Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System
Fractional-order derivative has been shown an adequate tool to the study of so-called "anomalous" social and physical behaviors, in reflecting their non-local, frequency- and history-dependent properties, and it has been used to model practical systems in engineering successfully, includin...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-2010-0566 |
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| author | Z.H. Wang M. L. Du |
| author_facet | Z.H. Wang M. L. Du |
| author_sort | Z.H. Wang |
| collection | DOAJ |
| description | Fractional-order derivative has been shown an adequate tool to the study of so-called "anomalous" social and physical behaviors, in reflecting their non-local, frequency- and history-dependent properties, and it has been used to model practical systems in engineering successfully, including the famous Bagley-Torvik equation modeling forced motion of a rigid plate immersed in Newtonian fluid. The solutions of the initial value problems of linear fractional differential equations are usually expressed in terms of Mittag-Leffler functions or some other kind of power series. Such forms of solutions are not good for engineers not only in understanding the solutions but also in investigation. This paper proves that for the linear SDOF oscillator with a damping described by fractional-order derivative whose order is between 1 and 2, the solution of its initial value problem free of external excitation consists of two parts, the first one is the 'eigenfunction expansion' that is similar to the case without fractional-order derivative, and the second one is a definite integral that is independent of the eigenvalues (or characteristic roots). The integral disappears in the classical linear oscillator and it can be neglected from the solution when stationary solution is addressed. Moreover, the response of the fractionally damped oscillator under harmonic excitation is calculated in a similar way, and it is found that the fractional damping with order between 1 and 2 can be used to produce oscillation with large amplitude as well as to suppress oscillation, depending on the ratio of the excitation frequency and the natural frequency. |
| format | Article |
| id | doaj-art-d82b40c1eb6d403ea2570d9409750d04 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-d82b40c1eb6d403ea2570d9409750d042025-02-03T05:45:22ZengWileyShock and Vibration1070-96221875-92032011-01-01181-225726810.3233/SAV-2010-0566Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration SystemZ.H. Wang0M. L. Du1Institute of Science, PLA University of Science and Technology, 211101 Nanjing, ChinaInstitute of Science, PLA University of Science and Technology, 211101 Nanjing, ChinaFractional-order derivative has been shown an adequate tool to the study of so-called "anomalous" social and physical behaviors, in reflecting their non-local, frequency- and history-dependent properties, and it has been used to model practical systems in engineering successfully, including the famous Bagley-Torvik equation modeling forced motion of a rigid plate immersed in Newtonian fluid. The solutions of the initial value problems of linear fractional differential equations are usually expressed in terms of Mittag-Leffler functions or some other kind of power series. Such forms of solutions are not good for engineers not only in understanding the solutions but also in investigation. This paper proves that for the linear SDOF oscillator with a damping described by fractional-order derivative whose order is between 1 and 2, the solution of its initial value problem free of external excitation consists of two parts, the first one is the 'eigenfunction expansion' that is similar to the case without fractional-order derivative, and the second one is a definite integral that is independent of the eigenvalues (or characteristic roots). The integral disappears in the classical linear oscillator and it can be neglected from the solution when stationary solution is addressed. Moreover, the response of the fractionally damped oscillator under harmonic excitation is calculated in a similar way, and it is found that the fractional damping with order between 1 and 2 can be used to produce oscillation with large amplitude as well as to suppress oscillation, depending on the ratio of the excitation frequency and the natural frequency.http://dx.doi.org/10.3233/SAV-2010-0566 |
| spellingShingle | Z.H. Wang M. L. Du Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System Shock and Vibration |
| title | Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System |
| title_full | Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System |
| title_fullStr | Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System |
| title_full_unstemmed | Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System |
| title_short | Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System |
| title_sort | asymptotical behavior of the solution of a sdof linear fractionally damped vibration system |
| url | http://dx.doi.org/10.3233/SAV-2010-0566 |
| work_keys_str_mv | AT zhwang asymptoticalbehaviorofthesolutionofasdoflinearfractionallydampedvibrationsystem AT mldu asymptoticalbehaviorofthesolutionofasdoflinearfractionallydampedvibrationsystem |