This paper proposes a novel design for a flow-induced vibration-based energy harvester, consisting of an elastic L-shaped beam, with an inherent nonlinearity in its structural stiffness as an alternative to the classical cantilever beam used in conventional fluidic energy harvester designs. The L-shaped beam supports a prism at its tip and undergoes large-amplitude galloping oscillations. The results from wind tunnel experiments show that by replacing a conventional linear structure that supports the prism with a nonlinear one, the high frequency flow components, shed from the tip prism, were capable of exciting the oscillations of the structure at higher harmonics of the main resonance, thus enhancing the power density of the energy harvester. As a result of improved power density values, the proposed harvester design holds great potential to be used as advanced space-efficient energy harvesters.

When a flexible or flexibly mounted structure is placed in fluid flow, it can oscillate due to flow forces. This oscillation is called Flow-Induced Vibration (FIV). If a flexibly mounted body with a circular cross section is placed in fluid flow, when the frequency of vortex shedding locks in with the system’s natural frequency as a result of an increase in the flow velocity, a region of relatively large-amplitude oscillations, known as the lock-in region, is observed. This is called Vortex-Induced Vibration (VIV). Vortex-induced vibration of a cylinder placed in flow is a canonical problem in fluid–structure interaction studies due to the fundamental questions associated with it, as well as the wide range of industrial applications, such as wind or marine energy harvesting. Work by several investigators has helped to elucidate the fundamentals of VIV.1–8 

A cylinder allowed to oscillate perpendicular to the flow direction constitutes a geometrically symmetric system. Several studies exist on flow-induced vibration of bluff bodies where the structure’s symmetry is broken.9–16 Once the structure is asymmetric, its orientation with respect to the direction of the incoming flow plays a major role in the response of the system. In the case of asymmetric structures placed in flow, besides VIV-type responses, galloping-type responses are observed. Galloping is the term normally used for flow-induced oscillations with a large amplitude and a low frequency,17,18 where “low” is in comparison with the shedding frequency. Prisms with square, rectangular, triangular, or D-shaped cross sections are among the structures prone to galloping instability.19–24 Interactions between the flow forces and the solid body of the prism have been widely studied based on both experimental methods and numerical simulations,10,11,25–31 where it has been shown that depending on the angle of attack, the prism can undergo VIV, galloping, or combination of both. A recent experimental study11 on the FIV of a triangular prism has shown that at angles of attack where a VIV response was observed, the shedding frequency was synchronized with the oscillation frequency, and there existed a 1:1 synchronization between the flow forces and the oscillation frequencies. At angles of attack where the VIV response switched to a galloping response, the 1:1 synchronization was replaced by combined 1:2 and 1:3 synchronizations. The 1:2 and 1:3 synchronizations imply the existence of high frequency vortex shedding in the wake of the prism, resulting in flow force frequencies that are twice and thrice the oscillation frequency, while the oscillations still occur at the main harmonic frequency. Through this study, it was shown that the high frequency vortex shedding patterns match the dominant frequency contents of the flow forces measured during the galloping response.

In studies of flow-induced vibration and, in particular, FIV-based energy harvesting applications, the structure that provides stiffness in the system is typically modeled as either a linear mechanical spring or a weakly nonlinear cantilever beam. The most common fluidic energy harvester consists of a piezoelectric cantilever beam, clamped at one end, connected to a rigid prism with a specific cross section at the free end, placed in flow.21,22,32–39 Large-amplitude oscillations occur when the vortex formation frequency in the wake of the prism is relatively close to the beam’s modal frequencies (usually, the first mode). The oscillations cause large strain near the clamped end. The strain produces a voltage difference in the piezoelectric patches, and a circuit converts the electric potential to a current. In such FIV studies of prismatic structures, despite the existence of high frequency flow forces, the linear structure only oscillates at its main harmonic, close to the natural frequency of the system.

There are, however, a whole class of nonlinear structures in which higher harmonics of the main frequency can be excited in addition to the primary resonance.40,41 This, in particular, becomes very important in FIV-based energy harvesters, where the harvester can use the higher harmonic flow forces available from the vortices that are shed from the bluff body and translate them into large-amplitude oscillations, which in return can improve the efficiency of the harvester. Nonlinearities can inherently exist in a harvester due to its geometric or material properties. Usually, they arise from the nonlinear strain due to large deformations.41,42 The key feature of a nonlinear structure is that the system’s natural frequency is no longer constant, and the frequency becomes a function of the vibration amplitude. Structural nonlinearities can extend the coupling between the excitation and a harmonic oscillator to a wider range of frequencies.43–48 This has prompted many researchers to exploit them to enhance harvester operation bandwidth.49–56 

However, recent studies on nonlinear fluidic energy harvesters have only focused on the response of the system around the main resonance;51,57–59 the effects of higher harmonic resonances and how they can extend the frequency bandwidth of the harvesters have not been investigated. In the present study, we introduce a flexible L-shaped harvester as an alternative to the classical cantilever beam used in FIV-based energy harvester designs. Studies on the nonlinear dynamics of a flexible L-shaped beam60–63 have shown that with proper geometric parameter selection, the beam can be tuned to have its first two bending mode natural frequencies with a ratio of 2:1 or 3:1. For an L-shaped beam, due to the presence of quadratic and cubic nonlinearities, energy exchange between the first two structural modes of vibration may occur,60,64 where the large-amplitude oscillations at the first bending mode can be excited by high frequency flow forces originating from high frequency vortex shedding. This is important in energy harvesting applications as the oscillations in the fundamental mode have much higher contributions to power generation compared to oscillations at other higher modes of the system.63,64 Based on this unique feature of L-shaped beams, in this present work, we will investigate the possibility of improving the energy extraction efficiency of the fluidic energy harvesters using an inherently nonlinear L-shaped beam in the structural design of the harvester.

In this study, we performed a series of experiments with the goal of observing high frequency flow-induced vibration of a square prism, supported by nonlinear springs, which have the form of an L-shaped beam. A schematic of the setup is shown in Fig. 1, which consists of an L-shaped beam, clamped at the base, supporting a square cross-section prism at its tip such that one side of the prism was perpendicular to the incoming flow. The beam was made out of stainless steel, with dimensions of L1 = 100.2 mm, L2 = 103.4 mm, B = 25.6 mm, and a thickness of 0.6 mm. The square cross-section tip prism had a side width of a = 19 mm and height of H = 127 mm. The tip prism mass was 18.3 g. With the selected geometrical parameters of the system, the ratio between the first two bending mode natural frequencies was 3:1. The bending mode frequency values were measured by exciting the L-shaped beam from its base with a sine sweep input, generated by using an electromagnetic LDS® shaker. The L-shaped beam was placed in the test section of a subsonic wind tunnel with a test section of 0.5 m × 0.38 m × 1.27 m and a turbulence intensity of less than 2% for up to a velocity of 15 m/s. The flow velocity was increased from zero, in small steps, to a maximum of U = 13.30 m/s, which corresponds to a Reynolds number of Re = 17 038. When the flow velocity, U, exceeded a critical value, the beam underwent steady-state oscillations. At each step, the steady-state tip displacement of the beams was measured by using a laser displacement sensor (Panasonic HL-G125) running in tandem with a series of piezoelectric transducer patches (FLDT1-028K) attached to the beam. Four piezoelectric patches were evenly distributed and fully covered the full length of the beam. At each step, the steady-state voltage produced in the piezoelectric patches was measured. The measurements were conducted with the sampling frequency of 1 kHz and frequency resolution of 0.03 Hz. The data from the series of four piezoelectric sensors attached to the beam were used to study how contributions of high frequency oscillations vary at different locations along the beam. This information is essential for informed placement of piezoelectric patches along the beam for optimal energy extraction. A high-speed camera (Flare 2 MP), placed below the test section of the wind tunnel, was used to capture the motion of the beam during oscillations. The setup enabled recording of videos at 800 frames per second at 2048 × 350 pixel resolution. The recordings were used to extract the orientation of the square prism with respect to the direction of the incoming flow (angle of attack) during oscillations. The prism’s displacement was measured in the wind speed range of U = 0.08 m/s–13.30 m/s, which corresponds to a reduced velocity range of U* = 0.55–89.53, where the reduced velocity is defined as U* = U/fna, with U, fn, and a being the air velocity, the first bending mode natural frequency of the system, and the width of the prism in the direction perpendicular to the flow, respectively. Decay tests were conducted by giving the tip of the beam an initial displacement and then recording the amplitude of oscillations for a minimum of 50 cycles of oscillations. Decay tests yielded in a first bending mode natural frequency of fn = 7.8 Hz and a structural damping ratio of ζ = 0.002. Damping ratio was measured before and after the piezoelectric patches were attached to the beam, and the results showed no significant change in the measured damping ratio values. For comparison, a cantilever clamped beam with the same material and width was considered, as well. The cantilever beam had the length of L = 168.3 mm selected such that both the L-shaped and cantilever beams have the same first bending mode natural frequency of fn = 7.8 Hz. Similar piezoelectric transducer patches as those used in the L-shaped beam were used to measure the FIV response of the cantilever beam over the range of flow velocities tested. The piezoelectric transducers were evenly distributed and fully covered the entire length of the cantilever beam.

FIG. 1.

Schematics of the proposed L-shaped energy harvester. The square cross-section prism held at the tip of the beam serves as a high frequency vortex generator as it undergoes galloping oscillations.

FIG. 1.

Schematics of the proposed L-shaped energy harvester. The square cross-section prism held at the tip of the beam serves as a high frequency vortex generator as it undergoes galloping oscillations.

Close modal

Figure 2 shows the normalized time histories and their corresponding frequency contents (FFT plots) of the voltage measurements at piezoelectric patches for the L-shaped beam compared to those of the cantilever beam as the beams experienced flow-induced vibration at a sample flow velocity of U = 7.1 m/s. The measurements are readings from two piezoelectric patches at two points along the length of the beams: one close to the clamped end and the other close to the free end of the beam (piezoelectric patches No. 1 and No. 4 in Fig. 1). The measured voltage values are normalized by the signal’s peak values at each sensor’s location. The frequency plots for both the L-shaped and cantilever beams exhibit distinct frequency contents: while the frequency content mainly consists of a main peak at a frequency close to the first bending mode natural frequency of the structure in both systems (fn = 7.8 Hz), for the L-shaped beam, contributions from the second and third harmonics are dominantly observed at this sample velocity [Figs. 2(f) and 2(h)]. High frequency flow forces at odd integers of the main harmonics have been previously observed in the galloping of an elastically mounted square prism.10,28 The present measurements are the evidence of the high harmonic oscillations existing in the flow-induced vibration response of a square prism supported by a nonlinear structure.

FIG. 2.

Normalized time histories and their corresponding frequency contents (FFT plots) of the flow-induced vibration response for the cantilever beam [(a)–(d)] compared to the L-shaped beam [(e)–(h)] at a sample wind speed of U = 7.1 m/s.

FIG. 2.

Normalized time histories and their corresponding frequency contents (FFT plots) of the flow-induced vibration response for the cantilever beam [(a)–(d)] compared to the L-shaped beam [(e)–(h)] at a sample wind speed of U = 7.1 m/s.

Close modal

To study how the frequency content of the FIV response changes with respect to the reduced velocity for both the L-shaped and cantilever beams, we consider relative contributions of the higher harmonic frequency components compared to the main frequency. Figure 3 shows the normalized amplitudes of the voltage recorded in the piezoelectric patches at the first three harmonics over the range of reduced flow velocities tested. For each piezoelectric patch, frequency contributions are calculated by averaging the voltage concentrated around each harmonic in a frequency bandwidth of ±0.5 Hz of that harmonic. Voltage values are normalized by the maximum value of the voltage recorded at the main harmonics of the corresponding piezoelectric patch within the range of reduced velocities tested. The results show that while the main frequency contributions are similar for both the L-shaped and cantilever beams, the higher harmonic contributions (second and third harmonics) are much larger for the L-shaped beam compared to the cantilever beam at every piezoelectric patch location, over the range of flow velocities tested. In addition, contributions from the higher harmonics of the main frequency attenuated as the flow velocity was increased, regardless of the location of the piezoelectric patch along the beam. Contributions from higher harmonics become more apparent at the location of piezoelectric patches close to the free end of the beams.

FIG. 3.

Normalized amplitudes of the voltage recorded in the piezoelectric patches (Piezo No. 1 close to the fixed end and Piezo No. 4 close to the free end of the beam) at the first three frequency harmonics (H1H3) vs reduced flow velocity.

FIG. 3.

Normalized amplitudes of the voltage recorded in the piezoelectric patches (Piezo No. 1 close to the fixed end and Piezo No. 4 close to the free end of the beam) at the first three frequency harmonics (H1H3) vs reduced flow velocity.

Close modal

The harvester’s performance is quantified through the efficiency of converting flow energy to strain energy of the harvester during vibrations. In this study, power density values have been calculated as a metric to evaluate the flow-to-structural energy conversion efficiency of the nonlinear L-shaped beam. The power density is defined as output power per unit volume of the entire harvester. To evaluate the performance of the nonlinear harvester, the power density values for the L-shaped beam are compared to those of the conventional linear cantilever harvester. The power density, in general, for continuous structures, such as the elastic beams used in this research, is calculated as65,66

PB12mpω0Lqtipϕtipϕ(x)ω2dxVolume,
(1)

where mp is the mass per unit length of the beam, ω is the system’s oscillation frequency in radians per second, L is the beam’s length, and qtip is the oscillation amplitude of tip displacement of the beam, measured experimentally through a laser displacement sensor. ϕ is the mass-normalized mode shape, and ϕtip represents the value of the mass-normalized mode shape at the tip of the beam. Shape functions available for the L-shaped beam are used in this study to approximate the mass-normalized mode shape of the L-shaped beam in its first mode of vibration.67 The mass-normalized mode shapes of the conventional cantilever beam are available in the literature.42,68 The volume considered in power density calculations is the total volume swept by the harvester, which is defined as the product of frontal area swept by the prism as it vibrates and multiplied by the length of the beam perpendicular to the frontal swept area. The volume is calculated as Volume = 2HLqtip, where H is the prism’s height, L′ is the length of the beam perpendicular to the swept area, and qtip is the oscillation amplitude of tip displacement of the beam, measured experimentally through a laser displacement sensor.

Figure 4 shows the power density values calculated for the L-shaped beam, compared to those of the cantilever beam. It is shown that the power density values calculated at each reduced velocity for the L-shaped beam surpass those of the cantilever beam during the entire range of reduced velocities tested. Attenuated power density values of the L-shaped beam are expressed in terms of percentage of improvement in the power density, calculated as

%Improvement=(PBLPBC)/PBC,
(2)

where PBL and PBC are the power density values calculated for the L-shaped and cantilever beams, respectively. Figure 4 (right vertical axis) shows augmented power density values for the L-shaped beam compared to the cantilever beam, over the entire reduced velocities tested. The improvement percentage starts at values close to 40% at a low reduced velocity of U* ≈ 10. As the flow velocity increases, the improvement percentage gradually decreases to values around 20% in the reduced velocity range of U* ≈ 10–40, and it stays around the same values for the reduced velocity ranges of U* ≈ 40–80. At higher reduced velocities above U* ≈ 80, the improvement percentage gradually increases to values close to 35%. It appears that at high reduced velocity values above U* ≈ 80, the power density values calculated for the cantilever beam reach a plateau value, while those of the L-shaped beam keep gradually increasing. The observed increase in power density values for the L-shaped beam in this range of reduced velocities corresponds well with the gradual increase in the percentage of improvement in power density from values around 20% at U* ≈ 80 to 35% at the maximum reduced velocity tested. Studying angles of attack (α) for both the cantilever and L-shaped beams using high-speed camera recordings shows that as the reduced velocity increases, the angle of attack at the local maxima of oscillation gradually increases for both the cantilever and L-shaped beams. The increase in angle of attack happens at a slower rate for the L-shaped beam. For example, at a sample reduced velocity of U* = 89.53, the angle of attack has the value of α = 27° for the cantilever beam and α = 20° for the L-shaped beam at the local maxima of the oscillations. Larger angles of attack, which occur in FIV of cantilever beam, mean that the prism is moving far away from the configuration where the flat face of the square prism faces the flow (α = 0°) to diamond configuration, where the sharp edge is facing the flow (α = 45°). Previous studies on the effects of angle of attack on the FIV response of a square prism have shown that the prism with the diamond configuration undergoes VIV with relatively smaller amplitudes of oscillation compared to large-amplitude galloping oscillation observed for the prism with flat end facing the flow.10,23 We believe that the increase in power density values at high reduced velocity values above U* ≈ 80 for the L-shaped beam can be related to the observed smaller angles of attack compared to the cantilever one, but further detailed analysis of the wake needs to be conducted to fully understand the ruling mechanism in place. Using power density as a performance metric in this study enables us to compare the level of energy harvested per volume that the harvester occupies. It should be noted that the power density values calculated are based on the set of system parameters selected for this study, such as the length of the beam, width, and height of the tip oscillator. The power density values can change based on the system configuration, and further parametric studies can investigate the design of the system to achieve optimal energy harvesting capacities for such nonlinear FIV-based harvester. Nevertheless, this does not affect the fact that large contributions of the higher harmonics exist in the oscillations of a nonlinear L-shaped beam with a prismatic tip oscillator that leads to an augmented performance of the nonlinear FIV-based energy harvester over a wide range of flow velocities. The surpassed power density values of the L-shaped beam make this design a promising FIV-based harvester design for more space-efficient energy harvesting applications. While our study was designed to focus specifically on the existence of large contributions of high frequency oscillations in the flow-induced vibration response of inherently nonlinear systems, the framework can be used for future research in this area to fully benefit from nonlinear phenomena, such as modal interactions, to further enhance the performance of nonlinear FIV-based energy harvesters.

FIG. 4.

Power density values and percentage of improvement in power density vs reduced flow velocity for the L-shaped and cantilever beams.

FIG. 4.

Power density values and percentage of improvement in power density vs reduced flow velocity for the L-shaped and cantilever beams.

Close modal

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This study was supported by the James R. Myers endowment fund provided to the Department of Engineering Technology at Miami University. The experiments and data collection were conducted at Miami University, the previous institute of the first and second authors.

1.
T.
Sarpkaya
, “
A critical review of the intrinsic nature of vortex-induced vibrations
,”
J. Fluids Struct.
19
,
389
447
(
2004
).
2.
C. H. K.
Williamson
and
R.
Govardhan
, “
Vortex-induced vibrations
,”
Annu. Rev. Fluid Mech.
36
,
413
455
(
2004
).
3.
P. W.
Bearman
, “
Circular cylinder wakes and vortex-induced vibrations
,”
J. Fluids Struct.
27
,
648
658
(
2011
).
4.
P. W.
Bearman
, “
Vortex shedding from oscillating bluff bodies
,”
Annu. Rev. Fluid Mech.
16
,
195
222
(
1984
).
5.
J. S.
Leontini
,
B. E.
Stewart
,
M. C.
Thompson
, and
K.
Hourigan
, “
Wake state and energy transitions of an oscillating cylinder at low Reynolds number
,”
Phys. Fluids
18
,
067101
(
2006
).
6.
J. M.
Dahl
,
F. S.
Hover
,
M. S.
Triantafyllou
,
S.
Dong
, and
G. E.
Karniadakis
, “
Resonant vibrations of bluff bodies cause multivortex shedding and high frequency forces
,”
Phys. Rev. Lett.
99
,
144503
(
2007
).
7.
D.
Lo Jacono
,
E.
Konstantinidis
,
J.
Leontini
,
J.
Zhao
, and
J.
Sheridan
, “
Excitation and damping fluid forces on a cylinder undergoing vortex-induced vibration
,”
Front. Phys.
7
,
185
(
2019
).
8.
S.
Mittal
 et al, “
Lock-in in vortex-induced vibration
,”
J. Fluid Mech.
794
,
565
594
(
2016
).
9.
B.
Seyed-Aghazadeh
and
Y.
Modarres-Sadeghi
, “
An experimental investigation of vortex-induced vibration of a rotating circular cylinder in the crossflow direction
,”
Phys. Fluids
27
,
067101
(
2015
).
10.
A.
Nemes
,
J.
Zhao
,
D.
Lo Jacono
, and
J.
Sheridan
, “
The interaction between flow-induced vibration mechanisms of a square cylinder with varying angles of attack
,”
J. Fluid Mech.
710
,
102
130
(
2012
).
11.
B.
Seyed-Aghazadeh
,
D. W.
Carlson
, and
Y.
Modarres-Sadeghi
, “
Vortex-induced vibration and galloping of prisms with triangular cross-sections
,”
J. Fluid Mech.
817
,
590
618
(
2017
).
12.
R.
Bourguet
and
D.
Lo Jacono
, “
Flow-induced vibrations of a rotating cylinder
,”
J. Fluid Mech.
740
,
342
380
(
2014
).
13.
K.
Sourav
,
D.
Kumar
, and
S.
Sen
, “
Vortex-induced vibrations of an elliptic cylinder of low mass ratio: Identification of new response branches
,”
Phys. Fluids
32
,
023605
(
2020
).
14.
Y.
Liang
,
L.
Tao
, and
L.
Xiao
, “
Energy transformation on flow-induced motions of multiple cylindrical structures with various corner shapes
,”
Phys. Fluids
32
,
027105
(
2020
).
15.
A.
Sareen
,
J.
Zhao
,
D.
Lo Jacono
,
J.
Sheridan
,
K.
Hourigan
, and
M. C.
Thompson
, “
Vortex-induced vibration of a rotating sphere
,”
J. Fluid Mech.
837
,
258
292
(
2018
).
16.
J.
Zhao
,
J.
Sheridan
,
K.
Hourigan
, and
M. C.
Thompson
, “
Flow-induced vibration of a cube orientated at different incidence angles
,”
J. Fluids Struct.
91
,
102701
(
2019
).
17.
G. V.
Parkinson
and
J. D.
Smith
, “
The square prism as an aeroelastic non-linear oscillator
,”
Q. J. Mech. Appl. Math.
17
,
225
239
(
1964
).
18.
M. P.
Païdoussis
,
S. J.
Price
, and
E.
De Langre
,
Fluid-structure Interactions: Cross-Flow-Induced Instabilities
(
Cambridge University Press
,
2010
).
19.
A.
Abdelkefi
, “
Aeroelastic energy harvesting: A review
,”
Int. J. Eng. Sci.
100
,
112
135
(
2016
).
20.
V. J.
Ovejas
and
A.
Cuadras
, “
Multimodal piezoelectric wind energy harvesters
,”
Smart Mater. Struct.
20
,
085030
(
2011
).
21.
H. D.
Akaydin
,
N.
Elvin
, and
Y.
Andreopoulos
, “
Energy harvesting from highly unsteady fluid flows using piezoelectric materials
,”
J. Intell. Mater. Syst. Struct.
21
,
1263
1278
(
2010
).
22.
H.
Akayd
n
ı,
N.
Elvin
, and
Y.
Andreopoulos
, “
Wake of a cylinder: A paradigm for energy harvesting with piezoelectric materials
,”
Exp. Fluids
49
,
291
304
(
2010
).
23.
X.
Li
,
Z.
Lyu
,
J.
Kou
, and
W.
Zhang
, “
Mode competition in galloping of a square cylinder at low Reynolds number
,”
J. Fluid Mech.
867
,
516
555
(
2019
).
24.
J.
Zhao
,
K.
Hourigan
, and
M. C.
Thompson
, “
Flow-induced vibration of d-section cylinders: An afterbody is not essential for vortex-induced vibration
,”
J. Fluid Mech.
851
,
317
343
(
2018
).
25.
G. V.
Parkinson
and
M. A.
Wawzonek
, “
Some considerations of combined effects of galloping and vortex resonance
,”
J. Wind Eng. Ind. Aerodyn.
8
,
135
143
(
1981
).
26.
S.
Deniz
and
T.
Staubli
, “
Oscillating rectangular and octagonal profiles: Interaction of leading- and trailing-edge vortex formation
,”
J. Fluids Struct.
11
,
3
31
(
1997
).
27.
M.
Zhao
,
L.
Cheng
, and
T.
Zhou
, “
Numerical simulation of vortex-induced vibration of a square cylinder at a low Reynolds number
,”
Phys. Fluids
25
,
023603
(
2013
).
28.
J.
Zhao
,
J. S.
Leontini
,
D.
Lo Jacono
, and
J.
Sheridan
, “
Fluid-structure interaction of a square cylinder at different angles of attack
,”
J. Fluid Mech.
747
,
688
721
(
2014
).
29.
A.
Joly
,
S.
Etienne
, and
D.
Pelletier
, “
Galloping of square cylinders in cross-flow at low Reynolds numbers
,”
J. Fluids Struct.
28
,
232
243
(
2012
).
30.
J.
Zhao
,
A.
Nemes
,
D.
Lo Jacono
, and
J.
Sheridan
, “
Branch/mode competition in the flow-induced vibration of a square cylinder
,”
Philos. Trans. R. Soc., A
376
,
20170243
(
2018
).
31.
C.
Mannini
,
A. M.
Marra
,
T.
Massai
, and
G.
Bartoli
, “
Interference of vortex-induced vibration and transverse galloping for a rectangular cylinder
,”
J. Fluids Struct.
66
,
403
423
(
2016
).
32.
A.
Abdelkefi
,
M. R.
Hajj
, and
A. H.
Nayfeh
, “
Power harvesting from transverse galloping of square cylinder
,”
Nonlinear Dyn.
70
,
1355
1363
(
2012
).
33.
X.
He
,
X.
Yang
, and
S.
Jiang
, “
Enhancement of wind energy harvesting by interaction between vortex-induced vibration and galloping
,”
Appl. Phys. Lett.
112
,
033901
(
2018
).
34.
H.
Liu
,
S.
Zhang
,
R.
Kathiresan
,
T.
Kobayashi
, and
C.
Lee
, “
Development of piezoelectric microcantilever flow sensor with wind-driven energy harvesting capability
,”
Appl. Phys. Lett.
100
,
223905
(
2012
).
35.
H. L.
Dai
,
A.
Abdelkefi
,
Y.
Yang
, and
L.
Wang
, “
Orientation of bluff body for designing efficient energy harvesters from vortex-induced vibrations
,”
Appl. Phys. Lett.
108
,
053902
(
2016
).
36.
G.
Hu
,
K. T.
Tse
,
K. C. S.
Kwok
,
J.
Song
, and
Y.
Lyu
, “
Aerodynamic modification to a circular cylinder to enhance the piezoelectric wind energy harvesting
,”
Appl. Phys. Lett.
109
,
193902
(
2016
).
37.
J.
Song
,
G.
Hu
,
K. T.
Tse
,
S. W.
Li
, and
K. C. S.
Kwok
, “
Performance of a circular cylinder piezoelectric wind energy harvester fitted with a splitter plate
,”
Appl. Phys. Lett.
111
,
223903
(
2017
).
38.
A.
Abdelkefi
,
A.
Hasanyan
,
J.
Montgomery
,
D.
Hall
, and
M. R.
Hajj
, “
Incident flow effects on the performance of piezoelectric energy harvesters from galloping vibrations
,”
Theor. Appl. Mech. Lett.
4
,
022002
(
2014
).
39.
R.
Naseer
,
H.
Dai
,
A.
Abdelkefi
, and
L.
Wang
, “
Comparative study of piezoelectric vortex-induced vibration-based energy harvesters with multi-stability characteristics
,”
Energies
13
,
71
(
2020
).
40.
L. A.
Wong
and
J. C.
Chen
, “
Nonlinear and chaotic behavior of structural system investigated by wavelet transform techniques
,”
Int. J. Non-Linear Mech.
36
,
221
235
(
2001
).
41.
R. E.
Mickens
,
An Introduction to Nonlinear Oscillations
(
CUP Archive
,
1981
).
42.
A. H.
Nayfeh
and
P. F.
Pai
,
Linear and Nonlinear Structural Mechanics
(
John Wiley & Sons
,
2008
).
43.
A. F.
Arrieta
,
P.
Hagedorn
,
A.
Erturk
, and
D. J.
Inman
, “
A piezoelectric bistable plate for nonlinear broadband energy harvesting
,”
Appl. Phys. Lett.
97
,
104102
(
2010
).
44.
L.
Gammaitoni
,
I.
Neri
, and
H.
Vocca
, “
Nonlinear oscillators for vibration energy harvesting
,”
Appl. Phys. Lett.
94
,
164102
(
2009
).
45.
X.
Wang
,
C.
Chen
,
N.
Wang
,
H.
San
,
Y.
Yu
,
E.
Halvorsen
, and
X.
Chen
, “
A frequency and bandwidth tunable piezoelectric vibration energy harvester using multiple nonlinear techniques
,”
Appl. Energy
190
,
368
375
(
2017
).
46.
S. D.
Nguyen
and
E.
Halvorsen
, “
Nonlinear springs for bandwidth-tolerant vibration energy harvesting
,”
J. Microelectromech. Syst.
20
,
1225
1227
(
2011
).
47.
E.
Wang
,
W.
Xu
,
X.
Gao
,
L.
Liu
,
Q.
Xiao
, and
K.
Ramesh
, “
The effect of cubic stiffness nonlinearity on the vortex-induced vibration of a circular cylinder at low Reynolds numbers
,”
Ocean Eng.
173
,
12
27
(
2019
).
48.
Z.
Yuan
,
W.
Liu
,
S.
Zhang
,
Q.
Zhu
, and
G.
Hu
, “
Bandwidth broadening through stiffness merging using the nonlinear cantilever generator
,”
Mech. Syst. Signal Process.
132
,
1
17
(
2019
).
49.
A. H.
Alhadidi
and
M. F.
Daqaq
, “
A broadband bi-stable flow energy harvester based on the wake-galloping phenomenon
,”
Appl. Phys. Lett.
109
,
033904
(
2016
).
50.
H.
Dai
,
A.
Abdelkefi
, and
L.
Wang
, “
Theoretical modeling and nonlinear analysis of piezoelectric energy harvesting from vortex-induced vibrations
,”
J. Intell. Mater. Syst. Struct.
25
,
1861
1874
(
2014
).
51.
A. W.
Mackowski
and
C. H. K.
Williamson
, “
An experimental investigation of vortex-induced vibration with nonlinear restoring forces
,”
Phys. Fluids
25
,
087101
(
2013
).
52.
S.
Zhou
and
L.
Zuo
, “
Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting
,”
Commun. Nonlinear Sci. Numer. Simul.
61
,
271
284
(
2018
).
53.
Z.
Yan
and
A.
Abdelkefi
, “
Nonlinear characterization of concurrent energy harvesting from galloping and base excitations
,”
Nonlinear Dyn.
77
,
1171
1189
(
2014
).
54.
K.
Li
,
Z.
Yang
,
Y.
Gu
,
S.
He
, and
S.
Zhou
, “
Nonlinear magnetic-coupled flutter-based aeroelastic energy harvester: Modeling, simulation and experimental verification
,”
Smart Mater. Struct.
28
,
015020
(
2018
).
55.
L.
Zhao
, “
Concurrent wind and base vibration energy harvesting with a broadband bistable aeroelastic energy harvester
,”
IOP Conf. Ser.: Mater. Sci. Eng.
531
,
012081
(
2019
).
56.
R. L.
Harne
and
K. W.
Wang
, “
A review of the recent research on vibration energy harvesting via bistable systems
,”
Smart Mater. Struct.
22
,
023001
(
2013
).
57.
H.
Sun
,
C.
Ma
, and
M. M.
Bernitsas
, “
Hydrokinetic power conversion using flow induced vibrations with cubic restoring force
,”
Energy
153
,
490
508
(
2018
).
58.
H.
Sun
and
M. M.
Bernitsas
, “
Bio-inspired adaptive damping in hydrokinetic energy harnessing using flow-induced oscillations
,”
Energy
176
,
940
960
(
2019
).
59.
C.
Ma
,
H.
Sun
,
G.
Nowakowski
,
E.
Mauer
, and
M. M.
Bernitsas
, “
Nonlinear piecewise restoring force in hydrokinetic power conversion using flow induced motions of single cylinder
,”
Ocean Eng.
128
,
1
12
(
2016
).
60.
A.
Erturk
,
J. M.
Renno
, and
D. J.
Inman
, “
Modeling of piezoelectric energy harvesting from an L-shaped beam-mass structure with an application to uavs
,”
J. Intell. Mater. Syst. Struct.
20
,
529
544
(
2009
).
61.
H.
Samandari
and
E.
Cigeroglu
,
Nonlinear Dynamics
(
Springer
,
2016
), Vol. 1, pp.
145
154
.
62.
F.
Georgiades
,
J.
Warminski
, and
M. P.
Cartmell
, “
Towards linear modal analysis for an L-shaped beam: Equations of motion
,”
Mech. Res. Commun.
47
,
50
60
(
2013
).
63.
T.-J.
Yu
,
W.
Zhang
, and
X.-D.
Yang
, “
Nonlinear dynamics of flexible l-shaped beam based on exact modes truncation
,”
Int. J. Bifurcation Chaos
27
,
1750035
(
2017
).
64.
R. L.
Harne
,
A.
Sun
, and
K. W.
Wang
, “
Leveraging nonlinear saturation-based phenomena in an L-shaped vibration energy harvesting system
,”
J. Sound Vib.
363
,
517
531
(
2016
).
65.
J. A.
Dunnmon
,
S. C.
Stanton
,
B. P.
Mann
, and
E. H.
Dowell
, “
Power extraction from aeroelastic limit cycle oscillations
,”
J. Fluids Struct.
27
,
1182
1198
(
2011
).
66.
J. M.
McCarthy
,
S.
Watkins
,
A.
Deivasigamani
, and
S. J.
John
, “
Fluttering energy harvesters in the wind: A review
,”
J. Sound Vib.
361
,
355
377
(
2016
).
67.
M.
Gürgöze
, “
On the dynamic analysis of a flexible L-shaped structure
,”
J. Sound Vib.
211
,
683
688
(
1998
).
68.
S. S.
Rao
,
Vibration of Continuous Systems
(
Wiley Online Library
,
2007
), Vol. 464.