Modal analysis of an aluminum beam coated with damaged and porous functionally graded material

Abstract

In this paper modal analysis is performed on a symmetric aluminum beam which is coated with functionally graded material containing porosities. A polynomial function is used to vary the density and elasticity through the thickness of the coating, while the effective elastic modulus and density are found with classical lamination theory. To achieve a truthful modeling the gradually changing mechanical properties of the coating are modeled as 25 layers of material, while each individual layer is isotropic and homogeneous. MATLAB is used to write a finite element code and Timoshenko beam elements are used to include shear deformation effects. To show the influences of crack location, crack depth, porosity and the polynomial function index on the natural beam frequencies a parametric study is conducted. Multiple boundary conditions were also considered and it was found that the natural frequency values were significantly affected by the studied parameters.

Determination of natural frequencies of non‐uniform aluminum beams coated with functionally graded material

“…The stiffness and mass matrices of the original Timoshenko element are derived either with energy methods or by analytical derivation [32, 33]. However, in order for the model to allow for axial loads and deformations, the stiffness and mass matrix of a bar element is included to obtain the following stiffness and mass matrices of the modified Timoshenko beam element, as detailed in earlier research [21, 25]. The present approach can be summarized simply, Equations (7–: $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr \left[K\right]=\rho Aa\left[\begin{matrix}{{2}\over{3}} & & & {{1}\over{3}} & & \\ & {m}_{1} & {m}_{2} & & {m}_{3} & {m}_{4} \\ & {m}_{2} & {m}_{5} & & -{m}_{4} & {m}_{6} \\ {{1}\over{3}} & & & {{2}\over{3}} & & \\ & {m}_{3} & -{m}_{4} & & {m}_{1} & -{m}_{2} \\ & {m}_{4} & {m}_{6} & & -{m}_{2} & {m}_{5} \end{matrix} \right]+\hfill\cr {{\rho {I}_{z}}\over{30a{\left(1+3\beta \right)}^{2}}}\left[\begin{matrix} & & & ...$$…”

“…The stiffness and mass matrices of the original Timoshenko element are derived either with energy methods or by analytical derivation [32,33]. However, in order for the model to allow for axial loads and deformations, the stiffness and mass matrix of a bar element is included to obtain the following stiffness and mass matrices of the modified Timoshenko beam element, as detailed in earlier research [21,25]. The present approach can be summarized simply, Equations (7-10):…”

“…While research on regular beams covered with functionally graded materials is rare, there are no studies on the natural frequencies of irregular non-uniform beams, coated with functionally graded materials [18][19][20][21][22][23][24]. Some research is done on nano beams and coated beams as well [25][26][27][28][29][30].…”