Free vibration of FGM layered beams by various theories and finite elements (original) (raw)
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Free vibration analysis of layered functionally graded beams with experimental validation
Materials & Design
An improved third order shear deformation theory is employed to formulate a governing equation for predicting free vibration of layered functionally graded beams. The Ritz method is adopted to solve the governing equation for various types of boundary conditions and the frequency results are validated by some available and experimental results. A multi-step sequential infiltration technique is used to fabricate the layered functionally graded beams for vibration testing. For the first time, a simple mathematical model, based on a power law distribution, is introduced to approximate material volume fraction of the layered beams. The details of layered beam fabrication according to the infiltration technique, microstructure and volume fraction analysis as well as vibration experimental set up are included and described in this investigation. Aspects which affect natural frequencies, such as material compositions, thickness ratio, and boundary conditions, are then taken into consideration. The impact on frequency of added mass is presented and discussed.
Archives of Materials Science and Engineering, 2021
In this study, the free vibration analysis of functionally graded materials (FGMs) sandwich beams having different core metals and thicknesses is considered. The variation of material through the thickness of functionally graded beams follows the power-law distribution. The displacement field is based on the classical beam theory. The wide applications of functionally graded materials (FGMs) sandwich structures in automotive, marine construction, transportation, and aerospace industries have attracted much attention, because of its excellent bending rigidity, low specific weight, and distinguished vibration characteristics. Design/methodology/approach: A mathematical formulation for a sandwich beam comprised of FG core with two layers of ceramic and metal, while the face sheets are made of homogenous material has been derived based on the Euler-Bernoulli beam theory. Findings: The main objective of this work is to obtain the natural frequencies of the FG sandwich beam considering different parameters. Research limitations/implications: The important parameters are the gradient index, slenderness ratio, core metal type, and end support conditions. The finite element analysis (FEA), combined with commercial Ansys software 2021 R1, is used to verify the accuracy of the obtained analytical solution results. Practical implications: It was found that the natural frequency parameters, the mode shapes, and the dynamic response are considerably affected by the index of volume fraction, the ratio as well as face FGM core constituents. Finally, the beam thickness was dividing into frequent numbers of layers to examine the impact of many layers' effect on the obtained results. Originality/value: It is concluded, that the increase in the number of layers prompts an increment within the frequency parameter results' accuracy for the selected models. Numerical results are compared to those obtained from the analytical solution. It is found that the dimensionless fundamental frequency decreases as the material gradient index increases, and there is a good agreement between two solutions with a maximum error percentage of no more than 5%.
Free Vibration Analysis of a Functionally Graded Beam with Finite Elements Method
In this study, free vibration analysis of a functionally graded (FG) beam under various boundary conditions is carried out. Natural frequencies of the FG beam are analysed by using Finite Elements method. The system of equations of motion is derived by using Lagrange's equations with the assumption of Euler-Bernoulli beam theory. The material properties of the beam are assumed to vary through thickness according to power-law form. Different boundary conditions are attained by applying different stiffness values to the springs connected at the ends. The model is validated by comparing the results with previous studies. The effects of various material distributions and spring support values on the natural frequen-cy parameters of the FG beam are discussed in detail.
Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam
Journal of Aerospace Technology and Management
Euler, Timoshenko and high shear deformation theories to analyze the free vibration of the functionally graded (FG) beam were developed. The mechanical properties of this beam were assumed to differ in thickness direction according to the model of a power-law distribution. The principle of Hamilton was used to find equations of motion. For free vibration, the analytical solution of these equations was presented using the Navier method. The effect of power index, aspect ratio, modulus ratio, and deformation theories on dimensionless frequency were studied numerically by Ansys software and analytically according to different beam theories using the Fortran program. The obtained results from these programs were compared with each other and with some previous research. Results showed an excellent agreement with the previous research. The numerical and analytical results showed that the use of this new FG beam model especially based on first and high shear deformation theories leads to the reduction of dimensionless frequency. It may be concluded that, the including of shear's effect leads to a decrease in the dimensionless frequency. From the modeling and analysis of this model, it is possible to know what is the appropriate design for this FG beam model to reduce the vibration.
Free vibration analysis of functionally graded beams with simply supported edges
Materials and Design, 2007
In this study, free vibration analysis of a functionally graded (FG) beam under various boundary conditions is carried out. Natural frequencies of the FG beam are analysed by using Finite Elements method. The system of equations of motion is derived by using Lagrange's equations with the assumption of Euler-Bernoulli beam theory. The material properties of the beam are assumed to vary through thickness according to power-law form. Different boundary conditions are attained by applying different stiffness values to the springs connected at the ends. The model is validated by comparing the results with previous studies. The effects of various material distributions and spring support values on the natural frequency parameters of the FG beam are discussed in detail.
Science and Engineering of Composite Materials, 2015
Free vibration behavior of short beams made of axially layered functionally graded material (FGM) was investigated experimentally and numerically. Beams, which have gradation of the material properties in the axial direction, are fabricated by powder metallurgy technique using different weight fractions of aluminum and silicon carbide powders. In order to determine elasticity modulus of axially layered functionally graded (FG) beams, homogeneous beams containing different weight fractions of Al (aluminum) and SiC (silicon carbide) are produced, and these homogeneous beams are subjected to tensile tests. Density of each homogeneous layer is also calculated experimentally. After determination of the mechanical properties of each layer of the FG beams, they are modeled in a finite element program (ANSYS) according to Timoshenko beam theory, and free vibration analyses are performed. Fundamental frequencies of the axially layered FG beams produced are also calculated experimentally. FG ...
Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach
Applied Mechanics
A Dynamic Finite Element (DFE) method for coupled axial–flexural undamped free vibration analysis of functionally graded beams is developed and subsequently used to investigate the system’s natural frequencies and mode shapes. The formulation is based on the Euler–Bernoulli beam theory and material grading is assumed to follow a power law variation through the thickness direction. Using the closed-form solutions to the uncoupled segments of the system’s governing differential equations as the basis functions of approximation space, the dynamic, frequency-dependent, trigonometric interpolation functions are developed. The interpolation functions are used with the weighted residual method to develop the DFE of the system. The resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies. Example elements using DFE, Finite Element Method (FEM) and the Dynamic Stiffness Method (DSM) are implemented in MATLAB for testing, verification, and validation....
Free vibration analysis of functionally graded beams using a higher-order shear deformation theory
2017
This paper presents an analytical solution to the free vibration analysis of functionally graded beams by using a refined hyperbolic shear deformation theory in which the stretching effect is included. The modulus of elasticity of beams is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. Equations of motion are derived from Hamilton's principle and Navier-type analytical solutions for simply supported beams are compared with the existing solutions to verify the validity of the developed theory. Numerical results are obtained to investigate the effects of the power-law index and sideto-thickness ratio on the natural frequencies. It can be concluded that the present theories are not only accurate but also simple in predicting the free vibration responses of FG beams.
Free vibration characteristics of a functionally graded beam by finite element method
Applied Mathematical Modelling, 2011
This paper presents the dynamic characteristics of functionally graded beam with material graduation in axially or transversally through the thickness based on the power law. The present model is more effective for replacing the non-uniform geometrical beam with axially or transversally uniform geometrical graded beam. The system of equations of motion is derived by using the principle of virtual work under the assumptions of the Euler-Bernoulli beam theory. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Numerical results are presented in both tabular and graphical forms to figure out the effects of different material distribution, slenderness ratios, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of the beam.
Implementation of numerical approximations in studying vibration of functionally graded beams
International Journal of Dynamics and Control, 2017
In this investigation, a brief review on three efficient computational techniques viz. Finite Element Method, Differential Quadrature Method and Rayleigh-Ritz Method along with their mathematical formulation to study free vibration of thin Functionally Graded (FG) beams subject to various classical boundary supports have been presented. The deformation of FG beam is based on the framework of classical beam theory. Three different FG beam constituents assumed in this study are Al/Al 2 O 3 , Al/ZrO 2 and SUS304/Si 3 N 4 , in which the first component is meant for the metal constituent and the second for ceramic constituent respectively. The material properties of FG beam are assumed to vary continuously along thickness direction in a power-law form. The objective is to outline exemplary works carried out by various researchers on the concerned problem and also to find the effect of volume fraction of FG constituents on natural frequencies. The natural frequencies of different FG beams under four sets of classical edge supports have been evaluated along with two-dimensional mode shapes after finding the convergence with reference to concerned numerical methods and validation with available literature. Keywords Vibration • Functionally graded beam • FEM • DQM • RRM 1 Introduction Functionally Graded (FG) materials are emerging advanced composites in recent decade for their thermal resistance properties, which was first discovered by a group of material scientists in Japan to withstand a huge temperature fluctuation across a very thin cross-section in a space-plane project. Major components of FG composites are metal and ceramic materials, in which the constituent properties vary spatially along thickness direction in a specific mathematical pattern. As their microstructure has not yet been revealed, the mechanics and governing equations related to homogeneous case are assumed to be true for elastic FG composites. Studying dynamics of functionally graded beam is one of the interesting problems in current era and literature related Karan K. Pradhan-SERB National Post-Doctoral Fellow.