Research Papers

Chaotic Behavior and Feedback Control of Magnetorheological Suspension System With Fractional-Order Derivative

[+] Author and Article Information
Chengyuan Zhang

College of Automotive Engineering,
Chongqing University,
Chongqing 401331, China
e-mail: enzozcy@cqu.edu.cn

Jian Xiao

College of Mathematics and Statistics,
Chongqing University,
Chongqing 401331, China
e-mail: xj4448@126.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 4, 2017; final manuscript received September 7, 2017; published online November 1, 2017. Assoc. Editor: Corina Sandu.

J. Comput. Nonlinear Dynam 13(2), 021007 (Nov 01, 2017) (9 pages) Paper No: CND-17-1250; doi: 10.1115/1.4037931 History: Received June 04, 2017; Revised September 07, 2017

The fractional differential equations of the single-degree-of-freedom (DOF) quarter vehicle with a magnetorheological (MR) suspension system under the excitation of sine are established, and the numerical solution is acquired based on the predictor–corrector method. The analysis of phase trajectory, time domain response, and Poincaré section shows that the nonlinear dynamic characteristics between fractional and integer-order suspension systems are quite different, which proves the superiority of using fractional order to describe the physical properties. By discussing the influence of each parameter on the vibration, the range of parameters to avoid the chaotic vibration is obtained. The variable feedback control is used to control the chaotic vibration effectively.

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Fig. 1

Single DOF quarter-car suspension model

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Fig. 2

Time domain response, phase trajectory, Poincaré section, and Lyapunov exponents of the integer-order differential suspension system: (a) time domain response, (b) phase trajectory, (c) Poincaré section, and (d) Lyapunov exponents

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Fig. 3

Bifurcation diagram of the suspension system: (a) x–q chaotic bifurcation diagram of suspension system and (b) Dqx–q chaotic bifurcation diagram of suspension system

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Fig. 4

The phase trajectory, time domain response, and Poincaré section of MR suspension under different fractional orders (=0.600,0.974, 0.976, 0.990, 0.997,1.000): (a) phase trajectory q = 0.60, (b) time domain response q = 0.60, (c) Poincaré section q = 0.60, (d) phase trajectory q = 0.974, (e) time domain response q = 0.974, (f) Poincaré section q = 0.974, (g) phase trajectory q = 0.976, (h) time domain response q = 0.976, (i) Poincaré section q = 0.976, (j) phase trajectory q = 0.990, (k) time domain response q = 0.990, (l), (m) phase trajectory q = 0.997, (n) time domain response q = 0.997, (o) Poincaré section q = 0.997, (p) phase trajectory q = 1.000, (q) time domain response q = 1.000, and (r) Poincaré section q = 1.000

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Fig. 5

x−ω bifurcation diagram of the suspension system

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Fig. 6

x−k1 bifurcation diagram of the suspension system

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Fig. 7

x−k2 bifurcation diagram of the suspension system

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Fig. 8

x−c1 bifurcation diagram of the suspension system

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Fig. 9

x−c2 bifurcation diagram of the suspension system

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Fig. 10

Chaos control based on system parameter adjustment: (a) phase trajectory (controlled), (b) time domain response (controlled), and (c) Poincare section (controlled)

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Fig. 11

Chaos control based on variable feedback: (a) phase trajectory controlled, (b) time domain response controlled, and (c) Poincare section controlled



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