INVERSE PROBLEM FOR A TIMOSHENKO BEAM WITH AN ADDITIONAL VISCOELASTIC SUPPORT UNDER NONSTATIONARY DEFORMATION

Abstract

The non-stationary loading of a mechanical system consisting of a beam hinged at the edges and an additional support installed in the span of the beam is considered. The deformation of the beam is modeled on the basis of Timoshenko's hypotheses, taking into account the influence of rotatory inertia and shear. The deformation of the beam is described by a system of partial differential equations, which is solved analytically by means of expansion of the unknown functions into the relevant Fourier series and further use of the Laplace integral transformation. It is assumed that the additional support has linear-elastic and linear-viscous components, and the displacements coincide at the point where the additional support is connected to the beam. The reaction between the beam and the additional support is replaced by an external unknown concentrated force applied to the beam, which varies in time. The law of time variation of this unknown reaction is determined by solving the Volterra integral equation. The inverse problem of deformable solid mechanics is solved, that is, it is assumed that the deflection at a point of the beam with the additional support is known, whereas the law of time variation of the external impulse load causing the deflection is unknown. The application point of the external load and the point of the additional support connection are considered to be known and do not change in the process of deformation (when obtaining the solution of the problem it was supposed that these could be any points of the beam except for its ends). The described inverse problem is reduced to a system of two Volterra integral equations of the first kind with regard to the unknowns of the external disturbing load and reaction between the plate and the additional support, which is solved by analytical and numerical method. Analytical relations and calculation results for specific numerical parameters are given. The results obtained in this work can be used for indirect measurement of impulse and shock loads acting on beams with additional supports, for which not only elastic but also linear-viscous characteristics are taken into account.