phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2013 - 3 issue » pp. 114-129

Archive of Issues

Israel Ashdod
Year
2013
Issue
3
Pages
114-129
<<
>>
Section Mechanics
Title On the determination of loading and fixing for one end of a rod according to its natural frequencies of oscillation
Author(-s) Akhtyamov A.M.ab, Muftakhov A.V.c, Akhtyamova A.A.a
Affiliations Bashkir State Universitya, Institute of Mechanics, Ufa Centre of the Russian Academy of Sciencesb, Shamoon College of Engineeringc
Abstract The identification problem of fixing conditions for a beam according to five natural frequencies of its vibrations is considered. On the basis of Plucker's conditions arising at the restoration of a matrix according to its minors of the maximal order, the set of well-posedness of the problem is constructed and the correctness according to A.N. Tikhonov is proved. We have found an explicit solution to the problem of the identification matrix of the boundary conditions, the above solution is written out in terms of the characteristic determinant for the corresponding spectral problem. The corresponding examples are provided.
Keywords eigenvalues, inverse problem, natural frequencies, beam, concentrated inertial element
UDC 534.113
MSC 35Q74, 74J25, 34B09
DOI 10.20537/vm130309
Received 19 February 2013
Language Russian
Citation Akhtyamov A.M., Muftakhov A.V., Akhtyamova A.A. On the determination of loading and fixing for one end of a rod according to its natural frequencies of oscillation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 114-129.
References
  1. Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach (Methods for solving ill-posed problems), Moscow: Nauka, 1974, 224 p.
  2. Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya (The theory of linear ill-posed problems and its applications), Moscow: Nauka, 1978, 200 p.
  3. Lavrent'ev M.M., Romanov V.G., Shishatskii S.P. Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-posed problems of mathematical physics and analysis), Moscow: Nauka, 1980, 288 p.
  4. Tikhonov A.N., Goncharskii A.V., Stepanov V.V., Yagola A.G. Chislennye metody resheniya nekorrektnykh zadach (Numerical methods for solving ill-posed problems), Moscow: Nauka, 1990, 232 p.
  5. Tikhonov A.N., Leonov A.S., Yagola A.G. Nelineinye nekorrektnye zadachi (Nonlinear ill-posed problems), Moscow: Nauka, 1995.
  6. Lavrent'ev M.M., Reznitskaya K.Kh., Yakhno V.G. Odnomernye obratnye zadachi matematicheskoi fiziki (One-dimensional inverse problems of mathematical physics), Novosibirsk: Nauka, 1982.
  7. Lavrent'ev M.M. Teoriya operatorov i nekorrektnye zadachi (The theory of operators and ill-posed problems), Novosibirsk: Izd. Inst. Mat., 1999.
  8. Strutt J.W. (Lord Rayleigh). The Theory of Sound, vol. I. Translated under the title Teoriya zvuka, tom 1, Moscow-Leningrad: Gostekhizdat, 1940.
  9. Kollatz L. Zadachi na sobstvennye znacheniya (s tehnicheskimi prilozhenijami) (Eigenvalue problems (with technical applications)), Moscow: Nauka, 1968. 503 p.
  10. Vibratsii v tekhnike: Spravochnik. Kolebaniya lineinykh sistem (Vibrations in technics: the handbook. Oscillation of linear systems, vol. 1), Moscow: Mashinostroenie, 1978, 352 p.
  11. Gontkevich V.S. Sobstvennye kolebaniya plastinok i obolochek (Characteristic oscillations of plates and shells), Kiev: Naukova dumka, 1964, 288 p.
  12. Akulenko L.D., Nesterov S.V. The frequency-parametric analysis of characteristic oscillations of non-uniform rods, Prikl. Mat. Mekh., 2003, vol. 67, no. 4, pp. 588-602.
  13. Gladwell G.M.L. Obratnye zadachi teorii kolebanii (Inverse problems of the theory of oscillations), Moscow-Izhevsk: Regular and Chaotic Dynamic, Instutute of Computer Science, 2008, 608 p.
  14. Gladwell G.M.L., Movahhedy M. Reconstruction of mass-spring system from spectral data. I: Theory, Inverse Problems in Engineering, 1995, vol. 1, issue 2, pp. 179-189.
  15. Movahhedy M., Ismail F., Gladwell G.M.L. Reconstruction of mass-spring system from spectral data. II: Experiment, Inverse Problems in Engineering, 1995, vol. 1, issue 4, pp. 315-327. DOI: 10.1080/174159795088027588
  16. Morassi A., Dilena M. On point mass identification in rods and beams from minimal frequency measurements, Inverse Problems in Engineering, 2002, vol. 10, issue 3, pp. 183–-201. DOI: 10.1080/10682760290010378
  17. Vatul'yan A.O. Obratnye zadachi v mekhanike deformiruemogo tverdogo tela (Inverse problems in the mechanic of deformable solid body), Moscow: Fizmatlit, 2007, 224 p.
  18. Il'gamov M.A., Khakimov A.G. Diagnostics of fastening and damages of a beam on elastic pillars, Kontrol'. Diagnostika, Moscow: Izd. Dom “Spektr”, 2010, no. 9, pp. 57-63.
  19. Il'gamov M.A. Diagnostics of damages of a vertical bar, Tr. Inst. Mekh. Ufim. Nauch. Tsentra Ross. Akad. Nauk, Ufa: Gilem, 2007, vol. 5, pp. 201-211.
  20. Akhatov I.Sh., Akhtyamov A.M. Determination of the form of attachment of a rod using the natural frequencies of its flexural oscillations, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 2, pp. 283-290.
  21. Akhtyamov A.M., Mouftakhov A.V., Yamilova L.S. Identification of the type and parameters of fastening from the natural frequencies of the fastened rod, Acoustical Physics, 2008, vol. 54, no. 2, pp. 146-152.
  22. 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, Journal of Applied and Industrial Mathematics, 2010, vol. 4, no. 1, pp. 1-5.
  23. Akhtyamov A.M. On the uniqueness of the solution of an inverse spectral problem, Differential Equations, 2003, vol. 39, no. 8, pp. 1061-1066.
  24. Akhtyamov A.M., Safina G.F. Vibration-proof conduit fastening, Journal of Applied Mechanics and Technical Physics, 2008, vol. 49, no. 1, pp. 114-121.
  25. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M. Obratnaya zadacha Shturma-Liuvillya s neraspadayushchimisya kraevymi usloviyami (Inverse Sturm-Liouville problem with nonseparated boundary conditions), Moscow: Moscow State University, 2009, 184 p.
  26. Ahtyamov A.M. Teoriya identifikatsii kraevykh uslovii i ee prilozheniya (Theory of identification of boundary conditions and its applications), Moscow: Fizmatlit, 2009, 272 p.
  27. Naimark M.A. Lineinye differentsial'nye operatory (Linear differential operators), Moscow: Nauka, 1969, 526 p.
  28. Postnikov M.M. Lineinaya algebra i differentsial'naya geometriya (Linear algebra and differential geometry), Moscow: Nauka, 1979, 312 p.
  29. Mamford D.B. Algebraicheskaya geometriya. 1. Kompleksnye mnogoobraziya (Algebraic geometry. 1. Complex varieties), Moscow: Mir, 1979.
  30. Hodge W.V.D., Pedoe D. Methods of Algebraic Geometry, Cambridge: Cambridge Univ. Press, 1994, viii+440 p.
  31. Finikov S. P. Teoriya par kongruentsii (Theory of pairs of congruence), Moscow: Gostekhizdat, 1956, 443 p.
Full text
/doc/issues-2013/13-03-09.pdf
<< Previous article
Next article >>