Nonlinear Free Vibration Analysis Research Articles

Nonlinear free and forced vibration analysis of Timoshenko nanobeams based on Mindlin's second strain gradient theory

The size-dependent geometrically nonlinear free and forced vibration behaviors of nanoscale beams are studied in this article using a numerical approach. The size effects are captured using Mindlin's second strain gradient theory (SSGT) in which the second- and third-order derivatives of displacement components are taken into account in the strain energy density. The basic relations are first derived using the Timoshenko beam theory and SSGT. Then, the variational differential quadrature (VDQ) method is adopted to solve the obtained governing equations in the context of variational formulation. Comprehensive numerical results are given to investigate the influences thickness-to-lattice parameter ratio on the nonlinear free and forced vibrations of nanobeams under different types of end conditions. Also, comparisons are made between the predictions of SSGT and the first strain gradient theory as well as the classical elasticity theory.

Linear and nonlinear free vibration analysis of laminated functionally graded shallow shells with complex plan form and different boundary conditions

Linear and geometrically nonlinear vibrations of the three-layered functionally graded shallow shells with a complex form of the base are studied. It is assumed that outer and inner layers are made of functionally graded materials (FGM) or an isotropic material (metal or ceramic). The first-order shear deformation theory of shallow shells (FSDT) is employed. Effective material properties of layers are varied along the thickness according to a power law. To calculate different mechanical characteristics for different types of lamination schemes, analytical expressions are obtained. The linear problem is solved by combining the Ritz and the R-functions method. Linearization of the nonlinear problem is carried out by a novel original approach. Namely, the initial problem is reduced to solving a sequence of linear problems including those vibrations related to linear, special type of the elasticity problem and the nonlinear system of ordinary differential equations. Validation of the proposed method and the developed software has been examined on test problems for shallow shells with rectangular plan form and different boundary conditions. In order to demonstrate possibilities of the proposed approach, new results for laminated FGM shallow shells of the complex form of the base are presented. Effects of different material distributions, lamination schemes, curvatures, boundary conditions, and geometrical parameters on natural frequencies and backbone curves are reported and analyzed.

A C1 continuous hexahedral element for nonlinear vibration analysis of nano-plates with circular cutout based on 3D strain gradient theory

Abstract The main objective of the present study is to develop a hexahedral C 1 continuous element for the three-dimensional nonlinear free vibration analysis of rectangular nano-plates with circular cutout based on the strain gradient elasticity theory. In the proposed element, in addition to the values of displacement components, the corresponding first, mixed second and mixed third order derivatives are taken into account as nodal values. Since arbitrary-shaped hexahedral elements are considered, spatial derivatives are presented using isoparametric formulation. To derive the nonlinear governing equations, the matrix form of kinetic and strain energies are written based on the three-dimensional strain gradient elasticity theory which can be reduced to the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST). The accuracy and convergence of the numerical results are first investigated. In the parametric study, the nonlinear free vibration response of nano-plate is studied for different length scale parameters, geometrical factors and boundary conditions.

Non-linear free and forced vibration analysis of sandwich nano-beam with FG-CNTRC face-sheets based on nonlocal strain gradient theory

In this paper, the nonlinear free and forced vibration responses of sandwich nano-beams with three various functionally graded (FG) patterns of reinforced carbon nanotubes (CNTs) face-sheets are investigated. The sandwich nano-beam is resting on nonlinear Visco-elastic foundation and is subjected to thermal and electrical loads. The nonlinear governing equations of motion are derived for an Euler-Bernoulli beam based on Hamilton principle and von Karman nonlinear relation. To analyze nonlinear vibration, Galerkin\' decomposition technique is employed to convert the governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Furthermore, the Multiple Times Scale (MTS) method is employed to find approximate solution for the nonlinear time, frequency and forced responses of the sandwich nano-beam. Comparison between results of this paper and previous published paper shows that our numerical results are in good agreement with literature. In addition, the nonlinear frequency, force response and nonlinear damping time response is carefully studied. The influences of important parameters such as nonlocal parameter, volume fraction of the CNTs, different patterns of CNTs, length scale parameter, Visco-Pasternak foundation parameter, applied voltage, longitudinal magnetic field and temperature change are investigated on the various responses. One can conclude that frequency of FG-AV pattern is greater than other used patterns.

Mode switching phenomenon in geometrically nonlinear free vibration analysis of in-plane inhomogeneous plates on elastic foundation

Abstract Effect of geometric nonlinearity onfree vibration behaviour of a non-uniform in-plane inhomogeneousplate on elastic foundation is carried out with an emphasis on mode switching phenomenon. The formulation is semianalytic displacement based and it is carried out in two distinct steps. First, the static problem is solved to find out the unknown displacement field by using minimum total potential energy principle. Secondly, subsequent dynamic problem is set up as an eigenvalue problem on the basis of the known displacement field. The governing set of equations in dynamic problem is obtained by using Hamilton’s principle. In static analysis, unknown co-efficient of the governing equations are solved using an iterative method, which is direct substitution with relaxation method. The dynamic problem is solved with the help of intrinsic Matlab solver. The results of the present method are validated with existing data. Backbone curve corresponding to different combinations of systemparameters are presented in non-dimensional plane.Mode switching is observed to occur in certain specific situation. The linear and nonlinear mode shapes are also furnished to support the presence of switching phenomenon.

Geometrically Nonlinear Free Vibration Analysis of Composite Rotor Blades with Advanced Tip Shapes

Geometrically Nonlinear Free Vibration Analysis of Composite Rotor Blades with Advanced Tip Shapes

Nonlinear free vibration analysis of defective FG nanobeams embedded in elastic medium

Defects in atomic structure can deteriorate the mechanical properties of nanomaterials, which is crucial to the potential application in nano electromechanical systems. This work studies the effect of the material defects on nonlinear vibration behavior of embedded functionally graded (FG) nanobeams. Employing the nonlocal strain gradient theory, the size-dependent governing equations accounting for the geometric nonlinearity and elastic medium are derived and the analytical solution for nonlinear frequency is presented. To quantify the material defects, the concept of defect degree is freshly introduced and then the defective FG nanobeam model is developed. Two methods, i.e., sensitivity based interval analysis method and iterative algorithm based interval analysis method, are proposed to solve this model in the framework of interval analysis theory. After model verification, the detailed parametric investigations are carried out to understand the combined effects of the material defects and size-dependent parameter, elastic medium as well as power-law index on the vibration frequency of FG nanobeam. Numerical results validate some significant conclusions which will offer guidance in the design of nanodevices.

Nonlinear free vibration analysis of functionally graded plate resting on elastic foundation in thermal environment using higher order shear deformation theory

This paper deals with nonlinear vibration analysis of functionally graded plate resting on Pasternak elastic foundation in thermal environment. A mathematical model is developed based on higher-order shear deformation theory using Green-Lagrange type nonlinearity. The model includes all the nonlinear terms to obtain a general form and to present the original flexure of the plate. The material properties are considered as temperature dependent and graded along thickness direction obeying a simple power law distribution in terms of volume fraction of the constituents. The compression/traction free condition is employed to obtain the simplified model with seven parameters instead of nine parameters. The plate model has been discretized into C0 eight noded quadratic elements with seven degrees of freedom per node. The governing equation of the functionally graded plate has been derived using Hamilton’s principle and is solved using direct iterative method. The present model is validated by comparing the obtained results with those published in the literature. The effects of volume fraction index, aspect ratio, thickness ratio, support conditions, elastic foundation modulus and temperature on the nonlinear frequencies of the functionally graded plates are discussed. It has been found that the intermediate material property does not necessarily give intermediate nonlinear frequency

Nonlinear free and forced vibration analysis of microbeams resting on the nonlinear orthotropic visco-Pasternak foundation with different boundary conditions

Using the modified couple stress theory and Euler-Bernoulli beam theory, this paper studies nonlinear vibration analysis of microbeams resting on the nonlinear orthotropic visco-Pasternak foundation. Using the Hamilton\'s principle, the set of the governing equations are derived and solved numerically using differential quadrature method (DQM), Newark beta method and arc-length technique for all kind of the boundary conditions. First convergence and accuracy of the presented solution are demonstrated and then effects of radius of gyration, Poisson\'s ratio, small scale parameters, temperature changes and coefficients of the foundation on the linear and nonlinear natural frequencies and dynamic response of the microbeam are investigated.

Geometrically nonlinear free and forced vibrations analysis of clamped-clamped functionally graded beams with multicracks

Geometrically nonlinear free and forced vibrations of clampedclamped Functionally Graded beams with multi-cracks, located at different positions, based on the equivalent rotational spring model of crack and the transfer matrix method for beams is investigated. The FG beam properties are supposed to vary continuously through the thickness direction. The theoretical model is based on the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. A homogenization procedure, taking into account the presence of the crack, is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. Upon assuming harmonic motion, the discretized expressions for the total strain and kinetic energies of the beam are derived, and through application of Hamilton’s principle and spectral analysis, the problem is reduced to a nonlinear algebraic system solved using an approximate explicit method developed previously (second formulation) to obtain numerically the FG multi-cracked beam nonlinear fundamental mode and the corresponding backbone curves for a wide range of vibration amplitudes. The numerical results presented show the effect of the number of cracks, the crack depths and locations, and the volume fraction on the beam nonlinear dynamic response.

Nonlinear free and forced vibration of Euler-Bernoulli beams resting on intermediate flexible supports

This paper deals with the geometrically nonlinear free and forced vibration analysis of a multi-span Euler Bernoulli beam resting on arbitrary number N of flexible supports, denoted as BNIFS, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. First, the linear case is examined to obtain the mode shapes used as trial functions in the nonlinear analysis. The beam bending vibration equation is first written in each span. Then, the continuity requirements at each elastic support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. Numerical results are given to illustrate the effects of the support stiffness and locations on the natural frequencies and mode shapes of the BNIFS. The nonlinear theory is then developed, based on the Hamilton’s principle and spectral analysis. The nonlinear beam transverse displacement function is defined as a linear combination of the linear modes calculated before. The problem is reduced to solution of a non-linear algebraic system using numerical or analytical methods. The nonlinear algebraic system is solved using an explicit method developed previously (second formulation) leading to the amplitude dependent nonlinear fundamental mode of the BNIFS.

Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system

Investigation of critical speeds and nonlinear free vibration analysis of a highly flexible rotor system have been studied. The rotary inertia and gyroscopic effect combined with inextensible geometric condition for pinned-guided shaft element have been taken into account to develop the governing equation of motion. The closed-form mathematical expressions have been derived for determining both linear and nonlinear natural frequencies and their behavioral patterns have been simultaneously demonstrated through time histories, FFTs and Poincare’s maps upon changing the control parameters. The nonlinear natural frequencies have been found to be higher by about 4–6% as compared to the findings obtained via linear analysis. For lower spin speed of the shaft, both linear and nonlinear forward natural frequencies have been found to be dominant. The influence of rotary inertia on natural frequencies by considering Euler and Rayleigh beam elements, separately has been reported. The system response has been observed to be either periodic or quasi-periodic depending on the spin speed. Outcomes which were not explored earlier enable significant theoretical understanding of free vibration analysis and whirling speeds of rotating system which are of great practical importance for investigating further dynamic performance.

Thermoelectrically induced nonlinear free vibration analysis of piezo laminated composite conical shell panel with random fiber orientation

Abstract This paper presents the free vibration response of piezo laminated composite geometrically nonlinear conical shell panel subjected to a thermo-electrical loading. The temperature field is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E2 only. The material properties are assumed to be independent of the temperature and the electric field. The basic formulation is based on higher order shear deformation plate theory (HSDT) with von-Karman nonlinearity. A C0 nonlinear finite element method based on direct iterative approach is outlined and applied to solve nonlinear generalized eigenvalue problem. Parametric studies are carried out to examine the effect of amplitude ratios, stacking sequences, cone angles, piezoelectric layers, applied voltages, circumferential length to thickness ratios, change in temperatures and support boundary conditions on the nonlinear natural frequency of laminated conical shell panels. The present outlined approach has been validated with those available results in the literature.

Nonlinear Axisymmetric Free Vibration Analysis of Liquid-Filled Spherical Shell With Volume Constraint

Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

Nonlinear free vibration analysis of thermally induced FG-CNTRC annular plates: Asymmetric versus axisymmetric study

Due to axisymmetric fundamental vibrational mode shape, the most studies on the large-amplitude free vibration of annular plates have been presented based on the axisymmetric formulation. However, the initial thermal loading can change the vibration behavior of annular plates. To analyze this effect, the nonlinear free vibration of carbon nanotube (CNT) reinforced composite annular plates is investigated under thermal loading based on the asymmetric formulation. The material properties of the CNT-reinforced composites are assumed to vary continuously along the thickness direction and estimated through a micromechanical model by which the temperature-dependency is taken into account. The governing equations are derived on the basis of first-order shear deformation theory, and an efficient numerical variational method is employed to solve the problem. In this regard, the quadratic form of the energy functional is directly discretized using numerical differential and integral operators, and the pseudo-arc length continuation method is then applied to find the frequency response of the structure. The numerical results of asymmetric and axisymmetric formulations are compared and it is observed that in the presence of initial thermal loading, the axisymmetric analysis leads to inaccurate results and complete asymmetric formulation should be considered.

Nonlinear Free Vibration Analysis of Micro-beams Resting on Viscoelastic Foundation Based on the Modified Couple Stress Theory

AbstractIn this paper, nonlinear free vibration analysis of micro-beams resting on the viscoelastic foundation is investigated by the use of the modified couple stress theory, which is able to capture the size effects for structures in micron and sub-micron scales. To this aim, the gov-erning equation of motion and the boundary conditions are derived using the Euler–Bernoulli beam and the Hamilton’s principle. The Galerkin method is employed to solve the governing nonlinear differential equation and obtain the frequency-amplitude algebraic equation. Final-ly, the effects of different parameters, such as the mode number, aspect ratio of length to height, the normalized length scale parameter and foundation parameters on the natural fre-quency-amplitude curves of doubly simply supported beams are studied.

Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation

The purpose of this research is to study the nonlinear free vibration and post-buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams resting on a nonlinear elastic foundation. Uniformly and functionally graded distributions of single walled carbon nanotubes as reinforcing phase are considered in the polymeric matrix. The modified form of rule of mixture is used to estimate the material properties of CNTRC beams. The governing equations are derived employing Euler-Bernoulli beam theory along with energy method and Hamilton\'s principle. Applying von Karman\'s strain-displacement assumptions, the geometric nonlinearity is taken into consideration. The developed governing equations with quadratic and cubic nonlinearities are solved using variational iteration method (VIM) and the analytical expressions and numerical results are obtained for vibration and stability analysis of nanocomposite beams. The presented comparative results are indicative for the reliability, accuracy and fast convergence rate of the solution. Eventually, the effects of different parameters, such as foundation stiffness, volume fraction and distributions of carbon nanotubes, slenderness ratio, vibration amplitude, coefficients of elastic foundation and boundary conditions on the nonlinear frequencies, vibration response and post-buckling loads of FG-CNTRC beams are examined. The developed analytical solution provides direct insight into parametric studies of particular parameters of the problem.

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

ABSTRACTIn this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.

Nonlinear Free Vibration Analysis of Fluid Conveying FG Tube Via Energy Balance Method

Hereby, a fluid conveying tube made by FG (functionally graded) materials was considered. For which a higher-order beam model for tubes which can satisfy the shear stress boundary conditions on the inner and outer surfaces was applied. Through the Hamilton’s principle, the governing equations were gained. By keeping the first order mode shape and using energy balance method, the linear and nonlinear frequencies were obtained. Numerical simulation justified the correctness of these results. Discussions on the relationship between the frequencies and the FG tube’s parameter were also performed.

Nonlinear Free Vibration Analysis of Laminated Carbon/Epoxy Curved Panels

Nonlinear frequency responses of the laminated carbon/epoxy composite curved shell panels have been investigated numerically and validated with in-house experimentation. The nonlinear responses have been computed numerically via customised computer code developed in MATLAB environment with the help of current mathematical model in conjunction with the direct iterative method. The mathematical model of the layered composite structure derived using various shear deformable kinematic models (two higher-order theories) in association with Green-Lagrange nonlinear strains. The current model includes all the nonlinear higher-order strain terms in the formulation to achieve generality. Further, the modal test has been conducted experimentally to evaluate the desired frequency values and are extracted via the transformed signals using fast Fourier transform technique. In addition, the results are computed using the simulation model developed in commercial finite element package (ANSYS) via batch input technique. Finally, numerical examples are solved for different geometrical configurations and discussed the effects of other design parameters (thickness ratio, curvature ratio and constraint condition) on the fundamental linear and nonlinear frequency responses in details.