Abstract <p>A mathematical model is proposed for longitudinal oscillations in a cylindrical sample, taking into account cubic nonlinearity in the extended Kelvin–Voigt model for a viscoelastic body. The effect of nonlinear terms describing the elastic damping properties of the material exerted on the longitudinal oscillations of the sample is analyzed. The solution to the obtained nonlinear equations is constructed numerically with the use of the Galerkin method. The effect of nonlinearities has been estimated by the amplitude–frequency characteristics in the range of the first resonant frequency of the sample at different amplitudes of kinematic excitation. An algorithm for identifying the parameters of the mathematical model based on the experimental results is proposed. and a criterion for a quantitative assessment of the identifying accuracy of the sought parameters is formulated.</p>