Gradient model of free longitudinal vibrations of an elastic rod

The problem of finding the spectrum of eigenvalues and eigenmodes of longitudinal oscillations of an elastic rod taking into account scale effects is investigated. Based on the non-classical theory of elasticity of Aifantis and the variational principle of Hamilton-Ostrogradsky, gradient models of free longitudinal oscillations of a rod with different boundary conditions are obtained. The following classical boundary conditions are considered: a) one end of the rod is clamped, and the other is free; b) both ends of the rod are rigidly fixed; c) both ends are free. The statements of the gradient models include the fourth-order equation of motion with a small gradient parameter at the highest derivative, as well as classical and non-classical boundary conditions. To solve singularly perturbed problems, the asymptotic Vishik-Lyusternik method is used. The expression for the amplitude of the rod displacement is presented as a sum of three functions, where the first term of the sum describes the change in the amplitude of oscillations far from the boundaries of the rod (classical solution), the second and third terms are the boundary layer functions. Each of the functions in the expression for the displacement is represented as an expansion in a small parameter. Restricting ourselves to two approximations in the Vishik-Lyusternik method, we obtain asymptotic corrections to the natural frequencies and modes of vibration of the rod for a small value of the scale parameter. We obtain exact values of the spectrum of eigenvalues in the gradient formulation by solving frequency equations that are obtained from the equality to zero of the determinant of the system of algebraic equations when the boundary conditions are satisfied. To check the accuracy of the asymptotic formulas, we compare the eigenvalues obtained by solving the frequency equation and by the asymptotic formulas for different values of the scale parameter. The applicability limits of the asymptotic approach are determined. The influence of the scale parameter on the values of the eigenvalues is investigated.