Abstract

The nonlinear and flexible effects on the continuum dynamics of rocker–rocker flexible mechanism are investigated. An experimental rocker–rocker mechanism with flexible coupler was first established. The flexible mechanism which incorporats the buckling motion of flexible coupler is acted as a double-well oscillator. The mechanism was actuated by electromagnet and measured by charge-coupled device (CCD) visual system. Rich dynamic behavior such as complex period, amplitude modulation, and chaos in the intrawell and interwell oscillations were observed. For investigating nonlinear dynamics, the dynamic behavior was analyzed through identification of linear and nonlinear lumped models. Both time-domain and frequency-domain approaches were carried out in identifying linear time-invariant model. Averaging multiple models were employed for the time-domain identification of linear model. The identification of nonlinear model was undertaken by the extension of the two-stage linear identification scheme. The response identification in input space was analyzed by utilizing semi-analytical harmonic balance method. The important boundary of chaotic response in operation was investigated by the proposed energy-well criterion, Melnikov's criterion, Moon's criterion, as well as Szemplińska-Stupnicka and Rudowski's criterion.

References

1.
Erdman
,
A. G.
, (ed.),
1993
,
Modern Kinematics, Developments in the Last Forty Years
,
Wiley, New York
.
2.
Wasfy
,
T. M.
, and
Noor
,
A. K.
,
2003
, “
Computational Strategies for Flexible Multibody Systems
,”
ASME Appl. Mech. Rev.
,
56
(
6
), pp.
553
613
.10.1115/1.1590354
3.
Eberhard
,
P.
, and
Schiehlen
,
W.
,
2006
, “
Computational Dynamics of Multibody Systems: History. Formalisms, and Applications
,”
ASME J. Comput. Nonlinear Dyn.
,
1
(
1
), pp.
3
12
.10.1115/1.1961875
4.
Shabana
,
A. A.
,
Bauchau
,
O. A.
, and
Hulbert
,
G. M.
,
2007
, “
Integration of Large Deformation Finite Element and Multibody System Algorithms
,”
ASME J. Comput. Nonlinear Dyn.
,
2
(
4
), pp.
351
359
.10.1115/1.2756075
5.
Viscomi
,
B. V.
, and
Ayre
,
R. S.
,
1971
, “
Nonlinear Dynamic Response of Elastic Slider-Crank Mechanism
,”
ASME J. Eng. Ind.
,
93
(
1
), pp.
251
262
.10.1115/1.3427883
6.
Jasinski
,
P. W.
,
Lee
,
H. C.
, and
Sandor
,
G. N.
,
1970
, “
Stability and Steady-State Vibrations in a High-Speed Slider-Crank Mechanism
,”
ASME, J. Appl. Mech.
,
37
(
4
), pp.
1069
1076
.10.1115/1.3408660
7.
Jasinski
,
P. W.
,
Lee
,
H. C.
, and
Sandor
,
G. N.
,
1971
, “
Vibrations of Elastic Connecting Rod of a High-Speed Slider-Crank Mechanism
,”
ASME J. Eng. Ind.
,
93
(
2
), pp.
636
644
.10.1115/1.3427974
8.
Winfrey
,
R. C.
,
1972
, “
Dynamic Analysis of Elastic Link Mechanisms by Reduction of Coordinates
,”
ASME J. Eng. Ind.
,
94
(
2
), pp.
577
582
.10.1115/1.3428197
9.
Imam
,
I.
,
Sandor
,
G. N.
, and
Kramer
,
S. N.
,
1973
, “
Deflection and Stress Analysis in High-Speed Planar Mechanism With Elastic Links
,”
ASME J. Eng. Ind.
,
95
(
2
), pp.
541
548
.10.1115/1.3438188
10.
Thompson
,
B. S.
, and
Sung
,
C. K.
,
1986
, “
A Survey of Finite Element Techniques for Mechanism Design
,”
Mech. Mach. Theory
,
21
(
4
), pp.
351
359
.10.1016/0094-114X(86)90057-1
11.
Chu
,
S. C.
, and
Pan
,
K. C.
,
1975
, “
Dynamic Response of a High-Speed Slider-Crank Mechanism With an Elastic Connecting Rod
,”
ASME J. Eng. Ind.
,
97
(
2
), pp.
542
550
.10.1115/1.3438618
12.
Sutherland
,
G. H.
,
1976
, “
Analytical and Experimental Investigation of a High-Speed Elastic-Membered Linkage
,”
ASME J. Eng. Ind.
,
98
(
3
), pp.
788
794
.10.1115/1.3439030
13.
Sadler
,
J. P.
, and
Sandor
,
G. N.
,
1973
, “
A Lumped Parameter Approach to Vibration and Stress Analysis of Elastic Linkages
,”
ASME J. Eng. Ind.
,
95
(
2
), pp.
549
557
.10.1115/1.3438189
14.
Sandor
,
G. N.
, and
Zhuang
,
X.
,
1985
, “
A Linearized Lumped Parameter Approach to Vibration and Stress Analysis of Elastic Linkages
,”
Mech. Mach. Theory
,
20
(
5
), pp.
427
437
.10.1016/0094-114X(85)90047-3
15.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
Wiley
,
New York
.
16.
Howell
,
L. L.
,
Magleby
,
S. P.
, and
Olsen
,
B. M.
, (ed.),
2013
,
Handbook of Compliant Mechanisms
,
Wiley
, Hoboken,
NJ
.
17.
Beroz
,
J.
,
Awtar
,
S.
, and
Hart
,
A. J.
,
2014
, “
Extensible-Link Kinematic Model for Characterizing and Optimizing Compliant Mechanism Motion
,”
ASME J. Mech. Des.
,
136
(
3
), p.
031008
.10.1115/1.4026269
18.
Burns
,
R. H.
, and
Crossley
,
F. R. E.
,
1966
, “
Structural Permutations of Flexible Link Mechanisms
,”
ASME
Paper No. 66-MECH-5.10.1115/66-MECH-5
19.
Neubauer
,
A. H.
,
Cohen
,
R.
, and
Hall
,
A. S.
,
1966
, “
An Analytical Study of the Dynamics of an Elastic Linkage
,”
ASME J. Eng. Ind.
,
88
(
3
), pp.
311
317
.10.1115/1.3670951
20.
Burns
,
R. H.
, and
Crossley
,
F. R. E.
,
1968
, “
Kinetostatic Synthesis of Flexible Link Mechanisms
,”
ASME
Paper No. 68-MECH-36.10.1115/68-MECH-36
21.
Erdman
,
A. G.
,
Sandor
,
G. N.
, and
Oakberg
,
R. G.
,
1972
, “
General Method for Kineto-Elastodynamic Analysis and Synthesis of Mechanisms
,”
ASME J. Eng. Ind.
,
94
(
4
), pp.
1193
1205
.10.1115/1.3428335
22.
Chen
,
J. S.
, and
Huang
,
C. L.
,
2001
, “
Dynamic Analysis of Flexible Slider-Crank Mechanisms With Non-Linear Finite Element Method
,”
J. Sound Vib.
,
246
(
3
), pp.
389
402
.10.1006/jsvi.2001.3673
23.
Shoup
,
T. E.
, and
McLarnan
,
C. W.
,
1971
, “
A Survey of Flexible Link Mechanism Having Lower Pairs
,”
J. Mech.
,
6
(
1
), pp.
97
105
.10.1016/0022-2569(71)90009-7
24.
Lowen
,
G. G.
, and
Jandrasits
,
W. G.
,
1972
, “
Survey of Investigations Into the Dynamic Behavior of Mechanisms Containing Links With Distributed Mass and Elasticity
,”
Mech. Mach. Theory
,
7
(
1
), pp.
3
17
.10.1016/0094-114X(72)90012-2
25.
Erdman
,
A. G.
, and
Sandor
,
G. N.
,
1972
, “
Kineto-Elastodynamic—A Review of the State-of-the-Art and Trends
,”
Mech. Mach. Theory
,
7
(
1
), pp.
19
33
.10.1016/0094-114X(72)90013-4
26.
Lowen
,
G. G.
, and
Chassapis
,
C.
,
1986
, “
The Elastic Behavior of Linkages: An Update
,”
Mech. Mach. Theory
,
21
(
1
), pp.
33
42
.10.1016/0094-114X(86)90028-5
27.
Dwivedy
,
S. K.
, and
Eberhard
,
P.
,
2006
, “
Dynamic Analysis of Flexible Manipulators, a Literature Review
,”
Mech. Mach. Theory
,
41
(
7
), pp.
749
777
.10.1016/j.mechmachtheory.2006.01.014
28.
Isermann
,
R.
, and
Münchhof
,
M.
,
2011
,
Identification of Dynamic Systems
,
Springer
, Berlin.
29.
Alexander
,
R. M.
, and
Lawrence
,
K. L.
,
1974
, “
An Experimental Investigation of the Dynamic Response of an Elastic Mechanism
,”
ASME J. Eng. Ind.
,
96
(
1
), pp.
268
274
.10.1115/1.3438309
30.
Alexander
,
R. M.
, and
Lawrence
,
K. L.
,
1975
, “
Experimentally Determined Dynamic Strains in an Elastic Mechanism
,”
ASME J. Eng. Ind.
,
97
(
3
), pp.
791
794
.10.1115/1.3438679
31.
Golebiewski
,
E. P.
, and
Sadler
,
J. P.
,
1976
, “
Analytical and Experimental Investigation of Elastic Slider-Crank Mechanisms
,”
ASME J. Eng. Ind.
,
98
(
4
), pp.
1266
1271
.10.1115/1.3439097
32.
Jandrasits
,
W. G.
, and
Lowen
,
G. G.
,
1979
, “
The Elastic-Dynamic Behavior of a Counterweighted Rocker Link With an Overhanging Endmass in a Four-Bar Linkage—Part II: Application and Experiment
,”
ASME J. Mech. Des.
,
101
(
1
), pp.
89
98
.10.1115/1.3454029
33.
Stamps
,
F. R.
, and
Bagci
,
C.
,
1983
, “
Dynamics of Planar, Elastic, High-Speed Mechanisms Considering Three-Dimensional Off-Set Geometry: Analytical and Experimental Investigations
,”
ASME J. Mech. Trans. Auto. Des.
,
105
(
3
), pp.
498
510
.10.1115/1.3267388
34.
Turcic
,
D. A.
,
Midha
,
A.
, and
Bosnik
,
J. R.
,
1984
, “
Dynamic Analysis of Elastic Mechanism Systems—Part II: Applications and Experimental Results
,”
ASME J. Dyn. Syst. Meas., Contro
l.,
106
(
4
), pp.
255
260
.10.1115/1.3140682
35.
Turcic
,
D. A.
, and
Midha
,
A.
,
1984
, “
Dynamic Analysis of Elastic Mechanism Systems—Part I: Applications
,”
ASME J. Dyn. Syst. Meas., Control.
,
106
(
4
), pp.
249
254
.10.1115/1.3140681
36.
Sung
,
C. K.
,
Thompson
,
B. S.
,
Xing
,
T. M.
, and
Wang
,
C. H.
,
1986
, “
An Experimental Study on the Nonlinear Elastodynamic Response of Linkage Mechanisms
,”
Mech. Mach. Theory
,
21
(
2
), pp.
121
132
.10.1016/0094-114X(86)90002-9
37.
Liou
,
F. W.
, and
Peng
,
K. C.
,
1993
, “
Experimental Frequency Response Analysis of Flexible Mechanisms
,”
Mech. Mach. Theory
,
28
(
1
), pp.
73
81
.10.1016/0094-114X(93)90048-Z
38.
Halbig
,
D.
, and
Beale
,
D.
,
1995
, “
Experimental Observations of a Flexible Slider Crank Mechanism at Very High Speeds
,”
Nonlinear Dyn.
,
7
(
3
), pp.
365
384
.10.1007/BF00046309
39.
Yang
,
K.-H.
, and
Park
,
Y.-S.
,
1998
, “
Dynamic Stability Analysis of a Flexible Four-Bar Mechanism and Its Experimental Investigation
,”
Mech. Mach. Theory
,
33
(
3
), pp.
307
320
.10.1016/S0094-114X(97)00048-7
40.
Gasparetto
,
A.
,
2004
, “
On the Modeling of Flexible-Link Planar Mechanisms: Experimental Validation of an Accurate Dynamic Model
,”
ASME J. Dyn. Syst. Meas., Control.
,
126
(
2
), pp.
365
375
.10.1115/1.1767856
41.
Chang
,
L. W.
, and
Hamilton
,
J. F.
,
1991
, “
The Kinematics of Robotic Manipulators With Flexible Links Using an Equivalent Rigid Link System (ERLS) Model
,”
ASME J. Dyn. Syst. Meas., Control.
,
113
(
1
), pp.
48
53
.10.1115/1.2896358
42.
Boyle
,
C.
,
Howell
,
L. L.
,
Magleby
,
S. P.
, and
Evans
,
M. S.
,
2003
, “
Dynamic Modeling of Compliant Constant-Force Compression Mechanism
,”
Mech. Mach. Theory
,
38
(
12
), pp.
1469
1487
.10.1016/S0094-114X(03)00098-3
43.
Tadjbakhsh
,
I. G.
, and
Younis
,
C. J.
,
1986
, “
Dynamic Stability of the Flexible Connecting Rod of a Slider Crank Mechanism
,”
ASME J. Mech., Trans., Autom.
,
108
(
4
), pp.
487
496
.10.1115/1.3258760
44.
Hsieh
,
S.
, and
Shaw
,
S. W.
,
1993
, “
Dynamic Stability and Nonlinear Resonance of a Flexible Connecting Rod: Continuous Parameter Model
,”
Nonlinear Dyn.
,
4
(
6
), pp.
573
603
.10.1007/BF00162233
45.
Hsieh
,
S.
, and
Shaw
,
S. W.
,
1994
, “
The Dynamic Stability and Non-Linear Resonance of a Flexible Connecting Rod: Single Mode Model
,”
J. Sound Vib.
,
170
(
1
), pp.
25
49
.10.1006/jsvi.1994.1045
46.
Badlani
,
M.
, and
Kleinhenz
,
W.
,
1979
, “
Dynamic Stability of Elastic Mechanisms
,”
ASME J. Mech. Des.
,
101
(
1
), pp.
149
153
.10.1115/1.3454014
47.
Zhu
,
Z. G.
, and
Chen
,
Y.
,
1983
, “
The Stability of the Motion of a Connecting Rod
,”
ASME J. Mech., Trans., Autom.
,
105
(
4
), pp.
637
640
.10.1115/1.3258527
48.
Badlani
,
M.
, and
Midha
,
A.
,
1982
, “
Member Initial Curvature Effects on the Elastic Slider-Crank Mechanism Response
,”
ASME J. Mech. Des.
,
104
(
1
), pp.
159
167
.10.1115/1.3256306
49.
Jandrasits
,
W. G.
, and
Lowen
,
G. G.
,
1979
, “
The Elastic-Dynamic Behavior of a Counterweighted Rocker Link With an Overhanging Endmass in a Four-Bar Linkage—Part I: Theory
,”
ASME J. Mech. Des.
,
101
(
1
), pp.
77
88
.10.1115/1.3454028
50.
Tadjbakhsh
,
I. G.
, and
Younis
,
C. J.
,
1985
, “
Effect of Flexibility on the Dynamic Stability of a Four-Bar Mechanism
,”
Int. J. Mech. Sci.
,
27
(
11–12
), pp.
813
822
.10.1016/0020-7403(85)90012-8
51.
Seneviratne
,
L. D.
, and
Earles
,
S. W. E.
,
1992
, “
Chaotic Behaviour Exhibited During Contact Loss in a Clearance Joint of a Four-Bar Mechanism
,”
Mech. Mach. Theory
,
27
(
3
), pp.
307
321
.10.1016/0094-114X(92)90021-9
52.
Rhee
,
J.
, and
Akay
,
A.
,
1996
, “
Dynamic Response of a Revolute Joint With Clearance
,”
Mech. Mach. Theory
,
31
(
1
), pp.
121
134
.10.1016/0094-114X(95)00061-3
53.
Farahanchi
,
F.
, and
Shaw
,
S. W.
,
1994
, “
Chaotic and Periodic Dynamics of a Slider-Crank Mechanism With Slider Clearance
,”
Mech. Mach. Theory
,
177
(
3
), pp.
307
324
.10.1006/jsvi.1994.1436
54.
Gu
,
P.
, and
Dubowsky
,
S.
,
1998
, “
The Design Implications of Chaotic and Near-Chaotic Vibrations in Machines
,”
ASME
Paper No. DET98/MECH-5849.10.1115/DET98/MECH-5849
55.
Khemili
,
I.
, and
Romdhane
,
L.
,
2008
, “
Dynamic Analysis of a Flexible Slider-Crank Mechanism With Clearance
,”
Euro. J. Mech. A/Solids
,
27
(
5
), pp.
882
898
.10.1016/j.euromechsol.2007.12.004
56.
Dupac
,
M.
, and
Beale
,
D. G.
,
2010
, “
Dynamic Analysis of a Flexible Linkage Mechanism With Cracks and Clearance
,”
Mech. Mach. Theory
,
45
(
12
), pp.
1909
1923
.10.1016/j.mechmachtheory.2010.07.006
57.
Rahmanian
,
S.
, and
Ghazavi
,
M. R.
,
2015
, “
Bifurcation in Planar Slider–Crank Mechanism With Revolute Clearance Joint
,”
Mech. Mach. Theory
,
91
, pp.
86
101
.10.1016/j.mechmachtheory.2015.04.008
58.
Shaw
,
S. W.
, and
Holmes
,
P. J.
,
1983
, “
A Periodically Forced Piecewise Linear Oscillator
,”
J. Sound Vib.
,
90
(
1
), pp.
129
155
.10.1016/0022-460X(83)90407-8
59.
Lin
,
R. M.
, and
Ewins
,
D. J.
,
1993
, “
Chaotic Vibration of Mechanical Systems With Backlash
,”
Mech. Syst. Sig. Proc.
,
7
(
3
), pp.
257
272
.10.1006/mssp.1993.1012
60.
Kovacic
,
I.
, and
Brennan
,
M. J.
, Eds.,
2011
,
The Duffing Equation, Nonlinear Oscillators and Their Behaviour
,
Wiley
, Chichester, UK.
61.
Thompson
,
J. M. T.
, and
Stewart
,
H. B.
,
1986
,
Nonlinear Dynamics and Chaos
,
Wiley
,
New York
.
62.
Holmes
,
P. J.
,
1979
, “
A Nonlinear Oscillator With a Strange Attractor
,”
Philos. Trans. R. Soc., London
,
292
(
1394
), pp.
419
448
.10.1098/rsta.1979.0068
63.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1986
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
.
64.
Moon
,
F. C.
,
1987
,
Chaotic Vibrations: An Introduction for Applied Scientists and Engineers
,
Wiley
,
New York
.
65.
Wu
,
Z.
,
Harne
,
R. L.
, and
Wang
,
K. W.
,
2015
, “
Excitation-Induced Stability in a Bistable Duffing Oscillator: Analysis and Experiments
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(1), p.
011016
.10.1115/1.4026974
66.
Abhyankar
,
N. S.
,
Hall
, II
,
E. K.
, and
Hanagud
,
S. V.
,
1993
, “
Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations
,”
ASME J. Appl. Mech.
,
60
(
1
), pp.
167
174
.10.1115/1.2900741
67.
Ramu
,
S. A.
,
Sankar
,
T. S.
, and
Ganesan
,
R.
,
1994
, “
Bifurcations, Catastrophes and Chaos in a Pre-Buckled Beam
,”
Int. J. Non-Linear Mech.
,
29
(
3
), pp.
449
462
.10.1016/0020-7462(94)90014-0
68.
Chen
,
J. S.
, and
Chian
,
C. H.
,
2001
, “
Effects of Crank Length on the Dynamic Behavior of a Flexible Connecting Road
,”
ASME J. Vib. Acoust.
,
123
(
3
), pp.
318
323
.10.1115/1.1368882
69.
Liu
,
B. Y.
,
Zhang
,
D. S.
,
Guo
,
J.
, and
Zhu
,
C. A.
,
2016
, “
Vision-Based Displacement Measurement Sensor Using Modified Taylor Approximation Approach
,”
Opt. Eng.
,
55
(
11
), p.
114103
.10.1117/1.OE.55.11.114103
70.
Wolf
,
A.
,
Swift
,
J. B.
,
Swinney
,
H. L.
, and
Vastano
,
J. A.
,
1985
, “
Determining Lyapunov Exponents From a Time Series
,”
Phys. D
,
16
(
3
), pp.
285
317
.10.1016/0167-2789(85)90011-9
71.
Chen
,
J. S.
, and
Chian
,
C. H.
,
2003
, “
On the Nonlinear Response of a Flexible Connecting Rod
,”
ASME J. Mech. Des.
,
125
(
4
), pp.
757
763
.10.1115/1.1631571
72.
Chang
,
R. J.
, and
Liu
,
Z. Y.
,
2017
, “
Identification of Viscoelastic Model of Four-Wire Suspension Systems in Optical Pickup Actuator
,”
Trans. Can. Soc. Mech. Eng.
,
41
(
5
), pp.
731
744
.10.1139/tcsme-2017-508
73.
Szemplińska-Stupnicka
,
W.
, and
Rudowski
,
J.
,
1992
, “
Local Methods in Predicting Occurrence of Chaos in Two-Well Potential Systems: Superharmonic Frequency Range
,”
J. Sound Vib.
,
152
(
1
), pp.
57
72
.10.1016/0022-460X(92)90065-6
74.
Szemplińska-Stupnicka
,
W.
,
1995
, “
The Analytical Predictive Criteria for Chaos and Escape in Nonlinear Oscillators: A Survey
,”
Nonlinear Dyn.
,
7
(
2
), pp.
129
147
.10.1007/BF00053705
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