On the influence of surface stresses and inertia on the natural low-frequency vibrations of an elastic ultrathin strip-beam

A differential equation is derived that describes free long-wave vibrations of a low-dimensional elastic isotropic strip-beam, taking into account effects on free surfaces. Boundary conditions on external surfaces are formulated within the framework of the Gurtin – Murdoch surface theory of elasticity, which takes into account surface inertia and shear stresses, including residual ones. Additional geometric dimensions are introduced, associated with the face surfaces, which are assumed to be small compared to the main geometric dimension  — the wavelength. The ratio of the thickness of the ultrathin strip to the wavelength of bending vibrations is considered as the main small parameter. Using the method of asymptotic integration of two-dimensional equations of the theory of elasticity over the thickness of the strip-beam, relations for displacements and stresses in the volume of the strip were obtained in explicit form. The main result of the paper is a differential equation for low-frequency vibrations of a beam, which takes into account surface effects and generalizes the well-known equations of beam theory. It is shown that the presence of surface stresses leads to an increase in natural frequencies from the lower spectrum, while taking into account surface inertia, as well as transverse shears in volume, leads to a decrease in frequencies.