Identification of forces in the supports of multi-span beams

Аuthors

Grishakin V. T.

Moscow Automobile and Road Construction State Technical University (MADI), MADI, 64, Leningradsky Prospect, Moscow, 125319, Russia

e-mail: grichacin@yandex.ru

Abstract

The article considers transverse vibrations of conservative mechanical systems in the form of hinged beams of the refined S.P. Timoshenko theory with deformable intermediate supports. Deformation processes in beams resulted through the concentrated inertial forces application moving at a constant speed. Solution of the direct problem (the problem of determining parameters of the stress-strain state of a beam from a given value of the moving force) was obtained in the form of expansion in Fourier series, followed by the application of operational calculus. The inverse problem of the deformable solid body mechanics, which consisted in identifying the forces in the supports according to the given deflections under the moving force, was considered on the example of the mechanical system under study as well. As long as the inverse problem posed in this way relates to the ill-posed ones (an insignificant change in the initial data corresponds to an arbitrarily large change in the calculation results), A.N. Tikhonov’s regularizing algorithm was applied to obtain a stable solution. A technique for the regularization parameter α computing is presented. Both the results of the beam deflections computing at the point of the moving force application and the results of identifying the forces in the supports of a six-span continuous beam structure, obtained with the same stiffness values of all five elastic supports, and with a change in the stiffness of one of them (the second one) are presented. Such numerical experiment was performed to illustrate the possibility of employing the proposed method, which opens up as a result of the proposed technique application, for example, in construction practice to identify defects in the supports of structures that perceive moving loads without stopping movement along them. All relevant information on the state of structures can be obtained employing a vehicle equipped with the necessary sensors. For the said two computational cases, the time dependences of the forces in the supports, obtained both as the result of solving direct and inverse problems, are presented in the form of graphs of the corresponding functions. For the considered cases of the structure loading, the results of the largest displacements computing in the most loaded span of the beam and the maximum elastic forces arising in the most loaded support are presented as well.

Keywords:

multi-span beam, movable load, identification, regularization, displacement, deformation

References

1. Panovko Ya.G., Gubanova I.I. Ustoichivost’ i kolebaniya uprugikh sistem (Stability and oscillations of elastic systems), Moscow, Nauka, 1967, 419 p.

2. Krylov A.N. Vibratsii sudov (Vibrations of ships), Moscow, ONTI. Glavnaya redaktsiya sudostroitel’noi literatury, 1936, 404 p.

3. Timoshenko S.P. Kolebaniya v inzhenernom dele (Fluctuations in engineering), Moscow, Fizmatgiz, 1967, 444 p.

4. Bondar’ N.G., Kazei I.I., Lesokhin B.F., Koz’min Yu.G. Dinamika zheleznodorozhnykh mostov (Dynamics of railway bridges), Moscow, Transport, 1965, 412 p.

5. Zemlyanitsyn V.D. Trudy Sibirskogo avtomobil’no-dorozhnogo instituta, 1975, no. 54, pp. 154-159.

6. Kitaev K.E. Trudy MIIT, 1952, no. 76, pp. 123-124.

7. Licari I.S., Wilson E.N. Dynamic response of a beam subjected to a moving forcing system, Proc. 4-th US National Congress of Applied Mechanics, Berkeley, 1962, vol. 1, pp. 419-425.

8. Bolotin V.V. Trudy MIIT, 1950, no. 74, pp. 269-296.

9. Bolotin V.V. Izvestiya AN SSSR. Mekhanika i mashinostroenie, 1961, no. 4, pp. 109-115.

10. Vol’per D.B., Morgaevskii A.B. Issledovaniya po teorii sooruzhenii, 1965, no. 14, pp. 73-91.

11. Morgaevskii A.B. Issledovaniya po teorii sooruzhenii, 1965, no. 14, pp. 67-72.

12. Filippov A.P. Izvestiya AN SSSR. Mekhanika i mashinostroenie, 1964, no. 4, pp. 120-125.

13. Filippov A.P. Kolebaniya deformiruemykh sistem (Vibrations of deformable systems), Moscow, Mashinostroenie, 1970, 736 p.

14. Dmitriev A.S. Problemy mashinostroeniya, 1986, no. 25, pp. 43-50.

15. Kokhmanyuk S.S., Dmitriev A.S., Shelud’ko G.A. et al. Dinamika konstruktsii pri vozdeistvii kratkovremennykh nagruzok (Dynamics of structures under the influence of short—term loads), Kiev, Naukova dumka, 1989, 304 p.

16. Kokhmanyuk S.S., Ol’shanskii V.P. Issledovaniya po teorii sooruzhenii, 1975, no. 21, pp. 157-160.

17. Kokhmanyuk S.S., Yanyutin E.G., Romanenko L.G. Kolebaniya deformiruemykh sistem pri impul’snykh i podvizhnykh nagruzkakh (Vibrations of deformable systems under pulsed and mobile loads), Kiev, Naukova dumka, 1980, 232 p.

18. Galiullin A.S. Obratnye zadachi dinamiki (Inverse problems of dynamics), Moscow, Nauka, 1981, 144 p.

19. Groetsch C.W. Inverse Problems in the Mathematical Sciences, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA, 1993, 154 p. URL: https://link.springer.com/book/10.1007/978-3-322-99202-4

20. Ditkin V.A., Prudnikov A.P. Integral’nye preobrazovaniya i operatsionnoe ischislenie (Integral transformations and operational calculus), Moscow, GIFML, 1961, 524 p.

21. Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach (Methods of solving incorrect problems), Moscow, Nauka, 1986, 288 p.

22. Borshevetskii S.A. Trudy MAI, 2023, no. 128. URL: https://trudymai.ru/eng/published.php?ID=171384. DOI: 10.34759/trd-2023-128-03

23. Levitskii D.Yu., Fedotenkov G.V. Trudy MAI, 2022, no. 125. URL: https://trudymai.ru/eng/published.php?ID=168157. DOI: 10.34759/trd-2022-125-05

24. Firsanov V.V., Vo A.Kh. Trudy MAI, 2018, no. 102. URL: https://trudymai.ru/eng/published.php?ID=98866

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