Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems II: Large Amplitude Oscillations

Elena Hernandez, Octavio Manero, Fernando Bautista and Juan Paulo Garcia-Sandoval
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Elena Hernandez: Departamento de Ingeniería Química, Universidad de Guadalajara, Guadalajara 44430, Mexico
Octavio Manero: Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de Mexico, CDMX, Mexico City 04510, Mexico
Fernando Bautista: Departamento de Física, Universidad de Guadalajara, Guadalajara 44430, Mexico
Juan Paulo Garcia-Sandoval: Departamento de Ingeniería Química, Universidad de Guadalajara, Guadalajara 44430, Mexico

Mathematics, 2022, vol. 10, issue 15, 1-21

Abstract: This work is the second in a series of articles that deal with analytical solutions of nonlinear dynamical systems under oscillatory input that may exhibit harmonic frequencies. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods, because in most cases analytical solutions such as the harmonic balance series solution turn out to be difficult, if not impossible, as they are based on an infinite series of trigonometric functions with harmonic frequencies. The key contribution of the analytic matrix methods reported in the present series of articles is to work with the invariant submanifold of the problem and to propose the solution as infinite power series of the oscillatory input; this procedure is a direct one that speeds up the computations compared to traditional series solution methods. The method reported in the first contribution of this series allows for the computation of the analytical solution only for small and medium amplitudes of the oscillatory input, because these series may diverge when large amplitudes are applied. Therefore, the analytic matrix method reported here, which is a reconfiguration of the method proposed in the first contribution in this series, allows the solving of problems in the regime of large-amplitude oscillations where the contributions of the high order harmonics affect the amplitudes of the low order harmonics, leading to amplitude- and frequency-dependent coefficients for the infinite series of trigonometric function expansion.

Keywords: frequency response; series method; nonlinear dynamical systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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