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\title{Multilayer Electro Magneto Elastic Actuator for Nanomechanics}
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             \author[1]{SM  Afonin}

             \affil[1]{  National Research University of Electronic Technology (MIET)}

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\date{\small \em Received: 13 December 2018 Accepted: 1 January 2019 Published: 15 January 2019}

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\begin{abstract}
        


In this paper, we consider the generalized structural diagram of the multilayer electro magneto elastic actuator or the multilayer piezo actuator for the nanomechanics in contrast to Cady and Mason?s electrical equivalent circuits for the calculation of the piezo transmitter and receiver, the vibration piezo motor. From the solution of the general equation of the electro magneto elasticity, the matrix equation for the equivalent quadripole of the multilayer electro magneto elastic actuator, and its boundary conditions we receive the generalized structuralparametric model, the structural diagram and matrix transfer function of the multilayer actuator.

\end{abstract}


\keywords{multilayer electro magneto elastic actuator, multilayer piezo actuator, equivalent quadripole, structural diagram, structural-parametric model.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
t present, we use the multilayer electro magneto elastic actuator on the piezoelectric, piezomagnetic, electrostriction, and magnetostriction effects for precise alignment in the range of movement from nanometers to tens of micrometers in nanomechanics systems for nanotechnology and adaptive optics. We receive the parametric structural schematic diagram of the multilayer piezo actuator for nanomechanics in contrast to Cady and Mason's electrical equivalent circuits for the calculation of the piezo transmitter, the piezo receiver, and the vibration piezo motor  {\ref [1 -12]}. The parametric structural schematic diagram of the multilayer electro magneto elastic actuator is obtained with the mechanical parameters of displacement and force  {\ref [14 ?20]}.The piezo actuator use for actuation of mechanisms, systems, or management based on the piezo effect, and convert electrical signals into mechanical movement or force. The investigation of the static and dynamic characteristics of the multilayer piezo actuator is necessary for the calculation of nanomechatronics systems. We apply multilayer piezo actuator in nanotechnology for scanning tunneling microscopy and atomic force microscopy  {\ref [6 ? 32]}. 
\section[{II. Parametric Structural Schematic Diagram}]{II. Parametric Structural Schematic Diagram}\par
In this paper, we have the parametric structural schematic diagram and the matrix transfer function of the multilayer electro magneto elastic actuator for the nanomechanics from its the structural-parametric model. In general, the equation for the electro magneto elasticity of the multilayer electro magneto elastic actuator \hyperref[b13]{[12,}\hyperref[b15]{14,}\hyperref[b17]{16,}\hyperref[b33]{31]} has the form ( ) ( )t , x T s t S j ij m mi i ? + ? ? =\textbf{(1)}\par
where( ) x t , x S i ? ? ? =\par
is the relative displacement along axis i of the cross-section of the actuator, therefore, we obtainH , D , E = ?\par
the generalized control parameter in the form m E in Figure \hyperref[fig_0]{1} for the voltage control, m D for the current control, m H for the magnetic field strength control along axis m, j T is the mechanical stress along axis j, mi ? is the coefficient of electro magneto elasticity, for example, mi d piezo module or magnetostriction coefficient, ? ij s is the elastic compliance with const = ? , For example, we consider the matrix equation for the Laplace transforms of the forces and the displacements \hyperref[b17]{[16]} at the input and output ends of k the piezo layer of the multilayer piezo actuator from n the piezo layers. We drew the equivalent T-shaped quadripole of k the piezo layer in Figure \hyperref[fig_1]{2}.  ? are the Laplace transforms of the displacements at input and output ends of k the piezo layer in Figure \hyperref[fig_1]{2}.( ) ( ) ( ) ( ) p Z p Z Z p F k k inp k 1 2 2 1 + ? + ? + ? = (2) ( ) ( ) ( ) ( ) p Z Z p Z p F k k out k 1 2 1 2 + ? + + ? ? = ? where ( ) ? ?? ? = ij s S Z th 0 1 , ( ) ?? ? = ? sh\par
Accordingly, we have for Figure \hyperref[fig_1]{2} the Laplace transforms the following system of the equations for k the piezo layer in the form( ) ( ) ( ) p Z Z Z p F Z Z p F k out k inp k 1 2 1 1 2 1 2 1 + ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? + = ? (3) ( ) ( ) ( ) p Z Z p F Z p k out k k 1 2 1 1 1 1 + ? ? ? ? ? ? ? ? ? + + = ?\par
the matrix equation for k the piezo layer ( ) ( ) [ ] ( ) ( ) ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? + p p F M p[ ] ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? + + = ? ? ? ? ? ? = 2 1 2 1 1 1 2 1 22 21 12 11 1 1 2 1 Z Z Z Z Z Z Z Z m m m m M (5)\par
Where ( )?? = + = = ch 1 2 1 22 11 Z Z m m , ( ) ?? = ? ? ? ? ? ? ? ? + = sh 2 0 1 1 1 12 Z Z Z Z m , (\textbf{)}0 2 21 sh 1 Z Z m ?? = = , ? ? = ij s S Z 0 0\par
For the multilayer piezo actuator the Laplace transforms the displacement The force on the output face for the k the piezo layer equal in magnitude and opposite in direction to the force on the input face for k+1 the piezo layer( ) ( ) p F p F inp k out k 1 + ? =\textbf{(6)}\par
From equation (  {\ref 3}) the matrix equation for n the piezo layers( ) ( ) [ ] ( ) ( ) ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? + p p F M p p F n out n n inp 1 1 1 (7)\par
with the matrix of the multilayer piezo actuator Figure \hyperref[fig_0]{1} a in the form [ ] ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? = n Z n n Z n M n ch sh sh ch 0 0\par
Accordingly, in general, the matrix for the equivalent quadripole of the multilayer electro magneto elastic actuator Figure \hyperref[fig_0]{1} a-? has the following form[ ] ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? = l Z l l Z l M n ch sh sh ch 0 0\par
Therefore, we have from the equation (  {\ref 7}) the equivalent quadripole of the multilayer piezo actuator on Figure \hyperref[fig_0]{1} a-? for the longitudinal piezo effect with length of the multilayer piezo actuator ? = n l , for the transverse piezo effect with nh l = , for the shift piezo effect with nb l = , where b , h , ? are the thickness, the height, the width for k the piezo layer.\par
Equations of the forces acting on the faces of the multilayer piezo actuator    at 0 = x , ( ) ( ) ( ) p p M p F S p , T j 1 2 1 1 0 0 ? + = (8) at l x = , ( ) ( ) ( ) p p M p F S s , l T jd , d , d g , g , g d , d , d v mi , ? ? ? ? ? = ? 1 3 1 3 1 3 H , H D , D E , E m , ? ? ? ? ? = ? H H H D D D E E E ij s ,? ? ? ? ? = ? H D E c c c c , ? ? ? ? ? ? ? ? = ? H D E , ? ? ? ? ? ? = nb nh n l . ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? ? p p l l p p F p M p m mi ij 2 1 1 2 1 1 ch sh 1 1 (10) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? ? p p l l p p F p M p m mi ij 1 2 
\section[{S s E E = ?}]{S s E E = ?}\par
The structural-parametric model of the multilayer piezo actuator for the transverse piezo effect has the form For the shift piezo effect the Laplace transform of the caused force The structural-parametric model of the multilayer piezo actuator for the shift piezo effect has the form We drew the structural schematic diagram of the actuator from the generalized structural-parametric model of the multilayer electro magneto elastic actuator for the nanomechanics.( ) ( ) E s p E S d p F 55 1 0 15 = (\textbf{15}) 0 55 55 S s E E = ? ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 2 1 3 33 33 1 2 1 1 ch sh 1 1 (12) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 1 2 3 33 33 2 2 2 2 ch sh 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 2 1 3 31 11 1 2 1 1 ch sh 1 1 (14) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 1 2 
\section[{III.}]{III.} 
\section[{Matrix Transfer Function}]{Matrix Transfer Function}\par
From equation \hyperref[b11]{(10)} we receive the matrix transfer function of the multilayer electro magneto elastic actuator with n the layers in the following form Therefore, in general, we have the matrix transfer function of the multilayer electro magneto elastic actuator in the form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? p F p F p p W p W p W p W p W p( ) [ ] ( ) [ ] ( ) [ ] p P p W p = ? (18) ( ) [ ] ( ) ( ) ? ? ? ? ? ? ? ? = ? p p p 2 1 ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ? ? ? ? ? ? = p W p W p W p W p W p W p W( ) ( ) ( ) ( ) [ ] ij ij mi m A l p M p p p W 2 th 2 2 1 11 ? ? + ? ? = ? ? = ? 0 S s ij ij ? ? = ? ( ) ( ) ( ) [ ] \{ \} ( ) ( ) ( ) [ ] 2 2 2 2 1 3 2 1 4 2 2 1 2 1 th th ? + ? + + ? ? ? + + ? ? + + ? = ? ? ? ? ? ? c p p c l M M p l c M M p M M A ij ij ij ij ( ) ( ) ( ) ( ) [ ] ij ij ij m A l p M p p p W 2 th 2 1 2 21 ? ? + ? ? = ? ? = ? ( ) ( ) ( ) ( ) [ ] ij ij ij A l p M p F p p W ? ? + ? ? ? = ? = ? ? th 2 2 1 1 12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ij ij A l p F p p W p F p p W ? ? ? = ? = = ? = ? sh 1 2 22 2 1 13 ( ) ( ) ( ) ( ) [ ] ij ij ij A l p M p F p p W ? ? + ? ? ? = ? = ? ? th 2 1 2 2 23\par
We receive the generalized parametric structural schematic diagram and the generalized matrix transfer function from the generalized structural-parametric model of the multilayer electro magneto elastic actuator to calculate its static and dynamic characteristics for the nanomechanics. Let us consider, for example, the voltage-controlled multilayer piezo actuator for the longitudinal piezo effect with the inertial load1 M m << , 2 M m << and ( ) ( ) 0 2 1 = = t F t F\par
and the static displacements of its faces in the following form( ) ( ) ( )( ) p U p pW t m p t ? = ? = ? ? ? ? ? ? ? 11 0 0 1 1 lim lim thus ( ) ( ) 2 1 2 33 1 M M M nU d m + = ? ? and ( ) ( ) ( )( ) p U p pW t m p t ? = ? = ? ? ? ? ? ? ? 21 0 0 2 2 lim lim thus ( ) ( ) 2 1 1 33 2 M M M nU d m + = ? ?\par
where m U is the amplitude of the voltage, m is the mass of the multilayer piezo actuator, = 480 nm. We derive the transfer function with concentrated parameters of the multilayer piezo actuator for the longitudinal piezo effect with the voltage control for the elastic-inertial load and one fixed face and its structural diagram in Figure  {\ref 4}.( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 2 1 1 15 55 1 2 1 1 ch sh 1 1 (16) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = ? p p l l p E d p F p M p E 1 2 1 15 55 2 2 2 2 ch sh 1 1 ( ) ( ) ( ) (\par
) ( )   The discrepancy between the experimental data and calculation results is no more than 5\%.1\par
IV. 
\section[{Results and Discussions}]{Results and Discussions}\par
We have the equations for the generalized structural-parametric model and the generalized structural diagram of the multilayer electro magneto elastic actuator. We receive the matrix transfer functions and the structural diagram of the multilayer electro magneto elastic actuator from the set of equations describing the structural parametric model of the multilayer actuator for the nanomechanics. The solution of the matrix equation for the equivalent quadripole of the multilayer electro magneto elastic actuator with the Laplace transform is used for the construction the structural diagram of the multilayer actuator.\par
As a result of the joint solution of the general equation of electro magneto elasticity, the matrix equation for the equivalent quadripole of the multilayer electro magneto elastic actuator, and its boundary conditions we receive the generalized structuralparametric model and the generalized structural diagram of the multilayer actuator. 
\section[{V. ONCLUSION}]{V. ONCLUSION}\par
We consider the generalized structuralparametric model, the structural diagram and the matrix transfer function of the multilayer electro magneto elastic actuator for the nanomechanics. We receive the structural diagram of the multilayer piezo actuator for the transverse, longitudinal, and shift piezo effects.\par
From the general equation of electro magneto elasticity, the force that causes the deformation, the system of the equations for the equivalent quadripole of the multilayer actuator, and the forces on its faces, we obtain generalized structural-parametric model and generalized structural diagram of the multilayer electro magneto elastic actuator for the nanomechanics.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
        			\footnote{Multilayer Electro Magneto Elastic Actuator for Nanomechanics} 			\footnote{© 2019 Global Journals} 			\footnote{Year 2019 © 2019 Global JournalsC} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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