The Analysis of Influence of High-Frequency Vibrations on the Nonlinear Model of a Construction
Authors: Tushev O.N., Markianov A.V. | Published: 12.10.2016 |
Published in issue: #10(679)/2016 | |
Category: Calculation and Design of Machinery | |
Keywords: random vibrations, statistical linearization, integro-power series, the expectation, correlation function, fundamental matrix, multiplicative integral |
The authors analyze the effect of high-frequency additive and multiplicative non-stationary or stationary vibrations on a nonlinear mechanical system with a finite number of degrees of freedom. The influences are defined within the correlation theory. It is considered that only the «slow motion» determined by mathematical expectation is of interest. High-frequency vibrations of phase coordinates are small and may be omitted in the calculations. For example, this is done when false readings («walks») of the pendulum accelerometer are analyzed at fast vibrations. First, the initial equation of motion is statistically linearized. To find the solution, the phase coordinate vector is represented as an integro-power series by the matrix containing the aligned random vibrations, and members up to quadratic ones are considered, inclusively. As a result, a convenient explicit dependence of the expectation vector on the elements of the correlation matrix is obtained. Using this equation, it is possible to establish a hierarchy with regards to their contribution to the «error» (vibration component of the solution). The fundamental matrix of the linearized system for the organization of the computing procedure is represented in the form of a multiplicative integral.
References
[1] Gottwald G., Harlim J. The role of additive and multiplicative noise in filtering complex dynamics systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013, № 469, pp. 96–112.
[2] Li C., Duan J. Impact of correlated noises on additive dynamical systems. Mathematical Problems in Engineering, 2014, article no. 678976. Available at: http://dx.doi.org/10.1155/2014/678976 (accessed 24 April 2016).
[3] Poliak B.T., Khlebnikov M.V., Shcherbakov P.S. Nelineinye sistemy s ogranichennymi ili mul’tiplikativnymi vozmushcheniiami [Nonlinear systems with limited or multiplicative perturbations. Problems of stability and control]. Problemy ustoichivosti i upravleniia. Sb. nauch. st., posviashchennyi 80-letiiu akademika V.M. Matrosova [Problems of stability and control. Collection of articles dedicated to the 80th anniversary of academician V.M. Matrosov]. Moscow, Fizmatlit publ., 2013, pp. 270–299.
[4] Blekhman I.I. Vibratsionnaia mekhanika [Mechanical Vibration]. Moscow, Fizmatlit publ., 1994. 394 p.
[5] Gusev A.S. Veroiatnostnye metody v mekhanike mashin i konstruktsii [Probabilistic methods in the mechanics of machines and structures]. Moscow, Bauman Press, 2009. 224 p.
[6] Svetlitskii V.A. Stokhasticheskaia mekhanika i teoriia nadezhnosti [Stochastic mechanics and the theory of reliability]. Moscow, Bauman Press, 2002. 504 p.
[7] Makarov R.N., Shkarupa E.V. Stochastic algorithms with Hermit cubic spline interpolation for global estimation of solutions of boundary value problems. SIAM Journal of Scientific Computing, 2007, vol. 3, no. 1, pp. 169–188.
[8] Zaitsev S.E., Tushev O.N. Otsenka vliianiia sluchainykh additivnykh i mul’tiplikativnykh vibratsii na dinamicheskoe povedenie sistemy [Assessing the impact of random additive and multiplicative vibration on the dynamic behavior of the system]. Izvestiia RAN. MTT [Mechanics of Solids]. 2001, no. 6, pp. 163–167.
[9] Kazakov I.E. Statisticheskaia teoriia sistem uravneniia v prostranstve sostoianii [Statistical theory of the equation systems in the state space]. Moscow, Nauka publ., 1975. 432 p.
[10] Gontmakher F.R. Teoriia matrits [The theory of matrices]. Moscow, Nauka publ., 1967. 432 p.