Formulation and study of the problem of optimal excitation of plate oscillations
A model problem of harmonic oscillations of a hinged plate, that is is under the influence of a certain number of point concentrated forces, is considered. The plate model is considered to satisfy Kirchhoff's conditions. The main task of the consideration is to determine the optimal characteristics of excitation - the number of forces, coordinates of their application, amplitudes and phases. The optimality criterion is constructed as the standard deviation of the complex deflections from a given profile function. With the given excitation characteristics, the problem of determining the vibrations is solved in the form of a superposition of the Green functions with singularities at the points of application of forces. The Green function is constructed as a Fourier series by a circular coordinate. By using Parseval equality in L2, the objective function of the optimization problem is represented as a combination of linear and Hermitian forms with respect to complex amplitudes of forces whose matrices are nonlinear (and not convex) dependent on the coordinates of singular points. A complete study of the objective function is performed. Sufficient conditions are determined for reducing the dimension of the control space by analytical determination of the amplitudes of forces. Expressions were obtained to calculate the gradients of the objective function by angular and radial coordinates. A partial case of grouping of excitation forces on concentric circles is considered, that leads to the degeneration of the problem.
Key words: harmonic plate oscillations, optimal excitation.
Pages of the article in the issue: 62-65
Language of the article: Ukrainian
BAZHANOV, V.L and GOLDENBLAT, I.I. i drugie. (1970) Plastini i obolochki iz stekloplasticov. Moskva: Visshaya shkola.
DONNELL, L.H. (1976) Beams, Plates, and Shell. New York: McCraw-Hill Book Company.
ZRAZHEVSKY, G.М. and ZRAZHEVSKA V.F. (2017) Optimizaziyniy pidhid do rozrahunku garmonichnih kolivan krugloiy plastini. Bull. of T. Shevchenko National University of Kyiv Ser.: Phys. & Math. N 3 рр. 63-66.
TIMOSHENKO, S. ( 1989) Theory of Plates and Shells. New York: McCraw-Hill Book Company.
LAWSON, C.L. and HANSON, R.J. (1974) Solving Least Square Problems. New Jersey: Prentice-Hall.
COURANT, R. and HILBERT, D. (1962) Methods of Mathematical Physics, Volume II. New York: Wiley-Interscience.
STRANG G. (2016) Linear Algebra and its Applications. Cambridge: Wellesley- Cambridge Press.
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).