Application of the method of moments for the correlation analysis of the dynamics of finite element models of structures

The calculation of the probability characteristics of structures on the basis of a multidimensional model for non-stationary modes of operation is of great importance. The developed method makes it possible to calculate these characteristics with engineering accuracy. In this paper, an external quasi-stationary random action is considered. The moment characteristics of the phase coordinates of a linear model of the structure under quasi-stationary additive actions are determined. To reduce the order of the system of equations, the modal truncation is used. The dissipation matrix is assumed to be proportional to the mass and stiffness matrices. The system of equations of the second order in modal coordinates is reduced to the vector equation of the Cauchy canonical normal form using the shaping filter equation that converts the white noise in real random processes. The filter equation is constructed by applying the transformation that is valid for stationary random processes whose spectral density has a rational structure. The well-known equations of the method of moments in terms of the vector of expectations and the matrix of correlation moments are used, which makes it possible to accurately solve non-stationary and stationary problems within the framework of the correlation theory. The expectations and variances of the displacements of a frame in a transient process are calculated by the finite element method using the mode superposition. It is shown that even a small number of series members can provide good accuracy.

References

[1] Vibratsii v tekhnike. Spravochnik. T. 1. Kolebaniia lineinykh sistem [Vibration technology. Handbook. Vol. 1. Oscillations of linear systems]. Ed. Bolotin V.V. Moscow, Mashinostroenie publ., 1978. 352 p.

[2] Svetlitskii V.A. Statisticheskaia mekhanika i teoriia nadezhnosti [Statistical mechanics and the theory of reliability]. Moscow, Bauman Press, 2002. 504 p.

[3] 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.

[4] Tushev O.N., Berezovskii A.V. Opredelenie spektral’nykh plotnostei dinamicheskikh kharakteristik nelineinoi modeli konstruktsii [Determination of the spectral densities of the dynamic characteristics of a nonlinear model of the design]. Problemy mashinostroeniia i nadezhnosti mashin [Journal of Machinery Manufacture and Reliability]. 2013, no. 1, pp. 18–27.

[5] Dmitriev S.N., Khamidullin R.K. Utochnennaia formula dlia vychisleniia koeffitsientov peredatochnoi matritsy v zadachakh statisticheskoi dinamiki [Improved formula for calculating coefficients of transfer matrix in statistical dynamics problems]. Nauka i obrazovanie. MGTU im. N.E. Baumana [Science and Education. Bauman MSTU]. 2013, no. 3. Available at: http://technomag.bmstu.ru/doc/543198.html (accessed 15 September 2014). Doi: 10.7463/0313.0543198.

[6] Shcheglov G.A. Ispol’zovanie vortonov dlia rascheta kolebanii balki v prostranstvennom potoke [Wharton use to calculate the vibrations of the beam in the spatial flow]. Problemy mashinostroeniia i nadezhnosti mashin [Journal of Machinery Manufacture and Reliability]. 2009, no. 4, pp. 8–12.

[7] Manevitch L.I. The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables. Nonlinear Dynamics, 2001, vol. 25, no. 1-3, pp. 95–109.

[8] Sodagar S., Honarvar F., Sinclair A. Multiple Scattering of Obliquely Incident Acoustic Wave from a Grating Immersed Cylindrical Shells. Applied Acoustics, 2011, vol. 72, no. 1, pp. 1–10.

[9] Breslavsky I.D., Avramov K.V. Two modes nonresonant interaction for rectangular plate with geometrical nonlinearity. Nonlinear Dynamics, 2012, vol. 69, no. 1-2, pp. 285–294.

[10] Hay T.A., Cheung T.Y., Hamilton M.F., Ilinskii Y.A. Frequency response of bubble pulsations in tubes with arbitrary wall impedance. The Journal of the Acoustical Society of America, 2008, no. 123, pp. 12–20.

[11] Duncan D.B. Response of linear time-dependent systems to random input. Journal of Applied Physics, 1953, no. 24(5), pp. 609–611.

[12] Kazakov I.E. Statisticheskaia teoriia sistem upravleniia v prostranstve sostoianii [Statistical theory of control systems in state space]. Moscow, Nauka publ., 1975. 432 p.