Harmonic 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.

Vibration stability of a lifting pipe transporting solid–liquid two phase flow

A new mathematical model is developed to investigate the vibration stability of a lifting pipe induced by solid–liquid two phase flow. Three different slip models and the momentum conservation theorem are introduced to describe the slip characteristics of solid particles in two phase flow, which are coupled with the pipe vibration equations to establish a new governing equation and solved using the harmonic differential quadrature method. The mathematical model is validated by published data for the single-liquid flow case. Numerical results show that the slip model has noticeable effects on two phase flow-induced pipe vibration performance. The transport concentration of solid particles has a significant effect on the stability characteristic; however, the effect of particle size is very limited. The pipe flow mass ratio and the gravity coefficient are also key parameters affecting two phase flow-induced pipe vibration performance.

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.

Dynamic stability, and absorbed energy measurements in a sandwich structure resting on the novel frictional viscoelastic torsional substrate and horizontal friction force

Like metallic materials, fiber metal laminate (FML) lets the material has a simple metal structure behavior but with such advantages as specialized strength properties, impact, fire resistance, weight savings, corrosion resistance, and metal fatigue. The main focus of this article is to indicate the dynamic response of circular size-dependent circular plates made of FML material via 3D-elasticity theories. The size-dependent structure is simulated by coupling the modified nonlocal couple stress and 3D-elasticity theories (MNCS-3DET). For accurate modeling of the elastic substrate, torsional and linear spring and Pasternak foundation are modeled using the equations of horizontal friction force and torsional interaction. Harmonic differential quadrature method and Chebyshev–Gauss–Lobatto grid distribution points are coupled for solving the partial differential equations of the current micro/nano-structure. Novelties of this work consider the effects of MCS-3DET, torsional interaction, horizontal friction force, and FML material in the construction of the circular micro/nanoplate. In the results section, the outputs are verified with other published articles, and the outcomes show that the MCS-3DET parameter, elastic foundation coefficient, horizontal friction force coefficient, and geometry have an essential role in the dynamic stability of the FML micro/nano-structure.

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.

Seismic analysis and optimization of concrete bridge under the moving train utilizing numerical methods and adaptive improved harmony search algorithm

Abstract In this paper, the simultaneous actions of moving force and seismic load on the dynamic displacement and optimization of the concrete bridge are studied. The sinusoidal shear deformation beam theory is employed for the modelling of the concrete bridge mathematically. The structural damping of the concrete bridge is assumed by the Kelvin–Voigt theory. Utilizing the method of energy and Hamilton’s law, the equations of motion are obtained. Three mixed numerical methods, including the integral quadrature, harmonic differential quadrature method, and Newmark technique, are presented for the numerical outcomes of the differential equations. Utilizing adaptive improved harmony search, improved harmony search, harmony search, and global harmony search algorithms, the optimization process of the concrete bridge is examined. The mentioned algorithm is improved adaptively by utilizing dynamic deflection. The harmony memory is corrected at first and second adjustments, respectively, based on emotional bandwidth and step size randomly. The optimum conditions of the concrete bridge are evaluated with various harmony existing search methods. The role of multiple parameters, including the velocity and acceleration of moving load, length and thickness of bridge, boundary conditions, and the amplitude of carrying load, in the dynamic displacement of the bridge is studied. The numerical results indicate that with increasing the velocity and acceleration of the moving train, the dynamic displacement of the concrete bridge increases. In addition, with increasing the length of the bridge, the time of maximum deflection (i.e. when the train is in the middle of the bridge) is increased. It is concluded for the concrete bridge under the seismic load that the optimum values of the bridge’s length and thickness are decreased (about 24%) and increased (about 21%), respectively. The optimum values of amplitude, velocity, and acceleration of moving train are decreased, respectively, about 34%, 33%, and 29% in the case of the concrete bridge under the earthquake load. In addition, the optimum length of the concrete bridge is decreased significantly, with increasing the moving load amplitude, velocity, and acceleration.

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.

Nonlinear dynamics simulation of the composite reinforced annular system using harmonic differential quadrature framework

In this research, for the first time, a nonlinear dynamic model for large oscillation and the disk’s chaotic responses is presented. The material of the annual disk is composite layer reinforced by graphene oxide (GO) nanofillers and macro carbon fibers (CFs) and covered with a piezoelectric layer. The kinematic and nonlinear constitutive dynamic equations of the disk isare obtained by the use of the Von-Karman nonlinear theory alongside with the Hamilton’s principle. The derived nonlinear equations coupled with boundary conditions are solved numerically employing harmonic differential quadrature method (HDQM). The multiple scales method also used in the evaluation of primary resonance of the disk. The results show that inside to outside radii ratio, thickness to radius ratio as geometrical parameters, and loading condition including applied voltage and harmonic load have a substantial impact the nonlinear large oscillation, and chaotic motion of the composite disk covered with piezoelectric layer. The most interesting and significant outcome of this article is that with increasing the thickness of the piezoelectric layer, 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 disk increases. Moreover, as the CF’s weight fraction increases, the system’s motion and dynamics change from chaotic to semi-harmonic.

Vibration analysis of an axially functionally graded material non-prismatic beam under axial thermal variation in humid environment

The vibration analysis of an axially functionally graded material non-prismatic Timoshenko beam under axial thermal variation in humid environment is carried out using the harmonic differential quadrature method. In this modeling, the length and width of the beam remains constant whereas thickness of the beam is linearly varied along the axis of the beam. The material properties are temperature dependent and are assumed to be varied continuously along the axial direction according to power law distribution. Three types of temperature variations are considered in this study, that is, uniform temperature rise, linear temperature rise, and non-linear temperature rise. The temperature of the beam remains constant under uniform temperature rise condition and it is varied linearly and nonlinearly along the length of beam for rest of the conditions. The beam is subjected to uniform moisture concentration to impose humidity. Hamiltonian’s approach is used to derive the governing equations of motion. The resultant governing equations are then solved using the harmonic differential quadrature method to obtain the natural frequencies of the axially functionally graded material non-prismatic beam. The results obtained using the harmonic differential quadrature method are compared with results obtained for special cases. The effects of thermal variation, humidity, non-homogeneity parameter, and end conditions on natural frequencies of the non-prismatic beam are reported.

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 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.

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.

On the use of differential quadrature-three-term conjugate finite-step length methods for reliability analysis of steel fiber-reinforced sinusoidal rupture beams

The reliability of a complex mathematical problem is presented in this work. The problem is a rupture concrete beam reinforced by steel fibers with various orientation angle. The structure is simulated by sinusoidal shear deformation theory and energy method. Harmonic differential quadrature method is applied for the solution of the problem under the buckling load. The reliability analysis of the mentioned system is studied utilizing a three-term conjugate finite-step length (TCFS) approach. The TCFS is formulated utilizing the conjugate gradient method with a limited scalar parameter to reach the numerical stabilization. The implicit buckling limit state function of the structure includes different parameters of length to thickness ratio of the beam, volume percent and orientation angle of steel fibers, rupture constant and boundary conditions. The results show that the failure probabilities of the studied structure may be decreased by increasing the rupture constant. In addition, the orientation angle of zero is the best choice for steel fiber in the concrete beam.

Effect of Winkler Foundation on Radially Symmetric Vibrations of Bi-Directional FGM Non-Uniform Mindlin’s Circular Plate Subjected to In-Plane Peripheral Loading

An analysis has been presented of the effect of elastic foundation and uniform in-plane peripheral loading on the natural frequencies and mode shapes of circular plates of varying thickness exhibiting bi-directional functionally graded characteristics, on the basis of first order shear deformation theory. The material properties of the plate are varying following a power-law in both the radial and transverse directions. The numerical solutions of the coupled differential equations leading the motion of simply supported and clamped plates acquired by using Hamilton’s principle, is attained by harmonic differential quadrature method. The effect of different plate parameters namely gradient index, heterogeneity parameter, density parameter, taper parameter and thickness parameter is illustrated on the vibration characteristics for the first three modes of vibration for various values of in-plane peripheral loading parameter together with foundation parameter. Critical buckling loads in compression are calculated for both the boundary conditions by putting the frequencies to zero. The reliability of the present technique is confirmed by comparing the results with exact values and results of published work.

Free axial vibration of cracked axially functionally graded nanoscale rods incorporating surface effect

This work aims to study effects of the crack and the surface energy on the free longitudinal vibration of axially functionally graded nanorods. The surface energy parameters considered are the surface stress, the surface density, and the surface Lame constants. The cracked nanorod is modelled by dividing it into two parts connected by a linear spring in which its stiffness is related to the crack severity. The surface and bulk material properties are considered to vary in the length direction according to the power law distribution. Hamilton\'s principle is implemented to derive the governing equation of motion and boundary conditions. Considering the surface stress causes that the derived governing equation of motion becomes non-homogeneous while this was not the case in works that only the surface density and the surface Lame constants were considered. To extract the frequencies of nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using the harmonic differential quadrature method. Since the present work considers effects of all the surface energy parameters, it can be claimed that this is a comprehensive work in this regard.

Dynamic snap-through of shallow spherical shells subjected to thermal shock

Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.

Nonlinear Vibration Behavior of Rapidly Heated Temperature-Dependent FGM Shallow Spherical Shells

In the present research, thermally induced vibration of shallow spherical shells made of functionally graded materials is investigated. The thermomechanical properties of the FGM media are assumed to be temperature dependent. Based on the uncoupled thermoelasticity theory, the one-dimensional heat conduction equation is established. The top and bottom surfaces of the shell are subjected to several types of rapid heating boundary condition. Because of the temperature dependency of the material properties, solution of nonlinear heat conduction equation needs a numerical method. In the first step, generalized differential quadrature method (GDQM) is applied to discretize the heat conduction equation across the shell thickness. Afterward, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson scheme. The obtained thermal force and thermal moment resultants at each time step are applied to the equations of motion. The axisymmetric equations of motion, using the first-order shear deformation theory (FSDT) based on the Kármán type of geometrical nonlinearity, are derived with the aid of the Hamilton principle. Using the harmonic differential quadrature method, shell domain is divided into a number of nodal points, and differential equations are turned into a system of ordinary differential equations. To obtain the displacement components at any time, time marching scheme based on the Newmark method is implemented. The obtained highly nonlinear algebraic equations are solved by means of the Newton–Raphson method. Comparison studies are performed to validate the formulation and solution method of the present research. Various examples are illustrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the shell, temperature dependency, shell opening angle, in-plane boundary conditions, type of thermal boundary conditions, and geometrical nonlinearity on thermally induced response of the FGM shell under thermal shock.

Forced vibration analysis of concrete slabs reinforced by agglomerated SiO2 nanoparticles based on numerical methods

Abstract In this research, forced vibration analysis of a concrete slab reinforced by SiO2 nanoparticles is investigated. The Mori-Tanaka low is used for obtaining the material properties of nano-composite structure and considering agglomeration effects. By means of third order shear deformation theory or Reddy theory, the total potential energy of system is calculated. Also, using Hamilton’s principle, the coupled motion equations are obtained. Based on the harmonic differential quadrature method (HDQM), finite element method (FEM) and Newmark method, the frequency response of system is calculated. The effects of volume percent and agglomeration of SiO2 nanoparticles, boundary conditions and geometrical parameters of structure are shown on the frequency response of system. Results show that with growing the volume percent of SiO2 nanoparticles up to 0.37, the linear frequency of the structure is increased and the maximum dynamic deflection is decreased.

Elastic effects on vibration of bilayer graphene sheets incorporating integrated VdWs interactions

The following study addresses the free vibration analysis of a bilayer graphene sheet (BLGS) embedded in an elastic medium in the presence of shear and tensile-compressive effects of van der Waals (vdWs) interactions. To ascertain the contribution of each force, the effects are considered separately and simultaneously. To model the geometry of the BLGS, the sandwich plate theory and the Hamilton’s principle are considered to derive the governing equations of motion. The Harmonic differential quadrature method is applied to solve the coupled equations and obtain the natural frequencies and related mode shapes. The results reveal that the contribution of tensile-compressive modulus of elastic medium is the most in changing the frequency of BLGSs.