In equation (5), seeking vibration frequencies, on can assume ( , ) ( )sin( ) Y S ξ τ ξ ντ = and harmonic external forcing sin( ) f f ντ = . This leads to: 2 2 2 2 1 1 2 1 3 4 2 2 2 2 ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ) S S K f f F f f S F ξ ξ ξ ξ ν ξ ξ ξ ξ ξ ξ ∂ ∂ ∂ + − = ∂ ∂ ∂ (6) where 0 0 1 4 2 0 0 0 E I K AL ρ = Ω , 1 2 2 0 0 0 F F AL ρ = Ω and 2 2 0 0 0 f F AL ρ = Ω . Given the properties of the FGM and equations (2), i.e., 1( ) f ξ , 3( ) f ξ, 2( ) f ξ and 4( ) f ξ , numerical solutions for equation (6) can, in principle, be obtained. FGM I is taken to be a composite made from aluminum and silicon carbide. The properties of the material are given in Table 1 and are taken from reference [12]. Aluminum / Silicon Carbide 0E (GPa) 105.197 0ρ( 3 / kg m ) 2710.000 a 1.14568 m 1.00000 n 0.17611 υ 0.33 Table 1 – Material properties for Al / SiC FGM FGM II The second FGM model utilized assumes a composition derived from of a mixture of two materials, with the material variation given by a power-law gradient (see, for instance, reference [13]). The material properties are given by: ( ) ( )( ) b t b x E x E E E L λ = + − , ( ) ( )( ) b t b x x L λ ρ ρ ρ ρ = + − (7) where λ is a positive constant describing the volume fraction, which can be determined experimentally ([13]). The subscripts b and t refer to the value of the parameter at 0 x = and x L= , respectively. These values are the ones for the “pure” materials involved in the composition of the FGM and are obtained from tables or manufacturer’s specifications. Here, the FGM is taken to be a mixture of aluminum and steel. Taking 1 b t E Eγ = and 2 b t ρ γ ρ = , one can write: 1 1 ( ) (1 ) t E x E λ γ γ ξ ⎡ ⎤ = ⎣ + − ⎦, 2 2 ( ) (1 ) t x λ ρ ρ γ γ ξ ⎡ ⎤ = ⎣ + − ⎦ (8) Equation (6) can still be used with the following modifications: 29
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