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Lower efficiencies induce higher energy costs and pose a barrier to wave energy devices’ commercial applications. Therefore, the efficiency enhancement of wave energy converters has received much attention in recent decades. The reported research presents the double snap-through mechanism applied to a hemispheric point absorber type wave energy converter (WEC) to improve the energy absorption performance. The double snap-through mechanism comprises four oblique springs mounted in an X-configuration. This provides the WEC with different dynamic stability behaviors depending on the particular geometric and physical parameters employed. The efficiency of these different WEC behaviors (linear, bistable, and tristable) was initially evaluated under the action of regular waves. The results for bistable or tristable responses indicated significant improvements in the WEC’s energy capture efficiency. Furthermore, the WEC frequency bandwidth was shown to be significantly enlarged when the tristable mode was in operation. However, the corresponding tristable trajectory showed intra-well behavior in the middle potential well, which induced a more severe low-energy absorption when a small wave amplitude acted on the WEC compared to when the bistable WEC was employed. Nevertheless, positive effects were observed when appropriate initial conditions were imposed. The results also showed that for bistable or tristable responses, a suitable spring stiffness may cause the buoy to oscillate in high energy modes.
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Evaluation of the Double Snap-Through Mechanism
on the Wave Energy ConvertersPerformance
Bingqi Liu
&Carlos Levi
&Segen F. Estefen
&Zhijia Wu
&Menglan Duan
#The Author(s) 2021
Lower efficiencies induce higher energy costs and pose a barrier to wave energy devicescommercial applications. Therefore, the
efficiency enhancement of wave energy converters has received much attention in recent decades. The reported research presents
the double snap-through mechanism applied to a hemispheric point absorber type wave energy converter (WEC) to improve the
energy absorption performance. The double snap-through mechanism comprises four oblique springs mounted in an X-config-
uration. This provides the WEC with different dynamic stability behaviors depending on the particular geometric and physical
parameters employed. The efficiency of these different WEC behaviors (linear, bistable, and tristable) was initially evaluated
under the action of regular waves. The results for bistable or tristable responses indicated significant improvementsin the WECs
energy capture efficiency. Furthermore, the WEC frequency bandwidth was shown to be significantly enlarged when the tristable
mode was in operation. However, the corresponding tristable trajectory showed intra-well behavior in the middle potential well,
which induced a more severe low-energy absorption when a small wave amplitude acted on the WEC compared to when the
bistable WEC was employed. Nevertheless, positive effects were observed when appropriate initial conditions were imposed.
The results also showed that for bistable or tristable responses, a suitable spring stiffness may cause the buoy to oscillate in high
energy modes.
Keywords Wave energy converter .Point absorber .Double snap-through mechanism .Bistable dynamic behavior .Tristable
dynamic behavior
1 Introduction
The continual increase in energy consumption globally is es-
timated as 9954 Mtoe 115765 TWh in 2020 (Enerdata
2021), unpredictable oil and gas market fluctuations, and the
accelerating pressure to reduce greenhouse gas emissions has
resulted in significant pressure to increase the energy genera-
tion from renewable sources. In the last few decades, renew-
able energy has begun to play an increasingly significant role
in replacing conventional sources. In this respect, wave energy
can provide continually available high-density energy (esti-
mated as 146 TWh/year) (Edenhofer et al. 2011), and it has
thus become a competitive renewable energy source. Also,
recent investigations have shown that upper-ocean warming,
mainly due to anthropogenic global warming, has caused sig-
nificant increases in ocean wave heights, and global wave
energy has been increasing at around 0.4% annually since
1948 (Reguero et al. 2019).
The technological race to harvest wave energy has resulted
in the design, construction, and testing of various wave energy
Article Highlights
The energy capture efficiency of the WEC has been significantly
improved in bistable or tristable mode.
The WEC frequency bandwidth is shown to be significantly enlarged in
the tristable mode.
Compared to the bistable WEC, the intra-well behavior in the middle
potential well of the tristable WEC induced a more severe low-energy
absorption under small wave amplitude excitations.
Positive effects on the low-energy absorption problem were observed
when appropriate initial conditions were imposed in tristable WEC.
*Segen F. Estefen
Ocean Engineering Department, COPPE, Federal University of Rio
de Janeiro, Rio de Janeiro 21945-970, Brazil
China Ship Scientific Research Center, Wuxi 214082, China
College of Safety and Ocean Engineering, China University of
Petroleum-Beijing, Beijing 102249, China
Journal of Marine Science and Application
converters (WECs) since the early 1980s (Czech and Bauer
2012; Falcão 2010; Shadman et al. 2019). In relation to their
geometry, WECs are usually classified as being either termi-
nator, attenuator, or point absorber(PA) types (Al Shami et al.
2019). As the PA type is mechanically simple and requires
lower capital and maintenance costs, it has attracted consider-
able attention from the wave energy technical community
(Astariz and Iglesias 2015). PA types also have vertically
symmetrical geometric features, which enable them to extract
energy independently from the direction of the incoming wave
(Budal and Falnes 1982). Relatively small bodies, such as
those used in a typical PA design, are also more convenient
for working in the arrays used in offshore wave energy farms,
and they can even be easily integrated with offshore wind
farms (Anvari-Moghaddam et al. 2020). Such combined char-
acteristics indicate the potential for the PA-WEC to be com-
mercialized for use in the electricity market, or as a comple-
mentary power supply for oil and gas offshore plants and other
offshore facilities.
However, the PA-WEC has a small characteristic length
(usually smaller than the local predominant wavelength),
which means that its operational bandwidth is narrower than
that of other WEC concepts (e.g., terminator and attenuator
devices). Therefore, the WEC-PA can only extract a signifi-
cant amount of energy from wave components within a rea-
sonably narrow wave frequency resonant band. If the PAs
natural frequency lies out of the predominant sea wave fre-
quency range, it undergoes an abrupt drop in its conversion
efficiency. Therefore, PA-WECs need to be either significant-
ly enlarged or be operated out of their resonant conditions.
To overcome such disadvantages, and to enhance their en-
ergy absorption efficiency, some alternative approaches have
been proposed. For example, one direct and straightforward
approach aimed to optimize buoy geometry at a given deploy-
ment site (Goggins and Finnegan 2014; Shadman et al. 2018);
however, the results showed limited efficiency gains. It is
evident, therefore, that to increase the potential efficiency
gains to adequate levels, alternative control strategies that
adopt either passive or active techniques need to be incorpo-
rated into the PA design, and the following schemes have been
proposed: complex conjugate control (Maria-Arenas et al.
2019), damping control (Garcia-Rosa et al. 2017; Jin et al.
2019; Rodríguez et al. 2019), latching control (Babarit and
Clément 2006;Falcão2008; Henriques et al. 2016), and mod-
el predictive control (MPC) (Andersen et al. 2015; Faedo et al.
2017;LiandBelmont2014a,2014b). However, of these, bi-
directional damping reactive power increases the complex-
ity of the Power Take Off (PTO) mechanism, which
impacts heavily on the cost of the device, and the other
control strategies require accurate and a priori predic-
tions of the wave excitation force acting on the PA;
these impose heavy burdens and are major obstacles
for enabling practical use of the device.
In light of such practical difficulties, many investigations
are now focusing on the use of non-reactive and passive-target
approaches (Wu et al. 2018,2019; Younesian and Alam
2017). By taking the well-succeeded solution adopted in the
vibration isolation problem and vibration energy harvesting
(Ramlan et al. 2010), the nonlinear stiffness system (NSS)
has been attracted increasing attention as a promising feasible,
efficient, and simple solution that enables wave energy con-
version (Zhang et al. 2014; Zhang and Yang 2015).
The bistable system configuration, which is known as a
snap-through mechanism,is a classical NSS that provides
a nonlinear restoring force through the unique configuration of
certain mechanical compression components, such as springs,
pneumatic actuators, buckling beams, and magnets (Wei and
Jing 2017). It has bistable dynamic behavior that provides two
statically stable states and one unstable state. If the excitation
frequency is lower than the natural frequency of the device,
the potential escaping phenomenon of the bistable behavior
causes the vibration energy harvester to absorb significantly
more energy (Harne 2017). In addition, its nonlinear stiffness
characteristics effectively broaden the frequency response
bandwidth, which is beneficial for harvesting in an irregular
excitation environment (Daqaq et al. 2014).
In recent years, the NSS has been attracting attention from
researchers investigating ocean wave energy conversion tech-
nology, mainly due to its ability to modulate natural frequen-
cies, which shifts it into the local dominant frequency range of
the ocean waves. In addition, its characteristic bandwidth en-
largement could be particularly beneficial when dealing with
the typical random behavior of real sea conditions.
However, currently only a few studies have discussed the
implementation of NSS in a PA design, either numerically or
experimentally. For example, Zhang et al. (2014)proposeda
nonlinear snap-through Power Take Off (PTO) system
consisting of two symmetrically oblique springs, see Figure 1a,
which provided a bistable mechanism. The authors conducted an
extensive analysis and comparison using both regular and
irregular waves and found that nonlinear WEC captures
relatively larger amounts of wave energy, even when using
devices with small dimensions with small amplitude waves. In
addition, Todalshaug (2015) patented a WEC with a negative
stiffness mechanism that uses either mechanical springs or pneu-
matic cylinders. The pneumatic configuration, see Figure 1b,was
then patented under the commercial name of WaveSpringand
used in the CorPower buoy prototype (CorPower), and to vali-
date this innovative concept, WaveSpringwas successfully
tested in a 1/16 scale model (Todalshaug et al. 2016). The exper-
imental results showed that the WEC responded efficiently if
tuned to work within the resonant range, and an increase in its
frequency bandwidth was found. Prior dry testing and ocean
deployment was then conducted using a 1/2 scale WEC to verify
the promising characteristics of the WaveSpringindustrial pro-
totype performance.
Journal of Marine Science and Application
In addition to the above-mentioned conventional bistable
system, other nonlinear stiffness systems have been intro-
duced into WEC technology. Zhang et al. (2018)proposeda
novel adaptive bistable mechanism (see Figure 1c) that in-
volves adopting two additional auxiliary springs for automat-
ically adjusting the potential function for lowering the poten-
tial barrier near the unstable equilibrium position. The time-
varying potential feature helps to avoid the possible occur-
rence of low-energy absorption, which can occur in the con-
ventional bistable WEC. In addition, the adaptive bistable
WEC can contain an even wider frequency bandwidth, if the
right system parameters are selected. As an alternative to the
conventional bistable potential adaptive, Younesian and Alam
(2017) proposed a multi-stable mechanism composed of two
oblique rigid links and two oblique springs (see Figure 1d)
with additional potential wells introduced into the dynamic
response. By adjusting the angle and initial spring lengths,
the mechanical system can switch frommonostable to bistable
or tristable states, accordingly. Their results have shown that
bistable and tristable modes significantly improve the WEC
efficiency, and the natural frequency can be shifted to a higher
frequency range. However, to achieve the tristable mode, the
length of the rigid links needs to be at least three times the
length of that of the initial spring, which means that it requires
a larger operational space. In addition, the WEC responses and
the effects of the system parameters in the tristable mode re-
main uncertain and have not been completely clarified.
Based on the above information, the present study intro-
duces a simpler and more compact double snap-through
mechanism that can be applied to work in a PA-WEC, enables
the occurrence of multi-stable states, and requires only a rela-
tively compact space. The main focus of this study is
conducting a motion characteristic analysis. In addition, com-
pared to previous studies mentioned above, greater insights
into the wave amplitude effect are provided.
2 Point Absorber WEC (PA-WEC) Featuring
a Double Snap-Through Mechanism
2.1 Description of Double Snap-Through Mechanism
Figure 2shows a schematic diagram of the configuration of a
WEC equipped with the proposed double snap-through mech-
anism. The hemispherical buoy is constrained to oscillate in
heave motion only, and the PTO is placed on the seabed. The
double snap-through mechanism comprises four oblique
springs in an X-shaped layout. One end of each spring is
connected to a seabed fixed support structure, and the other
follows the buoy-PTO pole connector. A similar mechanism
was proposed by Li et al. (2019), but its operations were
limited to a bistable mode only. The present study investigates
the full potential of the nonlinear stiffness mechanism in a
Figure 1 Schematic illustration of some of the nonlinear stiffness mechanisms previously proposed in literature: (a) conventional bistable mechanism,
(b) pneumatic bistable mechanism, (c) adaptive bistable mechanism, and (d) multi-stable mechanism
Figure 2 Schematic configuration of hemispherical point absorber with
the proposed double snap-through mechanism
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
much broader perspective that includes an analysis of its
tristable behavior.
In Figure 2, the vertical and horizontal distances between
the spring fixed end supports are shown as 2aand 2b,respec-
tively, and the spring free length and stiffness are Land K,
For any given vertical displacement, z, at any instant,
t, the vertical component of the force developed by the
double snap-through mechanism can be given by
and the associate potential energy stored by the springs at any
time, t,is
U¼2Kz22KL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2KL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qþ4KL ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2.2 Mathematical Model
Applying the well-known Cummins equation (Cummins
1962), in the time domain, the governing equation of the
hemispherical PA heave motion with double snap-through
mechanism is given as follows:
¼fWtðÞþfPTO tðÞ ð3Þ
where m= 2/3πR
ρis the buoys physical mass; A
is the
buoys added mass at an infinite frequency; ρis the water
density; Ris the hemispherical buoy radius; z(t), ˙
ztðÞ,and ::
ztðÞare the heave displacement, velocity, and acceleration at
time, t, respectively; and K
(t) is an impulse response function
defined in the kernel of the convolution term and in consider-
ation of the fluid memory effect of the radiation force. In
Ogilvie (1964), K
(t) relates to the radiation damping coeffi-
cient B(ω)asfollows:
0BωðÞcos ωtðÞdtð4Þ
Furthermore, f
(t) is the restoring force:
fRtðÞ¼CWLztðÞþfMtðÞ ð5Þ
where C
is the linear hydrostatic restoring coeffi-
cient, gis gravity, and f
(t) is the vertical component of the
force induced by the double snap-through mechanism used as
given by Eq. (1).
On the right hand side of Eq. (3), f
(t) represents the ver-
tical wave excitation force acting on the buoy. For regular
waves, at any given wave frequency, ω, and wave amplitude,
A, the corresponding force at any time, t, can be expressed as
(Falnes 2002):
rsin ωtðÞ ð6Þ
By considering a linear damper PTO (with a damping co-
efficient C), the PTO force can be calculated by:
fPTO tðÞ¼C
ztðÞ ð7Þ
To accelerate the computational processing time involved
to solve Eq. (3), the convolution integral can be solved
through the state space model defined below (Wu et al. 2018):
where X=[x
is the n
order state vector; A
and C
are the constant state space matrices that are calculated
through the frequency domain identification (FDI) method
discussed in Pérez and Fossen (2008,2009); vis the buoy
vertical velocity; and μis the convolution term.
Substituting Eqs. (8)and(9)intoEq.(3), the state space
model of the complete system is established as:
The power capture efficiency is represented by the capture
width ratio as defined in Eq. (11)(Falnes2002):
2RPwave ð11Þ
where P
is the average absorbed power,
is the wave energy flux per meter of wave front,
Pwave ¼ρg2A2
and the constant values ρ,g,andRareusedtodefinethe
non-dimensional parameters described below:
Journal of Marine Science and Application
The hydrodynamic coefficients for the hemispherical buoy
can be obtained as discussed by Hulme (1982). The added
mass at an infinite frequency is equal to A*
¼0.5, and the
non-dimensional radiation damping coefficient, B*, is shown
in Figure 3.
The solution can be obtained by applying the 4
order FDI
to define the matrices A
and the 4
order classical
Runge-Kutta method to solve the resultant ordinary differen-
tial equations (ODE) expressed by Eq. (10).
2.3 Static Analysis of the Multi-stable Mechanism
Based on the shape of the potential energy curve, U,as
expressed by Eq. (2), the dynamic behavior of the double
snap-through mechanism can be classified into three catego-
ries according to the characteristics of the respective curves:
&Monostable, involving a single potential well with one
stable equilibrium (one global minimum) (see Figure 4a)
&Bistable, involving a double potential well with two stable
equilibria (two minima) and with one unstable equilibrium
(one local maximum) in the middle (see Figure 4b)
&Tristable, involving a triple potential well with three stable
equilibria (three local minima) separated by two unstable
equilibria (two local maxima) (see Figure 4c)
To define the shapes of the bistable and tristable potential
curves shown in Figure 4, four parameters are given as fol-
lows: Δz*
p1, representing half of the displacement difference
between two outer stable equilibria for the bistable and
tristable modes; Δz*
p2, representing half of the displacement
difference between the two unstable equilibria for the tristable
mode; and ΔU*
p1and ΔU*
p2representing the difference be-
tween the potential energy for stable and unstable equilibria.
Also known as a potential barrier, this defines the threshold
value for the escaping energy level (ΔU*
p1for the bistable
mode; ΔU*
p2for the tristable mode).
Figure 5presents a stability distribution diagram of the
double snap-through mechanism. For a given combination
of spring physical characteristics (K*, L*), there exists a spe-
cific combination of geometric parameters (a*, b*) that deter-
mine its stability characteristics. For instance, if we take the
horizontal distance as b*= 0.4, the system stability moves
from bistable (orange region) to tristable (green region) then
to monostable (red region), if a* continues to increase.
The displacement difference and the potential barrier cor-
responding to the stable and unstable equilibria are two key
factors used in the analysis of the device performance.
Figure 6presents the results for the displacement difference
and potential barrier in the (a*, b*) domain. In this respect,
Figure 6a and b show that the (outer) stable equilibrium po-
sition, Δz*
p1, and the (outer) potential barrier, ΔU*
a*andb* decrease. For the middle potential well, as shown in
Figure 6c, the unstable equilibrium position, Δz*
have a greater relationship with a*,while the potential barrier,
p2, shows the opposite tendency with respect to ΔU*
Figure 7shows the influence of spring stiffness, K*, on the
potential curve behavior. The curves shown in Figure 7were
obtained for a given set of geometrical parameters (a* = 0.4,
b* = 0.3) to illustrate the tristable stability mode (triple poten-
tial well). The depth of the potential curve is seen to change
when the spring stiffness, K*, is adjusted. The results from
Figure 7show that with an increase in the value of K*, the
potential well depth becomes deeper, and this corresponds to
higher potential barrier values.
3 Analysis of WEC Efficiency
An evaluation of the effect of employing the double snap-
through mechanism on the operational of the WEC was con-
ducted to compare its performance with the linear WEC.
Figure 8shows the power capture width ratio for the linear
WEC in the (ω*, C*) parametric space. To enable the intuitive
comparison discussed in the following sections, the upper
limit of the capture width ratio of the color grading scale
was set as Ω= 1.5. Complying with the correct physics of
the linear WEC, the numerical results clearly show that the
maximum value of the capture width ratio occurs at a reso-
nance frequency of ω* = 1.0. The optimal capture width ratio
is Ω= 0.49, and the corresponding operational conditions are
defined by C* = 0.25. A similar analysis is discussed in Zhang
et al. (2018) and Zhang et al. (2019b).
For the 2.5 m radius hemisphere used here, the optimal
conditions are C* = 0.25 (corresponding to the damping
Figure 3 Non-dimensional radiation damping coefficient: hemispherical
buoy in heave motion (Hulme 1982)
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
coefficient, C= 16611 kg/s) and the WEC natural period
corresponding to 3.2 s (ω* = 1.0). As most of the oceans
energy is concentrated in waves lying within the period range
of 5.0 s to 15.0 s (ω* = 0.21 to 0.63) (Falnes 2002), the linear
WEC natural period (ω* = 1.0) is out of the energetic wave
range, which results in a very poor performance.
The results obtained from the linear WEC performance
were used to validate the numerical tools developed here,
through a code-to-code verification strategy.
In the following sections, the discussion focuses on possible
ways of applying the double snap-through mechanism to im-
prove the WEC performance. In Section 3.1, the WEC motion
characteristics resulting from the use of the double snap-
through mechanism to induce bistable and tristable WEC
modes generated data that are subsequently analyzed using time
series, phase portrait, and frequency spectra. In Sections 3.2 to
3.5, parametric studies show the effects of varying wave and
WEC parameters on its power capture performance.
3.1 WEC Motion Characteristics
To illustrate the performance of bistable and tristable WEC
modes, the following set of parameters were selected in the
following text: (a) a* = 0.30, b* = 0.50, K* = 1.0, L* = 1.0, A*
= 0.20, and C* = 0.25, leading to a bistable mode, and (b) a*=
0.37, b* = 0.37, K* = 1.0, L* = 1.0, A* = 0.20, and C* = 0.25,
leading to a tristable mode.
Figure 9shows the capture width ratio as a function of the
wave frequency for each one of the defined WEC modes.
Different from the smooth and continuous linear WEC curve,
the figure clearly shows that the bistable and tristable WEC
curves are split into three distinct regions: the blue curve on
and a set of randomly distributed points between the two
branches (chaotic zone). The marked points P1-P5 were select-
ed to illustrate the different possible responses of the double
snap-through mechanism: P1 and P2 both lie on the left branch
curves and show high inter-well motions (as described in
Figure 10); P3 lies inside the chaotic zones and shows a chaotic
motion (as described in Figure 11); P4 was defined for the
bistable mode and it lies inside the chaotic zone; for the tristable
mode, P4 lies on the right branch curve and indicates distinct
responses for each mode; and P5 lies on the right branch curves
and presents intra-well motions (as described in Figure 12).
Figure 9shows that the maximum capture width ratio of
both bistable and tristable WECs is Ω= 1.31; this result is
267% greater than the maximum value of the linear WEC (Ω
Further, the optimal wave frequencies for bistable (ω*=
0.55) and tristable (ω* = 0.59) are much lower than that for
linear WEC (ω* = 1.00), and bothof them are in the frequency
range of typical high-energy ocean waves (ω* = 0.21 to 0.63).
Zhang et al. (2019a) defines the operational frequency band-
width as the range in which the energy power capture ratio is
greater than half of the linear WEC maximum value. Based on
such a criterion, the results in Figure 9show that the frequency
bandwidths for linear, bistable, and tristable WECs are Δω*=
0.4, 0.6, and 0.95, respectively. Therefore, the bistable and
tristable WEC frequency bandwidths are 150% and 240% great-
er than the linear WEC range, which demonstrates their intrinsic
ability to broaden the operational frequency bandwidth, particu-
larly in the case of the tristable WEC.
Figure 4 Potential energy curves: (a) monostable, (b) bistable, and (c)tristable
Figure 5 Double snap-through mechanism: stability distribution regions
(K* = 1.0, L* = 1.0)
Journal of Marine Science and Application
Figure 6 Stable and unstable equilibria (K* = 1.0, L*=1.0):(a) bistable displacement difference (Δz
*), (b) bistable potential barrier (ΔU
*), (c)
tristable displacement difference (Δz
*), and (d) tristable potential barrier (ΔU
Figure 7 The effect of spring stiffness (K*) on tristable potential curves
(a* = 0.4, b*=0.3,K* = 1.0, L* = 1.0)
Figure 8 Energy capture width ratio: linear WEC in the (ω*, C*)
parametric space
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
Furthermore, Figure 9a and b show that the left branchesof
both the bistable and tristable WEC provide consistently
higher efficiencies than the linear WEC, thereby providing
the much-needed efficiency enhancement. However, the right
branch and chaotic zone of the bistable WEC (blue curve in
Figure 9a) delivers a rather poorer performance and a more
limited frequency bandwidth than the linear WEC responses
(red curve in Figure 9a). With respect to the tristable WEC
(blue curve in Figure 9b), if ω* > 1.0, the right branch of the
tristable curve in Figure 9b (blue curve) shows that the oper-
ation of the tristable WEC is very close to that of the linear
WEC (red curve), whereas if 0.75 < ω* < 1.0 (left branch), the
results show a superior performance for the tristable WEC,
and thus great efficiency gain possibilities.
Solutions covering a given frequency range of incident waves
acting on the linear, bistable, and tristable WEC responses illus-
trate the response behaviors corresponding to these three regions,
and the underlying reasons for these are thus discussed.
The results in Figures 10,11,and12 show the heave dis-
placement time history and the heave displacement phase and
velocity phase portraits. The figures also include results ofthe
spectral curves that were generated by a fast Fourier transform
analysis of the corresponding time series.
Figure 10 shows the motion characteristics of the linear,
bistable, and tristable modes for lower incident wave frequen-
cies (i.e., P1: ω* = 0.15 and P2: ω* = 0.55). From Figure 10b
e, it is evident that the bistable and tristable mechanisms result
in buoy responses that have larger amplitudes compared to the
linear responses shown in Figure 10a, but they are neverthe-
less periodic. If ω* = 0.15 (Figure 10 b and d), the buoy
oscillates either within one of the two potential wells
(bistable) or in the two outer potential wells (tristable) several
cycles prior to the snap-through behavior occurs, as previous-
ly indicated by Zhang et al. (2019a).
With respect to the latching control strategy, latching
occurs if the buoy velocity is zero and releases it if the
excitation wave force reaches its maximum value.
Looking back at the response curves in Figure 10 b
and d, the buoy was also trapped within the potential
well, and it escaped when the maximum or minimum
excitation forces occurred. This indicates that at low
frequencies, the bistable and tristable mechanisms re-
spond in the same way as those of the latch strategy.
By increasing the incident wave frequency toward mod-
erate values, such as ω* = 0.55 (as shown in Figure 10
c and e), larger amplitude periodic inter-well responses
occur between the two potential wells, which means that
the buoy reciprocates harmonically between the two
equilibrium points.
However, with an increase in the wave frequency, the
bistable and tristable WEC responses do not always manage
to cross their corresponding potential barriers. In such cases,
the WEC loses stability and oscillates chaotically around the
potential wells. Such a behavior is observed in the results for
P3 (ω* = 0.61) shown in Figure 11 b and c. The corresponding
frequency spectra of the chaotic motions contain a wide range
of frequency components, but they are still dominated by the
incident wave frequency component.
If ω* = 0.86 (P4), the bistable WEC also oscillates in the
chaotic zone and provides a low-energy output (see
Figure 12b). However, with the tristable WEC, one single
Figure 9 Wave energy capture width ratio versus wave frequency. (a) Bistable: a* = 0.30, b* = 0.50, K* = 1.0, L* = 1.0, A* = 0.20, C* = 0.25. (b)
Tristable: a* = 0.37, b* = 0.37, K* = 1.0, L* = 1.0, A* = 0.20, C* = 0.25
Figure 10 WEC motion characteristics (ω* = 0.15 and 0.55): (a) linear,
(b)bistable,(c) bistable, (d) tristable, and (e) Tristable. Non-dimensional
variables as defined in Eq. (14): t*istime,z* is heave displacement, ω*is
the incident wave frequency, A*
zis heave amplitude, and v*= dz*/dt*is
heave velocity
Journal of Marine Science and Application
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
inter-welloscillation mode occurs around the middle potential
well, and a relatively higher energy capture width ratio then
develops, which provides evidence of the benefit of introduc-
ing an extra potential well. For a relatively higher wave fre-
quency (P5: ω* = 1.22), the buoy does not cross the potential
barrier, and it becomes trapped in one of the potential wells
with a periodic intra-well oscillation (see Figure 12 d and e).
3.2 Effect of PTO Damping
Figure 13 shows a 2D color plot of the wave energy capture
width ratio in the (ω*, C*) domain. The upper limit of the
color scale is the same as that used in Figure 8(Ω=1.5).It
is evident that the optimal operational conditions for the
bistable mode are ω* = 0.56 and C* = 0.33, while the optimal
operational conditions for the tristable mode are ω* = 0.53 and
C* = 0.36. A comparison with the linear WEC (see Figure 8)
shows that the bistable and tristable responses reach higher
efficiencies (much higher than the optimal value of the linear
WEC: Ω= 0.49) in a broader wave frequency bandwidth and
a larger PTO damping range. However, if the PTO damping
parameter (C*) is increased, the optimal capture width ratio
firstly increases, reaches its maximum, and then begins to
decrease. In addition, the optimal operation frequency shifts
to lower values.
The white dashed line in Figure 13 corresponds with
the bistable and tristable WEC bifurcation diagram
shown in Figure 9(C* = 0.25). For the bistable WEC
(see Figure 13a), the high efficiency (red) region relates
to the left branch, and a sharp drop is then seen when
moving toward the chaotic zone and the right branch. In
contrast with the bistable response, there is no sharp
drop transition in the tristable response, and a signifi-
cantly high energy capture width ratio occurs over a
broader region in relation to the superior performance
defined by the right branch. This result proves that an
extra potential well (the middle potential well) can en-
large the response frequency bandwidth.
Figure 11 WEC motion characteristics (ω* = 0.61): (a) linear, (b) bistable, and (c)bistable
Journal of Marine Science and Application
Figure 12 WEC motion characteristics (ω* = 0.86 and 1.22): (a)linear,(b)bistable,(c) tristable, (d)bistable,and(e) tristable
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
3.3 Effect of Wave Amplitude
Figure 14 illustrates the variation in the wave energy capture
width ratio as a function of the wave amplitude and frequency
for both bistable and tristable responses. The optimal PTO
damping obtained through the results presented in Figure 13
(i.e., C* = 0.33, for bistable and C* = 0.36, for tristable) was
employed for the analysis of the influence of the wave ampli-
tude discussed in this section. During displacements of the
buoy with large amplitudes, the negative stiffness in both the
bistable and the tristable responses has relatively little influ-
ence. Therefore, as shown in Figure 14a for the bistable and
Figure 14b for the tristable cases, the peak energy capture
width ratio decreases as the wave amplitude increases. Also,
as shown in Figure 14a, the bistable WEC response captures
only a small amount of energy when the wave amplitude is
small (A
< 0.05) (in Figure 14, represented by the regions
below the white dashed line). This low-energy-absorption lim-
itation under low amplitude excitation conditions was previ-
ously observed, and is mentioned in. Thephysical reasoning is
that the small amount of energy carried by the small amplitude
waves is not sufficient to enable the buoy to cross the potential
barrier. This traps the buoy either within one of the bistable
potential wells or within the middle potential well in the
tristable case. In Figure 14b, which shows the results for the
tristable WEC, the low-energy-absorption limitation is even
more evident. The capture width ratio shows a very sharp drop
when a decrease in the wave amplitude occurs (A
< 0.17)
(represented by the region below the white dashed line), and
the buoy is trapped within the middle potential well.
In the above analysis, it is of note that the initial condition
(initial displacement, z
*, and initial velocity, v
*) of the
spring joint point (referred to as point O
in Figure 1)iszero
for both the bistable and tristable WECs (i.e., z
0). Therefore, different initial conditions were considered to
Figure 13 Wave energy capture width ratio in (ω*,C*) domain (K*=
1.0, L* = 1.0, A* = 0.2). (a) Bistable: a* = 0.30, b*=0.50.(b) Tristable:
a* = 0.37, b*=0.37
Figure 14 Wave energy capture width ratio in (ω*, A*) domain (K*=
1.0, L* = 1.0). (a)Bistable:a* = 0.30, b* = 0.50, C* = 0.33. (b)Tristable:
a* = 0.37, b* = 0.37, C* = 0.36
Journal of Marine Science and Application
investigate their effect on the responses of bistable and
tristable WECs. Figure 15 a presents the results for the bistable
WEC under different combinations of initial conditions at a small
wave amplitude (A* = 0.04), represented by the region below the
white dashed line in Figure 14a. Although the initial conditions
produce different responses in the transient phase, the buoy dis-
placement stabilizes after a few cycles, which indicates that the
initial conditions have a minimal effect on the response.
For the tristable WEC, the selected wave amplitude corre-
sponds to A* = 0.10. The results shown in Figure 15b indicate
that if the initial conditions are defined by z
0, the buoy overpasses the potential barriers and reaches a
high amplitude inter-well oscillation. However, other initial
conditions can trap the buoy within the middle potential well,
leading to small amplitude intra-well oscillations.
Figure 16 illustrates the energy capture width ratio in the
(ω*, A*) domain for the tristable WEC with initial conditions
defined by z
* = 0.4 and v
* = 0. A comparison with the
tristable WEC performance with zero initial conditions (see
Figure 14(b)) shows that the efficiency in the region below the
white dashed line is greatly enhanced and the low-energy-
absorption limitation is significantly alleviated. In addition,
there is limited change in the region above the white dashed
line, which shows no change in the energy capture width ratio,
and there is thus no detrimental effect on the tristable WECs
3.4 Effect of Spring End Positions
As described in Section 2.3, the combined parameters, a*and
b*, which define the spring end positions, determine whether
the system operates in a monostable, bistable, or tristable
mode. Two different waves (i.e., ω* = 0.53, A*=0.2andω*
=0.53,A* = 0.5) were chosen to investigate the influence of
geometry on the WECs performance, and the corresponding
results are shown in Figure 17.
The results indicate the WEC efficiency can be significant-
ly improved in both the bistable and tristable modes when a
suitable (a*, b*) combination is employed, whereas the use of
the monostable WEC always leads always to a poorer
In addition, the efficiency of the tristable WEC under rela-
tively higher amplitude conditions (see Figure 17b)issuperior
to that of the bistable WEC. Furthermore, compared with the
results shown Figure 6, the high-efficiency region in Figure 17
features a similar trend for both the displacement difference
and the potential barrier, which implies that these two factors
co-determine the WECs performance under certain external
Figure 15 Heave time history for different initial conditions. (a) Bistable:
A* = 0.04. (b) Tristable: A* = 0.10
Figure 16 Wave energy capture width ratio: tristable in (ω*, A*) domain
with initial conditions (z
* = 0.4, v
B. Liu et al.: Evaluation of the Double Snap-Through Mechanism on the Wave Energy ConvertersPerformance
3.5 Effect of Spring Stiffness
Figure 7showed that spring stiffness, K*, affects the intensity
of the depth of the potential curve associated with the double
snap-through mechanism. Further analysis is conducted in this
section to discuss the influence of spring stiffness on the wave
energy capture width ratio using the results presented in
Figure 18.
These results show a significant improvement in the WEC
efficiency that is obtained when a suitable K* range is
employed. For a small spring stiffness (i.e., K* = 0.5), the
potential curve depth is shallower, and this corresponds with
the minor negative stiffness introduced by the double snap-
through mechanism. As a result, the buoy trajectory (see blue
curve in Figure 19) follows an elliptical shape that is similar to
that of the linear WEC (K* = 0). However, when a suitable
spring stiffness is used (i.e., K* = 1.0), the buoy can cross the
potential barrier(s) and oscillate along a trajectory that has
reasonably high energy. If the spring stiffness is too high
(i.e., K* = 2.0), the threshold of the potential barrier becomes
too large, and the buoy easily becomes trapped and forms a
small amplitude oscillation within a single potential well.
4 Conclusions
This study applied a double snap-through mechanism to a
hemispherical point absorber wave energy converter (PA-
WEC) to evaluate the potential enhancement of its energy
absorption efficiency. The practical configuration devised
for the mechanism featured four symmetrical oblique springs
with an X-shape layout.
By adjusting the PAs geometric and physical parameters,
different multi-stable WEC dynamic behaviors were obtained.
A general mathematical formulation based on non-
dimensional parameters was used to calculate the time history
Figure 17 Wave energy capture width ratio in the (a*, b*) domain (C*=
0.36, K* = 1.0, L* = 1.0). (a)ω* = 0.53, A* = 0.2. (b)ω* = 0.53, A* = 0.5
Figure 18 Wave energy capture width ratio in (ω*, K*) domain (C*=
0.36, L*=1.0,A* = 0.2). (a) Bistable: a* = 0.30, b* = 0.50. (b) Tristable:
a* = 0.37, b*=0.37
Journal of Marine Science and Application
of the response for the multi-stable systems operating in reg-
ular waves. An analysis of the results included extensive com-
parisons between the performances of linear, bistable, and
tristable WEC modes and included an evaluation of the influ-
ence of certain practical parameters on the WEC energy cap-
ture width ratio.
Based on the above-mentioned analysis, the main conclu-
sions are listed as follows:
1) The double snap-through mechanism may cause the
WEC to undergo both bistable and tristable dynamic be-
haviors, depending on the suitable combination of rele-
vant parameters. The resulting effect may significantly
improve the energy capture efficiency of the system.
2) Compared with the bistable WEC, the tristable WEC fea-
tures a broader frequency bandwidth, which is associated
with the presence of the middle potential well.
3) A limitation of the WECs efficiency was identified in
relation to the possible occurrence of low-energy absorp-
tion under small wave amplitude excitations. This oc-
curred with the bistable WEC, and it was even more ob-
vious for the tristable WEC. However, for the tristable
WEC, such a limitation may be attenuated by imposing
appropriate initial conditions.
4) The geometric parameters (i.e., a*andb*) determine the
dynamic characteristics of the system and invoke the use
of either monostable, bistable, or tristable modes.
5) A large spring stiffness may cause the buoy to operate in
one potential well, and a small spring stiffness can cause
linear behavior. However, a suitable spring stiffness en-
ables the buoy to oscillate within a reasonably high ener-
gy trajectory.
Acknowledgements The authors acknowledge the financial support by
the Carlos Chagas Filho Foundation-FAPERJ (Grant No. E-26/202.600/
2019) to the research on Ocean Renewable Energy conducted in the
Subsea Technology Lab-COPPE/Federal University of Rio de Janeiro.
Funding This study is supported by the China ScholarshipCouncil under
Grant No. 201600090258, the National Key Research and Development
Program of China under Grant No. 2016YFC0303700, and the 111
Project under Grant No. B18054.
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... Based on the shapes of the potential energy curves, the bistable mechanisms can be divided into four categories, i.e., traditional bistable, multi-stable, adaptive bistable, and improved bistability, the details of which are listed in Table 3. [52] Ref. [65] Ref. [54] Ref. [76] Refs. [27], [34]- [38], [40]- [45], [31], [47], [65]- [66] [32], [54] [39], [46], [49], [76] [48], [50]- [53], [55]- [64] Younesian and Alam [65] proposed a bistable mechanism with two rigid links and two oblique stiffnesses. By adjusting the geometric parameters of the bitable mechanism, this structure can be achieved in different multi-stable states from bistable to four-stable, and the potential barrier can be reduced by increasing the number of steady states. ...
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In this work a three dimensional computational fluid dynamic (CFD) model has been constructed based on a 1/50 scale heaving point absorber wave energy converter (PAWEC). The CFD model is validated first via wave tank tests and then is applied in this study to investigate the joint effects of device geometry and power take-off (PTO) damping on wave energy absorption. Three PAWEC devices are studied with the following geometrical designs: a cylindrical flat-bottom device (CL); a hemispherical streamlined bottom design (CH) and a 90°-conical streamlined bottom structure (CC). A PTO force via varying damping coefficient is applied to compare the power conversion performances of the aforementioned devices. Free decay, wave-PAWEC interaction and power absorption tests are conducted via the CFD model. The results show that for CH and CC designs the added mass and hydrodynamic damping decrease by up to 60% compared with the CL device. Moreover, the CC design is the best of the three structures since its amplitude response increases by up to 100% compared with the CL. Applying an appropriate PTO damping to the CC device prominently increases the achievable optimal power by up to 70% under both regular and irregular waves (compared with the CL device).
Mechanical restoration of a point absorber (PA) wave energy converter (WEC) featured a nonlinear stiffness system built by conventional mechanical compression springs (NSMech). Numerical simulations conducted here considered irregular waves in a specific sea site. Results showed considerable improvement for a given suitable spring configuration (length and stiffness parameters). However, practical implementation of NSMech imposed limiting constraints on the geometrical and physical characteristics of the mechanical compression springs. Such constraints limited very much the feasible region of NSMech configurations, restricting significantly applications for wave energy converter. One alternative approach using pneumatic cylinder springs proved to overcome such a limitation aggregating obvious advantages with fewer elements and bringing more enhancement to the WEC performance.
The adaptive bistable point absorber wave energy converter (WEC) concept proposed by Zhang et al. (Applied Energy 2018(228), pp. 450–467) is further investigated by exploring the underlying mechanism and sensitivity related to the broadband energy harvesting of this device. The mechanism can be explained in two different ways: (i) the broadband power capture is due to the fact that the adaptive bistable mechanism introduces both negative (at low wave frequencies) and positive (at high wave frequencies) stiffness to the device; and (ii) an inherent ‘phase control’ feature is observed especially at low wave frequency region. The variation of initial conditions of floater motion only has an observable influence at intermediate wave frequencies corresponding to chaotic response of the floater. The broad frequency bandwidth can be realized through different combinations of main and auxiliary springs' stiffness. The effects of wave amplitude on broadband energy harvesting can be categorized according to an optimum amplitude (if existing) into a gradual reduction for a larger wave amplitude and a quick drop for a smaller amplitude.