Optimal Control of Elastically Connected Mixed Complex Systems. Department of Mathematics and Statistics, American University of Sharjah

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1 Optimal Control of Elastically Connected Mixed Complex Systems Ismail Kucuk and Ibrahim Sadek Department of Mathematics and Statistics, American University of Sharjah Sharjah, UAE Abstract The problem of damping out the oscillations of an elastically connected rectangular plate-membrane system by point-wise actuators is considered. Mixed-type structures are widely used in many branches of modern civil, mechanical, and aerospace engineering. The basic control problem is to minimize the deflection and the velocity of displacements in a given period of time with the minimum possible expenditure of actuators. A quadratic performance is chosen as the cost functional which comprises the functionals of the defection, velocity and the distributed actuators of the two coupled structure. Necessary conditions of optimality are obtained from a variational approach and formulated in the form of independent integral equations which lead to explicit expressions for the point-wise actuators. The results are applied to a specific problem, and numerical results are presented to demonstrate the effectiveness of the proposed control mechanism. The numerical results are obtained using Maple. Keywords Optimal Control, Plate, Membrane, Integral Equations, Elasticity 3 Introduction Mechanical structures are modeled as onedimensional simple continuous systems such as a string and beam or two dimensional simple continuous systems such as membrane and plate. The simple models are used in complex continuous systems such as an elastically connected double-solid system that consists of two elastics solids bounded continuously through a Winkler elastic layer []. The vibration analysis of complex continuous systems is one of the tools used in civil and mechanical engineering for theoretical and practical applications of the problems raising in engineering. Complex continuous systems of special types are studied in depth in [, 5] for transverse vibration theory. Our interest in this study is motivated by the problems considered in [4, 5]. In these papers, the control of vibrations in an elastically connected double-beam system is investigated. An analytical method based on the maximum principle in [6] is applied to the double beam system in [4, 5]. The method consists of the basic control problem of minimizing a weighted sum of objective functional under given constraints, where the maximum principle is used to solve an initial-terminal-boundaryvalue problem in a system of partial differential equations [6]. In this paper, we analyze the free transverse vibrations of plate-membrane complex twodimensional continuous system from the optimal control point of view. We suggest a new system where we include actuators as control parameters in order to prevent any unwanted resonance. The optimally controlled system is observed trough the performance index functional that consists of control actuators. Necessary conditions of optimality leads us to coupled non-homogeneous Fredholm integral equations with degenerate kernel. After decoupling the integral equations, we obtain 4N 4N system of linear equations that results with the optimal actuators since the variational approach is considered. The advantage of using variational approach is that one needs to solve system of linear equations rather than solving a partial differential equation. The proposed approach is illustrated by a numerical example in which the two discrete control actuators are applied to an elastically connected rectangular plate-membrane system. Numerical results indicate that the dynamical response of the mixed system can be reduced substantially in a given period of time. This can be achieved by selecting a suitable value of the membrane tension and applying an actuator in the domain of the plate. 4 Formulation of the Problem Free transverse vibrations of an elastically connected rectangular plate-membrane see Figure

2 with discretely distributed actuators are described by the Kirchhoff-Love plate theory in the following form of differential equations [3]: f t p p p p f t x, y x, y k f t p p 4 p p x, y p p x,y x, y 4 4 y N wx,y,t f t 3 f t 5 Figure : An elastically connected rectangular plate-membrane complex system with actuators. m ẅ + D w + kw w f j δx x p j, y yp j m ẅ N w + kw w f j δx x m j, y yj m where w i w i x, y, t is the transverse plate membrane displacement; x p j and x m j are from [, a], and y p j and yj m are from [, b]; f ij t L [, t f ] is distributed control actuators; D is the flexural rigidity of the plate; E is Young s modulus of the elastic plate; k is the stiffness modulus of a Winkler elastic layer; The subscripts i and are referring to plate and membrane, respectively, and t f is terminal time. D E h 3 γ, m i ρ i h i, ẇ i w i t, i,. The boundary conditions for the simply supported plate and membrane are introduced as: w i, y, t w i a, y, t w x,, t w i x, b, t, x 3a w,y,t x w a,y,t x 3b w x,,t y w x,b,t y. 3c The initial conditions are taken as: w i x, y, w i x, y, ẇ i x, y, v i x, y i,. The performance index function is taken as J f,..., f N, f,..., f N J F 4 where J F b a { µ w x, y, t f + µ ẇ x, y, t f + µ 3 wx, y, t f + µ 4 ẇx, y, t f dxdy+ tf N ɛ i fi t + N α i fi t dt i i Problem: that 5 Find an optimal f ij L [, t f ] such J F t J F t, f ij t L [, t f ] subject to Solution method of vibration problem The equations in and with boundary conditions in 3 can be solved by the eigenfunction expansions; assuming the solutions in the form of w x, y, t w x, y, t Φ x, y {{ ϕ m xψ n y T t 6 Φ x, ytt ϕ m xψ n ytt 7 Φ x, ytt where T t and T t are unknown time functions. Complete orthonormal sets of eigenfunctions of the operator Lw w x + w y 8

3 are obtained as { { ϕ m x m a sina mx m { ψ n y n { b sinb ny n 9 where a m a mπ; m,,... and b n b nπ; n,,... If one substitutes the solutions 6 and 7 into and, we obtain the following second order differential equation in time with the initial conditions given in 4 as follows T t + Γ T t Γ T t F t, T t + Γ T t Γ T t F t. The abbreviations introduced in equations are also used hereafter that are defined as Γ D k + k ; Γ m N k + k ; m Γ i k m i, i, ; F t m F t m k a m + b n f j tφ x p j, yp j, f j tφ x m j, yj m. To solve the system of second order differential equations, we transform it into first order system of differential equations: where Y y y y 3 y 4 dy dt AY + F t ; A Γ Γ ; Γ Γ F t F t. F t Finally, the solutions of system of second order differential equations of time in can be written as where X i T t T t T t T t b j X j t, b j X j t, 3 b 3j X j t, 4 b 4j X j t. 5 t t e λit e λis G i sds + c i e λit 6 where c is are constants to be determined by the proper form of initial conditions, and Gt B F t. Here the colus of B are the eigenvectors of A, and λ i s are the eigenvalues of A. Finally, the deflections in plate and membrane are obtained as w x, y, t w x, y, t Φ x, y Φ x, y 6 Necessary Conditions b j X j t, b 3j X j t. 7 First, we observe the following fact with help of orthonormality of the eigenfunctions b a [ wi x, y, t N f dxdy T i t f ] 8 Solutions of and in 7, respectively, and the obtained result in 8 rewrite the performance index as J F N µ i i b j X j t f + ɛi fi t + α ifi t dt i 9

4 and first variation of 9 becomes δ fk { N µ i i b j X j t f b ij e λjt f s G j Φ x p k, yp k f k + ɛ k f k s f k sds. Since is true for all variations of f k, we observe the following for fixed k,..., N [ N µ i i { b ij q [ e λjt f r G j N q f q rφ x p q, y p q N ] + G j f q rφ x m q, ym q ]dr + c j e λjt f [ N b ij e λjt f s G j Φ x p k, yp k ]+ ɛ k f k s where G j and G j are the terms of the coefficient matrices, and x p i, yp i and xm i, ym i are the points at which the control actuators are located for plate and membrane, respectively. We obtain the similar result from the variation of J with respect to f k for fixed k,..., N. After the variations we end up with coupled nonhomogeneous Fredholm integral equations with degenerate kernel. q i sf qr+ i i sf qr q e λit f r dr+ i i, sf qr If we define c q i and cq i P k s + ɛ k f k s, i sf qr+ e λit f r dr+ P k s + σ k f k s. a b as the integrals in, then s become q i q i c q i e λit f r f q rdr, c q i e λit f r f q rdr, i scq i + Kqk i scq i + P k s + ɛ k f k s, i scq i + Lqk i scq i + P k s + σ k f k s. 3a 3b 4a 4b If we multiply all terms of 4 by e λ lt f s and integrate from to t f, one obtains a system of linear equations in c q i and cq i : q i q i a lq ik cq i + blq ik cq i + h l k + ɛ k c l k, a lq ik cq i + blq ik cq i + h l k + ɛ k c l k, where k N and l 4, and a lq ik a lq ik h l k b lq ik b lq ik i seλ lt f s ds, ise λ lt f s ds, P k se λ lt f s ds, i seλ lt f s ds, ise λ lt f s ds, h l tf k P k seλ lt f s ds. 5a 5b 5 can be written in the following compact form of system of linear equations: A + IE C + A C + H O, B C + B + ISC + H O. 6

5 Here in 6, we obtain a system of linear equations for the unknown C and C that are defined as 4N colu block matrices. The matrices used in 6 can be written explicitly, and solutions are given as C A + IE A C 4n p B B + IS 4n p 4n p H H 4n p After finding C and C, we can compute f q and f q by substituting 3 into 4: f k ɛ k q i f k σ k q i N { P k s+ i scq i + Kqk i scq i N where k,..., N. N { P k s+ i scq i + Lqk i scq i Before Control After Control Reduction % N/A a 7b The initial conditions given in 8 allow us to study the behavior of the fundamental mode of the complex mixed continuous system. The values of the parameters characterizing the physical and geometrical properties of the plate-membrane system from reference [3] are used in the numerical calculations: a, b, k 4, E 8, h., h 4 3, m 5, m, t f 5, γ.3, ρ 5 3, ρ.5. In Table, the effect of the forces acting on both plate and membrane at the points x p, yp.3, and x m, ym.7,.4 are presented. It is observed from the Table that the system achieves a substantial reduction in the energy when only one controller is applied to plate with N. The system is uncontrollable when the two controllers are applied on both plate and membrane, or only the membrane. It is also observed that the changing of the membrane tension N has an evident influence on the frequencies of the system and can be used to suppress excessive vibration amplitudes, whilst other geometrical and physical parameters of the system can remain unchanged. For one actuator applied on the plate with membrane tension N, Figure and Figure 3 show that the peak displacement of the plate and membrane is reduced substantially as compared to the uncontrolled plate and membrane, respectively. It follows from Figure 4 that varying N gives different vibration reductions in the plate as well as in the membrane..5 w N Table : Energy in the system & N 7 Numerical Example To illustrate theoretical considerations presented, the behavior of uncontrolled and controlled platemembrane system is investigated. Moreover, the effect of various problem parameters on the control of the motion in the plate-membrane system are discussed. For the simplicity of the analysis, it is assumed that the plate-membrane system is subjected to the initial conditions 4 of the form : t After Control Before Control w i x, y, Φ x, y, ẇ i x, y, i,. 8 Figure : The deflection w of the plate when µ µ, and µ 3 µ 4. is obtained for uncontrolled and control cases.

6 w N t After Control Before Control Figure 3: The deflection w of the membrane when µ µ, and µ 3 µ 4. is obtained for uncontrolled and control cases. w t t N N N Figure 4: The deflection w of the plate after the control is obtained for different N s when µ µ, and µ 3 µ Conclusion In this paper, control vibrations of an elastically connected rectangular plate-membrane system is solved analytically. The vibratory system model is composed of a thin plate, a massless elastic layer medelled as a homogeneous Winkle-type foundation, and a parallel membrane stretched uniformly by suitable constant tensions applied at the edges. A method is proposed to damp the undesirable vibrations in the structure actively by means of control actuators applied in the domain of the structures. Upon the use of the eigenfunctions expansion method the control of a distributed parameter system is reduced to the control of a lumped parameter system. Necessary conditions of the optimal control lumped parameter system are derived in the from of degenerated integral equations which lead to explicit expressions for the point-wise controllers. A numerical example is presented to demonstrate the effectiveness of the method. Numerical results indicate that the dynamic response of the mixed system can be reduced drastically in a given period of time. It is also noted that the reduction in the dynamic response is a function of the membrane tension N. Moreover, the system is optimally controllable when an actuator is applied on the plate for a chosen N and an appropriate actuator location. Since we do not seek the optimal solution, we are satisfied with the reduction achieved in the energy. The obtained system is a sub-optimal project []. References [] A. V. Cherkaev and I. Kucuk. Detecting stress fields in an optimal structure part I: Two-dimensional case and analyzer. Structural and Multidisciplinary Optimization, 6-: 5, January 4. [] Z. Oniszczuk. Vibration analysis of the compound continous systems with elastic constraints. Technical report, Publishing house of Rzeszow University of Technnology, 997. In Polish. [3] Z. Oniszczuk. Free transverse vibrations of an elastically connected rectangular platemembrane complex sytem. Journal of Sound and Vibration, 64:37 47, 3. [4] I. Sadek, T. Abualrub, and M. Abukhaled. Optimal boundary control of systems of elastically connected paralle beams. Dynamic of Continous, Discrete and Impulsive Systems. To appear. [5] I. Sadek, M. Abukhaled, and T. Abualrub. Optimal pointwise control for a paralel system of euler-bernoulli beams. Journal of Computational and Applied Mathematics, 37:83 95,. [6] J. M. Sloss, J. B. Jr, and I. Sadek. A maximum principle for non-conservative self-adjoint systems. IMA Journal of Mathematical Control and Information, 6:99 6, 989.