2026 , Volume 31, № 3, p.5-23

In this paper we address the inverse problem of simultaneous identification for all four parameters of the fractional-differential generalization of the standard viscoelastic body model based on the analysis of the acoustic response to steady-state oscillations of an inhomogeneous viscoelastic beam. The problem is considered in two formulations: oscillations in a beam rigidly clamped at one end in the first formulation are caused by a bending force, while in the second one they are caused by a bending moment applied to the other end. In both formulations, either the displacement values at the end of the beam in a certain frequency range or the values of the angle of rotation of the end section serve as additional information for the inverse problem. Since the beam is inhomogeneous, it is assumed that two of the four parameters of the model, instantaneous and long-term moduli are functions of the longitudinal coordinate, while the other two: relaxation time and fractional parameter are constant. For both formulations, we performed sensitivity analysis of the acoustic response for each parameter in two ways, firstly by using computational experiments on alternately varying each of the parameters while fixing the others and the second approach was based on the analysis of the Frechet derivatives of the response function for each parameter. Using the results of sensitivity analysis for solving inverse problems, we constructed an iterative process based on the linearization method, supplemented by elements of the projection approach, which allows simultaneous determination of corrections for two sought functions and two constant parameters. An additional check of the conformity of the corrections to the model constraints, which we performed at each step, allows significant improvement of the convergence for the iterative process.In the last section, we present the results of computational experiments to determine sets of functions of different types and constants of different orders for both problem formulations.