Active vibration control of piezoelectric plates

The control of thermally induced vibrations of a rectangular plate is investigated. An optimization problem is formulated to determine the control voltage needed to perform vibration suppression with least control effort. By eigenfunction expansion, the optimal control problem will be converted from a distributed to a lumped parameter system. By utilizing the variational theory, an explicit optimal control criterion will be derived.


Introduction
The dynamics and the stability of a structure are likely to be effected as a result of rapid temperature variation.The change in temperature experienced by a spacecraft emerging from a shadow entering the sunlight is an example of a dynamic load experienced by a structure.The sudden exposure to heat leads to thermal vibrations, which needs to be damped.Structures subject to thermally induced vibrations have been discussed by many authors.For example, the piezo thermoelastic effects of distributed piezoelectric sensors and actuators in higher dimension structures such as plates and shells are discussed in [?,?,?,?].Sadek  In this paper, we consider a three-layer plate consisting of a thermal host layer and two outer piezothermoelastic layers as in Figure 1.The plate is assumed to be exposed to a sudden rise in temperature over one face, which will cause an undesired vibration.The main objective of this research article is to a e-mail: mabukhaled@aus.eduminimize a weighted quadratic functional of the dynamic responses of the smart thin plate in a prescribed terminal time using continuous piezoelectric patches (voltages ±V i ), with the least possible expenditure of control forces.The solution method is a combination of modal expansion and variational approaches.The modal expansion approach is used to eliminate the space parameter and hence convert the problem form an optimal control of a distributed parameter system to an optimal control of a lumped parameter system.The variational approach will be employed to derive an explicit optimal control criterion.The resulting set of ordinary differential equations will be solved to determine the corresponding displacement and velocity.

Optimal control problem
The equation of motion governing the non dimensional regular plate is given by where Here ζ denotes the strain-rate damping coefficient, D denotes the effective flexural stiffness, μ is mass per unit area, Γ 1 is the electric moment due to electric field applied to piezoelectric layers, v(t) is the applied voltage to the piezoelectric patch (see [?] and the references therein for further details).The boundary conditions for the simply supported plate are given by where Ω = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b} and ∂Ω = {(x, y) : x = 0, a and y = 0, b}.The initial conditions are given by w(x, y, 0) = w 0 (x, y), w t (x, y, 0) = w 1 (x, y).(3) , where Ω t = (0, t f ) be the set of admissible controls for a given terminal time t f .Consider the performance index where μ 1 , μ 2 , and μ 3 are nonnegative weighting factors satisfying μ 1 + μ 2 ≥ 0 and μ 3 > 0. The second integral on the right hand side of ( 4) is a penalty term in the control energy.The optimal control problem is now stated as follows: Determine the optimal control v * (t) ∈ U ad so that and subject to the governing PDE (1) and the boundary and initial condition ( 2)-(3).

Control problem in modal space
Let u(x, y, t) = w(x, y, t) + φ(y)ψ(t), where φ(y) = y 3 (y−l) 6l 2 .Substituting (6) into equation ( 1) gives (7) where with new boundary and initial conditions given by u(x, y, t) = u xx (x, y, t) = 0 for x = 0, a u(x, y, t) = u yy (x, y, t) = 0 for y = 0, b u(x, y, 0) = w 0 (x, y) + φ(y)ψ(0) and the performance index becomes The optimal control problem is now restated as follows: Determine the optimal control v * (t) ∈ U ad so that and such that equations ( 7)-( 9) hold.The expansion where in which converts equation (7) to the following set of ordinary differential equations where The general solution of equation ( 15) is given by where and in which λmn = λ 2 mn 1 − λ 2 mn ζ 2 .Using initial conditions (9), we obtain where Under expansion (12), the performance index (4) becomes MATEC Web of Conferences The optimal control v * (t) in U ad is determined such that J N [v * (t)] is minimum subject to (19)-(20).The necessary condition for the control v(t) to be optimal is that δ v J N [v(t)] = 0, and therefore, That is where  Upon solving system (26) for x and y, and substituting these values into equation ( 24), an explicit solution for the optimal control v(t) is readily obtained.

Summary
Active vibration control by the application of piezoelectric materials was applied to a thermally induced simply supported rectangular plate.Through a combination of Galerkin and variational approaches, the problem was set as an optimal control problem of a lumped parameter system and then the task of determining the corresponding displacement and velocity was reduced to just solving a set of ordinary differential equations.
et al [?], discussed vibrations suppression of a thermoelastic beam subject to sudden heat input.A thermoelastic beam controlled by an active control exercised by point actuators is examined in [?].Friswell et al discussed a closed loop control to suppress the vibrations of peizo thermoelastic beam [?].In [?], numerical studies are presented for thermally induced vibrations of peizo-thermo-viscoelastic composite beam subjected to a transient thermal load using coupled finite element method.