Methods to reduce non-linear mechanical systems for instability computation

• Published in: Archives of computational methods in engineering Vol. 11; no. 3; pp. 257 - 344
• Main Authors: Sinou, J J, Thouverez, F, Jézéquel, L
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01-01-2004; Springer Verlag
• Subjects: Complexity; Degrees of freedom; Dynamic stability; Dynamical Systems; Economic models; Eigenvalues; Engineering Sciences; Equations of motion; Fourier series; Harmonic balance method; Linearization; Mathematical models; Mathematics; Mechanical systems; Mechanics; Methods; Motion stability; Nonlinear dynamics; Nonlinear equations; Nonlinear programming; Nonlinear systems; Nonlinearity; Parameters; Performance evaluation; Perturbation; Physics; Polynomials; Reduced order models; Runge-Kutta method; Steady state; Structural mechanics; Structural stability; Studies; Time domain analysis; Vibrations
• ISSN: 1134-3060; 1886-1784
• DOI: 10.1007/BF02736228

Non-linear dynamical structures depending on control parameters are encountered in many areas of science and engineering. In the study of non-linear dynamical systems depending on a given control parameter, the stability analysis and the associated non-linear behaviour in a near-critical steady-state equilibrium point are two of the most important points; they make it possible to validate and characterize the non-linear structures. Stability is investigated by determining eigenvalues of the linearized perturbation equations about each steady-state operating point, or by calculating the Jacobian of the system at the equilibrium points. While the conditions and the values of the parameters which cause instability can be investigated by using linearized equations of motion studies of the non-linear behaviour of vibration problems, on the other hand, require the complete non-linear expressions of systems. Due to the complexity of non-linear systems and to save time, simplifications and reductions in the mathematical complexity of the non-linear equations are usually required. The principal idea for these non-linear methods is to reduce the order of the system and eliminate as many non-linearities as possible in the system of equations. In this paper, a study devoted to evaluating the instability phenomena in non-linear models is presented. It outlines stability analysis and gives a non-linear strategy by constructing a reduced order model and simplifying the non-linearities, based on three non-linear methods: the centre manifold concept, the rational approximants and the Alternating Frequency/Time domain method. The computational procedures to determine the reduced and simplified system via the centre manifold approach and the fractional approximants, as well as the approximation of the responses as a Fourier series via the harmonic balance method, are presented and discussed. These non-linear methods for calculating the dynamical behaviour of non-linear systems with several degrees-of-freedom and non-linearities are tested in the case of mechanical systems with many degrees-of-freedom possessing polynomial non-linearities. Results obtained are compared with those estimated by a classical Runge-Kutta integration procedure. Moreover, an extension of the centre manifold approach using rational approximants is proposed and used to explore the dynamics of non-linear systems, by extending the domain of convergence of the non-linear reduced system and evaluating its performance and suitability.[PUBLICATION ABSTRACT]