Response behavior of bi-stable point wave energy absorbers under harmonic wave excitations

Abstract

To expand the narrow response bandwidth of linear point wave energy absorbers (PWAs), a few research studies have recently proposed incorporating a bi-stable restoring force in the design of the absorber. Such studies have relied on numerical simulations to demonstrate the improved bandwidth of the bi-stable absorbers. In this work, we aim to understand how the shape of the bi-stable restoring force influences the effective bandwidth of the absorber. To this end, we use perturbation methods to obtain an approximate analytical solution of the nonlinear differential equations governing the complex motion of the absorber under harmonic wave excitations. The approximate solution is validated against a numerical solution obtained via direct integration of the equations of motion. Using a local stability analysis of the equations governing the slow modulation of the amplitude and phase of the response, the loci of the different bifurcation points are determined as function of the wave frequency and amplitude. Those bifurcation points are then used to define an effective bandwidth of the absorber. The influence of the shape of the restoring force on the effective bandwidth is also characterized by generating design maps that can be used to predict the kind of response behavior (small amplitude periodic, large amplitude periodic, or aperiodic) for any given combination of wave amplitude and frequency. Such maps are critical toward designing efficient bi-stable PWAs for known wave conditions.

Appendices

Appendix A: Eigensystem realization algorithm

Consider the following single-input single-output discrete-time dynamical system:

$$\begin{aligned} \begin{aligned} \mathbf{x }_{k+1}&=\mathbf{A }\mathbf{x }_k + \mathbf{B }u_k,\\ h_{k}&= \mathbf{C }\mathbf{x }_k + \mathbf{D } u_k, \end{aligned} \end{aligned}$$

(A.1)

and a discrete-time scalar input u:

$$\begin{aligned} u_{k}^{\delta } \equiv u^{\delta } (k \Delta t)= {\left\{ \begin{array}{ll} {1} &{} \text {if } k=0\\ 0 &{} \text {if } k=1,...\infty \end{array}\right. } \end{aligned}$$

(A.2)

According to linear system theory, the discrete-time impulse response data $h_{k}^{\delta }$ could be expressed as:

$$\begin{aligned} h_{k}^{\delta } \equiv h^{\delta }(k \Delta t)=\mathbf{C } \mathbf{A }^{k} \mathbf{B }, \quad (k=0,1,...., \infty ) \end{aligned}$$

(A.3)

In our analysis, the discrete-time impulse response data are obtained through substituting the radiation damping coefficients $B(\omega _i)$ into Eq. (3). The next step is to proceed by constructing the generalized Hankel matrix $\mathbf{H }_{r \times s} (g)$ for $g=0,1$ which consists of r rows and s columns, generated by stacking time-shifted impulse response data in the following order:

$$\begin{aligned} \mathbf{H }_{r \times s} (g) = \begin{pmatrix} h_{g}^{\delta } &{} h_{g+1}^{\delta } &{} \dots &{} h_{g+s-1}^{\delta } \\ h_{g+1}^{\delta } &{} h_{g+2}^{\delta } &{} \dots &{} h_{g+s}^{\delta } \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ h_{g+r-1}^{\delta } &{} h_{g+r}^{\delta } &{} \dots &{} h_{g+r+s-2}^{\delta } \end{pmatrix} \end{aligned}$$

(A.4)

Using Eq. (A.3), we can express the generalized Hankel matrix in Eq. (A.4) in terms of the realized state-space matrices $\mathbf{A }_r$, $\mathbf{B }_r$ and $\mathbf{C }_r$ as:

$$\begin{aligned}&\mathbf{H }_{r \times s} (g) \nonumber \\&\quad =\begin{pmatrix} \mathbf{C }_r \mathbf{A }_r^{g} \mathbf{B }_r &{} \mathbf{C }_r \mathbf{A }_r^{g+1} \mathbf{B }_r &{} \dots &{} \mathbf{C }_r \mathbf{A }_r^{g+s-1} \mathbf{B }_r \\ \mathbf{C }_r \mathbf{A }_r^{g+1} \mathbf{B }_r &{} \mathbf{C }_r \mathbf{A }_r^{g+2} \mathbf{B }_r &{} \dots &{} \mathbf{C }_r \mathbf{A }_r^{g+s} \mathbf{B }_r \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{C }_r \mathbf{A }_r^{g+r-1} \mathbf{B }_r &{} \mathbf{C }_r \mathbf{A }_r^{g+r} \mathbf{B }_r &{} \dots &{} \mathbf{C }_r \mathbf{A }_r^{g+r+s-2} \mathbf{B }_r \end{pmatrix}\nonumber \\ \end{aligned}$$

(A.5)

which could be reduced as:

$$\begin{aligned} \mathbf{H }_{r \times s} (g) = {\mathcal {O}} \mathbf{A }_r^{g} {\mathcal {C}} \end{aligned}$$

(A.6)

where

$$\begin{aligned} {\mathcal {O}}&=\left( \mathbf{C }_r \quad \mathbf{C }_r \mathbf{A }_r \quad \dots \quad \mathbf{C }_r \mathbf{A }_r^{r-1}\right) ^{T}\\ {\mathcal {C}}&=\left( \mathbf{B }_r \quad \mathbf{A }_r \mathbf{B }_r \quad \dots \quad \mathbf{A }_r^{r-1} \mathbf{B }_r\right) ^{T} \end{aligned}$$

are, respectively, the generalized observability and controllability matrices, with observability and controllability indices of r and s. Upon taking the singular value decomposition (SVD) for the first Hankel matrix $\mathbf{H }_{r \times s} (0)$, we get the following definition:

$$\begin{aligned} \begin{aligned} \mathbf{H }_{r \times s} (0)&= \mathbf{U } \Sigma \mathbf{V }^{T}\\&= \begin{pmatrix} \tilde{\mathbf{U }} &{} \mathbf{U }_{t}\\ \end{pmatrix} \begin{pmatrix} \tilde{\Sigma } &{} 0\\ 0 &{} \Sigma _{t} \\ \end{pmatrix} \begin{pmatrix} \tilde{\mathbf{V }}^T\\ \tilde{\mathbf{V }}_{t}^T \\ \end{pmatrix} \\&\approx \tilde{\mathbf{U }} \tilde{\Sigma } \tilde{\mathbf{V }}^T \end{aligned} \end{aligned}$$

(A.7)

where

$$\begin{aligned} \tilde{\mathbf{U }}^T \tilde{\mathbf{U }}&=\mathbf{I }\\ \tilde{\mathbf{V }}^T \tilde{\mathbf{V }}&=\mathbf{I }\\ \tilde{\Sigma }&= \begin{pmatrix} \sigma _1 &{} &{} &{} \\ &{} \sigma _2 &{} &{} \\ &{} &{} \ddots &{} \\ &{} &{} &{} \sigma _N\\ \end{pmatrix} \end{aligned}$$

The diagonal matrix $\tilde{\Sigma }$ which is constructed from the first $N \times N$ block of $\Sigma$ contains the dominant singular values $\sigma _i$ in the following order $(\sigma _1 \ge \sigma _2 \ge \dots \ge \sigma _N \ge 0)$, while $\Sigma _t$ contains the small truncated singular values. This truncation step is very vital for reducing the order of the realized state-space model; such that the realized dynamics matrix $A_r$ has size N. Also, vectors in $\tilde{\mathbf{U }}$ and $\tilde{\mathbf{V }}^T$ contain the dominant modes associated with the singular values retained in $\tilde{\Sigma }$. As a result, the product $\tilde{\mathbf{U }} \tilde{\Sigma } \tilde{\mathbf{V }}^T$ is considered to be a faithful representation of the original Hankel matrix $\mathbf{H }$ for the smallest size of $\tilde{\Sigma }$.

It follows from Eq. (A.6) that

$$\begin{aligned} \begin{aligned} \mathbf{H }_{r \times s} (0)&= \tilde{\mathbf{U }} \tilde{\Sigma } \tilde{\mathbf{V }}^T\\&= \left( \tilde{\mathbf{U }} \tilde{\Sigma }^{\frac{1}{2}} \right) \left( \tilde{\Sigma }^{\frac{1}{2}} \tilde{\mathbf{V }}^T \right) = {\mathcal {O}} {\mathcal {C}} \end{aligned} \end{aligned}$$

(A.8)

Using the above balanced decomposition of $\mathbf{H }_{r \times s} (0)$, we can write:

$$\begin{aligned} {\mathcal {O}} = \tilde{\mathbf{U }} \tilde{\Sigma }^{\frac{1}{2}} \qquad and \qquad {\mathcal {C}} = \tilde{\Sigma }^{\frac{1}{2}} \tilde{\mathbf{V }}^T \end{aligned}$$

Also from Eq. (A.6), we can express the second Hankel matrix $\mathbf{H }_{r \times s} (1)$ as:

$$\begin{aligned} \begin{aligned} \mathbf{H }_{r \times s} (1)&= {\mathcal {O}} \mathbf{A }_r {\mathcal {C}} \\&= \left( \tilde{\mathbf{U }} \tilde{\Sigma }^{\frac{1}{2}}\right) \mathbf{A }_r \left( \tilde{\Sigma }^{\frac{1}{2}} \tilde{\mathbf{V }}^T\right) \end{aligned} \end{aligned}$$

(A.9)

Using the properties of $\mathbf{U }$ and $\mathbf{V }$, we can write:

$$\begin{aligned} \tilde{\Sigma }^{\frac{1}{2}} \mathbf{A }_r \tilde{\Sigma }^{\frac{1}{2}} = \tilde{\mathbf{U }}^T \mathbf{H }_{r \times s} (1) \tilde{\mathbf{V }} \end{aligned}$$

(A.10)

It follows that the matrix $\mathbf{A }_r$ could be obtained through:

$$\begin{aligned} \mathbf{A }_r=\tilde{\Sigma }^{-\frac{1}{2}} \tilde{\mathbf{U }}^T \mathbf{H }_{r \times s} (1) \tilde{\mathbf{V }} \tilde{\Sigma }^{-\frac{1}{2}} \end{aligned}$$

(A.11)

Let:

$$\begin{aligned} \mathbf{E }^T_1 = \begin{pmatrix} 1 &{} 0 &{} \dots &{} 0 \\ \end{pmatrix} \qquad \mathbf{E }^T_2= \begin{pmatrix} 1 &{} 0 &{} \dots &{} 0 \\ \end{pmatrix} \end{aligned}$$

where $\mathbf{E }^T_1$ and $\mathbf{E }^T_2$ are, respectively, $1 \times r$ and $1 \times s$ vectors. We use that along with Equation (A.5) to write the following balanced expression for $h_k^\delta$:

$$\begin{aligned} \begin{aligned} h_k^\delta&= \mathbf{E }^T_1 \mathbf{H }_{r \times s} (g) \mathbf{E }^T_2 \\&= \mathbf{E }^T_1 ({\mathcal {O}} \mathbf{A }_r^{g} {\mathcal {C}}) \mathbf{E }_2 \\&= (\mathbf{E }^T_1 \tilde{\mathbf{U }} \tilde{\Sigma }^{\frac{1}{2}}) (\tilde{\Sigma }^{-\frac{1}{2}} \tilde{\mathbf{U }}^{T} \mathbf{H }_{r \times s} (1) \tilde{\mathbf{V }} \tilde{\Sigma }^{-\frac{1}{2}})^{g} (\tilde{\Sigma }^{\frac{1}{2}} \tilde{\mathbf{V }}^{T} \mathbf{E }_2) \\&\equiv \mathbf{C }_r \mathbf{A }_r^{g} \mathbf{B }_r \end{aligned}\nonumber \\ \end{aligned}$$

(A.12)

We use the decomposed expression above to obtain the reduced input and output matrices $\mathbf{B }_r$ and $\mathbf{C }_r$ as:

$$\begin{aligned} \mathbf{B }_r=\tilde{\Sigma }^{\frac{1}{2}} \tilde{\mathbf{V }}^{T} \mathbf{E }_2 \end{aligned}$$

(A.13)

$$\begin{aligned} \mathbf{C }_r=\mathbf{E }^T_1 \tilde{\mathbf{U }} \tilde{\Sigma }^{\frac{1}{2}} \end{aligned}$$

(A.14)

Appendix B: Approximating the convolution integral

To find an approximate analytical representation of the convolution integral, we first use the fact that the output, z(t), of a linear dynamical system can be expressed as a convolution integral between its input u(t), and an impulse response function h(t), in the form

$$\begin{aligned} \mathbf{z }(t)=\int _0^t h(t-\tau ) \mathbf{u }(\tau ) d\tau \approx \mathbf{C }_r \mathbf{x }(t). \end{aligned}$$

(B.1)

Here, $\mathbf{z}(t) \in {\mathbb {R}}^q$ is the output vector, $\mathbf{x}(t) \in {\mathbb {R}}^n$ is the state vector and $\mathbf{C }_r$ is a $q \times n$ output matrix governed by the following linear state-space equation

$$\begin{aligned} \mathbf{x }'(t)&=\mathbf{A }_r\mathbf{x }(t)+\mathbf{B }_r\mathbf{u }(t). \end{aligned}$$

(B.2)

Using the eigensystem realization algorithm (ERA) detailed in Ref. [26], the convolution integral can be expressed in terms of the realized state-space matrices $\mathbf{A }_r, \mathbf{B }_r$ and $\mathbf{C }_r$ as:

$$\begin{aligned} h(t)=\left( M \sqrt{\frac{g}{R}} \right) \mathbf{C }_r e^{\mathbf{A }_rt} \mathbf{B }_r, \end{aligned}$$

(B.3)

where the realized state-space matrices are :

$$\begin{aligned} \mathbf{A }_r= & {} 0.8 \begin{pmatrix} -1 &{} 1 &{} 1\\ -1 &{} 0 &{} 0\\ -1 &{} 0 &{} -2 \end{pmatrix}, \\ \mathbf{B }_r= & {} \begin{pmatrix} -0.48&-0.02&-0.22 \end{pmatrix}^T, \end{aligned}$$

and

$$\begin{aligned} \mathbf{C }_r = \begin{pmatrix} -0.46&0&0.18 \end{pmatrix}. \end{aligned}$$

Note that the numerical values appearing in the realized state-space matrices are general for any spherical buoy of radius, $\textit{R}$. For more details on the ERA procedure, the interested reader can refer to Appendix A.

The matrix exponential $e^{\mathbf{A }_rt}$ in Equation (B.3) can be further expressed in the following form

$$\begin{aligned} e^{\mathbf{A }_rt}={\mathcal {L}}^{-1}\left( s\mathbf{I } - \mathbf{A }_r\right) ^{-1}. \end{aligned}$$

(B.4)

Here, ${\mathcal {L}}^{-1}$ is the inverse Laplace transform, and I is the identity matrix. Substituting Equation (B.4) into Eq. (B.3), we obtain the following analytical expression for h(t):

$$\begin{aligned} h(t)=\left( M \sqrt{\frac{g}{R}} \right) e^{-\mu t}\left( \uplambda _1 + \uplambda _2 \cos (\mu t) + \uplambda _3 \sin (\mu t)\right) ,\nonumber \\ \end{aligned}$$

(B.5)

where $\mu , \uplambda _1, \uplambda _2$, and $\uplambda _3$ are constants listed in Table 1, and are valid for any spherical buoy of radius, R.

Appendix C: Parametric terms constants

$$\begin{aligned} G_0&=\omega _o^2-\frac{\eta ^2}{\omega _o^2}a_o^2+\frac{3\gamma }{2}a_o^2+\frac{3 \gamma \eta ^2}{4 \omega _o^4}a_o^4 + \frac{\gamma \eta ^2}{24 \omega _o^4}a_o^4 \end{aligned}$$

(C.1)

$$\begin{aligned} G_1&= 2\eta a_o - \frac{5 \gamma \eta }{2 \omega _o^2} a_o^3 \end{aligned}$$

(C.2)

$$\begin{aligned} G_2&= \frac{\eta ^2}{3 \omega _o^2}a_o^2 + \frac{3 \gamma }{2} a_o^2 - \frac{\gamma \eta ^2}{2\omega _o^4}a_o^4 \end{aligned}$$

(C.3)

$$\begin{aligned} G_3&= \frac{\gamma \eta }{2 \omega _o^2} a_o^3 \end{aligned}$$

(C.4)

$$\begin{aligned} G_4&= \frac{\gamma \eta ^2}{24 \omega _o^4} a_o^4 \end{aligned}$$

(C.5)

$$\begin{aligned} K_0&=-\omega _n^2 + 3\gamma \left( \frac{R_1}{2}+\frac{R_3^2}{2}+\frac{R_5^2}{2}\right) \end{aligned}$$

(C.6)

$$\begin{aligned} K_2&=3\gamma \left( \frac{R_1^2}{2}+R_1 R_3+R_3 R_5 \right) \end{aligned}$$

(C.7)

$$\begin{aligned} K_4&=3\gamma \left( R_1 R_3+R_1 R_5 \right) \end{aligned}$$

(C.8)

$$\begin{aligned} K_6&=3\gamma \left( \frac{R_3^2}{2}+R_1 R_5\right) \end{aligned}$$

(C.9)

$$\begin{aligned} K_8&=3\gamma \left( R_3 R_5 \right) \end{aligned}$$

(C.10)

$$\begin{aligned} K_{10}&=3\gamma \left( \frac{R_5^2}{2}\right) \end{aligned}$$

(C.11)

where:

$$\begin{aligned} R_1&= a_o \end{aligned}$$

(C.12)

$$\begin{aligned} R_3&= \frac{\gamma }{32 \Omega ^2} a_o^3 + \frac{3 \gamma ^2}{1024 \Omega ^4} a_o^5 \end{aligned}$$

(C.13)

$$\begin{aligned} R_5&=\frac{\gamma ^2}{1024 \Omega ^4} a_o^5 \end{aligned}$$

(C.14)