Free Vibration Response Research Articles

Free-vibration response of functionally graded porous shell structures in thermal environments with temperature-dependent material properties

Free-vibration response of functionally graded porous shell structures in thermal environments with temperature-dependent material properties

On the dynamics of 3D nonlocal solids

The stress-driven nonlocal continuum theory is a well-established integral elasticity formulation. Modelling the elastic strain via a convolution integral between stress and an averaging kernel, the theory leads to an integro-differential structural problem for arbitrary 3D nonlocal solids. Integral elasticity solutions are mainly available for 1D solids by adopting a bi-exponential kernel and reverting the boundary-value integro-differential problem to an equivalent differential formulation involving both classical and constitutive boundary conditions. A general computational framework to implement the stress-driven theory for 3D nonlocal solids is still missing. This paper tackles this issue, solving the relevant integro-differential problem by a two-field finite-element approach involving displacements and stresses. The approach can handle the dynamics of complex small-size structures of arbitrary shape. Focusing on free vibration responses, two examples are presented: a 2D rectangular plate, serving as benchmark to demonstrate size effects captured by stress-driven nonlocal elasticity, and a 3D solid modelling a typical dual microcantilever resonator with overhang.

Investigate the bending and free vibration responses of multi-directional functionally graded plates with variable thickness based on isogeometric analysis

This study investigates the static bending and free vibration behaviour of Multi-directional Functionally Graded (MFGM) plates having a variable thickness. With this regard, the third-order shear deformation plate theory is used to describe the kinematic relation. Material properties of MFGM are assumed to be graded in spatial direction within the plate’s volume and the effective material properties are calculated based on the power-law approach. The governing equations are derived based on Hamilton’s principle and the Isogeometric Analysis (IGA) is employed as a discretization tool to develop system equations. Several numerical examples are conducted to verify the accuracy of the proposed approach and investigate the influence of different geometrical and material parameters on the behaviour of MFGM plates

A novel formulation for the weak quadrature element method for solving vibration of strain gradient graded nonlinear nanobeams

A novel formulation of the weak form quadrature element method, referred to as the locally adaptive weak quadrature element method, is proposed to develop elements for nonlinear graded strain gradient Timoshenko and Euler–Bernoulli nanobeams. The equations of motion are obtained based on Hamilton principle while accounting for the position of the physical neutral axis. The proposed elements use Gauss quadrature points to ensure full integration of the variational statement. The proposed formulation develops matrices based on the differential quadrature method which employs Lagrange-based polynomials. These matrices can be modified to accommodate any number of extra derivative degrees of freedom including third-order beams and higher-order strain gradient beams without requiring an entirely new formulation. The performance of the proposed method is evaluated based on the free vibration response of the linear and nonlinear strain gradient Timoshenko and Euler–Bernoulli nanobeams. Both linear and nonlinear frequencies are evaluated for a large number of configurations and boundary conditions. It is shown that the proposed formulation results in good accuracy and an improved convergence speed as compared to the locally adaptive quadrature element method and other weak quadrature element methods available in the literature.

Free Vibration Response of Rotating Hybrid Composite Conical Shell Under Hygrothermal Conditions

Free Vibration Response of Rotating Hybrid Composite Conical Shell Under Hygrothermal Conditions

Modeling and Analysis of Radial Electromagnetic Force and Vibration Characteristics Based on Deflection Dual-Stator Switched Reluctance Generator.

In this paper, a mechanical model of the deflection dual-stator switched reluctance generator (DDSRG) is developed, and the advantages of the dual-stator structure for the deflecting motion are analyzed. Secondly, the spatio-temporal and spatial distribution characteristics of the inhomogeneous electromagnetic force are derived analytically and further verified by fast Fourier transform (FFT).Thirdly, the spatial and temporal distributions of electromagnetic forces of DDSRG are calculated based on finite element software, and the distributions of electromagnetic forces under different motion states are analyzed. By combining the analysis of modal analysis and harmonic response analysis, the free mode and vibration response acceleration variation laws of the internal and external stator are determined. The results show that the order of electromagnetic forces on the stator at rated speed is mainly 8 times the fundamental frequency, and the modal vibration order is more violent in the order of 2–7. Finally, the experimental platform of DDSRG is built, and the vibration characteristics are tested to verify the validity and accuracy of the proposed simulation results.

Free and Forced Vibration Analyses of Functionally Graded Graphene-Nanoplatelet-Reinforced Beams Based on the Finite Element Method.

The finite element method (FEM) is used to investigate the free and forced vibration characteristics of functionally graded graphene-nanoplatelet-reinforced composite (FG-GPLRC) beams. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary continuously along the beam thickness according to a linear, parabolic, or uniform pattern. For the FG-GPLRC beam, the modified Halpin–Tsai micromechanics model is used to calculate the effective Young’s modulus, and the rule of mixture is used to determine the effective Poisson’s ratio and mass density. Based on the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory, finite element formulations are derived to analyze the free and forced vibration characteristics of FG-GPLRC beams. A two-node beam element with six degrees of freedom is adopted to discretize the beam, and the corresponding stiffness matrix and mass matrix containing information on the variation of material properties can be derived. On this basis, the natural frequencies and the response amplitudes under external forces are calculated by the FEM. The performance of the proposed FEM is assessed, with some numerical results obtained by layering method and available in published literature. The comparison results show that the proposed FEM is capable of analyzing an FG-GPLRC beam. A detailed parametric investigation is carried out to study the effects of GPL weight fraction, distribution pattern, and dimensions on the free and forced vibration responses of the beam. Numerical results show that the above-mentioned effects play an important role with respect to the vibration behaviors of the beam.

Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.

Forced and free vibration of magnetostrictive laminated sandwich beams on Kerr’s foundations

The supporting system’s influence on the dynamic behavior of the frequencies and deflection of the smart laminated composite sandwich beam with functionally graded faces is investigated according to a higher-order shear deformation theory. The graphite fiber constituent and the matrix material are distributed through the cross-section of the beam faces based on power law gradation. Kerr’s foundation is proposed to use as a variable support system for the studied structure. The proposed structure is subjected to a magnetic field and an external force in x direction. For controlling the system, a simple feedback gain control with constant distribution is applied to the embedded Terfenol-D actuators. To obtain and solve the associated dynamic equations of the studied system, Hamilton’s principle and Navier’s approach are utilized. Several numerical examples are carried out to show the forced/ free vibration response of the system with detailed illustrations of the different geometrical parameters variation.

An Assumed Strain Finite Element for Composite Plates Analysis

This paper deals with the formulation of a novel four-node quadrilateral finite element based on the first-order shear deformation theory and using the strain-based approach for static and free vibration responses of composite plates. The developed element has five degrees of freedom per node and its displacement functions contain higher-order terms that satisfy the rigid body modes and are based on assumed strains satisfying compatibility equations. The efficiency of the present element is demonstrated by applying it to sandwich and symmetric/antisymmetric laminated composite plates with various shapes, boundary conditions, side-to-thickness ratios, stacking sequences, fiber orientation angles, and different types of loads. The obtained results are compared with published analytical and numerical solutions that have proved excellent accuracy and efficiency in predicting bending and free vibration of laminated composite plates and have shown that this element is free of shear locking.

Nonlinear free vibration analysis of functionally graded plates and shell panels using quasi-3D higher order shear deformation theory

This paper presents the nonlinear free vibration behavior and response of functionally graded plates and shell panels under various boundary conditions. The displacement formulation followed uses a sinusoidal non-polynomial quasi-3D higher order shear deformation theory with six variables and, the parabolic transverse stress distribution functions are used to model stresses in the thickness direction. Present formulation employs the Green-Lagrange nonlinear strains along with Sander’s approximation to account for larger flexural response which could be experienced by the structure. An eight noded C0 continuous isoparametric finite element is used for numerical implementation of nonlinear finite element solver developed in MATLAB. Developed numerical formulation is employed to study and analyze the nonlinear free vibration response of the spherical, cylindrical and hyperboloid shell panels and the obtained results are seen to be consistent and good agreement with the literature thus establishing the effectiveness and accuracy of the present work.

Free Vibration Response of FG Porous Joined Hemispherical–Cylindrical–Hemispherical Shell Vessels Reinforced by Graphene Platelet

In this research, the natural frequency behavior of functionally graded (FG) porous joined hemispherical–cylindrical–hemispherical shell vessels reinforced by graphene platelet (GPLs) has been studied for the first time. Three various types of porosity distribution are assumed through the thickness direction of shell vessel. In the two types of porosity patterns, a pattern of porosities in metal matrix is symmetric and the other one is uniform. Besides, five GPL patterns are assumed for dispersing of GPLs in metal matrix. Extended role of mixture and Tsai-Halpin is used to determine the mass density and Young’s modulus of elasticity of the structure, respectively. By employing 3D elasticity theory, Hamilton’s Principal and FEM in conjunction with Rayleigh–Ritz method, the governing equations of motion of the joined shell vessel are obtained and natural frequencies are extracted. The impact of various factors such as coefficient of porosity, several porosity patterns along with different GPLs distributions and weight fraction of graphene nanofillers on natural frequency behavior of FG porous joined hemispherical–cylindrical–hemispherical shell vessels reinforced by GPLs nanofillers has been reported for the first time.

Vibration Response Analysis of Tapered Porous FGM Plate Resting on Elastic Foundation

This paper attempts to demonstrate the vibration response of porous Functionally Graded Material (FGM) plate with variable thickness. For the first time, the porous tapered FGM plate being considered is mathematically modeled and assumed to be resting on a linear, parabolic, sinusoidal, and exponential varying Winkler’s elastic foundation. The even and uneven porosity distribution as a micro-defect is assumed in a tapered FGM plate that varies according to well-defined mathematical rules. The tapered FGM plate, across the edges, is supported with various boundary conditions. Simple Power (P-) and Sigmoid (S-) Law have been chosen for the homogenization of material properties that are tailored in the thickness direction. First-order shear deformation theory (FSDT) is applied to describe the displacement function for computing the strain field. The variational approach has been used to establish the formulation for free vibration response. The equation of motion has been derived using Hamilton’s principle and solved by implementing Galerkin Vlasov’s method. Parametric studies on elastic foundations have been done to explore the results and relevance to the real problems. It was observed that variable foundation has a significant effect on the tapered FGM plate rather than the tapered homogenous plate. In addition, it was witnessed that the effect of variable foundation effect diminishes as a constraint on the edges of the tapered FGM plate increases. Also, some benchmark results based on the porosity effect and the influence of variable elastic foundations are exhibited in this study for future reference.

On the static, vibration, and transient responses of micro-plates made of materials with different microstructures

In the present paper, for the first time, theoretical analysis for static, dynamic, and transient responses of micro-plates made of different materials were comprehensively performed. The constitutive relations of analyzed materials satisfy assumptions of the modified couple stress theory. Additionally, values of length scale parameters of studied materials are taken from well-established experimental results presented in the literature. A generalized model of micro-plates made from Cu, Ni, Ti, and Epoxy satisfies assumptions of Reissner–Mindlin plate theory. The formulated generalized boundary value problem is solved by analytical and the radial basis function finite difference (RBF-FD) methods. A convergence study is performed to show the advantages and correctness of the obtained numerical solution. The meshless numerical method is easy to implement and it provides accurate results that are in excellent agreement with analytical solutions. Effects of material parameters, boundary conditions, and assumptions of classical and the modified couple stress theory on the static bending, free vibration and transient responses of micro-plates are presented and discussed.

Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory

In the present study, size-dependent static and free torsional vibration responses of functionally graded porous nanotubes are examined using Fourier sine series and Stokes’ transformation for the first time. The boundary conditions of functionally graded porous nano-sized tubes are defined by the two elastic torsional springs at the both ends. A power law rule is utilized to describe the distribution of functionally graded material and this distribution is considered through the radius of nanotube. The governing equations of the mechanical response of porous nanotubes with elastic boundary conditions and subjected to torsion are accomplished via the modified couple stress theory. The purpose of the presented work is to construct an eigen value solution including the small scale parameter based on the modified couple stress theory, torsional spring coefficients representing the boundaries of the porous nanotubes, functionally graded index caused by power law rule and porosity volume fraction.

On the flexoelectric effect on size-dependent static and free vibration responses of functionally graded piezo-flexoelectric cylindrical shells

The flexoelectric effect depends on the strain gradient, which naturally appears in intelligent structures at micro and nano scales. Therefore, the flexoelectric phenomenon significantly affects electro-mechanical behavior of structures in these scales. In this study, the first-order shear deformation theory and the reformulated flexoelectric theory are applied to derive coupling equations of motion for a size-dependent static and dynamic responses of multilayered composite shell with simply supported and clamped edges. Constitutive equations are defined based on assumption that the internal energy function depends not only on strains and polarization tensors but also on strains and polarization gradient tensors. The piezo-flexoelectric structure is made of an isotropic functionally graded core and two orthotropic layers. Reynolds transport theorem and variational Hamilton principle are used to derive equations of motion expressed by forces and displacements, and corresponding boundary conditions. In order to check the correctness of the formulation, and solution of the problem obtained by Galerkin method, static and free vibrations analyses of macro-/micro-/nanoshell are investigated. Diagrams of deflection and electrical potential variations of the structure under the load are obtained for the direct flexoelectric effect. Additionally, the influence of various parameters, e.g., geometric properties, size effect, power-law index, and flexoelectric layer thickness, on the deflection, potential function, and natural frequency of the microshell have been investigated. The numerical results and discussion present that above parameters significantly affect a mechanical behavior of the microshell under consideration.

Radial basis collocation method for free vibration analysis of elastically supported porous bi-directional FGM plate under various types of porosity distribution

ABSTRACT In this paper, the radial basis collocation method (RBCM) for the free vibration response of elastically supported porous bi-directional functionally graded (BDPFG) plate is investigated in a framework of higher-order shear deformation theory (HSDT). The material properties of BDPFG plates are assumed to be varying continuously in the length and thickness direction. The material properties are evaluated based on Voigt’s micro-mechanical model considering power law distribution method with arbitrary power index. Governing differential equations (GDEs) of the porous BFGM plate area achieved utilising Hamilton’s principle and multiquadric radial basis function (MQRBF) is used for discretizing the GDEs. To incorporate porosity in the BDPFG plate, five types of porosity distribution models are carried out and analysed. The present results are validated with the available published results in the literature to certify the efficacy and accuracy. The influence of grading index in the length and thickness direction, porosity distribution, porosity index, span to thickness ratio, and elastic foundation over natural frequency of bi-directional BDPFG plate is investigated which is the main contribution.

The numerical calculation method based on equivalent frequency-independent time-domain damping model

In order to realize the time-domain analysis based on hysteretic damping model, the frequency-independent time-domain damping model of single degree of freedom (SDOF) system is constructed. Based on the assumed relationship of vibration responses, the equivalent frequency-independent time-domain damping model in complex domain and real domain are proposed. The characteristic that the dissipated energy in each cycle is not related to the vibration frequency of external excitation is retained for the two equivalent damping models. Combined with Newmark- β method, the corresponding numerical methods are obtained. The numerical examples show that the free vibration responses are stably convergent based on equivalent damping models. The numerical results of vibration responses of SDOF system due to earthquake wave have high calculation accuracy. Compared with equivalent frequency-independent time-domain damping model in real domain, the computational accuracy of equivalent frequency-independent time-domain damping model in complex domain is higher, and the computational efficiency is lower.

A semi-analytical method for the dynamic characteristics of stiffened plate with general boundary conditions

In this paper, the free, steady and transient vibration response of stiffened plate with general boundary conditions are studied through theoretical and experimental comparisons. Firstly, the structural energy functional is established in the framework of the first-order shear deformation theory (FSDT), the stiffened plate is divided into several equal parts according to the domain decomposition method (DDM). Next, the artificial springs are introduced to ensure the continuity of segments and diversified boundary conditions. The unified Jacobi orthogonal polynomials and trigonometric series are expanded to represents the displacement field along different directions, and the free, steady and transient vibration characteristics of stiffened plate can be obtained by using Rayleigh–Ritz method. Results for various boundary conditions, structure parameter are presented, the validity and accuracy of the proposed method is verified by comparing with the existing literatures and experiment, the arbitrary impulse excitation load and damped vibration are considered. It is clearly shown that the accurate solutions can be obtained by using proposed method and the research can supply technical support for the vibration control of stiffened plate structure.

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.