Cite this article as:

Bogachev I. V., Vatulyan A. O., Dudarev V. V., Lapina P. A., Nedin R. D. Identification of Properties of Inhomogeneous Plate in the Framework of the Timoshenko Model. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2017, vol. 17, iss. 4, pp. 419-430 DOI: 10.18500/1816-9791-2017-17-4-419-430


Identification of Properties of Inhomogeneous Plate in the Framework of the Timoshenko Model


We consider an inverse problem on identification of properties of an inhomogeneous circular plate for the Timoshenko model. The identification procedure is based on the analysis of
acoustical response at some point of the plate in the given set of frequencies. The vibrations are caused by a uniformly distributed load applied to the upper face of the plate. We have derived the oscillation equations for a symmetric circular plate and formulated the boundary conditions in the dimensionless form. To solve the inverse problem on a reconstruction of the inhomogeneous bending stiffness function, we have developed a special solving technique called the ‘algebraization method’ based on a decomposition of the sought-for functions by systems of linearly independent functions. After substitution of these decompositions in the original motion equations, the inverse problem is reduced to solving a system of linear equations with respect to the expansion coefficients for the deflection function and the normal rotation angle, and subsequent solving of a system of nonlinear equations with respect to the expansion coefficients for the bending stiffness function. The method developed is illustrated by a series of computational experiments on a reconstruction of monotonic and non-monotonic functions showing its efficiency.


1. Vatulyan A. O. Obratnye zadachi v mekhanike deformiruemogo tverdogo tela [Inverse problems in mechanics of solids]. Moscow, Fizmatlit, 2007. 223 p. (in Russian).
2. Timoshenko S. P., Voynovskiy-Kriger S. Plastinki i obolochki [The plates and the shells]. Moscow, Fizmatgiz, 1963. 635 p. (in Russian).
3. Grigoluk E. I., Selezov I. T. Neklassicheskie teorii kolebanij sterzhnej plastin i obolochek [Nonclassical theories of vibrations of plates of plates and shells]. Moscow, VINITI, 1973.
272 p. (in Russian).
4. Tovstik P. Е. On the non-classic models of beams, plates and shells. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2008, vol. 8, iss. 3, pp. 72–85 (in Russian).
5. Tovstik P. Е., Tovstik T. P. Two-dimensional model of an anisotropic non-uniform material plate. Proceedings of the Russian Academy of Sciences. Mechanics of solids, 2017, no. 2,
pp. 32–45 (in Russian).
6. Endo M. Study on an alternative deformation concept for the Timoshenko beam and Mindlin plate models. Intern. J. Engineering Sci., 2015, vol. 7, pp. 32–48. DOI:
7. Kovalev V. A. Dynamics of multilayered thermoviscoelastic plates. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2009, vol. 9, iss. 4, pt. 1, pp. 61–78 (in Russian).
8. Guk N. A., Stepanova N. I. Identification of the geometry and elastic properties of rigid inclusions in a thin plate. Eastern-European Journal of Enterprise Technologies, 2016,
vol. 2, no. 7(80), pp. 4–9 (in Russian). DOI:  10.15587/1729-4061.2016.64395
9. Ablitzer F., Pezerat C., Lascoup B., Brocail J. Identification of the flexural stiffness parameters of an orthotropic plate from the local dynamic equilibrium without a priori
Механика 429 knowledge of the principal directions. J. Sound and Vibration, 2017, vol. 404, pp. 31–46.
DOI: 10.1016/j.jsv.2017.05.037.
10. Pierron F., Gu X. Towards the design of a new standard for composite stiffness identification. Composites Part A: Applied Science and Manufacturing, 2016, vol. 91, pt. 2,
pp. 448–460. DOI: 10.1016/j.compositesa.2016.03.026.
11. Bogachev I. V., Vatul’yan A. O., Yavruyan O. V. Reconstruction of the stiffness of an inhomogeneous elastic plate. Acoustical physics, 2016, vol. 62, no. 3, pp. 377–382. DOI:
12. Fridman L. I., Morgachev K. S. Construction and implementation of solutions to problems of nonstationary plate oscillations (Tymoshenko model). Vestnik of Samara State
University. Natural Science Series, 2006, vol. 42, no. 2, pp. 92–102 (in Russian).
13. Fletcher K. Chislennye metody na osnove metoda Galerkina [Numerical methods based on the Galerkin method]. Moscow, Mir, 1988. 352 p. (in Russian).

Short text (in English): 
Full text: