Efficient Numerical Method For Free Vibration Analysis Of Functionally Graded Beams With Stochastic Property

Functionally grade materials(FGM),a new type of composite materials,were widely used in civil,mechanical and aerospace engineering applications due to their excellent mechanical properties and the capability of resistance for thermal and corrosion.Generally,the properties of FGM show gradient variation along the spatial coordinates,and there are large discrepancies of mechanical models between FGM and classical homogeneous material or laminated composite material.Thus classical material based mechanical analysis methods are not suitable for analysis of FGM.New methods for mechanical behavior analysis of FGM structures need to be proposed.The main contents include:(1)Free vibration of axially functionally graded beam with certain properties.By applying the Jacobi orthogonal polynomial based spectral expansion to approximate trial function,the characteristic equation of axially FGM beam is derived.Natural frequencies for free vibration can be evaluated by solving the eigenvalue in characteristic equation.The effectiveness of the proposed method is validated in numerical examples.In addition,the relative error analysis of natural frequency is discussed based on various Jacobi polynomials and the number of expansions.The effects of geometrical and material parameters on natural frequency are studied.(2)Free vibration of axially functionally graded beam with stochastic properties.Based on the stochastic material properties of axially FGM represented by random field,the free vibration governing equation of Euler beam is derived,and then a method that based on the approximation of Legendre orthogonal polynomial expansion is proposed to solve the equation.Finally,the characteristic equations of stochastic natural frequencies are obtained.The stochastic natural frequencies of axially FGM beam are evaluated based on chaos polynomial response surface model with various dimension of cross-section and gradient parameter of FGM.Whereas,the probability and statistical analysis of natural frequency indicates that variations of structural dimension and material properties can lead to the change of distribution of stochastic natural frequency.(3)Free vibration of radially functionally graded beam with stochastic propeffies.Based on the higher-order theory and random field,the stochastic dynamic stiffness matrix of radially FGM beam is derived by spectral element method.In order to carry out the statistical analysis of stochastic natural frequency,the sparse Gegenbauer polynomial response surface model is proposed.Then the effectiveness of proposed methods is validated based on stochastic natural frequency analysis of homogeneous beam.The influences of correlation length,coefficients of variation,covariance model and marginal probability distribution on stochastic natural frequency are studied in numerical examples.