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1 Beam with connecting circuit
The constitutive law is
[ D S ] = [ e T d d t s E ] [ E T ] \left[\begin{array}{l}
\mathbf{D} \\
\mathbf{S}
\end{array}\right]=\left[\begin{array}{ll}
\mathbf{e}^T & \mathbf{d} \\
\mathbf{d}_t & \mathbf{s}^E
\end{array}\right]\left[\begin{array}{l}
\mathbf{E} \\
\mathbf{T}
\end{array}\right] [ D S ] = [ e T d t d s E ] [ E T ] Assume the field in the piezo patch is uniform, we have the relation after Laplace transform
V ( s ) = L ⋅ E ( s ) , I ( s ) = s A ⋅ D ( s ) \mathbf{V}(s)=\mathbf{L} \cdot \mathbf{E}(s), \quad \mathbf{I}(s)=s \mathbf{A} \cdot \mathbf{D}(s) V ( s ) = L ⋅ E ( s ) , I ( s ) = s A ⋅ D ( s ) Then the constitutive law can be transformed as
[ I S ] = [ s A ε T L − 1 s A d d t L − 1 s E ] [ V T ] \left[\begin{array}{l}
\mathbf{I} \\
\mathbf{S}
\end{array}\right]=\left[\begin{array}{cc}
s \mathbf{A} \varepsilon^T \mathbf{L}^{-1} & s\mathrm{\boldsymbol{Ad}} \\
\mathbf{d}_t \mathbf{L}^{-1} & \mathbf{s}^{\mathrm{E}}
\end{array}\right]\left[\begin{array}{l}
\mathbf{V} \\
\mathbf{T}
\end{array}\right] [ I S ] = [ s A ε T L − 1 d t L − 1 s Ad s E ] [ V T ] We define the capacity of piezo patch as
A i ε i T / L i = C p i T A_i \varepsilon_i^T / L_i=C_{p i}^T A i ε i T / L i = C p i T Then Eq. (3) can be rewritten as
[ I S ] = [ s C p T s A d d t L − 1 s E ] [ V T ] = [ Y D ( s ) s A d d t L − 1 s E ] [ V T ] , \left[\begin{array}{c}
\mathbf{I} \\
\mathbf{S}
\end{array}\right]=\left[\begin{array}{cc}
s \mathbf{C}_p^T & s \mathbf{A d} \\
\mathbf{d}_t \mathbf{L}^{-1} & \mathbf{s}^E
\end{array}\right]\left[\begin{array}{l}
\mathbf{V} \\
\mathbf{T}
\end{array}\right]=\left[\begin{array}{cc}
\mathbf{Y}^D(s) & s \mathbf{A d} \\
\mathbf{d}_t \mathbf{L}^{-1} & \mathbf{s}^E
\end{array}\right]\left[\begin{array}{l}
\mathbf{V} \\
\mathbf{T}
\end{array}\right], [ I S ] = [ s C p T d t L − 1 s Ad s E ] [ V T ] = [ Y D ( s ) d t L − 1 s Ad s E ] [ V T ] , where Y D ( s ) \mathbf{Y}^D(s) Y D ( s )
For shunted piezoelectric applications,
[ I S ] = [ Y E L s A d d t L − 1 S E ] [ V T ] where Y E L = Y D + Y S U \left[\begin{array}{l}
\mathbf{I} \\
\mathbf{S}
\end{array}\right]=\left[\begin{array}{cc}
\mathbf{Y}^{E L} & s \mathbf{A d} \\
\mathbf{d}_{\mathbf{t}} \mathbf{L}^{-1} & \mathbf{S}^E
\end{array}\right]\left[\begin{array}{l}
\mathbf{V} \\
\mathbf{T}
\end{array}\right] \text { where } \quad \mathbf{Y}^{E L}=\mathbf{Y}^D+\mathbf{Y}^{S U} [ I S ] = [ Y E L d t L − 1 s Ad S E ] [ V T ] where Y E L = Y D + Y S U The first equation gives
V = ( Z E L ) I − ( Z E L s A d ) T \mathbf{V}=\left(\mathbf{Z}^{E L}\right) \mathbf{I}-\left(\mathbf{Z}^{E L} s \mathbf{A d}\right) \mathbf{T} V = ( Z E L ) I − ( Z E L s Ad ) T and inserting into second equation gives
S = [ s E − d t L − 1 Z E L s A d ] T + [ d t L − 1 Z E L ] I . \mathbf{S}=\left[\mathbf{s}^E-\mathbf{d}_t \mathbf{L}^{-1} \mathbf{Z}^{E L} s \mathbf{A d}\right] \mathbf{T}+\left[\mathbf{d}_t \mathbf{L}^{-1} \mathbf{Z}^{E L}\right] \mathbf{I} . S = [ s E − d t L − 1 Z E L s Ad ] T + [ d t L − 1 Z E L ] I . The mechanical compliance is
s S U = [ s E − d t L − 1 Z E L s A d ] . \mathbf{s}^{S U}=\left[\mathbf{s}^E-\mathbf{d}_t \mathbf{L}^{-1} \mathbf{Z}^{E L} \boldsymbol{s A d}\right] . s S U = [ s E − d t L − 1 Z E L sAd ] . Upon noting that with constant stress
Z E ( s ) = 0 = short circuit electrical impedance, Z D ( s ) = ( C p T s ) − 1 = open circuit electrical impedance, \begin{gathered}
\mathbf{Z}^E(\mathrm{~s})=\mathbf{0}=\text { short circuit electrical impedance, } \\
\mathbf{Z}^D(s)=\left(\mathbf{C}_p^T s\right)^{-1}=\text { open circuit electrical impedance, }
\end{gathered} Z E ( s ) = 0 = short circuit electrical impedance, Z D ( s ) = ( C p T s ) − 1 = open circuit electrical impedance, and that
s L − 1 ε T A = C p T s , s \mathbf{L}^{-1} \boldsymbol{\varepsilon}^T \mathbf{A}=\mathbf{C}_p^T s, s L − 1 ε T A = C p T s , equation (16) can be put in the form
s S U = [ s E − d t Z ‾ E L ( ε T ) − 1 d ] , \mathbf{s}^{S U}=\left[\mathbf{s}^E-\mathbf{d}_{\mathbf{t}} \overline{\mathbf{Z}}^{E L}\left(\boldsymbol{\varepsilon}^T\right)^{-1} \mathbf{d}\right], s S U = [ s E − d t Z E L ( ε T ) − 1 d ] , where the matrix of non-dimensional electrical impedances is defined as
Z ‾ E L = Z E L ( Z D ) − 1 = ( s C p T + Y S U ) − 1 s C p T \overline{\mathbf{Z}}^{E L}=\mathbf{Z}^{E L}\left(\mathbf{Z}^D\right)^{-1}=\left(s \mathbf{C}_p^T+\mathbf{Y}^{S U}\right)^{-1} s \mathbf{C}_p^T Z E L = Z E L ( Z D ) − 1 = ( s C p T + Y S U ) − 1 s C p T Here, we introduce the electromechanical coupling coefficients
shear, k 15 = d 15 / s 55 E ε 1 T = k 24 , transverse, k 31 = d 31 / s 11 E ε 3 T = k 32 , longitudinal, k 33 = d 33 / s 33 E ε 3 T , \begin{gathered}
\text { shear, } k_{15}=d_{15} / \sqrt{s_{55}^E \varepsilon_1^T}=k_{24}, \quad \text { transverse, } k_{31}=d_{31} / \sqrt{s_{11}^E \varepsilon_3^T}=k_{32}, \\
\text { longitudinal, } k_{33}=d_{33} / \sqrt{s_{33}^E \varepsilon_3^T},
\end{gathered} shear, k 15 = d 15 / s 55 E ε 1 T = k 24 , transverse, k 31 = d 31 / s 11 E ε 3 T = k 32 , longitudinal, k 33 = d 33 / s 33 E ε 3 T , or, in the notation used, for force in the j j j i i i
k i j = d i j / s j j E ε i T . k_{i j}=d_{i j} / \sqrt{s_{j j}^E \varepsilon_i^T} \text {. } k ij = d ij / s j j E ε i T . Substituting equation (25) into (23) gives
s j i S U = s j j E [ 1 − k i j 2 Z ˉ i E L ] . s_{j i}^{S U}=s_{j j}^E\left[1-k_{i j}^2 \bar{Z}_i^{E L}\right] . s j i S U = s j j E [ 1 − k ij 2 Z ˉ i E L ] . 1.1 Shunted with negative capacity
For our problem, the electric field is along z z z x x x
s 33 S U = s 33 E [ 1 − k 31 2 Z ˉ 1 E L ] = s 33 E [ 1 − s C p T k 31 2 s C p T + Y S U ] = s 33 E [ s C p T ( 1 − k 31 2 ) + Y S U s C p T + Y S U ] s_{33}^{S U} = s_{33}^E\left[1-k_{31}^2 \bar{Z}_1^{E L}\right] = s_{33}^E\left[1- \frac{ s C_p^T k_{31}^2}{s C_p^T+Y^{S U}}\right] = s_{33}^E \left[\frac{ s C_p^T (1-k_{31}^2)+Y^{SU}}{s C_p^T+Y^{S U}}\right] s 33 S U = s 33 E [ 1 − k 31 2 Z ˉ 1 E L ] = s 33 E [ 1 − s C p T + Y S U s C p T k 31 2 ] = s 33 E [ s C p T + Y S U s C p T ( 1 − k 31 2 ) + Y S U ] Then we have
E p S U ( ω ) = E p E i ω C p T + Y S U i ω C p T ( 1 − k 31 2 ) + Y S U E_{\mathrm{p}}^{\mathrm{SU}}(\omega)=E_{\mathrm{p}}^E \frac{\mathrm{i} \omega C_{\mathrm{p}}^T+Y^{\mathrm{SU}}}{\mathrm{i} \omega C_{\mathrm{p}}^T\left(1-k_{31}^2\right)+Y^{\mathrm{SU}}} E p SU ( ω ) = E p E i ω C p T ( 1 − k 31 2 ) + Y SU i ω C p T + Y SU where s = i ω s=i\omega s = iω
If shunted with a negative capacity, we have
Y S U = − i ω C ′ = − i ω H C 0 Y^{SU} = -i\omega C'=-i\omega H C_0 Y S U = − iω C ′ = − iω H C 0 So the shunting Young’s modulus is (see reference [1])
E p S U ( ω ) = E p E C ′ − C p T C ′ − C p T ( 1 − k 31 2 ) E_{\mathrm{p}}^{\mathrm{SU}}(\omega)=E_{\mathrm{p}}^E \frac{C' - C_{\mathrm{p}}^T}{C' - C_{\mathrm{p}}^T\left(1-k_{31}^2\right)} E p SU ( ω ) = E p E C ′ − C p T ( 1 − k 31 2 ) C ′ − C p T or
E p S U ( ω ) = E p E C ′ − C p T C ′ − C p S E_{\mathrm{p}}^{\mathrm{SU}}(\omega)=E_{\mathrm{p}}^E \frac{C' - C_{\mathrm{p}}^T}{C' - C_{\mathrm{p}}^S} E p SU ( ω ) = E p E C ′ − C p S C ′ − C p T where C p S = ( 1 − k 31 2 ) C p T , C p T = A ε 33 T / d C_p^S = (1-k_{31}^2)C_p^T, \ C_{p}^T = A \varepsilon_{33}^T / d C p S = ( 1 − k 31 2 ) C p T , C p T = A ε 33 T / d
Finally,
E e q = E b I b + 2 E p S U I p I b + 2 I p I b = b H 3 12 , I p = b h p 3 12 + b h p ( H 2 + h p 2 ) 2 \begin{aligned}
&E_{e q}=\frac{E_b I_b+2 E_p^{S U} I_p}{I_b+2 I_p}\\
&I_b=\frac{b H^3}{12}, \quad I_p=\frac{b h_p^3}{12}+b h_p\left(\frac{H}{2}+\frac{h_p}{2}\right)^2
\end{aligned} E e q = I b + 2 I p E b I b + 2 E p S U I p I b = 12 b H 3 , I p = 12 b h p 3 + b h p ( 2 H + 2 h p ) 2 1.2 Shunting circuit
Circuit diagram
The impedance is derived in the following picture.
Derivation of impedance
Now we have
Z 2 = 1 j ω C + 1 / R 0 , Z 3 = R 1 , Z 4 = R 2 Z_2 = \frac{1}{j\omega C + 1/R_0}, \quad Z_3 = R_1, \quad Z_4 = R_2 Z 2 = j ω C + 1/ R 0 1 , Z 3 = R 1 , Z 4 = R 2 Then we have impedance
Z in = − R 1 R 2 ( j ω C + 1 / R 0 ) = 1 j ω C N Z_{\text{in}} =- \frac{R_1}{R_2\left(j\omega C + 1/R_0\right)} = \frac{1}{j\omega C_N} Z in = − R 2 ( j ω C + 1/ R 0 ) R 1 = j ω C N 1 where R 0 R_0 R 0 1 / R 0 1/R_0 1/ R 0
C N = − R 2 R 1 C C_N = -\frac{R_2}{R_1} C C N = − R 1 R 2 C 2 Reference
[1] Hagood, N. W., & Von Flotow, A. (1991). Damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of sound and vibration , 146 (2), 243-268.
[2] Beck, B. S., Cunefare, K. A., & Collet, M. (2013). The power output and efficiency of a negative capacitance shunt for vibration control of a flexural system. Smart Materials and Structures , 22 (6), 065009.
[3] Trainiti, G., Xia, Y., Marconi, J., Cazzulani, G., Erturk, A., & Ruzzene, M. (2019). Time-periodic stiffness modulation in elastic metamaterials for selective wave filtering: Theory and experiment. Physical review letters , 122 (12), 124301.
