Generalized Differential Quadrature Research Articles

Free vibrational behavior of functionally graded carbon nanotube composite fluid-filled cylindrical-conical shell: Analytical and experimental investigation

This investigation analyzes the effects of internal static fluid on the free vibrational behavior of joined shells made of carbon nanotube (CNT) fiber‐reinforced composite. Various types of functionally graded CNTs composite with temperature dependence are considered. Classic shell theory and Love’s assumptions combined with Hamilton’s principle are used to obtain motion equations. Bernoulli equation, velocity potential, and impermeability condition are employed to the coupling effects of fluid pressure with motion equations of shells. Continuity conditions in the joint point of conical and cylindrical sections are supposed to couple equations of two parts. The generalized differential quadrature (GDQ) method is employed to solve equations for various boundary conditions. Experimental modal tests on a sample made by the 3-D printer and prior others’ research are used to validate the accuracy of results. Consequently, parametric studies are expressed, such as longitudinal and circumferential mode numbers, semi-vertex angle, distributions of CNTs, CNTs fraction volumes, fluid density, shell thickness, and operating temperature. According to the results, although fluid decreases the frequency generally, the process of changing frequency in terms of other parameters is similar.

Geometrically nonlinear response of FGM joined conical–conical shells subjected to thermal shock

Geometrically nonlinear thermally induced vibrations of functionally graded material (FGM) joined conical–conical shells is analyzed in the current research. Thermo-mechanical properties of the shell are assumed to be temperature and position dependent. The system of joined conical–conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and the Crank–Nicolson methods. This equation is nonlinear since the thermo-mechanical properties of the shell are temperature dependent. The total functional of the shell is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of FGM single conical shell. Parametric studies are conducted to examine the influences of conical geometry, material composition, temperature dependence, in-plane and out-of-plane mechanical, various configuration of joined shell system, and thermal boundary conditions.

ICAE 2021 - Preface

One of the applicable shell structures related to underwater systems, covering riser pipes, ice-breaking equipment, underwear noise migrations, ballast water components, and gliders, can be considered Linked Hemispherical-Conical Shell (LHCS) structures. Accordingly, the ensuing investigation is provided to compute the Natural Frequency Values (NFVs) of the LHCS, with the opening angle at starting part of the hemispherical structure (which will be studied for the first time) enhanced by graphene-based nanocomposite materials containing Graphene Nano-Platelet (GNP) and Graphene Oxide-Powder (GOP) nanocomposites. For this reason, the Halpin-Tsai approach and the rule of the mixture are engrossed in determining the productive functionally graded mechanical criteria of the graphene-based nanocomposite materials. To obtain the main formulations of the LHCS, Donnell's assumptions and First-Order Shear Deformation Theory (FOSDT) are also engaged. Additionally, Hamilton's technique is involved in finding the kinetic equations coupled with the hemispherical and the conical parts of the LHCS. Next, the distinguished approach, Generalized Differential Quadrature (GDQ) program, is applied to discrete the kinetic equations. Subsequently, the standard eigenvalue problem determines the NFVs related to the GNP and GOP nanocomposites LHCSs. At termination, the exactness of the suggested scheme is confirmed by comparing the provided outputs with the NFVs of the submitted work (which can be used as the benchmark) with responses determined by FEM commercial software and the only benchmark study that exists in the literature. Furthermore, some novel cases are solved to estimate the effects of different geometric criteria of the LHCS, such as the opening angle of the hemispherical, the Half-Apex (HA) angle of the conical segment, the thickness related to the structure, and also the effect of GNP and GOP scattering models spread in conjunction with the thickness of the LHCS on the NFVs coupled with the GNP and GNP nanocomposites LHCS.

Wave frequency responses estimate of the nanocomposite linked hemispherical-conical shell underwater-like bodies with the impacts of two types of graphene-based nanofillers

One of the applicable shell structures related to underwater systems, covering riser pipes, ice-breaking equipment, underwear noise migrations, ballast water components, and gliders, can be considered Linked Hemispherical-Conical Shell (LHCS) structures. Accordingly, the ensuing investigation is provided to compute the Natural Frequency Values (NFVs) of the LHCS, with the opening angle at starting part of the hemispherical structure (which will be studied for the first time) enhanced by graphene-based nanocomposite materials containing Graphene Nano-Platelet (GNP) and Graphene Oxide-Powder (GOP) nanocomposites. For this reason, the Halpin-Tsai approach and the rule of the mixture are engrossed in determining the productive functionally graded mechanical criteria of the graphene-based nanocomposite materials. To obtain the main formulations of the LHCS, Donnell's assumptions and First-Order Shear Deformation Theory (FOSDT) are also engaged. Additionally, Hamilton's technique is involved in finding the kinetic equations coupled with the hemispherical and the conical parts of the LHCS. Next, the distinguished approach, Generalized Differential Quadrature (GDQ) program, is applied to discrete the kinetic equations. Subsequently, the standard eigenvalue problem determines the NFVs related to the GNP and GOP nanocomposites LHCSs. At termination, the exactness of the suggested scheme is confirmed by comparing the provided outputs with the NFVs of the submitted work (which can be used as the benchmark) with responses determined by FEM commercial software and the only benchmark study that exists in the literature. Furthermore, some novel cases are solved to estimate the effects of different geometric criteria of the LHCS, such as the opening angle of the hemispherical, the Half-Apex (HA) angle of the conical segment, the thickness related to the structure, and also the effect of GNP and GOP scattering models spread in conjunction with the thickness of the LHCS on the NFVs coupled with the GNP and GNP nanocomposites LHCS.

On the vibrational analysis of combined paraboloidal-conical air vehicle segment shell-type structures

For the first time, the analysis of the free vibration performance related to the Combined Paraboloidal-Conical Shell (CPCS) structures is evaluated in this paper. To support, the First Shear Deformation Theory (FSDT) and General Shell Hypothesis (GSH) are involved in discovering the primary relationships related to the CPCS's shell segments. In addition, Hamilton's tactic determines the Governing Differential Motion Equations (GDMEs) affiliated with each shell segment of the CPCS. Further, the well-structured technique labeled the Generalized Differential Quadrature (GDQ) procedure is implemented to discretize the GDMEs. Furthermore, the eigenvalue determination is employed to derive the Natural Frequency Parameters (NFPs) of the CPCS structures. Additionally, because no study has been engaged in this structure in the literature, the NFPs measured by the proposed framework are contrasted with the NFPs found by FEM-based commercial software to credit the submitted methodology. In addition, the maximum error is observed to be equal to less than 0.5%. Based on this, the outputs of this paper can be used as the benchmark. Ultimately, by planning and solving novel examples, the effects of the geometry characteristics related to the CPCS are determined on the NFPs of the structure.

Free-damped vibration analysis of Graphene Nano-Platelet nanocomposite Joined Conical-Conical-Cylindrical Shell marine-like structures

The oncoming paper analyzes the Natural frequency (NF) of the most purposeful structures in ocean engineering as submerged, offshore, marine structures, gliders bodies, and shaft cowls labeled Joined Conical-Conical-Cylindrical Shell (JCCCS) structure. For this aim, 30 various geometry types related to the JCCCS structures are investigated in this work. In addition, Graphene Nano-Platelet (GNP) is applied to make a GNP-Nanocomposite (GNPN) material and enhance the vibrational performance associated with the JCCCS. For this purpose, Halpin-Tsai and the rule of mixture hypotheses are realized to discover the mechanical marks related to the GNPN. To access the core relationships associated with the JCCCS structures, the First Shear Deformation Theory (FSDT) and Donnell's simplifications are employed. Furthermore, Hamilton's scheme is involved determining the Differential Motion Equations (DMEs) coupled with the JCCCS structures. Also, the well-organized and distinguished technique branded the Generalized Differential Quadrature (GDQ) strategy is selected to discretize DMEs related to the JCCCS structures. The standard eigenvalues technique is organized to access the NFs related to the JCCCS structure. The NFs determined from the following method are compared to FE-based NFs related to the JCCCS structures to verify the upcoming procedure. Afterward, numerous novel examples are solved, which can be used as benchmarks to illustrate the various geometrical and material features and Boundary Conditions (BCs) on the NFs coupled with the isotropic-JCCCS and GNPN-JCCCS structures.

A comparative study on the effect of SMAs and CNTs on the vibration of composite plates exposed to thermal environments

The main purpose of this study is to compare the effects of two advanced materials, namely shape memory alloys (SMAs) and carbon nanotubes (CNTs) on the fundamental natural frequencies of rectangular graphite fiber/epoxy (GF/EP) composite plates. The performance of each of these materials is evaluated for different geometric conditions and boundary conditions as well as operating temperatures. Moreover, the glass transition temperature is considered as a limiting factor in the study. The governing equations are derived based on first-order shear deformation theory (FSDT) and then solved by generalized differential quadrature (GDQ) technique. The effective thickness-to-width ratio and temperature ranges are determined for both SMA wires and CNTs. The results of this study show that in some conditions, the use of CNTs is preferable over SMA wires and vice versa. Moreover, unlike SMA wires, the positive effect of the CNTs is limited to a certain extent. Furthermore, the numerical results indicate that depending on the different parameters, the use of SMAs may have constructive or destructive effects on the structural vibrations. Interestingly, it is shown that in some circumstances, the co-incorporation of SMAs and CNTs leads to a significant enhancement in the natural frequencies of the structure.

On the frequencies of graphene nanoplatelet agglomerated nanocomposite paired paraboloidal-cylindrical shells under arbitrary boundary conditions

This paper evaluates the vibrational behavior of the complex types of nanocomposites paired shells with arbitrary support conditions. The paired shell comprises two segments, including a paraboloid and a cylinder. In addition, arbitrary support conditions are incorporated and investigated by thirty-six different cases of boundary conditions. The influence of using agglomerated Graphene Nano-Platelets (GNP) for reinforcing the polymer matrix on the natural frequency of the nanocomposite shells is studied by conducting the Eshelby-Mori-Tanaka scheme as an efficient homogenization procedure. Furthermore, the impact of the GNP accumulation throughout the polymer matrix is also considered in formulation by defining agglomeration factors. To attain the displacement fields of the shell structure, the three-dimensional (3D) shell approach and First-order Shear Deformation Theory (FSDT) are engaged. Afterward, Hamilton's principle determines the structure's total strain and kinetic energies. By applying the variational calculus, the governing differential equations of the structural model are available. In this step, a robust and reputable approximated numerical solution method, namely Generalized Differential Quadrature (GDQ), is used to discretize the shell structure by defining several grid points. Finally, the established eigenvalue problem is solved numerically to capture the structure's natural frequency. The authors validate their formulation by comparing the results obtained by the present procedure and the outputs of the Finite Element Method (FEM). Some more functional, intricate, and new nanocomposite Paired Paraboloidal-Cylindrical Shells (PPCS) are considered to be analyzed for achieving the associated natural frequency.

Critical speed and frequency behavior of rotating joined FG-CNTRC conical-conical shells

This paper provides information on critical speed and frequency behavior of rotating joined functionally graded carbon nanotube reinforced (FG-CNTRC) conical-conical shells. Furthermore, frequency bifurcation as a consequence of rotational effects is investigated. The shell is assumed to be composed of an isotropic polymer matrix that is reinforced by uniformly or functionally distributions of carbon nanotubes (CNTs). The structure’s kinematics originates from the first order shear deformation theory (FSDT). To derive spin-based motion equations, Coriolis and centrifugal forces together with initial hoop tensions are propounded through the Hamilton’s principle. Moreover, matching and boundary conditions are ascertained to supplement the governing equations. To discretize the governing equations meridionally, the generalized differential quadrature (GDQ) technique is employed. After validity checking, some significant parameters like cones’ angles, rotation speed, edge supports, CNT volume fractions and dispersion patterns that play effective roles on the critical speed and vibration characteristics are examined. It is revealed that by enhancing spinning speed, frequencies of different FG patterns converge. In addition, difference between backward and forward frequencies decreases as the circumferential mode number increases.

Application of Chebyshev-based GDQ and Newmark methods to viscothermoelasticity responses of FG composite annular systems

This study explores the thermo-viscoelasticity response of annular disks made of a functionally graded graphene platelet reinforced composite (FG-GPLRC). Random orientation is considered for GPL nanofillers. A power-law model with a controller index is also used to determine the GPL volume fraction distribution along the radial direction. The effective thermomechanical properties of the nanocomposite medium are calculated through the one-point and two-point correlation homogenization processes. The Kelvin-Voigt technique is employed to design the viscosity nature of the system. Coupled thermoelasticity is considered to attain the governing formulation because the considered disk is subjected to a sudden high temperature. Furthermore, by incorporating generalized thermoelasticity based on the Lord-Shulman concept, the physical structure of the problem is altered. The transient governing equations are solved based on a hybrid approach. The spatial dependency is solved using a Chebyshev-based generalized differential quadrature (GDQ) method, while the time dependency is calculated using an average acceleration-based Newmark technique. After confirming the described formulation and method with previous research, different parametric samples are given to explain more about the behavior of viscoelastic nanocomposite disks that are heated quickly and rotated at a constant rate.

On vibrational-based numerical simulation of a jet engine cowl shell-like structure

The present research proposes an applicable and impressive analytical model of Aircraft Engine Cowl (AEC) due to auspicate its frequency parameter. For the first time, the authors configured the structure of AEC by joining two well-known shapes of shell, including elliptical and conical. This procedure is implemented by obtaining the governing motion organization of differential equations affiliated with the Joined Elliptical-Conical Shells (JECS) in line with first-order shear deformation theory (FSDT) and General Shell Formulation (GSF). Moreover, Functionally Graded Materials (FGMs) with various spreading forms along the structure’s thickness are used to enhance the dynamic performance of AEC. It is worth mentioning that the recognized Rule of Mixture (RM) is engaged for achieving the equivalent mechanical characteristics of FGM. Finally, a well-known semi-analytical Generalized Differential Quadrature (GDQ) procedure is hired to extract the resulting system of motion equations to access the free vibration responses of AEC with diverse Boundary Conditions (BCs). It is worth mentioning that an applicable and interesting numerical model is done to inspect the remnants of geometric and material features on the oscillation demeanor of AEC structures. It will be verified that the projected procedure can be helpful for AEC’s vibration analysis and extendable for other types of studies to predict the AEC structure's static and dynamic treatment.

Thermoelastic wave propagation damping in a hollow FG-GPLRC cylinder with the spinning motion

This analysis deals with generalized thermoelasticity via Lord–Shulman’s theory for hollow cylinders reinforced with graphene platelets (GPL). The viscosity effect is also considered for applied materials. The second-order correlation homogenization techniques are utilized to obtain the equivalent thermo-mechanical properties. The hollow cylinder with spinning motion is modeled based on the linear thermoelastic constitutive law. The GPLs are functionally distributed along the cylinder’s radius based upon a power-law function. The Kelvin–Voigt type of viscosity behavior which is a differential state of viscous materials, is employed. The energy equation is attained based on the coupled and generalized framework. After obtaining the motion and energy equations in dimensionless form, the generalized differential quadrature (GDQ) and Newmark time marching methods are implemented to extract the temporal evolution and wave propagation of temperature, displacement, and stresses that occurred in the structure. After validating the responses with the available theoretical article, some numerical results are demonstrated to examine the effect of thermo-mechanical properties on the thermoelastic behavior of chosen nanocomposite cylinder.

Size-dependent free vibration and buckling analysis of magneto-electro-thermo-elastic nanoplates based on the third-order shear deformable nonlocal plate model

In this paper, a size-dependent nanoplate model is developed to describe the free vibration and buckling behaviors of magneto-electro-thermo-elastic (METE) rectangular nanoplates. It is assumed that the METE nanoplate is subjected to external electric voltage, external magnetic potential, and uniform temperature rise. The nonlocal elasticity theory along with the third-order shear deformation plate theory is employed for the size-dependent mathematical modeling of nanoplates. The presented model has two advantages over available models: (1) the need for the correction factor is bypassed and (2) it can be successfully applied to thick nanoplates. The governing equations and the corresponding boundary conditions are derived using the Hamilton’s principle which are then discretized on the space domain based on the generalized differential quadrature (GDQ) method. Afterward, an efficient numerical Galerkin procedure is adopted to reduce the discretized equations into Duffing-type ordinary differential equations. Numerical results are presented to examine the influences of nonlocal parameter, length-to-thickness ratio, temperature rise, external electric potential, external magnetic potential and type of boundary condition on the free vibration, and buckling behaviors of METE nanoplates. It is revealed that frequency and critical buckling load of nanoplates are dependent on magneto-electro-mechanical loadings, whereas they are less dependent on thermal loading.

Analysis of vibration characteristics of auxetic sandwich cylindrical shells resting on elastic foundation

The main objective of this research work is focused on the vibration analysis of auxetic sandwich cylindrical shell structures resting on an elastic foundation. In the analysis, the sandwich shell structure is composed of three layers in which the middle layer consists of auxetic material with a negative Poisson’s ratio, and the two skin layers are isotropic homogeneous materials. The motion equation is extracted according to the first-order shear deformation theory (FSDT) and the Hamilton principle. The governing equations of coupled partial differential equations are solved by the generalized differential quadrature (GDQ) method, and the natural frequencies are determined. By comparing the experimental results with the numerical results calculated by commercial finite element software, the validity of the proposed theoretical model is verified. Finally, the influences of geometrical parameters and elastic foundation on the vibration behavior of sandwich shell structures have been investigated.

Dynamic analysis of electrorheological fluid sandwich cylindrical shells with functionally graded face sheets using a semi-analytical approach

The present research is performed to calculate the natural frequencies, loss factors, and associated mode shapes of a sandwich cylinder with moderately thick functionally graded (FG) face sheets and an electrorheological (ER) fluid core. Each face sheet is assumed to be made of FG materials, and its displacement field is estimated based on the first-order shear deformation theory, like the ER constrained layer. A suitable displacement continuity condition is considered between layers. The ER fluid used in the central middle is analyzed in the pre-yield area and considered electric field dependent. Hamilton’s principle is used to acquire the motion equations related to each layer and appropriate boundary conditions. A computational code is implemented based on the combination of trigonometric expansion and the generalized differential quadrature (GDQ) methods to extract the complex eigenvalue problem for the vibration and damping of the assumed structure. After carrying out a comprehensive verification study, some novel numerical results are obtained. In the parametric study, the effects of the applied electric field, the FG power-law index, various boundary conditions, and structure geometry to control the natural frequencies and modal loss factors, as well as mode shapes, are investigated.

Coupled Newmark beta and GDQ methods with a hybrid adaptive neuro-fuzzy for electromechanical energy absorption of microsystem

In the present paper, the size-dependent vibration in addition to the energy absorption of a three-layered nanoplate which contains a functionally graded core as well as two piezoelectric layers as face sheets. The system is resting on a Winkler-Pasternak viscoelastic foundation and is under a shock load resulted from a blast. The Mindlin plate theory together with Hamilton's principle is incorporated to extract the formulation as well as boundary condition. Also, the nonlocal strain gradient theory is employed in order to capture the impact of small size. Then, Generalized Differential Quadrature (GDQ) Method is utilized as the solution method to discretize the domain. Additionally, the Newmark-beta is used to solve the forced vibration as well as energy absorption. The formulation and solution procedures are validated by other results from papers. Then, different variables affecting the free and forced vibration along with the energy absorption of the plate is investigated. This paper can be a proper basis for further studies in area of energy absorption and forced vibration of small-sized plates.

Stress distribution in laminated composite tubular joints with damaged adhesive layer under torsion

ABSTRACT This study investigates the effect of geometric and mechanical properties of the adherends and adhesive layer on the shear stress distribution in a damaged adhesively bonded tubular lap joint (laminated composite to steel) under pure torsion. Using first-order shear deformation theory, the equilibrium equations are deduced and solved by the generalized differential quadrature (GDQ) method, results of which were compared and verified with those of finite element solution. Results show that for the steel and [±15]10 composite adherends, increasing the elastic modulus of the adhesive layer increases the maximum shear stresses in the adhesive layer while increasing the joint length beyond its so-called effective bond length has no additional effect on the peak shear stress in the adhesive. Additionally, selecting a quasi-isotropic lamination setup instead of [±15]10 reduces the peak shear stress, where the use of an orthotropic tube with the lamination sequence of ±45° further reduces this magnitude. Additionally, depending on the size and location of a void or debond in the adhesive layer, the peak shear stresses can gain higher values, with almost similar behaviors for the two defects. Nonetheless, for defects far enough from the adhesive ends, their effects on the adhesive peak shear stress seemed to be negligible.

Free vibration analysis of functionally graded graphene nanocomposite beams partially in contact with fluid

This paper presents the free vibration analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) beams partially in contact with fluid based on the first-order shear deformation theory (FSDT). By considering the 3D-random orientation of graphene platelets (GPLs), the 3D Halpin-Tsai model is introduced to calculate the elastic modulus of graphene nanocomposites. The fluid velocity potential and hydrodynamic loading are determined by virtue of variable separation method. The equations of motion are derived by using Hamilton’s principle and the natural frequencies are determined by the multidomain generalized differential quadrature (GDQ) approach together with an iterative scheme. Parametric studies are carried out to evaluate the influences of GPL distribution pattern, weight fraction and dimension, fluid depth and density, beam geometry and end supports on the free vibration behaviour of FG-GPLRC beam-fluid interaction systems. Results show that the beam-fluid interaction reduces the fundamental frequency most, but it hardly affects the first order vibration mode.

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.

Natural frequency analysis of FG-GOP/ polymer nanocomposite spheroid and ellipsoid doubly curved shells reinforced by transversely-isotropic carbon fibers

This investigation aims to determine the frequency responses associated with three-phase nanocomposite polymer-Graphene Oxide Powder (GOP)-carbon fiber spheroid and ellipsoid doubly-curved shells. In detail, the GOP nano-material is engaged to increase polymeric matrix mechanical features and form Crossbreed Matrix (CM) as well. For this reason, the rule of the mixture and the Halpin-Tsai schemes are recruited to pick up mechanical features associated with CM material. Next, carbon fibers are used to enhance mechanical criteria CM material and produce three-phase material as well. Accordingly, the Halpin-Tsai homogenization scheme is devoted to finding mechanical criteria connected to three-phase material. Afterwards, the equations of motion linked to spheroid and ellipsoid shells are mined hiring Hamilton's principle. Then, the compelling method named Generalized Differential Quadrature (GDQ) strategy is allocated to gap the basic motion equations associated with mentioned shells. At next, the standard eigenvalue computation is allocated to extract frequency responses related to the shells. To verify the offered procedure, one example is premeditated and compared with FEM commercial software. At last, various issues are arranged to excavate the trace of material mechanical and geometrical features related to the three-phase nanocomposite spheroid and ellipsoid doubly curved shells.