Abstract

In this paper, we consider the application of the piezoelectric energy harvesting using a portal frame structure of two-degrees-of-freedom. The piezoelectric material is considered as a linear device using a capacitive mathematical model. The portal structure is of two-degrees-of-freedom considering with quadratic coupling between the first and second modes of vibration. 2:1 internal resonance between the first and second modes is set, which is a particular condition of this type of system due to the appearance of a saturation phenomenon. As this phenomenon causes the system to start vibrating from the second mode and, at steady-state, vibrates at the first mode, the objective of this work is to verify the energy uptake, considering the different positioning of a piezoelectric material, which is coupled to the supported beam and/or to the column. In addition, the structure is excited by a nonideal DC motor with a limited power supply. The results show a considerably nonlinear behavior due to the nonideal motor, and, with the saturation phenomenon, it is more efficient to collect energy by coupling the PZT to the column. The investigation of the stability of the system due to the piezoelectric coefficient Θ is also taken into account, which is carried out by numerical tools as phase planes, Poincare maps, bifurcation diagrams, and 0–1 test.

References

References
1.
Rocha
,
R. T.
,
Balthazar
,
J. M.
,
Tusset
,
A. M.
, and
Quinn
,
D. D.
,
2018
, “
An Analytical Approximated Solution and Numerical Simulations of a Non-Ideal System With Saturation Phenomenon
,”
Nonlinear Dyn.
,
94
(
1
), pp.
429
442
.10.1007/s11071-018-4369-9
2.
Stephen
,
N. G.
,
2006
, “
On Energy Harvesting From Ambient Vibration
,”
J. Sound Vib.
,
293
(
1–2
), pp.
409
425
.10.1016/j.jsv.2005.10.003
3.
Syta
,
A.
,
Bowen
,
C. R.
,
Kim
,
H. A.
,
Rysak
,
A.
, and
Litak
,
G.
,
2015
, “
Experimental Analysis of the Dynamical Response of Energy Harvesting Devices Based on Bistable Laminated Plates
,”
Meccanica
,
50
(
8
), pp.
1961
1970
.10.1007/s11012-015-0140-1
4.
Triplett
,
A.
, and
Quinn
,
D. D.
,
2009
, “
The Effect of Nonlinear Piezoelectric Coupling on Vibration-Based Energy Harvesting
,”
J. Intell. Mater. Syst. Struct.
,
20
(
16
), pp.
1959
1967
.10.1177/1045389X09343218
5.
Daqaq
,
M. F.
,
Masana
,
R.
,
Erturk
,
A.
, and
Quinn
,
D. D.
,
2014
, “
On the Role of Nonlinearities in Vibratory Energy Harvesting: A Critical Review and Discussion
,”
ASME Appl. Mech. Rev.
,
66
(
4
), pp. 040801.10.1115/1.4026278
6.
Iliuk
,
I.
,
Balthazar
,
J. M.
,
Tusset
,
A. M.
,
Piqueira
,
J. R. C.
,
de Pontes
,
B. R.
,
Felix
,
J. L. P.
, and
Bueno
,
A. M.
,
2013
, “
A Non-Ideal Portal Frame Energy Harvester Controlled Using a Pendulum
,”
Eur. Phys. J. Spec. Top.
,
222
(
7
), pp.
1575
1586
.10.1140/epjst/e2013-01946-4
7.
Iliuk
,
I.
,
Balthazar
,
J. M.
,
Tusset
,
A. M.
,
Piqueira
,
J. R. C.
,
Pontes
,
B. R.
, Jr.
,
Felix
,
J. L. P.
, and
Bueno
,
A. M.
,
2014
, “
Application of Passive Control to Energy Harvester Efficiency Using a Non-Ideal Portal Frame Structural Support System
,”
J. Intell. Mater. Syst. Struct.
,
25
(
4
), pp.
417
429
.10.1177/1045389X13500570
8.
Pereira
,
T. L.
,
De Paula
,
A. S.
,
Fabro
,
A. T.
, and
Savi
,
M. A.
,
2019
, “
Random Effects in a Nonlinear Vibration-Based Piezoelectric Energy Harvesting System
,”
Int. J. Bifurcation Chaos
,
29
(
04
), p.
1950046
.10.1142/S0218127419500469
9.
Cellular
,
A.
,
da Silva Monteiro
,
L. L.
, and
Savi
,
M. A.
,
2018
, “
Numerical Investigation of Nonlinear Mechanical and Constitutive Effects on Piezoelectric Vibration-Based Energy Harvesting
,”
TM—Tech. Mess.
,
85
(
9
), pp.
565
579
.10.1515/teme-2017-0070
10.
Remick
,
K.
,
Quinn
,
D. D.
,
McFarland
,
D. M.
,
Bergman
,
L.
, and
Vakakis
,
A.
,
2016
, “
High-Frequency Vibration Energy Harvesting From Impulsive Excitation Utilizing Intentional Dynamic Instability Caused by Strong Nonlinearity
,”
J. Sound Vib.
,
370
, pp.
259
279
.10.1016/j.jsv.2016.01.051
11.
Erturk
,
A.
, and
Quinn
,
D. D.
,
2011
,
Piezoelectric Energy Harvesting
, John
Wiley and Sons
,
Chichester, UK
.
12.
Sugino
,
C.
,
Ruzzene
,
M.
, and
Erturk
,
A.
,
2020
, “
An Analytical Framework for Locally Resonant Piezoelectric Metamaterial Plates
,”
Int. J. Solids Struct.
,
182–183
, pp.
281
294
.10.1016/j.ijsolstr.2019.08.011
13.
Chiacchiari
,
S.
,
Romeo
,
F.
,
McFarland
,
D. M.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2019
, “
Vibration Energy Harvesting From Impulsive Excitations Via a Bistable Nonlinear Attachment-Experimental Study
,”
Mech. Syst. Signal Process.
,
125
, pp.
185
201
.10.1016/j.ymssp.2018.06.058
14.
Tan
,
D.
,
Yavarow
,
P.
, and
Erturk
,
A.
,
2018
, “
Resonant Nonlinearities of Piezoelectric Macro-Fiber Composite Cantilevers With Interdigitated Electrodes in Energy Harvesting
,”
Nonlinear Dyn.
,
92
(
4
), pp.
1935
1945
.10.1007/s11071-018-4172-7
15.
Quinn
,
D. D.
,
2007
, “
Resonant Dynamics in Strongly Nonlinear Systems
,”
Nonlinear Dyn.
,
49
(
3
), pp.
361
373
.10.1007/s11071-006-9126-9
16.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
John Wiley and Sons
,
Weinheim, Germany
, p.
720
.
17.
Nayfeh
,
A. H.
,
Mook
,
D. T.
, and
Marshall
,
L. R.
,
1973
, “
Nonlinear Coupling of Pitch and Roll Modes in Ship Motions
,”
J. Hydrodyn.
,
7
(
4
), pp.
145
152
.10.2514/3.62949
18.
Haddow
,
A. G.
,
Barr
,
A. D. S.
, and
Mook
,
A. B. D.
,
1984
, “
Theoretical and Experimental Study of Modal Interaction in a Two-Degree-of-Freedom Structure
,”
J. Sound Vib.
,
97
(
3
), pp.
451
473
.10.1016/0022-460X(84)90272-4
19.
Shaw
,
S. W.
, and
Balachandran
,
B.
,
2008
, “
A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008
,”
Jpn. Soc. Mech. Eng.
,
2
(
3
), pp.
611
640
.10.1299/jsdd.2.611
20.
Marcelo Tusset
,
A.
,
Piccirillo
,
V.
,
Bueno
,
A. M.
,
Manoel Balthazar
,
J.
,
Sado
,
D.
,
Felix
,
J. L. P.
, and
Brasil
,
R. M. L. R. D. F.
,
2016
, “
Chaos Control and Sensitivity Analysis of a Double Pendulum Arm Excited by an RLC Circuit Based Nonlinear Shaker
,”
J. Vib. Control
,
22
(
17
), pp.
3621
3637
.10.1177/1077546314564782
21.
Lee
,
Y.
,
Frank Pai
,
P.
, and
Feng
,
Z.
,
2008
, “
Nonlinear Complex Response of a Parametrically Excited Tuning Fork
,”
Mech. Syst. Signal Process.
,
22
(
5
), pp.
1146
1156
.10.1016/j.ymssp.2007.11.015
22.
Pai
,
P. F.
, and
Schulz
,
M. J.
,
2000
, “
A Refined Nonlinear Vibration Absorber
,”
Int. J. Mech. Sci.
,
42
(
3
), pp.
537
560
.10.1016/S0020-7403(98)00135-0
23.
Felix
,
J. L. P.
,
Balthazar
,
J. M.
, and
Brasil
,
R. M.
,
2005
, “
On Saturation Control of a Non-Ideal Vibrating Portal Frame Foundation Type Shear-Building
,”
J. Vib. Control
,
11
(
1
), pp.
121
136
.10.1177/1077546305047656
24.
Shoeybi
,
M.
, and
Ghorashi
,
M.
,
2005
, “
Control of a Nonlinear System Using the Saturation Phenomenon
,”
Nonlinear Dyn.
,
42
(
2
), pp.
113
136
.10.1007/s11071-005-4123-y
25.
Warminski
,
J.
,
Cartmell
,
M. P.
,
Mitura
,
A.
, and
Bochenski
,
M.
,
2013
, “
Active Vibration Control of a Nonlinear Beam With Self-and External Excitations
,”
Shock Vib.
,
20
(
6
), pp.
1033
1047
.10.1155/2013/792795
26.
Zhao, Y. Y., and Xu, J., 2012. “Using the Delayed Feedback Control and Saturation Control to Suppress the Vibration of the Dynamical System,”
Nonlinear Dyn.
, 67(1), pp. 735–753. 10.1007/s11071-011-0023-5
27.
Tusset
,
A. M.
,
Janzen
,
F. C.
,
Piccirillo
,
V.
,
Rocha
,
R. T.
,
Balthazar
,
J. M.
, and
Litak
,
G.
,
2018
, “
On Nonlinear Dynamics of a Parametrically Excited Pendulum Using Both Active Control and Passive Rotational (MR) Damper
,”
J. Vib. Control
,
24
(
9
), pp.
1587
1599
.10.1177/1077546317714882
28.
Oueini
,
S. S.
,
Nayfeh
,
A. H.
, and
Golnaraghi
,
M. F.
,
1997
, “
A Theoretical and Experimental Implementation of a Control Method Based on Saturation
,”
Nonlinear Dyn.
,
13
(
2
), pp.
189
202
.10.1023/A:1008207124935
29.
Pai
,
P. F.
,
Wen
,
B.
,
Naser
,
A. S.
, and
Schulz
,
M. J.
,
1998
, “
Structural Vibration Control Using Pzt Patches and Nonlinear Phenomena
,”
J. Sound Vib.
,
215
(
2
), pp.
273
296
.10.1006/jsvi.1998.1612
30.
Pratt
,
J. R.
,
Oueini
,
S. S.
, and
Nayfeh
,
A. H.
,
1999
, “
Terfenol-d Nonlinear Vibration Absorber
,”
J. Intell. Mater. Syst. Struct.
,
10
(
1
), pp.
29
35
.10.1177/1045389X9901000104
31.
Hall
,
B. D.
,
Mook
,
D. T.
,
Nayfeh
,
A. H.
, and
Preidikman
,
S.
,
2001
, “
Novel Strategy for Suppressing the Utter Oscillations of Aircraft Wings
,”
AIAA J.
,
39
(
10
), pp.
1843
1850
.10.2514/2.1190
32.
Ashour
,
O. N.
, and
Nayfeh
,
A. H.
,
2002
, “
Adaptive Control of Flexible Structures Using a Nonlinear Vibration Absorber
,”
Nonlinear Dyn.
,
28
(
3/4
), pp.
309
322
.10.1023/A:1015622630382
33.
Golnaraghi
,
M. F.
,
1991
, “
Vibration Suppression of Flexible Structures Using Internal Resonance
,”
Mech. Res. Commun.
,
18
(
2–3
), pp.
135
143
.10.1016/0093-6413(91)90042-U
34.
Nayfeh
,
A. H.
,
2000
,
Nonlinear Interactions: Analytical, Computational, and Experimental Methods
(Wiley Series in Nonlinear Science),
Wiley-Interscience
,
New York
.
35.
Mook
,
D. T.
,
Marshall
,
L. R.
, and
Nayfeh
,
A. H.
,
1974
, “
Subharmonic and Superharmonic Resonances in the Pitch and Roll Modes of Ship Motions
,”
J. Hydrodyn.
,
8
(
1
), pp.
32
40
.10.2514/3.62973
36.
Nayfeh
,
A. H.
,
1988
, “
Numerical-Perturbation Methods in Mechanics
,”
Comput. Struct.
,
30
(
1–2
), pp.
185
204
.10.1016/0045-7949(88)90226-X
37.
Balachandran
,
B.
, and
Nayfeh
,
A. H.
,
1990
, “
Nonlinear Motions of Beam-Mass Structure
,”
Nonlinear Dyn.
,
1
(
1
), pp.
39
61
.10.1007/BF01857584
38.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
2008
,
Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods
, John
Wiley and Sons
,
Weinheim, Germany
.
39.
Mook
,
D. T.
,
Plaut
,
R. H.
, and
HaQuang
,
N.
,
1985
, “
The Influence of an Internal Resonance on Nonlinear Structural Vibrations Under Subharmonic Resonance Conditions
,”
J. Sound Vib.
,
102
(
4
), pp.
473
492
.10.1016/S0022-460X(85)80108-5
40.
Alfredsson
,
K. S.
,
Josefson
,
B. L.
, and
Wilson
,
M. A.
,
1996
, “
Use of the Energy Flow Concept in Vibration Design
,”
AIAA J.
,
34
(
6
), pp.
1250
1255
.10.2514/3.13220
41.
Amin
,
T. M. F.
,
Huda
,
M. Q.
,
Tulip
,
J.
, and
Jäger
,
W.
,
2014
, “
A Virtual Pivot Point MEMS Actuator With Externally Mounted Mirror: Design, Fabrication and Characterization
,”
Sens. Transducers
,
183
(
12
), pp.
65
71
.https://search.proquest.com/docview/1645170155/fulltext/91960DA2B647CFPQ
42.
Felix
,
J. L. P.
,
Balthazar
,
J. M.
,
Rocha
,
R. T.
,
Tusset
,
A. M.
, and
Janzen
,
F. C.
,
2018
, “
On Vibration Mitigation and Energy Harvesting of a Non-Ideal System With Autoparametric Vibration Absorber System
,”
Meccanica
,
53
(
13
), pp.
3177
3188
.10.1007/s11012-018-0881-8
43.
Rocha
,
R. T.
,
Balthazar
,
J. M.
,
Tusset
,
A. M.
,
Piccirillo
,
V.
, and
Felix
,
J. L. P.
,
2017
, “
Nonlinear Piezoelectric Vibration Energy Harvesting From a Portal Frame With Two-to-One Internal Resonance
,”
Meccanica
,
52
(
11–12
), pp.
2583
2602
.10.1007/s11012-017-0633-1
44.
Rocha
,
R. T.
,
Haura Junior
,
R.
,
Lenz
,
W. B.
,
Ribeiro
,
M. A.
,
Tusset
,
A. M.
,
Balthazar
,
J. M.
, and
Jarzebowska
,
E.
,
2019
, “
Investigation of Energy Harvesting in a 2DOFs Portal Frame by Means of the Positioning of the Piezoelectric Material
,”
15th International Conference, Dynamical Systems—Theory and Applications
, Theoretical Approaches in Nonlinear Dynamical Systems, Lodz, Poland, Dec. 2–5, pp.
453
463
.
45.
Lee
,
W. K.
, and
Cho
,
D. S.
,
2000
, “
Damping Effect of a Randomly Excited Autoparametric System
,”
J. Sound Vib.
,
236
(
1
), pp.
23
31
.10.1006/jsvi.2000.2965
46.
Rocha
,
R. T.
,
Balthazar
,
J. M.
,
Tusset
,
A. M.
,
Piccirillo
,
V.
, and
Felix
,
J. L. P.
,
2016
, “
Comments on Energy Harvesting on a 2:1 Internal Resonance Portal Frame Support Structure, Using a Nonlinear-Energy Sink as a Passive Controller
,”
Int. Rev. Mech. Eng. (IREME)
,
10
(
3
), pp.
147
156
.10.15866/ireme.v10i3.8795
47.
Cveticanin
,
L.
, and
Zukovic
,
M.
,
2015
, “
Motion of a Motor-Structure Non-Ideal System
,”
Eur. J. Mech.-A/Solids
,
53
, pp.
229
240
.10.1016/j.euromechsol.2015.05.003
48.
Cveticanin
,
L.
, and
Zukovic
,
M.
,
2015
, “
Non-Ideal Mechanical System With an Oscillator With Rational Nonlinearity
,”
J. Vib. Control
,
21
(
11
), pp.
2149
2164
.10.1177/1077546313508297
49.
Preumont
,
A.
,
2006
,
Mechatronics: Dynamics of Electro-Mechanical and Piezoelectric Systems
, Vol.
136
,
Springer Science & Business Media
,
Berlin
.
50.
Younis
,
M. I.
,
2011
, “
MEMS Linear and Nonlinear Statics and Dynamics
,”
Microsystems
,
Springer US
,
Boston, MA
.
51.
Thompson
,
R. L.
, and
Soares
,
E. J.
,
2016
, “
Viscoplastic Dimensionless Numbers
,”
J. Non-Newtonian Fluid Mech.
,
238
, pp.
57
64
.10.1016/j.jnnfm.2016.05.001
52.
Bernardini
,
D.
, and
Litak
,
G.
,
2016
, “
An Overview of 0–1 Test for Chaos
,”
J. Braz. Soc. Mech. Sci. Eng.
,
38
(
5
), pp.
1433
1450
.10.1007/s40430-015-0453-y
53.
Bernardini
,
D.
,
Rega
,
G.
,
Litak
,
G.
, and
Syta
,
A.
,
2013
, “
Identification of Regular and Chaotic Isothermal Trajectories of a Shape Memory Oscillator Using the 0–1 Test
,”
Proc. Inst. Mech. Eng. Part K
,
227
(
1
), pp.
17
22
.10.1177/1464419312447498
54.
Litak
,
G.
,
Syta
,
A.
, and
Wiercigroch
,
M.
,
2009
, “
Identification of Chaos in a Cutting Process by the 0–1 Test
,”
Chaos Solitons Fractals
,
40
(
5
), pp.
2095
2101
.10.1016/j.chaos.2007.09.093
55.
Sun
,
K.
,
Liu
,
X.
, and
Zhu
,
C.
,
2010
, “
The 0-1 Test Algorithm for Chaos and Its Applications
,”
Chin Phys B
,
19
(
11
), p.
110510
.10.1088/1674-1056/19/11/110510
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