Free Vibration Response Research Articles

Vibration and nonlinear dynamic response of nanocomposite multi-layer solar panel resting on elastic foundations

This paper dealt with the nonlinear dynamic response and vibration of multilayer organic solar panels subjected to mechanical and thermal loadings. The nanocomposite multilayer organic solar panel contains five layers which are manufactured by Al, P3HT: PCBM, PEDOT: PSS, Glass and Graphene which is a flexible replacement for indium tin oxide as a transparent and conducting electrode in the construction of solar panel. Based on classical theory, the basic equations are established by taking into account the effect of elastic foundations, von Kármán nonlinearity terms, and initial imperfection. The closed form expressions of the frequency–amplitude curves and natural frequency are obtained by using the Galerkin and Runger–Kutta methods. The numerical results illustrate positive effect of elastic foundations, negative effect of temperature increment and initial imperfection, considerable influence of geometrical parameters as well as excitation force on the free vibration and dynamic response of multilayer organic solar panel.

Navier solution for static and free vibration analysis of sandwich plate with auxetic honeycomb core resting on pasternak elastic foundation

This study investigates the bending behaviour and free vibration response of a rectangular sandwich plate with auxetic honeycomb core and functionally graded material (FGM) face sheets. Governing equations are established from Hamilton’s principle in the framework of the first-order deformation theory (FSDT). The Navier solution has been utilized to determine the deflection and the natural frequency of the simply supported sandwich plate. The reliability and accuracy of the proposed model have been validated via verification studies. The numerical investigations are conducted to investigate the influence of geometric parameters of auxetic material, volume fraction index of FGM, the thickness ratio of layers, and elastic foundation stiffness on deflection and fundamental frequency of the FG-Auxetic sandwich plate.

Additively Manufactured Lattice Truss Sandwich Cylinder and Free Vibration Behaviors

In aerospace engineering, it is attractive to combine additive manufacturing (AM) with lightweight design to reduce the weight of primary load-bearing structures. In this paper, a double-layer pyramidal lattice truss sandwich cylinder was designed and printed by selective laser melting (SLM) method, which has advantages in integrally manufacturing end frames and bolt holes. With diameter of 0.493 m, this cylinder is one of the lattice truss sandwich structures with the largest diameter at present. Free vibration testing was carried out to reveal the free vibration responses of the cylinder. The first-order vibration turns from a circle to an oval in the plane of the cross section. The fundamental frequency is 133.9 Hz. An equivalent shell free vibration theory was proposed and finite element models (FEMs) based on discrete model and equivalent shell model were applied to predict the natural frequencies and mode shapes of the cylinder. Advantages of the AM method in making light and stiff large-sized aerospace structures are confirmed through this research.

Static and Free Vibration Analyses of Single-Walled Carbon Nanotube (SWCNT)-Substrate Medium Systems.

This paper proposes a novel nanobar–substrate medium model for static and free vibration analyses of single-walled carbon nanotube (SWCNT) systems embedded in the elastic substrate medium. The modified strain-gradient elasticity theory is utilized to account for the material small-scale effect, while the Gurtin–Murdoch surface theory is employed to represent the surface energy effect. The Winkler foundation model is assigned to consider the interactive mechanism between the nanobar and its surrounding substrate medium. Hamilton’s principle is used to consistently derive the system governing equation, initial conditions, and classical as well as non-classical boundary conditions. Two numerical simulations are employed to demonstrate the essence of the material small-scale effect, the surface energy effect, and the surrounding substrate medium on static and free vibration responses of single-walled carbon nanotube (SWCNT)–substrate medium systems. The simulation results show that the material small-scale effect, the surface energy effect, and the interaction between the substrate and the structure led to a system-stiffness enhancement both in static and free vibration analyses.

Response Sensitivity of Centrifugal Pendulum Vibration Absorbers to Symmetry-Breaking Absorber Imperfections

This paper details an analytical investigation on free and forced vibration response sensitivity of centrifugal pendulum vibration absorbers (CPVA) to symmetry-breaking absorber imperfections. Minor variations in the absorbers’ design due to manufacturing tolerances, errors as well as operational wear and tear tend to break cyclic-symmetry and lead to asynchronous absorber oscillations, unbalanced rotor translations and thus significant vibration amplification. Therefore, an in-depth understanding on the sensitivity of CPVA's vibration response to absorber imperfections is needed to devise an improved absorber tuning strategy for robust vibration and noise reduction from realistic vehicle transmissions and drivelines. In this paper, free and forced vibration response sensitivity of generalized CPVA systems are analytically derived for absorber spacing and mass variations using the perturbation-based eigensensitivity technique in conjunction with modal superposition. The methodology applied in this study is realized through a linearized planar dynamic model, which includes both torsional and translational motions of the rotor along with oscillations of N equally-spaced baseline identical absorbers. Distinct mode distortions and degenerate mode splitting from the known baseline modal structure (with translational, rotational and absorber modes) are investigated for systems with asymmetric absorber spacing and absorber mass variations. The baseline modal structure is modified differently due to having unequal absorber masses or absorber spacing errors, and these differences are analytically derived and demonstrated. Moreover, it was identified that CPVA systems do not exhibit mode splitting with the absorber mass and spacing deviations. Response alterations and localized response patterns due to asymmetric absorber spacing and mass variations are identified and visualized through a case study. Results define a new parametric lower limit for the overtuning of absorbers to avoid localized absorber response and vibration amplification due to absorber imperfections. Additionally, the outcomes are used to locate unique excitation orders, which would help distinguish the alterations in system response due to absorber spacing and mass from each other and serve as a diagnostic tool.

A unified formulation of various shell theories for the analysis of laminated composite spherical shells

This study investigates the static and free vibration responses of orthotropic laminated composite spherical shells using various refined shear deformation theories. Displacement-based refined shear deformation theories are presented herein for the analysis of laminated composite spherical shells via unified mathematical formulations. Equations of motion associated with the present theory are derived within the framework of Hamilton's principle. Analytical solutions for the static and free vibration problems of laminated spherical shells are obtained using Navier's technique for the simply supported boundary conditions. Few higher order and classical theories are recovered from the present unified formulation; however, many other theories can be recovered from the present unified formulation. The numerical results are obtained for symmetric as well as anti-symmetric laminated shells. The present results are compared with previously published results and 3-D elasticity solution. From the numerical results, it is concluded that the present theories are in good agreement with other higher order theories and 3-D solutions.

Nonlinear Bending and Vibration Analysis of Imperfect Functionally Graded Microplate with Porosities Resting on Elastic Foundation Via the Modified Couple Stress Theory

This paper represents the nonlinear bending and free vibration analysis of a simply supported imperfect functionally graded (FG) microplate resting on an elastic foundation based on the modified couple stress theory and the Kirchhoff plate theory (KPT) together with the von-Kármán’s geometrical nonlinearity. The FG microplates with even and uneven distributions of porosities are considered. Analytical solutions for the nonlinear bending and free vibration are obtained. Comparing the obtained results with the published one in the literature shows the accuracy of the current solutions. Numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power-law index and the elastic foundation on the nonlinear bending and free vibration responses of the FG microplate.

New HSDT for free vibration analysis of elastically supported porous bidirectional functionally graded sandwich plate using collocation method

The present paper deals with the free vibration response of elastically supported porous bidirectional functionally graded soft-core sandwich rectangular plate with proposed new tangent shape function-based higher-order transverse shear deformation theory. The theory is accomplished to uphold the continuity of transverse shear stresses and zero shear stress conditions on both the extreme surfaces of a plate. The bidirectional gradation varies through the thickness ( z-axis) and length ( x-axis) directions. The effective material properties are calculated via the modified power-law. The governing differential equations (GDEs) of motion are acquired via the Hamilton principle. The inverse multi-quadric radial basis function-based Collocation method is implemented to discretize the GDEs obtained by strong form formulation. The current theory and method’s accuracy and efficacy are validated by comparing the present results with existing results in the literature. Studies show that the results of the proposed theory are in close agreement with 3D approach. Numerical results are obtained with a different scheme of sandwich layers for free vibration analysis. Effects of grading index, porosity fraction, two parameters foundation, different scheme, and the span to thickness ratio on free vibration response have been discussed. New results for free vibration response in porous media are presented with different skin materials.

Free vibration response of cnt-reinforced multiscale functionally graded plates using the modified shear deformation theory

ABSTRACT This article represents the free vibration analysis of multiscale functionally graded plates, reinforced with a carbon nanotube. The modified third-order shear deformation theory with a variation in transverse displacement is employed in the present work. MATLAB code is developed for C0 finite element (FE) formulation. A rectangular nine-noded element containing 117 nodal unknowns at each element was considered. The top and bottom transverse shear stress of the plate is considered zero. No shear correction factor is required in the present model. In this study, the FGM plates are assumed to be reinforced with MWCNT and SWCNT and also graded with fibre material throughout the thickness. The effective elastic properties of the multiscale composite plate are estimated by the combination of the Halpin–Tsai equation and the homogenisation scheme. The effects of various parameters such as the volume fraction of fibre material, a/h ratio, aspect ratio and weight percentage of both types of CNTs on frequency are studied.

Stereovision-based method for free vibration measurement of a mine hoisting rope

Currently, there are no suitable means to measure the composite vibration response of mine hoisting rope. However, machine vision-based measurement technology is a potential way to solve this problem. Therefore, a non-contact and non-intrusive stereovision method is proposed to obtain the 2D vibration displacement of a moving hoisting rope. In this methodology, a novel 3D digital image correlation (DIC) method that combines digital image processing (DIP) algorithm with 2D-DIC algorithm is developed, and the method can simplify the procedure of rope target stereo matching in two synchronous image sequences. In order to confirm the correctness of the stereovision method, some free vibration responses of a rope are measured in a hoist experimental system. Test results demonstrate that the stereovision measurement technique can keep satisfactory consistency with the laser displacement sensor for vibration data identification. The method is verified to have high adaptability to different lighting conditions and be reasonable to measure the vibration parameters of dynamic mine hoisting rope.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.

Flapwise and chordwise free vibration analysis of a rotating laminated composite beam

In this paper, the flapwise and chordwise free vibrations of a rotating laminated composite beam are studied. To determine the potential energy of the rotating system, two different approaches are presented to evaluate the centrifugal force. Using Hamilton’s principle, the governing differential equations of motion for the rotating system are obtained. In order to solve these equations for the free vibration problem, the finite element method is utilized. The effects of some parameters, such as the hub radius, angular velocity, layups configuration, and material anisotropy, on the vibrational characteristics are investigated in detail. It is anticipated that the results of this paper can be utilized for a more accurate prediction of the free vibration response of rotating composite beams.

Comparison of beam theories for characterisation of a NREL wind turbine blade flap-wise vibration

The harsh weather conditions offshore wind turbine blades are working under have necessitated the need for a more effective approach for detecting their failures. Dynamic characteristics e.g., vibration, carry useful information related to blades structural health. Critical analysis of the blade’s free vibration represents one of the fundamental steps for assessing structural dynamics. When a rotating blade deflects, either in the plane of rotation or perpendicular to it, the centrifugal force exerts inertia force along the blade span, which increases the blade’s natural frequencies compared with the stationary ones. However, the influence of different blade parameters on the flap-wise vibrations is not very well understood. In this paper, the effects of such parameters on dynamic characteristics of National Renewable Energy Laboratory (NREL) 5-MW wind turbine blades are investigated using different beam theories. The examined models have been used to determine the natural frequencies and mode shapes of the National Renewable Energy Laboratory (NREL) 5-MW wind turbine. Results demonstrate that increasing angular velocity significantly impacts the natural frequencies and mode shapes. The rotary inertia is found to influence the free vibration responses of the studied blades. Moreover, increasing hub radius, pre-cone and pitch angles are found to have less influence on the natural frequencies. Mode shape comparisons were also carried out using MSF (modal scale factor) and MAC (modal assurance criteria). Compared to the other investigated methods, Bernoulli’s based algorithms are found to produce less accurate results.

Nonlocal Free Vibrations of Metallic FGM Beams

This work aims to analyse the free-vibration response of functionally graded, simply supported beams with different gradient directions, taking into account nonlocal effects. To this purpose, the first-order shear deformation theory and the nonlocal elasticity theory of Eringen are used, in order to assess the influence of size dependency effects on the free-vibration responses of those beams. The influence of other factors such as the aspect ratio of the beams and the evolution of the constituents’ mixture through the beam thickness and along its length is also considered. In this last case, a mixture distribution is proposed, accounting for the boundary conditions’ characteristics. The finite element model is first verified against existing alternative solutions, to assess and illustrate its performance. Based on the conclusions achieved, a set of parametric studies is then developed. The results are discussed considering the material distribution profiles, and conclusions are drawn with respect to their relative performance under the analysed conditions.

Vibration analysis of laminated open cylindrical shell coupled with rectangular plates

The first order shear deformation theory (FSDT) is used to formulate a generalized model of laminated open cylindrical shell coupled with rectangular plates (LOSPS). On this basis, characteristics of free vibration and steady state response of the combination are investigated. The technique of artificial springs is used to simulate the coupling relationships between the adjacent substructures and boundary edges, and the specific coupling formulas between the adjacent substructures are presented. Then, an improved Fourier series are introduced as the displacement admissible functions of LOSPS, where the corresponding unknown coefficients of the displacement components are derived with the Rayleigh-Ritz method. Convergence and good accuracy of the proposed method are demonstrated by means of a several numerical examples. Moreover, comprehensive vibration analyses with respect to the influence mechanism of some key parameters are implemented, and corresponding new results are obtained. The present work of innovativeness can be treated as the reference for works in relative fields.

A four node quadrilateral shell element for free vibration response of functionally graded spherical shell panels

Abstract The present work involves the free vibration analysis of functionally graded spherical shells using a four-node quadrilateral shell element. This four-node, shell element has seven DOF (Degrees of Freedom) per node namely; three displacements, two rotations of mid-plane, and two transverse shear strain com-ponents. The element is developed by modifying the discrete Kirchhoff quadri-lateral shell element based on Reddy’s third-order theory developed earlier by the authors for the cylindrical shell. By transforming all the degrees of freedom from local to global coordinates using a transformation matrix converts the plate bending element into a flat shell element, resulting in the present formulation being suitable to give results for thin shells as well as for thick shells. In the present work functionally graded spherical shell panels with simply supported as well as clamped boundary condition, with various radius to span ratios and with different volume fraction indices are analyzed for free vibration response. The power law property variation through the thickness is considered in this study. To assess the performance of the developed element, the results, of non-dimen-sionalised frequencies are compared with the results presented in the literature. Comparison of results shows that the developed element yields quite accurate results even with the coarser mesh, which indicates the computational efficiency. It is also seen that the percentage difference between the present results and the other results available in the literature is less than 2 in most of the cases.

Free vibration response of auxetic honeycomb sandwich plates using an improved higher-order ES-MITC3 element and artificial neural network

This article presents the combination of mixed interpolation of tensorial components technique of triangular elements and the edge-based smoothed finite element method (ES-MITC3) developed on the higher-order shear deformation theory (HSDT). The free vibration behavior of auxetic honeycomb sandwich plates with the functionally graded material (FGM) skin layers generated by the proposed program is trained and predicted by an artificial neural network (ANN) model using Matlab coding for reducing computational time and saving computer resources, the proposed approach is expected suitable for practical engineering problems. The strain field of edge-based smoothed triangular elements is approximated by using the generalized C0-HSDT and Lagrange shape functions. Ultra-light features of sandwich plates are obtained by using the auxetic honeycomb core layer (negative Poisson’s ratio) and reinforced by FGM skin layers. The present results are compared with other published studies to verify the accuracy and reliability. Besides, the ANN model is also built to train obtained results and accurately predict the natural frequency of structures without running the code. Moreover, the effects of parameters such as geometrical and material properties (especially the auxetic parameters) on the free vibration behavior of auxetic honeycomb sandwich plates with FGM skin layers are investigated in detail.

Dynamic characteristics of composite micro lattice plates comprises of graphene nanoplatelets based on MCST theory

This paper investigates the free vibration characteristics of a stiffened micro lattice plate with the graphene nanoplatelets (GPLs) using the modified couple stress theory and sinusoidal shear deformation theory. The governing equations of motion are derived using the Hamilton’s principle. Solution procedure is developed based on Navier’s technique for simply-supported boundary conditions. The effect of geometrical parameters of the lattice structures and GPL weight fraction on the vibration response of the functionally graded (FG) micro lattice plates is analyzed. The numerical study reveals that by increasing the cell density, the natural frequency of the FG micro lattice plate approaches to the simple FG micro plate. Also, the effect of the size, distribution pattern of GPLs and material length scale parameter on the free vibration response of the FG GPLs-reinforced micro lattice plate is demonstrated in this study.

Analysis and optimal control of smart damping for porous functionally graded magneto-electro-elastic plate using smoothed FEM and metaheuristic algorithm

This paper proposed an effective numerical approach for free vibration analysis and optimal control of the porous functionally graded magneto-electro-elastic (PFGMEE) plates integrated with the active constrained layer damping (ACLD). Specifically, in order to analyze the free vibration characteristic of the PFGMEE plates based on the layer-wise shear deformation theory, a cell-based smoothed discrete shear gap (CS-DSG3) method is applied. The accuracy and convergence of the CS-DSG3 method are demonstrated by comparing its results with those available in the literature. Additionally, due to the coupling characteristic between magnetic, electric, and elastic fields, the coupled response of the PFGMEE plate is also considered. Furthermore, in order to obtain the optimal performance of the PFGMEE plates with the ACLD treatment, a combination of the CS-DSG3 method with an adaptive elitist differential evolution (aeDE) algorithm is used to determine the optimal placement of ACLD patches and the piezoelectric fiber orientation angle of the PFGMEE plates. The effect of several boundary conditions and porous distribution types on the free vibration response and the optimal control solution of the PFGMEE plates is also examined.

Nonlocal nonlinear static/dynamic snap-through buckling and vibration of thermally post-buckled imperfect functionally graded circular nanoplates

The bi-stability characteristics of the post-buckled plates have demonstrated extensive potential applications for energy harvesting and vibration isolation. In this article, at first, thermal post-buckling, nonlinear bending, and vibration behavior of the functionally graded (FG) circular nanoplates in bifurcation buckling are investigated according to nonlocal elasticity theory. Then, the nonlinear static/dynamic snapping phenomena and free vibration responses of the thermally post-buckled nanoplate subjected to static and sudden types of mechanical load are presented. For this aim, the equations of motion in conjunction with the von-Kármán nonlinearity and geometrical imperfection are established in the framework of Hamilton's principle. In addition, two distinct cases of temperature distribution as well as two types of edge conditions are taken into consideration. The equations of motion are discretized using the Chebyshev-Ritz procedure along with three different numerical algorithms. To evaluate the nonlinear dynamic snap-through buckling, the Newmark time integration scheme is adopted. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. A set of parametric studies is presented to provide an insight into influences of the nonlocal parameter, geometrical imperfection, gradient index, and edge supports on the static and dynamic characteristics of the nanosystem.