Vibration Of Composite Beams Research Articles

Review of Nonlocal-in-Time Damping Models in the Dynamics of Structures

In the present paper, the nonlocal-in-time damping models, called “damping-with-memory” models, are reviewed. Since such models do not involve the distribution along the longitudinal coordinate, they are easily adjustable for the FEM (Finite Element Model) algorithm, which is a big advantage due to the fact that FEM is the most-used method in engineering calculations. Within damping-with-memory models, the internal damping of a structure at the current time, is assumed to be dependent not only on the instant strain-rate magnitude or displacement-velocity magnitude but also on the strain-rate or velocity values along the previous time history. The greater the gap between the two time points, the lower the influence that one of them has on the other. To implement a composite beam vibration simulation involving damping with memory, the equation of motion of a structure written in the matrix form could be solved using the central difference method. The models constructed could be calibrated based on 3D numerical simulation data with the least squares method. It has been shown that the results obtained using the implementation of a calibrated damping-with-memory model within the 1D finite-element beam algorithm are in good correlation with those given by the 3D-FEM numerical simulation data.

Dynamic response of FG-CNTRC beams subjected to a moving mass

This article presents the forced vibration of composite beams reinforced by single-walled carbon nanotubes (SWCNTs) and subjected to a moving mass. Considering the distribution of carbon nanotubes such as uniform (UD-CNT), functionally graded Λ (FGΛ-CNT) and X (FGX-CNT), three different beams are studied. Based on a third-order shear deformation theory (TSDT), the motion equations of the beams are derived using Hamilton's principle. Including mass interaction forces, the motion equations are transformed into a finite element equation in which a two-node beam element with eight degrees of freedom is utilized. To improve the efficiency of the beam element, the transverse shear rotation is employed as an independent variable in the derivation of the beam element. The vibration characteristics, including the dynamic magnification factors and the time histories for mid-span deflections are computed by using the Newmark method. Numerical result reveal that the vibration of the beams is clearly influenced of the CNT reinforcement, and the dynamic magnification is significantly decreased by increasing the CNT volume fraction. It is also shown that the FGX-CNT beam is the best in dynamic resistance in terms of the lowest dynamic deflection and dynamic magnification factors. The effects of the total volume fraction and the moving load velocity on the dynamic behaviour of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) beams are examined in detail and highlighted.

Experimental and finite element analysis of PPF controller effectiveness in composite beam vibration suppression

In this paper the problem of vibration reduction is considered. Generally, mechanical vibrations occurring during the operation of a system are undesirable and may have a negative effect on its reliability. A finite element model of a single active blade is developed using the Abaqus software. This structure consists of a multi-layer glass-epoxy composite beam with an embedded macro fiber composite (MFC) piezoelectric actuator. For vibration control the use of a positive position feedback (PPF) controller is proposed. To include the PPF controller in the Abaqus software, a special subroutine is created. The developed control algorithm code makes it possible to solve an additional differential equation by the fourth order RungeKutta method. A numerical dynamic analysis is performed by the implicit procedure. The beam responses with and without controller activation are compared. The control subsystem model also includes the hysteresis phenomenon of the piezoelectric actuator. Numerical findings regarding the PPF controller’s effectiveness are verified experimentally.

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.

An analytical model for flexural vibration of composite beams with shear slip based on third order deformation kinematics

In this paper, an analytical solution is derived for accurately predicting the free and forced vibration responses due to time varying as well as moving loads of two layered composite beams which consist of two different materials. For this purpose, the third order deformation kinematics is used in the proposed model, and it allows to take a parabolic (third order) variation of the longitudinal displacement along the beam depth for the two material layers. The partial shear interaction between the two material layers produced by the deformability of shear connectors joining these layers in the form of shear slip at their interface is modelled using distributed shear springs along the beam length. Hamilton’s principle is applied to derive the governing equations of the dynamic system which are solved analytically using a Navier type solution technique. Moreover, a two-dimensional (2D) finite element model is built up in ABAQUS for the validation of the proposed analytical model. Some parametric studies are conducted to investigate the effect of shear deformation on the forced vibration response of a two-layered composite beam partial with different shear stiffness, span-to-depth ratio and elastic-to-shear modulus ratio.

NONLOCAL IN TIME MODEL OF MATERIAL DAMPING IN COMPOSITE STRUCTURAL ELEMENTS DYNAMIC ANALYSIS

In this paper the problem of numerical simulation of composite bending elements dynamic considering internal (material) damping. For this purpose the nonlocal in time damping model, called damping with memory, is proposed as an alternative to the classic local Kelvin-Voigt model. Damping with memory makes damping forces not only dependent on the instant value of the strain rate, but also on the previous history of the vibration process. Since finite element analysis is the most common method of structural analysis, the nonlocal damping model is integrated into FEA algorithm. The FEA dynamic equilibrium equation is solved using the explicit scheme. The damping matrix was developed using the stationary full energy requirement. One-dimensional nonlocal in time model was implemented in MATLAB software package. The results of three-dimensional numerical simulation of the composite beam vibration obtained in SIMULIA Abaqus were used for model calibration. The obtained results were compared to the results based on classic Kelvin-Voight damping model.

Analysis of the nonlinear forced vibration and stability of composite beams using the reduced-order model

The purpose of this paper is to analyze the nonlinear vibration of composite beams and the stability of the solution by using the reduced-order model and the incremental harmonic balance (IHB) method. Timoshenko beam theory is used to indicate the displacement of the beam element. Each nodal point has three degrees of freedom. Simplified homogenized beam theory is used to calculate the equivalent moduli of each ply of the composite beam. Element matrices are created using the weak form quadrature element method, and the equation of motion at the element is created using Lagrange’s equation. A system matrix is created using the element matrix assemble rule of the finite element method. In order to reduce the calculation time, a reduced-order model is used. The nonlinear forced vibration equation is solved using the IHB method. The results calculated using the non-reduced-order model and the reduced-order model are compared, and the results are very close. Based on this, the reduced-order model is used to analyze the nonlinear vibrations of composite beams at the first resonance point, and stability tests are conducted for the calculated solutions using the multivariable Floquet theory.

COMPUTER SIMULATION OF STRUCTURAL VIBRATION DAMPING WITH ALLOWANCE FOR NONLOCAL PROPERTIES

Different approaches to the computer simulation of damping properties of structural elements, which are made of materials with complicated physical structure, such as composites and nano-materials, are considered in this paper. In such cases application of classical models, e.g. ones based on the Kelvin-Voight hypothesis, can lead to the results that are not even close to the real composite structure behavior. The main point of the proposed approaches is delocalization of the damping effects in space and time. The described nonlocal damping models are more flexible in comparison to classic ones. The model calibration based on the experiment data allows to determine the optimum value of the characteristic parameter of nonlocal model using the least square method. The results of three-dimensional numerical simulation of the composite beam vibration were used for model calibration. The numerical simulation was implemented in SIMULIA Abaqus software. The material was considered as orthotropic; its parameters were picked up according to the physical properties of the real composite material. The developed beam vibration models considering nonlocal damping were created in MATLAB. The obtained results were compared to the results based on classic Kelvin-Voight damping 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.

Effect of large displacements on the linearized vibration of composite beams

Natural frequencies and mode shapes are functions of the equilibrium state. In the large displacement regime, pre-stresses may modify significantly the modal behaviour of structures. In this work, a geometrical nonlinear total Lagrangian formulation that includes cross-sectional deformations is developed to analyse the vibration modes of composite beams structures in the nonlinear regime. Equations of motion are solved around nonlinear static equilibrium states, which are identified using a Newton–Raphson algorithm along with a path-following method of arc-length type. Different boundary conditions and stacking sequences are analysed. It is shown that vibration modes are strongly modified by nonlinear phenomena. Moreover, models that do not describe those effects accurately may results in misleading results, especially if compression is dominant. In fact, results show a crossing phenomenon in the post-buckling regime of an asymmetric cross-ply beam, whereas it is completely unforeseen by the linearized analysis.

Verification of Vibration of Composite Beam with Crack by using Finite Element Method and Experimental Method

This paper presents evaluation of analytical and experimental result discussion for vibration analysis of composite beam with crack. Beams and beam like elements are principal constituents of many mechanical structures and used widely in high speed machinery, aircraft and light weight structures. Finite Element Analysis is carried out using ANSYS 14.5. The model of beam is prepared and used for Finite Element Analysis. The model analysis is used to determine the natural frequencies and mode shapes of a structure. The experimental analysis is conducted by using a Fast Fourier Transform (FFT) device which is used to detect the potential faults and checking the status of the machine through the vibration analysis.

Vibration of composite beams with deformable shear connection: Mathematical modelling and numerical solution using general-purpose software for two-point boundary value problems

In this paper, we develop from first principles a linear one-dimensional mathematical model for the dynamic response of planar composite beams with deformable shear connection, in which the inertias associated with the axial motion of the components and with the bending rotations of their cross-sections are consistently taken into account. To the best of our knowledge, the paper by Auclair et al. [J. Sound Vib. 366 (2016) 230–247] is the only work which heretofore considered these two effects. However, we point out and correct a flaw in the derivations of these authors. To solve the eigenproblem defining the natural vibration frequencies of the composite beam and the corresponding vibration modes, we advocate an untried approach – the eigenproblem is converted into a “standard” inhomogeneous boundary value problem, which can be solved using freely available state-of-the-art mathematical software. For benchmarking purposes, the formulation of a conventional displacement-based conforming finite element is also briefly addressed. The effect of rotational and axial inertias on the natural vibration frequencies is found to be marginal and considerably less pronounced than the predictions of Auclair and coworkers. In disagreement with the findings of these authors, it appears legitimate to ignore the above-mentioned effect in a full dynamic analysis of composite beams, at least within the practical range of span-to-depth ratios.

Experimental analysis of active–passive vibration control on thin-walled composite beam

In the study presented, active and passive vibration control techniques for a thin-walled composite beam were developed and experimentally analysed. An MFC (Macro Fibre Composite) actuator was used, together with the magnetic forces between two permanent cuboidal magnets (passive vibration control) for vibration damping applications. After the experiments conducted with active vibration control (MFC actuator), it was found that in all cases, the vibration damping of the analysed system was much higher at its resonant frequency. Nevertheless, due to the experimental results, it could be concluded that active and passive vibration control systems were efficient for thin-walled composite beam vibration amplitude control. Methodology for thin-walled composite beam vibration control analysis was developed and presented in the article.

Vibration Analysis of Composite Beams with Sinusoidal Periodically Varying Interfaces

Abstract As an increasing variety of composite materials with complex interfaces are emerging, we develop a theory to investigate composite beams and shed some light on new physical insights into composite beams with sinusoidal periodically varying interfaces. For the natural vibration of composite beams with continuous or periodically varying interfaces, the governing equation has been derived according to the generalised Hamiltonian principle. For composite beams having different boundary conditions, we transform the governing equations into integral equations and solve them by using the sinusoidal functions as test functions as well as the basis of the vibration modes. Due to the orthogonality of the sinusoidal functions, expansion coefficients in closed form can be found. Therefore, the proposed iterative schemes, with the help of the Rayleigh quotient and boundary functions, can quickly find the eigenvalues and free vibration modes. The obtained natural frequencies agree well with those obtained using the finite element method. In addition, the proposed method can be extended easily to laminated composite beams in more general cases or complex components and geometries in vibration engineering. The effects of different material properties of the upper and lower components and varying interface geometry function on the frequency of the composite beams are examined. According to our investigation, the natural frequency of a laminated beam with a continuous or periodically varying interface can be changed by altering the density or elastic modulus. We also show the responses of the frequencies of the components to the varying periodic interface.

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.

Transverse Vibrations of Viscoelastic Sandwich Beams via Galerkin-Based State-Space Approach

AbstractA new state-space model is formulated for the dynamic analysis of sandwich beams that are made of two thin elastic layers continuously joined by a shear-type viscoelastic (VE) core. The model can accommodate different boundary conditions for each outer layer and accounts for the rate-dependent constitutive law of the core through additional state variables. The mathematical derivation is presented with the Standard Linear Solid (SLS) model (i.e., a primary elastic spring in parallel with a single Maxwell element) and then extended to the generalized Maxwell (GM) model. The kinematics equations are developed by means of Galerkin-type approximations for the fields of axial and transverse displacements in the outer layers, and imposing the pertinent compatibility conditions at the interface with the core. Numerical examples demonstrate the accuracy and versatility of the proposed approach, which endeavors to represent the effects of the VE memory on the vibration of composite beams.

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.

Free vibration and postbuckling of laminated composite Timoshenko beams

Abstract A numerical solution method was developed to investigate the postbuckling behavior and vibrations around the buckled configurations of symmetrically and unsymmetrically laminated composite Timoshenko beams subject to different boundary conditions. The Hamilton principle was employed to derive the governing equations and corresponding boundary conditions which are then discretized by introducing a set of matrix differential operators. The pseudo-arc-length continuation method was used to solve the postbuckling problem. To study the free vibration that takes place around the buckled configurations, the corresponding eigenvalue problem was solved by means of the postbuckling configuration modes obtained in the previous step. The static bifurcation diagrams for composite beams with different lay-up laminates are given, and it is shown that the lay-up configuration considerably affects the magnitude of critical buckling load and postbuckling behavior. The study of the vibrations of composite beams with different laminations around the buckled configurations indicates that the natural frequency in the prebuckling domain increases as the stiffness of a beam increases, while there is no specific relation between the lay-up lamination and natural frequency in the postbuckling domain which necessitates conducting an accurate analysis in this area.