Vibration suppression of a micro beam subjected to magneto-electric load

Abstract

Keywords: Magneto-electric; Maximum principle; Micro beam; Vibration; Piezoelectric actuator

Introduction

The definition, magneto-electro-elastic solid, is widely used to address to a kind of smart materials have capacity to transform reversibly their properties to respond external excitation such as temperature, moisture, stress, electric or magnetic fields. Since last two decades, magneto-­electroelastic (MEE) composites have gained great importance due to their ability of transforming one form of energy to another, having simple geometry and economic design and being useful in smart or intelligent structure applications. Much studies are done for examining on several properties of Magneto-electro-elastic structures. In [2], the state-­ vector method is applied to examine the free vibration of

MEE laminate plates. The natural frequencies and corresponding mode shapes are calculated and compared with results existing in the literature. [3] aimed computationally investigatation for effects of thermal and mechanical loads on dynamic characteristics of intelligent composite structures. In [4], the numerical calculations are made for investigating the nonlinear vibration, nonlinear bending, and magneto-­electric potential distributions through the thickness of the beam in different thermal environmental conditions. In [5], the non-local theory solution to a three-­ dimensional rectangular permeable crack in MEE materials is showed by using the generalized Almansi’s theorem and the Schmidt approach. The problems are modeled through

*Corresponding author. *E-mail address: k.yildirim@alparslan.edu.tr This paper was recommended for publication in revised form by Regional Editor Mustafa Düldül Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

Fourier transform as three pairs of dual integral equations, in which the unknown variables are the jumps of elastic displacement, electric and magnetic potential jumps across the crack surfaces. In [6], the effect of hygrothermal environment on the free vibration characteristics of (MEE) plates is presented by using finite element method. For this aim, higher order shear deformation theory is employed to evaluate the displacement fields. In [7], In order to estimate the first natural frequency and the critical angular velocity of a thermo-electro-magneto-elastic single-layer cylindrical nano-shell resting on a Winkler foundation, the governing equations of motion are derived based on the Hamiltonian Principle and by using first shear deformation theories in conjunction with modified couple stress theory. The effects of the centrifugal acceleration are considered in the formulation. In [8], Exact solutions are obtained for three-­ dimensional, anisotropic, linearly magneto-electroelastic, simply-supported, and multilayered rectangular plates subjecting to static loadings. While the homogeneous solutions are obtained in terms of a new and simple formalism that resemble the Stroh formalism, solutions for multilayered plates are presented in terms of the propagator matrix. In [9], analytical solutions are presented for free vibrations of three-dimensional, linear an isotropic, MEE multilayered rectangular plates under simply supported edge condition and the dispersion equation that characterizes the relationship between the natural frequency and wave number can be obtained in a simple form. In [10], Free vibration studies of multiphase and layered MEE beam is carried out. In-plane plate finite-element analysis is used to obtain the behavior of MEE beam. In [11], a partial mixed layerwise finite element model for adaptive plate structures is presented. Static analysis of MEE laminated plate structures is considered. The mixed finite element formulation is presented by considering a Reissner mixed variational principle. In [12], authors considered size-dependent geometrically nonlinear free vibration of magneto-electro-thermo elastic (METE) nanoplates using the nonlocal elasticity theory. The mathematical formulation is developed based on the first-order shear deformation plate theory, von Krmn-type of kinematic nonlinearity and nonlocal elasticity theory. In [16], Thermo-electro-magneto-mechanical bending analysis of a sandwich nanoplate is presented by Kirchhoffs plate theory and nonlocal theory. The sandwich nanoplate includes an elastic nano-core and two piezomagnetic face-sheets actuated by applied electric and magnetic potentials. The governing equations for the electro-magneto-mechanical bending are derived in terms of the displacement components and electric and magnetic potentials. Then, the problem is solved analytically by using Naviers method. In [17], Thermo-electro-mechanical transient analysis of a sandwich nanoplate is studied in this paper. The sandwich nanoplate consists of a KelvinVoigt viscoelastic nanoplate and two integrated piezoelectric face sheets resting on a visco-Pasternak foundation. The sandwich nanoplate

is subjected to thermal and mechanical loads, and the piezoelectric face sheets are subjected to an applied electric potential. Twovariable sinusoidal shear deformation plate theory is used for the description of the displacement components. The governing equations of motion are derived using Hamiltons principle by calculation of strain and kinetic energies and energy due to external forces. In [15], authors proposed the cell-based smoothed finite element method (CS-FEM) for superior calculations, in which the strain smoothing technique is introduced into FEM. Also, it is shown that CSFEM possesses high accuracy, low mesh restriction, much less computational-cost than FEM, and stronger handling ability when encountering strong mesh distortions and large deformations. In [18], Wave propagation analysis of a nanobeam made of functionally graded MEE materials with rectangular cross section rest on Visco-Pasternak foundation is studied in this paper. For modeling the axial, rotation and transverse deformations, Timoshenko beam model is used. Fundamental MEE equations of the model are derived for a general functionally graded beam excited to electric and magnetic potentials. Surface elasticity is employed for more confident modeling the behavior of nanobeam. Using Hamilton principle and calculation of kinetic and strain energies, the equations of motion are derived. In [19], A dynamic solution is presented for the propagation of harmonic waves in inhomogeneous (functionally graded) MEE composite plates. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The Legendre orthogonal polynomial series expansion approach is employed to determine the wave propagating characteristics in the plates. In [13], authors presented for the first time the thermal effect on dynamic characteristics of MEE intelligent structures by combining the coupled multiphysic (CP) CS-FEM with the modified Newmark method. The precision and converging of CPCS-FEM were proved by the comparing with FEM. In [20], The general solution of three-dimensional problems in transversely isotropic MEE media is obtained through five newly introduced potential functions. The displacements, electric potential, magnetic potential, stresses, electric displacements and magnetic inductions can all be expressed concisely in terms of the five potential functions, all of which are harmonic. The derived general solution is then applied to find the fundamental solution for a generalized dislocation and also to derive Green’s functions for a half-space MEE solid. In [14], authors aimed to conduct the research on studying the hygrothermal effect on the MEE-based structure using CS-FEM. In [21], Two independent state equations are established for transversely isotropic magneto-electro-elastic media by introducing proper stress and displacement functions. The free vibration problem of simply supported rectangular plates with general inhomogeneous (functionally graded) material properties along the thickness direction is then

considered. An approximate laminate model is employed to transform the state equations with variable coefficients to the ones with constant coefficients. As understood from literature review, there are very few and limited studies interested on vibration control of MEE micro beam. The original contribution of this paper and the reason making this paper is important is that vibrating control of MEE micro beam by using piezoelectric patch actuator is firstly considered in this paper and maximum principle is employed for determining the control voltage function optimally. In particularly, in the present paper, active vibration control of a MEE micro beam is taken into account and vibrations based on magneto-electric load are suppressed by means of piezoelectric patch actuators bonded on the surface of the micro beam. Before control actuation of the system, wellposedness and controllability properties of the System are stated and proved by a lemma and a theorem. The main aim of this study is to minimize the performance index functional of the system, which is defined a modified kinetic energy of the system, by using minimum control voltage function to be applied to piezoelectric patch. For determining the control voltage function, maximum principle is employed due to existence of Heaviside function in the equation of motion. By means of maximum principle, optimal control problem is transformed to a solving of system of partial differential equations including state and adjoint variables and they are linked to each other via terminal-boundary-initial conditions. The solution of the system is obtained by means of MATLAB and obtained results are presented in tables and graphical forms for indicating the effectiveness of the introduced control actuation.

Definition Of The Problem

Consider a five-layer structure which consists of a central host layer, which is a MEE microbeam, and both

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

side of the MEE micro beam is partially covered by elastic but non conductive rubber and two piezoelectric path actuators are perfectly deployed on these rubbers on the both side of a MEE micro beam as it is seen in Fig. 1. Initially, it is assumed that micro beam subjecting to magneto-electric load is undeformed and at rest position. Piezoelectric patch actuators are put on the same points of the micro beam for effective control actuation. The vibrations in a magneto-electro-elastic micro beam subjecting to magneto-­electric load is given by as follows [1]; DIuXXXX − ζ uXX + ρ AuTT = q( X ) + C(T , X )

where state variable u transverse displacement at (T , X ) ∈Ω= T ∈ (0,T f ), X ∈ (0, l ) , T is time variable, Tf is predetermined terminal time, X is space variable, l is the length of the MME microbeam, D = B + ea + fb in which B is the elasticity coefficient, e is the piezoelectric coefficient, a is height of MEE micro beam, f is piezo magnetic coefficient, b is the width of MEE microbeam, ζ = eδ1 + fδ2, in which δ1 and δ2 are electric potential and magnetic potential, respectively, between lower and upper surfaces of MEE microbeam, q(X) is external magneto-electric load function, C(T,X) = κV(T)(H″ (X – X1) – H″(X1 – X2)) in which κ is the elasticity coefficient of the rubber, V is the control voltage function to be applied to piezoelectric actuator, H is Heaviside function, (X – X2) is the location points of piezoelectric actuators. Eq. (1) is subject to the following initial conditions

and boundary moment conditions U(T,X) = 0, UXX(T,X) = 0 at X = 0,l.

Figure 1. A MEE microbeam diagram with patches in a thermal environment.

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

Because of the right hand-side of Eq. (5) is zero, we observe

∂u ( X ) 2   ∈ L (0, l ) , u0 ( X )∈ H 1 (0, l ) =u0 ( X )∈ L2 (0, l ) : 0 ∂X   (4b) u1 ( X )∈L2 (0, l )

dε (T ) = 0= ,That is ε (T ) constant dt Taking the initial conditions defined by Eq. (7) into consideration, we obtain that

l where L2 (Ω) denote the Hilbert space of real-valued 1  ∂4 ∂2 2 ε (0) DI ( W ) − ζ (W )2 square-integrable functions defined in the domain Ω in the =  2 ∫0  ∂X 4 ∂X 2 Lebesque sense with norm and usual inner product defined by (10) 2  ∂W   +ρ A 0.   |T =0 dX = = η 2 η ,η , ρ ,η= Ω ∫ρηdΩ.  ∂T   Ω

Then, the system under consideration has a solution [23]. Lemma 1. With Eq.4, the system defined by Eqs. (1)–(3) has a unique solution. Proof. Let us assume that in the same physical conditions, u1 and u2 are two different solutions to the system under consideration. Then, the difference W(T,X) = u1(T,X) – u2(T,X)

Then, it follows from Eqs. (7)–(10) that W(T,X) is identically equal to zero in Ω that is u1 = u2. Hence, proof is completed. For convenience, let us introduce nondimensional parameters as follows; x =

(11) l4 l4 , f (x ) = q= V (T ) . ξ = ( x ) , Ѵ(T ) satisfies the following homogeneous equation DI DI DI DIWXXXX – ζWXX + ρAWTT = 0 (5) In the light of Eq. (11), nondimensional equation of and following homogeneous boundary and initial con- motion is stated as follows; ditions, respectively. wxxxx – ξwxx + wtt = f(x) + C(t,x) (12) W(T,X) = Wxx(T,X) = 0 at X = 0,l (6) in which w is the transversal displacement at (t,x) W(T,X) = WT(T,X) = 0 at T = 0. ∈ Ω = (0,tf) × (0,1) and C(t,x) = κѴ(t)(H″(x - x1) – Let us show that W(T,X) is identically equal to zero in H″(x – x2)). Eq. (12) is subjected to following boundary Ω(T , X ). Then, consider the energy integral as follows; conditions

2 l 1  ∂4 ∂2  ∂W   2 2 ε= (T ) ∫  DI 4 (W ) − ζ 2 (W ) + ρ A    dX 2 0  ∂X ∂X  ∂T  

(8) and let us show that ε(T) is independent of T. Differentiating ε(T) with respect to T, we obtain l

Applying integration by parts and using boundary conditions given by Eq. (6), one observes Eq. (9) as follows; dε (T ) = dt

 ∂ 4W ∂ 2W ∂ 2W  ∂W DI − ζ + ρ A dX .  ∫0  ∂X 4 ∂X 2 ∂T 2  ∂T

Optimal Control Problem

The main goal of this study is to determine voltage function optimally to be applied to piezoelectric actuator which minimizes a given performance index at a predetermined terminal time tf with a minimum expenditure of control voltage energy. Before defining the performance index functional of the system, let us define the admissible control voltage function set as follows; Ѵad = {Ѵ(t) |Ѵ ∈ L2(Ω),

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

Then, the performance index functional of the system is given by as follows: l

2 2 2 I (Ѵ (t )) = ∫ ϑ1w (t f , x ) + ϑ2wt (t f , x ) dx + ϑ3 ∫Ѵ (t )dt (16)

in which ϑ1, ϑ2 ≥ 0, ϑ1 + ϑ2 ≠ 0 and ϑ3 > 0 are weighting constants. First integral on the left-hand side in Eq. (16) represents the modified dynamics response of the magneto-electro micro beam. First and second terms in this integral are quadratic functional of the displacement and velocity of the beam, respectively. Second term on the left-hand side in Eq. (16) is the measure of the total control voltage energy on the (0,tf) Then, optimal control problem is stated as follows; min I Ѵo (t ) = I (Ѵ) Ѵ ∈Ѵad

subject to the Eqs. (12)–(14). In order to achieve the Maximum principle for obtaining optimal control voltage function, let us introduce an adjoint variable ν ∈ L*, L* is the dual to L2(Ω) and has the same norm and inner product like in L2(Ω) Adjoint system corresponding to Eqs. (12)– (14) is expressed as follows; νxxxx – ξνxx + νtt = 0

∆w = w – wo, ∆wt = wt – wto. The operator φ(∆w) = ∆C(t,x) is subject to the following homogeneous boundary and initial conditions, respectively; ∆w(t,x) = ∆wxx(t,x) = 0 at x = 0,1

Now, note that the following functional Ω ∫∫ {I + I }d= Ω ∫∫ {ν∆C(t , x )}dΩ ∫∫ {νϕ (∆w ) − ∆wϕ *}d= 1

Applying the integration by parts to each term in the I1 and I2 in Eq. (28) and employing boundary and terminal conditions, respectively, given by Eq. (19) and Eq. (20), Eq. (28) becomes as follows;

= 2 ∫ ϑ1w(t f , x )∆w(t f , x ) + ϑ2wt (t f , x )∆wt (t f , x ) dx (31) 0

The deflections in the state variable and its derivatives with respect to the time variable are defined by

Theorem 1. (Maximum principle) The maximization problem is presented as follows;

The existence and uniqueness of the solutions to adjoint system defined by Eqs. (18)–(20) is shown similar to Eqs. (1)–(2). Note that is the unique solution to system given by Eqs. (12)–(14) and also convexity properties of the performance index functional guarantees the uniqueness of the control voltage function V(t). Hence, it is easy to observe that the system under consideration has a unique state and control function. Then, the system is called as observable, which equal to controllable [22]. Namely, the system Eqs. (12)-(14) is controllable. Maximum principle is derived a necessary condition for the optimal control function in terms of Hamiltonian functional. In case of some convexity assumptions, satisfied by Eq. (16), on performance index functional of the system, maximum principle is also sufficient condition for optimal control function. Let us derive the maximum principle as follows;

where Ѵo(t) is the optimal control voltage function. Proof. Before starting to proof, let us define following operator

subjected to following boundary and terminal conditions, respectively; ν(t,x) = 0,

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

For the right-hand side of Eq. (31), employe the properties of dirac-delta function H ′′( x − x1 ) = δ ′( x − x1 ),

= − g ′( x1 ), x1 ∈ (0,1) in which prime denotes spatial derivative. In the light of Eq. (32), the right hand-side of Eq. (31) is observed

which leads to κѴ(t)(νx (x2,t) – νx(x1,t)) + ϑ3Ѵ2(t) 2 ≥ κѴo(t)(νx(x2,t) – νx(x1,t)) + ϑ3Ѵo (t)

∫∫ κν (t , x ) ∆ Ѵ(t )  H ′′(x − x ) − H ′′(x − x ) dΩ 1

Focus on the difference of the performance index functional ∆I Ѵ(t ) = I Ѵ (t ) − I Ѵo(t ) =

Expanding w2(tf,x) and wt2(tf,x) in Taylor series around 2 w (tf,x) and wto (tf,x), yields o2

w2(tf,x) – w (tf,x) = 2wo(tf,x)∆w(tf,x) + r, wt2(tf,x) – wto (tf,x) = 2wto(tf,x)∆wt(tf,x) + rt

where r = 2(∆w)2 + higher order and positive terms > 0 and rt = 2(∆wt)2 + higher order and positive terms > 0. Substituting Eq. (35) into Eq. (34) gives Ѵ(t ) ∆I =

In this section, in order to indicate the effectiveness and robustness of the introduced control algorithm for damping excessive vibrations in a magneto-electro-elastic micro beam by optimally determined control voltage function to be applied to piezoelectric patch actuator, obtained theoretical results are presented in tables and graphical forms by solving following system of equations linked by terminal-initial-boundary conditions via MATLAB. wxxxx – ξwxx + wtt = f(x) + C(t,x), = C (t , x ) κ Ѵ(t ) ( H ′′( x − x1 ) − H ′′( x − x 2 )) , Ѵ (t ) = κ

Numerical Results And Discussions

∆I Ѵ (t ) ≥ ∫κ∆Ѵ(t ) (ν x ( x 2 , t ) − ν x ( x1 , t )) dt

Before expounding the results in graphs and tables, consider the optimal control voltage function given by Eq. (39), in which, it is easy to see that as the value of ϑ3 is decreasing, the value of the control voltage function is increasing. As a conclusion of this status, dynamic response of the beam given by first integral on the left side

Sigma J Eng Nat Sci, Vol. 39, No. 4, pp. 404–413, December, 2021

of the Eq. (16) is minimized by using minimum control voltage energy. Effectiveness of the introduced control actuation is examined in two cases. Both of two cases, tf is taken into account as 5. Weighted coefficients are taken into account as ϑ1,2 = 1 and ϑ3 =103 and ϑ3 =10–3 for uncontrolled and controlled case, respectively. All figures are plotted in the middle point of the MEE micro beam, x =

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