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Research Papers

Lyapunov Functions and Sliding Mode Control for Two Degrees-of-Freedom and Multidegrees-of-Freedom Fractional Oscillators

[+] Author and Article Information
Youan Zhang

Institute of Technology,
Yantai Nanshan University,
Yantai 265713, China
e-mail: zhangya63@sina.com

Jian Yuan

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: yuanjianscar@gmail.com

Jingmao Liu

Shandong Nanshan International Flight Co., Ltd.,
Yantai 265713, China
e-mail: liujingmao@nanshan.com.cn

Bao Shi

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: baoshi781@sohu.com

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 3, 2016; final manuscript received September 5, 2016; published online November 23, 2016. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 139(1), 011014 (Nov 23, 2016) (7 pages) Paper No: VIB-16-1245; doi: 10.1115/1.4034843 History: Received May 03, 2016; Revised September 05, 2016

This paper addresses the Lyapunov functions and sliding mode control design for two degrees-of-freedom (2DOF) and multidegrees-of-freedom (MDOF) fractional oscillators. First, differential equations of motion for 2DOF fractional oscillators are established by adopting the fractional Kelvin–Voigt constitute relation for viscoelastic materials. Second, a Lyapunov function candidate for 2DOF fractional oscillators is suggested, which includes the potential energy stored in fractional derivatives. Third, the differential equations of motion for 2DOF fractional oscillators are transformed into noncommensurate fractional state equations with six dimensions by introducing state variables with physical significance. Sliding mode control design and adaptive sliding mode control design are proposed based on the noncommensurate fractional state equations. Furthermore, the above results are generalized to MDOF fractional oscillators. Finally, numerical simulations are carried out to validate the above control designs.

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Figures

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Fig. 1

Mechanical model for 2DOF fractional oscillators

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Fig. 2

Mechanical model for MDOF fractional oscillators

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Fig. 3

Vibration response of 2DOF fractional oscillators

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Fig. 4

State response of the sliding mode control system

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Fig. 5

State response of the adaptive sliding mode control system

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