Vibration analysis of functionally graded porous shear deformable tubes excited by moving distributed loads

The present work focuses on the free and forced vibrations of the functionally graded (FG) porous tubes subjected to the continuously distributed loads, which move along their axial directions with a constant velocity. The modulus of elasticity of porous composite is assumed to be graded in the radial direction of the tube. The open-cell metal foam provides a typical mechanical feature to determine the relationship between coefficients of density and porosity. A refined beam theory in conjunction with the Lagrange method is employed to derive the governing equations of motion. The Newmark-β method is adopted to obtain the response of the tube in the time domain. Two types of graded porosity distributions are taken into account in the present work. A simply-supported FG porous tube is considered as an example to illustrate the effects of the porosity distribution, geometrical parameters, and load length on the vibration behaviors of the FG porous tube.