Structural Scheme of Electro Magneto Elastic Actuator for Nanotechnology and Nanoscience

The electro magneto elastic actuator on the piezoelectric, electrostriction, magnetostriction effects is used in the mechatronics control systems for nanotechnology and nanoscience [1- 6]. At work we consider the method of mathematical physics for the solution of the wave equation of the electro magneto elastic actuator with using the Laplace transform [7-14]. We have the structural-parametric model, the structural scheme and the matrix transfer function the electro magneto elastic actuator from equation of the magneto elastic elasticity and the decision of the wave equation [15-20]. The dynamic and static characteristics of the electro magneto elastic actuator are obtained for designing the mechatronics control systems of nanotechnology and nanoscience regarding its physical parameters and external load [21- 30].

Structural scheme

The method of the mathematical physics with the Laplace transform is applied for the solution the wave equation. The structural scheme of the electro magneto elastic actuator for nanotechnology and nanoscience is changed from Cady and Mason electrical equivalent circuits [5-8]. The equation of the electro magneto elasticity [5,8,12] has the following form

where S_i is the relative displacement along axis i of the cross section of the piezo actuator, Ψ_m = {E_m, D_m, H_m} is the control parameter, E_m is the electric field strength for the voltage control along axis m, D_m is the electric induction for the current control along axis m, H_m is the magnetic field strength for the magneto control along axis m, T_j is the mechanical stress along axis j, v_mi is the electro magneto elastic module, for example, the piezo module, S^Ψ_ij is the elastic compliance for the control parameter, Ψ = const and the indexes i=1, 2, … , 6; j=1, 2, … , 6; m=1, 2, 3. The main size along axis i for the electro magneto elastic actuator is determined us the working length l = {δ,h,b} in form the thickness, the height or the width for the longitudinal, transverse or shift piezo effect.

For the construction the structural scheme of the electro magneto elastic actuator is used the wave equation [8,12,14] for the wave propagation in the long line with damping but without distortions. With using Laplace transform is obtained the linear ordinary secondorder differential equation. The problem for the partial differential equation of hyperbolic type using the Laplace transform is reduced to the simpler problem [6,8,14] for the linear ordinary differential equation.

Where, Ξ(x, p) is the Laplace transform of the displacement of the section of the electro magneto elastic actuator γ = p/c^Ψ + α , is the propagation coefficient, c^Ψ is the sound speed for the control parameter, Ψ = const , α is the damping coefficient. Using method of the mathematical physics with the Laplace transform [6,12,23,26] for the solution of the wave equation, the equation of the electro magneto elasticity, the boundary conditions, we have the structural-parametric model and the structural scheme of actuator for nanotechnology and nanoscience on (Figure 1) in the following form

Figure 1:Structural scheme of electro magneto elastic actuator for nanotechnology and nanoscience.

where

v_mi is the electro magneto elastic module: d_mi is the piezo module at the voltage-controlled piezo actuator or the magneto strictive coefficient for the magneto strictive actuator, g_mi is the piezo module at the current-controlled piezo actuator, S₀ is the cross section area, M1, M2 are the mass of the loads 1, 2,Ξ₁(p), Ξ₂(p)and F₁(p), F₂(p) are the Laplace transforms of the appropriate displacements and the forces on the faces 1, 2.

For nanotechnology and nanoscience, the structural scheme of the voltage controlled, or current controlled piezo actuator are obtained from its structural-parametric model. From the structural parametric model and the structural scheme of the electro magneto elastic actuator we have the transfer function, the static and dynamic characteristics of the actuator for nanotechnology and nanoscience.

Transfer function

The matrix transfer function [5,18,21,26] of the electro magneto elastic actuator for nanotechnology and nanoscience is received from its structural-parametric model in the form

(Ξ(p)) = (W(p)) (P(p))

where (Ξ(p)) is the column-matrix of the Laplace transforms of the displacements for the faces 1,2 of the electro magneto elastic actuator, (W(p)) is the matrix transfer function, (P(p)) the columnmatrix of the Laplace transforms of the control parameter and the forces. The transfer function of the voltage-controlled transverse piezo actuator is obtained for the elastic-inertial load at M₁ →∞ , m < M₂ and the approximation the hyperbolic cotangent by two terms of the power series in the form

where U(p) is the Laplace transform of the voltage on the plates of the piezo actuator, k_t is the transfer coefficient, T_t is the time constant. ξ_t is the damping coefficient of the piezo actuator. Therefore for the piezo actuator from PZT with the elastic-inertial load at d₃₁= 2.5∙10^-10 m/V, h/δ =20, M₂ = 4 kg, C^E₁₁ = 2.10⁷ N/m , Ce = 0.5∙10⁷ N/m we obtain values the transfer coefficient kt =4 nm/V and the time constant of the piezo actuator T_t = 0.4∙10-3 s. For calculations the mechatronics control systems in nanotechnology and nanoscience with the electro magneto elastic actuator its transfer functions are obtained.

The method of mathematical physics for the solution of the wave equation of the actuator with Laplace transform is used for receiving the structural-parametric model and the structural scheme of the electro magneto elastic actuator for nanotechnology and nanoscience. The structural-parametric model, the structural scheme and the transfer functions of the electro magneto elastic actuator for nanotechnology and nanoscience are described the characteristics of the electro elastic actuator about its physical parameters, external load.

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