Active Control Optimization for Minimizing the Dynamic Response of Functionally Graded Plate

A control optimization problem for functionally graded (FG) plates is presented using a first-order shear deformation plate theory including through-the-thickness normal strain effect. The aim of the optimization is to minimize the vibrational response of FG plate with constraints on the control energy used in the damping process. An active control optimization is presented to determine the optimal level of a closed loop control function. Plate thickness and a homogeneity parameter of FG plates are used as design variables. Numerical results for the optimal control force and the total energy for a simply supported FG plate are given. The influence of through-the-thickness normal strain effect on the accuracy of the obtained results is illustrated. The effectiveness of the present control and design procedure is examined.


Introduction
The rapid development in various industrial fields requires new materials that can serve useful functions under certain conditions.In recent years, functionally graded materials (FGMs) have gained considerable attention in many engineering applications, and they are considered as a potential structural material for future high-speed spacecraft and power generation industries.Several studies have been performed to analyze the behavior of functionally graded plates [1].
In the aerospace industry and many other engineering applications, the suppression of excessive vibrations occurring in large structures represents one of the most pressing and difficult problems facing structural designers.An effective means of suppressing excessive vibrations is by active structural control.Thus, there is need for new light materials possessing a high degree of flexibility and with very low natural damping.These factors motivated the development for more accurate tools of analysis and rigorous design methods [2][3][4][5][6][7].A series of publications has appeared concerning the vibration control and design optimization of composite structures.Grandhi [8] studied the structural and control optimization of space structures.Sloss et al. [9] presented an integrated approach to solve multiobjective design and control optimization for many composite structures.Adali et al. [10,11], Sloss et al. [12], Sadek et al. [13], and Fares et al. [14][15][16] have studied the optimal design and control optimization of composite laminated plates and shells subjected to thermomechanical loadings.
In another work of Fares et al. [17], nonlinear optimization schemes are presented to solve multiobjective design and control problems for composite laminated plates.Other studies on this topic may be found in [18,19].
Many studies indicate that the inclusion of the normal strain and the transverse shear effects are very important for predicting accurate vibrational responses of composite plates and shells, particularly, for those made of FGMs; see, for example, [20].In the work [20], it is showed that the neglecting of these effects may cause errors in predicting the deflections and frequencies in FG plates reaching 30%.However, most studies related to the topic of design and control optimization of composite plates and shells were carried out based on classical theories which neglect the shear deformation and normal strain effects.
The current work deals with the active control of the dynamic response of a FG orthotropic rectangular plate.The present formulation is based on a consistent first-order 2 International Journal of Aerospace Engineering plate theory including the shear deformation and throughthe-thickness normal strain effects [20].The design and control objective are to minimize the dynamic response with minimum expenditure of control energy.The total elastic energy is taken as a measure of the dynamic response.The expenditure of the control energy is limited by attaching to the control objective a functional including the control force.The necessary and sufficient condition for optimal stabilization in the Liapunov-Bellman sense [21] is used to determine the optimal control force and controlled deflections.Numerical examples are given to assess the efficiency of the present design control approach in the damping process for FG plate.

Formulation and Basic Equations
Consider an elastic plate of uniform thickness ℎ, length , and width .The plate is composed of an orthotropic elastic material varying in the thickness direction.Let a Cartesian coordinates system ( 1 ,  2 ,  3 ) be taken such that the midplane coincides with the  1  2 plane and the plate occupies the following region: Let the upper surface of the plate ( 3 = ℎ/2) be subjected to a transverse distributed load ( 1 ,  2 ,  3 ) which may be taken as a control force.
The present formulation is based on a first-order theory including a normal strain and shear deformation effects derived using a mixed variational approach.This theory preserves the transverse shear stresses which vanish on the top and the bottom of the plate surfaces; therefore, the condition of introducing shear correction factors is not required.The displacement field is taken in the form [20]: where ( 1 ,  2 , ) are the displacements along  1 ,  2 , and  3 directions, respectively, ( 01 ,  02 ,  0 ) are the displacements of a point on the mid-plane, and ( 1 ,  2 ) are the slopes in the  1  3 and  2  3 planes due to bending only (slope rotations).
The constants  and  may take the values 0 or 1 to study the effect of the normal strain on the control process.The infinitesimal strains   associated with displacements (2) are given by where the suffix (,) denotes partial differentiation with respect to the coordinate   .Moreover, the stresses are taken in the following form: (,  = 1, 2) , The governing equations of this system are determined using the dynamic modified version of the mixed variational principle of Ressiner [20]: ( International Journal of Aerospace Engineering 3 Moreover, the constitutive equations are The stress and moment resultants   ,   , and   in ( 5) and (6a) and (6b) can be expressed as (1,  3 )    3 , (,  = 1, 2, 3) and the stiffness coefficients   ,   , and   are given in Appendix A.
In the present control problem, all edges of the plate are taken simply supported (SSSS), and then, the following conditions must be satisfied at the plate edges:

The Optimal Design and Control Problem
The objective of the present control problem is to minimize the dynamic response of the plate in a specified time 0 ≤  ≤  ≤ ∞ with the minimum possible expenditure of force ( 1 ,  2 , ), when the plate is subjected to the following initial disturbances: The dynamic response of the plate is measured by a cost functional including the energy of the plate which is a function of displacements, its derivatives, and the velocity.The control force ( 1 ,  2 , ) is introduced into the formulation by taking a performance index which comprises a weighted sum of the energy (cost functional) and penalty functional involving the control energy.Then, the mathematical formulation of the control problem can be reduced to determine the optimization variable  that minimizes the following control objective: where  1 ,  2 , and  3 are positive constant weighting factors and  1 and  2 represent the strain and the kinetic energies of the plate, respectively.The functional  3 is a penalty term involving the control function  ∈  2 , where  2 denotes the set of all bounded square integrable functions on the domain of solution {0 ≤  1 ≤ , 0 ≤  2 ≤ , 0 ≤  ≤  ≤ ∞}.Note that, the functional  is a quadratic and, hence, differentiable and positive definite on the domain of solution.
It is obvious that the solution of the present optimal control problem requires finding a solution of the initial boundary-value problem ( 5), (6a), (6b), (8), and ( 9) that minimizes the functional (10).Moreover, the design optimization procedure aims to minimize the actively controlled dynamic responses using geometry and material parameters as design variables.

Solution Procedure
The displacement functions ( 01 ,  02 ,  0 ,  1 ,  2 ,  1 ,  2 ) and the mechanical load  may be taken for simply supported plate in the form: where   (), Ψ  (),   (), and   () are unknown functions of the time .Using (6a) and (6b) we can get the governing equations ( 5) in terms of the displacements ( 01 ,  02 ,  0 ,  1 ,  2 ,  1 ,  2 ).For these equations, the in-plane inertia terms are neglected.Substituting expressions (11) into the resulting equations, we get the following time equations: where   is the kronecker delta and the coefficients  1 ,  2 , Ψ 1 , Ψ 2 ,  0 ,  1 ,  2 ,   are given in Appendix B. Solving the system of ( 12), one gets an equation of the time-dependent functions   and  0 only, where Δ 1 , Δ 2 , and Δ 3 are given in Appendix B. Again, using (2), ( 3), (4), and (11) with the objective functional (10), we get where the coefficients   ( = 1, 2, . . ., 6) are given in Appendix B. Now the optimal control problem is to find the control function  opt  () that minimizes the functional (14); that is, that is, The minimization of the dynamic response using the control force  0 can be carried out using the Liapunov-Bellman method [21] which is considered an effective approach for the solution.The necessary and sufficient condition for minimization of the functional (14) provided that the Liapunov function  is a positive definite, then Substituting expressions ( 13) and ( 14) into ( 17) one gets This condition gives that Substituting the Liapunov function (18) into the previous expression, we can get the minimum control force  opt  in the form: Again, using (18) with the expression (19), we obtain Using (21) with the last expression and equating the coefficients of Ẇ2 0 ,  2 0 , and  0 Ẇ 0 by zeroes, the following system of algebraic equations is obtained: where the coefficients   ( = 1, 2, . . ., 18) are given in Appendix C. Using ( 14) and ( 21), we obtain the following differential equation: The solution of (24) for which 2Ω 2  >   is given by where   and   are unknown coefficients which may be obtained from conditions (9) as (26) Using ( 14), (21), and (25), we can get the controlled displacement, the total energy, and the optimal control force.Then we complete the minimization process by determining the optimal design of the plate using geometric and material parameters as design variables.

Homogenization of Material Properties and the Design Objective
Consider that a FG plate is made of two randomly distributed isotropic constituents (ceramic and metal) and the macroscopic response of the composite is isotropic.The matrix phase, denoted by the subscript 2, is reinforced by spherical particles as a particulate phase, denoted by the subscript 1.The homogeneity of this composition is taken to vary only in the thickness direction.The effective elastic moduli of the graded composite may be evaluated from the volume fraction of the constituents  1 and  2 .The effective mass density  at a point is given by where  + 1 and  − 1 are, respectively, the volume fractions of the particulate ceramic phase on the top and the bottom surfaces of the plate.The parameter  indicates the volume fraction profile through the thickness.In this case, the elastic coefficients   become where  is Young's modulus, ] is Poisson's ratio,  is shear modulus, and  is the effective bulk modulus.The Mori and Tanaka [22] scheme is used to estimate the effective moduli for the regions of the graded microstructure which have a well-defined continuous matrix and a discontinuous particulate phase.The locally effective bulk modulus  and the effective shear modulus  of the composite are given by The design procedure aims to determine the values of the design variable  (= /ℎ) and the index  that minimize the energy ; that is, where  opt is the optimal control force obtained from the previous control procedure.

Numerical Examples and Discussion
Numerical results are presented for the total energy  and the control force  for simply supported plates.Let the constituent materials of the FG plate be aluminum (Al) as Tables 1 and 2 contain numerical results for the controlled total energy  and the control force  opt of simply supported Al/ZrO 2 FG plate with their counterparts J and qopt which are calculated neglecting the effect of the normal strains ( =  = 0).To study the influence of the normal strain effect on the control process, the following relative errors for the control force  and the total energy  are presented in Tables 1, 2, and 3 for various values of the ratios , / and the index .In these tables, Note that the relative errors in the control force decrease with the increase of ratio , and these errors increase with increasing the index  of the volume fraction profile, so that the errors reach more than 22% for plates with / = 1 and  = 10.This means that the effect of the normal strain is weak for thin FG plates with  < 10.Moreover, the inclusion of the effect of normal strain in the formulation with the cases  > 1 is required for obtaining more accurate results.This may be explained by the fact that, with increasing the index , the homogeneity of the plate differs rapidly through the plate thickness.This requires the inclusion of the normal strain effect in the formulation to give more accurate description for the deformation process through the thickness.Table 3 contains similar numerical results as in Tables 1 and 2 for second mode  = 2,  = 1.These results show that the relative errors increase with the increase of the mode numbers  and  confirming the importance of the inclusion of the normal strain effect in the formulation of the problem to obtain more accurate results.In general, the effect of the normal strain on the accuracy of the numerical results of the total energy  is less significant than that on the control force .change of ratios , / and the index .This indicates that these design variables may play an important role in reducing the vibrational response of the FG plates.Figure 3 shows -curves and Ĵ-curves against the time parameter .These curves show that the present design control procedure reduces the expenditure of the control energy and the time of the controlling process.Figure 4 contains curves for the control force  and the total energy Ĵ against the index number  for different values of the thickness ratio .These curves confirm the previous discussion on the efficiency of the present procedure in damping process, where the use of the thickness ratio  as design variable reduces significantly the level of the total energy and decreases the consumed control energy.Figure 5 contains -curves and Ĵ-curves against the volume fraction  + 1 .Note that the design procedure over the plate thickness is more efficient than that over the volume fractional profile, while the design procedure over both  and  is the most efficient.In general the present control and design procedure strongly improves the performance of the FG plates.
International Journal of Aerospace Engineering

Conclusion
A structural and control optimization procedure is presented to minimize the vibrational response of FG plates.A firstorder plate theory including the normal strain and shear deformation effects is used.Optimal level of closed-loop control force, plate thickness, and the volume fractional index are determined.The influence of the normal strain effect on the accuracy of the obtained results is illustrated.It is found that the inclusion of the normal stain effect is required, so that neglecting the normal stain effect leads to errors reaching more than 22% for square plates with homogeneity index  > 1.The presented results show that the present control and design procedure has a high efficiency in improving the performance of the FG plates.

A. The Definition of the Parameters in (6a) and (6b)
The stiffness coefficients   ,   , and   in (6a) and (6b) are International Journal of Aerospace Engineering B. The Definition of the Parameters in (12) and ( 14)

Figure 1 :Figure 2 :
Figure 1: Effect of the thickness ratio  and the index  on curve for (a) the control force  opt , (b) for the energy Ĵ, with ℎ = 0.02 m, / = 1.

Figures 1 and 2 Figure 3 :Figure 4 :
Figure 3: Effect of the index  on curves for (a) the optimal control force  opt , (b) for the energy Ĵ, against the time , ℎ = 0.02 m and / = 1.

Figure 5 :
Figure 5: Effect of the thickness ratio  and volume fraction  + 1 on curves of the control force  opt and the energy Ĵ for SSSS plates with  = 4 and / = 1.