Electromagnetelastic Actuator for Nanomechanics

Global Journal of Researches in Engineering: A Mechanical and Mechanics Engineering Volume 18 Issue 2 Version 1.0 Year 2018 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN:2249-4596 Print ISSN:0975-5861

Electromagnetelastic Actuator for Nanomechanics By Afonin SM National Research University of Electronic Technology Abstract- From the structural-parametric model of the electromagnetoelastic actuator we obtain the parametric structural schematic diagram and the matrix transfer function, the characteristics of the electromagnetoelastic actuator for the nanomechanics. The generalized parametric structural schematic diagram, the matrix transfer function of the electromagnetoelastic actuator is described with using its physical parameters and external load. Keywords: electromagnetoelastic actuator; structural-parametric model; nanodisplacement; piezoactuator; parametric structural schematic diagram; matrix transfer function. GJRE-A Classification: FOR Code: 100799

ElectromagnetelasticActuatorforNanomechanics

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Electromagnetelastic Actuator for Nanomechanics

Afonin SM

Abstract- From the structural-parametric model of the model of the electroelastic actuator was determined in electromagnetoelastic actuator we obtain the parametric contrast electrical equivalent circuit for calculation of structural schematic diagram and the matrix transfer function, piezoelectric transmitter and receiver [9 −12]. In [8, 27] the characteristics of the electromagnetoelastic actuator for was used the transfer functions of the piezoactuator for the nanomechanics. The generalized parametric structural the decision problem absolute stability conditions for a schematic diagram, the matrix transfer function of the 201 electromagnetoelastic actuator is described with using its system controlling the deformation of the electromagnetoelastic actuator. The elastic compliances physical parameters and external load. Year Keywords: electromagnetoelastic actuator; structural- and the mechanical and adjusting characteristics of the

piezoactuator were found in [18, 21 − 23, 28, 29] for 19 parametric model; nanodisplacement; piezoactuator; parametric structural schematic diagram; matrix transfer calculation its transfer functions and the structural- function. parametric models. The structural-parametric model of the multilayer and compound piezoactuator was I. Introduction determined in [18−20]. In this paper is solving the problem of building the generalized structural he electromagnetoelastic actuator for piezoelectric, parametric model and the generalized parametric piezomagnetic, electrostriction, magnetostriction structural schematic diagram of the electro- effects is used for the precise adjustment in the T magnetoelastic actuator for using the equation of nanomechanics, the nanotechnology, the adaptive electromagnetoelasticity. optics [1−32]. The piezoactuator on the inverse piezoeffect is serves for the actuation of mechanisms or II. Structural-Parametric Model I II Version XVIII Issue the management, converts the electrical signals into the displacement and the force [1−8]. The piezoactuator The general structural-parametric model and olume for the nanomechanics is provided the displacement the parametric structural schematic diagram of the V )

electromagnetoelastic actuator are obtained. In the

from nanometers to tens of micrometers, a force to 1000 N. The piezoactuator is used in the nanomechanics and electroelastic actuator are presented six stress ( the nanotechnology for the scanning tunneling components T1 , T2 , T3 , T4 , T5 , T6 , the components T1 microscopes, the scanning force microscopes and the − T3 are related to extension-compression stresses, T4 atomic force microscopes [14−32]. − T6 to shear stresses. For the electroelastic actuator In the present paper the generalized structural- its deformation corresponds to stressed state. In parametric model and the generalized parametric piezoceramics PZT the matrix state equations [12, 14] structural schematic diagram of the electro- connected the electric and elastic variables have the magnetoelastic actuator are constructed by solving the form two equations, then the first equation describes the equation of the electromagnetolasticity, the wave direct piezoelectric effect, the second - the inverse equation with the Laplace transform, the boundary Researches in Engineering piezoelectric effect conditions on loaded working surfaces of the actuator, the strains along the coordinate axes. The transfer D = dT + εT E (1) functions and the parametric structural schematic = E TsS + dt E (2) diagrams of the piezoactuator are obtained from the where is the column matrix of electric generalized structural-parametric model. In [6, 7] was D obal Journal of obal Journal

induction; is the column matrix of relative l determined the solution of the wave equation of the S G piezoactuator. In the [14−16, 30] were obtained the deformations; T is the column matrix of mechanical structural-parametric models, the schematic diagrams stresses; E is the column matrix of electric field for simple piezoactuator and this models were strength; s E is the elastic compliance matrix for transformed to the structural-parametric model E = const ; εT is the matrix of dielectric constants for of the electromagnetoelastic actuator. The structural- T = const ; dt is the transposed matrix of the

Author: National Research University of Electronic Technology (Moscow piezoelectric modules. Institute of Electronic Technology, MIET), Moscow, Russia. The piezoactuator (piezoplate) has the following e-mail: [email protected] properties: δ is the thickness, h is the height, b is the

width, respectively l {δ= h,, b the length of the   (pd )−Ψ      mi m   1   1   piezoactuator for the longitudinal, transverse and shift 1(p) =Ξ  1(pF )+− Ψ  γ    2   χ    [ch(l )Ξγ ()()p − Ξ p ] piezoeffects. The direction of the polarization axis Р, i.e.,  1 pM   ij   1 2    sh(lγ)  the direction along which polarization was performed, is usually taken as the direction of axis 3. The equation of (6) the inverse piezoeffect for controlling voltage [6, 12] has  −Ψ  the form  mi m (pd )   1   1   (p) =Ξ   (pF )+−     Ψ 2  2  2 Ψ  γ  i = mi Ψm ( )+ ij j ()xTstdS ,t  χ    [ch(l )Ξγ ()()p − Ξ p ] (3)  2 pM   ij   2 1    sh(lγ)  ∂= ξ( ,xS t) ∂x , ( ) =Ψ ()() = tUtEt δ i m m where is the relative displacement of the Si Ψ  ,, ddd 153133  ,, EEE 133 Ψ sij   cross section of the piezoactuator along axis i, Ψm (t) is Where χij = dmi =  , , ggg 153133 , Ψm =  , , DDD 133 , S0   201 the control parameter along axis m, dmi is the coefficient  33, ,ddd 1531  3 , 3 , HHH 1 of the electromagnetolasticity (for example the  EEE s s ,, s551133 Year  Ψ piezomodule), Em (t) is the electric field strength along Ψ DDD E D H sij = s , ,ss 551133 , l {δ= h,, b , = {cc c ,, c ,

20 axis m, U (t) is the voltage between the electrodes of sH ,sH ,sH  33 11 55 Ψ actuator, sij is the elastic compliance for Ψ = const , Ψ E D H {γ=γ ,, γγ , dmi is the coefficient of the () is the mechanical stress along axis j and i, j = 1, j xT ,t electromagnetolasticity (for example the piezomodule or 2, … , 6; m = 1, 2, 3. The main siz e l {δ= h,, b for the the coefficient of the magnetostriction). piezoactuator is respectively, the thickness, the height, the width for the longitudinal, transverse, shift piezoeffects. For calculation of actuator is used the wave equation [6, 7, 12, 14] for the wave propagation in a long line with damping but without distortions. After

XVIII Issue II Version I II Version XVIII Issue Laplace transform is obtained the linear ordinary second-order differential equation with the parameter p,

olume whereupon the original problem for the partial differential V

) hyperbolic equation of type using the Laplace transform

( is reduced to the simpler problem [6, 13] for the linear ordinary differential equation 2Ξ( p,xd ) γ− 2Ξ( p,x ) = 0 (4) dx2 with its solution Ξ( p,x ) −xγ += BeCe xγ (5) where Ξ(x, p) is the Laplace transform of the displacement of the section of the piezoactuator, Figure 1: Generalized parametric structural schematic

Researches in Engineering Ψ Ψ diagram of the electromagnetoelastic actuator =γ p c + α is the propagation coefficient, c is the sound speed for Ψ = const , α is the damping coefficient The generalized transfer functions of the of the wave, Ψ is the control parameter: E is the electric electromagnetoelastic actuator are the ratio of the

ournal of ournal field strength for the voltage control, D is the electrical Laplace transform of the displacement of the face induction for the current control, H is the magnet field actuator and the Laplace transform of the

obal J strength. corresponding control parameter or the force at zero l

G From (3), (4), the boundary conditions on initial conditions. loaded surfaces, the strains along the axes the system III. Matrix Transfer Function of equations for the generalized structural-parametric model and the generalized parametric structural The matrix transfer function of the schematic diagram Figure 1 of the actuator are electromagnetoelastic actuator for the nanomedicine determined and the nanotechnology is deduced from its structural- parametric model (6) in the following form

Ψ ()p  piezoactuator are described with using its physical  Ξ (p)  ()()()pWpWpW   m   1   11 12 13  parameters and external load.   =    F 1 () p  (7) Ξ2 (p)  21()()()22 23 pWpWpW     F2 ()p  IV. Results and Discussions << << For m M1 and m M 2 the static The structural-parametric model and parametric displacement of the faces of the piezoactuator for the structural schematic diagrams of the voltage-controlled transverse piezo effect are obtained piezoactuator for the longitudinal, transverse and shift piezoeffects are determined from the generalized ( )UppW hUd 031011 M 2 1(∞ξ ) = lim = (8) p→0 structural-parametric model of the electro- δp δ(M + M 21 ) →α 0 magnetoelastic actuator with the replacement of the following parameters. ( )UppW hUd 031021 M1 2 (∞ξ ) = lim = (9) p→0 Ψ E EE δp δ(M1 + M 2 ) Ψm = { 3 E,E 3 ,E1 , mi = { 33 d,dd 31,d15 , ij = { 33 s,ss ,s5511 , = {δ h,l ,b α→0

For the piezoactuator from PZT under the The generalized structural-parametric model, 201 the generalized parametric structural schematic diagram transverse piezoeffect at m << M1 , m << M 2 , and the matrix transfer function of the electro- Year d = 2.5⋅10−10 m/V, h δ = 20 , U = 30 V, M = 2 kg, M = 8 31 1 2 magnetoelastic actuator are obtained from the solutions 21 kg the static displacements of the faces are determined of the equation of the electromagnetoelasticity, the ξ1(∞) = 120 nm, ξ2 (∞) = 30 nm, ξ1( )+∞ ξ2 ()∞ = 150 nm. Laplace transform and the linear ordinary differential For the approximation of the hyperbolic equation of the second order. cotangent by two terms of the power series in transfer From the generalized matrix transfer function of function (7) the following expressions of the transfer the electromagnetoelastic actuator after algebraic function of the piezoactuator is obtained for the elastic- transformations are constructed the matrix transfer inertial load at M1 → ∞ , m << M 2 under the transverse function of the piezoactuator for the longitudinal, piezoeffect transverse and shift piezoeffects. Ξ (p) hd δ (pW ) = 2 = 31 V. Conclusions ()pU (1+ E ) 2 2 + 2TpT(CC ξ p +1) e 11 t tt (10) The generalized structural-parametric model, I II Version XVIII Issue E 2 E  E E  the generalized parametric structural schematic t = MT 2 ()Ce+C11 , ξt = α 11 3 McCh ()Ce + C11 

  diagram, the matrix transfer function of the olume electromagnetoelastic actuator for the nanomechanics V where U (p) is the Laplace transform of the )

are obtained. voltage, T is the time constant and ξ is the damping ( t t The structural-parametric model, the matrix coefficient of the piezoactuator. The expression for the transfer function and the parametric structural schematic transient response of the voltage-controlled diagram of the piezoactuator for the transverse, piezoactuator for the elastic-inertial load is determined longitudinal, shift piezoeffects are obtained from the    ξtt  generalized structural-parametric model of the electro-  −   T  magnetoelastic actuator. From the solution of the  e t  (t) m 1−ξ=ξ sin(ωtt + ϕt ) equation of the electromagnetolasticity, the wave  2   1 ξ− t    (11) equation with the Laplace transform and the     deformations along the axes the generalized structural- parametric model and the generalized parametric Researches in Engineering 2  2  31(hd δ)Um 1− ξt  1− ξt  structural schematic diagram of the electro- ξm = , ωt = , ϕ = arctg 1+ C C E T t  ξ  magnetoelastic actuator are constructed for the control e 11 t  t  systems in the nanomechanics. The deformations of the

Where ξm the steady-state value of actuator are described by using the matrix transfer

function of the electromagnetoelastic actuator. of obal Journal displacement of the piezoactuator is, U m is the l G amplitude of the voltage. For the voltage-controlled References Références Referencias piezoactuator from the piezoceramics PZT under the transverse piezoelectric effect for the elastic-inertial load 1. Schultz J, Ueda J, Asada H (2017) Cellular Actuators. Butterworth-Heinemann Publisher, M1 → ∞ , m << M 2 and input voltage with amplitude Oxford, 382 p. U = 25 V at d .2 ⋅= 105 −10 m/V, h δ = 20 , M = 4 kg, m 31 2 2. Uchino K (1997) Piezoelectric actuator and E 7 7 C11 = 2 ⋅10 N/m, e 0.C 5⋅= 10 H/m are obtained values ultrasonic motors. Kluwer Academic Publisher, −3 Boston, MA, 1997, 347 p. ξm =100 nm, Tt = 0.4⋅10 c. The characteristics of the

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