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In this paper, the effect of the porosity coefficient and its distribution on the natural frequencies and mode shapes of a thin and thick cylindrical shell made of asymmetric porous materials is investigated  analytically . The mechanical properties of the porous shell are non-uniform in the thickness and isotropic in the circumferential directions. The displacement field is assumed based on the first-order Mirsky–Hermann theory by considering the effects of transverse normal strain and rotary inertia. For more accurate stress resultants, the trapezoidal shape factor is considered in the formulation. The constitutive relations follow Biot’s theory for the porous materials and the kinematic of the problem is linear. The governing equations include four coupled partial differential equations that are solved using an analytical method and they are compared with the finite element analysis. By the sensitivity analysis, the effects of porosity distribution and geometry of the cylinder on the natural frequencies and mode shapes are studied. The results show that the largest frequency corresponds to the material which has the maximum density inside of the cylinder. Abbreviations:  FSDT, first-order shear-deformation theory; 2D, two-dimensional; 3D, three-dimensional; FG, functionally graded; FGM, functionally graded materials; DQ, differential quadrature; MDEW, maximum density in outer (external) wall; MDIW, maximum density in inner wall; C.C, complex conjugate (of proceeding terms)

Effect of porosity distribution on natural frequencies of thin and thick cylinders based on Mirsky–Hermann’s shear deformation theory