Free Vibration Of Composite Beams Research Articles

Transverse stretching vibration analysis of cross-ply composite and sandwich beams by means of an equivalent single-layer theory

Due to largely different stiffness between the face sheets and the core, the vibration modes corresponding to the transverse stretching displacement will occur in the sandwich plates. The third-order models will encounter difficulty to reasonably produce these frequencies corresponding to transverse stretching displacement modes, which will influence engineering design. Therefore, a global-local higher-order theory including transverse stretching effect has been proposed to study transverse stretching vibration, which can meet in advance the continuity conditions and the free surface conditions of transverse shear stresses. In the light of Hamilton’s principle, the equation of motion is presented, and analytical solutions of the simply-supported beams can be acquired by using Navier’s technique. In order to verify the capability of the proposed model, free vibration of composite beams is firstly investigated and the results have been compared to two-dimensional elasticity solutions. After the proposed model is assessed by two-dimensional elasticity solutions, the proposed model is extended to study dynamic behaviors corresponding to the transverse stretching displacement vibration modes, which will be verified by utilizing the three-dimensional finite element (3 D-FEM). Furthermore, variation of natural frequencies of the sandwich beams is also investigated by altering the thickness of the face sheets, which can further illustrate the capability of the proposed model.

Forced and free vibrations of composite beams solved by an energetic boundary functions collocation method

For the analyze of forced and free vibrations of composites with periodically varying interfaces, we develop an energetic boundary function method. Three types of beams under different homogeneous boundary conditions and a sequence of boundary functions for each type beam are given, which automatically satisfy the homogeneous boundary conditions. Then we derive an energy equation and construct the energetic boundary functions by further satisfying the energy equation. The solution of forced vibration is expressed in terms of energetic boundary functions as the numerical bases, and then we can derive the linear system to decide the expansion coefficients by a simple collocation technique. To find out the natural frequencies of composite beams we iteratively solve the varying linear systems which include the unknown natural frequencies, and at each iteration we employ the Rayleigh quotient to calculate the natural frequencies until convergence. Through numerical tests we find that the energetic boundary function methods are highly efficient and very accurate for solving the proposed vibration problems of composite beams.

Free vibration analysis of Timoshenko beams reinforced by BNNTs and a comparison with CNT-reinforced composite

This paper studies free vibration of composite beams reinforced by boron nitride nanotubes (BNNTs), lying on an elastic foundation. The beam matrix is taken as poly methyl methacrylate. The nanotubes have different layouts in the matrix (aligned and randomly oriented) and are also distributed through the beam thickness as one uniform and three different functionally graded distributions. A micromechanical model is applied to estimate the properties of BNNT-reinforced composite (BNNTRC). Equations of motion are achieved by means of Hamilton’s principle as well as Timoshenko beam theory. Natural frequencies are gained by introducing generalized differential quadrature method to the equations. Investigating the influence of different parameters as diverse as nanotube volume fraction, nanotubes distribution type, the elastic foundation, boundary conditions and the beam’s slenderness ratio on the natural frequency, reveals that any changes in aforementioned parameters has a significant impact on the natural frequency. Furthermore, the results for aligned BNNTRC beam are measured against the results of similar CNT-reinforced beam in the literature, illustrating that natural frequency of BNNTRC beam is lower than CNTRC beam. Finally, the comparison between natural frequencies of two BNNTRC beams with two different alignments of BNNTs, illustrates their different behavior due to different factors.

Free vibration of axially loaded composite beams using a four-unknown shear and normal deformation theory

This paper presents free vibration of composite beams under axial load using a four-unknown shear and normal deformation theory. The constitutive equation is reduced from the 3D stress-strain relations of orthotropic lamina. The governing differential equations of motion are derived using the Hamilton’s principle. A two-node C1 beam element is developed by using a mixed interpolation with linear and Hermite-cubic polynomials for unknown variables. Numerical results are computed and compared with those available in the literature and commercial finite element software (ANSYS and ABAQUS). The comparison study illustrates the effects of normal strain, lay-ups and Poisson’s ratio on the natural frequencies and load-frequency curves of composite beams.

An upper bound theory to approximate the natural frequencies and parameters identification of composite beams

For the free vibration of composite beams and non-uniform beams we propose a new upper bound theory to approximate the first few natural frequencies. The Rayleigh quotient is expressed in terms of boundary functions, instead of that in terms of eigenfunctions. The boundary function satisfies all boundary conditions of the given beam, and is at least the fourth-order polynomial with leading coefficient to be one. We prove that the maximality of the Rayleigh quotient in the space of the kth order boundary functions is equivalent to the orthogonality of the kth order boundary function to lower order optimal boundary functions. Hence, we can easily find the kth order natural frequency through an orthogonalization technique provided. When the first three natural frequencies are compared with the exact or numerically found ones, good results are obtained, which confirm the applicability of the present upper bound theory. We address the inverse problems of composite beam equations, where we use the orthogonal system of boundary functions as bases to expand the unknown functions and derive linear algebraic equations to determine the expansion coefficients. As a consequence, we can fast and accurately estimate the unknown rigidity function and planar inertial function with the help of the first three natural frequencies, and the supplemented measured data of recovered function on two boundaries. The robustness of the present inversion methods is demonstrated by numerical examples.

Free vibration analysis of composite beams with overlapping delaminations under axial compressive loading

Analytical solutions have been developed to study the free vibrations of composite beams with two overlapping delaminations under axial compressive load. The delaminated beam is analyzed as seven Euler–Bernoulli beams connected at the delamination boundaries. The continuity and equilibrium conditions are satisfied between the adjoining regions of the beams. Lower and upper bounds of the natural frequencies of the delaminated beams are identified by assuming totally ‘free’ and totally ‘constrained’ deformations of the delaminated layers, respectively. These solutions can serve as generalized solutions for vibration and buckling of delaminated composites. The relation between the square of the natural frequency of the delaminated beam and the axial compressive load is also investigated. Results show a linear relation between the square of the “constrained mode” and “free mode” frequencies of the simply supported beam and the axial compressive load. A non-linear relation is observed between the square of the “free mode” frequencies of a clamped–clamped beam and axial compressive load due to the opening of the delaminated layers in the “free mode” mode shape of the beam.

Coupled free vibration of composite beams with asymmetric cross-sections

A set of linear differential equations of motion for coupled free vibrations of composite beams with asymmetric cross-sections is obtained in this paper based on Euler–Bernoulli beam theory. More coupling terms are included in the governing equations; and hence, both even and odd order spatial derivatives are included. All these features lead to much difficulty in solving the resultant equations. The axially unloaded condition of F1=0, instead of e=0, is adopted to simplify the equations of motion for the case of inextensional beams. An algorithm is developed for solving the resultant equations to obtain natural frequencies and modes of the beams. Numerical examples for validation show that the equations of motion obtained in this paper are correct and the corresponding algorithm is effective and easy to use.

Approximate analytical solutions for the nonlinear free vibrations of composite beams in buckling

We present approximate analytical solutions for the nonlinear free vibrations of symmetrically or asymmetrically laminated composite beams in prebuckling and postbuckling. Simply supported and clamped–clamped boundary conditions are considered. Galerkin’s discretization is used to obtain the nonlinear ordinary differential equations governing the large-amplitude vibrations of composite beams in prebuckling and postbuckling, which are found to be of the same form. The variational method of He [20,21] is used to derive an approximate analytical solution for the nonlinear natural frequency and the nonlinear load–deflection relation. Results obtained by using the proposed analytical solution is compared with the finite element results available in the literature and a good agreement has been obtained. Numerical results to show the variation of the nonlinear natural frequency with the applied axial load for a variety of composite laminates are presented. The contribution of the amplitude of vibration on the nonlinear load–deflection response and the nonlinear natural frequency is found to be significant.

Vibration of Beams with a Single Delamination under Axial Loading

In this work analytical solutions are developed to study the free vibration of composite beams under axial loading. The beam with a single delamination is modeled as four interconnected Euler-Bernoulli beams using the delamination as their boundary. The continuity and the equilibrium conditions are satisfied between the adjoining beams. The studies show that the sizes and the locations of the delaminations significantly influence the natural frequencies and mode shapes of the beam. A monotonic relation between the natural frequency and the axial load is predicted.

Corrigendum to “Postbuckling and free vibrations of composite beams” [Compos. Struct. 88 (2009) 636–642

Corrigendum to “Postbuckling and free vibrations of composite beams” [Compos. Struct. 88 (2009) 636–642

Postbuckling and free vibrations of composite beams

Abstract An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.

Geometrically non-linear approximations on stability and free vibration of composite beams

Second-order formulations based on moderate rotations are often used in finite element models because this approximation facilitates their implementation. However, in some cases this approximation is not enough to obtain all couplings among bending, twisting and stretching deformations for a beam member. Therefore, a simplified second-order formulation may lead to the subsequent loss of some significant terms in the non-linear strains. Without these terms the beam model may predict inaccurate results when the effect of large rotations is important. The present paper investigates the influence of large rotations on the buckling and free vibration behavior of thin-walled composite beam. To carry out this analysis it is necessary to use a geometrically higher-order non-linear beam theory. A distinctive feature of the present study over others available in the literature is that this beam model incorporates, in a full form, the effects of shear flexibility (bending and warping shear). The numerical results show that second-order formulation overestimates the buckling load and natural frequency values when the effect of large rotation is important, and this effect depends on some factors such as load condition and stacking sequences.

Investigation of free vibrations of composite beams by using the finite-difference method

The free-vibration behavior of symmetrically laminated fiber-reinforced composite beams with different boundary conditions is examined. The effects of shear deformation and rotary inertia, separately and/or in combination, on the free-vibration properties of the beams are investigated. The finite-difference method is used to solve the partial differential equations describing the free-vibration motion in each case. The effect of shear deformation on the natural frequencies is considerable, especially for higher frequencies, whereas the influence of rotary inertia is less significant. The study includes comparisons with results available in the literature. In addition, the impact of such factors as the span/depth ratio, fiber orientation, stacking sequence, and material type on free vibrations of the composite beams is investigated.

The influence of delamination on free vibrations of composite beams on Pasternak soil

Free vibrations of delaminated beams resting on Pasternak soil are analysed. Differential stretching and bending-extension coupling are considered in the formulation. The influence of soil parameters, size and location of the delamination on the frequencies and mode shapes is investigated. Some numerical examples and comparisons are presented.

Two-dimensional solutions for orthotropic materials by the state space method

A state space method for two-dimensional problems of orthotropic materials is proposed which provides an alternative method to investigate the mechanical behavior of homogeneous and laminated beams. The method is illustrated for a homogenous/laminated beam subjected to time-dependent transverse loads. A particular solution for a straight-crested wave propagating along an infinite beam is obtained. The bending, buckling and free vibrations of composite beams are also studied. Numerical results are presented and compared with those available in the literature.

Free vibration analysis of composite beams with overlapping delaminations

The free vibration of composite beams with two overlapping delaminations have been solved analytically without resorting to numerical approximation. The delaminated beam is analyzed as seven interconnected Euler–Bernoulli beams using the delaminations as their boundaries. The continuity and equilibrium conditions are satisfied between the adjoining regions of the beams. Lower and upper bounds of the natural frequencies of the delaminated beams are identified by assuming totally ‘free’ and totally ‘constrained’ deformations of the delaminated layers, respectively. The influence of the delaminations on the natural frequencies and mode shapes of the cantilever and clamped–clamped beams are investigated. Results show the dominating influence of the longer delamination on the natural frequency of the beam. Similar trends are observed for the second mode frequency and mode shape of the cantilever beam and the fundamental frequency and mode shape of the clamped–clamped beam. Comparison with analytical and experimental results reported in the literature verifies the validity of the present solution.

Effect of material coupling on wave vibration of composite Euler–Bernoulli beam structures

Fiber-reinforced composite materials are an important development in the continuing search for lightweight materials for great strength and stiffness. Lighter structures are usually more prone to vibrations. Furthermore, due to material coupling, composite beams normally vibrate with coupled vibration modes, which complicates vibration analysis. This paper presents the effect of coupling between bending and torsional deformations on vibrations of composite Euler–Bernoulli beams from a wave vibration standpoint. In the study, it is found that the torsional mode is only unaffected by the material coupling at low frequencies. The flexural modes are found affected over the entire frequency band. Unlike their uncoupled metallic counterparts, the wavenumbers for propagating and decaying wave components are no longer the same due to material coupling. Free vibration analysis is performed from a wave standpoint. Free vibration of composite beam with the coupling effect taken into account is of great importance. This is because the free vibration characteristics can be designed favorably by suitable choice of ply orientations and stacking sequence—one of the unique features of composite structures. Numerical examples for which comparative results are available in the literature are presented.

Free vibration analysis of composite beams with two non-overlapping delaminations

An analytical solution to the free vibration of composite beams with two non-overlapping delaminations is presented. The delaminated beam is modeled as seven interconnected Euler–Bernoulli beams using the delaminations as their boundaries. The continuity and the equilibrium conditions are satisfied between adjoining beams. The analysis includes the differential stretching between the delaminated layers and the bending–extension coupling. The results of the present model agree well with the analytical and experimental data reported in the literature. Parametric studies show that the sizes and locations of the delaminations have significant effect on the natural frequencies and mode shapes. These results provide useful information in the study of the free vibration of delaminated composite beams.

Dynamics of anisotropic composite cantilevers weakened by multiple transverse open cracks

In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks as used in this paper have been derived via Hamilton's variational principle in conjunction with Timoshenko's beam model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a single crack. It should be mentioned that to the best of the authors' knowledge, with the exception of the present study, the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.

Free vibration of composite beams featuring interlaminar bonding imperfections and exposed to thermomechanical loading

In this paper, the implications played by sliding imperfections on the static response, buckling and free vibration behavior and on the frequency-load-temperature interaction of simply-supported laminated plates in cylindrical bending are investigated and pertinent conclusions outlined. To this end, a recently developed theory for multilayered beam-plates with slightly weakened interfaces, which fulfils (i) the shear traction continuity requirements at the interfaces, (ii) the free-shear condition at the upper and lower surfaces and (iii) incorporates the effect of imperfect bonding by featuring the presence of interlayer jumps of tangential displacements, is used. Within this framework, a number of numerical results by a closed-form approach, relative to symmetric and unsymmetric lamination schemes and various length-to-thickness ratios are supplied which illustrates the effects of the interfacial weakness. The obtained results reveal the detrimental effects induced by the bonding flaws upon eigenfrequencies, which appear the most sensitive, buckling loads and deflections.