Aeroelastic metastructure for simultaneously suppressing wind-induced vibration and energy harvesting under wind flows and base excitations

We consider two configurations for comparison, distinguished primarily by the characteristics of the masses of the LRs. Specifically, in Configuration I, the locally resonating masses are simply regular masses subjected to no external forces, exhibiting an electro-mechanical coupling behaviour; while in Configuration II, the locally resonating masses consist of bluff bodies with equal weights, which also experience aerodynamic galloping forces applied to them, displaying an aero-electro-mechanical coupling behaviour. From a more practical perspective, Configuration I is designed for internal installation, inside the flexible supporting beams or columns; while Configuration II is appropriate for scenarios where external installation exposed to ambient environment is permissible. For the mechanical-to-electrical energy conversion, piezoelectric transduction is incorporated by inserting piezoelectric elements to the LRs.

Figures 1(b) and (c) depicts the physical model of the proposed aeroelastic galloping metastructure, which consists of a long chain including n lumped masses that are elastically connected, representing the primary structure, and periodical electromechanical locally resonating oscillators attached to each primary mass to serve as both vibration absorbers and energy harvesters. The LRs absorb and dissipate a portion of the vibration energy from the primary structure while converting the remaining portion of the transferred energy into electricity through electromechanical coupling. The primary structure is subjected to aerodynamic galloping excitation induced by a bluff body at its free end, which results in an aerodynamic galloping force acting on the lumped mass positioned at the free end of the primary chain, representing the location of the bluff body. In scenarios involving concurrent wind and base excitations, the fixed end of the structure will further be subjected to external base vibration. Since our focus is on the power conversion efficiency of the harvesters as well as the impact of the coupling strength and backward electrical damping on the system's dynamic responses, rather than the sophisticated power extraction and regulation mechanisms, therefore, we employ a simple AC circuit consisting of a simple resistive load connected to the piezoelectric element in each cell. For the purpose of proof-of-concept, prototypes of Configuration I and Configuration II (with three LRs, as an example, shown in figure 11) are fabricated. Which wind tunnel and base vibration experiments are carried out, with details given in section 4.

The governing equations of the system for two configurations can be expressed, respectively, as follows, by considering the aero-electro-mechanical coupling between the mechanical primary structure, LRs, piezoelectric elements connected to the AC circuits, and the external aerodynamic loading: for Configuration I,

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {m_1}{{\ddot u}_{11}} + {c_1}\left({{\dot u}_{11}} - {{\dot u}_{12}}\right) + {k_1}\left({u_{11}} - {u_{12}}\right) + {c_2}\left({{\dot u}_{11}} - {{\dot u}_{21}}\right) + {k_2}\left({u_{11}} - {u_{21}}\right) - \theta {v_1} = {F_1} \hfill \\ {m_1}{{\ddot u}_{1i}} + {c_1}\left(2{{\dot u}_{1i}} - {{\dot u}_{1\left(i + 1\right)}} - {{\dot u}_{1\left(i - 1\right)}}\right) + {k_1}\left(2{u_{1i}} - {u_{1\left(i + 1\right)}} - {u_{1\left(i - 1\right)}}\right) + {c_2}\left({{\dot u}_{1i}} - {{\dot u}_{2i}}\right) + {k_2}\left({u_{1i}} - {u_{2i}}\right) - \theta {v_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 2,\ldots,n - 1\right) \hfill \\ \end{gathered} \\ {{m_1}{{\ddot u}_{1n}} + {c_1}\left(2{{\dot u}_{1n}} - {{\dot u}_{1\left(n - 1\right)}} - {{\dot u}_b}\right) + {k_1}\left(2{u_{1n}} - {u_{1\left(n - 1\right)}} - {u_b}\right) + {c_2}\left({{\dot u}_{1n}} - {{\dot u}_{2n}}\right) + {k_2}\left({u_{1n}} - {u_{2n}}\right) - \theta {v_n} = 0{\kern 1pt} {\kern 1pt} } \\ {{m_2}{{\ddot u}_{2i}} + {c_2}\left({{\dot u}_{2i}} - {{\dot u}_{1i}}\right) + {k_2}\left({u_{2i}} - {u_{1i}}\right) + \theta {v_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \\ {{v_i}/{R_L} + {C_P}{{\dot v}_i} - \theta \left({{\dot u}_{2i}} - {{\dot u}_{1i}}\right) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \end{array}} \right.\end{align} \tag{ 1 }$$

and for Configuration II

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {m_1}{{\ddot u}_{11}} + {c_1}\left({{\dot u}_{11}} - {{\dot u}_{12}}\right) + {k_1}\left({u_{11}} - {u_{12}}\right) + {c_2}\left({{\dot u}_{11}} - {{\dot u}_{21}}\right) + {k_2}\left({u_{11}} - {u_{21}}\right) - \theta {v_1} = {F_1} \hfill \\ {m_1}{{\ddot u}_{1i}} + {c_1}\left(2{{\dot u}_{1i}} - {{\dot u}_{1\left(i + 1\right)}} - {{\dot u}_{1\left(i - 1\right)}}\right) + {k_1}\left(2{u_{1i}} - {u_{1\left(i + 1\right)}} - {u_{1\left(i - 1\right)}}\right) + {c_2}\left({{\dot u}_{1i}} - {{\dot u}_{2i}}\right) + {k_2}\left({u_{1i}} - {u_{2i}}\right) - \theta {v_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 2,\ldots,n - 1\right) \hfill \\ \end{gathered} \\ {{m_1}{{\ddot u}_{1n}} + {c_1}\left(2{{\dot u}_{1n}} - {{\dot u}_{1\left(n - 1\right)}} - {{\dot u}_b}\right) + {k_1}\left(2{u_{1n}} - {u_{1\left(n - 1\right)}} - {u_b}\right) + {c_2}\left({{\dot u}_{1n}} - {{\dot u}_{2n}}\right) + {k_2}\left({u_{1n}} - {u_{2n}}\right) - \theta {v_n} = 0{\kern 1pt} {\kern 1pt} } \\ {{m_2}{{\ddot u}_{2i}} + {c_2}\left({{\dot u}_{2i}} - {{\dot u}_{1i}}\right) + {k_2}\left({u_{2i}} - {u_{1i}}\right) + \theta {v_i} = {F_{2i}}} \\ {{v_i}/{R_L} + {C_P}{{\dot v}_i} - \theta \left({{\dot u}_{2i}} - {{\dot u}_{1i}}\right) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \end{array}} \right.\end{align} \tag{ 2 }$$

where m₁ , c₁ , k₁ are the mass, damping, stiffness of each cell on the primary chain, respectively; u_1i represents the displacement of the ith primary mass; m₂ , c₂ , k₂ are the mass, damping, stiffness of each LR; u_2i is the displacement of the ith LR; v_i is output voltage of the piezoelectric element on the ith oscillator; θ is the electromechanical coupling; R_L is the connected loading resistance in the circuit; and C_P is the capacitance of the piezoelectric element. The displacement of the base excitation is given by ${u_b} = {y_0}\text{cos} ({\Omega _b}t)$. F₁ and F_2i are the aerodynamic forces due to galloping, exerted on m₁ at the free end of the primary chain and m₂ in the ith cell, respectively. The aerodynamic model is built based on the quasi-steady hypothesis [61], given by

$$\begin{align}{F_1} = \frac{1}{2}\rho {h_1}{L_1}{U^2}\sum\limits_{j = 1}^3 {{A_j}{{\left(\frac{{{{\dot u}_{11}}}}{U}\right)}^j}} \end{align} \tag{ 3 }$$

$$\begin{align}{F_{2i}} = \frac{1}{2}\rho {h_2}{L_2}{U^2}\sum\limits_{j = 1}^3 {{B_j}{{\left(\frac{{{{\dot u}_{2i}}}}{U}\right)}^j}} \end{align} \tag{ 4 }$$

where ρ is the air density; A_j and B_j are aerodynamic coefficients; h₁ and h₂ are the heights of windward side of bluff bodies; L₁ and L₂ are the lengths of windward side of bluff bodies, where subscript 1 and 2 represent bluff body 1 on the primary structure and bluff body 2 on the LRs, respectively.

The governing equation is further nondimensionalized based on the following dimensionless parameters,

$$\begin{align*} & {\Omega _1} = \sqrt {\frac{{{k_1}}}{{{m_1}}}}, {\Omega _2} = \sqrt {\frac{{{k_2}}}{{{m_2}}}}, \alpha = \frac{{{m_2}}}{{{m_1}}}, \lambda = \frac{{{\Omega _2}}}{{{\Omega _1}}}, {\zeta _1} = \frac{{{c_1}}}{{2{m_1}{\Omega _1}}},\\ & {\zeta _2} = \frac{{{c_2}}}{{2{m_2}{\Omega _2}}}, {\omega _b} = \frac{{{\Omega _b}}}{{{\Omega _1}}}, {p_i} = \frac{{{u_{1i}}}}{{{h_1}}}, {q_i} = \frac{{{u_{2i}}}}{{{h_1}}}, {V_i} = \frac{{{C_P}{v_i}}}{{\theta {h_1}}},\\ & r = {R_L}{C_P}{\Omega _1}, z = \frac{y}{{{h_1}}}, {k_e} = \frac{{{\theta ^2}}}{{{C_P}{k_1}}}, {m_{a1}} = \frac{{{m_1}}}{{\rho h_1^2{L_1}}}, {m_{a2}} = \frac{{{m_2}}}{{\rho h_2^2{L_2}}},\\ & {u_{a1}} = \frac{U}{{{\Omega _1}{h_1}}}, {f_{11}} = \frac{{{A_1}}}{{2{m_{a1}}}}, {f_{13}} = \frac{{{A_3}}}{{2{m_{a1}}}}, {f_{21}} = \frac{{{B_1}}}{{2{m_{a2}}}}, {f_{23}} = \frac{{{B_3}}}{{2{m_{a2}}}},\\ & \tau = {\Omega _1}t.\end{align*}$$

and the dimensionless form of the governing equation is given by the following form for two Configurations: for Configuration I

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\ddot p}_1} + 2{\zeta _1}\left({{\dot p}_1} - {{\dot p}_2}\right) + \left({p_1} - {p_2}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_1} - {{\dot q}_1}\right) + \alpha {\lambda ^2}\left({p_1} - {q_1}\right) - {k_e}{V_1} = {f_{11}}{u_{a1}}{{\dot p}_1} + \frac{{{f_{13}}}}{{{u_{a1}}}}\dot p_1^3 \hfill \\ {{\ddot p}_i} + 2{\zeta _1}\left(2{{\dot p}_i} - {{\dot p}_{\left(i + 1\right)}} - {{\dot p}_{\left(i - 1\right)}}\right) + \left(2{p_i} - {p_{\left(i + 1\right)}} - {p_{\left(i - 1\right)}}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_i} - {{\dot q}_i}\right) \hfill \\ + \alpha {\lambda ^2}\left({p_i} - {q_i}\right) - {k_e}{V_i} = 0{\kern 1pt} {\kern 1pt} \left(i = 2,\ldots,n - 1\right) \hfill \\ \end{gathered} \\ {{{\ddot p}_n} + 2{\zeta _1}\left(2{{\dot p}_n} - {{\dot p}_{\left(i - 1\right)}} - {{\dot z}_b}\right) + \left(2{p_n} - {p_{\left(n - 1\right)}} - {z_b}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_n} - {{\dot q}_n}\right) + \alpha {\lambda ^2}\left({p_n} - {q_n}\right) - {k_e}{V_n} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {\alpha {{\ddot q}_i} + 2\alpha \lambda {\zeta _2}\left({{\dot q}_i} - {{\dot p}_i}\right) + \alpha {\lambda ^2}\left({p_i} - {q_i}\right) + {k_e}{V_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \\ {{V_i}/r + {{\dot V}_i} - \left({{\dot q}_i} - {{\dot p}_i}\right) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \end{array}} \right.\end{align} \tag{ 5 }$$

and for Configuration II

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\ddot p}_1} + 2{\zeta _1}\left({{\dot p}_1} - {{\dot p}_2}\right) + \left({p_1} - {p_2}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_1} - {{\dot q}_1}\right) + \alpha {\lambda ^2}\left({p_1} - {q_1}\right) - {k_e}{V_1} = {f_{11}}{u_{a1}}{{\dot p}_1} + \frac{{{f_{13}}}}{{{u_{a1}}}}\dot p_1^3 \hfill \\ {{\ddot p}_i} + 2{\zeta _1}\left(2{{\dot p}_i} - {{\dot p}_{\left(i + 1\right)}} - {{\dot p}_{\left(i - 1\right)}}\right) + \left(2{p_i} - {p_{\left(i + 1\right)}} - {p_{\left(i - 1\right)}}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_i} - {{\dot q}_i}\right) \hfill \\ + \alpha {\lambda ^2}\left({p_i} - {q_i}\right) - {k_e}{V_i} = 0{\kern 1pt} {\kern 1pt} \left(i = 2,\ldots,n - 1\right) \hfill \\ \end{gathered} \\ {{{\ddot p}_n} + 2{\zeta _1}\left(2{{\dot p}_n} - {{\dot p}_{\left(i - 1\right)}} - {{\dot z}_b}\right) + \left(2{p_n} - {p_{\left(n - 1\right)}} - {z_b}\right) + 2\alpha \lambda {\zeta _2}\left({{\dot p}_n} - {{\dot q}_n}\right) + \alpha {\lambda ^2}\left({p_n} - {q_n}\right) - {k_e}{V_n} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {\alpha {{\ddot q}_i} + 2\alpha \lambda {\zeta _2}\left({{\dot q}_i} - {{\dot p}_i}\right) + \alpha {\lambda ^2}\left({p_i} - {q_i}\right) + {k_e}{V_i} = {f_{21}}\frac{{{u_{a1}}}}{\lambda }{{\dot q}_1} + {f_{23}}\frac{\lambda }{{{u_{a1}}}}\dot q_1^3{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \\ {{V_i}/r + {{\dot V}_i} - \left({{\dot q}_i} - {{\dot p}_i}\right) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left(i = 1,2,\ldots,n\right)} \end{array}} \right..\end{align} \tag{ 6 }$$

By introducing the state vector S_i ,

$$\begin{align}{{\mathbf{S}}_i} = \left[ {\begin{array}{*{20}{c}} {{s_{i1}}} \\ {{s_{i2}}} \\ {{s_{i3}}} \\ {{s_{i4}}} \\ {{s_{i5}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{p_i}} \\ {{{\dot p}_i}} \\ {{q_i}} \\ {{{\dot q}_i}} \\ {Vi} \end{array}} \right].\end{align} \tag{ 7 }$$

The governing equations can be transformed into the following state-space form:

$$\begin{align}{{\mathbf{S}}_i} = {{\mathbf{C}}_i}{{\mathbf{S}}_i}{{ + }}{{\mathbf{D}}_i}\,\,\,\,\,\,\left(i = 1,\,2,\,\ldots,\,n\right).\end{align} \tag{ 8 }$$

In equation (8), the coefficient matrices are given by: for Configuration I,

$$\begin{align}{{\mathbf{C}}_i} = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0 \\[2pt] { - k\left(i\right) - 1 - \alpha {\lambda ^2}}&{{f_{11}}{u_{a1}} \cdot g\left(i\right) + 2{\zeta _1}\left[ - k\left(i\right) - 1\right] - 2\alpha \lambda {\zeta _2}}&{\alpha {\lambda ^2}}&{2\alpha \lambda {\zeta _2}}&{{k_e}} \\[2pt] 0&0&0&1&0 \\[2pt] {{\lambda ^2}}&{2\lambda {\zeta _2}}&{ - {\lambda ^2}}&{ - 2\lambda {\zeta _2}}&{ - {k_e}/\alpha } \\[2pt] 0&{ - 1}&0&1&{ - 1/r} \end{array}} \right]\end{align} \tag{ 9 }$$

$$\begin{align}{{\mathbf{D}}_i} = \left[ {\begin{array}{*{20}{c}} 0 \\[2pt] {g\left(i\right) \cdot \frac{{{f_{13}}}}{{{u_{a1}}}}s_{12}^3 + 2{\zeta _1}\left[l\left(i\right){s_{\left(i + 1\right)2}} + k\left(i\right){s_{\left(i - 1\right)2}} + m\left(i\right){{\dot z}_b}\right] + l\left(i\right){s_{\left(i + 1\right)1}} + k\left(i\right){s_{\left(i - 1\right)1}} + m\left(i\right){z_b}} \\[2pt] 0 \\[2pt] 0 \\[2pt] 0 \end{array}} \right]\end{align} \tag{ 10 }$$

and for Configuration II,

$$\begin{align}{{\mathbf{C}}_i} = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0 \\[2pt] { - k\left(i\right) - 1 - \alpha {\lambda ^2}}&{{f_{11}}{u_{a1}} \cdot g\left(i\right) + 2{\zeta _1}\left[ - k\left(i\right) - 1\right] - 2\alpha \lambda {\zeta _2}}&{\alpha {\lambda ^2}}&{2\alpha \lambda {\zeta _2}}&{{k_e}} \\[2pt] 0&0&0&1&0 \\[2pt] {{\lambda ^2}}&{2\lambda {\zeta _2}}&{ - {\lambda ^2}}&{{f_{21}}\frac{{{u_{a1}}}}{\lambda } - 2\lambda {\zeta _2}}&{ - {k_e}/\alpha } \\[2pt] 0&{ - 1}&0&1&{ - 1/r} \end{array}} \right]\end{align} \tag{ 11 }$$

$$\begin{align}{{\mathbf{D}}_i} = \left[ {\begin{array}{*{20}{c}} 0 \\[2pt] {g\left(i\right) \cdot \frac{{{f_{13}}}}{{{u_{a1}}}}s_{12}^3 + 2{\zeta _1}\left[l\left(i\right){s_{\left(i + 1\right)2}} + k\left(i\right){s_{\left(i - 1\right)2}} + m\left(i\right){{\dot z}_b}\right] + l\left(i\right){s_{\left(i + 1\right)1}} + k\left(i\right){s_{\left(i - 1\right)1}} + m\left(i\right){z_b}} \\[2pt] 0 \\[2pt] {\lambda {f_{23}}/{u_{a1}} \cdot s_{i4}^3} \\[2pt] 0 \end{array}} \right].\end{align} \tag{ 12 }$$

In the above matrices, we have

$$\begin{align}\begin{gathered} g\left(i\right) = \left\{ \begin{gathered} 1\,\,\,\,\,\,\,\left(i = 1\right) \hfill \\ 0\,\,\left(2 \unicode{x2A7D} i \unicode{x2A7D} n\right) \hfill \\ \end{gathered} \right.;\;k\left(i\right) = \left\{ \begin{gathered} 0\,\,\,\,\,\,\,\left(i = 1\right) \hfill \\ 1\,\,\left(2 \unicode{x2A7D} i \unicode{x2A7D} n\right) \hfill \\ \end{gathered} \right. \hfill \\ l\left(i\right) = \left\{ \begin{gathered} 0\,\,\,\,\,\,\,\left(i = n\right) \hfill \\ 1\,\,\left(1 \unicode{x2A7D} i \unicode{x2A7D} n - 1\right) \hfill \\ \end{gathered} \right.;\;m\left(i\right) = \left\{ \begin{gathered} 0\,\,\left(1 \unicode{x2A7D} i \unicode{x2A7D} n - 1\right) \hfill \\ 1\,\,\,\left(i = n\right) \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} .\end{align} \tag{ 13 }$$

Equation (8) can be numerically solved using the Runge–Kutta algorithm to obtain the mechanical displacement responses and electrical outputs.

The cut-in wind speed u_cr can be obtained through analysing the linearized governing equations. Because the aerodynamic loading will induce an equivalent negative damping effect [44], galloping takes place when the negative damping overcomes the positive damping, which is the sum of intrinsic mechanical damping and the backward electrical damping resulting from the piezoelectric energy harvesting process. Based on this principle, the cut-in wind speed is theoretically predicted by making the coupled damping equal to zero. The matrix C_i reflects that the galloping cut-in wind speed will be influenced by the load resistance r, and the cross-section geometry of the bluff body on both the primary structure and the LRs (geometry determines the values of f₁₁, f₁₂, f₂₁, and f₂₂). Introducing

$$\begin{align}\left| {{\beta _m}{\boldsymbol{I}} - {{\boldsymbol{C}}_{\boldsymbol{i}}}} \right| = 0.\end{align} \tag{ 14 }$$

The relationship of cut-in wind speed with load resistance for certain cross-section geometries of bluff bodies can be determined by examining the features of values of β_m , which are the eigenvalues of the matrix C_i. I is the identity matrix. For an aeroelastic metastructure containing n unit cells, there are 2n sets of eigenvalues, which are complex conjugates. The system is stable if the real parts of all sets of eigenvalues are negative; otherwise, the system transitions to unstable if the real parts of any one set of eigenvalues become positive. In the following numerical investigations, unless specified otherwise, the parameters are taken as α= 0.4, λ= 0.3, ζ₁ = ζ₂ = 0.01, k_e = 7.7296 × 10⁻⁵ , r= 1.644, f₁₁ = 0.015, and f₁₃= −0.1685. Moreover, for Configuration I, the aerodynamic coefficients in the LRs are set to f₂₁ = 0 and f₂₃ = 0; for configuration II, they are set to f₂₁ = 0.0375 and f₂₃= −0.4213. For better consistency with the experiments, the number of cells is set to n= 3. The relationship of cut-in wind speed u_cr with load resistance r is plotted in figure 2. u_cr of Configuration II is lower than that of Configuration I; more importantly, compared to the original primary structure without LRs, Configuration II is capable of reducing the cut-in speed, which is a major component for mitigating wind-induced vibration. However, the overall galloping suppression performance needs to be evaluated by further comparing the steady-state oscillation amplitude of the system beyond the cut-in speed, which will be discussed in the following section. By enlarging figure 2(a), it is found in figures 2(b) and (c) that U_cr of the two configurations both slightly vary with r, which increase firstly and then decrease with increasing r. Maximum U_cr are obtained as r approximates 1, indicating the electrical damping reaches maximum.

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Firstly, the dynamic and electrical responses of the two configurations are examined under pure wind excitation. The average power output from each unit cell is calculated by V_{i RMS} ²/r, where V_{i RMS} is the root mean square (RMS) generated voltage in the ith cell.

The findings shown in figure 3 demonstrate that the two proposed aeroelastic metastructure configurations both exhibit effective performance in suppressing the vibration of the primary structure as well as generating power. Notably, Configuration I, without the locally resonating bluff body in each cell, considerably increases the cut-in wind speed compared to the pure primary structure without the LRs, suggesting that it can effectively attenuate galloping vibrations at low wind speeds. On the other hand, Configuration II that includes bluff bodies in the locally resonating unit cells, despite having a slightly lower cut-in wind speed, shows strong potential in suppressing galloping vibration at high wind speeds. Beyond the cut-in wind speed, steady-state limit-cycle oscillations (LCOs) occur. It is seen that Configuration II yields a much smaller slope of displacement-wind speed variation than that of the pure primary structure, further benefits the galloping suppression. Figure 3(b) displays the relationship between the vibration frequency and reduced wind speed. For Configuration I, the vibration frequency remains constant at approximately 0.51, which is close to the second natural frequency of the metastructure system, resulting in an out-of-phase motion. Conversely, configuration II exhibits a vibration frequency of approximately 0.27, close to the first natural frequency of the system, resulting in an in-phase motion. The output power from the 1st unit cell of the two configurations is shown in figure 3(c), where Configuration II exhibits a higher output power than Configuration I in the considered wind speed range. In addition, phase trajectories are plotted for showing the formation of LCO and its evolution process. The bigger the covered area of phase trajectories is, the larger the oscillation displacement is. As shown in figures 3(d)–(f), LCOs are observed as the wind speed surpasses the corresponding cut-in wind speeds of the configuration. Both Configuration I and II can effectively mitigate the galloping amplitude of the primary structure.

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The influences of key system parameters, including the frequency ratio, mass ratio, load resistance and electromechanical coupling strength, on the dynamic behaviours of the proposed aeroelastic metastructure as well as the energy transfer and harvesting efficiency are subsequently investigated. Figure 4 investigates the effects of frequency ratio $\lambda = {\Omega _2}/{\Omega _1}$, defined as the ratio of the natural frequency of the LR to that of the primary oscillator in each cell.

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Firstly, we evaluate the vibration suppression performance by examining the dimensionless RMS galloping displacement of the primary mass m₁ at the free end of the aeroelastic metastructure in the 1st cell (see figure 1), i.e. p_{1_RMS}. The variations of p_{1_RMS} with various λ are compared to that of the primary structure without the LRs, as shown in figure 4(a) for Configuration I and figure 4(b) for Configuration II. For Configuration I, it is seen that when the frequency ratio λ is below or equal to 0.3, the galloping displacement of the primary structure is effectively suppressed within the considered wind speed range (u_{a 1} ⩽ 1.82), meanwhile, the cut-in wind speed is significantly increased. The galloping suppression performance improves with the increase of λ. However, when λ increases to a medium value of 0.5, although the metastructure is able to attenuate vibration at low wind speeds and slightly increase the cut-in wind speed, the galloping suppression performance is degraded at high wind speeds, where the displacement unfavourably increases beyond the case without LRs. When λ further increases and exceeds 0.8, the aeroelastic metastructure fails to increase the cut-in speed or suppress the vibration displacement. In Configuration II, although the cut-in wind speed is not reduced, it effectively attenuates the vibration of the primary structure right after LCO manifests when λ ⩽0.3; moreover, similar to Configuration I, the galloping suppression performance also improves with increasing λ. Nevertheless, any λ beyond 0.5 is ineffective in suppressing galloping oscillation. Overall, in both aeroelastic metastructure configurations, the optimal frequency ratio should be carefully tuned to a small value up to 0.5 for effective galloping suppression. It is worth noting that this conclusion differs from the case with base vibration suppression metastructures [26, 47, 49], where the linearized natural frequency of the LRs should always be tuned to approach that of the primary structure in each cell, i.e. λ ≈ 1.0, for optimal vibration suppression capability.

The energy harvesting capability of the aeroelastic metastructure as the frequency ratio changes is investigated in figures 4(c) and (d). Generally, in both configurations, the harvester in the 1st cell at the free end of the chain produces the highest power compared to others. The total output power combinedly from all unit cells corresponds to No. 4 on the horizontal axis of 'number of the oscillator' for convenient plotting. The aeroelastic metastructures effectively generate power when λ gradually increases up to 0.5; however, beyond this value, the power output is minimal. This is as expected since any λ beyond 0.5 is ineffective in suppressing the oscillation of the primary structure, indicating the inability to transfer kinetic energy to the locally resonating harvesters for the subsequent vibratory-to-electrical power conversion. It is evident Configuration II outperforms Configuration I by generating almost three times more power. This is reasonable since the inclusion of bluff bodies in the LRs enhances the capability to capture power from wind flows.

It is interesting to see from figure 4(b) that at λ= 0.5, the galloping displacement of the primary structure in Configuraiton II notably decreases at a wind speed around u_{a 1} = 1.20 if the initial disturbance is small. The time histories of displacements of the primary mass m₁ and locally resonating mass m₂ in the free-end unit cell, i.e. p₁ and q₁, at different wind speeds are shown in figure 5, along with the power spectrums. When this jump down occurs, extra frequency components appear, and the oscillations of m₁ and m₂ become aperiodic, as shown in figure 5(b1–b2). Moreover, the aeroelastic metastructure experiences a sharp frequency shift from 0.338 to 0.486 at around u_{a 1} = 1.20, as presented in figure 5(d3). A further examination on the responses in figure 5(c1–c2) and (d1–d2) reveals that with a large initial disturbance, the oscillation of the system is dominated by the galloping of the primary bluff body; in contrast, with a small initial disturbance, the responses are dominated by the auxiliary bluff bodies in the unit cells. The later case results in more wind energy transfer to the auxiliary bluff bodies compared to the primary bluff body, leading to a signaficant suppression of the primary structure's oscillation at high wind speeds. However, we should stress that in practical scenarios, unavoidable gusts or turbulences are ubiquitous, posing challenges to achieve such a significant vibration suppression that requires a small initial disturbance.

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In figure 6, we investigate the effect of mass ratio $\alpha = {m_2}/{m_1}$ on the mechanical and electrical responses of the aeroelastic metastructure. For Configuration I (figure 6(a)), with the increase of α, the galloping suppression performance firstly improves then degrades. There is an optimal α= 1.0 where the galloping displacement is fully suppressed within the wind speed range of interest; when α is beyond 5.0, there is no benefit as the displacement even increases compared to the initial primary structure. In Configuration II (figure 6(b)), the optimal galloping suppression performance is achieved at a much smaller α = 0.2, with a substantially increased cut-in wind speed and reduced vibration amplitude. It is worth noting that with Configuration II, α should be kept at small values, which is another distinct characteristic as compared to the metastructures designed for base vibration suppression, where larger mass ratios improve the vibration suppression efficiency [18, 23, 47]. The power output variations in figures 6(c) and (d) show a monotonic increasing trend with increasing α. In Configuration I, the 1st cell at the free end of the system generates the highest power; while in Configuration II, it appears the distributed cells yield comparable power levels. Overall, if the function of energy harvesting is prioritised, α should be tuned to as small as possible.

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Figure 7 investigates the influence of the electromechanical coupling strength in the LRs on the performance of galloping suppression and wind power extraction. In Configuration I, the galloping amplitude is significantly affected by the coupling strength, as can be seen in figures 6(a) and (c). Increasing the coupling strength remarkably improves galloping suppression by decreasing the galloping amplitude as well as increasing the cut-in wind speed, as a result of the increased backward electrical damping. When k_e is increased beyond 3.865 × 10⁻⁴ (i.e. 5 × 7.73 × 10⁻⁵, figure 7(c)), the cut-in wind speed is dramatically increased by the large electrical damping, and galloping of the primary structure is fully attenuated to zero within the considered wind speed range. Conversely, the coupling strength only has minimal influence on the galloping amplitude in Configuration II, particularly at high wind speeds. It is deduced that the oscillation of the added bluff bodies in the unit cells primarily relies on the wind excitation and is not obviously impacted by the electrical damping.

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As for the generated power, there exists a respective optimal coupling strength at each wind speed in Configuration I for highest power output, either in individual cells (figure 7(e), Cell 1 as an example) or from the aeroelastic metastructure combinedly (figure 7(g)), similar to a typical galloping energy harvester as we analysed in [44]. Therefore, selection of a proper coupling strength for Configuration I depends on the priority of the intended function. On the contrary, the power output appears to monotonically increases with increasing coupling strength in Configuration II. This is because there is no longer a competitive relationship between the suppressed displacement amplitude and improved energy conversion efficiency under strong coupling. Therefore, with Configuration II, it is always favourable to maximize the electromechanical coupling.

The impact of load resistance on the galloping suppression and power extraction performances is analysed in figure 8. When a weak electromechanical coupling is employed, as shown in figures 8(a) and (b), it is observed that the loading resistance has minimal influence on the galloping suppression performance in the two configurations. However, by further varying the coupling strength in figures 8(c) and (d), it is found that tuning the load resistance to an optimal value under medium or strong coupling can enhance galloping suppression in both configurations. Since the impact of coupling strength on the behavior of Configuration I is more pronounced than that of Configuration II (figure 7), tuning to the optimal load resistance in Configuration I leads to a much more evident improvement in suppressing the displacement amplitude. At a strong coupling of 7.73 × 10⁻⁴ (i.e. 10 × 7.73 × 10⁻⁵, figure 8(c)), the displacement in Configuration I is suppressed to zero. As for the power output, in Configuration I, there exists one optimal load resistance that produces a power peak under weak coupling, and two of them that yields two power peaks of equal amplitudes when the coupling gets stronger, consistent with the typical piezoelectric galloping energy harvester characteristic [44]. However, in Configuration II, the load tuning task is simpler as there is always a single optimal load resistance, the value of which is not noticeably affected by the coupling strength.

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In this section, we explore the influence of mass ratio α and frequency ratio λ on the performance of vibration suppression and energy harvesting under concurrent wind and base excitations. The dimensionless base acceleration is fixed at 0.011 65, and the wind speed is constant at u_{a 1} = 1.82. Five cases are discussed. In figure 9(a1–a3), α is set to 0.4 while λ increases from 0.3 to 1.0. It is seen that, similar to the pure galloping excitation scenario, the vibration suppression degrades with increasing λ, in particular at off-resonance. In terms of the at-resonance synchronization region (i.e. lock-in region, where galloping frequency and base excitation frequency are locked into each other [62]), Configuration II is able to reduce the peak displacement when λ⩽ 0.5, although the location is shifted due to the addition of the LRs. In figure 9(a1, a4–a5), we fixe λ at 0.3 while vary α between 0.2 and 1.0. Again, like the pure galloping case, the overall vibration suppression performance degrades as α increases. Moreover, the highest power output is obtained at the smallest α and λ employed in the investigation (figure 9(b4)). Therefore, it is recommended that small α and λ should be chosen for improving vibration suppression and energy harvesting simultaneously under concurrent excitations.

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In order to gain further insights into the working mechanism, we consider the case with α= 0.4 and λ= 0.3 in figure 9 for Configuration II, and analyze the time histories and power spectrums. At the frequency ω_b = 0.25 and 0.50 (figures 10(b1) and (d1)), which match the first and second natural frequency of the aeroelastic metastructure, the response exhibits in-phase and out-of-phase motion, respectively, corresponding to the two displacement peaks in figure 9(a1). At these resonances, the oscillation frequency is predomenantly synchronized to the base excitation frequency, as evidenced in the power specturms given in figures 10(b2) and (d2). At off-resonance, as shown in figures 10(a1), (c1) and (e1), quasi-periodic vibration with strong modulated response phenomena [63] can be observed. The oscillation frequency is dominated at 0.266, which is the galloping frequency of the locally resonating bluff bodies, along with the associated 3rd harmonic frequency component at about 0.8, as seen in the corresponding FFT results. In particular, between the two synchrozation resonance peaks, the modulated response appears to lead to a displacement valley, dramatically suppressing the vibration of the primary structure (see figure 9(a1)).

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