Experimental and Analytical Study of Initial Condition Effects On Nonlinear Vibrations of Thin-Walled Beams

This article is concerned with the non-linear free vibration and transient response of laminated composite cylindrical and spherical shells with piezoelectric layers in thermal environments. The theoretical formulations are based on the first-order shear deformation theory and the von Karman-type non-linear kinematics. The analysis is carried out using the quadratic C 0 eight-noded isoparametric element. The governing non-linear equations are solved by using the direct iteration method for the eigenvalue problem for free vibration and the Newmark average acceleration method in time integration in conjunction with the modified Newton-Raphson iteration scheme for the transient analysis. The validity of the numerical model is demonstrated by comparing the present results with those available in the literature. The effects of temperature, voltage, curvature, thickness, number of layers and boundary conditions on the non-linear free vibration and transient response of piezoelectric laminated cylindrical and sp...

Functionally graded nanocomposites in which carbon nanotube (CNT) or graphene reinforcements are dispersed nonuniformly in the matrix have been considered as the new generation materials with great application potentials in various engineering areas such as aerospace, automobile, energy, biomedical and electronic industries. The mechanical analysis of structural elements made from such nanocomposites have therefore attracted increasing attention from both research and engineering communities due to their great importance in both theoretical and practical aspects.This thesis deals with the nonlinear behaviour and imperfection sensitivity of functionally graded nanocomposite structures reinforced with either CNTs or graphene platelets (GPLs). Their material properties are assumed to be functionally graded along the thickness direction, among which the effective material properties of functionally graded CNT-reinforced composites (FG-CNTRCs) are predicted by the extended rule of mixture, while that of functionally graded GPL-reinforced composites (FG-GPLRCs) are estimated by the modified Halpin-Tsai model. The research content of this thesis includes several aspects: 1) buckling and free vibration analysis of sandwich beams with FG-CNTRC face sheets, 2) imperfection sensitive analysis of FG-CNTRC beams, 3) thermo-electro-mechanical analysis of piezoelectric FG-CNTRC beams with geometric imperfections, 4) mechanical analysis of FG-GPLRC beam and plate structures under thermo-mechanical loading.Linear governing equations of sandwich beams with FG-CNTRC facesheets are derived within the framework of first-order shear deformation theory (FSDT). By using the differential quadrature (DQ) method, the buckling and free vibration behaviours of FG-CNTRC sandwich beams are investigated. A parametric study is conducted to show the effects of CNT volume fraction, core-to-facesheet thickness ratio, slenderness ratio, and end supports on the critical buckling load and natural frequencies. Numerical results for sandwich beams with uniformly distributed CNTRC (UD-CNTRC) face sheets are also provided for comparison. The results demonstrate that the sandwich beam with FG-CNTRC facesheets outperforms the beam with UD-CNTRC facesheets in terms of the buckling and vibration performances.For the imperfection sensitivity analysis, nonlinear governing equations and their dimensionless forms are deduced by using the principle of virtual displacements. A generic imperfection model in the form of the product of trigonometric and hyperbolic functions are used to describe the various possible geometric imperfections such as sine type, global, and localized imperfections. The imperfection sensitivity analysis covers several subjects for FG-CNTRC beams under different loading conditions. Among those, the compressive and thermal postbuckling are studied by means of the DQ-based Newton-Raphson technique, the nonlinear free vibration is analysed by using the Ritz method together with a standard iteration procedure. Comprehensive numerical results are presented for geometrically imperfect FG-CNTRC beams, with a particular focus on the influences of imperfection parameters such as half-wave number, location, and amplitude. The results indicate that geometric imperfections have important effects on the nonlinear postbuckling and free vibration behaviours.In the thermo-electro-mechanical analysis, the FG-CNTRC beams are integrated with surface-bonded piezoelectric layers and subjected to a combined action of a uniform temperature rise, a constant actuator voltage and an in-plane force. The effects of shear deformation and geometric imperfection are taken into consideration in theoretical formulations. The thermo-electro-mechanical postbuckling equilibrium path is traced by using the DQ method in conjunction with Newton-Raphson technique. The obtained postbuckling displacements are then included in the free vibration analysis of thermo-electro-mechanically postbuckled FG-CNTRC beams. Free vibration results of postbuckled FG-CNTRC beams with and without geometric imperfections are given and compared to highlight the influence of geometric imperfections. The effects of CNT distribution pattern and volume fraction, temperature rise, actuator voltage, in-plane force, and boundary condition are also discussed in detail.Based on the FSDT and von Karman geometric nonlinearity, the nonlinear dynamic governing equations are established for shear deformable FG-GPLRC beams and plates that are subjected to a uniform temperature rise and a periodic uniaxial in-plane force. By using the DQ method, the thermo-mechanical buckling and postbuckling, as well as free vibration are studied. Subsequently, the parametric instability of FG-GPLRC beams and plates under thermo-mechanical loading are investigated, and the principal unstable region is determined by employing the Bolotin’s method. Special attention is given to the effects of GPL distribution pattern, weight fraction, and geometry so as to best explore the potentials of GPLs towards the development of advanced lightweight composite structures. Numerical results indicate that the addition of a low content of GPLs significantly improves the mechanical properties and structural performances of polymer nanocompositestructures. Symmetric distribution with more GPLs near the surface layers and few GPLs close to the neutral plane is most desirable in terms of the reinforcing effect.As the result of extensive theoretical and numerical analysis by using the computer program packages developed in MATLAB, the present thesis makes valuable contributions to the knowledge base in the subject area by providing comprehensive first-ever-known results, which are helpful in better understanding the mechanical behaviour of such functionally graded nanocomposite structures.

Motivated by the evident lack of studies on the nonlinear thermal stability and vibration of functionally graded material (FGM) circular plates (CPs) including porosity and elastically restrained edge, this article investigates the combined influences of porosities, geometrical imperfection and tangential elastic constraint of boundary edge on the nonlinear axisymmetric free vibration and postbuckling behavior of porous FGM CPs subjected to uniform temperature rise. The pores are distributed into the FGM via even and uneven distribution types. Temperature dependence of material properties is included and effective properties of porous FGM are evaluated adopting a modified mixture rule. The CP is assumed to be under axisymmetric deformation and clamped at periphery. Motion and compatibility equations in terms of deflection and stress function are derived on the basis of the first-order shear deformation theory (FSDT) taking into account nonlinear strains in von Kármán sense and initial geometric imperfection. The derived equations are solved by using analytical solutions and Galerkin method. In the thermal postbuckling analysis, an iteration algorithm is employed to evaluate critical temperatures and postbuckling temperature–deflection curves. In the nonlinear free vibration analysis, fourth–order Runge–Kutta scheme is adopted to seek the frequencies of nonlinear free vibration. After verification, parametric studies are presented to assess numerous influences on the thermoelastic nonlinear vibration and postbuckling of porous FGM CPs. It is found that rigorous constraint of edge renders the thermal buckling resistance capacity weaker and the frequency nonlinearity stronger.

Nonlinear free and forced vibration of porous piezoelectric doubly-curved shells based on NUEF model

The nonlinear free vibration and transient response of laminated composite cylindrical and spherical shells with imperfection in hygrothermal environments is studied using the finite element method. The theoretical formulations are based on the first-order shear deformation theory and von Kármán-type nonlinear kinematics. An imperfection function capable of modeling a variety of sine type, global type, and localized type imperfections is used. The analysis is carried out using quadratic C0 eight-noded isoparametric element. The governing nonlinear equations are solved using the direct iteration method for the eigenvalue problem for free vibration and Newmark average acceleration method in the time integration in conjunction with modified Newton—Raphson iteration scheme for transient analysis. The validity of the model is demonstrated by comparing the present results with those available in the literature. The effects of moisture, temperature, amplitude of imperfection, and imperfection types on the nonlinear free vibration and transient responses of laminated composite shells are studied.

Two-degrees-of-freedom nonlinear free vibration analysis of magneto-electro-elastic plate based on high order shear deformation theory

This paper presents an analytical solution for nonlinear free and forced vibration response of smart laminated nano-composite beams resting on nonlinear elastic foundation and under external harmonic excitation. The structure is under a temperature change and an electric excitation through the piezoelectric layers. Different distribution patterns of the single walled aligned and straight carbon nanotubes (SWCNTs) through the thickness of the beam are considered. The beam complies with Euler-Bernoulli beam theory and von Kármán geometric nonlinearity. The nonlinearity is due to the mid-plane stretching of the beam and the nonlinear stiffness of the elastic foundation. The Multiple Time Scales perturbation scheme is used to perform the nonlinear dynamical analysis of functionally graded carbon nanotube-reinforced beams. Analytical expressions of the nonlinear natural frequencies, nonlinear dynamic response and frequency response of the system in the case of primary resonance have been presented. The effects of different parameters including applied voltage, temperature change, beam geometry, the volume fraction and distribution pattern of the carbon nanotubes on the nonlinear natural frequencies and frequency-response curves are presented. It is found that the volume fractions of SWCNTs as well as their distribution pattern significantly change the behavior of the system.

A symplectic Galerkin method for non-linear vibration of beams and plates

Studies of vibration of plates have matured and are a well-established branch of research in structural dynamics. They have a vast range of applications in engineering and technology. But not much work can be found on vibration analysis of Functionally Graded Materials (FGMs) as compared to isotropic and composite plates and shells. FGMs are those in which the volume fraction of the two or more constituent materials is varied, as a power-law distribution, continuously as a function of position along certain dimension(s) of the structure [1, 2]. From the perspective of finite element method (FEM) studies of FGM, Praveen and Reddy [3], studied the static and dynamic responses of functionally graded (FG) ceramic-metal plate accounting for the transverse shear deformation, rotary inertia and moderately large rotations in the Von-Karman sense, in which the effect of an imposed temperature field on the response of the FG plate was discussed in detail. Ng et al. [4] dealt with the parametric resonance of FG rectangular plates under harmonic in-plane loading. Ferreira and Batra [5] provided a global collocation method for natural frequencies of FG plates by a meshless method with first order shear deformation theory (FSDT). Woo et al. [6] presented an analytical solution for the nonlinear free vibration behavior of FGM plates, where the fundamental equations were obtained using the Von-Karman theory for large transverse deflection, and the solution was based in terms of mixed Fourier series. Zhao et al. [7] studied the free vibration analysis of metal and ceramic FG plates using the element-free kpRitz method. The FSDT was employed to account for the transverse shear strain and rotary inertia, mesh-free kernel particle functions were used to approximate the two-dimensional displacement fields and the eigen-equation was obtained by applying the Ritz procedure to the energy functional of the system. Batra and Jin [8] used the FSDT coupled with the FEM to study the free vibrations of an FG anisotropic rectangular plate with various edge conditions. Also, Batra and Aimmanee [9] studied a higher order shear and normal deformable plate theory by FEM. Many studies conducted on FGMs are related to the analysis of free vibration by applying FSDT (see [10-12] and the references there in). Other forms of shear deformation theory, such as the third order-shear deformation theory (TSDT) that accounts for the transverse effects, have been considered. Cheng and Batra [13]

Nonlinear vibration analysis of a double curved shallow sandwich shell in which the core made of three-phase nanocomposite and the two-outer layer of electromagnetic materials

ACOMPUTATIONAL method for design sensitivity analysis of an eigenvalue and an eigenvector of a beam under nonlinear forced vibration is presented in this paper. The nonlinear vibration problem is only analyzed once in the proposed method. The geometric nonlinearity of concern results from the large deflection of a beam. The finite element system equation for nonlinear vibration is symmetric. However, it is found that the equation for computing the design sensitivity of an eigenvector is linear and unsymmetric. A numerical example is included to validate the proposed computational procedure. Contents The nonlinear vibrations studied herein refer to the periodic (though not necessarily harmonic) and stable motions of a nonlinear system. Recently, Hou and Yuan1 addressed the design sensitivity analysis of eigenvalues and eigenvectors of a beam under nonlinear free vibration. In that paper, the longitudinal inertia was not considered in the nonlinear vibration formulation, so that the problem could be expressed in terms of the lateral deflection alone. In the present work the design sensitivity analysis has been extended successfully to a nonlinear forced vibration problem whose formulation includes the in-plane inertia and displacement. The variational formulation2 for the nonlinear forced vibrations of beams under harmonic excitation is given as

This article deals with the large amplitude free and forced vibration analysis of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) annular sector plates based on a numerical approach. The modified rule of mixture is used to estimate the material properties. The equations of motion are developed based on the first‐order shear deformation theory (FSDT) and the von Kármán geometric nonlinearity. First, the discretized form of energy functional of structure is given with the aid of variational differential quadrature method. Then, a time periodic discretization is performed and the frequency response of the nanocomposite plate is determined using the pseudo‐arc length continuation method. After verifying the correctness of the proposed approach, a comprehensive parametric study is presented to investigate the effects of important factors on the nonlinear vibration characteristics of the FG‐CNTRC annular sector plates. The results imply that the volume fraction and distribution type of nanotubes have considerable effects on the fundamental frequency as well as nonlinear frequency response curves. POLYM. COMPOS., 40:E1364–E1377, 2019. © 2018 Society of Plastics Engineers

This paper is devoted to investigate the nonlinear vibration characteristics and active control of composite lattice sandwich plates using piezoelectric actuator and sensor. Three types of the sandwich plates with pyramidal, tetrahedral and Kagome cores are considered. In the structural modeling, the von Karman large deflection theory is applied to establish the strain–displacement relations. The nonlinear equations of motion of the structures are derived by Hamilton’s principle with the assumed mode method. The nonlinear free and forced vibration responses of the lattice sandwich plates are calculated. The velocity feedback control (VFC) and H∞ control methods are applied to design the controller. The nonlinear vibration responses of the sandwich plates with pyramidal, tetrahedral and Kagome cores are compared. The influences of the ply angle of the laminated face sheets, the thicknesses of the lattice core and face sheets and the excitation amplitude on the nonlinear vibration behaviors of the sandwich plates are investigated. The correctness of the H∞ control algorithm is verified by comparing with the experiment results reported in the literature. The controlled nonlinear vibration response of the sandwich plate is computed and compared with that of the uncontrolled structural system. Numerical results indicate that the VFC and H∞ control methods can effectively suppress the large amplitude vibration of the composite lattice sandwich plates.

This paper is concerned with the analysis of the free and forced nonlinear vibrations of the suspended cable with thermal effects. By introducing the new thermal stressed configuration, the effect of temperature variation on the nonlinear vibration equations of motion is reflected by a non-dimensional cable tension variation factor. The partial differential equations of the planar motion are discretized to the ordinary equations via the Galerkin method, and the single-mode discretization is investigated. The Lindstedt–Poincare method and multiple scales method are applied to obtain the higher-order approximate solutions of the nonlinear free vibrations and primary resonances, respectively. Parametric investigations of temperature effects on the linear and nonlinear vibration characteristics of the suspended cable with different sag-to-span ratios and excitation amplitudes are performed. The results of the perturbation analysis show that the nonlinear free and forced vibration characteristics would be changed by the temperature variations qualitatively and quantitatively, depending on the sag-to-span ratio and the excitation amplitude. The asymmetric phenomena between the effects of warming and cooling conditions on the vibration characteristics can be observed. The crossover points between next two mode frequencies are shifted under the temperature variation, and these would significantly influence the internal resonances of the suspended cable. Finally, temperature effects on the time-history diagram of the cable axial total tension force are investigated.

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.