# Investigation of the semi-strip’s stress state in the case of steady-state oscillations

## DOI:

https://doi.org/10.17721/1812-5409.2019/1.11## Abstract

The elastic semi-strip under the dynamic load concentrated at the centre of the semi-strip’s short edge is considered. The lateral sides of the semi-strip are fixed. The case of steady-state oscillations is considered. The initial problem is reduced to the one-dimensional problem with the help of the semi-infinite sin-, cos-Fourier’s transform. The one-dimensional problem is formulated in the vector form. Its solution is constructed as a superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is found with the help of the matrix differential calculations. The partial solution is expressed through Green’s matrixfunction, which is constructed as the bilinear expansion. The inverse Fourier’s transform is applied to the derived expressions for the displacements. The solving of the initial problem is reduced to the solving of the singular integral equation. Its solution is searched as the series of the orthogonal Chebyshev polynomials of the second kind. The orthogonalization method is used for the solving of the singular integral equation. The stress-deformable state of the semi-strip is investigated regarding both the frequency of the applied load, and the load segment’s length.

* Key words*: semi-strip, steady-state oscillations, Fourier’s transform, Green’s matrix-function, singular integral equation.

* Pages of the article in the issue*: 54-57

** Language of the article**: Ukrainian

## References

TARTAKOVSKY, G. P. (1957) Dynamics of automatic gain control systems (in Russian). Moskow-Leningrad: Gosenergoizdat.

SELEZOV, I. T., KRIVONOS, Yu. G. and YAKOVLEV, V. V. (1985) Wave scattering by local heterogeneity in continuous mediums (in Russian). Kyiv: Naukova dumka.

MYKHASKIV, V. V. and KHAY, O. M. (2009) Interaction between rigid-disc inclusion and penny-shaped crack under elastic time-harmonic wave incidence. Int. J. Solids Struct. 46 (3–4). p. 602–616.

GOMILKO, A. M., GRINCHENKO, V. T. & MELESHKO, V. V. (1990) Method of homogeneous solutions and superposition in the mixed problem for an elastic half-strip. Soviet Appl. Mech. 26 (2). p. 193–202.

WÜNSCHE, M., SLADEK, J., SLADEK, V. & ZHANG, CH. (2017) Time-harmonic analysis of cracks in functionally graded piezoelectric materials. PAMM. 17 (1). p. 283-284.

KUBENKO, V. & YANCHEVSKII, I. (2015) Nonstationary load on the surface of an elastic half-strip. International Applied Mechanics. 51 (3), p. 303.

KOVALEV, V.A. & TARANOV, O.V. (2007) Analysis of exact and approximate solutions for the boundary layer near the conditional front of the Rayleigh surface wave in an elastic semi-strip (in Russian). Vestnik SamGu. 6 (56). p. 43-50.

ITOU, S. (1994) Transient dynamic stresses around two equal circular cavities in an infinite elastic strip. Archive of Applied Mechanics. 64 (3). p. 192-205.

VAYSFEL’D, N.D. & ZHURAVLOVA, Z.YU. (2015) On one new approach to the solving of an elasticity mixed plane problem for the semi-strip. Acta Mechanica. 226 (12). p. 4159-4172.

POPOV, G. YA., ABDIMANOV S. A. & EFIMOV V. V. (1999) Funkcii I matrici Grina odnomernih kraevih zadach. Almatu: Racah.

## Downloads

## How to Cite

*Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics*, (1), 54–57. https://doi.org/10.17721/1812-5409.2019/1.11

## Issue

## Section

## License

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).