Differential Quadrature Finite Element Research Articles

Sound Insulation Characteristic Analysis of Corrugated Core Sandwich Panel by Using Differential Quadrature Finite Element Method

This paper proposes a universal vibroacoustic coupling theoretical analysis model for investigating the sound insulation characteristic of corrugated core sandwich panel (CCSP) under different boundary conditions. The vibroacoustic coupling model consists of CCSP (structural field) and two three-dimensional acoustic cavities which simulate the incident sound field and accept the sound field, respectively. Based on first-order shear deformation plate and shell theories and acoustic theory, the dynamic model of CCSP and three-dimensional acoustic cavities are established by employing the differential quadrature finite element method in conjunction with the energy principle. The coupling of CCSP and three-dimensional acoustic cavities is realized by the velocity continuity condition at the acoustic–structure interfaces. The validation of the vibroacoustic coupling model is verified by comparing the results calculated by the established model with the corresponding results calculated by ABAQUS. The influence of model parameters on the vibroacoustic characteristic of vibroacoustic coupling model is investigated systematically for determining the critical structure parameters which have influence on the vibroacoustic behaviors. The effect of critical structure parameters on the sound insulation characteristic of CCSP is analyzed systematically for the optimization of the sound insulation performance of CCSP and provides theoretical basis and technique guidance.

Analysis of Vibration Characteristics for Rotating Braided Fiber-Reinforced Composite Annular Plates with Perforations.

In the current study, a comprehensive numerical model for analyzing the vibrational characteristics of braided fiber-reinforced composite (BFRC) rotating annular plate with perforations under diverse boundary constraints was introduced. This model employs the differential quadrature finite element method (DQFEM), which was developed based on the first-order shear deformation theory (FSDT) and coordinate transformation approach. The BFRC material, specifically a two-dimensional biaxial orthogonal fabric, was utilized to fabricate the annular plate with two distinct types of holes: circular and sector-shaped. The model's convergence, accuracy, numerical stability, and reliability were confirmed through comparative assessments utilizing data from the literature, from ABAQUS software, and from experimental findings. The analysis focuses on studying the influences of structural properties, material parameters, and boundary restraints on the frequencies of vibration for BFRC rotating annular plates with holes. This theoretical model helps provide scientific basis and technical guidance for the stability and lightweight design of rotating annular plates, such as rotor structures in aircraft engines.

Dynamic Modeling and Vibration Characteristic Analysis of Fiber Woven Composite Shaft–Disk Rotor with Weight-Reducing Holes

In order to achieve the goal of a lightweight shaft–disk rotor, this paper applies the fiber woven composite material to the disk structure, and at the same time considers the design of the weight-reducing holes on the porous disk. It introduces the domain decomposition and coordinate mapping technology for this, and then establishes the dynamics model of the fiber woven composite material shaft–disk rotor. The model is based on the differential quadrature finite element method, which is suitable for fiber woven composite rotors with arbitrary complex hole patterns. The validity of the model is verified by comparing the results with the literature, finite element simulation, and experiments, and the mechanism of the influence of the material parameters and pore parameters on the vibration characteristics of the system is investigated, which provides the data support and theoretical basis for the analysis of the dynamics of the fiber woven composite rotor.

Free vibration of functionally graded graphene platelets reinforced porous composite plate with a diamond-shaped hole and two cracks

This paper investigates the free vibration of the functionally graded graphene platelets reinforced porous composite (FG-GPLRPC) plate with a diamond-shaped hole and two cracks. Based on the Mindlin-Reissner plate theory and Hamilton principle, the governing equations and boundary conditions are derived. These equations are discretized using the generalized differential quadrature finite element method (GDQFEM), and the natural frequency is obtained by numerical solutions. The effects of the hole size, crack length and crack angle on the natural frequency are discussed. The results show that the natural frequency decreases with increasing crack angle and crack length.

Vibration frequency and mode localization characteristics of strain gradient variable-thickness microplates

Based on the modified strain gradient theory and Kirchhoff–Love assumptions, a free vibration model is established for variable-thickness microplates with three material length scale parameters, and the corresponding Euler–Lagrange equation is derived. Keeping the microplate volume constant, three kinds of thickness variations along the X-direction are considered: linear, parabolic, and cubic variations. Further, a C2-type four-node 36-degree-of-freedom variable-thickness differential quadrature finite element is developed to solve the resulting variable-coefficient boundary value problem by combining the Gauss–Lobatto and differential quadrature rules. Galerkin approximate solution for fully simply supported microplates is provided as a benchmark for numerical method verification. The convergence and accuracy of the model are verified through several examples. Finally, the vibration frequencies of the microplates under different thickness variation types, taper ratios, thickness ratios, material length scale parameters, and boundary conditions are examined through parametric analysis. Also, the influence of each factor on the mode localization of the microplates is quantitatively characterized by the modal assurance criterion (MAC) for the first time. The numerical results show that the thickness variation has a minor effect on the vibration frequency but a significant influence on the mode shape when the volume of the microplate is constant. The strain gradient, taper ratio, and discontinuous boundary have a remarkable effect on the distribution of higher-order mode shape contour lines and the regions where vibration localization occurs.

Investigation on the vibration mechanisms of a rotating FG-GPLRC shaft-disk-shell combined system

In this paper, a dynamics model of a rotating shaft-disk-shell (SDK) combined system is proposed to solve the vibration characteristics of a flexible rotor system. Unlike many previous researches that only studied shaft-disk systems, this paper simplifies the rotor into a combined structure of shaft, disk and cylindrical shell. The functionally graded graphene platelet (GPL) reinforced composites are innovatively applied in the disk to achieve the system lightweight. Timoshenko beam theory and first-order shear deformation theory are used to model the substructures. And the energy expressions of each substructure are derived based on the multi-body dynamics theory. Finally, the differential quadrature finite element method is used to discretize and reassemble the energy matrix to obtain the overall system’s motion differential equation. The model is proved to have excellent convergence and accuracy. Then the influence mechanism of GPL parameters, system geometry, etc. on the system’s coupled vibration characteristics is further explored, and important conclusions are drawn, like when the shell parameters h/R < 0.08 and l/R < 0.03, the system’s 3rd coupled natural frequency is insensitive to these parameters.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.

Effect of crack presence on the dynamic and buckling responses of bidirectional functionally graded beams based on quasi-3D beam model and differential quadrature finite element method

Effect of crack presence on the dynamic and buckling responses of bidirectional functionally graded beams based on quasi-3D beam model and differential quadrature finite element method

Buckling analysis of nanocomposite beams based on refined beam theory by considering the Agglomeration effect of CNTs

The present work deals with buckling load behavior of nanocomposite beams by considering the agglomeration effect of single-walled carbon nanotubes (CNTs) and different patterns CNTs in polymeric matrix. The material properties of nanocomposite beams are estimated using the Eshelby–Mori– Tanaka approach based on an equivalent fiber. The equations of motion are derived based on refined beam strain gradient theory, employing Hamilton’s principle and using Differential Quadrature Finite Element Method (DQFEM) derived from the differential quadrature method (DQM). The results are compared with analytical results in the literature. It is remarked that mechanical properties and therefore critical buckling loads of nanocomposite beam are seriously affected by CNTs agglomeration.

A study on the crack presence effect on dynamical behaviour of bi-directional compositionally imperfect material graded micro beams

This work uses the Quasi-3D (Q3D) shear and normal deformation beam concept in combination with the length scale capturing theory through the “Modified Couple Stress Theory” (MCST) and the novel computational approach namely, Differential Quadrature Finite Element Method (DQFEM) to investigate free vibratory conducts of an imperfect BD-FG microbeams with a crack. To obtain the relevant governing equations, the Lagrange's rule is utilized. “The differential quadrature” (DQ) and “Gauss-Lobatto quadrature” procedures for weighting coefficient matrices are used to easily construct the mass and stiffness matrices. The Gauss-Lobatto node scheme is used to determine the griding knots. In order to show the efficiency and exactness of the described numerical resolution method, the attained outcomes are authenticated with the similar results found in the literature. The effects of the crack position, crack depth, beam thickness by material length scale aspect, the volume fraction of material imperfection, edge supports state types, length through thickness factor, and the two-way material grading indexes on the nondimensional natural frequencies of 2D porous FG microbeams are inspected. This work will provide a numerical basis for the design of FG microstructures in the field of micromechanics. These findings can be used to construct porous FG microstructures in engineering.

An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.

Free Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Higher Order Refined Composite Beams Using Differential Quadrature Finite Element Method

Present paper deals on the free vibration investigation of carbon nanotube-reinforced composite (CNTs) beams, based on refined third order shear deformation finite element beam theory. The particularity of this model is that, it can capture shear deformation effect without using of any shear correction factor by satisfying shear stress free at free edges. The carbon nanotubes are supposed to be immersed in a polymeric matrix with functionally graded pattern across the thickness direction of the beam, and their material properties are evaluated using the rule of mixture. The differential equations of motion and related boundary conditions are extracted using Lagrange’s principle and solved employing a robust numerical tool called, Differential Quadrature Finite Element Method (DQFEM) for the first time, with high convergence speed, fast calculus performance as well as a good numerical stability. The obtained results have been validated with those available in literature, in order to show the correctness of the present model. Afterwards, a deep parametric study is performed to examine the effects of various geometrical and material parameters on the vibration behavior of FG-CNTs beams.

Accurate mechanical buckling analysis of couple stress-based skew thick microplates

We investigate, for the first time, the elastic buckling of skew thick microplates under combined in-plane shear and compressive loading within the framework of modified couple stress theory. The displacement field of the microplates is described by a two-variable refined shear deformation theory, wherein the transverse deflection is partitioned into bending and shear components, resulting in a simple and universal elastic buckling model. The Euler-Lagrange equation is used to obtain the equations governing the motion of the skew thick microplates. Obtaining an analytical buckling solution for general boundary supported skew microplates is challenging; therefore, a C1-type four-node 32-DOF differential quadrature finite element is developed. The element stiffness and geometric stiffness matrices are derived according to the minimum potential energy principle. Significant parametric studies are conducted with different geometrical dimensions, boundary edges, in-plane loadings, and material length scale parameters. The buckling characteristics of skew microplates are investigated using a combination of unequal biaxial compression loads and in-plane compression and shear loads. Numerical results imply that the buckling modes are influenced by the combination of size effects and skew angle and not by the buckling loads alone.

Dynamic modeling and analysis of thin-webbed spur gear pair

A dynamic model of spur gear pair combined with elastic shaft–disc–ring rotor is proposed to study the effect of web thickness on the dynamic characteristics. The shaft, disc and ring structures are simulated based on Timoshenko straight beam theory, Mindlin plate theory and Timoshenko curved beam theory, respectively. A universal method to derive the kinetic energy of rotating gear rotor is proposed based on multi-body dynamics theory and small deformation assumption. The elastic deformations of thin-web are included in the potential energy of time-varying mesh process. The equations of motion are established based on differential quadrature finite element method (DQFEM). The dynamic model of gear rotor is validated numerically by comparing to finite element (FE) model. The effects of web thickness on dynamic responses are studied by analyzing the dynamic deformations and stresses within web, which reveals that elasticity of web and shaft can reduce the vibration energy transferred to support structures, which contributes to vibration reduction of gear transmission system.

Vibration analysis of rotating functionally graded graphene platelet reinforced composite shaft-disc system under various boundary conditions

A unified dynamic model of a rotating shaft-disc system with multilaminar functional graded graphene platelet reinforced composite (FG-GPLRC) disc is established in order to study the vibration characteristics of the system. Based on first-order shear deformation theory, Timoshenko Beam theory and differential quadrature finite element method, the discrete energy matrices of shaft and disc are obtained. Then the total differential equation of the system is procured by assembling the shaft and disc element matrices. The accuracy and reliability of the model are verified by comparing with the software simulation results. Besides, the influence of boundary conditions, GPL distribution types, GPL weight, disc layers and physical dimensions on the system vibration characteristics is analyzed. It reflects that the GPL distribution types and boundary conditions will make a certain difference to the frequency parameters of shaft-disc system. Furthermore, the vibration characteristics of different GPL distribution types are differently sensitive to the variation in material and geometric parameters.

Application of a Modified Differential Quadrature Finite Element Method to Flexural Vibrations of Composite Laminates with Arbitrary Elastic Boundaries

This paper formulates a modified differential quadrature finite element method (DQFEM) by a combination of the standard DQFEM and the virtual boundary spring technique, which makes it easy to implement arbitrary elastic restraints by assigning reasonable values to the boundary spring stiffnesses. This new formulated method can offer a unified solution for flexural vibrations of composite laminates subjected to general elastic boundary combinations including all the classical cases. The influences of the number of Gauss–Lobatto nodes and the boundary spring stiffnesses on the convergence characteristics of natural frequencies are investigated, and some conclusions are drawn in terms of the minimum number of unilateral nodes required to generate convergent solutions and the optimal values of the boundary spring stiffnesses to simulate classical boundaries. Numerical examples are performed for composite laminates under various classical boundary conditions. Excellent accuracy, numerical stability, and reliability of the present method are demonstrated by comparisons with available exact and numerical solutions in open literatures. Additionally, for elastically constrained composite laminates, which are beyond the scope of most existing approaches, numerous new results obtained by the present method may serve as reference values for other research.

Vibration characteristic analysis of flexible rotor-bearing system subjected to external combined loads

This paper proposed a general five degrees of freedom dynamic model for investigating the vibration characteristics of flexible rotor-bearing system based on nonlinear elastic Hertz contact theory and Timoshenko beam theory by employing differential quadrature finite element and improved Newton-Raphson methods. The flexible rotor-bearing system is made up of stepped shaft and angular contact ball bearing (ACBB). First, the validation of the established model is verified by the existing literatures and finite element software ANSYS 18.2. Then, the dynamic behaviors of ACBB are discussed systematically for the investigations of the vibration behaviors of flexible rotor-bearing system provide reference. Finally, the vibration characteristics of flexible rotor-bearing system subjected to external combined loads are investigated from different perspectives thoroughly. The results show that the preload, external force, radial force and torque have important influence on the vibration behaviors of flexible rotor-bearing system.

Size-Dependent Elastic Buckling of Two-Variable Refined Microplates Embedded in Elastic Medium

This study applies a two-variable refined shear deformation theory (TVRSDT) and a modified couple stress theory (MCST) to develop a size-dependent elastic buckling model for microplates under combined axial compression and in-plane shear and resting on Winkler–Pasternak foundation. Our model incorporates two transverse displacement variables and one material length scale parameter (MLSP); it satisfies the zero-traction boundary conditions on the top and bottom surfaces of microplates, thereby circumventing the use of a shear correction factor. The equations of motion are determined via the Euler–Lagrange equation. A closed-form buckling solution is presented for microplates with four edges simply supported. To handle the nonsimply supported microplates, a thirty-two-degree-of-freedom (32-DOF) four-node differential quadrature (DQ) finite element with [Formula: see text]-continuity is proposed. The corresponding element matrices are obtained using the minimum potential energy principle. Comparison studies are conducted to exhibit the validity of the derived formulations. Finally, the present buckling model is employed for predicting the elastic buckling behaviors of microplates embedded in an elastic medium. The effects of in-plane loading patterns, boundary conditions, aspect ratio, length-to-thickness ratio, MLSP and elastic medium parameters on the buckling load and buckling mode are elucidated.

The coupling free vibration characteristics of a rotating functionally graded shaft-disk system in thermal field

The numerical model of a functionally graded (FG) shaft-disk rotor system with flexible shaft and flexible various-thickness disk as components is established to investigate the coupling free vibration behavior of this rotor system in thermal field. Considering the thermophysical properties of the material, the properties of functionally graded material vary continuously according to the power law with the thickness of the disk. Based on the theory of rotor dynamics, the kinetic energy expression and strain energy expression of rotating shaft-disk system are obtained. In the framework of Ritz method, the global control differential equation of the rotor system assembled by shaft and FG disk is established by using differential quadrature finite element method. On this basis, a set of numerical examples are presented to perform convergence, correctness and effectiveness verification of the current model. Moreover, the influence of temperature difference, rotating speed, geometric and material parameters on the thermoelastic coupling vibration characteristics of light-weight shaft-disk rotor is discussed in detail, which can help to improve the rotor vibration performance in design work and future research.