Abstract
This is the first on a series of articles that deal with nonlinear dynamical systems under oscillatory input that may exhibit harmonic and non-harmonic frequencies and possibly complex behavior in the form of chaos. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods because, most of the time, analytical solutions turn out to be difficult, if not impossible, since they are based on infinite series of trigonometric functions. The analytic matrix method reported here is a direct one that speeds up the solution processing compared to traditional series solution methods. In this method, we work with the invariant submanifold of the problem, and we propose a series solution that is equivalent to the harmonic balance series solution. However, the recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method. This method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a usual Bode plot where regions of small and medium amplitude oscillatory input are well described. We found that the identification of such regions requires both the amplitude as well as the frequency to be properly specified. In the second paper of the series, the method to solve problems in the field of large amplitudes will be addressed.
Highlights
Frequency response is a powerful technique to analyze the characteristics and properties of dynamical systems, describing the system response of a sinusoidal input with variable frequency
We present some criteria to identify different modes of behavior such as the small amplitude oscillatory (SAOs) and the medium amplitude oscillatory (MAOs) that arise in several frequency response technique (FRT) applications
The recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method
Summary
Frequency response is a powerful technique to analyze the characteristics and properties of dynamical systems, describing the system response of a sinusoidal input with variable frequency. In the literature, the application of the method of harmonic balance and other methods to analyze the behavior of nonlinear systems subjected to FRT has been mostly using numeric approaches to particular study cases (see, for instance, [3,10,11,12,13]), and solving equations with the methods described previously can be challenging since the analytic approach may not be solvable and the numerical one does not provide an exact solution. We present an analytic method for FRT with applications to nonlinear dynamical systems that is an improvement over the series solution methods available This method provides a structured procedure to obtain information of nonlinear systems subjected to periodic inputs for the cases where the output is stable and others where the system is not at equilibrium. The recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method
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