Stability of similar nonlinear normal modes under random excitation
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ORIGINAL PAPER
Stability of similar nonlinear normal modes under random excitation Y. V. Mikhlin . G. V. Rudnyeva
Received: 15 March 2020 / Accepted: 13 November 2020 Ó Springer Nature B.V. 2020
Abstract Two-DOF nonlinear system under stochastic excitation is considered. It is assumed that the system allows from two up to four nonlinear normal modes (NNMs) with rectilinear trajectories in the system configuration space. Influence of the random excitation to the NNMs stability is analyzed by using the analytical–numerical test, which is an implication of the well-known stability definition by Lyapunov. Boundary of the stability/instability regions is obtained in plane of the system parameters. Stability of the NNMs under deterministic chaos excitation is also considered. Keywords Nonlinear normal modes Stochastic excitation Lyapunov stability definition
1 Introduction Investigation of nonlinear normal modes (NNMs) is an important part of general analysis of dynamical systems. Different theoretical aspects of the NNMs theory and applications of the theory are presented in numerous publications, in particular, in reviews [1, 2]. NNMs having rectilinear trajectories in configuration Y. V. Mikhlin G. V. Rudnyeva (&) Department of Applied Mathematics, National Technical University ‘‘KhPI‘‘, Kharkiv, Ukraine e-mail: [email protected]
space (so-called similar nonlinear normal modes) were first found in some essential nonlinear systems by Rosenberg [3]. Numerous publications are dedicated to investigation of behavior of nonlinear dynamical systems under stochastic excitation. In this regard, different theoretical and numerical procedures are developed. We cite here only small part of publications on the subject and proposed approaches. It is used, in particular, the generalized Pontryagin equations [4, 5] which allow a closed-form solution in few simple cases only [6]. The reliability function which can be found by the Backward-Kolmogorov equation is considered in [7, 8]. The first-passage time distribution and the reliability function of a system under consideration can be determined by the stochastic averaging of the equations of motion in [9–11]. The statistical linearization is used in [12, 13]. The Galerkin projection is proposed in [14] to obtain approximate solution of the Backward-Kolmogorov partial differential equation. There are also numerous publications on computational approaches used in such problems, in particular, the Monte Carlo simulation [15, 16]. Other numerical schemes, such us the Wiener Path Integral [17–19], or probability density evolution schemes [20], the multi-dimensional FEM [21] et all, are also used. In publications [22, 23], the dynamics of aerospace vehicles and/or other structures subject to random excitation is investigated using a reduced order model in terms of its nonlinear normal modes. Evolution of NNMs is studied by the continuation
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Y. V. Mikhlin, G. V. Rudnyeva
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