Dynamic modeling and analysis of improved wave energy harvesters with novel X-structured nonlinear inerter mechanism

Abstract

In this paper, a new passive mechanical structure, that is, an X-structured wave energy conversion system with a nonlinear inerter is proposed. The JONSWAP wave spectrum is utilized to simulate irregular waves, and the dynamic equations are deduced by the energy method and one-dimensional linear potential wave theory. The system response is calculated to analyze the performance of the nonlinear system. The potential benefits of nonlinear inerter are discussed in-depth. Increased inertance will result in a gradual decrease in resonance frequency. Whereas, when the inertance is larger than a certain limit value, the resonance frequency will be almost unchanged. Additionally, increasing the inertance helps to suppress the displacement and velocity of the system. This allows higher power to be captured without causing major damage to the mechanical structure of the system. More importantly, increasing inertance is conducive to improving the maximum power generated. In general, a nonlinear WEC system will capture more power than its linear counterpart and the case with no inerter, especially when with a small initial installation parameter. However, when the initial installation parameter becomes large, the nonlinear inerter system will capture less power than the linear inerter system and inerter-free system. This study provides innovative and novel insight into designing nonlinear ocean energy converters exploiting geometrical nonlinearities.

Keywords

Nonlinear converters X-shaped structures nonlinear inerter performance enhancement irregular waves

Highlights

• A novel nonlinear-inerter-based X-structured ocean energy converter is for the first time proposed.

• The power generation with nonlinear inerters outperforms its linear counterpart and the case with no inerter.

• The initial installation parameter of the X-structure would also play an important role, for both the cases with and without nonlinear inerter.

• This study provides innovative and novel insight into designing nonlinear ocean energy converters exploiting geometrical nonlinearities.

1. Introduction

Wave energy is considered a sustainable power source due to its reliability, consistent availability, and abundant reserves (Gardner, 2015). The wave energy technology has been explored for many years and significant results have been achieved (Clement et al., 2002; Gunn and Stock-Williams, 2012). For wave energy converters (WECs) (Guo and Ringwood, 2021), thousands of models and theories have been proposed, including oscillating water columns, overtopping devices, and oscillating body wave energy converters (Zhang et al., 2022). Oscillating bodies, also known as point absorbers, show great promise as wave energy converters, particularly for offshore locations (Falcão, 2010).

In linear WEC systems, the converted power is typically low under non-resonant conditions, as real sea states do not conform to regular wave patterns (Zhao et al., 2023). This means that WEC and power-take-off (PTO) systems may not consistently reach their optimal operating conditions or resonate effectively. Consequently, researchers are exploring potential solutions to enhance the power capture performance of the oscillator WEC, including control methods (Paparella and Ringwood, 2016), hybrid approaches (Zhao et al., 2019), shape optimization (Shadmani et al., 2024), and PTO enhancement (Zou et al., 2019). Differently, this paper focuses on point absorbers to study how inerter nonlinearities improve their power performance.

Nonlinearities can greatly enhance vibrating systems in several ways (Fang et al., 2023; Yang et al., 2024). They can decrease resonant frequency without compromising load capacity (Yan et al., 2022), reduce resonant peaks while maintaining high-frequency vibration (Ding and Chen, 2019), and expand the frequency range of vibration suppression (Ibrahim, 2008). Utilizing these nonlinear characteristics can also greatly enhance the efficiency of energy harvesting (Tran et al., 2018). The X-structure/mechanism approach (Jing, 2022) has been proposed as a novel nonlinear mechanism, which drew inspiration from the skeletal systems of animals, and offers a dependable and versatile solution for applications that demand exceptional nonlinear stiffness and damping characteristics. In the realm of wave energy harvesting, X-structure devices may also have good potential when exploiting their nonlinearities.

Inerter is a passive mechanical component that consists of two nodes with distinct properties. Its terminals are directly proportional to the relative acceleration between them when a force is applied to them (Smith, 2020). A ratio known as inertance in kilograms is defined to express this proportionality. Through proper design, the inertance of an inerter can greatly surpass its actual weight (Zhang et al., 2019). Integrating inerter into X-structures may establish inertial coupling, thereby maximizing vibration transfer for optimal performance. However, the theoretical research on combining the excellent properties of the inerter to reduce the system’s natural frequency and regulate the system’s displacement and velocity with WEC systems is still very limited.

Although nonlinear stiffness mechanisms have been explored in some studies of nonlinear WEC systems, they are still rather simple in structure and do not take advantage of the X-structures in wave energy harvesting. Importantly, to the authors’ best knowledge, very few studies on WECs have been reported in terms of the nonlinear inerter mechanism. This makes the results of previous studies somewhat inadequate both in terms of power captured and frequency range. These are the main motivations of our present work.

In this paper, a new double-layer X-structure WEC system with an inerter, a spring, and an electromechanical harvester has been proposed. The X-structure within the system is specially designed, which could generate geometrically non-linearities (although the spring, the inerter, and the electromechanical harvester are all linear devices), especially the geometrically nonlinear inerter. It should be noted that the nonlinear inerter has the advantage of regulating the natural frequency of the system, limiting displacement and velocity of the floater, and increasing the power captured by the system. Comparing linear inerter and inerter-less situations, we found that, the nonlinear inerter solution is more advantageous, especially for the low-frequency case (i.e., irregular wave peak frequency is lower than the system resonance frequency).

The rest of the paper is organized as follows. The designs of the inerter-type nonlinear WEC system are presented in Section 2. In Section 3, dynamic modelings for the proposed system are obtained by using the JONSWAP spectrum and energy methods. In Section 4, system responses are calculated, and the nonlinear characteristics of the system are discussed in depth by comparing different parameters. Afterwards, conclusions are drawn in Section 5.

2. The proposed nonlinear inerter-type X-shaped wave energy converter

The X-structure-based wave energy harvesting system with geometrically nonlinear inerter arrangement is proposed. In this case, a linear inerter device is placed horizontally within the X-shaped structure, to trigger geometrically nonlinear property, as shown in Figure 1(a). To make comparisons, in Figure 1(b), a linear inerter device is placed vertically within the X-shaped structure. Compared with the horizontal one, it can be regarded as the counterpart linear case, that is, linear inerter arrangement.

Figure 1.

The inerter-type X-shaped ocean energy conversion systems: (a) Nonlinear inerter arrangement, (b) linear inerter arrangement.

The buoy on the water’s surface is a solid sphere with a uniform mass distribution that rises and falls with the wave motion. The mass of the sphere is $m$ , and the radius is $R$ . The buoy is rigidly connected to the bollard below. The bollard is also rigid. The PTO system, which is connected to the bollard, consists of a double-layer X-structure, which consists of six rods, an inerter, a spring, and an electromechanical harvester (they are all linear devices). Six bars form the basic frame of the X-structure. Two of the rods, AD and BC, are hinged together at the midpoints, connecting the upper and lower levels and their length is $2 l$ . The lowermost point of the entire PTO system is secured to the seabed and protected by a shield, and the electromechanical harvester is also connected horizontally between $C$ and $D$ . A linear spring with coefficient $k$ is placed horizontally between $A$ and $B$ . In the system shown in Figure 1(a), a linear inerter is connected horizontally between $A$ and $B$ . The inertance produced by the inerter is $b$ . In Figure 1(b) the same inerter is placed vertically between $F$ and $O$ . In addition, a structure with no inerter element at all will also be considered for comparing the influence of inerter.

The electrical power generation through the corresponding electromechanical harvester is regarded as the power created. Its damping coefficient is $C_{e}$ . For the magnet-coil type harvesters, some scholars modeled the electromechanical coupling coefficient $C_{e}$ using polynomial fitting (Kecik and Mitura, 2020; Mitura and Kecik, 2023, 2024). By using this approach, the coupling coefficient can be better justified as a more precise constant value for a certain range of displacement. However, for the present X-structure-based ocean converter, it is suitable to use a regenerative shock absorber as the mentioned electromechanical harvester. Generally, this type of regenerative shock absorber consists of a rack/pinion pair, a gearbox, and a generator motor (Li et al., 2012). The motion of the X-structure would drive the regenerative shock absorber to work, and the rotation of the generator motor would generate electricity. In this case, $C_{e}$ is the equivalent electrical damping of the electromagnetic generator unit, which is related to the external resistant load $R_{e}$ , the internal resistant load $R_{i}$ , and the inductance L (Wei et al., 2019), as shown in Figure 1. In detail, $C_{e} = \frac{{k_{g}}^{2} k_{e} k_{t}}{r^{2} η_{g} (R_{e} + R_{i})}$ would be the detailed expression of the equivalent electrical damping (Li et al., 2012), in which $k_{g}$ is the transmission ratio of the gear system, $η_{g}$ is the efficiency of the gear system, r is the radius of pinion gears, $k_{e}$ and $k_{t}$ are the voltage and torque constants of the generator.

In summary, the electromechanical coupling coefficient can be determined by the selection of the generator motor and the design of the gear transmission system. As a theoretical study, in this paper, a constant value of $C_{e}$ was assumed. In a future study, we will consider more detailed parameters to conduct some experimental validations to form a more comprehensive work.

The wave force will act directly on the buoy to drive the motion of the X-structure as well as the PTO system, as shown in Figure 2.

Figure 2.

Motion of the X-structure and PTO system driven by the wave.

In this work, we concentrate on the wave energy converter’s dynamic reaction and power collection capabilities in irregular waves. Because spherical buoys exhibit axial symmetry, only the Buoy’s heave motion is taken into account for simplicity, and motion in the direction of other degrees of freedom is disregarded. Because of the axisymmetric nature of the spherical buoy, only irregular waves incident in one direction are considered. It is also assumed that the water’s depth is unlimited.

3. Dynamic modeling of the inerter-type X-shaped wave energy converter

In this section, some geometric relationships are used to derive links between some key variables and then we use the Lagrange method to derive the dynamical equations of the structure. Irregular waves are generated by the superposition of numerous regular waves based on the rules of the JONSWAP spectrum. Also, the wave loads are divided into three components, which are added to the dynamic equations of the WEC system after obtaining their expressions separately. With these three steps, the dynamical equations of the inerter-type X-shaped WEC system can be fully obtained.

3.1. Dynamical equations of the inerter-type X-shaped WEC system

Note that the triangles $Δ F A B$ , $Δ O A B$ , $Δ O C D$ , and $Δ E C D$ are congruent because the pole lengths do not change. Figure 3 illustrates the geometric relationships of the X-structure during deformation.

Figure 3.

The geometric relationships within X-structure.

Based on geometric relationships, we have ( $y_{0}$ is both the original length of the spring and the initial distance between AB(CD))

y_{A B} = y_{C D} = 2 \sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}},

(1)

Δ y = y_{0} - y_{A B} = 2 \sqrt{l^{2} - z_{0}^{2}} - 2 \sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}},

(2)

{\dot{y}}_{A B} = {\dot{y}}_{C D} = - \frac{1}{2} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} .

(3)

3.1.1. Modeling of the horizontal inerter case

In Figure 1(a), the system’s kinetic energy is

T = \frac{1}{2} m {\dot{z}}^{2} + \frac{1}{2} b {\dot{y}}_{A B}^{2},

(4)

and the system’s potential energy is

U = \frac{1}{2} k {(Δ y)}^{2} .

(5)

Substituting them into the Lagrange equation

L = T - U,

(6)

\frac{d}{d t} (\frac{\partial L}{\partial \dot{z}}) - \frac{\partial L}{\partial z} = Q,

(7)

Q = C_{e} {\dot{y}}_{C D} = C_{e} {\dot{y}}_{A B},

(8)

in which Q represents the electromechanical force induced by the electromechanical harvester.

Considering equations (1)–(3), we have

\begin{array}{l} m \ddot{z} + \frac{b}{4} \ddot{z} \frac{{(z_{0} + \frac{1}{4} z)}^{2}}{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}} + \frac{1}{16} b l^{2} \frac{{\dot{z}}^{2} {(z_{0} + \frac{1}{4} z)}^{2}}{{[l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}]}^{2}} \\ + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} - (z_{0} + \frac{1}{4} z)] + \frac{1}{2} C_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} = 0 . \end{array}

(9)

3.1.2. Modeling of the vertical inerter case

In Figure 1(b), the system kinetic energy is

T = \frac{1}{2} m {\dot{z}}^{2} + \frac{1}{8} b {\dot{z}}^{2},

(10)

and the system potential energy equation remains the same. Substituting into the Lagrange equation. Considering equations (1)–(3), we have

[m + \frac{1}{4} b] \ddot{z} + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} - (z_{0} + \frac{1}{4} z)] + \frac{1}{2} C_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} = 0 .

(11)

3.1.3. Modelling of the inerter-less case

For X-structure WEC systems with only spring and electromechanical damping, the system kinetic energy is transformed into

T = \frac{1}{2} m {\dot{z}}^{2} .

(12)

Given that the WEC system’s potential energy and non-potential force remain the same, the final result is

m \ddot{z} + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} - (z_{0} + \frac{1}{4} z)] + \frac{1}{2} C_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} = 0 .

(13)

It should be mentioned that the above equation (equations (9), (11), and (13)) are the dynamical equation of the nonlinear X-structured inerter-type wave energy harvesting system without considering the wave force.

3.2. Descriptions of the wave forces

According to Nemoto et al. (Nemoto et al., 2022), the wave excitation can be divided into three parts, the excitation force $f_{e}$ , the hydro-static force $f_{a}$ , and the radiation force $f_{r}$ .

3.2.1. Hydro-static force

Regarding the buoyancy of the spherical buoy, it is specified that in equilibrium without any excitation, the draft line of the sphere passes through the center of the ball. Since the displacement of the buoy is much smaller than its radius, that is,

z ≪ R,

(14)

the buoyant force on the entire ball can be approximated by the hydro-static buoyant force on a cylinder. Its cross-sectional area is denoted by

S

. The calculation formula is as follows

S = π R^{2},

(15)

f_{a} = ρ g S z = ρ g π R^{2} z .

(16)

3.2.2. Wave excitation force

The real-world random wave response can be approximated as a superposition of multiple regular wave components. In this paper, the JONSWAP spectrum is utilized to simulate the irregular wave condition. The JONSWAP spectrum is expressed as (Hasselmann et al., 1973)

S (ω) = α H_{s}^{2} T_{p}^{- 4} ω^{- 5} \exp [- 1.25 {(T_{p} ω)}^{- 4}] χ^{\exp [- {(T_{p} ω - 1)}^{2} / (2 σ^{2})]},

(17)

where

S (ω)

is the spectrum density,

σ

and

α

are coefficients,

H_{s}

is the significant wave height,

T_{p}

is the spectral peak period (

T_{p} = \frac{1}{2 π ω_{p}}

), and

χ

is the peak lifting parameter.

A range of circular frequencies is needed to produce irregular waves with the JONSWAP spectrum. We define

ω_{n} = n Δ ω + ω_{0} (n = 0, 1, . . . . . ., M - 1),

(18)

where

ω_{0}

is the minimum circle frequency,

n

is the number of circle frequency,

M

denotes a total of

N

components, and

Δ ω

is the interval between two neighboring circle frequency elements.

The formula for the amplitude of the nth regular wave component is given by Zhang and Yang (2015), shown as below

A_{ω_{n}} = \sqrt{2 S (ω_{n}) Δ ω},

(19)

η_{n} = A_{ω_{n}} \cos (ω_{n} t + θ_{n}),

(20)

η = \sum_{n = 0}^{N - 1} η_{n} = \sum_{n = 0}^{N - 1} A_{ω_{n}} \cos (ω_{n} t + θ_{n}),

(21)

where

θ_{n}

is the random real number, which is used to model the irregular phase response of the irregular sea state and

η

is the relative displacement of the sea surface.

Theoretically, the wave excitation force caused by irregular waves can be superimposed by the wave excitation forces caused by countless regular waves with infinitely small amplitudes (Zhang et al., 2014). For the wave model in this paper, only the radiation potential and the incident potential in one direction need to be considered. The wave excitation force modulus $| f_{e} |$ of a spherical buoy is proportional to $A_{ω}$ , that is, $| f_{e} | = Γ (ω) A_{ω}$ , where $Γ_{z} (ω)$ is the excitation force coefficient and $A_{ω}$ is the amplitude of the regular wave (Falnes and Perlin, 2003). In the case of an axisymmetric body vibrating in deep water, Falnes and Perlin (2003) determined the excitation force coefficient to be

Γ_{z} (ω) = {(\frac{2 g^{3} ρ B_{z} (ω)}{ω^{3}})}^{1 / 2} .

(22)

The above equation is also known as the Haskind relationship (Haskind, 1958), where $B_{z} (ω)$ denotes the radiative damping coefficient of the floating body.

After obtaining the radiative damping coefficients generated during the oscillation of the floating body, the excitation force induced by the countless regular waves is summed up, and an expression for the excitation force is finally obtained as

f_{e} = \sum_{i} Γ_{z} (ω_{i}) A_{ω_{i}} \cos (ω_{i} t + θ_{i}) .

(23)

3.2.3. Wave radiation force

The magnitude of the wave radiative force depends on a kernel function of the heave oscillations and a radiative drag coefficient, which, according to Cummins (1962), can be expressed in the form of convolution integral as

f_{r} = \int_{0}^{t} k_{z} (t - τ) \dot{z} (τ) d τ,

(24)

k_{z} (t) = \frac{2}{π} \int_{0}^{\infty} B_{z} (ω) \cos ω t d ω .

(25)

Nevertheless, the analysis of the nonlinear issue is computationally demanding and difficult if the convolution integral is computed in the time domain. Different techniques can be used to estimate the convolution integral in order to prevent such issues.

Ogilvie (1964) determined the links between the time domain and frequency domain model parameters using the Fourier transform, which are listed in Appendix A.

In this case, the kernel function is transformed by the Fourier transform into

k_{z} (j ω) = \int_{0}^{\infty} k_{z} (t) e^{- j ω t} d ω = B_{z} (ω) + j ω [A_{z} (ω) - A_{\infty}],

(26)

in which

A_{\infty}

is the added mass coefficient at the infinite wave frequency.

According to Zhang and Yang (2015), the final form of the radiative force can be expressed by the state-space equation

\begin{array}{l} \dot{X} (t) = A X (t) + B \dot{z} (t), \\ f_{r} (t) = C X (t), \end{array}

(27)

in which

X

is referred to as the state vector and provides an internal description of the system. To be specific,

X

represents the system output when the input is the heave velocity

\dot{z} (t)

. The coefficients of the state-space equations are

A

B

, and

C

Obtaining the transfer function of the system by Laplace transforms

\hat{K} (s) = H (s) = \frac{Y (s)}{U (s)} = \frac{p_{n} s^{n} + p_{n - 1} s^{n - 1} + . . . + p_{0}}{s^{r} + q_{r - 1} s^{r - 1} + . . . + q_{0}} .

(28)

It’s worth noting that equation (28) is accurate only when the order of the system tends to infinity.

Restricting $s$ to only take imaginary values in equation (28) allows us to derive the model’s frequency response. By comparing these two equations (28) and (26), we can obtain the values of $A$ , $B$ , and $C$ by means of parametric approximations. Please see Appendix A for details.

When using the iterative method to obtain the coefficient matrices $A$ , $B$ , and $C$ , we must include some constraints to ensure the correctness and stability of the results. According to Perez and Fossen (2011), some constraints are chosen to be added to the calculation. The specific constraints selected are shown in Table 1 in Appendix A.

3.3. Ultimate dynamic equations of the conversion system

Finally, equations (9), (11), and (13) can be updated and written as follows, using the state space equations to construct the convolution integrals and accounting for the hydro-static and excitation forces. The initial conditions chosen for this paper can be found in Appendix B.

(1) Non-linear inerter arrangement:

{\begin{array}{c} \begin{array}{l} (m + A_{\infty}) \ddot{z} + \frac{b}{4} \ddot{z} \frac{{(z_{0} + \frac{1}{4} z)}^{2}}{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}} + \frac{1}{16} b l^{2} \frac{{\dot{z}}^{2} {(z_{0} + \frac{1}{4} z)}^{2}}{{[l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}]}^{2}} + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} \\ - (z_{0} + \frac{1}{4} z)] + ρ g S z + \frac{1}{2} c_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} + C X (t) = f_{e} \end{array} \\ \dot{X} (t) = A X (t) + B \dot{z} (t) \end{array}

(29)

(2) Linear inerter arrangement:

{\begin{array}{c} \begin{array}{l} [m + \frac{b}{4} + A_{\infty}] \ddot{z} + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} - (z_{0} + \frac{1}{4} z)] \\ + \frac{1}{2} c_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} + ρ g S z + C X (t) = f_{e} \end{array} \\ \dot{X} (t) = A X (t) + B \dot{z} (t) \end{array}

(30)

(3) Inerter-less model:

{\begin{array}{c} \begin{array}{l} [m + A_{\infty}] \ddot{z} + k [\frac{\sqrt{l^{2} - z_{0}^{2}} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} - (z_{0} + \frac{1}{4} z)] \\ + \frac{1}{2} c_{e} \frac{\dot{z} (z_{0} + \frac{1}{4} z)}{\sqrt{l^{2} - {(z_{0} + \frac{1}{4} z)}^{2}}} + ρ g S z + C X (t) = f_{e} \end{array} \\ \dot{X} (t) = A X (t) + B \dot{z} (t) \end{array}

(31)

4. Results and discussions

For the above-proposed system of nonlinear differential equations, in this paper, we use the adaptive 4^th-order Runge–Kutta method with transformable step size to solve the equations. After obtaining the displacement $z$ and the velocity $\dot{z}$ of the system, the power performance of the system can be evaluated.

4.1. Predicted power generation

Initial conditions are selected, and calculation time is set from 0 to $1600 s$ . The system parameters are set as: rod length $l = 2 m$ , inertance $b = 50000 kg$ , spring elasticity coefficient $k = 15000 N / m$ , electromechanical damping coefficient $c_{e} = 50000 N / (m / s)$ , significant wave height $H_{s} = 2 m$ , and peak frequency $ω_{p} = 1.3 rad / s$ .

The displacement, velocity, and acceleration of the lifting and sinking motion are obtained, as shown in Figure 4.

Figure 4.

Buoy displacement and velocity over time: (a) Displacement over time, (b) velocity over time.

First, we define an initial installation parameter as the ratio of the initial distance (equilibrium position) of $A B$ (or $C D$ ) to the length $l$ of the rod, as below

γ = \frac{y_{0}}{l} (0 < γ < 2)

(32)

Time-average power can be expressed as

P = \frac{1}{T} \int_{0}^{T} C_{e} {\dot{y}}_{C D}^{2} d t,

(33)

in which

{\dot{y}}_{C D}

is related to the displacement

z

and the velocity

\dot{z}

of the system (refer to equation (3)).

Also, instantaneous power is

p = C_{e} {\dot{y}}_{C D}^{2} .

(34)

The instantaneous power can be obtained, as shown in Figure 5. As seen in the figure, the system can reach a maximum instantaneous power of 14 kW, and the average power can be up to 1208 W, which is not even the limit of the system’s performance when it reaches resonance.

Figure 5.

Instantaneous power and average power (in order to eliminate the power instability caused by the transient effects of the system, the captured power generated in the first 100 seconds is ignored in the calculation).

4.2. Effect of inerter on average power

The inerter, as an inertial element attached within the X-structure, mainly affects the coefficients of ${\dot{z}}^{2}$ and $\ddot{z}$ in the dynamic equations. It will lead to a change in the resonance frequency, which in turn affects the amount of power captured by the WEC system.

To assess the impact of the inertance on the system’s collected power, an appropriate interval for the value of $b$ was selected, as shown in Figure 6. It should be mentioned that, as the parameters $k = 5000 N / m$ and $C_{e} = 5000 N / (m / s)$ are selected, we changed the value of $ω_{p}$ into 2 rad/s accordingly. As shown in Figure 6(a), in most cases, the energy capture power of nonlinear inerter is much higher than those of linear inerter and no inerter. But when the inertance in a non-linear inerter system is too large ( $b > 58500 kg$ ), its ability to capture power is greatly reduced. Also, according to Figure 6(b), when $0 < b < 50000 kg$ , the natural frequency of the system varies almost linearly with $b$ , but when $b > 50000 kg$ , the natural frequency no longer varies with $b$ or varies much less.

Figure 6.

Effect of inertance $b$ on the system ( $ω_{p} = 2 rad / s$ , $H_{s} = 2 m$ , $γ = 0.6$ ): (a) Effect of change in inertance on time-averaged power, (b) effect of inerter on system natural frequency ( $k = 5000 N / m$ , $C_{e} = 5000 N / (m / s)$ ).

The inerter can adjust the natural frequency of the system, and in actual sea conditions, the wave’s peak frequency will fluctuate continuously. Because of that, the tuned inerter WEC system can be used to follow the peak frequency with the change of the inertance by some control means, such as the static admittance control (Nemoto et al., 2022), the performance guaranteed control (Asai and Sugiura, 2021), and the maximum power point tracking algorithm (Han et al., 2022), to obtain a larger capture frequency.

When the peak frequency $ω_{p}$ approaches the resonant frequency of the system, the average power would be the largest. Herein, we define the largest average power as the maximum average power. As can be seen in Figure 7, increasing the inertance will contribute to the increase of the power generation. However, for linear systems, we can see that the enhancement is minimal.

Figure 7.

Effect of inertance on maximum average power ( $ω_{p} = 1.3 rad / s$ , $H_{s} = 2 m$ , $γ = 0.6$ ).

From Figure 8(a), for the nonlinear inerter design, it can be seen that increasing the inertance will lead to an decrease of resonance frequency of the system. Also, as the inertance increases, the average power generated at system resonance will increase, while the frequency range with higher values of captured power will become smaller.

Figure 8.

Effect of inertance on the power-frequency plot ( $H_{s} = 2 m$ , $γ = 0.4$ ): (a) Nonlinear inerter, (b) linear inerter.

When $b$ is very large, if the peak frequency is not near the resonant frequency of the system, the captured power will decrease dramatically, and it will even be lower than the captured power of the linear system. Therefore, increasing $b$ will not always benefit the system, and it requires consideration of the specific sea state and the combination of other components. It should be mentioned that, herein we use $γ = 0.4$ instead for the calculations of Figures 8(a) and (b). Compared to $γ = 0.6$ , $γ = 0.4$ is a smaller case, which can make the nonlinear inerter mechanism more effective.

For the linear inerter system, as shown in Figure 8(b), the larger the inertance is, the larger the captured power is in the low-frequency range. However, when the peak frequency exceeds a certain value, a larger inertance will result in a smaller captured power instead.

4.3. Significant wave height’s impact on average power

In the JONSWAP spectrum, the square of the significant wave height corresponds to the amplitude of the regular wave components, as equations (19)–(21) have shown. On the other hand, a linear superposition of a series of regular wave components results in an irregular wave sequence, indicating that the linear WEC power achieved is proportional to the square of the effective wave height.

For oscillating WECs with nonlinear PTO systems, it could be much different. To verify it, $ω_{p} = 1.37 rad / s$ (the linear inerter system and the inerter-free system have reached the resonance frequency, but the non-linear inerter system has not yet reached resonance.) was selected to examine how significant wave height affects the power collection efficiency. The results are shown in Figure 9.

Figure 9.

Effect of significant wave height on average power ( $ω_{p} = 1.37 rad / s$ , $γ = 0.6$ , $b = 10000 kg$ ).

As can be seen from the figure, although the nonlinear inerter system has not yet reached resonance, its captured power is still larger than the other two. A nonlinear inerter wave energy converter system’s energy absorption is not directly proportional to the square of the significant wave height. Although a linear inerter WEC system captures more power than inerter-free system, the difference between the two is relatively small.

4.4. Effects of initial installation parameter on average power

In this subsection, the WEC system’s average power is assessed by taking $γ = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8$ ( $∠ A F B \approx 11, 23, 35, 47, 60, 74, 89, 106, 128 degrees$ , see Figure 3), as well as setting the peak frequency $ω_{p} = 1.37 rad / s$ , the significant wave height $H_{s} = 2 m$ , and $l = 2 m$ .

It is evident from Figure 10 that when the peak frequency and effective wave height are kept constant, the nonlinear WEC system can obtain a relatively high power with a smaller $γ$ . In addition, the geometrically nonlinear inerter mechanism can greatly improve the system captured power. Namely, the nonlinear inerter design has a higher power output compared to the linear inerter and inerter-less designs, especially when $γ$ is smaller than 1.2.

Figure 10.

Average power versus initial installation parameter $γ$ for three different configurations ( $ω_{p} = 1.37 rad / s$ , $H_{s} = 2 m$ , $b = 10000 kg$ ).

In addition, it is worth stating that when $γ$ is too small, the displacement and velocity of the heaving motion of the nonlinear inerter system can be too large and even exceed the limits of the system, causing damage to the mechanical structure of the WEC system.

Taking the peak frequency $ω_{p} = 0.25, 0.5, . . . . . ., 4 rad / s$ with an interval up to $0.25 rad / s$ , the average power curves at various $γ$ can be obtained, as seen in Figures 11 –13. It is worth mentioning that the parameters are set as: the significant wave height $H_{s} = 2 m$ , $b = 20000 kg$ , and $l = 2 m$ , and the random phase $θ_{i}$ of the regular wave should be taken from the same set of data.

Figure 11.

Captured power versus peak frequency: Horizontal arrangement of inerter.

Figure 12.

Captured power versus peak frequency: Vertical arrangement of inerter.

Figure 13.

Captured power versus peak frequency: WEC system without inerter.

It has been known that, for WECs without an inerter, the resonant frequency of the WEC system varies significantly with the initial installation parameter $γ$ . This is because the initial installation parameter directly relates to the installation angle of the X-structure (Please see Figure 1). Our previous studies have reported that the resonant frequency of the X-structured system would vary with the installation angle (Li et al., 2018; Li and Jing, 2019, 2021). For the new cases with a nonlinear inerter, the inerter b and the initial installation parameter $γ$ would contribute to the resonant frequency of the WEC system simultaneously. In this case, the trend of the resonant frequency change has been reported in Figure 6(b). Herein, from Figures 11 –13, we can see that the average power obtained by the WEC system approaches the peak value when the peak frequency of the irregular waves approaches the system’s resonant frequency.

As shown in Figure 11, when the inerter is placed horizontally, in the low-frequency region (where the system’s resonance frequency is higher than the irregular waves’ peak frequency), the nonlinear WEC system captures more power with a smaller initial installation parameter $γ$ ; whereas in the high-frequency region (when the irregular waves’ peak frequency exceeds the system’s resonance frequency), the system’s collected power decreases with increasing initial installation parameter.

However, for systems where the inerter is placed vertically (Figure 12) and for systems without the inerter (Figure 13), regardless of how the peak frequency of the irregular waves varies, the system’s collected power increases with a smaller initial installation parameter $γ$ . Overall, regarding the power gathered by the WEC system, the non-linear inerter outperforms the linear inerter under the same wave circumstances and geometrical mechanism, whereas the linear inerter outperforms the no inerter system.

5. Conclusions

This study examines the oscillator WEC’s power capture capability in irregular waves, using a nonlinear inerter-type X-structure. First, we propose a new inerter enhanced X-shaped wave energy converter and then derive the dynamic equations of such structure using the JONSWAP spectrum and energy method. After solving the equations, the following inferences can be made in light of the numerical result.

(a) Changing the value of inertance can adjust the natural frequency of the WEC system, which is particularly noticeable for systems with smaller $k$ and $C_{e}$ . But there exists a limiting value of inertance, regardless of the values of $k$ and $C_{e}$ , and when $b$ is larger than this value, the natural frequency of the system will no longer vary with $b$ . In addition, increasing the inertance can improve the captured power at system resonance, especially for nonlinear inerter systems. However, increasing the value of the inertance will result in a narrower frequency capture range of the system, and its practicality will deteriorate, so a trade-off has to be considered.

(b) The power captured by the oscillator WEC with a nonlinear inerter-type X-structure reaches its maximum value when the peak frequency of the irregular wave approaches the nonlinear natural frequency of the WEC system.

(c) An increase in the significant wave height $H_{s}$ of the JONSWAP spectrum results in an increase in the power captured by the WEC system, while other parameters remain constant. This effect is more pronounced for the cases of X-structures with nonlinear inerters.

(d) For the X-structured WEC systems with nonlinear inerters, in the low-frequency region (where the system’s resonance frequency is higher than the irregular waves’ peak frequency), nonlinear WEC systems capture more power with a relatively small initial installation parameter. While in the high-frequency region (when the irregular waves’ peak frequency exceeds the system’s resonance frequency), as initial installation parameter decreases, the captured power of the system will decrease rapidly and can even fall below the capture power of a linear system. Importantly, for the X-structured WEC systems with and without linear inerter, the smaller the $γ$ , the higher the captured power of the system.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers and the editors for their relevant comments and useful suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (No. 12302025 and No. 12272323), the Natural Science Foundation of Sichuan Province of China (No. 2023NSFSC1294), the Fundamental Research Funds for the Central Universities (No. 2682023CX054), the International Postdoctoral Exchange Fellowship Program: Talent-Introduction Program (No. YJ20220310).

ORCID iD

Meng Li

Appendix

References

Asai

Sugiura

(2021) Numerical evaluation of a two-body point absorber wave energy converter with a tuned inerter. Renewable Energy 171: 217–226.

Clement

McCullen

Falcao

, et al. (2002) Wave energy in Europe: current status and perspectives. Renewable and Sustainable Energy Reviews 6(5): 405–431.

Cummins

(1962) The impulse response function and ship motion. Report 1661, Department of the Navy, David W. Taylor Model Basin, Hydromechanics Laboratory, Research and Development Report.

Ding

Chen

(2019) Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators. Nonlinear Dynamics 95(3): 2367–2382.

Falcão

AFO

(2010) Wave energy utilization: a review of the technologies. Renewable and Sustainable Energy Reviews 14(3): 899–918.

Falnes

Perlin

(2003) Ocean waves and oscillating systems: linear interactions including wave-energy extraction. Applied Mechanics Reviews 56(1): B3.

Fang

Chen

Zhao

, et al. (2023) Simultaneous broadband vibration isolation and energy harvesting at low frequencies with quasi-zero stiffness and nonlinear monostability. Journal of Sound and Vibration 553: 117684.

Gardner

(2015) The Outlook for Energy: A View to 2040. In 2015 EDI, Kiawah Island, SC, USA, 12–15 April 2015, pp. 37.

Gunn

Stock-Williams

(2012) Quantifying the global wave power resource. Renewable Energy 44: 296–304.

10.

Guo

Ringwood

(2021) Geometric optimisation of wave energy conversion devices: a survey. Applied Energy 297: 117100.

11.

Han

Maeda

Itakura

, et al. (2022) Experimental study on the wave energy harvesting performance of a small suspension catamaran exploiting the maximum power point tracking approach. Ocean Engineering 243: 110176.

12.

Haskind

(1958) Oscillation of a cascade of thin sections in incompressible flow. Journal of Applied Mathematics and Mechanics 22: 349–354.

13.

Hasselmann

Barnett

Bouws

, et al. (1973) Measurements of wind-wave growth and swell decay during the joint north sea wave project. Ergaenzungsheft zur Deutschen Hydrographischen Zeitschrift, Reihe A 8: 1–95.

14.

Ibrahim

(2008) Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration 314(3–5): 371–452.

15.

Jing

(2022) The X-structure/mechanism approach to beneficial nonlinear design in engineering. Applied Mathematics and Mechanics 43(7): 979–1000.

16.

Kecik

Mitura

(2020) Energy recovery from a pendulum tuned mass damper with two independent harvesting sources. International Journal of Mechanical Sciences 174: 105568.

17.

Jing

(2019) Novel tunable broadband piezoelectric harvesters for ultralow-frequency bridge vibration energy harvesting. Applied Energy 255: 113829.

18.

Jing

(2021) A bistable X-structured electromagnetic wave energy converter with a novel mechanical-motion-rectifier: design, analysis, and experimental tests. Energy Conversion and Management 244: 114466.

19.

Zuo

Luhrs

, et al. (2012) Electromagnetic energy-harvesting shock absorbers: design, modeling, and road tests. IEEE Transactions on Vehicular Technology 62(3): 1065–1074.

20.

Zhou

Jing

(2018) Improving low-frequency piezoelectric energy harvesting performance with novel X-structured harvesters. Nonlinear Dynamics 94: 1409–1428.

21.

Mitura

Kecik

(2023) Investigation of electromechanical coupling characteristic of a double magnet system. Materials Research Proceedings 30: 55–60.

22.

Mitura

Kecik

(2024) Energy recovery from a system with double magnet. Analytical approach. Archive of Mechanical Engineering 71(1): 73–85.

23.

Nemoto

Takino

Tsukamoto

, et al. (2022) Numerical study of a point absorber wave energy converter with tuned variable inerter. Ocean Engineering 257: 111696.

24.

Ogilvie

(1964) Recent progress toward the understanding and prediction of ship motions. In Proceedings of the 5th symposium on Navan hydrodynamics. Bergen, Norway, 3–79.

25.

Paparella

Ringwood

(2016) Optimal control of a three-body hinge-barge wave energy device using pseudospectral methods. IEEE Transactions on Sustainable Energy 8(1): 200–207.

26.

Perez

Fossen

(2011) Practical aspects of frequency-domain identification of dynamic models of marine structures from hydrodynamic data. Ocean Engineering 38(2-3): 426–435.

27.

Shadmani

Nikoo

Gandomi

, et al. (2024) Advancements in optimizing wave energy converter geometry utilizing metaheuristic algorithms. Renewable and Sustainable Energy Reviews 197: 114398.

28.

Smith

(2020) The inerter: a retrospective. Annual Review of Control, Robotics, and Autonomous Systems 3(1): 361–391.

29.

Tran

Ghayesh

Arjomandi

(2018) Ambient vibration energy harvesters: a review on nonlinear techniques for performance enhancement. International Journal of Engineering Science 127: 162–185.

30.

Wei

Zhang

, et al. (2019) A tunable nonlinear vibrational energy harvesting system with scissor-like structure. Mechanical Systems and Signal Processing 125: 202–214.

31.

Yan

Wei

, et al. (2022) Nonlinear compensation method for quasi-zero stiffness vibration isolation. Journal of Sound and Vibration 523: 116743.

32.

Yang

Liu

Luo

, et al. (2024) Improving the performance of nonlinear isolator through triboelectric nanogenerator damper integrating energy harvesting. Energy 293: 130722.

33.

Zhang

Yang

(2015) Power capture performance of an oscillating-body WEC with nonlinear snap through PTO systems in irregular waves. Applied Ocean Research 52: 261–273.

34.

Zhang

Yang

Xiao

(2014) Numerical study of an oscillating wave energy converter with nonlinear snap-through power-take-off systems in regular waves. In Twenty-fourth international ocean and polar engineering conference. Busan, Korea, 15–20 June, 522–527.

35.

Zhang

Cao

Pan

(2019) Inerter system and its state-of-the-art. Engineering Mechanics 36(10): 8–27.

36.

Zhang

Zhou

, et al. (2022) Recent advances in wave energy converters based on nonlinear stiffness mechanisms. Applied Mathematics and Mechanics 43(7): 1081–1108.

37.

Zhao

Zou

Yan

, et al. (2019) A water-proof magnetically coupled piezoelectric-electromagnetic hybrid wind energy harvester. Applied Energy 239: 735–746.

38.

Zhao

Zou

Xie

, et al. (2023) Mechanical intelligent wave energy harvesting and self-powered marine environment monitoring. Nano Energy 108: 108222.

39.

Zou

Zhao

Gao

, et al. (2019) Mechanical modulations for enhancing energy harvesting: principles, methods and applications. Applied Energy 255: 113871.