Harmonic Differential Quadrature Research Articles

Stability analysis for internal flow-induced vertical cantilevered pipe subject to multiple lumped masses

This study investigates the effects of both single and multiple lumped masses attached to a vertical cantilevered pipe conveying internal flow. By introducing delta and Heaviside functions, the governing equation of the vibration of a pipe with lumped masses is newly established based on Hamilton’s principle. The gravitational force of the lumped mass should be considered in the theoretical model but has been essentially ignored in previous research. The harmonic differential quadrature (HDQ) method is applied to solve the governing vibration equation. The newly developed model is validated through a comparison with published data. Numerical results show that the gravity force of the lumped mass plays an important role, and ignoring this force leads to inaccurate results in the investigation of lumped mass effects. The instability of the pipe is analyzed using this new model with various lumped mass weights, positions, and fluid mass ratios. The numerical results showed that lumped masses significantly affect the critical flow velocity, vibration frequency, and modal shapes. The conclusions are expected to supply valuable guidance to reduce pipe vibration using lumped masses in engineering applications.

Analysis of vibration stability of fluid conveying pipe on the two-parameter foundation with elastic support boundary conditions

In this study, the vibration stability of fluid conveying pipe resting on two-parameter foundation is investigated under four different elastic support boundary conditions. The harmonic differential quadrature (HDQ) method is applied to solve the governing vibration equation derived based on Euler–Bernoulli beam theory subject to the elastic foundation and boundary conditions. As a result, a general set of second-order ordinary differential equations emerges, and by appropriately setting the stiffness of the end springs, one can easily study the dynamics of various systems with classical or non-classical boundary conditions. The numerical simulations are conducted to study the pipe instability performance with respect to various boundary conditions, elastic support parameters, elastic foundation parameters and fluid mass ratios. The numerical model is validated by comparison with published data. It is found that the elastic support boundary conditions have significant effects on the stability of pipe resting on elastic foundation. The pipe stability performance is very sensitive to the two elastic foundation parameters. Larger fluid mass ratio enhances the pipe flutter stability performance but has no effects on the divergence.

Application of harmonic differential quadrature (HDQ) method for vibration analysis of pipes conveying fluid

Harmonic differential quadrature (HDQ) method is developed to analyze the one dimensional vibration problem of pipes conveying fluid with various boundary conditions. HDQ uses the trigonometric functions to formulate the harmonic test function and weighting coefficients are calculated explicitly. Unevenly spaced grid point distribution method is applied to discretize pipe domain. The high efficiency and accuracy of HDQ method are demonstrated by comparing with other methods for various internal flow induced pipe vibration problems. Numerical simulations show HDQ method holds better computational efficiency and accuracy than other numerical models. HDQ method is more suitable for rapid calculation of large number of numerical cases than other methods.

Mathematical modelling, numerical analysis and damage of dams subjected to hydrodynamic pressure

Mathematical modelling and numerical analysis of a concrete dam with combined effects of water-sediment-foundation-earthquake are the main and novel contributions of this study. For this purpose, a concrete dam is modeled by exponential shear deformation beam theory (ESDBT). The hydro-dynamic pressure of water due to earthquake load is obtained by Helmholtz equation assuming various boundary conditions of reservoir and wave reflection coefficient. In addition, the forces of foundation and sediment are assumed by vertical and axial springs. On the basis of Hamilton's principle, the final motion equations are derived and solved by Harmonic differential quadrature (HDQ), Integral Quadrature (IQ) and with Newmark methods for calculating the dynamic deflection of the concrete dam. The Drucker–Prager criterion is applied for damage analysis of the concrete dam. The influences of various important parameters such as wave reflection coefficient, sediment type and its depth as well as foundation are investigated on the dynamic deflection, hydrodynamic pressure as a function of water depth, time history of hydrodynamic pressure and Drucker–Prager damage index of the concrete dam. The results indicated that with enhancing the sediment depth, the dynamic deflection and hydrodynamic pressure are decreased. In addition, the hydrodynamic pressure increases the dynamic deflection of the concrete dam about 35% with respect to hydrostatics pressure. With increasing the wave reflection constant, the absolute of Drucker–Prager damage index and the safety of concrete dam are reduced at the crest, middle and heel of the concrete dam.

Free vibration analysis of fiber-metal laminated composite plates using differential, generalized and harmonic quadrature methods: experimental and numerical studies

PurposeThis study aims to demonstrate the numerical application of differential quadrature (DQ) methods and show the experimental application of free vibration analysis of fiber-metal laminated composite (FML) plates with various boundary conditions.Design/methodology/approachThe FMLs are hybrid structures consisting of fiber-reinforced polymer matrix composites such as carbon, glass, aramid and different metal sheets, and are currently widely used in the automobile, aircraft and aerospace industries. Thus, free vibration analysis of these hybrid materials is necessary for the design process. The governing equations of motion are derived based on the classical plate theory. The DQ, generalized DQ (GDQ) and harmonic DQ (HDQ) differential quadrature methods have been used to solve the governing equations of an FML composite plate numerically. The accuracy and convergence of the numerical model have been verified by comparing the results available in the published literature with the results obtained from these methods. Moreover, an experimental procedure has been performed in order to compare the results against those of the numerical methods.FindingsIt is noteworthy that a high degree of similarity and accuracy was observed between the numerical results obtained by the DQ methods and the experimental results. Thus, the present study validates the applicability of the DQ methods for designing the FML composite plates.Originality/valueIn this study, the advantages of the DQ methods have been demonstrated differently from previous studies on the vibration analysis of the FML plates.

A convergence behavior of HDQ and GDQ methods to analyze vibration characteristics of different configurations of FGM beams

In this investigation, a convergence behavior of harmonic differential quadrature (HDQ) and generalized differential quadrature (GDQ) methods is presented. The vibration behavior of different types of functionally graded material (FGM) beams are taken into consideration. The four types of material distributions are considered i.e., isotropic, thickness FGM, axial FGM and bi-direction FGM beam. The clamped-clamped boundary condition is considered in this study. It can be revealed that both numerical methods give well convergence.

Optimization and buckling of rupture building beams reinforced by steel fibers on the basis of adaptive improved harmony search-harmonic differential quadrature methods

The optimization and buckling analysis of the rupture beams reinforced by steel fibers are presented in this article. The structure is simulated by hyperbolic shear deformation theory (HSDT). Utilizing Harmonic differential quadrature (HDQ) method, the buckling load is calculated. Utilizing adaptive improved harmony search (AIHS) algorithm, the optimization process of the rupture beam is examined. The mentioned algorithm is improved adaptively utilizing dynamic variables. The harmony memory is corrected at first and second adjustments, respectively on the basis of dynamic bandwidth and step size randomly. The optimum conditions of the rupture beam are evaluated with various harmony existing search methods. The influences of various constants including beam length to thickness ratio, rupture parameter, steel fibers volume fraction, steel fibers orientation angle and boundary supports are shown on the buckling load. The outcomes indicate that the proposed AIHS is performed better with respect to other harmony search methods. Meanwhile, the buckling load is reduced by enhancing the rupture parameter. Furthermore, the buckling load of the rupture beam with horizontal steel fibers is better with respect to other orientation angles.

Free vibration of rotating graphene-reinforced laminated composite conical shells

The free vibration behaviors of rotating laminated composite conical shells reinforced with graphene sheets are investigated. Refined Halpin-Tsai micromechanical approach and Donnell shell theory are used for material properties and kinematic relations, respectively. Both uniform and functionally graded patterns of graphene sheets are assumed. The effects of rotation about revolution axis, Coriolis acceleration, centripetal acceleration and hoop tension are considered in the governing equations which are derived in an explicit differential form. Direct coupling boundary conditions with governing equations and harmonic differential quadrature are applied to discretize three-coupled PDEs with variable coefficients in spatial domain. To verify, natural frequencies of rotating homogenous cylindrical and conical shells, stationary functionally graded and graphene-reinforced composite cylindrical shells are compared with available results in literature and reliable accuracy is observed. Then, the influences of wave number, angular velocity, boundary condition, graphene distribution and geometrical parameters on natural frequencies are examined via parametric studies.

Effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam in HDQ analysis

The aim of this investigation is to study the effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam. The harmonic differential quadrature (HDQ) method is used to compute the natural frequency of Timoshenko beam. In this regards, three different types of meshing are used namely uniform meshing, normalized meshing and non-uniform. The results obtained using these meshing are compared with results obtained in previous published work under different types of boundary conditions. It is found that non-uniform meshing gives better convergence as well as good results as compare to results using uniform and normalized meshing under all boundary conditions.

Free vibration analysis of axially functionally graded tapered beam using harmonic differential quadrature method

Abstract In this work, the harmonic differential quadrature (HDQ) method is employed to investigate the vibration characteristics of axially functionally graded (AFG) tapered beam. The material parameters of beam are assumed to be varied continuously along beam length according to power law function. The Hamilton’s principle is used to obtain the equation of motion considering Timoshenko beam theory. Then the governing equations are solved using harmonic differential quadrature method. The non-dimensional frequencies are obtained for clamped–clamped type of boundary condition. The competency of the present method is then validated by the other available numerical techniques in the literature. The influence of non-homogeneity parameter, taper ratio and aspect ratio on the frequencies of the AFG tapered beam is discussed in detail.

Geometric Mapping for Non-Rectangular Plates with Micro/Nano or Macro Scaled under Different Effects

The main purpose of this study is to give a perspective via discrete singular convolution, differential quadrature (DQ) and harmonic differential quadrature (HDQ). For this purpose, DQ and HDQ methods are developed for the buckling, analysis of non-rectangular plates. Plates of, skew, shape is considered under axial loads. Furthermore, transformation formulations and some perspective for nano or macro scaled many problems with different effects discussed via discrete singular convolution and differential quadrature methods.

Free vibration analysis of laminated and FGM composite annular sector plates

In this article, the authors used two different numerical approaches for frequency response of annular sector and sector plates with functionally graded materials and laminated composite cases. First-order shear deformation (FSDT) and Love's conical shell theories are used for obtaining the annular sector plate equations via some suitable angles and geometric parameters. The method of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) have been used for numerical solution of the resulting governing equations for modal analysis. Simple power-law and four-parameter power law distributions in terms of the volume fractions of constituents have been used for FGM composites. Comparison and convergence study for the present numeral methods have been made via existing results available in the literature for sector and annular sector plates. Frequencies values for annular sector/sector plates have been obtained for different material and geometric parameters, boundary conditions and sector angles.

Efficacy of Harmonic Differential Quadrature method to vibration analysis of FGPM beam

The priority of this paper is to explore the computational characteristics of Harmonic Differential Quadrature (HDQ) method for free flexural vibration analysis of functionally graded piezoelectric material (FGPM) beam. The modified Timoshenko beam theory is used in this study where electric potential is assumed to have sinusoidal variation across the depth. The equations of motion and boundary conditions are derived by employing Hamilton‘s principle. The material properties are assumed to have a power law or sigmoid law variation across the depth. The available equations of motion are then solved using the Harmonic Differential Quadrature (HDQ) method to obtain the natural frequencies of the FGPM beam. The efficacy of the present method is validated by comparing the results with the previous published work. It is observed that the results obtained by HDQ method are at the same level of accuracy than those of previous analysis using GDQ method.

Vibration analysis of circular cylindrical panels with CNT reinforced and FGM composites

The aim of this study is to give a numerical solution for free vibration problem of functionally graded and carbon nanotube reinforced (CNTR) circular cylindrical panel. Based on the Love’s shell theory and first-order shear deformation theory (FSDT), related equations of motion have been obtained. Then, the methods of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) are used for numerical computations. Two different material properties are assumed to change continuously in the thickness direction according to the volume fraction power law and the general four-parameter power law distributions. Accuracy and convergence of the methods have been validated by comparing the obtained results with the existing results available in the open literature. Frequency values have been given for different material and geometric parameters, modes and boundary cases for circular cylindrical panels with FGM and CNTR cases. The effects of types of distributions of CNTR material are also discussed.

Comparisons of methods for solving static deflections of a thin annular plate

A highly accurate method for obtaining static deflections of a thin annular plate is helpful to effectively design the complicated structures with these plates. There have been numerous methods to achieve such a target. However, there is no direct technical literature for comparing these methods comprehensively. Therefore, the current study aims at performing comparison of three methods, optimization method (OM), finite element method (FEM), and harmonic differential quadrature (HDQ) method. Combining an instance, the comparisons give us insight into high accuracy and consistency of each other, showing high accuracy of the methods in this field. Compared with the results of FEM, the maximum error, less than 1%, demonstrates that the accuracy of the OM is high enough. Combining the small errors, the excellent stability of those brings a reliable method in this field. The maximum error and fluctuation drawn from the HDQ are evidently larger than those of the OM, and it is difficult to obtain results with higher accuracy based on the HDQ. Finally, the work described here suggests that the OM can be utilized to deal with such a complex problem in view of engineering and theory, and the HDQ method is more suitable to study the method for solving very complex partial differential systems of high order.

Numerical approaches for vibration response of annular and circular composite plates

In the present investigation, by using the two numerical methods, free vibration analysis of laminated annular and annular sector plates have been studied. In order to obtain the main equations two different shell theories such as Love\'s shell theory and first-order shear deformation theory (FSDT) have been used for modeling. After obtaining the fundamental equations in briefly, the methods of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) are used to solve the equation of motion. Accuracy, convergence and reliability of the present HDQ and DSC methods were tested by comparing the existing results obtained by different methods in the literature. The effects of some geometric and material properties of the plates are investigated via these two methods. The advantages and accuracy of the HDQ and DSC methods have also been examined with different grid numbers and shell theory. Some results for laminated annular plates and laminated circular plates were also been supplied.

A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods

In this paper, bending, buckling and buckling of embedded nano-sandwich plates are investigated based on refined zigzag theory (RZT), sinusoidal shear deformation theory (SSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT). In order to present a realistic model, the material properties of system are assumed viscoelastic using Kelvin–Voigt model. The elastic medium is simulated by orthotropic visco-Pasternak medium. Based on energy method and D'Alembert's principle, the derivation of the nonlinear equations of motion is presented. A novel numerical method namely as differential cubature (DC) method is applied for obtaining the static response, the natural frequencies and the buckling loads of nano-sandwich plates. The effects of different parameters such as nonlocal parameter, structural damping, viscoelastic foundation, geometrical parameters, stiffness of core and boundary conditions are shown on the bending, buckling and vibration behaviors of nanostructure. The accuracy of the proposed method is verified by comparing its numerical predictions with other published works as well as solution of system with differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. The numerical investigation shows that RZT is highly accurate in predicting the deflection, frequency and buckling load of nano-sandwich plates without requiring any shear correction factors.

Dynamic stability analysis of a pressurized FG-CNTRC cylindrical shell interacting with supersonic airflow

The objective of this research is to investigate the aeroelastic buckling and flutter instability of a pressurized functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shell subjected to supersonic airflow. The dynamic model of the FG-CNTRC cylindrical shell is established in accordance with the first-order shear deformation theory, Donnell kinematic theory along with the von Karman geometrical nonlinearity. The quasi-steady Krumhaar's modified piston theory by considering the effect of the panel curvature is used to estimate the aerodynamic pressure induced by the supersonic airflow. The dynamic equations are discretized using trigonometric expansion through the circumferential direction and harmonic differential quadrature (HDQ) method through the meridional direction. Effects of boundary conditions, geometrical parameters, volume fraction and distribution of CNTs and the Mach number on the flutter instability, onset of the buckling and deformation shapes of the cylindrical shell are put into evidence through a set of parametric studies. The simulation indicates that the critical flutter dynamic pressure may be significantly enhanced through functionally graded distribution of CNTs in a polymer matrix. Furthermore, it is found that presence of the aerodynamic pressure may completely change deformation shapes of the FG-CNTRC cylindrical shell.

Stability and accuracy of three Fourier expansion‐based strong form finite elements for the free vibration analysis of laminated composite plates

SummaryIn the present paper, strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates. In particular, the stability and accuracy of three different Fourier expansion‐based differential quadrature techniques are shown. These techniques are used to solve the partial differential system of equations inside each computational element. The three approaches are called harmonic differential quadrature, Fourier differential quadrature and improved Fourier expansion‐based differential quadrature methods. The improved Fourier expansion‐based differential quadrature method implements auxiliary functions in order to approximate functional derivatives up to the fourth order, with respect to the Fourier differential quadrature method that has a basis made of sines and cosines. All the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic. A study of such kind has never been proposed in the literature, and it could be useful as a reference for future investigation in this matter. Copyright © 2016 John Wiley & Sons, Ltd.

Harmonic Differential Quadrature Analysis of Soft-Core Sandwich Panels under Locally Distributed Loads

Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional (2D) elasticity theory without any pre-assumption on the displacement field. The top and bottom faces act like the elastic supports on the top and bottom edges of the core. The differential equations of the 2D core are then solved by the harmonic differential quadrature method (HDQM). To circumvent the difficulties in dealing with the locally distributed load by point discrete methods such as the HDQM, a general and rigorous way is proposed to treat the locally distributed load. Detailed formulations are provided. The static behavior of sandwich panels under different locally distributed loads is investigated. For verification, results are compared with data obtained by ABAQUS with very fine meshes. A high degree of accuracy on both displacement and stress has been observed.