Three-layered Sandwich Beam Research Articles

Analysis of bending vibrations of a three-layered pre-twisted sandwich beam with an exact dynamic stiffness matrix

This paper presents the novel development of the dynamic stiffness matrix for a three-layered symmetric pre-twisted sandwich beam (PTSB), aiming to investigate its free vibration characteristics. The outer layers of the beam are modeled using the Euler–Bernoulli theory, while the core is assumed to deform solely in shear. The boundary conditions and governing partial differential equations of motion are derived based on Hamilton's principle. By applying harmonic variations of displacements, the governing equations of motion are expressed as a tenth-order equation, which is solved to obtain the desired dynamic stiffness matrix. To compute the natural frequencies of in-plane and out-of-plane free vibration for both uniform and PTSBs, the Wittrick–Williams algorithm is employed. The computed frequencies are compared with the results obtained by other authors as well as those obtained from ABAQUS simulations. Various vibration modes of uniform and twisted sandwich beams are plotted and thoroughly discussed. Interestingly, contrary to straight symmetric sandwich beams, the results indicate that flexural displacements in pre-twisted symmetric sandwich beams exhibit coupling in two planes. Additionally, although there are minor changes in vibration frequencies, the mode shapes undergo significant transformations as the pre-twist angle is altered.

Nonlinear damping and forced vibration investigation of three-layered viscoelastic sandwich beams on nonlinear elastic foundation with interlaminar continuous shear stress Zig–Zag theories

Nonlinear damping and forced vibration investigation of three-layered viscoelastic sandwich beams on nonlinear elastic foundation with interlaminar continuous shear stress Zig–Zag theories

Analysis of tapered three-layered sandwich flexural members (Stobie poles) with partial shear interaction: Analytical model

This research paper presents an analytical model for predicting the load–deflection curves of three-layered sandwich (TLS) beams with interlayer slips, accounting for material nonlinearity in all three layers. An advanced 1D beam element with five degrees of freedom per node was developed to simulate the structural behaviour of TLS beams, with two additional degrees of freedom for slippage between adjacent layers. The Euler–Bernoulli kinematic assumptions were adopted, and the governing equations were derived based on equilibrium equations, followed by the derivation of the weak form equations. The finite element formulation was then developed, and the Newton–Raphson scheme was adopted for the solution of the nonlinear system of equations, with five-point Gaussian quadrature used for the integration over the length of the beam and trapezoidal integration used for the integration over the area of the cross section. The proposed model was applied to tapered steel-concrete-steel sandwich flexural elements known as Stobie poles. The model results were verified with full-scale experimental tests and numerical models, and the effects of slippage constitutive relationship, cross-section tapering, and mechanical properties of the core and skin layers were evaluated. The presented model provides a reliable tool for designing and analysing three-layered sandwich beams with partial shear interaction.

Out-of-plane free vibration analysis of three-layer sandwich beams using dynamic stiffness matrix

This paper has been devoted to exact calculation of the natural frequencies for out-of-plane free vibration of three-layer symmetric sandwich beams by using dynamic stiffness matrix method. Governing differential equations of motion have been derived using Euler-Bernoulli beam theory and St. Venant’s torsion theory, so that the effects of shear deformation in the face layers and rotary inertia are not taken into account and the cross-section is allowed to warp without restraint. Wittrick-Williams algorithm has been used to calculate the natural frequencies. Accuracy and ability of the dynamic stiffness matrix method have been illustrated in form of several examples. Results are verified through comparison with the natural frequencies obtained from ABAQUS software. Full independence of out-of-plane bending and torsional modes of vibration for three-layer symmetric sandwich beams has been proven.

Effects of magnetoelastic loads on free vibration characteristics of the magnetorheological-based sandwich beam

In this contribution, we have investigated the effects of magnetoelastic loads on free vibration characteristics of the magnetorheological-based sandwich beam. The considered sandwich beam consists of a magnetorheological core with elastic top and base layers. For these means, the structural governing equations are derived using the Hamilton principle and solved by the finite element method. The results are validated in comparison with the existing results in the literature. The effects of variation in the parameters such as magnetic field intensity and the thickness of the core and top layers on the deviation of the first natural frequency and the corresponding loss factor are studied as well. Finally, in order to provide deep insight, the effects of magnetoelastic loads on the dynamic behavior of the three-layered sandwich beam are examined through a comprehensive survey.

Dynamic analysis of laminated composite sandwich beam containing carbon nanotubes reinforced magnetorheological elastomer

In this study, dynamic analysis of three-layered sandwich beam with thin orthotropic skins and multiwall carbon nanotubes reinforced magnetorheological elastomer (MWCNT-MRE) core have been investigated. The laminates are made of glass fiber/epoxy composites with ply orientation of [0/90] s and are prepared using vacuum-assisted hand layup process. Experiments are performed to investigate the vibration characteristics of MWCNT-MRE and MRE sandwich structures by applying the various magnetic field intensities under different end conditions. Further, the laminated composite MWCNT-MRE and MRE sandwich beams are modeled using ABAQUS finite element software and validated through experimental results in terms of natural frequencies. Also, the present finite element model is validated with available literature in terms of natural frequencies. A very good agreement is observed between the results of the experimental method and those extracted from numerical simulation. Also, the influence of applied magnetic field levels, core thickness ratio, ply orientation, and various boundary conditions on the vibration behavior of laminated composite MWCNT-MRE and MRE sandwich beams are analyzed.

Boundary feedback stabilization of a Rao-Nakra sandwich beam

In this paper, we study the boundary feedback stabilization of a three-layer Rao-Nakra sandwich beam. Since we found a proper state space so that the energy of the system is dissipative. We then convert the beam system to an abstract first-order evolution equation with operator A. Afterwards, we obtain A is the infinitesimal generator of a C0 – semigroup of contractions by using semigroup theory. The boundary damping makes the system is stable of order 1/2, which is proved by using the frequency domain characterization of polynomial stability.

Uncertainty propagation and experimental verification of nonlinear viscoelastic sandwich beams

Abstract The nonlinear dynamic analysis of viscoelastic systems is not an easy task. In some cases, it is due to the difficulty in dealing with the temperature- and frequency-dependent properties of the viscoelastic substructure. Also, since the real-life viscoelastic treatments are characterized by inherent uncertainties affecting their efficiency, their handling in the nonlinear modeling is essential from an engineering point of view. In this contribution, a stochastic modeling based on the Karhunen-Loeve expansion is proposed for a three-layer sandwich beam having nonlinear behavior using the concept of complex modulus and shift factor. The nonlinear frequency responses were obtained by using an approximated harmonic balance method with the Galerkin bases. Due to the difficulty in solving the resulting complex nonlinear eigenproblem with a frequency-dependent stiffness, making the stochastic nonlinear analyses very costly, it is proposed an iterative reduction method to approximate the complex eigenmodes. The influence of the forcing amplitude and temperature on the computed nonlinear frequency responses has been confirmed by performing laboratory experiments with a three-layer sandwich beam specimen placed inside an environmental chamber. Also, a curve-fitting has been performed to calibrate the deterministic nonlinear model using optimization tools. The envelopes of nonlinear responses demonstrate the relevance of considering the uncertainties in design variables.

Stability Analysis of a Tapered Symmetric Sandwich Beam Resting on a Variable Pasternak Foundation

The static and dynamic stability analysis of a three-layered, tapered and symmetric sandwich beam resting on a variable Pasternak foundation and undergoing a periodic axial load has been carried out for two different boundary conditions by using a computational method. The governing equation of motion has been derived by using Hamilton’s principle along with generalized Galerkin’s method. The effects of elastic foundation parameter, core-loss factor, the ratio of length of the beam to the thickness of the elastic layer, the ratio of thickness of shear-layer of Pasternak foundation to the length of the beam, different modulus ratios, taper parameter, core thickness parameter, core-density parameter and geometric parameter on the non-dimensional static buckling load and on the regions of parametric instability are studied. This type of study will help the designers to achieve a system with high strength to weight ratio and better stability which are the desirable parameters for many modern engineering applications, such as in the attitude stability of spinning satellites, stability of helicopter components, stability of space vehicles etc.

Identification of frequency-dependent viscoelastic damped structures using an adjoint method

Viscoelastic parameter identification in sandwich structures (shear modulus and loss factor) is addressed in the present paper. Finite element modelling of three-layered viscoelastic sandwich beams is considered leading to the resolution of a non-linear eigenvalue problem for modal properties determination. An adjoint-based gradient method is developed to minimize the quadratic error between the finite element model and the experimental modal data (frequencies and loss factors). The discrete adjoint approach requires the resolution of a linear system to compute the gradient of the objective function. The method is successfully validated in comparison with central finite difference schemes for gradient's computation. Two use cases are used to illustrate the performance of the method in the case of viscoelastic sandwich beams parameter identification.

Dynamic instability and free vibration behavior of three-layered soft-cored sandwich beams on nonlinear elastic foundations

The purpose of the present work was to study the dynamic instability of a three-layered, symmetric sandwich beam subjected to a periodic axial load resting on nonlinear elastic foundation. A higher-order theory was used for analysis of sandwich beams with soft core on elastic foundations. In the higher-order theory, the Reddy\'s third-order theory was used for the face sheets and quadratic and cubic functions were assumed for transverse and in-plane displacements of the core, respectively. The elastic foundation was modeled as nonlinear\'s type. The dynamic instability regions and free vibration were investigated for simply supported conditions by Bolotin\'s method. The results showed that the responses of the dynamic instability of the system were influenced by the excitation frequency, the coefficients of foundation, the core thickness, the dynamic and static load factor. Comparison of the present results with the published results in the literature for the special case confirmed the accuracy of the proposed theory.

Vibrations of sandwich beams with tip mass: Numerical and experimental investigations

A numerical and experimental investigation of the natural frequencies and frequency response of a cantilevered three-layer sandwich beam carrying a tip-mass is presented. The tip-mass is such that its center of mass is offset from the point of attachment. The higher-order sandwich panel theory (HSAPT) and the Hamilton’s principle are employed in deriving the system governing equations of motion and boundary conditions, which are solved using the generalized differential quadrature (GDQ) method. The obtained results are compared with those obtained from the use of commercial finite element software ANSYS, and they are both experimentally validated. The results are generally in good agreement. Parametric studies are conducted to examine the influence of some design variables on system vibrational behavior.

Stiffness and damping characteristics of MR fluid-based sandwich beams: experimental study

The study investigates the behaviour of three-layered cantilever sandwich beams filled with magnetorheological fluids (MRFs) differing in the iron particle content by volume. Outer layers are made of aluminium, the space between them is sealed with silicone rubber. Two types of beams are considered: fully filled beams and partially filled beams, subjected to the magnetic field. The aim of the study is to determine stiffness and damping characteristics in relation to the magnetic field strength and the actual location where the magnetic field acts upon the beam. For this purpose, measurements have been taken of the beam free vibration response for various magnetic field strength levels and for various positions of the electromagnet located along the beam axis. Basing on the developed measurement data processing algorithm, the influence of the vibration amplitude on the natural frequency and a dimensionless damping coefficient have been determined. Finally, the equivalent natural frequency and the dimensionless damping coefficient have been derived accordingly, and the stiffness and damping ratio have been determined in function of the magnetic field strength, the electromagnet position and the MRF iron particle content by volume.

Free Vibration Analysis of Symmetric Sandwich Beams

The aim of the present work is to study the linear free symmetric vibration of three-layer sandwich beam using the energy method. The zigzag model is used to describe the displacement field. The theoretical model is based on the top and bottom layers behave as Euler-Bernoulli beams while the core layer as a Timoshenko beam. Based on Hamilton’s principle, the governing equation of motion sandwich beam is obtained in order to calculate the linear frequency parameters. Two types of boundary conditions simple supported-simple-supported (SS-SS) and clamped-clamped (C-C) under the influence of materials properties and geometrical parameters are studied. The validation of results is done by comparing with another studies, which available in the literature and found good agreement between the studies.

Variability Analysis of Modal Characteristics of Frequency-Dependent Visco-Elastic Three-Layered Sandwich Beams With Spatial Random Geometrical and Material Properties

Material and physical properties of a frequency-dependent visco-elastic sandwich beam are modeled as a set of spatial random fields and represented by means of the Karhunen–Loève expansion. Variability analysis of frequency and loss factor are performed. An efficient approach based on modal stability procedure (MSP) is used, the so-called Monte Carlo simulation (MCS)–MSP method. The latter provides very reliable results and allows to analyze the impact of the input variability of a high number of random spatial quantities on the output response. The effect of independent and correlated couples of spatial random fields is investigated. It is shown that the output variability is generally more important for damping than for natural frequencies. Moreover, it is demonstrated that the input variability in geometrical properties are the most impacting for damping and frequency. The influence of input coefficient of variation on output variability is also studied. It is shown that a negative correlation between the face and core thicknesses result in high levels of output variability, when one parameter increases as the other decreases.

Variability of dynamic responses of frequency dependent visco-elastic sandwich beams with material and physical properties modeled by spatial random fields

Probabilistic engineering mechanics aims to take into account uncertainties of input parameters of models and evaluating their impact onto the model responses. For frequency dependent visco-elastic sandwich structures, used in many industrial applications such as for vibration damping, the computation of the stochastic response can lead to a prohibitive computational cost. In this paper, an efficient methodology is proposed to analyze the output variability by a stochastic method based on fast Monte Carlo Simulation using a specific formulation obtained with the modal stability assumption. Geometrical and mechanical properties of a three-layered sandwich beam with a visco-elastic core made of 3M ISD112 damping polymer are represented by random fields to model the spatial variability. The validation of the formulation is studied, then the variability of the frequency response is evaluated for moderate and high variability levels. The benefits of the approach are also discussed.

Aeroelastic characteristics of magneto-rheological fluid sandwich beams in supersonic airflow

Supersonic aeroelastic instability of a three-layered sandwich beam of rectangular cross section with an adaptive magneto-rheological fluid (MRF) core layer is investigated. The panel is excited by an airflow along it’s longitudinal direction. The problem formulation is based on classical beam theory for the face layers, magnetic field dependent complex modulus approach for viscoelastic material model and the linear first-order piston theory for aerodynamic pressure. The classical Hamilton’s principle and the assumed mode method are used to set up the equations of motion. The validity of the derived formulation is confirmed through comparison with the available results in the literature. The effects of applied magnetic field, core layer thickness and constraining layer thickness on the critical aerodynamic pressure are studied. The onset of instability in terms of the critical value of the nondimensional aerodynamic pressure for the sandwich beam is calculated using the p-method scheme. Simply supported, clamped–clamped and clamped-free boundary conditions are considered. The results show that the magnetic field intensity and thickness ratios have significant effects on the instability bounds.

Parametric Instability of an Asymmetric Sandwich Beam with Thermal Gradient under Various Boundary Conditions by Computational Method

Dynamic stability of a three-layered asymmetric sandwich beam with viscoelastic core subjected to a periodic axial load and a steady, one-dimensional thermal gradient is considered under five different boundary conditions. Equations of motion are derived by using Hamilton's principle. A generalized Galerkin's method is used to reduce the non-dimensional equations of motion to a set of coupled Hill's equations with complex coefficients. The zones of parametric instability are obtained using Saito-Otomi conditions. The effects of shear parameter, core loss factor, core thickness parameter, thermal gradient on the regions of parametric instability are studied.

Multi-Objective Robust Active Vibration Control of an Arbitrary Thick Piezolaminated Beam

A multi-objective mixed / robust output feedback control synthesis with additional regional pole placement constraints in a linear matrix inequalities (LMI) framework is adopted for active vibration suppression of a simply supported, arbitrary thick, piezolaminated beam with continuously integrated sensor and actuator layers, in the face of high frequency unmodeled dynamics (residual uncertainties) and external disturbance. The structural formulation is based on a spatial state-space approach using the exact linear two-dimensional piezoelasticity theory and involving local/global transfer matrices. To assist control system design, subspace-based multivariable system identification is carried out by using the first four modes in the frequency range of 0–4000 rad/sec for control purpose, while the remaining high frequency (residual) modes in the control bandwidth are left as the uncertainty of modeling. The dynamic performance of vibration control system is demonstrated in both frequency and time domains for a three-layered sandwich beam including three pairs of piezoelectric actuator/sensor (Ba2NaNb5O15/PZT4) segments asymmetrically collocated on both sides of an orthotropic core for two different types of loading (i.e., impulsive load and random disturbance). The accuracy of dynamic analysis is established with the aid of a commercial finite element package and the data available in the literature.

Dynamic Analysis of Three-Layer Sandwich Beams with Thick Viscoelastic Damping Core for Finite Element Applications

This paper presents an analysis of the dynamic behaviour of constrained layer damping (CLD) beams with thick viscoelastic layer. A homogenised model for the flexural stiffness is formulated using Reddy-Bickford’s quadratic shear in each layer, and it is compared with Ross-Kerwin-Ungar (RKU) classical model, which considers a uniform shear deformation for the viscoelastic core. In order to analyse the efficiency of both models, a numerical application is accomplished and the provided results are compared with those of a 2D model using finite elements, which considers extensional and shear stress and longitudinal, transverse, and rotational inertias. The intermediate viscoelastic material is characterised by a fractional derivative model, with a frequency dependent complex modulus. Eigenvalues and eigenvectors are obtained from an iterative method avoiding the computational problems derived from the frequency dependence of the stiffness matrices. Also, frequency response functions are calculated. The results show that the new model provides better accuracy than the RKU one as the thickness of the core layer increases. In conclusion, a new model has been developed, being able to reproduce the mechanical behaviour of thick CLD beams, reducing storage needs and computational time compared with a 2D model, and improving the results from the RKU model.