The paper outlines a procedure to identify the space-and time-dependent external nonstationary load acting on a closed circular cylindrical shell of medium thickness. Time-dependent deflections at several points of the shell are used as input data to solve the inverse problem. Examples of numerical identification of various nonstationary loads, including moving ones are presented. The relationship between the external load and the stress-strain state of the shell is described by the Volterra equation of the first kind. The identification problem is solved using Tikhonov's regularization method and Apartsin's h-regularization method Keywords: cylindrical shell, nonstationary load, inverse problem, Volterra equation of the first kind, regularization methodIntroduction. Publications on nonstationary vibrations of elastic members are very numerous. Noteworthy are recent interesting studies in this field [7][8][9][10][11]14]. There are far fewer publications concerned with so-called inverse problems. The monographs [5,6] formulate several problems to determine the time-dependent components of the loads on various structural members, assuming that their spatial distribution is known and that the result of application of these loads (displacements or strains) in the form of functions of time at some points is known too. These monographs also give a detailed description of purely mathematical studies that actually provide a theoretical basis for solving inverse problems in mathematical physics and solid mechanics. There are also a few papers on inverse problems [12,13,15] which address the nonstationary behavior of deformed objects. The objective of the present paper is to recover not only the time-dependent component of the nonstationary load on a cylindrical shell, but also its distribution over the mid-surface of the shell.1. Let us identify an axisymmetric nonstationary transverse load arbitrarily distributed along the axis of a hinged cylindrical Timoshenko-type shell (Fig. 1). To solve the identification problem, we will use Tikhonov's regularization [3, 6] and Apartsin's h-regularization [1,2]. To solve the inverse, it is first necessary to solve the direct problem, which should be considered as the first, auxiliary stage in solving the identification problem.The reaction of a Timoshenko-type shell of medium thickness to an axisymmetric transverse load is modeled by a system of linear differential equations [4]:

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The paper presents a method to identify a system of several nonstationary independent transverse loads on a rectangular plate of medium thickness. The input data for solving the inverse problem are time-dependence of displacements or strains given at some points of the plate. Examples of numerical calculations to identify two or three loads are presented. The deformation of plates is modeled using a refined Timoshenko theory. Tikhonov's regularizing algorithm is used to solve the Volterra equations for the unknown loads Introduction. The need to ensure high efficiency of strength analysis of structural members contributes to the development of a branch of mechanics focused on solving direct and inverse problems to identify parameters of mechanical systems under static and dynamic loading [11][12][13][14][15][16][17].Of special interest are studies on ill-posed inverse problems. The possibility of solving such problems stems from the theory of ill-posed problems of mathematical physics, which is based on Tikhonov's fundamental works. This theory in combination with the wide use of computers made it possible to develop a method for solving ill-posed problems in solid mechanics [9, 10, 18], astrophysics [3], and thermal physics [5]. Serious results were obtained in [6] in constructing stable solutions by the averaging method for differential equations of motion with discontinuous right-hand sides.An important issue that needs to be dealt with in solving inverse problems of solids mechanics is insufficient data such as unknown impulsive loads on various structural members (plates and shells).1. Let us consider an elastic isotropic plate of medium thickness subject to several impulsive loads normal to the upper face. By an impulsive load is meant a nonstationary load acting over a time interval commensurable with the period of vibration at the lowest natural frequency. The behavior of the plate is examined over a finite time interval. We assume that the manner in which these loads vary is unknown but the areas of load application and their coordinates are specified. It is also assumed that these areas are simply connected (in the general case), and the normal loads within these areas are independent of the spatial coordinates.We choose a Cartesian coordinate system such that the midsurface of the plate lies in the plane xOy and is bounded by the straight lines x x l y = = = 0 0 , , , and y m = (Fig. 1), and the Oz-axis is normal to the plane xOy.