We now consider the deformation of an electroelastic actuator corresponding to its stress state. If electric field strength \(E\) is created in the piezo actuator, strain \(S\) and mechanical stress \(T\) will appear. Correspondingly, if mechanical stress \(T\) is created in the piezo actuator, we will observe the appearance of electrical induction \(D\) and electrical charge at the plates of the piezo actuator.
The electroelastic equations of the nanomechanical actuators in the case of the inverse and direct piezo effect take the general form [6–8]
$${{S}_{i}} = {{d}_{{mi}}}{{E}_{m}} + s_{{ij}}^{E}{{T}_{j}};$$
$${{D}_{m}} = {{d}_{{mi}}}{{T}_{i}} + \varepsilon _{{mk}}^{T}{{E}_{k}},$$
where \(i,\;j = 1,\;2,\;...\;,6\) and \(m,\;k = 1,\;2,\;3\); \({{S}_{i}}\) is the relative displacement of the actuator cross section along the i axis; \({{d}_{{mi}}}\) is the piezomodulus for a generalized piezo effect; \({{E}_{m}}\left( t \right) = {{U\left( t \right)} \mathord{\left/ {\vphantom {{U\left( t \right)} \delta }} \right. \kern-0em} \delta }\) is the electric field strength along the m axis; \(U\left( t \right)\) is the voltage at the actuator plates; t is the time; \(\delta \) is the actuator thickness; \(s_{{ij}}^{E}\) is the elastic pliability when \(E = {\text{const;}}\) \({{T}_{j}}\) is the mechanical stress along the j axis; \({{D}_{m}}\left( t \right)\) is the electrical induction along the m axis; and \({\varepsilon }_{{mk}}^{T}\) is the dielectric permittivity when \(T = {\text{const}}{\text{.}}\)
The equation for the forces on the face of the electroelastic actuator takes the form
$$T{{S}_{0}} = F + M\frac{{{{{\text{d}}}^{2}}\xi \left( {x,\,\,t} \right)}}{{{\text{d}}{{t}^{2}}}},$$
where \({{S}_{0}}\) is the actuator area; F is the external force on the actuator; and M is the mass being moved.
In calculations of the electroelastic actuator, we use the wave equation describing wave propagation in a long line with damping but no distortion [6–8, 12]. By Laplace transformation, the wave equation with partial derivatives of hyperbolic type is reduced to a linear differential equation with parameter p, where p is the Laplacian operator [9–12].
By applying the Laplace transformation to the wave equation, in the case of null initial conditions, we obtain a second-order linear differential equation in the form
$$\frac{{{{{\text{d}}}^{2}}\Xi \left( {x,p} \right)}}{{{\text{d}}{{x}^{2}}}} - {{\gamma }^{2}}\Xi \left( {x,p} \right) = 0.$$
Its solution is the function
$$\Xi \left( {x,p} \right) = C{{{\text{e}}}^{{ - x\gamma }}} + B{{{\text{e}}}^{{x\gamma }}},$$
(1)
where \(\Xi \left( {x,p} \right)\) is the Laplace transformation of the displacement of the actuator’s cross section; \(\gamma = {p \mathord{\left/ {\vphantom {p {{{c}^{E}}}}} \right. \kern-0em} {{{c}^{E}}}} + \alpha \) is the propagation coefficient; \({{c}^{E}}\) is the speed of sound in the actuator when \(E = {\text{const}}\); and α is the damping factor, taking account of the damping due to energy scattering with the heat losses in wave propagation.
The structural-parametric model of a piezo actuator with voltage control is derived by solving the second-order linear differential equation, the equation of the inverse piezo effect, and the equation for the forces at the actuator’s faces.
In solving the linear differential equation, the coefficients С and B are determined on the basis of the conditions
$$\Xi \left( {0,p} \right) = {{\Xi }_{{\text{1}}}}\left( p \right){\text{ when }}x = 0;$$
$$\Xi \left( {l,p} \right) = {{\Xi }_{{\text{2}}}}\left( p \right){\text{ when }}x = l.$$
We find that
$$C = {{\left( {{{\Xi }_{1}}{{{\text{e}}}^{{l\gamma }}} - {{\Xi }_{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Xi }_{1}}{{{\text{e}}}^{{l\gamma }}} - {{\Xi }_{2}}} \right)} {\left[ {2{\text{sinh}}\left( {l\gamma } \right)} \right]}}} \right. \kern-0em} {\left[ {2{\text{sinh}}\left( {l\gamma } \right)} \right]}};$$
$${\text{and}}\,\,B = {{\left( {{{\Xi }_{2}} - {{\Xi }_{1}}{{{\text{e}}}^{{ - l\gamma }}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Xi }_{2}} - {{\Xi }_{1}}{{{\text{e}}}^{{ - l\gamma }}}} \right)} {\left[ {2{\text{sinh}}\left( {l\gamma } \right)} \right]}}} \right. \kern-0em} {\left[ {2{\text{sinh}}\left( {l\gamma } \right)} \right]}}.$$
The solution of the linear differential equation takes the form
$$\begin{gathered} \Xi (x,p) \\ = {{\left\{ {{{\Xi }_{1}}\left( p \right){\text{sinh}}\left[ {\left( {l - x} \right)\gamma } \right] + {{\Xi }_{2}}\left( p \right){\text{sinh}}\left( {x\gamma } \right)} \right\}} \mathord{\left/ {\vphantom {{\left\{ {{{\Xi }_{1}}\left( p \right){\text{sinh}}\left[ {\left( {l - x} \right)\gamma } \right] + {{\Xi }_{2}}\left( p \right){\text{sinh}}\left( {x\gamma } \right)} \right\}} {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}. \\ \end{gathered} $$
The equations for the forces at the actuator’s faces are
$${{T}_{j}}\left( {0,p} \right){{S}_{0}} = {{F}_{1}}\left( p \right) + {{M}_{1}}{{p}^{2}}{{\Xi }_{1}}\left( p \right)\,\,{\text{when }}x = 0;$$
$${{T}_{j}}\left( {l,p} \right){{S}_{0}} = - {{F}_{2}}\left( p \right) - {{M}_{2}}{{p}^{2}}{{\Xi }_{2}}\left( p \right)\,\,{\text{when }}x = l.$$
We obtain a system of equations for the mechanical stress of the electroelastic actuator when x = 0 and x = l
$$\left\{ {\begin{array}{*{20}{c}} {{{T}_{j}}\left( {0,p} \right) = \frac{1}{{s_{{ij}}^{E}}}{{{\left. {\frac{{{\text{d}}\Xi \left( {x,p} \right)}}{{{\text{d}}x}}} \right|}}_{{x = 0}}} - \frac{{{{d}_{{mi}}}}}{{s_{{ij}}^{E}}}{{E}_{m}}\left( p \right){\text{;}}} \\ {{{T}_{j}}\left( {l,p} \right) = \frac{1}{{s_{{ij}}^{E}}}{{{\left. {\frac{{{\text{d}}\Xi \left( {x,p} \right)}}{{{\text{d}}x}}} \right|}}_{{x = l}}} - \frac{{{{d}_{{mi}}}}}{{s_{{ij}}^{E}}}{{E}_{m}}\left( p \right){\kern 1pt} .} \end{array}} \right.$$
(2)
On the basis of Eq. (2), we may write the following structural-parametric model of the electroelastic actuator with a generalized piezo effect, in the case of voltage control and the parametric structure in Fig. 1
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{1}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{E}}}} \right. \kern-0em} {\chi _{{ij}}^{E}}}} \right)\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \left. {\left. {{{ - }^{{}}}\left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{1}}\left( p \right) - {{\Xi }_{2}}\left( p \right)} \right]} \right]} \right\}; \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{2}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{E}}}} \right. \kern-0em} {\chi _{{ij}}^{E}}}} \right)\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \,\,\left. {\left. { - \left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{2}}\left( p \right) - {{\Xi }_{1}}{{{\left( p \right)}}^{{}}}} \right]} \right]} \right\}{\text{,}} \hfill \\ \end{gathered} \end{array}} \right.$$
(3)
where \(\chi _{{ij}}^{E} = {{s_{{ij}}^{E}} \mathord{\left/ {\vphantom {{s_{{ij}}^{E}} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}.\)
After appropriate transformations, we obtain
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{1}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{E}}}} \right. \kern-0em} {\chi _{{ij}}^{E}}}} \right)\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \left. {\left. {{{ - }^{{}}}{{\gamma {{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{1}}\left( p \right)} {{\text{tanh}}\left( {l\gamma } \right) + {{\gamma {{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{2}}\left( p \right)} {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right) + {{\gamma {{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{2}}\left( p \right)} {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}}}} \right]} \right\}{\text{;}} \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{2}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{E}}}} \right. \kern-0em} {\chi _{{ij}}^{E}}}} \right)\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \left. {\left. {{{ - }^{{}}}{{\gamma {{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{2}}\left( p \right)} {{\text{tanh}}\left( {l\gamma } \right) + {{\gamma {{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{1}}\left( p \right)} {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right) + {{\gamma {{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{\gamma {{\Xi }_{1}}\left( p \right)} {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}}}} \right]} \right\}. \hfill \\ \end{gathered} \end{array}} \right.$$
(4)
We may write Eq. (3) in the form
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{1}}\left( p \right) + C_{{ij}}^{E}l\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \left. {\left. {{{ - }^{{}}}\left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{1}}\left( p \right) - {{\Xi }_{2}}\left( p \right)} \right]} \right]} \right\}{\text{;}} \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{2}}\left( p \right) + C_{{ij}}^{E}l\left[ {d_{{mi}}^{{}}{{E}_{m}}\left( p \right)} \right.} \right. \hfill \\ \left. {\left. {{{ - }^{{}}}\left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{2}}\left( p \right) - {{\Xi }_{1}}\left( p \right)} \right]} \right]} \right\}{\text{,}} \hfill \\ \end{gathered} \end{array}} \right.$$
where \(C_{{ij}}^{E} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{ij}}^{E}l} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{E}l} \right)}} = {1 \mathord{\left/ {\vphantom {1 {\left( {\chi _{{ij}}^{E}l} \right)}}} \right. \kern-0em} {\left( {\chi _{{ij}}^{E}l} \right)}}\) is the rigidity of the actuator with a generalized piezo effect, in the case of voltage control.
On the basis of the structural-parametric model, we may determine the transfer functions of the actuator. Solution of Eq. (3) for the displacement of two faces in an actuator with a generalized piezo effect, in the case of voltage control, yields
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = {{W}_{{11}}}\left( p \right){{E}_{m}}\left( p \right) \hfill \\ + \,\,{{W}_{{12}}}\left( p \right){{F}_{1}}\left( p \right) + {{W}_{{13}}}\left( p \right){{F}_{2}}\left( p \right){\kern 1pt} {\text{;}} \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = {{W}_{{21}}}\left( p \right){{E}_{m}}\left( p \right) \hfill \\ + \,\,{{W}_{{22}}}\left( p \right){{F}_{1}}\left( p \right) + {{W}_{{23}}}\left( p \right){{F}_{2}}\left( p \right){\kern 1pt} . \hfill \\ \end{gathered} \end{array}} \right.$$
(5)
In Eq. (5), the generalized transfer functions take the form
$$\begin{gathered} {{W}_{{11}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{E}_{m}}\left( p \right)}}} \right. \kern-0em} {{{E}_{m}}\left( p \right)}} \\ = {{{{d}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{E}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{d}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{E}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{21}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{E}_{m}}\left( p \right)}}} \right. \kern-0em} {{{E}_{m}}\left( p \right)}} \\ = {{{{d}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{E}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{d}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{E}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{12}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. \kern-0em} {{{F}_{1}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{E}\left[ {{{M}_{2}}\chi _{{ij}}^{E}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{E}\left[ {{{M}_{2}}\chi _{{ij}}^{E}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{13}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} = {{W}_{{22}}}(p) \\ = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. \kern-0em} {{{F}_{1}}\left( p \right)}} = {{\left[ {{{\chi _{{ij}}^{E}\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{E}\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\chi _{{ij}}^{E}\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{E}\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{23}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{E}\left[ {{{M}_{1}}\chi _{{ij}}^{E}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{E}\left[ {{{M}_{1}}\chi _{{ij}}^{E}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}, \\ \end{gathered} $$
where \(\chi _{{ij}}^{E} = {{s_{{ij}}^{E}} \mathord{\left/ {\vphantom {{s_{{ij}}^{E}} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}\) and
$$\begin{gathered} {{A}_{{ij}}} = {{M}_{1}}{{M}_{2}}{{\left( {\chi _{{ij}}^{E}} \right)}^{2}}{{p}^{4}} \\ + \,\,\left\{ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{E}} \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{E}} {\left[ {{{c}^{E}}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right. \kern-0em} {\left[ {{{c}^{E}}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right\}{{p}^{3}} \\ + \,\,\left[ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{E}\alpha } \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{E}\alpha } {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}} + {1 \mathord{\left/ {\vphantom {1 {{{{\left( {{{c}^{E}}} \right)}}^{2}}}}} \right. \kern-0em} {{{{\left( {{{c}^{E}}} \right)}}^{2}}}}} \right]{{p}^{2}} \\ + \,\,{{2\alpha p} \mathord{\left/ {\vphantom {{2\alpha p} {{{c}^{E}}}}} \right. \kern-0em} {{{c}^{E}}}} + {{\alpha }^{2}}{\text{.}} \\ \end{gathered} $$
From Eq. (5), we obtain the matrix equation
$$\left( {\begin{array}{*{20}{c}} {{{\Xi }_{1}}\left( p \right)} \\ {{{\Xi }_{2}}\left( p \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{W}_{{11}}}\left( p \right)}&{{{W}_{{12}}}\left( p \right)}&{{{W}_{{13}}}\left( p \right)} \end{array}} \\ {\begin{array}{*{20}{c}} {{{W}_{{21}}}\left( p \right)}&{{{W}_{{22}}}\left( p \right)}&{{{W}_{{23}}}\left( p \right)} \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{E}_{m}}\left( p \right)} \\ {{{F}_{1}}\left( p \right)} \\ {{{F}_{2}}\left( p \right)} \end{array}} \right).$$
We now consider the influence of the actuator’s reaction associated with the counteremf in the direct generalized piezo effect with static deformation.
The maximum force \({{F}_{{{\text{max}}}}}\) and mechanical stress \({{T}_{{j{\text{max}}}}}\) developed by the piezo actuator with a generalized piezo effect in the case of a voltage source are as follows
$${{F}_{{{\text{max}}}}} = U\frac{1}{\delta }{{d}_{{mi}}}\frac{{{{S}_{0}}}}{{s_{{ij}}^{E}}};$$
$$\frac{{{{F}_{{{\text{max}}}}}}}{{{{S}_{0}}}}s_{{ij}}^{E} = {{E}_{m}}{{d}_{{mi}}};$$
$${{T}_{{j{\text{max}}}}}s_{{ij}}^{E} = {{E}_{m}}{{d}_{{mi}}}.$$
Hence
$${{T}_{{j\max }}} = {{{{E}_{m}}{{d}_{{mi}}}} \mathord{\left/ {\vphantom {{{{E}_{m}}{{d}_{{mi}}}} {s_{{ij}}^{E}}}} \right. \kern-0em} {s_{{ij}}^{E}}};$$
$${{F}_{{\max }}} = {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} \mathord{\left/ {\vphantom {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} {s_{{ij}}^{E}}}} \right. \kern-0em} {s_{{ij}}^{E}}}.$$
To simplify the equations, we use the electromechanical coupling coefficient [8, 10–12]
$${{k}_{{mi}}} = {{{{d}_{{mi}}}} \mathord{\left/ {\vphantom {{{{d}_{{mi}}}} {\sqrt {\varepsilon _{{mk}}^{T}s_{{ij}}^{E}} }}} \right. \kern-0em} {\sqrt {\varepsilon _{{mk}}^{T}s_{{ij}}^{E}} }}.$$
Then, for a piezo actuator of TsTS or PZT piezoceramic, we obtain the electromechanical coupling coefficients for the transverse, longitudinal, and shear piezo effects in the form
$${{k}_{{31}}} = {{{{d}_{{31}}}} \mathord{\left/ {\vphantom {{{{d}_{{31}}}} {\sqrt {\varepsilon _{{33}}^{T}s_{{11}}^{E}} }}} \right. \kern-0em} {\sqrt {\varepsilon _{{33}}^{T}s_{{11}}^{E}} }};$$
$${{k}_{{33}}} = {{{{d}_{{33}}}} \mathord{\left/ {\vphantom {{{{d}_{{33}}}} {\sqrt {\varepsilon _{{33}}^{T}s_{{33}}^{E}} }}} \right. \kern-0em} {\sqrt {\varepsilon _{{33}}^{T}s_{{33}}^{E}} }};$$
$${{k}_{{15}}} = {{{{d}_{{15}}}} \mathord{\left/ {\vphantom {{{{d}_{{15}}}} {\sqrt {\varepsilon _{{11}}^{T}s_{{55}}^{E}} }}} \right. \kern-0em} {\sqrt {\varepsilon _{{11}}^{T}s_{{55}}^{E}} }}.$$
The maximum force \({{F}_{{{\text{max}}}}}\) and maximum stress \({{T}_{{j{\text{max}}}}}\) developed by an actuator with a generalized piezo effect, in the case of voltage control, may then be derived
$$\left. \begin{gathered} {{F}_{{{\text{max}}}}} = U\frac{1}{\delta }{{d}_{{mi}}}\frac{{{{S}_{0}}}}{{s_{{ij}}^{E}}} + {{F}_{{{\text{max}}}}}\frac{1}{{{{S}_{0}}}}{{d}_{{mi}}}{{S}_{p}}\frac{1}{{{{\varepsilon _{{mk}}^{T}{{S}_{p}}} \mathord{\left/ {\vphantom {{\varepsilon _{{mk}}^{T}{{S}_{p}}} \delta }} \right. \kern-0em} \delta }}}\frac{1}{\delta }{{d}_{{mi}}}\frac{{{{S}_{0}}}}{{s_{{ij}}^{E}}}; \hfill \\ \frac{{{{F}_{{{\text{max}}}}}}}{{{{S}_{0}}}}s_{{ij}}^{E}\left( {1 - \frac{{d_{{mi}}^{2}}}{{\varepsilon _{{mk}}^{T}s_{{ij}}^{E}}}} \right) = {{E}_{m}}{{d}_{{mi}}}; \hfill \\ k_{{mi}}^{2} = \frac{{d_{{mi}}^{2}}}{{\varepsilon _{{mk}}^{T}s_{{ij}}^{E}}};\,\,\,\,{{T}_{{j{\text{max}}}}}\left( {1 - k_{{mi}}^{2}} \right)s_{{ij}}^{E} = {{E}_{m}}{{d}_{{mi}}}. \hfill \\ \end{gathered} \right\}$$
(6)
Hence, we may write
$$\left. \begin{gathered} {{T}_{{j{\text{max}}}}}s_{{ij}}^{D} = {{E}_{m}}{{d}_{{mi}}}; \hfill \\ s_{{ij}}^{D} = \left( {1 - k_{{mi}}^{2}} \right)s_{{ij}}^{E} = {{k}_{s}}s_{{ij}}^{E}; \hfill \\ {{k}_{s}} = 1 - k_{{mi}}^{2} = {{s_{{ij}}^{D}} \mathord{\left/ {\vphantom {{s_{{ij}}^{D}} {s_{{ij}}^{E}}}} \right. \kern-0em} {s_{{ij}}^{E}}},\,\,\,\,{{k}_{s}} > 0, \hfill \\ \end{gathered} \right\};$$
(7)
where \({{k}_{s}}\) characterizes the change in elastic pliability.
From Eqs. (6) and (7)
$${{F}_{{{\text{max}}}}} = {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} \mathord{\left/ {\vphantom {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} {\left( {s_{{ij}}^{E}{{k}_{s}}} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{E}{{k}_{s}}} \right)}} = {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} \mathord{\left/ {\vphantom {{{{E}_{m}}{{d}_{{mi}}}{{S}_{0}}} {s_{{ij}}^{D}}}} \right. \kern-0em} {s_{{ij}}^{D}}}{\kern 1pt} ;$$
$${{T}_{{j{\text{max}}}}} = {{{{E}_{m}}{{d}_{{mi}}}} \mathord{\left/ {\vphantom {{{{E}_{m}}{{d}_{{mi}}}} {s_{{ij}}^{D}}}} \right. \kern-0em} {s_{{ij}}^{D}}}{\kern 1pt} .$$
The actuator’s elastic pliability \({{s}_{{ij}}}\) satisfies the condition \(s_{{ij}}^{E} > {{s}_{{ij}}} > s_{{ij}}^{D},\) while \({{s_{{ij}}^{E}} \mathord{\left/ {\vphantom {{s_{{ij}}^{E}} {s_{{ij}}^{D}}}} \right. \kern-0em} {s_{{ij}}^{D}}} \leqslant 1,2\). Correspondingly, \(C_{{ij}}^{E} < \;C{}_{{ij}} < \;C_{{ij}}^{D}\), where \({{C}_{{ij}}} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {{{s}_{{ij}}}l} \right)}}} \right. \kern-0em} {\left( {{{s}_{{ij}}}l} \right)}}\) is the rigidity of the piezo actuator; \(C_{{ij}}^{E} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{ij}}^{E}l} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{E}l} \right)}}\) is its rigidity with voltage control; and \(C_{{ij}}^{D} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{ij}}^{D}l} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{D}l} \right)}}\) is its rigidity with current control [10]. With open electrodes, the rigidity of the piezo actuator is greater than with closed electrodes. With increase in resistance of the power source and the matching circuits, the elastic pliability of the actuator declines, while its rigidity increases.
For a piezo actuator with a generalized piezo effect in the case of a power source with finite resistance, taking of the direct piezo effect, we may write the maximum force of the actuator in the form
$${{F}_{{{\text{max}}}}} = U\frac{1}{\delta }{{d}_{{mi}}}\frac{{{{S}_{0}}}}{{s_{{ij}}^{E}}} + {{F}_{{{\text{max}}}}}\frac{1}{{{{S}_{0}}}}{{d}_{{mi}}}{{S}_{p}}\frac{1}{{{{\varepsilon _{{mk}}^{T}{{S}_{p}}} \mathord{\left/ {\vphantom {{\varepsilon _{{mk}}^{T}{{S}_{p}}} \delta }} \right. \kern-0em} \delta }}}{{k}_{u}}\frac{1}{\delta }{{d}_{{mi}}}\frac{{{{S}_{0}}}}{{s_{{ij}}^{E}}}.$$
Hence
$$\frac{{{{F}_{{{\text{max}}}}}}}{{{{S}_{0}}}}s_{{ij}}^{E}\left( {1 - \frac{{d_{{mi}}^{2}{{k}_{u}}}}{{\varepsilon _{{mk}}^{T}s_{{ij}}^{E}}}} \right) = {{E}_{m}}{{d}_{{mi}}};$$
$${{T}_{{j{\text{max}}}}}\left( {1 - k_{{mi}}^{2}{{k}_{u}}} \right)s_{{ij}}^{E} = {{E}_{m}}{{d}_{{mi}}},\,\,\,\,0 \leqslant {{k}_{u}} \leqslant 1,$$
where the coefficient \({{k}_{u}}\) characterizes the type of control (voltage or current control). In the case of current control, \({{\left. {{{k}_{u}}} \right|}_{{R \to \infty }}} = 1\); in the case of voltage control, \({{\left. {{{k}_{u}}} \right|}_{{R \to 0}}} = 0\).
The elastic pliability takes the form
$${{s}_{{ij}}} = \left( {1 - k_{{mi}}^{2}{{k}_{u}}} \right)s_{{ij}}^{E} = {{k}_{s}}s_{{ij}}^{E};$$
$${{k}_{s}} = 1 - k_{{mi}}^{2}{{k}_{u}},\,\,\,\,{{k}_{s}} > 0;$$
$$\begin{gathered} {{\left. {\left( {1 - k_{{mi}}^{2}} \right)} \right|}_{{R \to \infty }}} \leqslant {{k}_{s}} \leqslant {{\left. 1 \right|}_{{R \to 0}}}{\kern 1pt} ,\,\,\,\,{{\left. {{{k}_{s}}} \right|}_{{R \to \infty }}} = 1 - k_{{mi}}^{2}, \\ {{\left. {{{k}_{s}}} \right|}_{{R \to 0}}} = 1, \\ \end{gathered} $$
where the coefficient \({{k}_{s}}\) characterizes the change in elastic pliability.
For a piezo actuator with a generalized piezo effect in the case of a power source with finite resistance, when the structural-parametric model includes feedback with respect to the force (Fig. 2), we may write
$${{U}_{{F\alpha }}}\left( p \right) = \frac{{{{k}_{u}}\left( {l{\text{/}}\delta } \right){{d}_{{mi}}}}}{{{{C}_{0}}}}{{F}_{\alpha }}\left( p \right),\,\,\,\,\alpha = 1,2.$$
In the case of current control, when the resistance of the source is infinite, \({{\left. {{{k}_{u}}} \right|}_{{R \to \infty }}} = 1\).
For an electroelastic actuator with current control, the mechanical stress when x = 0 and x = l may be written in the form
$$\left\{ {\begin{array}{*{20}{c}} {{{T}_{j}}\left( {0,p} \right) = \frac{1}{{s_{{ij}}^{D}}}{{{\left. {\frac{{d\Xi \left( {x,p} \right)}}{{dx}}} \right|}}_{{x = 0}}} - \frac{{{{g}_{{mi}}}}}{{s_{{ij}}^{D}}}{{D}_{m}}\left( p \right){\kern 1pt} {\text{;}}} \\ {{{T}_{j}}\left( {l,p} \right) = \frac{1}{{s_{{ij}}^{D}}}{{{\left. {\frac{{d\Xi \left( {x,p} \right)}}{{dx}}} \right|}}_{{x = l}}} - \frac{{{{g}_{{mi}}}}}{{s_{{ij}}^{D}}}{{D}_{m}}\left( p \right),} \end{array}} \right.$$
and thus the structural-parametric model of an electroelastic actuator in the case of a generalized piezo effect and current control takes the form
$$\left\{ \begin{gathered} {{\Xi }_{1}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{1}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{D}}}} \right. \kern-0em} {\chi _{{ij}}^{D}}}} \right)\left[ {{{g}_{{mi}}}{{D}_{m}}\left( p \right)_{{}}^{{}}} \right.} \right. \hfill \\ \left. {\left. { - \,\,\left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]_{{}}^{{}}\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{1}}\left( p \right) - {{\Xi }_{2}}\left( p \right)} \right]} \right]} \right\}{\text{;}} \hfill \\ {{\Xi }_{2}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{2}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{D}}}} \right. \kern-0em} {\chi _{{ij}}^{D}}}} \right)\left[ {{{g}_{{mi}}}{{D}_{m}}\left( p \right)_{{}}^{{}}} \right.} \right. \hfill \\ \left. {\left. { - \,\,\left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]_{{}}^{{}}\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{2}}\left( p \right) - {{\Xi }_{1}}\left( p \right)} \right]} \right]} \right\}{\text{,}} \hfill \\ \end{gathered} \right.$$
(8)
where \(\chi _{{ij}}^{D} = {{s_{{ij}}^{D}} \mathord{\left/ {\vphantom {{s_{{ij}}^{D}} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}{\kern 1pt} .\)
From Eq. (8), we obtain
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = {{W}_{{11}}}\left( p \right){{D}_{m}}\left( p \right) \hfill \\ + \,\,{{W}_{{12}}}\left( p \right){{F}_{1}}\left( p \right) + {{W}_{{13}}}\left( p \right){{F}_{2}}\left( p \right){\text{;}} \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = {{W}_{{21}}}\left( p \right){{D}_{m}}\left( p \right) \hfill \\ + \,\,{{W}_{{22}}}\left( p \right){{F}_{1}}\left( p \right) + {{W}_{{23}}}\left( p \right){{F}_{2}}\left( p \right){\text{,}} \hfill \\ \end{gathered} \end{array}} \right.$$
where the transfer functions are
$$\begin{gathered} {{W}_{{11}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{D}_{m}}\left( p \right)}}} \right. \kern-0em} {{{D}_{m}}\left( p \right)}} \\ = {{{{g}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{D}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{g}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{D}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{21}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{D}_{m}}\left( p \right)}}} \right. \kern-0em} {{{D}_{m}}\left( p \right)}} \\ = {{{{g}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{D}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{g}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{D}{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{12}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. } {{{F}_{1}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{D}\left[ {{{M}_{2}}\chi _{{ij}}^{D}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. } {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{D}\left[ {{{M}_{2}}\chi _{{ij}}^{D}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. } {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. } {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{13}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} = {{W}_{{22}}}\left( p \right) \\ = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. \kern-0em} {{{F}_{1}}\left( p \right)}} = {{\left[ {{{\chi _{{ij}}^{D}\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{D}\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\chi _{{ij}}^{D}\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{D}\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{23}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{D}\left[ {{{M}_{1}}\chi _{{ij}}^{D}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{D}\left[ {{{M}_{1}}\chi _{{ij}}^{D}{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}. \\ \end{gathered} $$
Here \(\chi _{{ij}}^{D} = {{s_{{ij}}^{D}} \mathord{\left/ {\vphantom {{s_{{ij}}^{D}} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}\) and
$$\begin{gathered} {{A}_{{ij}}} = {{M}_{1}}{{M}_{2}}{{\left( {\chi _{{ij}}^{D}} \right)}^{2}}{{p}^{4}} \\ + \,\,\left\{ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{D}} \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{D}} {\left[ {{{c}^{D}}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right. \kern-0em} {\left[ {{{c}^{D}}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right\}{{p}^{3}} \\ + \left[ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{D}\alpha } \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{D}\alpha } {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}} + {1 \mathord{\left/ {\vphantom {1 {{{{\left( {{{c}^{D}}} \right)}}^{2}}}}} \right. \kern-0em} {{{{\left( {{{c}^{D}}} \right)}}^{2}}}}} \right]{{p}^{2}} \\ + \,\,{{2\alpha p} \mathord{\left/ {\vphantom {{2\alpha p} {{{c}^{D}}}}} \right. \kern-0em} {{{c}^{D}}}} + {{\alpha }^{2}}. \\ \end{gathered} $$
Introducing the control parameter \(\Psi = E,D\) for the actuator, we now write the transfer functions in the general form
$$\begin{gathered} {{W}_{{11}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{E}_{1}}\left( p \right)}}} \right. \kern-0em} {{{E}_{1}}\left( p \right)}} \\ = {{{{d}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{\Psi }{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{d}_{{mi}}}\left[ {{{M}_{2}}\chi _{{ij}}^{\Psi }{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{21}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{E}_{m}}\left( p \right)}}} \right. \kern-0em} {{{E}_{m}}\left( p \right)}} \\ = {{{{d}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{\Psi }{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} \mathord{\left/ {\vphantom {{{{d}_{{mi}}}\left[ {{{M}_{1}}\chi _{{ij}}^{\Psi }{{p}^{2}} + \gamma {\text{tanh}}\left( {{{l\gamma } \mathord{\left/ {\vphantom {{l\gamma } 2}} \right. \kern-0em} 2}} \right)} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{12}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. \kern-0em} {{{F}_{1}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{\Psi }\left[ {{{M}_{2}}\chi _{{ij}}^{\Psi }{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{\Psi }\left[ {{{M}_{2}}\chi _{{ij}}^{\Psi }{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{13}}}\left( p \right) = {{{{\Xi }_{1}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{1}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} = {{W}_{{22}}}\left( p \right) \\ = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{1}}\left( p \right)}}} \right. \kern-0em} {{{F}_{1}}\left( p \right)}} = {{\left[ {{{\chi _{{ij}}^{\Psi }\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{\Psi }\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\chi _{{ij}}^{\Psi }\gamma } \mathord{\left/ {\vphantom {{\chi _{{ij}}^{\Psi }\gamma } {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}{\kern 1pt} ; \\ \end{gathered} $$
$$\begin{gathered} {{W}_{{23}}}\left( p \right) = {{{{\Xi }_{2}}\left( p \right)} \mathord{\left/ {\vphantom {{{{\Xi }_{2}}\left( p \right)} {{{F}_{2}}\left( p \right)}}} \right. \kern-0em} {{{F}_{2}}\left( p \right)}} \\ = - {{\chi _{{ij}}^{\Psi }\left[ {{{M}_{1}}\chi _{{ij}}^{\Psi }{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} \mathord{\left/ {\vphantom {{\chi _{{ij}}^{\Psi }\left[ {{{M}_{1}}\chi _{{ij}}^{\Psi }{{p}^{2}} + {\gamma \mathord{\left/ {\vphantom {\gamma {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}}} \right]} {{{A}_{{ij}}}}}} \right. \kern-0em} {{{A}_{{ij}}}}}, \\ \end{gathered} $$
where \(\chi _{{ij}}^{\Psi } = {{s_{{ij}}^{\Psi }} \mathord{\left/ {\vphantom {{s_{{ij}}^{\Psi }} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}{\kern 1pt} ;\) and
$$\begin{gathered} {{A}_{{ij}}} = {{M}_{1}}{{M}_{2}}{{\left( {\chi _{{ij}}^{\Psi }} \right)}^{2}}{{p}^{4}} \\ + \,\,\left\{ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{\Psi }} \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{\Psi }} {\left[ {{{c}^{\Psi }}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right. \kern-0em} {\left[ {{{c}^{\Psi }}{\text{tanh}}\left( {l\gamma } \right)} \right]}}} \right\}{{p}^{3}} \\ + \,\,\left[ {{{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{\Psi }\alpha } \mathord{\left/ {\vphantom {{\left( {{{M}_{1}} + {{M}_{2}}} \right)\chi _{{ij}}^{\Psi }\alpha } {{\text{tanh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{tanh}}\left( {l\gamma } \right)}} + {1 \mathord{\left/ {\vphantom {1 {{{{\left( {{{c}^{\Psi }}} \right)}}^{2}}}}} \right. \kern-0em} {{{{\left( {{{c}^{\Psi }}} \right)}}^{2}}}}} \right]{{p}^{2}} \\ + \,\,{{2\alpha p} \mathord{\left/ {\vphantom {{2\alpha p} {{{c}^{\Psi }}}}} \right. \kern-0em} {{{c}^{\Psi }}}} + {{\alpha }^{{\text{2}}}}{\text{.}} \\ \end{gathered} $$
Hence, the structural-parametric model for an electroelastic actuator with a generalized piezo effect (Fig. 1) takes the form
$$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} {{\Xi }_{1}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{1}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{1}}\left( p \right) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{\Psi }}}} \right. \kern-0em} {\chi _{{ij}}^{\Psi }}}} \right)\left[ {{{d}_{{mi}}}{{\Psi }_{m}}\left( p \right)_{{}}^{{}}} \right.} \right. \hfill \\ \,\left. {\left. { - \left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]_{{}}^{{}}\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{1}}\left( p \right) - {{\Xi }_{2}}\left( p \right)} \right]} \right]} \right\}{\text{;}} \hfill \\ \end{gathered} \\ \begin{gathered} {{\Xi }_{2}}\left( p \right) = \left[ {{1 \mathord{\left/ {\vphantom {1 {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{M}_{2}}{{p}^{2}}} \right)}}} \right]\left\{ { - {{F}_{2}}(p) + \left( {{1 \mathord{\left/ {\vphantom {1 {\chi _{{ij}}^{\Psi }}}} \right. \kern-0em} {\chi _{{ij}}^{\Psi }}}} \right)\left[ {{{d}_{{mi}}}{{\Psi }_{m}}\left( p \right)_{{}}^{{}}} \right.} \right. \hfill \\ \,\left. {\left. { - \left[ {{\gamma \mathord{\left/ {\vphantom {\gamma {{\text{sinh}}\left( {l\gamma } \right)}}} \right. \kern-0em} {{\text{sinh}}\left( {l\gamma } \right)}}} \right]_{{}}^{{}}\left[ {{\text{cosh}}\left( {l\gamma } \right){{\Xi }_{2}}\left( p \right) - {{\Xi }_{1}}\left( p \right)} \right]} \right]} \right\}{\text{,}} \hfill \\ \end{gathered} \end{array}} \right.$$
(9)
where \(\chi _{{ij}}^{\Psi } = {{s_{{ij}}^{\Psi }} \mathord{\left/ {\vphantom {{s_{{ij}}^{\Psi }} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}.\)
To take account of the velocity of the actuator with a generalized piezo effect associated with the counteremf due to the direct piezo effect, we introduce feedback in the structural-parametric model
$${{U}_{{\dot {\Xi }\alpha }}}\left( p \right) = \frac{{{{d}_{{mi}}}{{S}_{0}}R}}{{\delta {{s}_{{ij}}}}}{{\mathop \Xi \limits^ \bullet }_{\alpha }}\left( p \right),\,\,\,\,\alpha = 1,2.$$
From Eq. (9), we obtain the transfer functions for an electroelastic actuator fixed at one end in the frequency range \(0 < \omega < 0.01{{{{c}^{\Psi }}} \mathord{\left/ {\vphantom {{{{c}^{\Psi }}} l}} \right. \kern-0em} l}\) when \({{{{M}_{2}}} \mathord{\left/ {\vphantom {{{{M}_{2}}} m}} \right. \kern-0em} m} \gg 1,\) in the case of an inertial load and voltage or current control
$${{W}_{{21}}}\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{{{E}_{m}}\left( p \right)}} = \frac{{{{d}_{{mi}}}l}}{{{{{\left( {T_{{ij}}^{\Psi }} \right)}}^{2}}{{p}^{2}} + 2T_{{ij}}^{\Psi }\xi _{{ij}}^{\Psi }p + 1}};$$
$${{W}_{{23}}}\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{{{F}_{2}}\left( p \right)}} = - \frac{{{1 \mathord{\left/ {\vphantom {1 {C_{{ij}}^{\Psi }}}} \right. \kern-0em} {C_{{ij}}^{\Psi }}}}}{{{{{\left( {T_{{ij}}^{\Psi }} \right)}}^{2}}{{p}^{2}} + 2T_{{ij}}^{\Psi }\xi i_{{ij}}^{\Psi }p + 1}};$$
$${{W}_{{21}}}\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{d}_{{mi}}}\left( {{l \mathord{\left/ {\vphantom {l \delta }} \right. \kern-0em} \delta }} \right)}}{{{{{\left( {T_{{ij}}^{\Psi }} \right)}}^{2}}{{p}^{2}} + 2T_{{ij}}^{\Psi }\xi _{{ij}}^{\Psi }p + 1}};$$
$$\begin{gathered} T_{{ij}}^{\Psi } = \sqrt {{{{{M}_{2}}s_{{ij}}^{\Psi }l} \mathord{\left/ {\vphantom {{{{M}_{2}}s_{{ij}}^{\Psi }l} {{{S}_{0}}}}} \right. \kern-0em} {{{S}_{0}}}}} = \sqrt {{{{{M}_{2}}} \mathord{\left/ {\vphantom {{{{M}_{2}}} {C_{{ij}}^{\Psi }}}} \right. \kern-0em} {C_{{ij}}^{\Psi }}}} ;\,\,\,\,C_{{ij}}^{\Psi } = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{ij}}^{\Psi }l} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{\Psi }l} \right)}}; \\ \xi _{{ij}}^{\Psi } = {{\alpha \delta \sqrt {{m \mathord{\left/ {\vphantom {m {{{M}_{2}}}}} \right. \kern-0em} {{{M}_{2}}}}} } \mathord{\left/ {\vphantom {{\alpha \delta \sqrt {{m \mathord{\left/ {\vphantom {m {{{M}_{2}}}}} \right. \kern-0em} {{{M}_{2}}}}} } 3}} \right. \kern-0em} 3}. \\ \end{gathered} $$
Here \(T_{{ij}}^{\Psi } = T_{{ij}}^{E},T_{{ij}}^{D},\) \(\xi _{{ij}}^{\Psi } = \xi _{{ij}}^{E},\xi _{{ij}}^{D},\) \(C_{{ij}}^{\Psi } = C_{{ij}}^{E},C_{{ij}}^{D},\) and \(s_{{ij}}^{\Psi } = s_{{ij}}^{E},s_{{ij}}^{D}\) are, respectively, the time constant, damping factors, rigidity, and elastic pliability of the actuator with control parameter \(\Psi = E{\text{,}}\,\,D\), where \(E\) and \(D\) correspond to voltage and current control, respectively.
In the case of a power source with finite resistance and a generalized piezo effect, taking account of the elastic pliability and rigidity of the electroelastic actuator, the feedback with respect to the force may be written in the form
$${{U}_{F}}\left( p \right) = \frac{{{{k}_{u}}\left( {l{\text{/}}\delta } \right){{d}_{{mi}}}}}{{{{C}_{0}}}}{{F}_{2}}\left( p \right).$$
For current control, with infinite source resistance, \({{\left. {{{k}_{u}}} \right|}_{{R \to \infty }}} = 1\); for voltage control, with zero source resistance, \({{\left. {{{k}_{u}}} \right|}_{{R \to 0}}} = 0\).
Correspondingly, for an electroelastic actuator with a generalized piezo effect, in the case of a power source with finite resistance, we introduce the following feedback in the structural-parametric model
$${{U}_{{\dot {\Xi }}}}\left( p \right) = \frac{{{{d}_{{mi}}}{{S}_{0}}R}}{{\delta {{s}_{{ij}}}}}{{\dot {\Xi }}_{2}}\left( p \right).$$
We now consider structures with distributed and point parameters for an electroelastic actuator fixed at one end with inertial elastic loading and voltage control, with finite source resistance. From Eqs. (4) and (5), as \({{M}_{1}} \to \infty ,\) we obtain a structure with distributed parameters (Fig. 3).
After replacing the hyperbolic cotangent with two terms of the power series and introducing the coefficient \({{k}_{d}}\) of the direct electroelastic effect and the coefficient \({{k}_{r}}\) of the inverse electroelastic effect in Fig. 3\(\left( {{{k}_{d}} = {{k}_{r}} = \frac{{{{d}_{{mi}}}{{S}_{0}}}}{{\delta {{s}_{{ij}}}}}} \right)\), we obtain a structure with point parameters as \({{M}_{1}} \to \infty \) (Fig. 4).
Transformation of Fig. 4 yields the structure with point parameters for the actuator in Fig. 5.
Taking account of the structure with point parameters, we find the transfer function in the form
$$W\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{k}_{r}}}}{{R{{C}_{0}}{{M}_{2}}{{p}^{3}} + \left( {{{M}_{2}} + R{{C}_{0}}{{k}_{\nu }}} \right){{p}^{2}} + \,\,\left( {{{k}_{\nu }} + R{{C}_{0}}C{}_{{ij}} + R{{C}_{0}}C{}_{e} + R{{k}_{r}}{{k}_{d}}} \right)p + C{}_{{ij}} + C{}_{e}}},$$
where \({{\Xi }_{2}}\left( p \right)\) and \(U\left( p \right)\) are Laplace transforms of the tip motion and the voltage at the actuator plates; \(C{}_{{ij}}\) is the rigidity of the electroelastic actuator (\(C_{{ij}}^{E} < C{}_{{ij}} < C_{{ij}}^{D}\)); and \({{k}_{\nu }}\) is its damping factor.
When \(R{{k}_{r}}{{k}_{d}} \ll {{k}_{\nu }}\) or \(Rk_{r}^{2} \ll {{k}_{\nu }}\) we may write
$$W\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{k}_{r}}}}{{\left( {R{{C}_{0}}p + 1} \right)\left( {{{M}_{2}}{{p}^{2}} + {{k}_{\nu }}p + C{}_{{ij}}\, + \,C{}_{e}} \right)}}.$$
When \(R = 0,\) we write the transfer function in the form
$$\begin{gathered} W\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{k}_{r}}}}{{{{M}_{2}}{{p}^{2}} + {{k}_{\nu }}p + C_{{ij}}^{E} + C{}_{e}}} \\ = \frac{{{{{{k}_{r}}} \mathord{\left/ {\vphantom {{{{k}_{r}}} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right. \kern-0em} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}}}{{\left( {\left( {{{{{M}_{2}}} \mathord{\left/ {\vphantom {{{{M}_{2}}} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right. \kern-0em} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right){{p}^{2}} + \left( {{{{{k}_{\nu }}} \mathord{\left/ {\vphantom {{{{k}_{\nu }}} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right. \kern-0em} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right)p + 1} \right)}}. \\ \end{gathered} $$
Hence
$$W\left( p \right) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{d}_{{mi}}}\left( {{l \mathord{\left/ {\vphantom {l \delta }} \right. \kern-0em} \delta }} \right)}}{{\left( {1 + {{{{C}_{e}}} \mathord{\left/ {\vphantom {{{{C}_{e}}} {C_{{ij}}^{E}}}} \right. \kern-0em} {C_{{ij}}^{E}}}} \right)\left( {T_{t}^{2}{{p}^{2}} + 2{{T}_{t}}{{\xi }_{t}}p + 1} \right)}},$$
where
$${{T}_{t}} = \sqrt {{{{{M}_{2}}} \mathord{\left/ {\vphantom {{{{M}_{2}}} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} \right. \kern-0em} {\left( {C_{{ij}}^{E} + C{}_{e}} \right)}}} ;$$
$${{\xi }_{t}} = {{{{k}_{\nu }}} \mathord{\left/ {\vphantom {{{{k}_{\nu }}} {\left( {2\left( {C_{{ij}}^{E} + C{}_{e}} \right)\sqrt {{{M}_{2}}\left( {C_{{ij}}^{E} + C{}_{e}} \right)} } \right)}}} \right. \kern-0em} {\left( {2\left( {C_{{ij}}^{E} + C{}_{e}} \right)\sqrt {{{M}_{2}}\left( {C_{{ij}}^{E} + C{}_{e}} \right)} } \right)}};$$
$${{C}_{{ij}}} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{ij}}^{E}l} \right)}}} \right. \kern-0em} {\left( {s_{{ij}}^{E}l} \right)}} = {1 \mathord{\left/ {\vphantom {1 {\left( {\chi _{{ij}}^{E}l} \right)}}} \right. \kern-0em} {\left( {\chi _{{ij}}^{E}l} \right)}}.$$
For the transfer function of an electroelastic actuator with a transvers piezo effect and voltage control, when \(R = 0\)
$$W(p) = \frac{{{{\Xi }_{2}}\left( p \right)}}{{U\left( p \right)}} = \frac{{{{d}_{{31}}}{h \mathord{\left/ {\vphantom {h \delta }} \right. \kern-0em} \delta }}}{{\left( {1 + {{{{C}_{e}}} \mathord{\left/ {\vphantom {{{{C}_{e}}} {C_{{11}}^{E}}}} \right. \kern-0em} {C_{{11}}^{E}}}} \right)\left( {T_{t}^{2}{{p}^{2}} + 2{{T}_{t}}{{\xi }_{t}}p + 1} \right)}}.$$
(10)
Where
$${{T}_{t}} = \sqrt {{{{{M}_{2}}} \mathord{\left/ {\vphantom {{{{M}_{2}}} {\left( {C_{{11}}^{E} + C{}_{e}} \right)}}} \right. \kern-0em} {\left( {C_{{11}}^{E} + C{}_{e}} \right)}}} ;$$
$${{\xi }_{t}} = {{\alpha {{h}^{2}}C_{{11}}^{E}} \mathord{\left/ {\vphantom {{\alpha {{h}^{2}}C_{{11}}^{E}} {\left( {3{{c}^{E}}\sqrt {M\left( {C_{{11}}^{E} + C{}_{e}} \right)} } \right)}}} \right. \kern-0em} {\left( {3{{c}^{E}}\sqrt {M\left( {C_{{11}}^{E} + C{}_{e}} \right)} } \right)}};$$
$$C_{{11}}^{E} = {{{{S}_{0}}} \mathord{\left/ {\vphantom {{{{S}_{0}}} {\left( {s_{{11}}^{E}h} \right)}}} \right. \kern-0em} {\left( {s_{{11}}^{E}h} \right)}} = {1 \mathord{\left/ {\vphantom {1 {\left( {\chi _{{11}}^{E}h} \right)}}} \right. \kern-0em} {\left( {\chi _{{11}}^{E}h} \right)}}.$$
In Eq. (10), \(\delta \) and \(h\) are the actuator’s thickness and height; and \({{T}_{t}}\) and \({{\xi }_{t}}\) are its time constant and damping coefficient.
From Eq. (10), by Laplace transformation, we determine the transient characteristic of an actuator with transvers piezo effect and voltage control
$$\xi \left( t \right) = {{\xi }_{m}}\left( {1 - \frac{{{{e}^{{ - \frac{{{{\xi }_{t}}t}}{{{{T}_{t}}}}}}}}}{{\sqrt {1 - \xi _{t}^{2}} }}{\text{sin}}\left( {{{\omega }_{t}}t + {{\varphi }_{t}}} \right)} \right).$$
Here
$$\begin{gathered} {{\xi }_{m}} = \frac{{{{d}_{{31}}}\left( {{l \mathord{\left/ {\vphantom {l \delta }} \right. \kern-0em} \delta }} \right){{U}_{m}}}}{{1 + {{{{C}_{e}}} \mathord{\left/ {\vphantom {{{{C}_{e}}} {C_{{11}}^{E}}}} \right. \kern-0em} {C_{{11}}^{E}}}}};\,\,\,\,{{\omega }_{t}} = {{\sqrt {1 - \xi _{t}^{2}} } \mathord{\left/ {\vphantom {{\sqrt {1 - \xi _{t}^{2}} } {{{T}_{t}}}}} \right. \kern-0em} {{{T}_{t}}}}; \\ {{\varphi }_{t}} = {\text{arctan}}\left( {{{\sqrt {1 - \xi _{t}^{2}} } \mathord{\left/ {\vphantom {{\sqrt {1 - \xi _{t}^{2}} } {{{\xi }_{t}}}}} \right. \kern-0em} {{{\xi }_{t}}}}} \right), \\ \end{gathered} $$
where \({{\xi }_{m}}\) is the steady displacement; and \({{U}_{m}}\) is the voltage amplitude.
For a piezo actuator of TsTS ceramic that is rigidly fixed at one end, with a transvers piezo effect and voltage control, in the case of an inertial elastic load, when \({{M}_{1}} \to \infty \) and \(m \ll {{M}_{2}},\) we find that \({{\xi }_{m}} = 160\) nm and \({{T}_{t}} = 0.4 \times {{10}^{{ - 3}}}\) s for a step voltage of amplitude \({{U}_{m}} = 50\) V and the following parameters: \({{d}_{{31}}} = 2 \times {{10}^{{ - 10}}}\) m/V; \({h \mathord{\left/ {\vphantom {h \delta }} \right. \kern-0em} \delta } = 20\); \({{M}_{2}} = 4\) kg; \(C_{{11}}^{E} = 2 \times {{10}^{7}}\) N/m; and \({{C}_{e}} = 0.5 \times {{10}^{7}}\) N/m.
