Boundary Vibration Control of Strain Gradient Timoshenko Micro-Cantilevers Using Piezoelectric Actuators
BBoundary Vibration Control of Strain GradientTimoshenko Micro-cantilevers Using PiezoelectricActuators
Amin Mehrvarz*, Hasan Salarieh*, Aria Alasty, and Ramin Vatankhah***Department of Mechanical Engineering, Sharif University of Technology**Department of Mechanical Engineering, Shiraz University
Abstract —In this paper, the problem of boundary control ofvibration in a clamped-free strain gradient Timoshenko micro-cantilever is studied. For getting systems closer to reality, theforce/moment exertion conditions should be modeled. To this end,a piezoelectric layer is laminated on one side of the beam and thecontrolling actuation is applied through the piezoelectric voltage.The beam and piezoelectric layer are coupled and modeled at thesame time and the dynamic equations and boundary conditionsof the system are achieved using the Hamilton principle. Toachieve the purpose of eliminating vibration of the system, thecontrol law is obtained from a Lyapunov function using LaSalle’sinvariant set theorem. The control law has a form of feedbackfrom the spatial derivatives of boundary states of the beam. Thefinite element method using the strain gradient Timoshenko beamelement has been used and then the simulation is performed toillustrate the impact of the proposed controller on the micro-beam.
Index Terms —Strain gradient Timoshenko micro-beam, Piezo-electrical actuator, PDE model, Boundary control
I. I
NTRODUCTION
Many mechanical systems are modeled by partial differ-ential equations with boundary conditions which are knownas continues systems. To investigate such systems, accuratemodeling is needed. This modeling is combined with a lot ofsimplification and as the level of simplification being lower,the model will being closer to reality [1]–[3].One of the important continuous systems in mechanical en-gineering is the micro-beam. Micro-beams have many applica-tions in transportation [4], MEMS and NEMS [5],atomic forcemicroscopy (AFM) [6], micro switches [7]–[10], mass sensors,micro-accelerometers, micro-mirrors [11], [12], grating lightvalves (GLV) [13]–[15] and cell contraction assays [16].According to the position of continues systems control,various controllers have been designed for different purposes.For example in micro switches, the purpose of control isposition control of the end of beam [17], in atomic forcemicroscopy the goal is controlling the vibration of the beamin their resonant mode [18] and in the static atomic forcemicroscopy the goal is controlling the shape and position ofthe beam [19].Controller design methods are different. A group of thesemethods convert the partial differential equations (PDE) toseveral ordinary differential equations (ODE) and then forthese equations, the controller is designed [20], [21]. In these methods, it is clear that the main system has been changed,and the infinite dimensions of the system are reduced tosome finite dimensions of freedom, so the obtained controlleris suitable when only a few specified modes of the systemdynamics are excited which may not be guaranteed in real-world applications. Another group of controllers is boundarycontrol methods that design the controller directly for theinfinite dimensional main system and do not change it [22]–[24]. These controllers have many functions such as marineriser [25]–[27] and robotic [28].After designing the controller, it is required that controlactuation being applied to the system. Various methods existfor this purpose. Electrostatic and piezoelectric actuators arethe most common ones. One of the applications of electrostaticactuators is in grating light valves (GLV) [29]. Besides, thepiezoelectric actuators are usually utilized for atomic forcemicroscopy, which nowadays is considered as the most effec-tive tool in surface topography [30], [31] and also they areused for energy harvesting [32].In this paper boundary control of a clamped-free straingradient Timoshenko micro-cantilever with considering theeffects of piezoelectric actuator is studied. In this state, it isassumed that a piezoelectric layer is ideally attached to oneside of the beam. In the second section, the dynamic equationsof the system are derived. A linear control law based on thetheory of boundary control is proposed to suppress the systemvibration, in section three. In the fourth section, the finiteelement method (FEM) is utilized for modeling the system.Simulation results before and after applying the control laware presented in the fifth section. Finally, the conclusion isgiven in the last section.II. D
YNAMICS M ODEL
The investigated beam is a clamped-free strain gradientTimoshenko micro-cantilever that a layer of piezoelectric isideally attached on it and shown in Fig.1. In this figure h b isthe beam thickness, h p is the piezoelectric thickness, L is thelength of the beam and b is the width of the beam.The Hamilton principle is used to obtain partial differentialequations with the boundary condition of the system that is a r X i v : . [ ee ss . SP ] M a y ig. 1. A schematic view of the beam with piezoelectric actuator and somegeometric parameters [21]. shown in (1). t (cid:90) t ( δT + δW ∗ e − δW m + δW nc ) dt = 0 (1)where T is the kinetic energy, W ∗ e is the potential co-energy, W m is the magnetic potential energy and W nc is the work ofnon-conservative forces [33].Also for the piezoelectric and beam, W ∗ e is obtained fromequation (2) [1], [33]. dW ∗ e = (cid:18) dW ∗ e dV (cid:19) p dV p + (cid:18) dW ∗ e dV (cid:19) b dV b = [ ( 12 E T εE + S T eE ) − (cid:16) σ ij ε ij + P i γ i + τ (1) jik η (1) ijk + m sij χ sij (cid:17) × ] dV p − (cid:16) σ ij ε ij + P i γ i + τ (1) jik η (1) ijk + m sij χ sij (cid:17) dV b (2)where E is the electric field vector that is shown in equation(3), ε is the permittivity matrix, S is the strain vector inengineering representation that is shown in equation (4), e isthe matrix of piezoelectric constants, ε ij is the strain tensor, γ i is the dilatation gradient tensor, η (1) ijk is the deviatoric stretchgradient tensor and χ sij is the symmetric rotation gradienttensor. Also σ ij is the classical stress tensor, P i τ jik and m ij are higher-order stresses, V is the total volume of materialand superscript p and b indicates that the regarded parameteris related to the piezo-layer or the beam. The relations givenin (2) are written using the Einstein notation for summation. E = (cid:2) E ( t ) (cid:3) T = (cid:104) u ( t ) h p (cid:105) T (3) S = (cid:2) − zα x β (cid:3) T (4)In the equation (3), u ( t ) is the piezoelectric voltage and inthe equation (4), z is the distance from the neutral axis, α denotes the rotation of line elements along the centerline dueto pure bending, and β is obtained from the equation (5). v x ( x, t ) = β ( x, t ) + α ( x, t ) (5)where x and t indicate the independent spatial (along thelength of the beam) and time variables, respectively, v rep-resents the lateral deflection and subscripts x and t indicatesderivative with respect to position and derivative with respectto time. The remaining equations are given from [1]. By replacing all the equation in (2), we have: W ∗ e = L (cid:90) [ (cid:18) ε u ( t ) h p − ze α x u ( t ) (cid:19) b p −
12 [ k p ( v xxx − β xx ) + k p ( v xx − β x ) + k p (2 v xx − β x ) + k p ( v xx − β x ) + k p β ] −
12 [ k b ( v xxx − β xx ) + k b ( v xx − β x ) + k b (2 v xx − β x ) + k b ( v xx − β x ) + k b β ]] dx (6)in the equation (6), k i , i = 1 , , ..., are defined as follows: k = µI (cid:0) l + l (cid:1) k = I (cid:0) k + µ (cid:1) + 2 µAl k = µAl k = µAl k = k s µA (7)where k , µ and k s are the bulk module, shear module andthe shear coefficient of the Timoshenko beam. l , l and l demonstrate the additional independent material parametersDefining the following parameters will simplify the govern-ing equations. A = ρ p h p b p + ρ p h p b p B = ρ p I p + ρ b I b C = k p + k b D = k p + k b E = k p + k b F = k p + k b G = k p + k b H = ze b p (8)where ρ is the density of the beam or piezoelectric.The first variant of the potential co-energy W ∗ e , takes thefollowing form: δW ∗ e = L (cid:90) [ − Hu ( t ) δα x − Cα xx δα xx − Dα x δα x − E × ( v xx + α x ) ( δv xx + δα x ) − F (2 α x − v xx ) ( 2 δα x − δv xx ) − G ( v x − α ) ( δv x − δα )] dx (9)The kinetic energy of the system is obtained from theequation (10). T = 12 L (cid:90) (cid:2) Av t + Bα t (cid:3) dx (10)The first variation of the kinetic energy T is shown in theequation (11). δT = L (cid:90) [ Av t δv t + Bα t δα t ] dx (11)It is assumed that the external force is equal to zero. W m = W nc = 0 (12)eplacing (9), (11) and (12) in to (1), we have: t (cid:90) t L (cid:90) [ Av t δv t + Bα t δα t − Hu ( t ) δα x − Cα xx δα xx − Dα x δα x − E ( v xx + α x ) ( δv xx + δα x ) − F ( 2 α x − v xx ) (2 δα x − δv xx ) − G ( v x − α ) ( δv x − δα )] dxdt = 0 (13)Using integration by parts on several terms of the equation(13), the following results are achieved: L (cid:90) (cid:104) ( Av t δv + Bα t δα ) | tt (cid:105) dx + t (cid:90) t L (cid:90) [ [ − Av tt − E × ( v xxxx + α xxx ) + F (2 α xxx − v xxxx ) + G ( v xx − α x ) ] δv + [ − Bα tt − Cα xxxx + Dα xx + E ( v xxx + α xx ) + 2 F (2 α xx − v xxx ) + G ( v x − α ) ] δα ] dxdt + t (cid:90) t [[ − Hu ( t ) + Cα xxx − Dα x − E ( v xx + α x ) − F × (2 α x − v xx ) + G ( v x − α ) ] δα + [ − Cα xx ] δα x + [ E ( v xxx + α xx ) − F (2 α xx − v xxx ) − G ( v x − α ) ] δv + [ − E ( v xx + α x ) + F (2 α x − v xx )] δv x ] | L dt = 0 (14)Equation (14) is equal to zero. Therefore all of its termsshould be equal to zero. So, the following equations areobtained: (cid:40) Av t δv | tt = 0 Bα t δα | tt = 0 (15) Av tt + ( E + F ) v xxxx + ( E − F ) α xxx − G ( v xx − α x ) = 0 Bα tt + Cα xxxx − ( E − F ) v xxx − ( D + E + 4 F ) × α xx − G ( v x − α ) = 0 (16) (( E + F ) v xxx + ( E − F ) α xx − G ( v x − α )) | ( L,t ) = 0(( E + F ) v xx + ( E − F ) α x ) | ( L,t ) = 0( Cα xxx − ( E − F ) v xx − ( D + E + 4 F ) α x ) | ( L,t ) = Hu ( t ) α xx | ( L,t ) = 0 v | (0 ,t ) = v x | (0 ,t ) = α | (0 ,t ) = α x | (0 ,t ) (17)Equation (16) is the partial differential equation of thesystem and equation (17) is the boundary condition of thesystem. Potential co-energy of the system by using (8) will beobtained as follow: U = 12 L (cid:90) [ ε b p h p u ( t ) − Hα x u ( t ) − C ( v xxx − β xx ) − D ( v xx − β x ) − E (2 v xx − β x ) − F ( v xx − β x ) − Gβ ] dx (18) In the next section, a boundary controller will be designedfor the obtained model.III. C ONTROLLER DESIGN
Many flexible systems are modeled using a linear PDEand a set of BCs. To achieve the control purposes of flexiblestructures, most engineers rely on discretizing the governingPDE into a set of ordinary differential equations (ODEs) [4],[34], [35]. This is because of the abundance of control designtechniques available for ODEs and mathematical complexitiesof boundary control of PDE models. For more clarification,in the field of vibration control of micro-cantilever beams, inreferences [20], [21], the Galerkin method and finite elementmethod were employed to change the governing PDE of theEulerBernoulli micro-beam to a set of ODEs, respectively.After that, a controller was designed for the resulting ODEmodel. Unfortunately, a stability result generated for a dis-cretized ODE model under a proposed control cannot begeneralized to the PDE model under the same control. Thatis, the neglected higher order modes could possibly destabilizethe mechanical system under a discretized model-based control(i.e. spillover instability). Also, some devices and instrumentssuch as strain gages are needed to feedback the vibrationinformation at different points of the object, and an observershould be used to estimate the required vibration informationbased on the measured data. However, in many applications,using the measurement instruments at the interior points of theobjects is impossible or at least very difficult [36].To eliminate the problems of both observation and controlspillover, many investigators have proposed boundary controlstrategies for PDE models of elastic systems (i.e. the controlinvolves only a few actuators placed at the boundary of media).The boundary controllers designed for the non-discretizedPDE models are often simple compensators which ensureclosed-loop stability for an infinite number of modes. Themost significant advantage of the boundary control is that itcan stabilize the motion of mechanical systems without usingin-domain aligned actuators. This novelty has an importantrole in the field of industry and engineering applications.
A. Boundary control
If any undesired initial condition or noisy excitation is ap-plied to the beam, the system may show unwanted vibrations.In this case, the boundary controller is designed to suppress thevibration and return the system to the equilibrium state. In ourdesign, feedback of boundary states is utilized and the voltageof piezoelectric is tuned based on the feedback to stabilize thevibration.Well-posedness of the closed-loop system for eliminat-ing the system vibration has great importance. Semigrouptechnique and operator theory for designing the controllershould be used. After that, benefitting from the Lyapunovstability method and the LaSalle’s invariant set theorem, theasymptotic stability of the closed-loop system will be proved.For boundary controller design and well-posedness analysis ofhe controlled system, the PDE model (16) should be writtenin the state-space, as shown in (19). X t = [ A ] × X (19)where X is defined as: X = vv t αα t (20)Also, the matrix A is defined as follow: [ A ] = a v b v a α b α (21)where a v = − ( E + F ) A ∂ ∂x + GA ∂ ∂x a α = − ( E − F ) A ∂ ∂x − GA ∂∂x b v = ( E − F ) B ∂ ∂x + GB ∂∂x b α = − CB ∂ ∂x + ( D + E +4 F ) B ∂ ∂x − GB (22)In the equation (19), the matrix A is the PDE operator.To achieve the controlling purpose (eliminating the systemvibration), proper functional space should be chosen and thecorresponding inner product should be defined by using thekinetic energy and potential co-energy without the terms ofthe electrical energy.The proper functional space is denoted by V which isdefined on the proper functional space (Ω) and is shown in(23). V = H (Ω) × L (Ω) × H (Ω) × L (Ω) (23)In (23), L p (Ω) is a Lebesgue space which is the space ofmeasurable functions whose L p norm is bounded, equation(24), and H k (Ω) is a Hilbert space that is defined in a Sobolevspace W k (Ω) in (25). (cid:90) Ω | f | p dµ p < ∞ (24) W k (Ω) ≡ H k (Ω) = (cid:8) f : D α f ∈ L (Ω) , f or all ≤ α ≤ k (cid:9) (25)In the equation (25), D α f is the α th-order weak derivativeof function [37].The corresponding inner product introduced on the Hilbertspace V has the following form: (cid:104) Y, Z (cid:105) = 12 (cid:90) Ω [ Aa b + Ba b + Ca xx b xx + Da x b x + E ( a xx + a x ) ( b xx + b x ) + F (2 a x − a xx ) ( 2 b x − b xx ) + G ( a x − a ) ( b x − b )] d Ω (26) In (26), Y = ( a , a , a , a ) , Z = ( b , b , b , b ) and i =1 , ..., are scalar-valued functions defined on Ω which aredefined in (27). a j , b j ∈ H (Ω) , j = 1 , a j , b j ∈ L (Ω) , j = 2 , (27)As mentioned, the inner product defined as the summationof kinetic energy (10) and potential co-energy (18) without theterms which correspond to electrical energy. The target of thisinvestigation is to show that the system (16) with boundaryconditions (17) under boundary feedbacks appeared in theequation (28) is well-posed and have an asymptotic decay rate. u ( t ) = k u α t ( L ) (28)In the equation (28), k u is the controller gain and has a positivevalue.The system equations (16) with boundary condition (17) inthe state space is summarized as (29). X t = [ A ] X Γ x =0 : v = v x = α = α x = 0Γ x = L : ( E + F ) v xxx ( L ) + ( E − F ) α xx ( L ) − G ( v x ( L ) − α ( L )) = 0( E + F ) v xx ( L ) + ( E − F ) α x ( L ) = 0 Cα xxx ( L ) − ( E − F ) v xx ( L ) − ( D + E + 4 F ) α x ( L ) = Hu ( t ) α xx ( L ) = 0 (29)From operator A and boundary conditions of the system in(29), the domain of the operator A is determined as (30). D ( A ) = H (Ω) × H (Ω) × H (Ω) × H (Ω) (30)where H (Ω) = (cid:8) f : f ∈ H (Ω) , f | Γ = f x | Γ (cid:9) (31)To illustrate the well-posedness of the controlled systemexpressed in (29), first it should be proved that the operator A is a dissipative operator. T heorem. The linear operator A whose domain isdefined in the equation (30) is dissipative. P roof.
From the definition of the inner product in theequation (26), a Lyapunov function is defined as (32). (cid:104)
X, X (cid:105) V = 12 L (cid:90) [ Av t + Bα t + Cα xx + Dα x + E ( v xx + α x ) + F (2 α x − v xx ) + G ( v x − α ) ] dx = E ( t ) (32)By taking the time derivatives of the Lyapunov function(32), we have: ddt (cid:104) X, X (cid:105) V = 2 (cid:104) X, AX (cid:105) V = L (cid:90) [ Av t v tt + Bα t α tt + Cα xx α xxt + Dα x α xt + E ( v xx + α x ) ( v xxt + α xt ) + F ( 2 α x − v xx ) (2 α xt − v xxt ) + G ( v x − α ) ( v xt − α t )] dx (33)y replacing v tt and α tt form equation (16), the followingequation is obtained: (cid:104) X, AX (cid:105) V = 12 L (cid:90) [ − v t [ ( E + F ) v xxxx + ( E − F ) × α xxx − G ( v xx − α x ) ] − α t [ Cα xxxx − ( E − F ) × v xxx − ( D + E + 4 F ) α xx − G ( v x − α ) ] + Cα xx α xxt + Dα x α xt + E ( v xx + α x ) ( v xxt + α xt ) + F ( 2 × α x − v xx ) (2 α xt − v xxt ) + G ( v x − α ) ( v xt − α t )] dx (34)By rearranging (34), we have (cid:104) X, AX (cid:105) V = 12 L (cid:90) [ E ( v xx v xxt − v xxxx v t ) + F ( v xx × v xxt − v xxxx v t ) + G ( v xx v t + v x v xt ) + E ( v xxx × α t − α xxx v t + α xt v xx + α x v xxt ) + 2 F ( α xxx v t − v xxx α t − α x v xxt − α xt v xx ) + G ( α t v x − α x v t − α t v x − αv xt ) + C ( α xx α xxt − α xxxx α t ) + D ( α xx α t + α x α xt ) + E ( α xx α t + α x α xt ) + 4 F ( α xx × α t + α x α xt )] dx (35)Performing some integration by parts on (35), the followingresults are achieved: (cid:104) X, AX (cid:105) V = 12 [ E ( − v xxx v t + v xx v xt ) + F ( − v xxx v t + v xx v xt ) + Gv x v t + E ( α t v xx + α x v xt − α xx v t ) + 2 × F ( α xx v t − α x v xt − α t v xx ) − Gαv t + C ( − α xxx α t + α xx α xt ) + Dα x α t + Eα x α t + 4 F α x α t ] L | (36)Factorizing the terms that have time derivatives results in (cid:104) X, AX (cid:105) V = 12 [ − v t ( Ev xxx + F v xxx − Gv x + Eα xx − F α xx + Gα ) + v xt ( Ev xx + F v xx + Eα x − F α x )+ α t ( Ev xx − F v xx − Cα xxx + Dα x + Eα x + 4 F α x )+ Cα xt α xx ] L | (37)Implementing boundary condition (17) into equation (37)yields, (cid:104) X, AX V (cid:105) = 12 [ − v t ( ( E + F ) v xxx + ( E − F ) α xx − G ( v x − α ) ) + v xt (( E + F ) v xx + ( E − F ) α x ) − α t ( Cα xxx − ( E − F ) v xx − ( D + E + 4 F ) α x )+ Cα xt α xx ] L | (38) By replacing (17) in (38) we have, (cid:104) X, AX (cid:105) V = − Hα t ( L ) u ( t ) (39)By replacing control law (28) in (39) the following relationis obtained. (cid:104) X, AX (cid:105) V = − Hk u α t ( L ) (40)According to (40), it is clear that for the closed-loop systemwe have, (cid:104) X, AX (cid:105) V ≤ (41)Thus, from the definition of the dissipative operators [38],the proof will be complete. (cid:110) In the following, the continuity of the operator ( γI − A ) − is checked to achieve the final purpose. T heorem. The operator ( γI − A ) − exists and it iscontinuous for any γ . P roof.
It is assumed that we have: ( γI − A ) X = X (42)For demonstrating the existence of the operator ( γI − A ) − it is sufficient to show that only one solution exists for (42).The result of theorem 1 (equation (40)) is used to obtain thefollowing relation. (cid:104) ( γI − A ) X, X (cid:105) V = (cid:104) γX, X (cid:105) V − (cid:104) AX, X (cid:105) V = γ (cid:104) X, X (cid:105) V + 12 Hk u α t ( L ) ≥ γ (cid:104) X, X (cid:105) V = γ (cid:107) X (cid:107) V (43)The above result is shown that the bilinear form q withthe definition of a ( u, v ) = (cid:104) ( γI − A ) u, v (cid:105) is coercive on theHilbert space V . Now, using the Lax-Milgram theorem, onecan easily prove equation 42 has a unique weak solution andso the operator exists ( γI − A ) − [39].In [37] it is shown that if the operator ( γI − A ) − isbounded, it will be continuous. So considering equation (43)that is obtained from the dissipativity of operator one canconclude that: γ (cid:107) X (cid:107) V ≤ (cid:104) ( γI − A ) X, X (cid:105) V = (cid:104) X , X (cid:105) V ≤ (cid:107) X (cid:107) V (cid:107) X (cid:107) V → (cid:107) X (cid:107) V ≥ γ (cid:107) X (cid:107) V (44)Since (cid:107) X (cid:107) V is bounded, (cid:107) X (cid:107) V is also bounded and asa result, because boundedness contains the continuity, theoperator ( γI − A ) − is continues and the proof is complete. (cid:110) In the following, according to the control purpose we showthat equation (29) is well-posed.
T heorem. Equation (29) with initial condition X ( t =0) ∈ D ( A ) is well-posed. P roof.
According to the definition of the functional do-main space, that is H (Ω) × H (Ω) × H (Ω) × H (Ω) ⊂ V ,it is clear that D ( A ) is dense in V . Also, it is clear that therange of ( γI − A ) is dense in V , it means: R ( γI − A ) = V (45)According to theorem 2, ( γI − A ) has a continuous inverse ( γI − A ) − for any γ > . Therefore, according to theefinition of the resolving set of an operator [38], γ is inthe resolving set of the operator A .As shown in theorem 1, it is demonstrated that the operator A is a dissipative operator. Therefore, according to the Lumer-Phillips theorem [40], equation (29) with control law (28) andinitial condition X ( t = 0) ∈ D ( A ) is well-posed. (cid:110) The asymptotic stability of the closed-loop system isachieved by using LaSalles invariant set theorem which isbased on the Lyapunov method. For using this theorem, itshould be shown that ( γI − A ) − is compact for any γ > [41]. T heorem. Operator ( γI − A ) − is compact for any γ > . P roof.
It is shown that the operator ( γI − A ) − for any γ > is bounded. This subject is shown in the proof oftheorem 2. Also, it is obvious that: ( γI − A ) − V ⊂ D ( A ) (46)According to Rellich-Kondrachov compact embedding the-orem [38], since the closure of ( γI − A ) − V is H (Ω) × H (Ω) × H (Ω) × H (Ω) and this space is compactly em-bedded in H (Ω) × L (Ω) × H (Ω) × L (Ω) [38], thereforethe compactness of the above-mentioned resolving is obtainedand the proof will be completed. (cid:110) According to these theorems, by using LaSalle’s invariantset theorem, the asymptotic stability of the closed loop systemwill be demonstrated.
T heorem. The system of equation (29) with controlfeedback (28) will asymptotically tend toward zero.
P roof.
According to selected Lyapunov function thatcontains some terms of kinetic energy and potential co-energyand according to the defined inner product, it was shown that E ( t ) = (cid:104) X, X (cid:105) V ≥ is positive definite. Also, it was shownin Theorem 1 that the time derivative of Lyapunov function isequal to: ˙ E ( t ) = − k k u α t ( L ) (47)Equation (47) shows only the convergence of α t ( t ) to zero,but one can use the LaSalle theorem to prove the asymptoticstability. So, Theorems 2-4 have been proved. It is clearfrom the above equation that ˙ E ( t ) ≤ and E ( t ) ≥ hasrequirements of a Lyapunov function. Therefore, Because ofcompactness of the resolving ( γI − A ) − proved in Theorem4, the LaSalles invariant set theorem [41] gives the asymptoticdecay rate of the controlled and the proof will be complete.IV. F INITE E LEMENT M ETHOD
In this section for system modeling, the finite elementmethod is provided. For modeling, strain gradient Timoshenkobeam element is selected. It is assumed that the element hastwo nodes and each node has (cid:2) v v x α α x (cid:3) variables andand have the following polynomial forms: (cid:26) v = c + c x + c x + c x α = c + c x + c x + c x (48) For using this element, at first the shape function shouldbe obtained and then with using the shape function, kineticand potential energy, mass, stiffness and force matrices arecalculated.Equation (48) can be written in a matrix form as (49). vv x αα x = x x x x x
00 0010 00 x x x x x c c c c c c c c = gC (49)where in (49), c i , i = 1 , ..., are constant and h matrix fortwo nodes x = 0 , L e has been calculated that L e is the lengthof the beam element. h = L e
100 0000 L e L e
00 0000 L e L e
00 00100010 000100 L e L e L e L e L e (50) c c c c c c c c = h − v v x α α x v v x α α x = h − q (51)Finally: vv x αα x = gh − q = N q (52)where N in (52) is the shape function of a Timoshenko beamelement that is defined as: N = gh − = H H x H H x
00 00 H H x H H x H H x H H x
00 00 H H x H H x (53)In (53), H i , i = 1 , ..., are obtained as follows: H = x L e − x L e + 1 H = x − x L e + x L e H = x L e − x L e H = x L e − x L e (54)ow by using kinetic energy and potential co-energy (10)and (18) and variational method [42], mass, stiffness and forcematrices will be obtained as (55), (56) and (57). M e = L e (cid:90) ( D T AD + D T BD ) dx (55) K e = L e (cid:90) ( B T CB + B T DB + B T EB + B T F B + B T GB ) dx (56) F e = L e (cid:90) − k U B T dx (57)In the above equations D i , B j , i = 1 , and j = 1 , ..., aredefined as: D = (cid:2) (cid:3) ND = (cid:2) (cid:3) NB = (cid:2) ∂∂x (cid:3) NB = (cid:2) (cid:3) NB = (cid:2) ∂∂x (cid:3) NB = (cid:2) − ∂∂x (cid:3) NB = (cid:2) − (cid:3) N (58)For making a model of the real system by assuming tennodes in the system, matrices M , K and F will be obtained byassembling the matrices given in equation (55), (56) and (57).Time evolution of the system will be obtained by numericalintegration of the following equation. [ M ] { ¨ q } + [ K ] { q } = { F } (59)In this section for showing the accuracy of the controllerdesigned by the boundary control method, replacing realvalues instead of parameters and using finite element modelingthat was presented in the previous section, the strain gradientTimoshenko micro-cantilever for two cases before and afterapplying the control actuator is simulated.First of all, it is required that the system parameters be-come nondimensionalized. The following nondimensionalizedvariables and parameters are utilized. ˜ x = xL ˜ L = LL ˜ b = bL ˜ h b = h b L ˜ h p = h p L ˜ ρ b = ρ b ρ b ˜ ρ p = ρ p ρ b ˜ I b = I b L ˜ I p = I p L ˜ c ij = c ij ρ b L ω ˜ e = e e ˜ c = c ρ b L ω ˜ c = c ρ b L ω ˜ u = ue ρ b L ω ˜ t = tω ˜ l = l L ˜ l = l L ˜ l = l L ˜ E b = E b E b ˜ E p = E p E b (60)Also, the physical characteristics of the beam and piezo-electric layer can be found in Table.I [21], [29], [43] and thegeometry of the piezoelectric and beam are given in Table.II[21]. Fig. 2. Response of the micro beam before control voltage exertion: (a) lateraldeflection v ( x, t ) , (b) rotation of line elements along the centerline α ( x, t ) . According to table 1, we have: e = (cid:88) d i c i = − . C / m (61)Also, the bulk module, shear module and the shear coef-ficient of the Timoshenko beam are obtained from equations(62), (63) and (64). K = E − υ ) (62) µ = E υ ) (63) K s = 5 + 5 υ υ (64)In the equation (28), a proper control gain obtained viatrial and error which has a suitable settling time and transientresponse is selected as k u = 0 . . By replacing this value inequations and considering ten nodes on the beam, the equation(59) is solved for two mentioned states in a distinct timeperiod. The results are shown in Fig.2 and Fig.3, and thecontroller voltage that was obtained in the equation (28) isshown in Fig.4. ig. 3. Response of the micro beam after control voltage exertion: (a) lateraldeflection v ( x, t ) , (b) rotation of line elements along the centerline α ( x, t ) .TABLE IM ATERIAL PROPERTIES OF SILICON DIOXIDE BEAM AND
PZT
ACTUATOR .Material SiO2 PZTDensity (Kg/m3) 2200 7700Poisson coefficient 0.17 0.31Young modulus of elasticity (GPa) 73 71Piezoelectric Constants (10-12C/N) - d31=175d33=400d55=580Relative permittivity 3.9 1700TABLE IIG
EOMETRICAL DIMENSIONS OF BEAM AND PIEZOELECTRIC LAYER ( ALLIN M )Beam length 90Beam thickness 10Beam width 30Piezoelectric length 150Piezoelectric thickness 10Piezoelectric width 30 Fig. 4. Control voltage u ( t ) = k u α t ( L ) . V. S
IMULATION
As it is clear, after applying the control action, the vibrationof the system caused by the non-zero initial displacement hasbeen suppressed and the system has become asymptoticallystable.Such vibration damping mechanism can be also realizedfor the acoustic wave utilizing the destructive interferences[44]. According to this simulation, the accuracy of controllaw and obtained equations can be confirmed. This methodof control is used for different applications such as vibrationcontrol of the fluid containers [45]. If strain gradient modelis used for flexible structure in side-wall of these containers,better results will be obtained.VI. C
ONCLUSION
In this paper strain gradient Timoshenko micro-cantileverwith a piezoelectric layer laminated on one side of the beamwas modeled and equations of the system with boundaryconditions were obtained in state space. Then well-posednessof equations were checked and by using the Lyapunov functionand LaSalles invariant set theorem, a control law for thestability of the system was proven. This control law forsuppressing the vibration of the system was achieved fromthe feedback of temporal derivatives of boundary states ofthe beam, and it was applied through exciting voltage ofthe piezoelectric layer. For showing accuracy of the designedcontroller, the simulation was done. In this work by using finiteelement method and strain gradient Timoshenko element, thesystem was modeled and by using numerical solution for twocases means closed-loop and open loop systems, the simulationwas performed which verified the achieved theoretical resultsof this work. R
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