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

Authors: M. E. Fares; M. Kh. Elmarghany; Doaa Atta
Journal: International Journal of Aerospace Engineering
 (2014)

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## Abstract

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.


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## Body

## 1. 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 [2–7]. 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. [14–16] 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 plate theory including the shear deformation and through-the-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.


## 2. Formulation and Basic Equations

Consider an elastic plate of uniform thicknessh, length a, and width b. The plate is composed of an orthotropic elastic material varying in the thickness direction. Let a Cartesian coordinates system (x1,x2,x3) be taken such that the mid-plane coincides with the x1x2 plane and the plate occupies the following region:
(1)0≤x1≤a,0≤x2≤b,-h2≤x3≤h2.Let the upper surface of the plate (x3=h/2) be subjected to a transverse distributed load q(x1,x2,x3) 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]:
(2)u1=u01+ψ1x3,u2=u02+ψ2x3,w=w0+αw1x3+βw2x32,
where (u1, u2, w) are the displacements along x1, x2, and x3 directions, respectively, (u01, u02, w0) are the displacements of a point on the mid-plane, and (ψ1, ψ2) are the slopes in the x1x3 and x2x3 planes due to bending only (slope rotations). The constants α and β may take the values 0 or 1 to study the effect of the normal strain on the control process.The infinitesimal strainsεij associated with displacements (2) are given by
(3)εii=u0i,i+ψi,ix3,εi3=ψi+w0,i+αw1,ix3+βw2,ix32,(i=1,2),ε33=αw1+2βw2x3,ε12=(u01,2+u02,1)+(ψ1,2+ψ2,1)x3,
where the suffix (,i) denotes partial differentiation with respect to the coordinate xi.Moreover, the stresses are taken in the following form:(4)σij=Nijh+12Mijh3x3,σi3=3Qi32h[1-(x3h/2)2],(i,j=1,2),σ33=(3N332h+30M33h3x3)[1-(x3h/2)2]+q14[(x3h/2)2-(x3h/2)-2](1+x3h/2)+q24[(x3h/2)2+(x3h/2)-2](1-x3h/2).The governing equations of this system are determined using the dynamic modified version of the mixed variational principle of Ressiner [20]:
(5)δu0i:Nii,i+Nij,j=I1u¨0i+I2ψ¨i,δψi:Mii,i+Mij,j-Qi3=I2u¨0i+I3ψ¨i,(i,j=1,2,i≠j),δw0:Q13,1+Q23,2+q=I1w¨0+αI2w¨1+βI3w¨2,δw1:hq-N33=I2w¨0+αI3w¨1+βI4w¨2,δw2:h2(Q13,1+Q23,2)20-2M33+9h2q20=I3w¨0+αI4w¨1+βI5w¨2.Moreover, the constitutive equations are(6a){N11N22N33N12M11M22M33M12}=[A11A12A130B11B12B130A12A22A230B12B22B230A13A23A330B-13B-23B330000A66000B66B11B12B-130D11D12D130B12B22B-230D12D22D230B13B23B330D13D23D330000B66000D66]-1[u01,1-F13u02,2-F23αw1-F33u01,2+u02,1ψ1,1-K13ψ2,2-K232βw2-K33ψ1,2+ψ2,1](6b)Qi3=1Akk[ψi+w0,i+(βh2w2,i20)+Hi],(i=1,2;k=i+(4i)).The stress and moment resultantsNij, Mij, and Qij in (5) and (6a) and (6b) can be expressed as
(7)(Nij,Mij)=∫-h/2h/2(1,x3)σijdx3,(i,j=1,2,3)(Qk3)=∫-h/2h/2σk3dx3,(k=1,2),
and the stiffness coefficients Aij, Bij, and Dij 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:(8)u01=w0=w1=w2=ψ2=N11=M11=0,x1=0,au02=w0=w1=w2=ψ1=N22=M22=0,x2=0,b.


## 3. 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 time0≤t≤τ≤∞ with the minimum possible expenditure of force q(x1,x2,t), when the plate is subjected to the following initial disturbances:
(9)w(x1,x2,0)=A-(x1,x2),w˙(x1,x2,0)=B-(x1,x2).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 forceq(x1,x2,t) 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 q that minimizes the following control objective:
(10)J=μ1J1+μ2J2+μ3J3,J1=12∫0∞∫0b∫0a∫-h/2h/2εijσijdx3dx1dx2dt,(i,j=1,2,3),J2=12∫0∞∫0b∫0a∫-h/2h/2ρ(u˙12+u˙22+w˙2)dx3dx1dx2dt,J3=∫0∞∫0b∫0aq2(x1,x2,t)dx1dx2dt,
where μ1, μ2, and μ3 are positive constant weighting factors and J1 and J2 represent the strain and the kinetic energies of the plate, respectively. The functional J3 is a penalty term involving the control function q∈L2, where L2 denotes the set of all bounded square integrable functions on the domain of solution {0≤x1≤a,0≤x2≤b,0≤t≤τ≤∞}. Note that, the functional J 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.


## 4. Solution Procedure

The displacement functions (u01, u02, w0, ψ1, ψ2, w1, w2) and the mechanical load q may be taken for simply supported plate in the form:
(11)(u0i,ψi)=∑m,n(Uimn(t),Ψimn(t))Xi,iXi+1,(i=1⟶i+1=2,i=2⟶i+1=1)(wj,q)=∑m,n(Wjmn(t),qmn(t))X1X2,(j=0,1,2)X1(x1)=sinμmx1,μm=mπa,X2(x2)=sinλnx2,λn=nπb,
where Uimn(t), Ψimn(t), Wjmn(t), and qmn(t) are unknown functions of the time t.Using (6a) and (6b) we can get the governing equations (5) in terms of the displacements (u01, u02, w0, ψ1, ψ2, w1, w2). For these equations, the in-plane inertia terms are neglected. Substituting expressions (11) into the resulting equations, we get the following time equations:
(12)U1iU1mn+U2iU2mn+Ψ1iΨ1mn+Ψ2iΨ21mn+W0iW0mn+W1iW1mn+W2iW2mn=W¨0mnW-i(δi3+δi6+δi7)+Qiqmn,(i=1,2,…,7),
where δij is the kronecker delta and the coefficients U1i, U2i, Ψ1i, Ψ2i, W0i, W1i, W2i, Qi are given in Appendix B. Solving the system of (12), one gets an equation of the time-dependent functions qmn and W0mn only,
(13)W¨0mn+ω2W0mn=lq0,ω2=Δ1Δ3,l=Δ2Δ3,
where Δ1, Δ2, and Δ3 are given in Appendix B.Again, using (2), (3), (4), and (11) with the objective functional (10), we get
(14)J=∑mn∫0∞J-mndt,J-mn=K1W0mn2+K2W0mnqmn+K3qmn2+K4W˙0mn2+K5W˙0mnq˙mn+K6q˙mn2,
where the coefficients Ki (i=1,2,…,6) are given in Appendix B.Now the optimal control problem is to find the control functionqmnopt(t) that minimizes the functional (14); that is,
(15)J(qmnopt)≤J(qmn),∀qmn(t)∈L2([0,∞));
that is,
(16)min⁡qmnJ=min⁡∑Jmn=∑min⁡qmn∈L2J.The minimization of the dynamic response using the control forceq0 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) in the Liapunov-Bellman sense is
(17)min⁡q[∂V-∂W0mnW˙0mn+∂V-∂W˙0mnW¨0mn+J-]=0;
provided that the Liapunov function V- is a positive definite, then
(18)V-=AW0mn2+2BW0mnW˙0mn+CW˙0mn2,A>0,C>0,AC>B2.
Substituting expressions (13) and (14) into (17) one gets
(19)min⁡q[∂V-∂W0mnW˙0mn+∂V-∂W˙0mn(lqmn-ω2W0mn)+K1W0mn2+K2W0mnqmn+K3qmn2∂V-∂W˙0mn+K4W˙0mn2+K5W˙0mnq˙mn+K6q˙mn2]=0.
This condition gives that
(20)l∂V-∂W˙0mn+K2W0mn+2K3qmnopt=0.Substituting the Liapunov function (18) into the previous expression, we can get the minimum control force qmnopt in the form:
(21)qmnopt=-12K3(2Bl+K2)W0mn-1K3ClW˙0mn.
Again, using (18) with the expression (19), we obtain
(22)min⁡q[(2AW0mnW˙0mn+2BW˙0mn2)+(2BW0mn+2CW˙0mn)(lqmn-ω2W0mn)+K1W0mn2+K2W0mnqmn+K3qmn2(2AW0mnW˙0mn+2BW˙0mn2)+K4W˙0mn2+K5W˙0mnq˙mn+K6q˙mn2]=0.Using (21) with the last expression and equating the coefficients of W˙0mn2, W0mn2, and W0mnW˙0mn by zeroes, the following system of algebraic equations is obtained:
(23)C2(a1B2+a2B+a3)+a4B2+a5B+a6=0,C2(a7C2+a8B+a9)+a10B2+a11B+a12=0,a13A+C(a14C2+a15BC2+a16B2+a17B+a18)=0,
where the coefficients ai (i=1,2,…,18) are given in Appendix C.Using (14) and (21), we obtain the following differential equation:
(24)W¨0mn+αmnW˙0mn+Ωmn2W0mn=0,αmn=-lc2,Ωmn2=(ω2-lc1),c1=-(2Bl+K2)(2K3),c2=-ClK3.
The solution of (24) for which 2Ωmn2>αmn is given by
(25)W0mnopt=e-αmnt/2[βmncos⁡(υmnt)+γmnsin(υmnt)],υmn=Ωmn2-αmn4,
where βmn and γmn are unknown coefficients which may be obtained from conditions (9) as
(26)γmn=(αmnβmn+2A-mn)(2υmn),(βmn,A-mn)=4abω2∫0b∫0a(A-,B-)X1X2dx1dx2.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.


## 5. 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 constituentsV1 and V2. The effective mass density ρ at a point is given by
(27)ρ=ρ1V1+ρ2V2,V1=V1-+(V1+-V1-)(12+x3h)p,
where V1+ and V1- are, respectively, the volume fractions of the particulate ceramic phase on the top and the bottom surfaces of the plate. The parameter p indicates the volume fraction profile through the thickness. In this case, the elastic coefficients Cij become
(28)C11=C22=C33=1E,C12=C13=C23=-νE,C44=C55=C66=1G,K=E3(1-2ν),G=E2(1+ν),
where E is Young’s modulus, ν is Poisson’s ratio, G is shear modulus, and K 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 K and the effective shear modulus G of the composite are given by
(29)K-k1k2-k1=V21+(1-V2)((k2-k1)/(k1+(4/3)G1)),G-G1G2-G1=V21+(1-V2)((G2-G1)/(G1+f1)),f1=G1(9k1+8G1)(6k1+12G1).The design procedure aims to determine the values of the design variabler (=a/h) and the index p that minimize the energy J; that is,
(30)J^(qopt,ropt,popt)=min⁡r,pJ(qopt,r,p),
where qopt is the optimal control force obtained from the previous control procedure.


## 6. Numerical Examples and Discussion

Numerical results are presented for the total energyJ and the control force q for simply supported plates. Let the constituent materials of the FG plate be aluminum (Al) as a matrix phase and zirconia as particulate phase with material and geometrical properties given by
(31)ZrO2:E1=200GPa,ν1=0.3,ρ1=5700kg/m3,Al:E2=70GPa,ν2=0.3,ρ2=2702kg/m3,a=b=0.5m,h=0.02m,μ1=μ2=μ4=1,μ3=0.1,A-(x1,x2)=0.2X1(x1)X2(x2),B-(x1,x2)=0,q-opt=a3qopt(E2h3),T=(hta2)E2ρ2.Tables1 and 2 contain numerical results for the controlled total energy J and the control force qopt of simply supported Al/ZrO2 FG plate with their counterparts J~ and q~opt which are calculated neglecting the effect of the normal strains (α=β=0). To study the influence of the normal strain effect on the control process, the following relative errors for the control force q and the total energy J are presented in Tables 1, 2, and 3 for various values of the ratios r, a/b and the index p. In these tables,
(32)q*=[(qopt-q~opt)qopt],J*=[(J-J~)J].Table 1
Comparison ofqopt, q~opt, J, and J~ of simply supported square Al/ZrO2 FG plates with m=n=1.
p
r
q
-
opt
q
~
opt
J
J
~
q
*
%
J
*
%
1
5
0.0312
0.0295
0.1391
0.1401
5.5
0.7
10
3.1358
3.0909
8.1959
8.2149
1.4
0.2
15
43.401
43.133
103.64
103.79
0.61
0.1
20
247.45
246.62
662.64
663.59
0.33
0.1
4
5
0.0344
0.0286
0.1062
0.1072
16.8
0.9
10
3.0272
2.8914
6.1625
6.1720
4.5
0.2
15
39.992
39.259
78.319
78.362
1.8
0.05
20
221.95
219.87
499.76
500.30
0.9
0.1
10
5
0.0367
0.0284
0.0948
0.09580
22.6
1.0
10
3.0335
2.8507
5.5484
5.5533
6.0
0.1
15
39.201
38.246
70.938
70.456
2.4
0.01
20
214.70
212.04
452.59
453.06
1.2
0.1Table 2
Comparison ofqopt, q~opt, J, and J~ of simply supported Al/ZrO2 FG plates with m=n=1, a/h=5.
p
a
/
b
q
-
opt
q
~
opt
J
J
~
q
*
%
J
*
%
1
1
0.0312
0.0295
0.1391
0.1401
5.5
0.7
2
0.062
0.059
0.014
0.014
4.5
1.33
3
0.021
0.020
0.003
0.003
4.8
1.73
4
1
0.344
0.286
0.106
0.107
16.8
0.94
2
0.069
0.060
0.011
0.011
11.9
2.64
3
0.023
0.021
0.003
0.003
10.1
2.28
10
1
0.367
0.284
0.095
0.096
22.7
1.06
2
0.071
0.059
0.009
0.009
15.9
2.85
3
0.024
0.021
0.002
0.002
11.7
2.84Table 3
Comparison ofqopt, q~opt, J, and J~ of simply supported square Al/ZrO2 FG plates with m=2,n=1.
p
a
/
h
q
-
opt
q
~
opt
J
J
~
q
*
%
J
*
%
1
5
0.0762
0.0599
0.0279
0.0282
21.5
1.3
10
5.9923
5.6012
1.3429
1.3489
6.5
0.5
15
88.508
85.897
14.884
14.912
3.0
0.2
4
5
0.1399
0.0603
0.0220
0.0223
56.9
1.7
10
6.5968
5.3641
1.0158
1.0205
18.7
0.5
15
81.835
80.583
11.203
11.201
8.5
0.02
10
5
0.2549
0.0597
0.0196
0.0199
76.6
1.7
10
7.0887
5.3134
0.9099
0.9134
25.0
0.4
15
89.668
79.529
10.088
10.070
11.3
0.2Note that the relative errors in the control force decrease with the increase of ratior, and these errors increase with increasing the index p of the volume fraction profile, so that the errors reach more than 22% for plates with a/b=1 and p=10. This means that the effect of the normal strain is weak for thin FG plates with r<10. Moreover, the inclusion of the effect of normal strain in the formulation with the cases p>1 is required for obtaining more accurate results. This may be explained by the fact that, with increasing the index p, 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 m=2, n=1. These results show that the relative errors increase with the increase of the mode numbers m and n 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 J is less significant than that on the control force q.Figures1 and 2 contain curves for the controlled force q and the total energy J^ (=100J) versus the ratios r and a/b for different values of index p for SSSS plates. These curves show that the volume fraction profile through the plate thickness has a significant effect on the control force and controlled energy, particularly for FG plates with r≥15 and square plates. Moreover, the curves show the high sensitivity of the total energy and control force of the FG plates versus the change of ratios r, a/b and the index p. This indicates that these design variables may play an important role in reducing the vibrational response of the FG plates. Figure 3 shows q-curves and J^-curves against the time parameter T. 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 q and the total energy J^ against the index number p for different values of the thickness ratio r. These curves confirm the previous discussion on the efficiency of the present procedure in damping process, where the use of the thickness ratio r as design variable reduces significantly the level of the total energy and decreases the consumed control energy. Figure 5 contains q-curves and J^-curves against the volume fraction V1+. 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 r and p is the most efficient. In general the present control and design procedure strongly improves the performance of the FG plates.Effect of the thickness ratior and the index p on curve for (a) the control force q-opt, (b) for the energy J^, with h=0.02 m, a/b=1.
(a)
(b)Effect of the aspect ratioa/b and the index p on curves for (a) the control force q-opt, (b) for the energy J^, with h=0.02 m, r=10.
(a)
(b)Effect of the indexp on curves for (a) the optimal control force q-opt, (b) for the energy J^, against the time T, h=0.02 m and a/b=1.
(a)
(b)Effect of the thickness ratior and the index p on curves of the control force q-opt and the energy J^ for SSSS plates with a/b=1.
(a)
(b)Effect of the thickness ratior and volume fraction V1+ on curves of the control force q-opt and the energy J^ for SSSS plates with p=4 and a/b=1.
(a)
(b)


## 7. Conclusion

A structural and control optimization procedure is presented to minimize the vibrational response of FG plates. A first-order 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 indexp>1. The presented results show that the present control and design procedure has a high efficiency in improving the performance of the FG plates.



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