Three-dimensional problems of controlling the dynamics of incompletely observed thick elastic plates. Part II. The case of discretely specified desired condition
DOI:
https://doi.org/10.17721/1812-5409.2025/2.32Keywords:
spatially distributed dynamical systems, spatial problems of elasticity theory, thick elastic plates, control problemsAbstract
For the first time, complex control problems for a thick elastic plate, the dynamics of which are described by Lame's equations in a bounded three-dimensional spatial domain over a finite time interval, are being solved. The essence of these problems lies in the fact that the field of elastodynamic displacements of the plate under the influence of a control function should attain a desired state, specified discretely in space and time, or remain in the minimally possible least-squares neighborhood of these values, given the known conditions. The control functions can be chosen from volumetric, initial, surface, and edge functions of external dynamic influences, which can affect the plate's dynamics either individually or in practically feasible combinations of several such influences. These problems are solved given the presence of discretely defined external dynamic observations of the initial, boundary, and edge conditions of the plate at known points, which belong to the domains of definition of the aforementioned conditions. In fact, these observations are discrete initial, boundary, and edge conditions that are complemented with Lame's equations. However, their quantity and quality can be arbitrary and will only affect the complexity of the computations. Based on the discrete initial, boundary, and edge conditions, along with the discrete desired states, and considering the integral form of the model, control functions are found as pseudosolutions of a system of algebraic equations or a system of integral equations. Analytical dependencies of the external dynamic control factors and the field of elastodynamic displacement of the points of the plate induced by them are also constructed and evaluated for accuracy and uniqueness. Partial cases of some possible combinations of control strategies for the dynamics of the thick elastic plate are also considered.
Pages of the article in the issue: 202 - 210
Language of the article: Ukrainian
References
Grigolyuk, E. I., & Selezov, I. T. (1973). Mechanics of solid deformable bodies. Vol. 5. Non-classical theories of oscillations of rods, plates and shells. VINITI [in Russian].
Grigorenko, Ya. M., & Grigorenko, A. Ya. (2013). Static and dynamic problems for anisotropic inhomogeneous shells with variable parameters and their numerical solution (review). International applied mechanics, 49(2), 123–193. https://doi.org/10.1007/s10778-013-0558-x
Grigorenko, Ya. M., Savula, Ya. G., & Mukha, I. S. (2000). Linear and nonlinear problems of the elastic deformation of complex shells and methods of their numerical solution. International applied mechanics, 36(8), 979–1000. https://doi.org/10.1023/A:1026645731095
Lur'e, A. I. (1964). Three-Dimensional Problems of the Theory of Elasticity. Interscience Publishers.
Nemysh, Y. N., & Khoma, I. Y. (1993). Stress-strain state of non-thin plates and shells. Generalized theory (survey). International applied mechanics, 29(11), 873–902.
Nigul, U. (1963). On the application of the symbolic method of A.I. Lurye in the three-dimensional theory of the dynamics of elastic plates. Bulletin of the Academy of Sciences of the Estonian SSR. Series of physical, mathematical and technical sciences, 12(2), 146–155 [in Russian].
Pichkur, V. V., & Sobchuk, V. V. (2021). Mathematical model and control design of a functionally stable technological process. Journal of Optimization, Differential Equations and Their Applications (JODEA), 29(1), 32–41. http://dx.doi.org/10.15421/142102
Pichkur, V., Sobchuk, V., Cherniy, D., & Ryzhov, A. (2024). Functional stability of production processes as control problem of discrete systems with change of state vector dimension. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, 78(1), 105–110. https://doi.org/10.17721/1812-5409.2024/1.21
Selezov, I. T. (1961). On the hypotheses underlying the refined equations of transverse plate vibrations and some features of these equations. Applied mechanics, 7, 538–546 [in Ukrainian].
Selezov, I. T. (2018). Development and application of the Cauchy–Poisson method to layer elastodynamics and the Timoshenko equation. Cybernetics and Systems Analysis, 54, 434–442. https://doi.org/10.1007/s10559-018-0044-x
Sheremetyev, M. P., & Pelekh, B. L. (1964). Towards the construction of a refined theory of plates. Engineering Journal, 4(3), 504–509 [in Russian].
Stoyan, V. A. (2016). Methods of mathematical modeling in problems of dynamics of thick elastic plates. VPTs Kyivs'kyi Universitet [in Ukrainian].
Stoyan, V. A. (2024). Distributed spatiotemporal systems under uncertainty. VPTs Kyivs'kyi Universitet [in Ukrainian].
Stoyan, V. A. (2018). Three-dimensional integral mathematical models of the dynamics of thick elastic plates. Cybernetics and Systems Analysis, 54(2), 232–241. https://doi.org/10.1007/s10559-018-0024-1
Stoyan, V. A., Cherniy, D. I., & Voloshchuk, S. D. (2024). Three-dimensional problems of controlling the dynamics of incompletely observed thick elastic plates. Part II. The case of discretely specified desired condition. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, 79(2), 65–71 [in Ukrainian].
Timoshenko, S., & Woinowsky-Krieger, S. (1987). Theory of plates and shells. McGraw-Hill.
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