Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations
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 wi...
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| Format: | Article |
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MDPI AG
2021-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/24/3287 |
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| author | Elena Hernandez Octavio Manero Fernando Bautista Juan Paulo Garcia-Sandoval |
| author_facet | Elena Hernandez Octavio Manero Fernando Bautista Juan Paulo Garcia-Sandoval |
| author_sort | Elena Hernandez |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T03:37:05Z |
| format | Article |
| id | doaj.art-515eeac0228a4f12ae3426c36e3b4650 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:37:05Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-515eeac0228a4f12ae3426c36e3b46502023-11-23T09:26:58ZengMDPI AGMathematics2227-73902021-12-01924328710.3390/math9243287Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude OscillationsElena Hernandez0Octavio Manero1Fernando Bautista2Juan Paulo Garcia-Sandoval3Departamento de Ingeniería Química, Universidad de Guadalajara, Guadalajara 44430, MexicoInstituto de Investigaciones en Materiales, Universidad Autónoma de México, Mexico City 04510, MexicoDepartamento de Física, Universidad de Guadalajara, Guadalajara 44430, MexicoDepartamento de Ingeniería Química, Universidad de Guadalajara, Guadalajara 44430, MexicoThis 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.https://www.mdpi.com/2227-7390/9/24/3287frequency responseseries methodnonlinear dynamical systems |
| spellingShingle | Elena Hernandez Octavio Manero Fernando Bautista Juan Paulo Garcia-Sandoval Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations Mathematics frequency response series method nonlinear dynamical systems |
| title | Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations |
| title_full | Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations |
| title_fullStr | Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations |
| title_full_unstemmed | Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations |
| title_short | Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations |
| title_sort | analytic matrix method for frequency response techniques applied to nonlinear dynamical systems i small and medium amplitude oscillations |
| topic | frequency response series method nonlinear dynamical systems |
| url | https://www.mdpi.com/2227-7390/9/24/3287 |
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