Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

The main objective of this study is to investigate the free vibrational frequencies of constrained nonlocal Rayleigh nanobeams consisting of functionally graded material with four different porosity distributions embedded in a Winkler foundation under the influence of rotary inertia and deformable springs. For this purpose, an efficient analytical solution is presented which includes the properties of the material distribution based on the power-law rule, rotary inertia and deformable spring effects. The presented nanobeams have two sets of end conditions, in fact all possible combinations of elastic and rigid boundary conditions are applied to the boundaries. The advantage of such models is that specific support conditions (rigid or deformable) can be considered. In this work, sets of four equations of infinite series are derived for the force boundary conditions using Fourier series and the Stokes' transform. Then, two different eigenvalue problems are formulated by excluding the coefficients presented for the analytical solution. The eigenvalues of the problems give the vibration frequencies. To the best of the authors' knowledge, no previous work has been presented that examines the four different porosity distributions considered in this study together with nonlocal elasticity, rotary inertia, Winkler foundation and deformable boundaries. Various studies are performed on the effects of Winkler parameter, porosity parameter, nonlocal parameter, various rigid and deformable boundary conditions and the rotary inertia. The nonlocal parameter and the rotary inertia have a decreasing effect on the frequencies, while the Winkler foundation parameter increases the frequencies. Also, the type of porosity distribution and the stiffness of the deformable springs have significant effects on the frequencies.