Application of Haar wavelet discretization and differential quadrature methods for free vibration of functionally graded micro-beam with porosity using modified couple stress theory

The present investigation is aimed at the implementation of Haar Wavelet Discretization Method (HWDM) and Differential Quadrature Method (DQM) on the free vibration of a Functionally Graded (FG) micro-beam with uniformly distributed porosity along the thickness. As per the power-law exponent model, the material properties such as Young's modulus, and mass density are varied along the thickness of the FG micro-beam and the beam is made up of Aluminum (Al) as metal constituent and Alumina (Al2O3) as ceramic constituent. Modified couple stress theory is employed to capture the small scale effect and pointwise convergence studies for HWDM as well as DQM have also been carried out to exhibit the effectiveness of the methods with respect to the undertaken problem. The results obtained by both methods are compared to demonstrate the accuracy of the present model, revealing excellent accuracy. The effect of power-law exponent, porosity volume fraction index, and thickness to material length scale parameter on the natural frequencies is thoroughly investigated with proper physical explanations for Hinged-Hinged (H-H), Clamped-Hinged (C-H), Clamped-Clamped (C-C), and Clamped-Free (C-F) boundary conditions. Further, mode shapes are also plotted for qualitatively assessing the dynamics of the structural component.