Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations

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.

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

“…In our previous work [9], it was assumed that the invariant submanifold, ξ(z), can be described by the following series:…”

“…Then, applying the method proposed in [9], the following series solution with the form in (6) can be obtained:…”

“…. ., are proportional to A 2k−1 /ω 3k−2 ; as such, in [9] it was shown that an approximated limit to guarantee the convergence of series (12) is that the frequency-amplitude relation must satisfy ω 3 > A 2 . When this limit is not fulfilled, the solution ( 12) is no longer suitable to describe the dynamical behavior and the series may diverge, as the series coefficients for high order powers are not convergent and may have the same order of magnitude as the terms with low order power.…”

“…In contrast, a different system may display distorted waves with the presence of multiple peaks corresponding to the harmonics and subharmonics [8]. In the first paper in this series [9], we presented an analytical method that simplifies traditional approaches to obtaining the solution using the harmonic balance series method. Such an analytical method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a typical Bode plot.…”

“…This reconsideration proved useful in obtaining information on highly nonlinear systems subjected to periodic inputs in cases where the output is stable but exhibits multiple subharmonics. In this paper, we present an analytic method for FRT with applications to nonlinear dynamical systems that adapts the analytic matrix method for FRT we previously reported [9]. With this, we then provide a complementary structured procedure to obtain information on nonlinear systems subjected to periodic inputs in LAOs.…”