Identification of the parameters of a rod with a longitudinal rectangular groove using two spectra of natural frequencies of bending vibrations

Cover Page

Cite item

Full Text

Abstract

The inverse coefficient problem involves determining the geometric parameters of a longitudinal rectangular groove based on the natural frequencies of the bending vibrations of a rectangular rod. It is assumed that the groove does not extend along the entire length of the rod, but rather from a certain point to the right end. To solve the problem, the rod with the longitudinal groove is modeled as two sections: the first section without a groove and the second section with a groove.
Mating conditions are applied at the connection point, where deflection values, rotation angles, bending moments, and shear forces are equated. The behavior of the natural frequencies of bending vibrations when changing the length of the groove was investigated. A solution method is proposed that allows for determining the required parameters based on a finite number of natural frequencies of bending vibrations. It is shown that the solution is unambiguous when using frequency spectra with respect to mutually perpendicular axes.

About the authors

Ilnur M. Utyashev

Mavlyutov Institute of Mechanics, Ufa Centre RAS

Author for correspondence.
Email: utyashevim@mail.ru
ORCID iD: 0000-0002-2342-0492
SPIN-code: 7856-5351
Scopus Author ID: 56966700200
ResearcherId: J-1064-2018
https://www.mathnet.ru/person139311

Cand. Phys. & Math. Sci.; Researcher; Lab. of Solid Mechanics

Russian Federation, 450054, Ufa, prosp. Oktyabrya, 71

Alfir F. Fatkhelislamov

Ufa University of Science and Technology

Email: alfir93@mail.ru
ORCID iD: 0000-0002-8494-9592
https://www.mathnet.ru/person203542

Senior Lecturer; Dep. of Information Security Management

Russian Federation, 450076, Ufa, Zaki Validi st., 32

References

  1. Shakirzyanov R. A., Shakirzyanov F. R. Dinamika i ustoichivost’ sooruzhenii [Dynamics and Stability of Structures]. Moscow, IP Ar Media, 2022, 119 pp (In Russian). DOI: https://doi.org/10.23682/116444
  2. Akulenko L. D., Baidulov V. G., Georgievskii D. V., Nesterov S. V. Evolution of natural frequencies of longitudinal vibrations of a bar as its cross-section defect increases, Mech. Solids, 2017, vol. 52, no. 6, pp. 708–714. EDN: XYGSVN. DOI: https://doi.org/10.3103/S0025654417060103.
  3. Nesterov S. V., Baidulov V. G. Evolution of natural frequencies and forms of flexural vibrations of the rod with the increase of the defect cross section, Proc. Geosred., 2018, no. 4, pp. 1174–1179 (In Russian). EDN: YSJIJN.
  4. Nusratullina L. R., Pavlov V. P. Transverse vibrations of a rod with variable cross section and calculating its natural frequencies and shapes, Inform. Technol. Probl. Solutions, 2019, no. 3, pp. 37–42 (In Russian). EDN: LRPOQR.
  5. Bukenov M., Ibrayev A., Zhussupova D., Azimova D. Numerical solution of a problem on bending oscillation of a rod, Bulletin Karaganda Univ. Math., 2019, no. 2, pp. 32–36. EDN: AJRJBB. DOI: https://doi.org/10.31489/2017M2/32-36.
  6. Akulenko L. D., Gavrikov A. A., Nesterov S. V. Identification of cross-section defects of the rod by using eigenfrequencies and features of the shape of longitudinal oscillations, Mech. Solids, 2019, vol. 54, no. 8, pp. 1208–1215. EDN: GKJAEX. DOI: https://doi.org/10.3103/S0025654419080119.
  7. Popov A. L., Sadovskiy S. A. On the conformity of theoretical models of longitudinal rod vibrations with ring defects experimental data, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2021, vol. 25, no. 1, pp. 97–110 (In Russian). EDN: JXMCLM. DOI: https://doi.org/10.14498/vsgtu1827.
  8. Akhtyamov A. M., Salyakhova E. V. Does the presence of a cavity in the rod always change the natural frequencies?, Techn. Acoustics, 2011, vol. 11, 7 (In Russian). EDN: ONGQLJ.
  9. Utyashev I.M., Fatkhelislamov A. F. Identification of the length of the longitudinal notch of the rod by the natural frequencies of bending vibrations, Sist. Upravl. Inform. Tekhnol., 2022, vol. 4, no. 90, pp. 19–22 (In Russian). EDN: FJCOQH. DOI: https://doi.org/10.36622/VSTU.2022.90.4.004.
  10. Rice J. R., Levy N. The part-through surface crack in an elastic plate, J. Appl. Mech., 1972, vol. 39, no. 1, pp. 185–194. DOI: https://doi.org/10.1115/1.3422609.
  11. Freund L. B., Herrmann G. Dynamic fracture of a beam or plate in plane bending, J. Appl. Mech., 1976, vol. 43, no. 1, pp. 112–116. DOI: https://doi.org/10.1115/1.3423760.
  12. Narkis Y. Identification of crack location in vibrating simply-supported beames, J. Sound Vibrations, 1994, vol. 172, no. 4, pp. 549–558. DOI: https://doi.org/10.1006/jsvi.1994.1195.
  13. Akhtyamov A. M., Ilgamov M. A. Flexural model for a notched beam: Direct and inverse problems, J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 1, pp. 132–141. EDN: RFFRAF. DOI: https://doi.org/10.1134/S0021894413010161.
  14. Vatulyan A. O., Osipov A. V. Transverse vibrations of beam with localized heterogeneities, Vestn. Donsk. Gos. Tekhn. Univ., 2012, vol. 12, no. 8, pp. 34–40 (In Russian). EDN: QADMKN.
  15. Il’gamov M.A., Khakimov A.G. Diagnosis of damage of a cantilever beam with a notch, Russ. J. Nondestruct. Test, 2009, vol. 45, no. 6, pp. 430–435. EDN: MWWMPV. DOI: https://doi.org/10.1134/S1061830909060072.
  16. Akhtyamov A. M., Urmancheev C. F. Determination of the parameters of a rigid body clamped at an end of a beam from the natural frequencies of vibrations, J. Appl. Ind. Math., 2010, vol. 4, no. 1, pp. 1–5. DOI: https://doi.org/10.1134/S1990478910010011.
  17. Utyashev I. M., Akhtyamov A. M. Determination of the boundary conditions for string fastening using natural vibration frequencies in a medium with a variable asymmetric elasticity coefficient, J. Appl. Mech. Techn. Phys., 2018, vol. 59, no. 4, pp. 755–761. EDN: YBIXLF. DOI: https://doi.org/10.1134/S0021894418040235.
  18. Vibratsii v tekhnike [Vibrations in the Technics], vol. 1, Kolebaniia lineinykh sistem [Oscillations of Linear Systems], ed. V. V. Bolotin. Moscow, Mashinostroenie, 1978, 352 pp.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Figure 1. Image of a rod with a longitudinal groove

Download (52KB)
Indexing metadata
3. Figure 2. Dependence of $\lambda_{1}$ on the groove length

Download (64KB)
Indexing metadata
4. Figure 3. Dependence of $\lambda_{2}$ on the groove length

Download (65KB)
Indexing metadata
5. Figure 4. Dependence of $\lambda_{3}$ on the groove length

Download (65KB)
Indexing metadata

Copyright (c) 2024 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).