Complex Modulus Approach Research Articles

Magnetic and vibrational amplitude dependences of MRE grid composite sandwich plates

In this work, both magnetic and vibration amplitude dependent properties of magnetorheological elastomer (MRE) grid composite sandwich plates (MREGCSPs) are investigated. Initially, to prove such a nonlinearly dependent phenomenon, a series of characterization tests are performed on the MREGCSP specimens with different magnetic field intensities and base vibrational amplitudes. Then, using the Jones-Nelson nonlinear material theory, the strain energy density function method, the complex modulus approach, and the Biot-Savart law, the nonlinear material assumption of MRE is defined. A theoretical model consisting of an MRE grid core and two fiber-reinforced polymer (FRP) skins is also proposed to obtain the solutions for the nonlinear frequency, damping, and vibration response parameters, which is based on the modified first-order shear deformation theory, the energy principle, the eigenvalue increment method, the Newmark- beta approach, etc. Finally, a detailed comparison of magnetic and vibrational amplitude dependent natural frequencies, loss factors, and vibration responses is performed to confirm the effectiveness and superiority of the current model over a linear model. This provides a solid basis to reveal the nonlinear dynamic mechanism of the studied smart structure subjected to complex excitation loads. Also, some practical suggestions are summarized for improving its active vibration control capability.

Theoretical and experimental investigations of vibration and damping behaviors of carbon fiber-reinforced composite thin shells with partial bolt looseness constraints

In this paper, the vibration and damping behaviors of fiber-reinforced composite thin shells under partial bolt looseness boundary conditions are investigated both theoretically and experimentally. Based on Love's shell theory, the complex modulus approach, and Hooke's law, a theoretical model of fiber-reinforced composite thin shells is constructed. A non-uniform artificial spring technique is proposed to accurately describe the non-uniform looseness in the bolt constraint positions and their interval locations, in which the two kinds of artificial springs, namely the “main spring” and “secondary spring”, are taken into account separately. After the natural characteristics, damping properties, and vibration responses are solved by the Rayleigh-Ritz method, strain energy approach, and Duhamel integral method, detailed measurements are undertaken on a CA 500 carbon/epoxy composite shell specimen to examine vibration and damping characteristics as well as to validate the proposed model. By comparing the theoretical results with the experimental ones, the maximum calculation errors of natural frequencies, resonant responses, and damping ratios are within an acceptable level. In addition, the influences of partial bolt looseness constraints on the vibration characteristics of such shell structures are also investigated here.

Thermal buckling and free vibration of viscoelastic functionally graded sandwich shells with tunable auxetic honeycomb core

In this paper, thermal buckling and free vibration of viscoelastic functionally graded sandwich shell with tunable auxetic honeycomb core is investigated. The sandwich shell has functionally graded material (FGM) skins at top and bottom and tunable auxetic honeycomb core at the middle. The viscoelastic behavior of the skins and core are modeled according to complex modulus approach in combination with the elastic–viscoelastic correspondence principle. Poisson's ratio of the auxetic honeycomb core is tunable. It has positive or negative value, which depends on an angle parameter of the unit cell of the auxetic honeycomb core. The equations of motion for the viscoelastic FGM sandwich shell with tunable auxetic honeycomb core are derived using Hamilton's principle. Analytical solution for simply supported shell is obtained using Navier's solution procedure. The effects of the geometry parameters, temperature rise, parameters of the tunable auxetic honeycomb core are studied.

Flutter analysis of sandwich panel with the constrained viscoelastic layer considering the effects of the imperfection and the core thickness deformation

The supersonic flutter analysis of panels treated with the viscoelastic constrained layer damping (CLD) is performed with a special focus on the effects of the initial geometric imperfection and the core through-the-thickness deformation on the flutter boundaries. For kinematic modeling of the CLD core, a higher-order shear deformation theory is used where the through-the-thickness deformation is also taken into account. The core viscoelastic property is also described using the frequency-dependent complex-modulus approach and the first-order Piston theory is employed to determine the aerodynamic pressure. The discretized equations of motion are then derived using the finite element method (FEM). The integrations required in the FEM for obtaining each element stiffness matrix are also facilitated by approximating the imperfection function using the quintic Hermite interpolation function. Moreover, due to the large dimension of the final global matrices, the model is reduced using the oscillatory damped modes of the panel, which greatly reduces the computation time needed for flutter analysis. Numerical studies are performed for two different viscoelastic materials (VEMs), which shows that an efficient flutter suppression could not always be achieved with any type of VEMs. The thickness deformation effect is also found to be most prominent when the CLD length is about 80% of the panel length. Moreover, the mode crossing phenomenon that occurs by increasing the imperfection amplitude is found to significantly reduce the flutter boundaries.

Optimum Design of Aft-swept Composite Wings for Aeroelastic Characteristics with Structural Damping

The present paper deals with the aeroelastic characteristics of aft-swept composite plate wings with structural damping. Based on the complex modulus approach, a minimum weight design of cantilevered laminated plates with a sweep-back angle under the flutter and divergence speed constraints is conducted by using a differential evolution in which lamination parameters are used as design variables.

Effect of carbon nanotube-reinforced magneto-electro-elastic facings on the pyrocoupled nonlinear deflection of viscoelastic sandwich skew plates in thermal environment

This work presents a finite-element-based numerical formulation to evaluate the nonlinear deflections of magneto-electro-elastic sandwich skew plates with a viscoelastic core and functionally graded carbon nanotube-reinforced magneto-electro-elastic face sheets. Meanwhile, the proposed formulation accommodates the geometrical skewness as well. The magneto-electro-elastic sandwich skew plate is operated in the thermal environment and subjected to various multiphysics loads, including electric and magnetic loads. The viscoelastic core is modelled via the complex modulus approach. Two different forms of viscoelastic cores, such as Dyad 606 and EC 2216, are considered in this study. Also, different thickness configurations of core and facing arrangements are taken into account. The plate kinematics is presumed through higher-order shear deformation theory, and von Karman's nonlinear strain displacement relations are incorporated. The global equations of motion are arrived at through the total potential energy principle and solved via the direct iterative method. Special attention is paid to assessing the influence of pyroeffects, coupling fields and electromagnetic boundary conditions on the nonlinear deflections of magneto-electro-elastic sandwich plates working in the thermal environment and subjected to electromagnetic loads, which is the first of its kind. Also, parametric studies dealing with the skew angles, carbon nanotube distributions and volume fractions, thickness ratio, and aspect ratio have been discussed. The results of this work are believed to be unique and serve as a guide for the design engineers towards developing sophisticated smart structures for various engineering applications.

Impact and Bandgap Characteristics of Periodic Rods With Viscoelastic Inserts and Local Resonators

Abstract A comprehensive theoretical and experimental study is presented of the bandgap behavior of periodic viscoelastic material (VEM) composites subjected to impact loading. The composites under consideration consist of an assembly of aluminum sections integrated with periodic inserts which are arranged in one-dimensional configurations. The investigated inserts are manufactured either from VEM only or VEM with local resonators (LR). A finite element model (FEM) is developed to predict the dynamics of this class of VEM composites by integrating the dynamics of the solid aluminum sections with those of VEM using the Golla-Hughes-Mctavish (GHM) mini-oscillator approach. The integrated model enables, for the first time, the accurate predictions of the bandgap characteristics of periodic viscoelastic composites unlike previous studies where the viscoelastic damping is modeled using the complex modulus approach with storage modulus and loss factor are assumed constants and independent of the frequency or the unrealistic and physically inaccurate Kelvin–Voigt viscous-damping models. The predictions of the developed FEM are validated against the predictions of the commercial finite element package ansys. Furthermore, the FEM predictions are checked experimentally using prototypes of the VEM composites with VEM and VEM/LR inserts. Comparisons are also established against the behavior of plain aluminum rods in an attempt to demonstrate the effectiveness of the proposed class of composites in mitigation of the structural response under impact loading. Close agreements are demonstrated between the theoretical predictions and the obtained experimental results.

Effect of temperature and moisture on free vibration characteristics of skew laminated hybrid composite and sandwich plates

This paper is concerned with the effect of variation in temperature and moisture concentration on free vibration response of skew laminated hybrid composite and sandwich plates. The coupled thermo-elastic and hygro-elastic finite element model is formulated using the first-order shear deformation theory (FSDT). Uniform temperature and moisture concentration rise is considered for the analysis. Soft-core viscoelastic materials are considered for the sandwich plates and are modeled using the complex modulus approach. Linear strain-displacement relations are used to develop a mechanical stiffness matrix, and the initial stress stiffness matrix is generated using non-linear strain-displacement relations to represent the non-mechanical stiffness matrix. Numerical examples for the generated finite element model are presented and discussed comprehensively to understand the effect of temperature, moisture concentration, skew angle, length to width ratio, length to thickness ratio, and boundary conditions on the vibration response of the laminated hybrid composite and sandwich plates. Further investigation is devoted to studying the influence of temperature and moisture concentration-dependent material properties, stacking sequence, core to face sheet thickness ratio, and fiber orientation on vibration behavioral response of sandwich and hybrid composite plates.

An Amplitude- and Temperature-Dependent Vibration Model of Fiber-Reinforced Composite Thin Plates in a Thermal Environment.

A thermal environment has a complex influence on the dynamic characteristics of fiber-reinforced composite materials and structures. It is challenging to consider the effects of high temperature and external vibration energy simultaneously on their nonlinear vibration response. In this research, the material nonlinearities, due to both the excitation amplitudes and the high temperatures, are studied for the first time, and a new nonlinear vibration model of fiber-reinforced composite thin plates in a thermal environment is proposed by introducing the nonlinear thermal and amplitude fitting coefficients simultaneously. Then, based on the classical laminated plate theory, the complex modulus approach, and the power function and the Ritz methods, dynamic governing equations in high-temperature environments are derived to solve the nonlinear natural frequencies and vibration responses and damping parameters. Moreover, the three-dimensional fitting curves of the elastic moduli and loss factors, excitation amplitudes, and temperature values are obtained so that the key nonlinear fitting coefficients in the amplitude- and temperature-dependent model can be identified. To validate this model, the experimental tests on CF130 carbon/epoxy composite thin plates are undertaken. It is found that the 3rd and 5th natural frequencies, vibration responses, and damping results obtained from the nonlinear model are consistent with the experimental measurements, and the mechanism of nonlinear thermal vibration behaviour is revealed.

Influence of active constrained layer damping on the coupled vibration response of functionally graded magneto-electro-elastic plates with skewed edges

This article makes the first attempt in assessing the influence of active constrained layer damping (ACLD) treatment towards precise control of frequency responses of functionally graded skew-magneto-electro-elastic (FGSMEE) plates by employing finite element methods. The materials are functionally graded across the thickness of the plate in terms of modest power-law distributions. The principal equations of motion of FGSMEE are derived via Hamilton’s principle and solved using condensation technique. The effect of ACLD patches are modelled by following the complex modulus approach (CMA). Additionally, distinctive emphasis is laid to evaluate the influence of geometrical skewness on the attenuation capabilities of the plate. The accuracy of the current analysis is corroborated with comparison of previous researches of similar kind. Additionally, a complete parametric study is directed to understand the combined impacts of various factors like coupling fields, patch location, fiber orientation of piezoelectric patch in association with skew angle and power-law index.

Modeling of the nonlinear dynamic degradation characteristics of fiber-reinforced composite thin plates in thermal environment

In this research, the nonlinear dynamic degradation characteristics of fiber-reinforced composite plates in thermal environment are investigated through a novel dynamic degradation model, which is established by introducing the thermal and time fitting coefficients simultaneously to express dynamic elastic moduli of such composite materials. Based on the classical laminated plate theory, the improved exponential function method, the complex modulus approach and the Ritz method, the dynamic equations are derived to solve the dynamic parameters. Besides, a particle swarm optimization algorithm is employed to iteratively calculate dynamic elastic moduli, and the nonlinear least squares technique in MATLAB software is utilized to draw the three-dimensional fitted surfaces of dynamic elastic moduli, degradation time and temperature data, so that the concerned fitting coefficients in the model developed can be identified. In order to validate the dynamic degradation model, experimental measurements of E120 carbon fiber/ FRD-YG-03 resin composite thin plates are undertaken. The first three natural frequencies, resonant responses and modal damping ratios obtained from the model are compared with experimental results at different degradation time and stabilized thermal environment, which are shown to be in good agreement. Also, the specific influences with and without consideration of the degradation behavior on the dynamic characteristics are investigated and evaluated.

Finite element simulation of controlled frequency response of skew multiphase magneto-electro-elastic plates

The linear frequency response of skew multiphase magneto-electro-elastic composite plate embedded with active constrained layer damping treatment has been studied. The volume fraction of piezoelectric fibres embedded in the piezomagnetic matrix significantly affects the coupling characteristic of this multiferroic material, and hence, the frequency of the skew multiphase magneto-electro-elastic plate is drastically altered. This study emphasizes on evaluating the influence of different volume fraction of barium titanate (BaTiO3) and cobalt ferrite (CoFe2O4) on the frequency characteristics of skew multiphase magneto-electro-elastic. In this regard, a finite element formulation has been proposed to assess the damped response of such skew multiphase magneto-electro-elastic plates. Incorporating the complex modulus approach, the constrained viscoelastic layer of the active constrained layer damping patch is modelled. In addition, the effect of geometrical skewness has also been investigated. Meanwhile, an exhaustive parametric study is carried out to analyse the influence of control gain, patch position and fibre orientation angle of piezoelectric composite.

Vibration control of skew magneto-electro-elastic plates using active constrained layer damping

In this article, the effect of active constrained layer damping (ACLD) on the linear frequency response characteristics of skew magneto-electro-elastic (SMEE) plates has been evaluated. The constraining and the constrained layer of ACLD treatment are considered to be made of 1–3 piezoelectric composite (PZC) and viscoelastic layer, respectively. The advantage of incorporating smart materials such as PZC in attenuating the vibrations of the SMEE plates is thoroughly investigated. In this regard, a three-dimensional finite element (FE) formulation is derived with the aid of the principle of virtual work considering the coupling between magnetic, electric and elastic fields. The SMEE plate kinematics is governed by layer-wise shear deformation theory. The Complex modulus approach is adopted in order to model the constrained layer of the ACLD patch. A special emphasis is placed on investigating the effect of various skew angles on the frequency response of SMEE plates with ACLD treatment. Meanwhile, an exhaustive parametric study is carried out to analyze the influence of control gain, stacking sequence, patch position, fiber orientation angle of PZC, aspect ratio, and coupling effects. The results confirm that these parameters in association with ACLD treatment and skew angle significantly affect the damping characteristics and hence the controlled frequency response of SMEE plates. Moreover, the numerical results of this research lay a strong platform in the field of vibration and control. Also, it encourages the optimized design and analysis of smart SMEE structures for the application of sensors and actuators.

Analysis of multilayered structures embedding viscoelastic layers by higher-order, and zig-zag plate elements

In this work, a number of higher-order plate elements and an improved theory based on a zig-zag power function have been applied to metallic and composite layered structures with viscoelastic layers. The kinematic field is written by using an arbitrary number of continuous piecewise polynomial functions. The polynomial expansion order of a generic subdomain is a combination of zig-zag power functions depending on the plate thickness coordinate. Damped free-vibrations and frequency response analysis are performed modelling the viscoelastic properties with the complex modulus approach, and taking into account different damping laws for the viscoelastic layers included a five-parameter fractional-derivative model. The governing equations are derived from the Principle of Virtual Displacements and solved using the Finite Element (FE) method, and both full and reduced-order formulation are taken into account. Furthermore, the Mixed Interpolated Tensorial Components (MITC) method is employed to contrast the shear locking phenomenon. The Carrera Unified Formulation (CUF) has been employed to derive the governing FE equations for the various theories considered in this paper in an unified form. Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present plate element. The results are reported in terms of frequencies, modal loss factors and frequency responses, and they are compared with solutions published in the literature and with solid finite element models.

Various refined theories applied to damped viscoelastic beams and circular rings

In this work, a number of enhanced finite beam elements has been tested considering layered structures with viscoelastic layers. The viscoelastic properties have been defined with the complex modulus approach, and the equations of motion have been derived using the principle of virtual displacement. Higher-order theories, based on equivalent single-layer and the layerwise approaches, have been obtained with the Carrera unified formulation, which makes it possible to generate an infinite number of kinematic approximations. Both Lagrange-type elements and higher-order zigzag theories have been developed within the layerwise approach. On the other hand, Taylor-like expansions and Murakami-type zigzag functions have been used to conceive the equivalent single-layer models. Numerical simulations have been performed considering symmetric and asymmetric laminated structures with rectangular and circular cross sections. The results are reported in terms of frequencies, modal loss factors, and frequency responses. The obtained results have been compared with solutions published in the literature and with solid finite element models. The accuracy of the different formulations has been found to be problem dependent to a great extent.

Layerwise Analyses of Compact and Thin-Walled Beams Made of Viscoelastic Materials

This paper evaluates the vibration characteristics of structures with viscoelastic materials. The mechanical properties of viscoelastic layers have been described with the complex modulus approach. The equations of motion are derived using the principle of virtual displacement (PVD), and they are solved through the finite element method (FEM). Higher-order beam elements have been derived with the Carrera Unified Formulation (CUF), which enables one to go beyond the assumptions of the classical one-dimensional (1D) theories. According to the layerwise approach, Lagrange-like polynomial expansions have been adopted to develop the kinematic assumptions. The complex nonlinear dynamic problem has been solved through an iterative technique in order to consider both constant and frequency-dependent material properties. The results have been reported in terms of frequencies and modal loss factors, and they have been compared with available results in the literature and numerical three-dimensional (3D) finite element (FE) solutions. The proposed beam elements have enabled bending, torsional, shell-like, and coupled mode shapes to be detected.

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.

Macro-composites with star-shaped inclusions for vibration damping in wind turbine blades

The work describes the numerical and experimental assessment of using biphase composite structures with non-classical shape inclusions. Star-shaped biphase cells have been designed, modeled and tested to evaluate the complex engineering constants corresponding to various deformation modes. A Finite Element homogenisation method using the complex modulus approach has been used to evaluate the variation of the storage moduli, loss factors and amounts of strain energy dissipated in the matrix versus the unit cell geometry parameters. Experimental results have been obtained on aluminium/cast epoxy sample using a shear dynamic test rig and a dynamic mechanical analyser. The results have been benchmarked against unit biphase composite configurations with cylindrical inclusions having the same contact surface between inclusion and matrix than the star-shaped reinforcements. The composite cells are intended for a possible use as structural damping units to be located in maximum nodal strain positions corresponding to specific wind turbine blade modes.

Investigation of the thickness variability and material heterogeneity effects on free vibration of the viscoelastic circular plates

The present study deals with free vibration analysis of variable thickness viscoelastic circular plates made of heterogeneous materials and resting on two-parameter elastic foundations in addition to their edge conditions. It is assumed that the viscoelastic material properties vary in the transverse and radial directions simultaneously. The complex modulus approach is employed in conjunction with the elastic–viscoelastic correspondence principle to obtain the solution. The governing equations are solved by means of a power series solution. Finally, a sensitivity analysis including evaluation of effects of various edge conditions, thickness variations, coefficients of the elastic foundation, and material loss factor and heterogeneity on the natural frequencies and modal loss factors is accomplished.

A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations

In the present paper, a power series solution is developed for free vibration and damping analyses of viscoelastic functionally graded plates with variable thickness on elastic foundations. It is assumed that the material properties of the functionally graded material (FGM) vary in the transverse and radial directions, simultaneously. Therefore, the presented solution can be employed for the transversely-graded and radially-graded viscoelastic circular plates, as special cases. In addition to the edge conditions, the plate may be resting on a two-parameter elastic foundation. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A sensitivity analysis including effects of various edge conditions, geometric parameters, coefficients of the elastic foundation, parameters of the functionally graded material, and material loss factor is carried out. In the present paper, concept of the complex modal stresses of the viscoelastic plates is discussed in detail.