Kinematically excited beam vibrations under random perturbations

DOI: 10.37153/2618-9283-2023-6-45-52

Authors:  
Rubric:     Theoretical and experimental studies   
Key words: stationary random process, spectral matrix, steady-state beam oscillations, initial conditions, boundary conditions, kinematically excited deterministic oscillations, random field of deviations, dispersion, kinematic perturbation model, vector stationary random process, stationary coupled components, standard deviations of displacements
Annotation:

There is numerous information in the technical and scientific literature that often the sources of forced kinematically excited oscillations have a clearly expressed random nature. Kinematically excited vibrations of beams are considered. Their sources are random disturbances. Steady (in a probabilistic sense) vibrations of beams are studied. As a consequence, the transverse vibrations of the beam will also be random, and it becomes necessary and expedient to switch to stochastic motion models.

Introduction. For a random perturbation process, a spectral matrix is specified. Initial conditions for the equations are not required. The boundary conditions are similar to those used for free and kinematically excited deterministic oscillations. The task is to find the spectral matrix of the random field of beam deviations and dispersion for a given spectral matrix. The question of determining the mathematical expectation is not raised. It is argued that it can be easily reduced to known deterministic problems.

Methods. To carry out specific calculations, a model of kinematic disturbances is used in the form of a vector N-dimensional stationary random process with stationary connected components that have hidden periodicities (characteristic frequencies). A test example was completed to determine the dispersion and standard deviation of displacements. The numerical integration step was adopted after numerical experiments. When performing calculations, the parity of the integrand was taken into account. The lower limit of integration is taken to be zero, followed by doubling of the final result.

Results. For small values of the broadband parameter, the results of the stochastic and deterministic problems differ little. Examples are presented that are stochastic analogs of harmonic oscillations. For beams under kinematic harmonic disturbances, test examples were performed to determine dispersions and standard deviations.

The discussion of the results. The results of calculations in the article are presented by curves having numbers that coincide with the numbers of the matrices. The first matrix corresponds to the absolute correlation of disturbances. The second matrix corresponds to the absolute correlation between the first and third, second and fourth disturbances, while these pairs are absolutely uncorrelated. The third matrix describes random perturbations when the indicated pairs, being perfectly positively correlated within, are absolutely perfectly negatively correlated between pairs. The fourth curve corresponds to the case of ideal correlation of the first three disturbances with each other, while the movements of the fourth support are in antiphase with them. In the latter case, the movements of the outer supports are in antiphase with the movements of the middle ones, which favors the greatest bending of the beam. Therefore, the elastic line has a curvature that is significantly greater than the others. Used Books:

1.         Akimov P.A., Belostockij T.B. i dr. Informatika v stroitel'stve (s osnovami matematicheskogo i komp'yuternogo modelirovaniya): uchebnoe posobie. Moskva: KNORUS, 2017. 420 p. [In Russian]

2.         Arutyunov S.K., Ovchinnikov I.N., Startsev V.A. Modelirovanie kolebanij balki pri sluchajnom vibronagruzhenii. Prikladnye problemy mekhaniki raketno-kosmicheskih sistem: Tez. dokl. vseros. konf., posvyashch. 40-letiyu so dnya osnovaniya kaf. "Aerokosm. sistemy" (SM-2) MGTU im. N. E. Baumana, Moskva, 5 dek. 2000. M.: Izd-vo MGTU. 2000, p. 66. [In Russian]

3.         Kulterbaev Kh.P., Pshenichkina V.A. Sluchajnye protsessy i kolebaniya stroitel'nyh konstruktsij i sooruzhenij. Volgogradskij gosud. arhit.-stroit. un-t. Volgograd: VolgGASU. 2006. 335 p. [In Russian]

4.         Kulterbaev Kh.P., Baragunova L.A., Shogenova M.M. Random Oscillations of the Vertical Continual and Discrete Rod under Seismic Influences. Materials Science Forum Submitted: 2018-06-13 ISSN: 1662-9752, vol. 931, pp. 66–71 Accepted: 2018-06-22 doi: 10.4028/www.scien-tific.net/MSF.931.66 Online: 2018-09-20© 2018 Trans Tech Publications, Switzerland.

5.    Zolotov A.B., Akimov P.A., Sidorov V.N. Matematicheskie metody v stroitel'noj mekhanike (s osnovami teorii obobshchyonnyh funkcij). M.: Izdatel'stvo ASV. 2008. 336 p. [In Russian]

6.    Kulterbaev Kh.P. Kinematicheski vozbuzhdaemye kolebaniya kontinual'no-diskretnoj mnogoprolyotnoj balki. Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo. No. 4, chast 2. Trudy X Vserossijskogo s’ezda po fundamentalnym problemam teoreticheskoj i prikladnoj mekhaniki. Izd-vo NNGU im. N.I. Lobachevskogo. 2011, pp. 198–200. [In Russian]

7.    Kul'terbaev Kh.P., Kaziev A.M. O vynuzhdennyh kolebaniyah rastyanutyh balok pri kombinirovannyh vozmushcheniyah. Nadyozhnost' i dolgovechnost' stroitel'nyh materialov i konstrukcij. Materialy III Mezhdunarodnoj nauchno-tekhnicheskoj konferentsii (27–29 marta 2003). CHast' II. Volgograd. 2003, pp. 15–18. [In Russian]

8.    Korchinskij I.L., Borodin L.A. i dr. Sejsmostojkoe stroitel'stvo zdanij. M.: Vysshaya shkola. 1971. 320 p. [In Russian]

9.    Klaf Z., Penzien Dzh. Dinamika sooruzhenij. M.: Strojizdat. 1979. 320 p. [In Russian]

10.    Rozanov Yu.A. Sluchajnye protsessy (kratkij kurs). M.: «Nauka». 1971. 288 p. [In Russian]

11.    Abdurashidov K.S., Eizenberg J.M. i dr. Sejsmostojkost' sooruzhenij. M.: Nauka.1989.192 p. [In Russian]

12.    Eizenberg J.M., Abakarov A.D. Analiz nadyozhnosti sistem sejsmozashchity s rezervnymi elementami, opisyvaemye «skhemoj gibeli». Stroitel'naya mekhanika i raschyot sooruzhenij. 1975, № 3, pp. 2–17. [In Russian]

13.    Barshtejn M.F., Borodachyov N.M., Blyumina L.H. i dr. Dinamicheskij raschyot sooruzhenij na special'nye vozdejstviya / red. Koreneva B.G., Rabinovicha I.M. M.: Strojizdat. 1981. 215 p. [In Russian]

14.    Bolotin V.V. Metod momentov v teorii sluchajnyh kolebanij mekhanicheskih system. Annotatsii dokladov IV Vsesoyuz. S’ezd po teor. i prikl. mekh. Kiev. 1976, may, 21–28. [In Russian]

15.    Bolotin V.V., Radin V.P., Chirkov V.P. Issledovanie povedeniya zdanij i sooruzhenij so snizheniem zhyostkosti pri sejsmicheskih vozdejstviyah. Izvestiya vuzov. Stroitel'stvo. 2003, no.7, pp. 6–10. [In Russian]

16.    Bolotin V.V. Sluchajnye kolebaniya uprugih sistem. M.: Nauka.1979. 335 p. [In Russian]

17.    Ventcel E.S., Ovcharov L.A. Teoriya sluchajnyh processov i eyo inzhenernye prilozheniya. M.: Vysshaya shkola. 2000. 383 p. [In Russian]

18.    Korchinskij I.L. Sejsmicheskie nagruzki na zdaniya i sooruzheniya. M.: Gosstrojizdat. 1959.78 p. [In Russian]

19.    Kulikov A.N. Raschet sooruzhenij na sejsmicheskie vozdejstviya i vetrovuyu nagruzku s pul'sacionnoj sostavlyayushchej. Volgogr. gos. arhit.-stroit. un-t; Volzh. in-t str-va i tekhnol. (filial) VolgGASU. Volgograd, VolgGASU. 2008. 91 p. [In Russian]