Free Vibration Analysis of a Functionally Graded Beam with Finite Elements Method (original) (raw)
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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.
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.
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.
2021
This paper studies the free vibration of simply supported functionally graded beam with material graduation transversally through the thickness using the power-law model. Two finite element models are proposed to calculate the first five frequency parameters of a simply supported FG beam. These models are shell and solid models and they are employed using the ANSYS APDL version 17.2. The two models have been verified with the previously published works and found a good agreement with them. Numerical results are presented in graphical forms to study the effects of the power-law index (i.e. material distribution), length-to-thickness ratio, and modulus ratio on the first five frequency parameter of the FG beam. The above mention effects play very important role on the free vibration of the beam. Index of power-law is a parameter that primarily has an effect on the FG beam frequency parameter. It was found that increasing the index of the power-law lead to frequency parameter increases...
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....
Nonlinear free vibration analysis of the functionally graded beams
2012
Nonlinear natural oscillations of beams made from functionally graded material (FGM) are studied in this paper. The equation of motion is derived according to the Euler-Bernoulli beam theory and von Karman geometric nonlinearity. Subsequently, Galerkin's solution technique is applied to obtain the corresponding ordinary differential equation (ODE) for the FGM beam. This equation represents a kind of a nonlinear ODE containing quadratic and cubic nonlinear terms. This nonlinear equation is then solved by means of three efficient approaches. Homotopy perturbation method is applied at the first stage and the corresponding frequency-amplitude relationship is obtained. Frequency-amplitude formulation and Harmonic balance method are then employed and the consequent frequency responses are determined. In addition, Parameter Expansion Method is utilized for evaluating the nonlinear vibration of the system. A parametric study is then conducted to evaluate the influence of the geometrical and mechanical properties of the FGM beam on its frequency responses. Different types of material properties and boundary conditions are taken into account and frequency responses of the system are evaluated for different gradient indexes. The frequency ratio (nonlinear to linear natural frequency) is obtained in terms of the initial amplitude and compared for different materials and end conditions.
Composite Structures, 2010
Exact solutions are presented to study the free vibration of a beam made of symmetric functionally graded materials. The formulation used is based on a unified higher order shear deformation theory. Material properties are taken to be temperature-dependent, and vary continuously through the thickness according to a power law distribution (P-FGM), or an exponential law distribution (E-FGM) or a sigmoid law distribution (S-FGM). The beam is assumed to be initially stressed by a temperature rise through the thickness. Temperature field is considered constant in xy plane of the beam. Hamilton's principle is used to derive the governing equations of motion. Free vibration frequencies are obtained by solving analytically a system of ordinary differential equations, for different boundary conditions.
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.
Effects of different shear deformation theories on free vibration of functionally graded beams
Free vibration of functionally graded (FG) beams subjected to all sets of boundary conditions is examined in the present article. Different higher-order shear deformation beam theories (SDBTs) have been incorporated for the free vibration response of FG beam. The material properties of FG beam are taken in thickness direction in power-law form and trial functions denoting the displacement components are expressed in algebraic polynomials. Rayleigh-Ritz method is used to estimate frequency parameters in order to handle to all sorts of boundary conditions at the edges by a simple way. Comparison of frequency parameters is made with the existing literature in special cases and new results are also provided after checking the convergence of frequency parameters.
Dynamic stiffness formulation and free vibration analysis of functionally graded beams
Composite Structures, 2013
Free vibration analysis of orthotropic plate has been investigated with the help of dynamic stiffness method using classical plate theory. The rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. Hamilton's principle is used to derive the governing differential equations of motion and natural boundary conditions in free vibration. The dynamic stiffness matrix is derived by relating the amplitudes of forces to those of the displacements at the plate ends. The Wittrick-Williams algorithm is used as the solution technique when applying the dynamic stiffness matrix to compute the natural frequencies and mode shapes.