Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 3, pp. 769-784, Warsaw 2012 50th Anniversary of JTAM OPTIMAL VIBRATION CONTROL OF CONICAL SHELLS WITH COLLOCATED HELICAL SENSOR/ACTUATOR PAIRS H. Li, S.D. Hu, H.S. Tzou Zhejiang University, School of Aeronautics and Astronautics, StrucTronic Systems and Control Lab, Zhejiang, P.R. China e-mail: hstzou@zju.edu.cn; Lhlihua@gmail.com Z.B. Chen Harbin Institute of Technology, School of Mechatronics Engineering, Harbin, P.R. China This paper focuses on the optimal vibration control of clamped-free conical shells using di- stributed helical piezoelectric sensor/actuator (S/A) pairs. Based on the independentmodal space control, the response of conical shell to external excitations is representedby the sum- mation of all participating natural modes and their respective modal participation factors, and eachmode canbe controlled independently. Themodal equation is transformed into the linear state space form.The linear quadratic (LQ) controllers are designed for each indepen- dent mode. The optimal gainmatrix is related to the ratio G∗ between the control voltage and sensing signal by the modal control force per unit voltage B2 and the sensing signal per unit displacement C1. Because B2 and C1 change with locations of the S/A pair, the optimal control effects, modal control forces and corresponding optimal control voltages are evaluated using two S/A pairs at different locations. The results indicate that the optimal control method is effective in vibration control of the shell. The optimal control effect also depends on the location of the S/Apair andmodal shapes aswell as themodal control force and input voltage. Key words: smart structure, conical shell, diagonal sensor/actuator, optimal vibration control 1. Introduction Thin plates and shells laminated with active sensing and actuation layers regulated by con- trol electronics are common configurations of smart structures and structronic systems. The laminated thin shells have desired characteristics, such as precision manipulation, light weight, highly integrated, size/energy savings, etc. Among commonly used smart materials, piezoelec- tricmaterials are widely used in laminated smart structures for vibration control, shape control, structural diagnosis, energy harvesting and so on (Chai et al., 2004, 2006; Tzou et al., 2003; Tzou and Fu, 1994). In vibration control applications, piezoelectric layers are attached to the substructure andwork on both direct and converse piezoelectric effects. In the direct mode, the piezoelectric patch performs as a sensor, and the sensing signals can be used for the controller. With the converse effect, the piezoelectric patches perform as actuators, and they generate the control force for structural actuation and control (Tzou, 1993). The conical shell iswidely used innozzles, hones, aerospace structures, submarines; for exam- ple, spacecraft adapters and noses of rockets and missiles. In these applications, conical shells are subjected to severe dynamic loads and, thus, active vibration control is necessary to protect the payload or guarantee shape precision of the shell structure. Dynamic and elastic behavior of conical shells have been investigated for decades. Leissa (1993) reviewed and summarized earlier work. Studies of vibration behavior of conical shells and conical panels as well as the 770 H. Li et al. multi-layered conical shells were carried out using the kernel particle (KP) Ritz method (Liew et al., 2005). Piezoelectric materials are prevalent for distributed sensing and active control of shells. The sensing signals and control actions of piezoelectric sensors and actuators are related to the location, size, shape and contributing modes. And certain vibration modes cannot be controlled by using fully distributed sensors and actuators due to the lack of observability and controllability (Tzou, 1993). The optimal control technology, as an effective control algorithm, has been adapted to the control of beamsandplates (Chellabi et al., 2009;ChenandShen, 1997;ToandChen, 2007). But the applications to distributed shell structures are still not well explored. Tzou andDing (2004) investigated the optimal control of precision paraboloidal shell structronic systems. A linear quadratic optimal state feedback controller was designed to suppress the transverse vibrations using the independent modal control method. Ray (2003) reported the active vibration control of a shell panel using the optimal control algorithm, and found that the piezoelectric actuators need higher control voltages as the panel width (central angle) decreases to achieve equivalent control effects. This study focuses on the optimal vibration control of conical shells using surfaced laminated revolving helical sensor/actuator pairs. Thin helical-shape sensors and actuators are collocated on both surfaces of the conical shells. This helical arrangement is used to avoid the observation of spillover. The equations of motion are given for conical shell first, followed by the derivation of modal vibration equations, and then transformed into the first-order linear state space form. The modal participation factor and its time derivative are chosen to be the state vector; the sensing signal andmodal control force are chosen as the system output vector and control input respectively. TheLQ controller is designed for each independentmode.The optimal gainmatrix is evaluated byminimizing the performance criterion function. Then case studies are performed to evaluate the control effectiveness. 2. Dynamics and actuation of conical shells Revolving helical distributed piezoelectric sensors and actuators (S/A) are laminated on the conical shell surface. In the control procedure, the dynamic response is detected by piezoelectric sensors, and the signals are subsequently used as the control input. The controller estimates the required control voltage based on the vibrational signal and then generates a control signal to the power amplifier (PA). The PA generates the accurate voltage to actuate the piezoelectric actuator. Thus, the vibration characteristics should be investigated before the controller design in the control procedure. It is assumed that the truncated conical shell of an revolution is made of isotropic material, and the shell is thin so that the Kirchhoff-Love assumptions are applicable (Tzou, 1993). Also, since the S/A layers are much thinner than the shell structure, their mass and elasticity are negligible, and only the piezoelectric sensing/actuation effects are considered. Figure 1 shows the conical shellmodel inwhich the tri-orthogonal coordinate system (x,ψ,3) locates on the neutral surface of the shell, where x is the longitudinal coordinate, ψ the circumferential coordinate, and α3 the thickness direction. The shell is defined from x1 (minor end) to x2 (major end) in the x coordinate, and from – h/2 to h/2 in the α3 coordinate, where h is shell thickness. The hatched area represents the helical piezoelectric S/A strips. The conical shell has semi-vertex angle β∗, the sensor stripe has thickness hS and the actuator has thickness ha. The equations of motion for a thin circular conical shell in linear vibration are given as (Soedel, 2004) Optimal vibration control of conical shells... 771 Fig. 1. Definition of conical shell with helical piezoelectric sensor/actuator ∂Nxx ∂x + 1 xsinβ∗ ∂Nψx ∂ψ + 1 x (Nxx −Nψψ)+qx = ρh ∂2ux ∂t2 ∂Nxψ ∂x + 2 x Nψx + 1 xsinβ∗ ∂Nψψ ∂ψ + 1 xtanβ∗ Qψ3+qψ = ρh ∂2uψ ∂t2 ∂Qx3 ∂x + 1 x Qx3+ 1 xsinβ∗ ∂Qψ3 ∂ψ + 1 xtanβ∗ Nψψ + q3 = ρh ∂2u3 ∂t2 (2.1) And the transverse shear forces Qi3 are defined as Qx3 = ∂Mxx ∂x + Mxx x + 1 xsinβ∗ ∂Mψx ∂ψ − Mψψ x Qψ3 = ∂Mxψ ∂x + 2 x Mψx + 1 xsinβ∗ ∂Mψψ ∂ψ (2.2) where Nij and Mij are the membrane forces and the bending moments respectively; ρ is the shellmassdensity; qi is themechanical load; ui is thedisplacement response in the i-th direction and i = x,ψ,3. In the vibration analysis of conical shells, themodal expansionmethod is used to synthesize the dynamic response using known mode shape functions. The amount of each modal parti- cipation in the total dynamic response is defined by a modal participation factor. The total dynamic response can be represented by the summation of all participating natural modes and their respective modal participation factors (Tzou, 1993) ui(x,ψ,t) = ∞∑ m=1 ηm(t)Uim(x,ψ) (2.3) where the subscript i = x,ψ,3 denotes the coordinate; ηm(t) is the modal participation factor; Uim(x,ψ) is the modal shape function of the m-th mode; and the subscript m = 1,2, . . . of a shell continuum. Therefore, imposing the modal orthogonality, transforming shell Eqs. (2.1) into themodal coordinate and adding themodal viscous damping yields the independentmodal vibration equation η̈m +2ζmωmη̇m +ω 2 mηm = Fm(t) (2.4) 772 H. Li et al. where ωm is the m-th natural frequency; ζm is the modal damping ratio; Fm(t) is the modal force which can be decomposed into the mechanical force F̂m and control force F̂ c m and their contributing components (T̂m)i,j Fm = F̂m + F̂ c m (2.5) and F̂m = 1 ρhNm ∫ x ∫ ψ ( 3∑ i=1 qiUim ) xsinβ∗ dψ dx F̂cm =(T̂m)x,mem +(T̂m)x,bend +(T̂m)ψ,mem +(T̂m)ψ,bend (2.6) where (T̂m)i,j is the modal control force components, i.e., the longitudinal membrane and ben- ding components ((T̂m)x,mem, (T̂m)x,bend) and the circumferential membrane and bending com- ponents ((T̂m)ψ,mem, (T̂m)ψ,bend)derived later.At the steady state, the response is alsoharmonic, but lagging behind by a phase angle ψm ηm(t)= η ∗ me j(ωt−ψm) (2.7) where j= √ −1; ω is the frequency of excitation; and η∗m is the responsemagnitude η∗m = Fm ω2m √ [1− (ω/ωm)2]3+4ζ2m(ω/ωm)2 ψm =tan −1 2ζ(ω/ωm) 1− (ω/ωm)2 (2.8) Both themodalmechanical force and themodal control force are functions of themode shape function. Themode shape functions depends on the boundary conditions in this study, because the mode shape function should satisfy the geometrical constrains at the boundaries. Then the constants in the modal functions are solved using the Rayleigh-Ritz method. In this study, the boundary conditions are chosen to be clamped at the major end and free at the minor end, i.e., similar to a nozzle setup. The modal functions for clamped-free conical shells are further assumed as (Leissa andKang, 1999) Uxm(x,ψ) = cos(nψ)(x−x2) I∑ i=0 Aix i Uψm(x,ψ)= sin(nψ)(x−x2) J∑ j=0 Bjx j U3m(x,ψ) = cos(nψ)(x−x2) K∑ k=0 Ckx k (2.9) where n is the circumferentialwave number. Ai,Bj and Ck are arbitrary coefficients determined by using theRayleigh-Ritz procedure. I, J and K are constants defining the order of themodal functions.Based on thismode shape functions, the four components, i.e., longitudinalmembrane and bending components and circumferential membrane and bending components of the modal control force in Eq. (2.6)2 are given as (Li et al., 2010a) Optimal vibration control of conical shells... 773 (T̂m)x,mem =− d31Ypφ a(t)sinβ∗ ρhNm · x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ){x[δ(x−x∗1)− δ(x−x∗2)]+1}(x−x2) I∑ i=0 Aix i dψ dx (T̂m)x,bend =− rad31Ypφ a(t)sinβ∗ ρhNm x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ) { 2[δ(x−x∗1)−δ(x−x∗2)] +x ∂ ∂x [δ(x−x∗1)− δ(x−x∗2)] } (x−x2) K∑ k=0 Ckx k dψ dx (T̂m)ψ,mem = d31Ypφ a(t) ρhNm { sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)(x−x2) I∑ i=0 Aix i dψ dx − x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)]sin(nψ)(x−x2) J∑ j=0 Bjx j dψ dx +cosβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)(x−x2) K∑ k=0 Ckx k dψ dx } (T̂m)ψ,bend = rad31Ypφ a(t) ρhNm { −1 tanβ∗ · x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)]sin(nψ) x−x2 x J∑ j=0 Bjx j dψ dx +sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)[δ(x−x∗1)− δ(x−x∗2)](x−x2) K∑ k=0 Ckx k dψ dx − 1 sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ) ∂ ∂ψ [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)] x−x2 x K∑ k=0 Ckx k dψ dx } (2.10) and Nm = x2∫ x1 ψ2∫ ψ1 ( 3∑ i=1 U2im ) xsinβ∗ dψ dx (2.11) where Yp is Young’s modulus of the piezoelectric actuator; r a is the distance between the shell neutral surface and themid-surface of the actuator layer; d31 is the piezoelectric strain constant; and φa(t) is the control voltage. Assuming that the piezoelectric layer and the shell structure have auniformthickness, and the thickness keeps identical over the actuator patch. It is assumed that only the transverse voltage φ3(x,ψ,t) is considered, i.e., φ a(t)= φ3(x,ψ,t). δ(·) is aDirac delta function. x∗i and ψ ∗ i are locations of the S/A pair. 774 H. Li et al. Themodal sensing signal of the clamped-free conical shell is expressed as φSm = hS Se xS 2∫ xS 1 ψS 2∫ ψS 1 (h31Sxx +h32Sψψ)xsinβ ∗ dψ dx (2.12) where Sxx and Sψψ are the strains of the m-th mode; h S is the sensor thickness; h31 and h32 are piezoelectric strain constants; and Se is the electrode area given as Se = xS 2∫ xS 1 ψS 2∫ ψS 1 xsinβ∗ dψ dx (2.13) Here the modal sensing signal is a time-domain response, because the strains Sxx and Sψψ are functions of the modal participation factor ηm(t) and mode shape functions Uim(x,ψ). The amplitudes of the modal strains are S∗xx and S ∗ ψψ, i.e., Sxx = ηm(t)S ∗ xx and Sψψ = ηm(t)S ∗ ψψ. Furthermore, the strains can be written as S∗xx = S 0∗ xx −rSxk∗xx and S∗ψψ = S0∗ψψ −rSψk∗ψψ, where rSi is the distance between the shell neutral surface and the mid-surface of the sensor layer. Accordingly, the sensing signal φSm can be factorized into four components, i.e., longitudinal membrane and bending components and circumferential membrane and bending components (Li et al., 2010b) φSm = φ S x,mem +φ S x,bend +φ S ψ,mem +φ S ψ,bend (2.14) and φSx,mem = ηm(t) h31h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 cos(nψ) ( I∑ i=0 (1+ i)Aix i −x2 I∑ i=0 iAix i−1 ) xsinβ∗ dψ dx φSx,bend = ηm(t) rSxh31h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 cos(nψ) ( K∑ k=0 (k+1)kCkx k−1 −x2 K∑ k=0 k(k−1)Ckxk−2 ) xsinβ∗ dψ dx φSψ,mem = ηm(t) h32h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 ( n sinβ∗ (x−x2) J∑ j=0 Bjx j−1+(x−x2) I∑ i=0 Aix i−1 (2.15) + 1 tanβ∗ (x−x2) K∑ k=0 Ckx k−1 ) cos(nψ)xsinβ∗ dψ dx φSψ,bend =−ηm(t) h32h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 (ncosβ∗ sin2β∗ (x−x2) J∑ j=0 Bjx j−2+ n2 sin2β∗ (x−x2) K∑ k=0 Ckx k−2 − K∑ k=0 (k+1)Ckx k−1+x2 K∑ k=0 kCkx k−2 ) cos(nψ)xsinβ∗ dψ dx Themodal control force and sensing signal are important parameters in the controller design. The sensing signal represents the dynamic response of the structure and it is usually used as the input signal to the controller. And the modal control force equations are used by the controller to evaluate the accurate control voltage to the actuator. The system equation in the state space and the controller design are presented next. Optimal vibration control of conical shells... 775 3. State space equations In this section, the modal vibration equation is transformed into the state space since the state space formulation is fundamental inmodern control techniques. The performance criterion function and optimal gain matrix will be derived next. The standard form of state space equations are given as (Tzou and Ding, 2004) ẋ=Ax+Bu+Ew y=Cx (3.1) where x is the state variable vector; u is the control input vector; w is the external excitation vector; y is the output vector; A, B, C and E are the system matrix, input matrix, output matrix and excitation matrix, respectively (Wang et al., 2006). Here the modal participation factor and its time derivative are chosen as the state variables (Tzou and Ding, 2004) x= { ηm(t) η̇m(t) } (3.2) According to Eq. (2.6)2, distributed control actions induced by the distributed actuators are converted into a single modal control force F̂cm. Therefore, the control input vector reduces into a single parameter u= φa (3.3) Theoutput vector is chosen to be the sensing signal inducedby the distributed sensor related to the m-th mode y= φSm (3.4) And themodal mechanical load is chosen to be the external excitation w= F̂m (3.5) Tokeep consistentwithEqs. (3.1)-(3.5), the correspondingcoefficientmatrices Band Ehave to be reduced to be single column vectors, respectively and the matrix C has to be reduced to be a single row vector. By substituting the constant matrix and system vectors in Eqs. (3.2)-(3.5), Eq. (2.4) can be transformed into the first order linear state space form as { η̇m(t) η̈m(t) } = [ 0 1 −ω2m −2ζmωm ]{ ηm(t) η̇m(t) } + { 0 B2 } φa(t)+ { 0 1 } F̂m(t) y= φSm = [ C1 0 ]{ηm(t) η̇m(t) } (3.6) where B2 and C1 are elements of the coefficient matrix B and C, respectively. B2 is the control modal force generated by the piezoelectric actuator with unit control voltage. And C1 is the sensing signal generated by the piezoelectric sensor with unit modal participation factor (ηm =1). According toEqs. (2.6)2 and (2.10), the ratio of the control force and the control input signal B2 becomes B2 = F̂cm φa = 1 φa [(T̂m)x,mem +(T̂m)x,bend +(T̂m)ψ,mem +(T̂m)ψ,bend] (3.7) 776 H. Li et al. where (T̂m)x,mem φa =−d31Yp sinβ ∗ ρhNm · x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ){x[δ(x−x∗1)− δ(x−x∗2)]+1}(x−x2) I∑ i=0 Aix i dψ dx (T̂m)x,bend φa =− rad31Yp sinβ ∗ ρhNm x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ) { 2[δ(x−x∗1)− δ(x−x∗2)] +x ∂ ∂x [δ(x−x∗1)− δ(x−x∗2)] } (x−x2) K∑ k=0 Ckx k dψ dx (T̂m)ψ,mem φa = d31Yp ρhNm { sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)(x−x2) I∑ i=0 Aix i dψ dx − x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)]sin(nψ)(x−x2) J∑ j=0 Bjx j dψ dx +cosβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)(x−x2) K∑ k=0 Ckx k dψ dx } (T̂m)ψ,bend φa = rad31Yp ρhNm { −1 tanβ∗ · x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)]sin(nψ) x−x2 x J∑ j=0 Bjx j dψ dx +sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ)[δ(x−x∗1)− δ(x−x∗2)](x−x2) K∑ k=0 Ckx k dψ dx − 1 sinβ∗ x∗ 2∫ x∗ 1 ψ∗ 2∫ ψ∗ 1 cos(nψ) ∂ ∂ψ [δ(ψ −ψ∗1)− δ(ψ −ψ∗2)] x−x2 x K∑ k=0 Ckx k dψ dx } (3.8) The coefficient C1 can be derived according to the signal expressions in Eqs. (2.14) and (2.15) C1 = φSm ηm = 1 ηm (φSx,mem +φ S x,bend +φ S ψ,mem +φ S ψ,bend) (3.9) where φSx,mem ηm = h31h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 cos(nψ) ( I∑ i=0 (1+ i)Aix i −x2 I∑ i=0 iAix i−1 ) xsinβ∗ dψ dx Optimal vibration control of conical shells... 777 φSx,bend ηm = rSxh31h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 cos(nψ) ( K∑ k=0 (k+1)kCkx k−1 −x2 K∑ k=0 k(k−1)Ckxk−2 ) xsinβ∗ dψ dx φSψ,mem ηm = h32h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 ( n sinβ∗ (x−x2) J∑ j=0 Bjx j−1+(x−x2) I∑ i=0 Aix i−1 (3.10) + 1 tanβ∗ (x−x2) K∑ k=0 Ckx k−1 ) cos(nψ)xsinβ∗ dψ dx φSψ,bend ηm =− h32h S Se xS 2∫ xS 1 ψS 2∫ ψS 1 (ncosβ∗ sin2β∗ (x−x2) J∑ j=0 Bjx j−2+ n2 sin2β∗ (x−x2) K∑ k=0 Ckx k−2 − K∑ k=0 (k+1)Ckx k−1+x2 K∑ k=0 kCkx k−2 ) cos(nψ)xsinβ∗ dψ dx Once the coefficientmatrices A,B,Cand E are known, the task left is to determine theoptimal controller u so as to minimize the objective quadric function. 4. Optimal control In this section, the focus is on the development of an optimal control law with the helical distributed piezoelectric S/A pair. The objective is to determine the control input functions that minimize the performance criterion given by J = ∞∫ o [xTQx+uTRu] dt (4.1) where Q is the state weighting matrix, R is the control weighting matrix, both are real sym- metric, positive-definite. Generally, to achieve quick vibration suppression, a larger value of the state weight matrix should be chosen. Also, to reduce the energy consumption, a larger value of R matrix should be chosen (Sethi and Song, 2005). G is the optimal gain matrix (usually diagonal). The superscriptTdenotes thematrix transpose. u is the control input vector defined, in the controller design, the control input vector is usually written as u=−Gx (4.2) here G is the optimal controller gain. This gain is a main parameter in active control and it is a function of the control weighting matrix R, system input vector B and the matrix M G=R−1BTM (4.3) where the matrix M is the solution to the Riccati equation MA+ATM−MBR−1BTM+Q=0 (4.4) 778 H. Li et al. In thisRiccati equation, the coefficientmatrices Q and R are theweightingmatrix inEq. (4.1). When Q and R are specified, the matrix M can be evaluated. The state weighting matrix is chosen as Q= [ 1 0 0 Q22 ] (4.5) where Q22 is the state weighting coefficient. According to Eq. (3.3), the control input vector reduces to a single parameter. Therefore, the control weighting matrix R in Eq. (4.1) reduces to a single parameter R. As mentioned before, in the distributed control of shell vibrations, the sensing signal from the sensor in thehelical S/Apair is used as the feedback signal. The optimal controller calculates the control voltage according to the sensing signals u= φa =−G∗ { φSm φ̇Sm } =−G∗C1 { ηm(t) η̇m(t) } =−C1G∗x (4.6) here G∗ is the gain between the control input u to the sensing signal φSm and its time de- rivative φ̇Sm. From Eqs. (4.2) and (4.6), one derives the relation between G and G ∗ written as G∗ = 1 C1 G (4.7) This optimal control scheme of conical shells using the revolving helical S/A pair is demon- strated in case studies presented next. 5. Case studies In this section, the optimal control of independent modes of a clamped-free conical shell is investigated by the use of the foregoing algorithm. The conical shell model is made of Plexiglas material. The hexagonal piezoelectric material polyvinylidene-fluoride (PVDF) is used for the sensors and actuators, then d31 = d32 and h31 = h32. (Note that the generic mathematical model and derivations can account for various shell and piezoelectric materials.) The helical S/A strips are divided into 10 segments or ten S/A pairs; each has its respective electrodes, as shown in Fig. 2. And the locations of the S/A pairs are defined as x1+ i−1 10 (x2−x1) < x < x1+ i 10 (x2−x1) i =1,2,3, . . . ,10 x−x2 x1−x2 α∗− w 2 < ψ < x−x2 x1−x2 α∗+ w 2 where α∗ = π/4 is the orientation angle of the diagonal strips in the circumferential direction and w is their width. The structural andmaterial parameters are listed in Table 1. Four cases are performed to evaluate the controller of the clamped-free conical shell: case 1 – optimal control ofmode (1,2); case 2 – optimal control ofmode (2,2); case 3 – optimal control of mode (1,3); and case 4 – optimal control of mode (2,3). The parameters in the modal functions are chosen as I = J = K =4. It is assumed that: 1) no external mechanical excitation is applied to the shell, 2) the structural damping effect of the conical shell is 1% for each mode, 3) the initial velocity is 0m/s, 4) the initial displacement of the reference point is 0.001m. Optimal vibration control of conical shells... 779 Fig. 2. Locations of S/A pair segments Table 1.Geometry andmaterial properties of the conical shell control system Parameters Plexiglas shell PVDF layer Units Location x1 =0.2 – m x2 =0.5 – m Semi-vertex angle β∗ = π/4 – rad Thickness h =2 ·10−3 hS = ha =0.5 ·10−3 m Width – w =0.2 rad Young’s modulus Y =3.1 ·109 Yp =2.0 ·109 N/m2 Poisson’s ratio µ =0.3 – – Mass density ρ =1190.0 ρp =1800.0 kg/m 3 Piezoelectric const. – h31 =4.32 ·108 V/m – d31 =2.3 ·10−11 C/N The reference points used in the four cases are the points on the shell surface where the vibra- tional displacement amplitude gets themaximal value. The corresponding initial modal partici- pation factors are evaluated via dividing the initial displacements by themodal shape functions. The snap-back response is used to evaluate the shell control effectiveness. To enhance the vibration control, the S/A segment at the optimal location is chosen for the vibration control for eachmode. Also, S/A-2 is chosen in the optimal control of eachmode. The selected S/A segments are listed in Table 2. Table 2. S/A segments chosen for the specific modal control m n 2 3 1 No. 2 and 5 No. 2 and 5 2 No. 2 and 9 No. 2 and 9 In the four cases, the control performances of both S/A segments are evaluated. The optimal controllers are firstly designed to satisfy the expected control performances and control voltage requirements. In this case, the control inputweighting coefficient R is chosen to be 1. The state weighting matrix Q is set as a diagonal positive-definite matrix, where Q11 = 1 and Q22 is the design parameter of the optimal controller with the expected efficiency and cost. Q22 is evaluated using the following empirical equation 780 H. Li et al. Q22 = [q1+q2cos(q3ωm,n)+q4 sin(q3ωm,n)] √ |BOpt2,mn/BSA22,mn| where q1 =5.72 ·104 q2 =−2.20 ·104 q3 =2.36 ·10−3 q4 =−7.57 ·104 and B Opt 2,mn is the ratio of the control force and the control input signal of the optimal S/A segment; and BSA22,mn is the ratio of the control force and the control input signal of the S/A-2 segment. Generally, higher control voltage results in better vibration suppression effects, but high voltage may lead to overload or damage to the control system.And this empirical equation is chosen to avoid the overload and to guarantee the control effects. Case 1: Optimal control of mode (1,2) The time domain displacement responses of the conical shell, control voltages, damping ratio andoptimal gains G∗ are solved for eachS/Asegment.Figure 3 shows thedisplacement response of the reference pointwith respect to time formode (1,2). The thin solid line is the displacement response of the conical shell without control; the dashed line is the displacement response with the optimal control using S/A-2; and the thick solid line is the displacement response with the optimal control using the optimal S/A segments (No. 5). In this mode, the optimal controller is effective to vibration control, and the optimal S/A segment generatesmore damping.Themodal ratio using S/A-2 is 0.317%, and the damping ratio using the optimal S/A segment is 0.47%. The optimal gain G∗ of S/A-2 is [2.62E-9,30.3], and it is [1.07E-9,9.92] for the optimal S/A segment. Figure 4 shows the control voltages for both S/A segments. Fig. 3. Displacement response of (1,2) mode using S/A-2 and S/A-5 (optimal) Fig. 4. Control voltages of S/A-2 and S/A-5 (optimal), mode (1,2) Optimal vibration control of conical shells... 781 Case 2: Optimal control of mode (2,2) In the control of mode (2,2), the optimal S/A segment (No. 9) is much more effective than S/A-2, as shown in Fig. 5. The displacement response of the conical shell reduces to zero at t = 0.025s using the optimal S/A-9, and the modal damping ratio is 4.74%. S/A-2 generates a damping ratio of 0.120%. The optimal gain G∗ for S/A-2 is [1.06E-8,26.3], and it is [9.73E- 9,2.61] for the optimal S/A-9.Atmode (2,2), the control voltages for S/A-2 and the optimal S/A haveoppositephases, as shown inFig. 6.This isbecause that the twoS/Asegmentshavedifferent locations. At thismode, the optimal S/A generate a positive actuation force (extensional modal control force) while applying positive voltage, but S/A-2 generates a negative modal control force (shrink force) with the same voltage. Therefore, to generate the same control effects, the S/A segments need opposite control voltage. Fig. 5. Displacement response of (2,2) mode using S/A-2 and S/A-9 (optimal) Fig. 6. Control voltages of S/A-2 and S/A-9 (optimal), mode (2,2) Case 3: Optimal control of mode (1,3) At mode (1,3), the modal damping ratio of optimal S/A segments is 2.29%, the modal damping ratio of S/A-2 is 0.948%, but the control voltage of the optimal S/A-5 is lower than the voltage of S/A-2, as shown inFigs. 7 and 8. The optimal gain G∗ of S/A-2 is [2.84E-9,21.3], and it is [−1.75E-8,−59.2] for the optimal S/A-9. Case 4: Optimal control of mode (2,3) Figure 9 shows the displacement response of the conical shell at mode (2,3). The optimal S/A reduces the vibration to zero at t = 0.03s, and the modal damping ratio is 4.94%. The modal damping ratio of S/A-2 is 1.11%. The optimal gain G∗ of S/A-2 is [3.51E-8,67.7], and the optimal gain G∗ for the optimal S/A-9 is [1.81E-8,7.25]. But the control voltage of S/A-9 782 H. Li et al. Fig. 7. Displacement response of (1,3) mode using S/A-2 and S/A-5 (optimal) Fig. 8. Control voltages of S/A-2 and S/A-5 (optimal), mode (1,3) Fig. 9. Displacement response of (2,3) mode using S/A-2 and S/A-9 (optimal) Fig. 10. Control voltages of S/A-2 and S/A-9 (optimal), mode (2,3) Optimal vibration control of conical shells... 783 is a little less than the voltage of S/A-2, as shown in Fig. 10. At this mode, the control voltages of the two S/A segments have opposite phases, for the same reason at mode (2,2). The proposed optimal controller is effective in the vibration control of conical shell using different S/A segments. But the control performancemainly depending on the S/A location and themode shape. In the numerical case, the S/A-5 ismore effective than others in m =1modes. S/A-5 suppresses the vibrationmore quickly and generates a highermodal damping ratio while the control voltages are close to each other. In m =2 modes, S/A-9 at the major end is more effective. For all evaluatedmodes, the element G∗1 is far less than the element G ∗ 2 of the optimal gain G∗. This reveals that the control voltage solved by the optimal controller ismainly related to themodal velocity η̇. The overall control effects depending on the S/A locations, modes and the control voltage. To get best control effects, the optimal actuator is desired when the control voltage is given. 6. Conclusions This study focused on the optimal control of conical shells using distributed revolving helical piezoelectric sensors and actuators. The helical piezoelectric strips were used as the sensors and actuators; the sensor strip was laminated on the inner surface of the conical shell and the actuator strip was laminated on the outer surface. The actuation and sensing equations of the helical sensors and actuators were given and the dynamic equations were translated into modal equation. The modal participation factor and its time differential were chosen as the system state vector, and the state space equation of the vibration equationwas established. The optimal controller was derived using a linear quadratic regulator with output feedback. Four cases were performed to evaluate the controller at four modes of the clamped-free conical shell. In each case, two S/A segments or pairs (S/A-2 and the optimal S/A pair) were chosen to study the performance differences of different S/A segments. For the same controller or control voltage, the control effects are related to the location of the S/A segments andmodes. The optimal S/A segment is No. 5 in the middle of the conical shell for m = 1 modes; for m =2modes the optimal S/A segment is No. 9 at the major end. The optimal control voltage is mainly related to the modal velocity. Therefore, the S/A locations for each mode as well as the weighting matrix and modal control force should be considered in the controller design for effective vibration control of shells. Acknowledgments This work was supported by China Postdoctoral Science Foundation (No. 20110491787) and the National Natural Science Foundation of China (No. 11102182 and 11172262). References 1. Chai W.K., Han Y., Higuchi K., Tzou H.S., 2006, Micro-actuation characteristics of rocket conical shell sections, Journal of Sound and Vibration, 293, 1/2, 286-298 2. Chai W.K., Smithmaitrie P., Tzou H.S., 2004, Neural potentials and micro-signals of non- linear deep and shallow conical shells,Mechanical Systems and Signal Processing, 18, 4, 959-975 3. 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Wang S.Y., Tai K., Quek S.T., 2006, Topology optimization of piezoelectric sensors/actuators for torsional vibration control of composite plates, Smart Materials and Structures, 15, 2, 253-269 Optymalne sterowanie drganiami powłok stożkowych za pomocą spiralnego skolokowanego układu czujnika i elementu wykonawczego Streszczenie W pracy skoncentrowano się na problemie optymalnego sterowania drganiami jednostronnie zamo- cowanych powłok stożkowych za pomocą spiralnie ułożonego układu piezoelektrycznego czujnika skolo- kowanego z elementem wykonawczym (S/A). W oparciu o wyniki płynące z rozwiązania przestrzennego zagadnienia sterowania modalnego, dynamiczną odpowiedź powłoki na wymuszenie zewnętrzne wyrażo- no sumą postaci własnych, jednocześnie stwierdzając, że można niezależnie ingerować w poszczególne postaci własne układu. Równanie modalne przetransformowano do liniowej formuły stanu. Zaprojekto- wano sterowniki liniowo-kwadratowe (LQ) niezależnie dla każdej postaci własnej. Optymalną macierz współczynnikówwzmocnienia skorelowano z transmitancją G∗ pomiędzy napięciem sterowania i sygnału czujnika poprzez jednostkową siłę sterującą odniesioną do napięcia B2 oraz sygnału czujnika w stosun- ku do przemieszczenia C1. Ponieważ B2 i C1 zmieniają się wraz ze zmianą położenia skolokowanych par S/A, przeprowadzono ewaluacjęmodalnych sił sterujących i odpowiadających im napięć dla różnych położeń układu sterowania. Wyniki badań potwierdziły efektywność optymalnego sterowania drgania- mi powłoki stożkowej. Zauważono także wrażliwość uzyskanego stopnia sterowania na położenie pary czujnika i elementu wykonawczego oraz wartości modalnej siły sterującej, jak i napięcia. Manuscript received March 21, 2012; accepted for print April 1, 2012