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Research Papers

Primary Resonance of Dry-Friction Oscillator With Fractional-Order Proportional-Integral-Derivative Controller of Velocity Feedback

[+] Author and Article Information
Yongjun Shen

School of Mechanical Engineering,
Shijiazhuang Tiedao University,
Shijiazhuang 050043, China
e-mail: shenyongjun@126.com

Jiangchuan Niu

School of Mechanical Engineering,
Shijiazhuang Tiedao University,
Shijiazhuang 050043, China
e-mail: menjc@163.com

Shaopu Yang

School of Mechanical Engineering,
Shijiazhuang Tiedao University,
Shijiazhuang 050043, China
e-mail: yangsp@stdu.edu.cn

Sujuan Li

School of Information Science and Technology,
Shijiazhuang Tiedao University,
Shijiazhuang 050043, China
e-mail: melsj@126.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 10, 2015; final manuscript received April 13, 2016; published online June 7, 2016. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 11(5), 051027 (Jun 07, 2016) (9 pages) Paper No: CND-15-1432; doi: 10.1115/1.4033443 History: Received December 10, 2015; Revised April 13, 2016

The classical mass-on-moving-belt model describing friction-induced vibration is studied. The primary resonance of dry-friction oscillator with fractional-order PID (FOPID) controller of velocity feedback is investigated by Krylov–Bogoliubov–Mitropolsky (KBM) asymptotic method, and the approximately analytical solution is obtained. The effects of the parameters in FOPID controller on dynamical properties are characterized by five equivalent parameters. Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response. The effects of dry friction on the dynamical properties are characterized in the form of the equivalent linear damping and nonlinear damping. The amplitude-frequency equation for steady-state solution associated with the stability condition is also studied. A comparison of the analytical solution with the numerical results is fulfilled, and their satisfactory agreement verifies the correctness of the approximately analytical results. Finally, the effects of the coefficients and orders in FOPID controller on the amplitude-frequency curves are analyzed, and the control performances of FOPID and traditional integer-order proportional-integral-derivative (PID) controllers are compared. The comparison results show that FOPID controller is better than traditional integer-order PID controller for controlling the primary resonance of dry-friction oscillator, when the coefficients of the two controllers are the same. This presents theoretical basis for scholars and engineers to design similar fractional-order controlled system.

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Figures

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Fig. 1

Closed-loop feedback of FOPID control system

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Fig. 2

Example of dry-friction force

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Fig. 3

Comparison between approximate analytical and numerical solution

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Fig. 4

Amplitude-frequency curves with different Kp

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Fig. 5

Amplitude-frequency curves with different Ki

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Fig. 6

Amplitude-frequency curves with different Kd

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Fig. 7

(a) Equivalent damping curve with λ, (b) resonant frequency curve with λ, (c) maximum response amplitude curve with λ

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Fig. 8

(a) Equivalent damping curve with δ, (b) resonant frequency curve with δ, (c) maximum response amplitude curve with δ

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Fig. 9

Comparison between the control performances of FOPID and integer-order controllers

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