Accounting the highest tones of oscillations when calculating the sensitivity of modes of their own to variations in the parameters of a mechanical system
Authors: Batseva O.D., Dmitriev S.N.
Published in issue: #7(79)/2018
DOI: 10.18698/2308-6033-2018-7-1785
Category: Mechanics | Chapter: Dynamics, Strength of Machines, Instruments, and Equipment
The article proposes the method for calculating the derivatives of numbers of their own and eigenvectors in a symmetric generalized eigenvalue problem in parameters of a mechanical system. Expansion in series of modes of their own is applied to calculate the derivatives forms of their own. The convergence of these series is investigated. The paper shows that the number of terms of the series turns out to be quite large for achieving an acceptable accuracy (in 1%). Formulas for the approximate accounting of the sum of all discarded terms are found from the analysis of the coefficients of the series. The solution of the auxiliary problem of the statistic loading of a mechanical system by a certain force vector is required for this, therefore the correction is called static. Accounting for it allows us to drastically reduce the number of terms of the series, providing acceptable accuracy
References
[1] Haug J.E., Kyung K.Ch., Komkov V. Design sensitivity analysis of structural systems. Mathematics in Science and Engineering, vol. 177. Academic Press INC., Orlando FL, 1986 [In Russ.: Haug J. E., Kyung K.Ch., Komkov V. Analiz chuvstvitelnosti pri proektirovanii konstruktsiy. Moscow, Mir Publ., 1998, 428 p.].
[2] Adelman H.M., Haftka R.T. Aerokosmicheskaya tekhnika — AIAA Journal: devoted to aerospace research and development, 1986, no. 12, pp. 77–90.
[3] Foks R.L., Kapur M.P. Raketnaya tekhnika i kosmonavtika — AIAA Journal: devoted to aerospace research and development, 1968, no. 12, pp. 227–230.
[4] Tushev O.N., Berezovskiy A.V. Vestnik MGTU im. N.E. Baumana. Ser. Mashinostroenie — Herald of the Bauman Moscow State Technical University. Series Mechanical Engineering, 2007, no. 1, pp. 35–40.
[5] Tushev O.N., Berezovskiy A.V. Izv. RAN. Problemy mashinostroeniya i nadezhnosti mashin — Journal of Machinery Manufacture and Reliability, 2013, no. 1, pp. 18–27.
[6] Bushuev A.Yu., Yakovlev D.O. Vestnik MGTU im. N.E. Baumana. Seriya: Estestvennye nauki — Herald of the Bauman Moscow State Technical University. Series Natural Sciences. 2011, Spec. iss. Mathematical Modeling, pp. 66–69.
[7] Bushuev A.YU. Funktsii chuvstvitelnosti v zadachakh matematicheskogo modelirovaniya sistem s raspredelennymi parametrami [Sensitivity functions in problems of mathematical modeling of systems with distributed parameters]. Aktualnye problemy nauki: IGUMO i IT kak issledovatelskiy tsentr [Proc. Actual problems of science: IGUMO and IT as a research center]. 2013, vol. 2, no. 1, pp. 61–67.
[8] MSC NASTRAN 2012. Design sensitivity and Optimization. User’s Guide, MSC Software Corporation, 2011, 718 с.
[9] Heylen W., Lammens S., Sas P. Modal Analysis Theory and Testing. Katholieke Universiteit Leuven, 1997, 340 p. [In Russ.: Heylen W., Lammens S., Sas P. Modalnyy analiz: teoriya i ispytaniya. Moscow, Novatest Publ., 2010, 319 p.].
[10] Boldyrev A.V. Optimizatsiya raspredeleniya materiala v kombiniro-vannykh aviatsionnykh konstruktsiyakh. Dis. na soiskanie uchenoy stepeni kand. tekhn. nauk [Optimization of material distribution in combined aviation structures. Dis. eng. sc. diss.]. Samara, SGAU, 2005, 159 p.
[11] Grabovskiy A.V., Tkachuk N.A., Tanchenko A.Yu., Tkachuk N.N., Mazur I.V. Vestnik Nats. tekhn. un-ta “KHPI”, Temat. vyp. Transportnoe mashinostroenie (NTU “KhPI” Bulletin, Them. iss. Transport engineering), 2015, no. 43 (1152), pp. 25–29.
[12] Grabovskiy A.V Vestnik Nats. tekhn. un-ta “KHPI”. Series Novі rіshennya v suchasnikh tekhnologіyakh (NTU “KhPI” Bulletin), 2017, no. 7 (1229), pp. 39–48.
[13] Pivovarova M.V., Besschetnov V.A. Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Aerokosmicheskaya tekhnika — PNRPU Aerospace Engineering Bulletin, 2017, no. 50, pp. 5–14.
[14] Zenkevich O. Metod konechnykh elementov v tekhnike [Finite element method in engineering]. Moscow, Mir Publ., 1975, 511 p.
[15] Grobov V.A. Teoriya kolebaniy mekhanicheskikh system [Theory of oscillations of mechanical systems]. Kiev, Vishcha shkola Publ., 1982, 182 p.
[16] Ilin M.M., Kolesnikov K.S., Saratov Yu.S. Teoriya kolebaniy [Theory of oscillations]. Moscow, BMSTU Publ., 2003, p. 271.
[17] Craig R., Chang C.-J. AIAA Journal, 1976, vol. 14, no. 11, pp. 1633–1635.
[18] Likhoded A.I., Sidorov V.V. Kosmonavtika i raketostroenie — Cosmonautics and rocket engineering, 2013, no. 2 (71), p. 20.