The structural model of the nano piezoengine is determined for applied biomechanics and biosciences. The structural scheme of the nano piezoengine is obtained. For calculation nano systems the structural model and scheme of the nano piezoengine are used, which reflect the conversion of electrical energy into mechanical energy of the control object. The matrix equation is constructed for the nano piezoengine in applied biomechanics and biosciences.

Keywords: nano piezoengine, structural model, applied biomechanics and biosciences

The nano piezoengine based on the inverse piezoeffect is used for applied biomechanics and biosciences, nanomedicine, nanobiology, microsurgery. The nano piezoengine is provided for applied biomechanics and biosciences in scanning probe microscopy, interferometers and adaptive optics, actively dampen vibrations, deform mirrors and the work with the genes.^1–6

For calculation nano systems the structural model and scheme of the nano piezoengine are used, which reflect the conversion of electrical energy into mechanical energy of the control object.^6–19

Structural model

For calculation the nano piezoengine on Figure 1 is determined the inverse piezoeffect.^1–49

$S_i= d_{ mi} E_m+s_{ ij}^{ E} T_j$

here $d_{ mi},s_{ ij}^{ E},E_m,T_j,S_i$  are piezomodule, elastic compliance, strength electric field, strength mechanical field, relative deformation.

Figure 1 Nano piezoengine.

For the nano piezoengine the differential equation is evaluated^4–56

$\frac{d^2 \Xi \left(x,s\right)}{d{x}^2} - {\gamma }^2 \Xi \left(x,s\right)=0$

here $\Xi \left(x,s\right),x,s,\gamma$  are the transform of deformation, the coordinate, the parameter of transform, the factor of propagation.

For the longitudinal piezoengine at $x=0$  the deformation $\Xi \left(0,s\right)={\Xi }_1\left(s\right)$ , at $x=\delta ,\Xi \left(\delta ,s\right)={\Xi }_2\left(s\right)$  are calculated.

Its solution is written^4–36

$\Xi \left(x,s\right)=\left\{ {\Xi }_1 \left(s\right) \text{sh} \left[ \left(\delta -x\right) \gamma \right]+{\Xi }_2\left(s\right)\text{sh}\left(x\gamma \right)\right\}/\text{sh}\left(\delta \gamma \right)$

For the nano longitudinal piezoengine in Figure 1 its relative displacement on 3 axes^1–29 has the form

$S_3= d_{33} E_3+s{ }_{33 }^{ E} T_3$

The system for the nano longitudinal piezoengine is obtained^11–31 for $x=0,x=\delta$

$T_3\left(0,s\right)=\frac{1}{s{ }_{33}^{ E}} {\frac{d \Xi \left(x,s\right)}{dx} |}_{x=0}-\frac{d_{ 33}}{s{ }_{33}^{ E}}{E}_3\left(s\right)$

$T_3 \left(\delta ,s\right) = \frac{1}{s{ }_{33 }^{ E}} {\frac{d \Xi \left(x,s\right)}{dx} |}_{x=\delta }- \frac{d_{33}}{s{ }_{33 }^{ E}} E_3 \left(s\right)$

The structural model is evaluated for applied biomechanics and biosciences

$\begin{array}{l}{\Xi }_1\left(s\right)={\left(M_1{s}^2\right)}^{-1}\left\{\begin{array}{l} -F_1\left(s\right)+{\left( {\chi }_{33}^E \right)}^{ -1}\\ \times \left[\begin{array}{l} d_{33} E_3 \left(s\right)-\left[\gamma / \text{sh} \left( \delta \gamma \right) \right] \\ \times \left[\text{ch}\left( \delta \gamma \right){\Xi }_1\left(s\right)-{\Xi }_2\left(s\right)\right]\end{array}\right] \end{array}\right\}\\ {\Xi }_2 \left(s\right) = {\left(M_2{s}^2\right)}^{-1} \left\{\begin{array}{l} - F_2 \left(s\right) + \left(\chi { }_{33}^{ E} \right){ }^{-1}\\ \times \left[\begin{array}{l} d_{33} E_3 \left(s\right) - \left[ \gamma / \text{sh} \left( \delta \gamma \right) \right]\\ \times \left[ \text{ch} \left( \delta \gamma \right) {\Xi }_2 \left( s \right) - {\Xi }_1 \left(s\right) \right] \end{array}\right] \end{array}\right\}\\ \chi { }_{33}^{ E} = s{ }_{33}^{ E} / S_0\end{array}$

here ${\Xi }_1\left(s\right),{\Xi }_2\left(s\right)$  - the transformations of displacements, $S_0$  - the area.

For the nano transverse piezoengine the expression of the transverse inverse piezoeffect^1–29

$S_1= d_{31} E_3+ s{ }_{11}^E T_1$

The system for the nano transverse piezoengine is determined for $x=0$  and $x=h$

$\begin{array}{l}{T}_1 \left(0,s\right)= \frac{1}{s{ }_{11}^E} {\frac{d \Xi \left(x,s\right)}{dx} |}_{x=0}- \frac{d_{ 31}}{s_{11}^{ E}} E_3 \left(s\right)\\ T_1 \left(h,s\right) = \frac{1}{s_{11}^E}{ \frac{d \Xi \left(x,s\right)}{dx} |}_{x=h}- \frac{d_{31}}{s_{11}^E} E_3 \left(s\right)\end{array}$

The structural model of the nano transverse piezoengine is calculated

$\begin{array}{l}{\Xi }_1 \left(s\right) = {\left( M_1{s}^2 \right)}^{-1} \left\{\begin{array}{l} - F_1 \left(s\right)+ {\left( \chi { }_{11}^{ E} \right)}^{-1}\\ \times \left[\begin{array}{l} d_{31} E_3\left(s\right) - \left[\gamma / \text{sh} \left( h\gamma \right) \right]\\ \times \left[\text{ch}\left(h\gamma \right) {\Xi }_1\left(s\right)- {\Xi }_2\left( s \right)\right]\end{array}\right] \end{array}\right\}\\ {\Xi }_{ 2} \left(s\right) = {\left( M_2 s^2 \right)}^{-1}\left\{\begin{array}{l} - F_2 \left(s\right) + \left( \chi { }_{11}^{ E} \right){ }^{-1}\\ \times \left[\begin{array}{l} d_{31} E_3\left(s\right)- \left[ \gamma / \text{sh} \left( h\gamma \right) \right]\\ \times \left[\text{ch}\left( h\gamma \right) {\Xi }_{ 2} \left( s \right)-\Xi { }_1 \left( s \right) \right] \end{array}\right] \end{array}\right\}\\ \chi { }_{11}^E = s{ }_{11 }^E/ S_0\end{array}$

For the nano shift piezoengine the expression of the shift inverse piezo effect^1–29

$S_5=d_{15} E_1+s{ }_{55}^{ E} T_5$

The system for the shift piezoengine is written at $x=0$  and $x=b$

$\begin{array}{l}{T}_5 \left(0, s\right)= \frac{1}{s{ }_{55}^{ E}} {\frac{d \Xi \left(x,s\right)}{dx} |}_{ x=0}- \frac{d_{ 15}}{s{ }_{55}^{ E}} E_1 \left(s\right)\\ T_5 \left(b,s\right)=\frac{1}{s{ }_{55}^{ E}}{\frac{d \Xi \left( x,s \right)}{ dx} |}_{ x=b} - \frac{d_{ 15}}{s{ }_{55}^E} E_1 \left(s\right)\end{array}$

The structural model is calculated

$\begin{array}{l}{\Xi }_1\left( s \right)= {\left( M_1 s^2 \right)}^{-1} \left\{\begin{array}{l} - F_{1 }\left( s \right)+ \left( \chi { }_{55}^{ E} \right){ }^{-1}\\ \times \left[\begin{array}{l}d{ }_{15} E{ }_1\left( s \right)- \left[ \gamma / \text{sh} \left( b \gamma \right) \right] \\ \times \left[\text{ch}\left(b\gamma \right){\Xi }_1\left( s \right)- \Xi { }_{2 }\left( s \right) \right]\end{array}\right] \end{array}\right\}\\ \Xi { }_{2 }\left( s \right)= \left( M_2{s}^2 \right){ }^{-1} \left\{\begin{array}{l} -F_2 \left( s \right)+ \left( \chi { }_{55}^E \right){ }^{-1}\\ \times \left[\begin{array}{l} d{ }_{15} E{ }_1 \left( s \right)- \left[ \gamma / \text{sh} \left( b\gamma \right) \right]\\ \times \left[\text{ch}\left( b\gamma \right){\Xi }_2\left(s\right)- \Xi { }_1 \left( s \right) \right] \end{array}\right] \end{array}\right\}\\ \chi { }_{55}^{ E} = s{ }_{55}^{ E} / S_{ 0}\end{array}$

At $x=0$  and $x=l$  for $l=\{\delta ,h,b$  the system in general is obtained

$\begin{array}{l}{T}_j\left(0,s\right)= \frac{1}{s_{ ij }^{\Psi }} {\frac{d \Xi \left(x,s\right)}{dx} |}_{x=0}- \frac{{\nu }_{ mi}}{s_{ ij }^{\Psi }} {\Psi }_m \left(s\right)\\ T_j\left(l,s\right)= \frac{1}{s_{ ij }^{\Psi }} {\frac{d \Xi \left(x,s\right)}{ dx } |}_{x=l}- \frac{{\nu }_{ mi}}{s_{ ij }^{\Psi }} {\Psi }_{m }\left(s\right)\end{array}$

The structural model and scheme of the nano piezoengine on Figure 2 are evaluated

${\Xi }_1\left(s\right)={\left(M_1{s}^2\right)}^{-1}\left\{\begin{array}{l}-F_{ 1 } \left(s\right)+\left(\chi { }_{ ij }^{ \Psi } \right){ }^{-1}\\ \times \left[\begin{array}{l}{\nu }_{ mi } {\Psi }_{ m }\left(s\right)-\left[ \gamma / \text{sh} \left(l\gamma \right) \right]\\ \times \left[\text{ch}\left(l\gamma \right){\Xi }_1\left(s\right)-{\Xi }_2\left(s\right)\right]\end{array}\right] \end{array}\right\}$

${\Xi }_{ 2}\left(s\right)={\left(M_2 s^2\right)}^{-1}\left\{\begin{array}{l} -F_{ 2}\left(s\right)+\left(\chi { }_{ ij }^{\Psi } \right){ }^{-1}\\ \times \left[\begin{array}{l}{\nu }_{mi}{\Psi }_m\left(s\right)-\left[ \gamma / \text{sh} \left( l\gamma \right) \right]\\ \times \left[ \text{ch} \left(l\gamma \right){\Xi }_{ 2 }\left(s\right)-{\Xi }_{ 1 }\left(s\right)\right]\end{array}\right]\end{array}\right\}$

$\chi { }_{ij}^{\Psi } = s{ }_{ij}^{\Psi } / S_0$

$v{ }_{mi} = \{\begin{array}{c} d_{ 33 }, d_{ 31} , d_{ 15} \\ g_{ 33} , g_{ 31 }, g_{ 15 }\end{array}$

$\Psi { }_{m }= \{\begin{array}{c} E_{ 3 }, E_{ 3 }, E_{ 1 }\\ D_{ 3 }, D_{ 3}, D_{ 1} \end{array}$

$s{ }_{ ij }^{\Psi } = \{\begin{array}{c} s_{ 33 }^{ E }, s{ }_{ 11 }^{ E }, s{ }_{ 55 }^{ E }\\ s_{ 33 }^{ D },s{ }_{ 11 }^{ D }, s{ }_{ 55 }^{ D }\end{array}$

$\gamma =\{{\gamma }^E,{\gamma }^D$

$c^{ \Psi }=\{c^{ E},c{ }^D$

Figure 2 In general scheme of nano piezoengine.

The displacements matrix is calculated

$\left(\begin{array}{c}{\Xi }_1\left(s\right)\\ {\Xi }_2\left(s\right)\end{array}\right)=\left(\begin{array}{c}\begin{array}{ccc}{W}_{ 11 }\left(s\right)& W_{ 12 }\left(s\right)& W_{ 13 }\left(s\right)\end{array}\\ \begin{array}{ccc}{W}_{ 21 }\left(s\right)& W_{ 22 }\left(s\right)& W_{ 23 }\left(s\right)\end{array}\end{array}\right)\left(\begin{array}{c}{\Psi }_m\left(s\right)\\ F_1\left(s\right)\\ F_2\left(s\right)\end{array}\right)$

$W_{11}\left(s\right)={\Xi }_1\left(s\right)/{\Psi }_m\left(s\right)={{\nu }_{ mi} \left[M_2\chi { }_{ ij }^{\Psi } s^2+\gamma \text{th} \left(l\gamma /2\right)\right]/A}_{ ij }$

$A_{ ij }=M_1{M}_2{\left(\chi { }_{ ij }^{\Psi } \right)}^2{s}^4+\left\{\left(M_1+M_2\right){\chi }_{ ij }^{\Psi }/\left[c^{\Psi } \text{th} \left(l\gamma \right)\right]\right\} s^3+\left[\left(M_1+M_2\right)\chi { }_{ ij }^{\Psi } \alpha / \text{th} \left(l\gamma \right)+1/{\left(c^{\Psi }\right)}^2\right]s^2+2\alpha s/c^{\Psi }+{\alpha }^2$

$W_{ 21 }\left(s\right)={\Xi }_2\left(s\right)/{\Psi }_m\left(s\right)={{\nu }_{ mi }\left[M_1\chi { }_{ ij }^{\Psi }{s}^2+\gamma \text{th} \left(l\gamma /2\right)\right]/A}_{ ij }$

$W_{ 12 }\left(s\right)={\Xi }_1\left(s\right)/F_{ 1}\left(s\right)=-{\chi { }_{ ij }^{\Psi } \left[M_{ 2} \chi { }_{ ij }^{\Psi } s^2+\gamma / \text{th} \left(l\gamma \right)\right]/ A}_{ ij }$

$W_{ 13 }\left(s\right)={\Xi }_1\left(s\right)/F_{ 2}\left(s\right)=W_{ 22 }\left(s\right)={\Xi }_2\left(s\right)/F_{ 1}\left(s\right)=\left[\chi { }_{ ij }^{ \Psi }\gamma / \text{sh} \left(l\gamma \right)\right]/ A{ }_{ ij }$

$W_{ 23 }\left(s\right)={\Xi }_2\left(s\right)/F_{ 2 }\left(s\right)=-{\chi { }_{ ij }^{\Psi }\left[M_{1 } \chi { }_{ ij }^{\Psi } s^2+\gamma / \text{th} \left(l\gamma \right)\right]/ A}_{ ij }$

The static longitudinal displacements are evaluated

$\begin{array}{l}{\xi }_1=d_{ 33 } U M_{2 } / \left(M_1+M_2\right)\\ {\xi }_2=d_{ 33 } U M_{ 1 } / \left(M_1+M_2\right)\end{array}$

For $d_{ 33 }$  = $4\cdot {10}^{-10}m/V$ , $U$  = 25 V, $M_1$  = 1 kg, $M_2$  = 4 kg the static displacements ${\xi }_1$  = 8 nm, $${\xi }_2$$  = 2 nm and ${\xi }_1+{\xi }_2$  = 10 nm are evaluated at error 10%.

The equation of the direct piezo effect is used^1–29

$\begin{array}{l}D{ }_m= d_{mi} T_i + \epsilon { }_{mk }^{ E}{E}_{ k}\\ k_{ d}= \frac{d_{ mi }{S}_0}{\delta s{ }_{ij}^E}\end{array}$

here $\epsilon { }_{mk}^{ E},D_m,k_d$  - the permittivity, the electric induction and the direct coefficient. The transform the voltage of feedback for the nano piezoengine on Figure 3 is calculated

$U_d\left(s\right)=\frac{d{ }_{mi }{S}_0 R}{\delta s{ }_{ij}^E} {\stackrel{•}{\Xi }}_{{}_n} \left(s\right)= k_d R {\stackrel{•}{\Xi }}_{{}_n} \left(s\right),n=1,2$

Figure 3 Scheme of nano piezoengine with back electromotive force.

For the nano piezoengine its static deformation is obtained.

For voltage control

$\begin{array}{l}{T}_{j\text{max}}= E_m d_{mi }/ s{ }_{ij}^E\\ F_{\text{max}}= E_{m } d_{ mi} S_{ 0 }/ s{ }_{ij}^E\end{array}$

For current control

$\begin{array}{l}{F}_{\text{max}}=\frac{U}{\delta } d{ }_{mi} \frac{S_{ 0}}{s{ }_{ij}^E}+ \frac{F{ }_{\text{max}}}{S_0} d_{ mi }S{ }_c \frac{1}{\epsilon { }_{mk}^{ T} S_{ c}/ \delta } \frac{1}{\delta } d_{ mi} \frac{S_{ 0}}{s{ }_{ij}^E}\\ \frac{F_{\text{max}}}{S_0} \left( 1- \frac{d{ }_{mi}^{ 2}}{\epsilon { }_{mk}^{ T} s{ }_{ij}^E} \right)s{ }_{ij}^E=E{ }_m d_{ mi }\\ T_{j\text{max} }\left( 1-k{ }_{mi}^{ 2} \right) s{ }_{ij}^E=E_{ m} d{ }_{mi}\\ k_{ mi} =d_{ mi }/\sqrt{ s{ }_{ij }^E \epsilon { }_{mk }^{ T}}\end{array}$

here $S_c,C_0,k_{ mi }$  - the sectional area of capacitor, the capacitance, and the coefficient of electromechanical coupling.

For current control of the nano piezoengine

$\begin{array}{l}{T}_{j \text{max}}=E{ }_m d_{ mi }/ s{ }_{ij}^D\\ F_{\text{max}}=E{ }_m d{ }_{mi} S_{ 0} / s{ }_{ij}^D\\ s{ }_{ ij }^{ D} = \left(1- k{ }_{ mi }^{ 2 }\right) s_{ ij }^{ E }\end{array}$

The mechanical characteristic of the nano piezoengine is obtained

$\begin{array}{l}\Delta l=\Delta l_{\text{max}}\left(1-F/F_{\text{max}}\right)\\ \Delta l_{ \text{max}}= \nu { }_{mi} {\Psi }_{ m }l\\ F_{\text{max}}=T_{j\text{ max}}{S}_0= \nu { }_{mi }\Psi { }_m S_{0 }/ s{ }_{ij }^{\Psi }\end{array}$

The expression of the mechanical characteristic of the nano transverse piezoengine is calculated

$\begin{array}{l}\Delta h=\Delta h_{\text{max}}\left(1-F/F_{\text{max}}\right)\\ \Delta h_{\text{max}}=d_{ 31 } E_{3}h\\ F_{ \text{max}}=d{ }_{3 1} E{ }_{3 }{S}_{ 0 }/ s{ }_{1 1}^{ E}\end{array}$

At $d_{ 3 1}$  = $2\cdot {10}^{-10}m/V$ , $E_3$  = 0.5∙10⁵ V/m, h = 2.5∙10^-2 m, $S_0$  = 1.5∙10^-5 m², $s{ }_{1 1}^E$  = 15∙10^-12 m²/N the parameters $\Delta h_{\text{max}}$  = 250 nm, $F_{\text{max}}$  = 10 N are obtained on Figure 4 at error 10%.

Figure 4 Mechanical characteristic of nano transverse piezoengine.

The deformation piezoengine at elastic load is obtained

$\begin{array}{l}\frac{\Delta l}{l} = \nu { }_{mi } {\Psi }_m- \frac{s{ }_{ij}^{\Psi } C_e}{S_0} \Delta l\\ F=C_e \Delta l\end{array}$

The control characteristic of the nano piezoengine is determined

$\begin{array}{l}\Delta l= \frac{{\nu }_{ mi } l {\Psi }_m}{1+ C{ }_{e }/ C_{ ij}^{ \Psi }}\\ s{ }_{ij } = k_{s }{s}_{ ij }^{ E},\left( 1- k{ }_{mi}^{ 2} \right)\le k_s\le 1\end{array}$

here $k_s$  the coefficients change of elastic compliance.

For the nano piezoengine its reverse and direct coefficients are calculated

$k_r=k_d= \frac{d_{ mi} S_0}{\delta s_{ ij }}$

By using the equation of load the scheme of the nano piezoengine with one fixed face on Figure 5 is calculated.

Figure 5 Scheme of nano piezoengine with one fixed face.

The expression on voltage for Figure 4 is calculated

$\begin{array}{l}W\left(s\right)={\Xi }_2\left(s\right)/U\left(s\right)=k_r/N\left(s\right)\\ N\left(s\right)=a_0 s^3+a_1 s^2+a_2 s+a_3\\ a_0=R C_0 M_2,a_1 =M_2+ R C_{0 } k_v\\ a_2=k_v+R C_0 C{}_{ij } + R C_{0 }C{}_e+ R k_{r }{k}_d,a_{ 3}= C{}_e + C{}_{ij }\end{array}$

here $k_v$  - the coefficient of damping.

For the nano transverse piezoengine for $R=0$  the expression on voltage is determined

$\begin{array}{l}W \left(s\right)= \frac{\Xi \left(s\right)}{U \left(s\right)}=\frac{k{ }_{31 }^{ U}}{T_t^2 s^2+2T_t {\xi }_t s+1}\\ k{ }_{31}^{ U} =d_{ 31 } \left(h/\delta \right)/\left( 1+C{ }_l/C{ }_{11}^{ E}\right)\\ T_t= \sqrt{M /\left( C{ }_l + C{ }_{1 1 }^{ E}\right)},{\omega }_t=1/ T_t\end{array}$

For $M$  = 2 kg, $C_l$  = $0.2\cdot {10}^{7}N/m$ , $C{ }_{11}^{ E}$  =$1.6\cdot {10}^{7}N/m$  the parameters $T_t$  = 0.33×10^-3 s, ${\omega }_t$  =$3 \cdot 10{ }^3 s{ }^{-1}$ are evaluated on Figure 6 at error 10%.

Figure 6 Bandwidth of nano transverse piezoengine.

The static displacement

$\Delta h=\frac{d_{ 31 }\left(h/\delta \right)U}{1+C_l /C{ }_{11}^{ E}}=k{ }_{31}^{ U} U$

For $d_{ 31}$  = $2\cdot {10}^{-10}m/V$ , $h/\delta$  = 24, $C_l /C{ }_{11}^{ E}$  = 0.1 the parameter $k{ }_{31}^{ U}$  = 4.4 nm/V is evaluated at error 10%.

For calculation nano systems the structural model and scheme of the nano piezoengine are used, which reflect the conversion of electrical energy into mechanical energy. The structural model and schemes of the nano piezoengine are obtained for applied biomechanics and biosciences. The matrix of the deformations of the nano piezoengine is constructed. The parameters of the nano piezoengine are determined for applied biomechanics and biosciences.

None.

None.

The authors declare that they have no conflict of interest.

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