•A new concept is developed for the mathematical modeling of oscillatory processes.•A differential equation of rod oscillations is derived based on the DPL model.•An exact solution is obtained for the modified equation of rod oscillations.•The phenomenological resistance and relaxation coefficients are identified.•New physical phenomena are discovered that are confirmed by experimental data. By introducing relaxation summands in the formula of Hooke’s law, in order to consider the finite propagation velocity of stress and deformations, an equation for damped oscillations of an elastic rod was obtained, including, as opposed to the well-known equation, the third derivative of motion in time as well as the mixed derivative of the spatial variable and time. By using Fourier’s method, its accurate analytical decision was found, the study of which showed that considering relaxation factors results in elimination of step-wise change of stresses and deformations. Comparison of the results of theoretical studies and experimental data applied to longitudinal oscillations of a rod fixed at one end showed that the amplitude and frequency of their oscillation coincided in a satisfactory manner.