Non-Linear Model Dynamics Analysis for Aerospace Engineering Structures Subjected to Non-Steady-State Random Loads

In aerospace engineering, it is customary to employ stochastic analysis methods at the design stage to investigate how the mechanical system responds to random external forces. This is relevant due to high reliability requirements for spacecraft. We developed a method for probabilistic estimation of the dynamic properties of a structure subjected simultaneously to external (additive) and parametric (multiplicative) vibrations. An ordinary non-linear vector differential equation describes the vibrations in the elastic structure. Non-linear position and velocity properties of kinematic pairs may have cusps and discontinuities. We assume that the probabilistic dispersions of respective phase coordinates are close to the normal distribution of probability density. The initial non-linear vibration equations are statistically linearised. The system of differential equations is not rewritten in the canonical form, which means that it is possible to carry out the probabilistic analysis of the system for any external non-steady-state effect. The fundamental matrix of the linearised system is used to find the expected value vector and the correlation function matrix of the phase coordinate vector. The solution consists of a matrix integro-power series containing linear and quadratic terms. Using the method makes it possible to assess the contribution of each external force component to the total result. We consider an example of a non-linear system responding to a stepwise non-steady-state external influence

The study was supported by the RFBR grant no. 20-08-01076а

References

[1] Safronov I. Angosat-1 has left the visibility range. Kommersantˮ, 2018, no. 6, р. 15 (in Russ.).

[2] Bondarenko A.Yu., Likhoded A.I., Malinin A.A., et al. Dynamic analysis of space structures, excited by external acceleration or force. Kosmonavtika i raketostroenie, 2017, no. 3, pp. 5--13 (in Russ.).

[3] Lipnitskiy Yu.M., Likhoded A.I., Sidorov V.V. Comparative load spectra analysis for structural elements affected with vibration and acoustic pressure impulses. Kosmonavtika i raketostroenie, 2007, no. 2, pp. 84--93 (in Russ.).

[4] Bondarenko A.Yu., Sidorov V.V. Methodical approach to ground testing of products of rocket and space technology under loads resulting from transient processes. Kosmonavtika i raketostroenie, 2016, no. 3, pp. 77--82 (in Russ.).

[5] Karp K.A., Evdokimenko V.N., Dineev V.G. Inzhenernye metody veroyatnostnogo analiza aviatsionnykh i kosmicheskikh system [Engineering methods of spacecraft and aircraft probabilistic analysis]. Moscow, Fizmatlit Publ., 2010.

[6] Zolkin S.N., Titov V.A. Verification of dynamic models of rocket and space technological products, based on a comparison of calculated and experimental amplitude-frequency characteristics. Kosmonavtika i raketostroenie, 2013, no. 2, pp. 28--34 (in Russ.).

[7] Svetlitskiy V.A. Statisticheskaya mekhanika i teoriya nadezhnosti [Statistic mechanics and reliability theory]. Moscow, Bauman MSTU Publ., 2004.

[8] Gusev A.S. Veroyatnostnye metody v mekhanike mashin i konstruktsiy [Probabilistic methods in machine and construction mechanics]. Moscow, Bauman MSTU Publ., 2009.

[9] Gottwald G., Harlim J. The role of additive and multiplicative noise in filtering complex dynamics systems. Proc. R. Soc. Lond. A: Math., Phys. Eng. Sc., 2013, vol. 469, no. 2155, pp. 96--112. DOI: 10.1098/rspa.2013.0096

[10] Zaytsev S.E., Tushev O.N. Impact assessment of random additive and multiplicative vibration on system dynamic behaviour. Izvestiya RAN. MTT, 2001, no. 6, pp. 163--167 (in Russ.).

[11] Blekhman I.I. Vibratsionnaya mekhanika [Vibrational mechanics]. Moscow, Fizmatlit Publ., 1994.

[12] Tushev O.N., Markianov A.V. The analysis of influence of high-frequency vibrations on the nonlinear model of a construction. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie [Proceedings of Higher Educational Institutions. Маchine Building], 2016, no. 10, pp. 32--38 (in Russ.). DOI: 10.18698/0536-1044-2016-10-32-38

[13] Kazakov I.E. Statisticheskaya teoriya sistem upravleniya v prostranstve sostoyaniy [Statistic theory of control system in state space]. Moscow, Nauka Publ., 1975.

[14] Kazakov I.E. Statisticheskie metody proektirovaniya sistem upravleniya [Statistic methods of control system engineering]. Moscow, Mashinostroenie Publ., 1969.

[15] Gantmakher F.R. Teoriya matrits [Matrix theory]. Moscow, Fizmatlit Publ., 2010.