Abstract

Vibration of micro-electromechanical systems (MEMS) is of growing interest for applications in vibration measurements, vibration energy harvesters, and vibration sensors. Among the structures and devices reported, a class of 3D structures formed by mechanically guided assembly is showing promising potentials, owing to the ability of controlled vibration behaviors (e.g., modes and natural frequencies) by reversibly changing the compressive strain. In addition, serpentine structures are good candidates for MEMS due to their ultra-low natural frequencies. Hence, we present a study on the vibration of the mechanically assembled 3D serpentine structures. A theoretical model is established to capture the vibration mechanism, and therefore, a simple analytical expression for the natural frequency is derived. On this basis, the influence of material/geometry parameters on the natural frequency is systematically discussed. The developed analytical model would provide a better understanding of vibration mechanism in mechanically assembled 3D structures

References

1.
Tseng
,
W. Y.
, and
Dugundji
,
J.
,
1971
, “
Nonlinear Vibrations of a Beam Under Harmonic Excitation
,”
ASME J. Appl. Mech.
,
38
(
2
), pp.
467
476
.
2.
Min
,
G. B.
, and
Eisley
,
J. G.
,
1972
, “
Nonlinear Vibration of Buckled Beams
,”
ASME J. Eng. Ind.
,
94
(
2
), pp.
637
645
.
3.
Gil-Santos
,
E.
,
Ramos
,
D.
,
Martinez
,
J.
,
Fernandez-Regulez
,
M.
,
Garcia
,
R.
,
San Paulo
,
A.
,
Calleja
,
M.
, and
Tamayo
,
J.
,
2010
, “
Nanomechanical Mass Sensing and Stiffness Spectrometry Based on Two-Dimensional Vibrations of Resonant Nanowires
,”
Nat. Nanotechnol.
,
5
(
9
), pp.
641
645
.
4.
Wang
,
H.
,
Ning
,
X.
,
Li
,
H.
,
Luan
,
H.
,
Xue
,
Y.
,
Yu
,
X.
,
Fan
,
Z.
, et al
,
2018
, “
Vibration of Mechanically-Assembled 3D Microstructures Formed by Compressive Buckling
,”
J. Mech. Phys. Solids
,
112
, pp.
187
208
.
5.
Han
,
M.
,
Wang
,
H.
,
Yang
,
Y.
,
Liang
,
C.
,
Bai
,
W.
,
Yan
,
Z.
,
Li
,
H.
, et al
,
2019
, “
Three-Dimensional Piezoelectric Polymer Microsystems for Vibrational Energy Harvesting, Robotic Interfaces and Biomedical Implants
,”
Nat. Electron.
,
2
(
1
), pp.
26
35
.
6.
Shui
,
L.
,
Jia
,
L.
,
Li
,
H.
,
Guo
,
J.
,
Guo
,
Z.
,
Liu
,
Y.
,
Liu
,
Z.
, and
Chen
,
X.
,
2020
, “
Rapid and Continuous Regulating Adhesion Strength by Mechanical Micro-Vibration
,”
Nat. Commun.
,
11
(
1
), p.
1583
.
7.
Yan
,
G.
,
Zou
,
H.-X.
,
Wang
,
S.
,
Zhao
,
L.-C.
,
Wu
,
Z.-Y.
, and
Zhang
,
W.-M.
,
2021
, “
Bio-Inspired Vibration Isolation: Methodology and Design
,”
ASME Appl. Mech. Rev.
,
73
(
2
), p.
020801
.
8.
Zhao
,
J.
,
Li
,
W.
,
Guo
,
X.
,
Wang
,
H.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2020
, “
Theoretical Modeling of Tunable Vibrations of Three-Dimensional Serpentine Structures for Simultaneous Measurement of Adherent Cell Mass and Modulus
,”
MRS Bull.
,
46
(
1
), pp.
107
114
.
9.
Park
,
K.
,
Millet
,
L. J.
,
Kim
,
N.
,
Li
,
H.
,
Jin
,
X.
,
Popescu
,
G.
,
Aluru
,
N. R.
,
Hsia
,
K. J.
, and
Bashir
,
R.
,
2010
, “
Measurement of Adherent Cell Mass and Growth
,”
Proc. Natl. Acad. Sci. USA
,
107
(
48
), pp.
20691
20696
.
10.
Martinez-Martin
,
D.
,
Flaschner
,
G.
,
Gaub
,
B.
,
Martin
,
S.
,
Newton
,
R.
,
Beerli
,
C.
,
Mercer
,
J.
,
Gerber
,
C.
, and
Muller
,
D. J.
,
2017
, “
Inertial Picobalance Reveals Fast Mass Fluctuations in Mammalian Cells
,”
Nature
,
550
(
7677
), pp.
500
505
.
11.
Cermak
,
N.
,
Olcum
,
S.
,
Delgado
,
F. F.
,
Wasserman
,
S. C.
,
Payer
,
K. R.
,
Murakami
,
M.
,
Knudsen
,
S. M.
, et al
,
2016
, “
High-Throughput Measurement of Single-Cell Growth Rates Using Serial Microfluidic Mass Sensor Arrays
,”
Nat. Biotechnol.
,
34
(
10
), pp.
1052
1059
.
12.
Kang
,
J. H.
,
Miettinen
,
T. P.
,
Chen
,
L.
,
Olcum
,
S.
,
Katsikis
,
G.
,
Doyle
,
P. S.
, and
Manalis
,
S. R.
,
2019
, “
Noninvasive Monitoring of Single-Cell Mechanics by Acoustic Scattering
,”
Nat. Methods
,
16
(
3
), pp.
263
269
.
13.
Gross
,
W.
, and
Kress
,
H.
,
2017
, “
Simultaneous Measurement of the Young's Modulus and the Poisson Ratio of Thin Elastic Layers
,”
Soft Matter
,
13
(
5
), pp.
1048
1055
.
14.
Ning
,
X.
,
Wang
,
H.
,
Yu
,
X.
,
Soares
,
J.
,
Yan
,
Z.
,
Nan
,
K.
,
Velarde
,
G.
, et al
,
2017
, “
Three-Dimensional Multiscale, Multistable, and Geometrically Diverse Microstructures With Tunable Vibrational Dynamics Assembled by Compressive Buckling
,”
Adv. Funct. Mater.
,
27
(
14
), p.
1605914
.
15.
Wang
,
H.
,
Wei
,
C.
,
Zhang
,
Y.
,
Ma
,
Y.
,
Chen
,
Y.
,
Wang
,
H.
, and
Feng
,
X.
,
2021
, “
Tunable Three-Dimensional Vibrational Structures for Concurrent Determination of Thin Film Modulus and Density
,”
ASME J. Appl. Mech.
,
89
(
3
), p.
031009
.
16.
Park
,
S.
,
Kim
,
H.
,
Vosgueritchian
,
M.
,
Cheon
,
S.
,
Kim
,
H.
,
Koo
,
J. H.
,
Kim
,
T. R.
, et al
,
2014
, “
Stretchable Energy-Harvesting Tactile Electronic Skin Capable of Differentiating Multiple Mechanical Stimuli Modes
,”
Adv. Mater.
,
26
(
43
), pp.
7324
7332
.
17.
Zhao
,
Y.
,
Gao
,
S.
,
Zhang
,
X.
,
Huo
,
W.
,
Xu
,
H.
,
Chen
,
C.
,
Li
,
J.
,
Xu
,
K.
, and
Huang
,
X.
,
2020
, “
Fully Flexible Electromagnetic Vibration Sensors With Annular Field Confinement Origami Magnetic Membranes
,”
Adv. Funct. Mater.
,
30
(
25
), p.
2001553
.
18.
Todd
,
N. P. M.
,
Rosengren
,
S. M.
, and
Colebatch
,
J. G.
,
2008
, “
Tuning and Sensitivity of the Human Vestibular System to Low-Frequency Vibration
,”
Neurosci. Lett.
,
444
(
1
), pp.
36
41
.
19.
Guo
,
X.
,
Avila
,
R.
,
Huang
,
Y.
, and
Xie
,
Z.
,
2021
, “
Flexible Electronics With Dynamic Interfaces for Biomedical Monitoring, Stimulation, and Characterization
,”
Int. J. Mech. System Dyn.
,
1
(
1
), pp.
52
70
.
20.
Xu
,
S.
,
Yan
,
Z.
,
Jang
,
K.-I.
,
Huang
,
W.
,
Fu
,
H.
,
Kim
,
J.
,
Wei
,
Z.
, et al
,
2015
, “
Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling
,”
Science
,
347
(
6218
), pp.
154
159
.
21.
Zhang
,
Y.
,
Yan
,
Z.
,
Nan
,
K.
,
Xiao
,
D.
,
Liu
,
Y.
,
Luan
,
H.
,
Fu
,
H.
, et al
,
2015
, “
A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes
,”
Proc. Natl. Acad. Sci. USA
,
112
(
38
), pp.
11757
11764
.
22.
Zhang
,
Y.
,
Zhang
,
F.
,
Yan
,
Z.
,
Ma
,
Q.
,
Li
,
X.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2017
, “
Printing, Folding and Assembly Methods for Forming 3D Mesostructures in Advanced Materials
,”
Nat. Rev. Mater.
,
2
(
4
), p.
17029
.
23.
Luo
,
G.
,
Fu
,
H.
,
Cheng
,
X.
,
Bai
,
K.
,
Shi
,
L.
,
He
,
X.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2019
, “
Mechanics of Bistable Cross-Shaped Structures Through Loading-Path Controlled 3D Assembly
,”
J. Mech. Phys. Solids
,
129
, pp.
261
277
.
24.
Pang
,
W.
,
Cheng
,
X.
,
Zhao
,
H.
,
Guo
,
X.
,
Ji
,
Z.
,
Li
,
G.
,
Liang
,
Y.
, et al
,
2020
, “
Electro-Mechanically Controlled Assembly of Reconfigurable 3D Mesostructures and Electronic Devices Based on Dielectric Elastomer Platforms
,”
Natl. Sci. Rev.
,
7
(
2
), pp.
342
354
.
25.
Xue
,
Z.
,
Jin
,
T.
,
Xu
,
S.
,
Bai
,
K.
,
He
,
Q.
,
Zhang
,
F.
,
Cheng
,
X.
, et al
,
2022
, “
Assembly of Complex 3D Structures and Electronics on Curved Surfaces
,”
Sci. Adv.
,
8
(
32
), p.
eabm6922
.
26.
Shuai
,
Y.
,
Zhao
,
J.
,
Bo
,
R.
,
Lan
,
Y.
,
Lv
,
Z.
, and
Zhang
,
Y.
,
2023
, “
A Wrinkling-Assisted Strategy for Controlled Interface Delamination in Mechanically-Guided 3D Assembly
,”
J. Mech. Phys. Solids
,
173
, p.
105203
.
27.
Fan
,
Z.
,
Hwang
,
K.-C.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2018
, “
A Double Perturbation Method of Postbuckling Analysis in 2D Curved Beams for Assembly of 3D Ribbon-Shaped Structures
,”
J. Mech. Phys. Solids
,
111
, pp.
215
238
.
28.
Fu
,
H.
,
Nan
,
K.
,
Bai
,
W.
,
Huang
,
W.
,
Bai
,
K.
,
Lu
,
L.
,
Zhou
,
C.
, et al
,
2018
, “
Morphable 3D Mesostructures and Microelectronic Devices by Multistable Buckling Mechanics
,”
Nat. Mater.
,
17
(
3
), pp.
268
276
.
29.
Ma
,
Q.
,
Cheng
,
H.
,
Jang
,
K.-I.
,
Luan
,
H.
,
Hwang
,
K.-C.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2016
, “
A Nonlinear Mechanics Model of Bio-Inspired Hierarchical Lattice Materials Consisting of Horseshoe Microstructures
,”
J. Mech. Phys. Solids
,
90
, pp.
179
202
.
30.
Yan
,
Z.
,
Zhang
,
F.
,
Liu
,
F.
,
Han
,
M.
,
Ou
,
D.
,
Liu
,
Y.
,
Lin
,
Q.
, et al
,
2016
, “
Mechanical Assembly of Complex, 3D Mesostructures From Releasable Multilayers of Advanced Materials
,”
Sci. Adv.
,
2
(
9
), p.
e1601014
.
31.
Cheng
,
X.
, and
Zhang
,
Y.
,
2019
, “
Micro/Nanoscale 3D Assembly by Rolling, Folding, Curving, and Buckling Approaches
,”
Adv. Mater.
,
31
(
36
), p.
e1901895
.
32.
Liu
,
Y.
,
Wang
,
X.
,
Xu
,
Y.
,
Xue
,
Z.
,
Zhang
,
Y.
,
Ning
,
X.
,
Cheng
,
X.
, et al
,
2019
, “
Harnessing the Interface Mechanics of Hard Films and Soft Substrates for 3D Assembly by Controlled Buckling
,”
Proc. Natl. Acad. Sci. USA
,
116
(
31
), pp.
15368
15377
.
33.
Fan
,
Z.
,
Yang
,
Y.
,
Zhang
,
F.
,
Xu
,
Z.
,
Zhao
,
H.
,
Wang
,
T.
,
Song
,
H.
,
Huang
,
Y.
,
Rogers
,
J. A.
, and
Zhang
,
Y.
,
2020
, “
Inverse Design Strategies for 3D Surfaces Formed by Mechanically Guided Assembly
,”
Adv. Mater.
,
32
(
14
), p.
e1908424
.
34.
Kim
,
B. H.
,
Li
,
K.
,
Kim
,
J.-T.
,
Park
,
Y.
,
Jang
,
H.
,
Wang
,
X.
,
Xie
,
Z.
, et al
,
2021
, “
Three-Dimensional Electronic Microfliers Inspired by Wind-Dispersed Seeds
,”
Nature
,
597
(
7877
), pp.
503
510
.
35.
Ehsani
,
H.
,
Boyd
,
J. D.
,
Wang
,
J.
, and
Grady
,
M. E.
,
2021
, “
Evolution of the Laser-Induced Spallation Technique in Film Adhesion Measurement
,”
ASME Appl. Mech. Rev.
,
73
(
3
), p.
030802
.
36.
Firooz
,
S.
,
Steinmann
,
P.
, and
Javili
,
A.
,
2021
, “
Homogenization of Composites With Extended General Interfaces: Comprehensive Review and Unified Modeling
,”
ASME Appl. Mech. Rev.
,
73
(
4
), p.
040802
.
37.
Yu
,
H.-H.
, and
Hutchinson
,
J. W.
,
2002
, “
Influence of Substrate Compliance on Buckling Delamination of Thin Films
,”
Int. J. Fract.
,
113
(
1
), pp.
39
55
.
38.
Vella
,
D.
,
Bico
,
J.
,
Boudaoud
,
A.
,
Roman
,
B.
, and
Reis
,
P. M.
,
2009
, “
The Macroscopic Delamination of Thin Films From Elastic Substrates
,”
Proc. Natl. Acad. Sci. USA
,
106
(
27
), pp.
10901
10906
.
39.
Mei
,
H.
,
Landis
,
C. M.
, and
Huang
,
R.
,
2011
, “
Concomitant Wrinkling and Buckle-Delamination of Elastic Thin Films on Compliant Substrates
,”
Mech. Mater.
,
43
(
11
), pp.
627
642
.
40.
Pan
,
K.
,
Ni
,
Y.
,
He
,
L.
, and
Huang
,
R.
,
2014
, “
Nonlinear Analysis of Compressed Elastic Thin Films on Elastic Substrates: From Wrinkling to Buckle-Delamination
,”
Int. J. Solids Struct.
,
51
(
21
), pp.
3715
3726
.
41.
Zhang
,
Q.
, and
Yin
,
J.
,
2018
, “
Spontaneous Buckling-Driven Periodic Delamination of Thin Films on Soft Substrates Under Large Compression
,”
J. Mech. Phys. Solids
,
118
, pp.
40
57
.
42.
Emam
,
S. A.
,
2013
, “
Approximate Analytical Solutions for the Nonlinear Free Vibrations of Composite Beams in Buckling
,”
Compos. Struct.
,
100
, pp.
186
194
.
43.
Emam
,
S. A.
, and
Nayfeh
,
A. H.
,
2004
, “
On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation
,”
Nonlinear Dyn.
,
35
(
1
), pp.
1
17
.
44.
Faghih Shojaei
,
M.
,
Ansari
,
R.
,
Mohammadi
,
V.
, and
Rouhi
,
H.
,
2014
, “
Nonlinear Forced Vibration Analysis of Postbuckled Beams
,”
Arch. Appl. Mech.
,
84
(
3
), pp.
421
440
.
45.
Lacarbonara
,
W.
,
Nayfeh
,
A. H.
, and
Kreider
,
W.
,
1998
, “
Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam
,”
Nonlinear Dyn.
,
17
(
2
), pp.
95
117
.
46.
Lestari
,
W.
, and
Hanagud
,
S.
,
2001
, “
Nonlinear Vibration of Buckled Beams: Some Exact Solutions
,”
Int. J. Solids Struct.
,
38
(
26–27
), pp.
4741
4757
.
47.
Nayfeh
,
A. H.
,
Kreider
,
W.
, and
Anderson
,
T. J.
,
1995
, “
Investigation of Natural Frequencies and Mode Shapes of Buckled Beams
,”
AIAA J.
,
33
(
6
), pp.
1121
1126
.
48.
Noijen
,
S. P. M.
,
Mallon
,
N. J.
,
Fey
,
R. H. B.
,
Nijmeijer
,
H.
, and
Zhang
,
G. Q.
,
2007
, “
Periodic Excitation of a Buckled Beam Using a Higher Order Semianalytic Approach
,”
Nonlinear Dyn.
,
50
(
1–2
), pp.
325
339
.
49.
Nayfeh
,
A. H.
, and
Emam
,
S. A.
,
2008
, “
Exact Solution and Stability of Postbuckling Configurations of Beams
,”
Nonlinear Dyn.
,
54
(
4
), pp.
395
408
.
50.
Li
,
S.
,
Han
,
M.
,
Rogers
,
J. A.
,
Zhang
,
Y.
,
Huang
,
Y.
, and
Wang
,
H.
,
2019
, “
Mechanics of Buckled Serpentine Structures Formed via Mechanics-Guided, Deterministic Three-Dimensional Assembly
,”
J. Mech. Phys. Solids
,
125
, pp.
736
748
.
51.
Zhao
,
J.
,
Zhang
,
F.
,
Guo
,
X.
,
Huang
,
Y.
,
Zhang
,
Y.
, and
Wang
,
H.
,
2021
, “
Torsional Deformation Dominated Buckling of Serpentine Structures to Form Three-Dimensional Architectures With Ultra-Low Rigidity
,”
J. Mech. Phys. Solids
,
155
, p.
104568
.
52.
Xu
,
S.
,
Zhang
,
Y.
,
Cho
,
J.
,
Lee
,
J.
,
Huang
,
X.
,
Jia
,
L.
,
Fan
,
J. A.
, et al
,
2013
, “
Stretchable Batteries With Self-Similar Serpentine Interconnects and Integrated Wireless Recharging Systems
,”
Nat. Commun.
,
4
(
1
), p.
1543
.
53.
Zhang
,
Y.
,
Xu
,
S.
,
Fu
,
H.
,
Lee
,
J.
,
Su
,
J.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2013
, “
Buckling in Serpentine Microstructures and Applications in Elastomer-Supported Ultra-Stretchable Electronics With High Areal Coverage
,”
Soft Matter
,
9
(
33
), pp.
8062
8070
.
You do not currently have access to this content.