Forced vibrations of a three-layer step-variable thickness circular plate under impact

M. V. Markova

Problemy fiziki, matematiki i tehniki, no. 3 (2022), pp. 28-36.

Résumé

The problem of forced oscillations of a three-layer circular plate with step-variable thickness of the outer layers is presented. The historical review of the theories that have been developed for the straining representation of a three-layer structure is presented. The deformation of the plate follows the zig-zag theory. In thin border layers of plate Kirchhoff’s hypotheses are valid. In a relatively thick in thickness medium layer Timoshenko’s hypothesis is fulfilled. The equations of motion are derived from Hamilton’s variational principle. A solution is constructed to determine the displacements during forced vibrations of a plate under impact. Numerical results of the obtained solution are presented. The influence of impact function on the oscillatory character is analyzed.

Détail

Keywords: circular three-layer plate, plates with step-variable thickness, forced vibration, stroke.

Bibliographie

[1] O. Bauchau, J. Craig, “Kirchhoff plate theory”, Structural analysis, 163 (2009), 819–914 | DOI

[2] S.P. Timoshenko, “On the correction for shear the differential equation for transverse vibrations of the prismatic bars”, Philosophical magazine and journal of science, 21:41 (1921), 744–746 | DOI

[3] V.V. Bolotin, Yu.N. Novichkov, Mekhanika mnogosloinykh konstruktsii, Mashinostroenie, M., 1980, 375 pp.

[4] V.V. Bolotin, “K teorii sloistykh plit”, Izvestiya AN SSSR. Mekhanika i mashinostroenie, 1963, no. 3, 65–72

[5] Yu.N. Novichkov, “Variatsionnye printsipy dinamiki i ustoichivosti mnogosloinykh obolochek”, Trudy Moskovskogo energeticheskogo in-ta. Dinamika i prochnost mashin, 1973, no. 164, 14–22

[6] E.I. Grigolyuk, P.P. Chulkov, Ustoichivost i kolebaniya trekhsloinykh obolochek, Mashinostroenie, M., 1973, 172 pp.

[7] E.I. Grigolyuk, P.P. Chulkov, “Nelineinye uravneniya tonkikh mnogosloinykh obolochek regulyarnogo stroeniya”, Inzhenernyi zhurnal. Mekhanika tverdogo tela, 1967, no. 1, 163–169

[8] V.A. Zorich, Matematicheskii analiz, v. I, izd. 6-e dopoln., MTsNMO, M., 2012, 710 pp.

[9] V. Novatskii, Teoriya uprugosti, Mir, M., 1975, 872 pp.

[10] M.V. Markova, D.V. Leonenko, “Postanovka nachalnokraevoi zadachi ob osesimmetrichnykh kolebaniyakh krugovoi trekhsloinoi plastiny peremennoi tolschiny”, mezhdunar. nauch.-tekhn. sb., Teoreticheskaya i prikladnaya mekhanika, 36, Minsk, 2022, 3–10

[11] M.V. Markova, “Vynuzhdennye kolebaniya krugovoi trekhsloinoi plastiny stupenchato-peremennoi tolschiny”, Izvestiya Gomelskogo gosudarstvennogo universiteta imeni F. Skoriny, 2022, no. 3 (132), 121–127

[12] E.I. Starovoitov i dr., “Deformirovanie stupenchatoi kompozitnoi balki v temperaturnom pole”, Inzhenerno-fizicheskii zhurnal, 88:4 (2015), 987–993

[13] E.I. Starovoitov, A.A. Poddubnyi, “Izgib trekhsloinogo sterzhnya so stupenchato-peremennoi granitsei, chastichno opertogo na uprugoe osnovanie”, Mekhanika mashin, mekhanizmov i materialov, 2011, no. 1 (14), 47–55

[14] E.I. Starovoitov, D.V. Leonenko, L.N. Rabinskii, Deformirovanie trekhsloinykh fizicheski nelineinykh sterzhnei, Izd-vo MAI, M., 2016, 184 pp.

[15] K.N. Tong, Teoriya mekhanicheskikh kolebanii, Mashgiz, M., 1963, 351 pp.

[16] I.G. Aramanovich, V.I. Levin, Uravneniya matematicheskoi fiziki, Nauka, M., 1969, 288 pp.

[17] M.V. Markova, “Inertsionnaya matematicheskaya model dinamicheskogo deformirovaniya krugovoi trekhsloinoi stupenchatoi plastiny”, Izvestiya Gomelskogo gosudarstvennogo universiteta imeni F. Skoriny, 2021, no. 6 (129), 164–170

[18] G. Beitman, A. Erdeii, Vysshie transtsendentnye funktsii, Nauka, M., 1974, 296 pp.

[19] G.N. Vatson, Teoriya Besselevykh funktsii, Izd-vo inostrannoi literatury, M., 1949, 799 pp.

[20] G.B. Arfken, H.J. Weber, Mathematical methods for physicists, 6th edition, Elsevier Inc., Oxford, 2005, 1182 pp. | MR | Zbl

[21] Yu.M. Pleskachevskii, E.I. Starovoitov, D.V. Leonenko, Mekhanika trekhsloinykh sterzhnei i plastin, svyazannykh s uprugim osnovaniem, monografiya, Fizmatlit, M., 2011, 560 pp.

[22] SP 5.04.01–2021. Stalnye konstruktsii, Vved. 29.07.2021, Minstroiarkhitektury, Minsk, 2021, 147 pp.

[23] A.M. Ivanov i dr., Stroitelnye konstruktsii s primeneniem plastmass. Primery proektirovaniya i rascheta, ucheb. posobie dlya inzh.-stroit. vuzov i fak., Vysshaya shkola, Moskva, 1968, 220 pp.