique to find the numerical results.

Analysis of natural vibration of truncated conical shells partially filled with fluid

AbstractIn this paper, we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid. The sloshing effect on the free surface of the fluid is neglected. The dynamic behavior of the elastic structure is investigated based on the classical shell theory, the constitutive relations of which represent a system of ordinary differential equations written for new unknowns. Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates. Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov's orthogonal sweep method. Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method. The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions. The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered shells) have been studied comprehensively. For conical straight and inverted shells, a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.