Investigation Of Free Vibrations Research Articles

Investigation of free vibration of piezoelectric actuator reinforced with functionally graded Boron Nitride nanotube using timoshenko beam model and differential quadrature method

The vibrational characteristics and natural frequencies of Boron Nitride Nanotube (BNNT)-reinforced piezoelectric beams, based on the Timoshenko model, have been studied for the first time in this paper. Four types of functionally graded (FG) distribution patterns of BNNT reinforcements were considered for the FG-BNNT-reinforced piezoelectric beam. It is assumed that the material properties of FG-BNNT-reinforced beams are estimated using the Representative Volume Element (RVE) micromechanical model. The FG-BNNT-reinforced piezoelectric beam with different Boundary Conditions (BCs) is exposed to electrical voltage. Coupled partial differential equations of the system have been derived with the aid of Hamilton's principle and then converted into corresponding dimensionless equations using dimensionless variables. The differential quadrature method (DQM) has been used to obtain the natural frequencies and shape modes. According to the convergence analysis, twelve grid points were enough to obtain an accurate result for the analysis. To confirm the accuracy and reliability of the present study, the obtained results were validated with frequencies from the literature. Also, the effects of BNNT volume fraction and its reinforcement pattern, as well as applied voltage, on dimensionless natural frequencies have been investigated for different BCs. The results show that applying electric loadings can be utilized as well as controlling parameters to modify the resonance frequency of the composite actuator. More specifically, it is shown that applying voltages to a FG-BNNT-reinforced piezoelectric beam changes the natural frequency for each reinforcement pattern, BCs, and mode numbers. Such ability could be employed when designing reinforced piezoelectric composites for vibrational applications.

Investigation of Free Vibrations of Stepped Nanobeam Embedded In Elastic Foundation

The free vibration of stepped nanobeams embedded in the elastic foundation was investigated using Eringen's nonlocal elasticity theory. It is fixed at the system ends with a simple-simple support. The stepped nanbeam’s equations of motion are obtained by using Hamilton's principle. Multi-time scale, which is the perturbation methods, was used for the analytical solution of the equations. To observe the effects of nano size effect, elastic basis coefficient and step location, natural frequencies of the first three modes of the system were obtained for different non-local parameter values, elastic foundation coefficients, step rates and step positions. In the results, it was seen that the non-local parameter had a negative effect on the natural frequency. The elastic foundation coefficient has been shown to reduce vibration amplitudes.

Free Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Higher Order Refined Composite Beams Using Differential Quadrature Finite Element Method

Present paper deals on the free vibration investigation of carbon nanotube-reinforced composite (CNTs) beams, based on refined third order shear deformation finite element beam theory. The particularity of this model is that, it can capture shear deformation effect without using of any shear correction factor by satisfying shear stress free at free edges. The carbon nanotubes are supposed to be immersed in a polymeric matrix with functionally graded pattern across the thickness direction of the beam, and their material properties are evaluated using the rule of mixture. The differential equations of motion and related boundary conditions are extracted using Lagrange’s principle and solved employing a robust numerical tool called, Differential Quadrature Finite Element Method (DQFEM) for the first time, with high convergence speed, fast calculus performance as well as a good numerical stability. The obtained results have been validated with those available in literature, in order to show the correctness of the present model. Afterwards, a deep parametric study is performed to examine the effects of various geometrical and material parameters on the vibration behavior of FG-CNTs beams.

Investigation of free vibration of FGP plates partially supported by elastic foundation using Finite Element method

Investigation of free vibration of FGP plates partially supported by elastic foundation using Finite Element method

Dynamic Responses Analysis of A 5MW NREL Wind Turbine Blade under Flap-Wise and Edge-Wise Vibrations

A wind turbine is subjected to a regime of varying loads. For example, each rotor revolution causes a complete gravity stress reversal in the low-speed shaft, and there are varying stresses from the out-of-plane loading cycle due to fluctuating wind load. Consequently, wind turbine blade design is governed by fatigue rather than ultimate load considerations. Previous studies have adopted many different beam theories, using different techniques and codes, to model the National Renewable Energy Laboratory (NREL) 5 MW offshore wind turbine blade. There are differences, from study to study, in the free vibration results and the dynamic response. Thecontribution of this study is to apply the code written by the authors to the different beam theories used with the aim of comparing the different beam theories presented in the literature and that developed by the authors. This paper reports the investigation of the effects of deformation parameters on the dynamic characteristics of the NREL 5 MW offshore wind turbine blades predicted by the different beam theories. The investigation of free vibrations is a fundamental step in the analysis of structural dynamics, and this study compares different computational structural methods and investigates their effect on the predicted dynamic response. The modalcharacteristics of every model examined have been combined with strip theory to determine the dynamic response of the blade.

Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Analytical and numerical investigation of free vibration for stepped beam with different materials

Abstract This work describes the application of classical Rayleigh method (CRM), modified Rayleigh method (MRM), and ANSYS finite element method (FEM) to calculate the natural frequency of non-homogenous cantilever beam. Two-step cantilever stepped beam was investigated through six studied cases. Each step has different material properties and the same cross section area. Results showed that the combination of materials is useful in order to increase the natural frequencies and reduce the weight of the beam at the same time when the cantilever beam is fixed by the side of the stronger material. There is a good agreement between the CRM and FEM for the region with length larger than half length of beam, on the other hand, there is an excellent agreement between the MRM and FEM for the region with length smaller than half length of beam.

Assessment on dynamic characterization of the cylindrical laminated composite shallow shell panel through experimental and numerical approach

In the present study, the dynamic characterization of the glass fiber-reinforced polymer (GFRP)-laminated composite cylindrical open shallow shell panels is explored with the experimental and numerical approach. In the finite element modeling (FEM), the governing equilibrium equation of the cylindrical shell panel is developed with higher-order shear deformation theory by considering the nine-noded rectangular elements. The convergence and validation study of the present FEM is accomplished with the available literature. The cylindrical laminated composite shallow shell panels are fabricated with a curvature radius of 0.8 m, and the experimentation on the free vibration investigation is performed with various boundary conditions; then, the experimental outcomes are compared with the present FEM to verify the effectiveness. The detailed parametric investigation is executed to explore the impact of curvature radius (R), boundary conditions, thickness ratio, aspect ratio (L/B), and stacking sequence on the structural response of the cylindrical laminated composite shallow shell panel. Transverse vibration response of the GFRP cylindrical laminated composite shallow shell panel is also performed with the various curvature ratios at clamped at all end conditions.

Free vibration of thin-walled open-section beams with functionally graded materials along the contour direction

An investigation of free vibration of a thin-walled beams with functionally graded materials (FGMs) is presented. The mechanical properties of thin-walled beam are assumed to vary along the contour direction of the thin-walled cross section. Governing differential equations are derived by means of Hamilton's principle. A finite element model is developed to formulate the problem. The natural frequencies and corresponding vibrational modes are obtained for thin-walled FG mono-symmetric I- and channel-sections beams with several different types of material distributions. A numerical comparison is carried out to show the validity of the proposed theory with available results in the literature. Furthermore, effects of gradual law and volume fraction of ceramic on the natural frequencies of beams are also studied.

Investigation of Free Vibrations and Stability of Functionally Graded Three-Layer Plates by Using the R-Functions Theory and Variational Methods

We propose a numerical-analytic method aimed at the investigation of free vibrations and stability of functionally graded sandwich-type plates and based on the refined Timoshenko-type theory of the first order. We considered various schemes of arrangement of the layers: (1) the filler is a functionally graded material and the top and bottom layers are made of isotropic materials and (2) vice versa, the filler is made of an isotropic material, whereas the top and bottom layers are made of functionally graded materials. The proposed method is based on the R-functions theory and the Ritz variational method. This method and the developed software take into account the presence of an inhomogeneous subcritical state of the plate. Analytic relations are obtained for the evaluation of the elastic constants and density of the material under the assumption that Poisson’s ratios are identical, for the components of functionally graded materials. The numerical results obtained for the eigenfrequencies and the critical load are compared with the available data and their good agreement is demonstrated. To illustrate the possibilities of the proposed approach, we perform the numerical analyses of plates of complex geometric shapes loaded in the middle plane. The influence of various geometric and mechanical parameters on the dynamic behavior of the plate, critical load, and the zones of dynamic instability is investigated.

Buckling and free vibrations of CNT-reinforced cross-ply laminated composite plates

This article deals with the investigation of free vibration and buckling behaviors of carbon nanotube (CNT)-reinforced cross-ply laminated composite plates. The plate kinematics is assumed to follow a first-order shear deformation theory (FSDT). After the coupled equations of motion and buckling are derived, the method of discrete singular convolution (DSC) is used for the numerical solution of the problems. Both the regularized Shannon’s and Lagrange’s delta kernels are used for spatial discretizing of the resulting governing equations of buckling and vibration of CNT-reinforced laminated plates. Natural frequencies and critical buckling loads are obtained for different cases. Wherever possible, the present DSC results are verified by comparing them with the existing analytical results available in the literature. Then, a detailed parametric study is performed to examine the effects of boundary conditions, CNT distributions and volume fraction, aspect ratio, length-to-width ratio, and number of layers on the frequencies and buckling loads.

Dynamic stiffness matrix of a conical bar using the Rayleigh-Love theory with applications

Based on the Rayleigh-Love theory, the dynamic stiffness matrix of a conical bar in longitudinal vibration is developed for the investigation of free vibration and response characteristics of such bars and their assemblies. First the governing differential equation of motion in free longitudinal vibration of a conical bar using the Rayleigh-Love theory which accounts for the inertia effects due to transverse or lateral deformations is derived by applying Hamilton's principle. Next, for harmonic oscillation, the governing differential equation is recast in the form of Legendre's equation, providing a series solution connected by integration constants. The expressions for the amplitudes of displacements and forces are then obtained by means of the series solution. Finally, the frequency dependent dynamic stiffness matrix is formulated by relating the amplitudes of forces to those of the corresponding displacements at the ends of the conical bar and thereby eliminating the integration constants. As an established solution technique, the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix when computing the natural frequencies and mode shapes of some illustrative examples. The theory is also applied to investigate the response of a cantilever conical Rayleigh-Love bar with a harmonically varying load applied at the tip. The results computed from the Rayleigh-Love model based dynamic stiffness theory are compared and contrasted with those computed from conventional classical theory with significant conclusions drawn.

Free vibration analysis of the MWCNT reinforced hybrid laminated composite sandwich beam

Abstract In this study, free vibration investigation of the hybrid laminated composite sandwich beam were performed with and without the reinforcement of multi walled carbon nanotube (MWCNT). MWCNT were dispersed with the epoxy through ultrasonic liquid processor to achieve a good surface contact between the epoxy and MWCNT. Sandwich beam specimens were fabricated with reinforcement of the MWCNT in the laminated composite skin and the structural integrity of the laminated hybrid composite sandwich beam were studied at the distinctive limit conditions. The influence of MWCNT in the sandwich composite structure were studied in the terms of natural frequencies and damping behavior. The infusion of the MWCNT in the laminated composite sandwich structures enhanced the better natural frequencies and damping ratio and this study exposes the impact of the MWCNT on the laminated composite structure

Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory

In this research paper, the free vibrational behavior of the simply supported FG nano-plate is studied using the nonlocal two variables integral refined plate theory. The present model takes into account the small scale effect. The effective\'s properties of the plate change according to the power law variation (P-FGM). The equations of motion of the system are determined and resolved via Hamilton\'s principle and Navier procedure, respectively. The validity and efficiency of the current model are confirmed by comparing the results with those given in the literature. At the last section, several numerical results are presented to show the various parameters influencing the vibrational behavior such as the small-scale effect, geometry ratio, material index and aspect ratio.

Analytical approach to investigation of free vibration of thin rectangular plate immersed in fluid, resting on Winkler and Pasternak foundations

Dynamic behaviour of free vibration of rectangular plate is investigated. This study considered an analytical approach to investigation of free vibration of thin rectangular plate immersed in fluid, resting on Winkler and Pasternak foundations.The governing nonlinear partial differential equation is analyzed using two-dimensional differential transform method. The accuracy of the analytical solutions obtained is verified with existing results in literature and confirmed in excellent agreement. Thereafter, the analytical solutions are used for investigation of effect of elastic foundation, fluid and aspect ratio on vibrating plate.From the result, it is observed that, increase elastic foundation parameters increases natural frequency, increase aspect ratio increase natural frequency, influence of fluid decreases the natural frequency of the plate.Hopefully the present study will contribute to existing knowledge in field of vibration.

An investigation of free vibrations of a strain gradient Timoshenko beams with the method of initial values

Investigated herein is the free vibrations of beams based on the strain gradient Timoshenko beam theory with the method of initial values. For the vibration of strain-gradient Timoshenko beam (SGTB), the sixth-order ordinary differential equation and three boundary conditions at each end have been obtained by using the Hamilton principle. The effect of the characteristic length on the frequencies of free vibrations is shown. The frequencies of the SGTB are compared to the frequencies of the strain gradient Euler beam (SGEB), classical Timoshenko beam (CTB) and classical Euler beam (CEB). It has been observed that the high-frequency values of conventional and strain-gradient beams are very different. This result can be used to determine the value of the material characteristic length for a nanobeam for which lengthscale effects are believed to be dominant.

Investigation of Free Vibrations of Three-Layered Circular Shell Supported by Annular Ribs of Rigidity

The construction of a mathematical model and the development of an algorithm of free vibrations investigation in the three-layered circular shell with a light-weight aggregate supported by annular rigidity ribs are considered in paper. The hypotheses of Kirchoff-Lyav are accepted for external bearing layers of shell and for aggregate there is accepted the linear law of tangential displacements change by thickness. The boundary conditions of a shell region closed between the ribs are established. Using the boundary transition, conditions along the lines of the ribs, taking into account and without deformations of displacement in the ribs, but without taking into account the torsional rigidity in the ribs are determined. The equation of motion of supported three-layered shell is obtained. The frequencies of free vibrations were investigated and values of parameter of the first frequency of free vibrations for a shell, supported by one and three rigidity ribs, were calculated. There are given values depending on the physical and mechanical properties of materials and geometric dimensions of the shell, the curvature parameter, and the rigidity parameter of an aggregate.

Experimental modal analysis using laser vibrometer and finite element modeling of milling machine arbor

Experimental modal analysis has received a lot of consideration in recent days for dynamic analysis of complicated engineering structure and machine tools. It is a process which is used to determine the inherent dynamic properties of structure such as damping, mode shapes and natural frequencies. The experimental modal analysis benefits in design of all types of structures including automotive components, aircraft components, spacecraft, computers and tennis rackets and golf clubs etc. The present study deals with the experimental investigation of free vibration of milling machine arbor. Experimental modal analysis of milling machine arbor is carried out by using laser vibrometer for measurement and impact hammer for excitation. The experimental results are compared with the numerical predictions obtained using finite element modeling software ANSYS workbench 14.5. The scope of present work is to understand and investigate the modal parameters of milling machine arbor in the form of natural frequency and mode shape for robust design.

Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of eringen theory using conformable fractional derivative and Galerkin method

The purpose of this paper is to study the free vibration and buckling of a Timoshenko nano-beam using the general form of the Eringen theory generalized based on the fractional derivatives. In this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method. Free vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams. The new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.