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.

References

References
1.
Sreekar Reddy
,
M. B. S.
,
Vigneshwar
,
P.
,
Sita Ram
,
M.
,
Raja Sekhar
,
D.
, and
Sai Harish
,
Y.
,
2017
, “
Comparative Optimization Study on Vehicle Suspension Parameters for Rider Comfort Based on RSM and GA
,”
Mater. Today
,
4
(
2 Part A
), pp.
1794
1803
.
2.
Bouazara
,
M.
,
Richard
,
M. J.
, and
Rakheja
,
S.
,
2006
, “
Safety and Comfort Analysis of a 3-D Vehicle Model With Optimal Non-Linear Active Seat Suspension
,”
J. Terramech.
,
43
(
2
), pp.
97
118
.
3.
Naik
,
R. D.
, and
Singru
,
P. M.
,
2011
, “
Resonance, Stability and Chaotic Vibration of a Quarter-Car Vehicle Model With Time-Delay Feedback
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
8
), pp.
3397
3410
.
4.
Fakhraei
,
J.
,
Khanlo
,
H.
, and
Dehghani
,
R.
,
2017
, “
Nonlinear Dynamic Behavior of a Heavy Articulated Vehicle With Magnetorheological Dampers
,”
ASME. J. Comput. Nonlinear Dynam
,
12
(
4
), p.
041017
.
5.
Abtahi
,
S. M.
,
2016
, “
Chaotic Study and Chaos Control in a Half-Vehicle Model With Semi-Active Suspension Using Discrete Optimal Ott-Grebogi-Yorke Method
,”
Proc. Inst. Mech. Eng. Part K
,
231
(
1
), pp. 148–155.
6.
Zhang
,
W.
,
Li
,
J.
,
Zhang
,
K.
, and
Cui
,
P.
,
2013
, “
Design of Magnetic Flux Feedback Controller in Hybrid Suspension System
,”
Math. Prob. Eng.
,
2013
, p. 712764.
7.
Chen
,
Q. T.
,
Huang
,
Y. J.
, and
Song
,
Y. R.
,
2014
, “
Modeling and Analysis of High-Order MR Vibration System Based on Fractional Order
,”
Chin. J. Sci. Instrum.
,
35
(
12
), pp.
2761
2771
.
8.
Turnip
,
A.
,
Park
,
S.
, and
Hong
,
K. S.
,
2010
, “
Sensitivity Control of a MR-Damper Semi-Active Suspension
,”
Int. J. Precis. Eng. Manuf.
,
11
(
2
), pp.
209
218
.
9.
Liem
,
D. T.
,
Truong
,
D. Q.
, and
Ahn
,
K. K.
,
2015
, “
Hysteresis Modeling of Magneto-Rheological Damper Using Self-Tuning Lyapunov-Based Fuzzy Approach
,”
Int. J. Precis. Eng. Manuf.
,
16
(
1
), pp.
31
41
.
10.
Mainardi
,
F.
,
2010
, “
Fractional Calculus and Waves in Linear Viscoelasticity
,”
Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models
,
Imperial College Press
, London.
11.
Wang
,
Z. H.
,
2011
, “
Fractional Calculus: A Mathematical Tool Describing the Memory Characteristics and Intermediate Processes
,”
Scientific Chinese
,
3
, pp. 76–78.
12.
Sun
,
H. L.
,
Jin
,
C.
,
Zhang
,
W. M.
,
Li
,
H.
, and
Tian
,
H. Y.
,
2014
, “
Modeling and Tests for a Hydro-Pneumatic Suspension Based on Fractional Calculus
,”
J. Vib. Shock
,
33
(
17
), pp.
167
172
; 190.
13.
Spanos
,
P. D.
, and
Evangelatos
,
G. I.
,
2010
, “
Response of a Nonlinear System With Restoring Forces Governed by Fractional Derivatives: Time Domain Simulation and Statistical Linearization Solution
,”
Soil Dyn. Earthquake Eng.
,
30
(
9
), pp.
811
821
.
14.
Machado
,
J. A. T.
,
Silva
,
M. F.
,
Barbosa
,
R. S.
,
Jesus
,
I. S.
,
Reis
,
C. M.
,
Marcos
,
M. G.
, and
Galhano
,
A. F.
,
2010
, “
Some Applications of Fractional Calculus in Engineering
,”
Math. Prob. Eng.
,
2010
, p.
639801
.
15.
Scherer
,
R.
,
Kalla
,
S. L.
,
Tang
,
Y.
, and
Huang
,
J.
,
2011
, “
The Grünwald–Letnikov Method for Fractional Differential Equations
,”
Comput. Math. Appl.
,
62
(
3
), pp.
902
917
.
16.
Chen
,
B. S.
, and
Huang
,
Y. J.
,
2009
, “
Application of Fractional Calculus on the Study of Magnetorheological Fluids' Characterization
,”
J. Huaqiao Univ.
,
5
, p. 003.
17.
Atan
,
O.
,
Chen
,
D.
, and
Türk
,
M.
,
2016
, “
Fractional Order PID and Application of Its Circuit Model
,”
J. Chin. Inst. Eng.
,
39
(
6
), pp.
695
703
.
18.
Dimeas
,
I.
,
Tsirimokou
,
G.
,
Psychalinos
,
C.
, and
Elwakil
,
A. S.
,
2017
, “
Experimental Verification of Fractional-Order Filters Using a Reconfigurable Fractional-Order Impedance Emulator
,”
J. Circuits Syst. Comput.
,
26
(
09
), p.
1750142
.
19.
Kai
,
D.
,
Ford
,
N. J.
, and
Freed
,
A. D.
,
2002
, “
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
,
29
(
1–4
), pp.
3
22
.
20.
Feng
,
Y. L.
,
Wang
,
J. H.
,
Rai
,
Z. P.
, and
Zhou
,
Z.
,
2014
, “
Estimation and Correction of Fractional Calculus and Its Application
,”
J. Nantong Univ. (Nat. Sci. Ed.)
,
13
(
1
), pp.
76
80
.
21.
Zhang
,
Q.
, and
Wang
,
B. H.
,
2002
, “
Chaos Control With Variable Feedback in Power System
,”
Electr. Power Autom, Equip.
,
23
(
11
), pp.
9
12
.
22.
Petráš
,
I.
,
2009
, “
Chaos in the Fractional-Order Volta's System: Modeling and Simulation
,”
Nonlinear Dyn.
,
57
(
1
), pp.
157
170
.
23.
Shojaeefard
,
M. H.
,
Khalkhali
,
A.
, and
Safarpour Erfani
,
P.
,
2014
, “
Multi-Objective Suspension Optimization of a 5-DOF Vehicle Vibration Model Excited by Random Road Profile
,”
Int. J. Adv. Des. Manuf. Technol.
,
7
(
1
), pp.
1
7
.http://www.sid.ir/en/VEWSSID/J_pdf/1000820140101.pdf
24.
Litak
,
G.
, and
Borowiec
,
M.
,
2006
, “
Nonlinear Vibration of a Quarter‐Car Model Excited by the Road Surface Profile
,”
Commun. Nonlinear Sci. Numer. Simul.
,
13
(
7
), pp.
1373
1383
.
25.
Houfek
,
M.
,
Sveda
,
P.
, and
Kratochvíl
,
C.
,
2011
, “
Identification of Deterministic Chaos in Dynamical Systems
,”
Mechatronics
,
Springer
,
Berlin
.
You do not currently have access to this content.