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.

References

References
1.
Seireg
,
A. A.
,
1998
,
Friction and Lubrication in Mechanical Design
,
Marcel Dekker
,
New York
.
2.
Ding
,
Q.
, and
Zhai
,
H. M.
,
2013
, “
The Advance in Researches of Friction Dynamics in Mechanic System
,”
Adv. Mech.
,
43
(
1
), pp.
112
131
.
3.
Berger
,
E. J.
,
2002
, “
Friction Modeling for Dynamic System Simulation
,”
ASME Appl. Mech. Rev.
,
55
(
6
), pp.
535
577
.
4.
Awrejcewicz
,
J.
, and
Olejnik
,
P.
,
2005
, “
Analysis of Dynamic Systems With Various Friction Laws
,”
ASME Appl. Mech. Rev.
,
58
(
1–6
), pp.
389
410
.
5.
Hartog
,
J. P. D.
,
1931
, “
Forced Vibrations With Combined Coulomb and Viscous Friction
,”
Trans. ASME
,
53
(
9
), pp.
107
115
.
6.
Awrejcewicz
,
J.
, and
Delfs
,
J.
,
1990
, “
Dynamics of a Self-Excited Stick-Slip Oscillator With Two Degrees of Freedom—Part I: Investigation of Equilibria
,”
Eur. J. Mech., A: Solids
,
9
(
4
), pp.
269
282
.
7.
Awrejcewicz
,
J.
, and
Delfs
,
J.
,
1990
, “
Dynamics of a Self-Excited Stick-Slip Oscillator With Two Degrees of Freedom—Part II: Slip-Stick, Slip-Slip, Stick-Slip Transitions, Periodic and Chaotic Orbits
,”
Eur. J. Mech., A: Solids
,
9
(
5
), pp.
397
418
.
8.
Awrejcewicz
,
J.
,
1990
, “
Parametric and Self-Excited Vibrations Induced by Friction in a System With Three Degrees of Freedom
,”
KSME J.
,
4
(
2
), pp.
156
166
.
9.
Thomsen
,
J. J.
,
1999
, “
Using Fast Vibrations to Quench Friction-Induced Oscillations
,”
J. Sound Vib.
,
228
(
5
), pp.
1079
1102
.
10.
Thomsen
,
J. J.
, and
Fidlin
,
A.
,
2003
, “
Analytical Approximations for Stick-Slip Vibration Amplitudes
,”
Int. J. Non-Linear Mech.
,
38
(
3
), pp.
389
403
.
11.
Liu
,
X. J.
,
Wang
,
D. J.
, and
Chen
,
Y. S.
,
1998
, “
Approximate Analytical Solution of the Self-Excited Vibration of Piecewise-Smooth Systems Induced by Dry Friction
,”
Acta Mech. Sin.
,
14
(
1
), pp.
78
84
.
12.
Saha
,
A.
, and
Wahi
,
P.
,
2014
, “
An Analytical Study of Time-Delayed Control of Friction-Induced Vibrations in a System With a Dynamic Friction Model
,”
Int. J. Non-Linear Mech.
,
63
, pp.
60
70
.
13.
Cheng
,
G.
, and
Zu
,
J. W.
,
2004
, “
Dynamics of a Dry Friction Oscillator Under Two-Frequency Excitations
,”
J. Sound Vib.
,
275
(
3–5
), pp.
591
603
.
14.
Petras
,
I.
,
2011
,
Fractional-Order Nonlinear Systems
,
Higher Education Press
,
Beijing
.
15.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations, Mathematics in Science and Engineering
,
Academic Press
,
New York
.
16.
Shen
,
Y. J.
,
Wei
,
P.
, and
Yang
,
S. P.
,
2014
, “
Primary Resonance of Fractional-Order Van der Pol Oscillator
,”
Nonlinear Dyn.
,
77
(
4
), pp.
1629
1642
.
17.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems
,”
Acta Mech.
,
120
(
1–4
), pp.
109
125
.
18.
Yang
,
S. P.
, and
Shen
,
Y. J.
,
2009
, “
Recent Advances in Dynamics and Control of Hysteretic Nonlinear Systems
,”
Chaos, Solitons Fractals
,
40
(
4
), pp.
1808
1822
.
19.
Li
,
C. P.
, and
Deng
,
W. H.
,
2007
, “
Remarks on Fractional Derivatives
,”
Appl. Math. Comput.
,
187
(
1
), pp.
777
784
.
20.
Cao
,
J. X.
,
Ding
,
H. F.
, and
Li
,
C. P.
,
2013
, “
Implicit Difference Schemes for Fractional Diffusion Equations
,”
Commun. Appl. Math. Comput.
,
27
(
1
), pp.
61
74
.
21.
Li
,
X. H.
, and
Hou
,
J. Y.
,
2016
, “
Bursting Phenomenon in a Piecewise Mechanical System With Parameter Perturbation in Stiffness
,”
Int. J. Non-Linear Mech.
,
81
, pp.
165
176
.
22.
Li
,
X. H.
,
Hou
,
J. Y.
, and
Chen
,
J. F.
,
2016
, “
An Analytical Method for Mathieu Oscillator Based on Method of Variation of Parameter
,”
Commun. Nonlinear Sci. Numer. Simul.
,
37
, pp.
326
353
.
23.
Chen
,
J. H.
, and
Chen
,
W. C.
,
2008
, “
Chaotic Dynamics of the Fractionally Damped Van der Pol Equation
,”
Chaos, Solitons Fractals
,
35
(
1
), pp.
188
198
.
24.
Song
,
C.
,
Cao
,
J. D.
, and
Liu
,
Y. Z.
,
2015
, “
Robust Consensus of Fractional-Order Multi-Agent Systems With Positive Real Uncertainty Via Second-Order Neighbors Information
,”
Neurocomputing
,
165
, pp.
293
299
.
25.
Chen
,
L. C.
,
Zhao
,
L.
,
Li
,
W.
, and
Zhao
,
J.
,
2016
, “
Bifurcation Control of Bounded Noise Excited Duffing Oscillator by a Weakly Fractional-Order PID Feedback Controller
,”
Nonlinear Dyn.
,
83
(
1
), pp.
529
539
.
26.
Zeng
,
Q. S.
,
Silva
,
D.
, and
Larence
,
W.
,
2012
, “
The Application of Fractional Order Control in an Industrial Fish Processing Machine
,”
Control Intell. Syst.
,
40
(
3
), pp.
177
185
.
27.
Podlubny
,
I.
,
1999
, “
Fractional-Order Systems and PIλDμ–Controllers
,”
IEEE Trans. Autom. Control
,
44
(
1
), pp.
208
214
.
28.
Zeng
,
G. Q.
,
Chen
,
J.
,
Dai
,
Y. X.
,
Li
,
L. M.
,
Zheng
,
C. W.
, and
Chen
,
M. R.
,
2015
, “
Design of Fractional Order PID Controller for Automatic Regulator Voltage System Based on Multi-Objective Extremal Optimization
,”
Neurocomputing
,
160
, pp.
173
184
.
29.
Zhong
,
J. P.
, and
Li
,
L. C.
,
2015
, “
Tuning Fractional-Order (PID mu)-D-Lambda Controllers for a Solid-Core Magnetic Bearing System
,”
IEEE Trans. Control Syst. Technol.
,
23
(
4
), pp.
1648
1656
.
30.
Chen
,
Y. Q.
,
Petras
,
I.
, and
Xue
,
D. Y.
,
2009
, “
Fractional Order Control—A Tutorial
,”
2009 Conference on American Control Conference
. St. Louis, MO, June 10–12, IEEE Press, pp.
1397
1411
.
31.
Saidi
,
B.
,
Amairi
,
M.
,
Najar
,
S.
, and
Aoun
,
M.
,
2015
, “
Bode Shaping-Based Design Methods of a Fractional Order PID Controller for Uncertain Systems
,”
Nonlinear Dyn.
,
80
(
4
), pp.
1817
1838
.
32.
Chen
,
L. C.
, and
Zhu
,
W. Q.
,
2011
, “
Stochastic Jump and Bifurcation of Duffing Oscillator With Fractional Derivative Damping Under Combined Harmonic and White Noise Excitations
,”
Int. J. Non-Linear Mech.
,
46
(
10
), pp.
1324
1329
.
33.
Chen
,
L. C.
,
Li
,
Z. S.
,
Zhuang
,
Q. J.
, and
Zhu
,
W. Q.
,
2013
, “
First-Passage Failure of Single-Degree-of-Freedom Nonlinear Oscillators With Fractional Derivative
,”
J. Vib. Control
,
19
(
14
), pp.
2154
2163
.
34.
Wang
,
Z.
, and
Zheng
,
Y. G.
,
2009
, “
The Optimal Form of the Fractional-Order Difference Feedbacks in Enhancing the Stability of a SDOF Vibration System
,”
J. Sound Vib.
,
326
(
3–5
), pp.
476
488
.
35.
Chen
,
Y. Q.
,
Bhaskaran
,
T.
, and
Xue
,
D. Y.
,
2008
, “
Practical Tuning Rule Development for Fractional Order Proportional and Integral Controllers
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
2
), p. 021403.
36.
Basiri
,
M. H.
, and
Tavazoei
,
M. S.
,
2015
, “
On Robust Control of Fractional Order Plants: Invariant Phase Margin
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p. 054504.
37.
Awrejcewicz
,
J.
,
2014
,
Ordinary Differential Equations and Mechanical Systems
,
Springer International Publishing
, Cham,
Switzerland
.
38.
Awrejcewicz
,
J.
, and
Krodkiewski
,
J.
,
1983
, “
Analysis of Self-Excited Vibrations Due to Nonlinear Friction in a System with Two Degrees of Freedom System
,”
Sci. Bull. Lodz Tech. Univ.
,
68
, pp.
21
28
.
39.
Awrejcewicz
,
J.
,
1987
, “
Analysis of Self-Excited Vibration in Mechanical System With Four Degrees of Freedom
,”
Sci. Bull. Lodz Tech. Univ.
,
72
, pp.
5
27
.
40.
Awrejcewicz
,
J.
, and
Holicke
,
M.
,
2006
, “
Analytical Prediction of Stick-Slip Chaos in a Double Self-Excited Duffing-Type Oscillator
,”
Math. Probl. Eng.
,
2006
, p.
70245
.
41.
Awrejcewicz
,
J.
, and
Holicke
,
M. M.
,
1999
, “
Melnikov's Method and Stick-Slip Chaotic Oscillations in Very Weekly Forced Mechanical Systems
,”
Int. J. Bifurcation Chaos
,
9
(
3
), pp.
505
518
.
42.
Kluge
,
P. N. V.
,
Germaine
,
D. K.
, and
Crepin
,
K. T.
,
2015
, “
Dry Friction With Various Frictions Laws: From Wave Modulated Orbit to Stick-Slip Modulated
,”
Mod. Mech. Eng.
,
5
(02), pp.
28
40
.
43.
Shen
,
Y. J.
,
Yang
,
S. P.
,
Xing
,
H. J.
, and
Gao
,
G. S.
,
2012
, “
Primary Resonance of Duffing Oscillator With Fractional-Order Derivative
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
7
), pp.
3092
3100
.
44.
Shen
,
Y. J.
,
Yang
,
S. P.
, and
Sui
,
C. Y.
,
2014
, “
Analysis on Limit Cycle of Fractional-Order Van der Pol Oscillator
,”
Chaos, Solitons Fractals
,
67
(
10
), pp.
94
102
.
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