Analysis of Geometrically Non-Linear Free Vibrations of Functional Graded Beams in a Thermal Environment (original) (raw)
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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.
Thermal Buckling and Free Vibration Analysis of Heated Functionally Graded Material Beams
Defence Science Journal, 2013
The effect of temperature dependency of material properties on thermal buckling and free vibration of functionally graded material (FGM) beams is studied. The FGM beam is assumed to be at a uniform through thickness temperature, above the ambient temperature. Finite element system of equations based on the first order shear deformation theory is developed. FGM beam with axially immovable ends having the classical boundary conditions is analysed. An exhaustive set of numerical results, in terms of buckling temperatures and frequencies, is presented, considering the temperature independent and temperature dependent material properties. The buckling temperature and fundamental frequency obtained using the temperature independent material properties is higher than that obtained by using the temperature dependent material properties, for all the material distributions, geometrical parameters in terms of length to thickness ratios and the boundary conditions considered. It is also observed that the frequencies of the FGM beam will reduce with the increase in temperature. This observation is applicable for the higher modes of vibration also. The necessity of considering the temperature dependency of material properties in determining thermal buckling and vibration characteristics of FGM beams is clearly demonstrated.
An analytical study on the nonlinear vibration of functionally graded beams
2010
Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.
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.
Response of Functionally Graded Beams of Different Material Combinations under Thermal Environment
In the present work, static and free vibration response of functionally graded beam is investigated under thermal environment using Finite Element method (FEM). The functionally graded (FG) material beam is considered to be composed of various material combinations like metal/metal, metal/ceramic and ceramic/ceramic. The material properties of the beam are assumed to be graded in the thickness direction according to a simple power law distribution. The deflection and mode shapes of the FG beams are obtained for Clamped-Free, Clamped-Clamped and Simply Supported boundary conditions under both mechanical and thermal load. Numerical results are obtained for the model in ANSYS software to show the influence of grading of materials, material constituents, boundary conditions, volume fraction and temperature conditions on the response of the FG beams.
Vibrational behavior of beams made of functionally graded materials by using a mixed formulation
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2020
This paper investigates the vibrational behavior of beams made of functionally graded materials using a mixed formulation. Unlike the other high order shear deformation theories (HSDTs), the proposed formulation is elaborated within a double field of displacements and stresses which offers the possibility of the development of low order linear elements with enhanced accuracy. As well as, the effect of the transverse shear strains and the zero condition of the transverse shear stresses on the top and bottom surfaces are verified. The material characteristics of the beams are described via a power law distribution in order to take into account the continuous variation of the volume fraction of its constituents along the thickness direction. Numerical simulations are conducted to show the influence of power law index, slenderness ratios, and boundary conditions on natural frequencies of functionally graded beams. Results demonstrate the efficiency and the applicability of the model bas...
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.
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.