Vibration attenuation and shape control of surface mounted, embedded smart beam

1(2012) 1 – 25 1 Vibration attenuation and shape control of surface mounted, embedded smart beam Abstract Active Vibration Control (AVC) using smart structure is used to reduce the vibration of a system by automatic modification of the system structural response. AVC is widely used, because of its wide and broad frequency response range, low additional mass, high adaptability and good efficiency. A lot of research has been done on Finite Element (FE) models for AVC based on Euler Bernoulli Beam Theory (EBT). In the present work Timoshenko Beam Theory (TBT) is used to model a smart cantilever beam with surface mounted sensors / actuators. A Periodic Output Feedback (POF) Controller has been designed and applied to control the first three modes of vibration of a flexible smart cantilever beam. The difficulties encountered in the usage of surface mounted piezoelectric patches in practical situations can be overcome by the use of embedded shear sensors / actuators. A mathematical model of a smart cantilever beam with embedded shear sensors and actuators is developed. A POF Controller has been designed and applied to control of vibration of a flexible smart cantilever beam and effect of actuator location on the performance of the controller is investigated. The mathematical modeling and control of a Multiple Input multiple Output (MIMO) systems with two sensors and two actuators have also been considered. Keywords AVC, FE, EBT, TBT, POF, MIMO, LTI, OLR, CLR 2 3 4 5 6 7 8 9 1 Vivek Rathi and Arshad Hussain Khan Mechanical Engineering Department, Aligarh Muslim University, Aligarh India, 202001 Received 17 May 2012; In revised form 24 May 2012 ∗ Author email: logonrathi@yahoo.co.in, arshad1976@rediffmail.com INTRODUCTION Undesired noise and vibrations have always been a major problem in many human activities and domains. From buildings to atomic force microscopes, all can be disturbed in their normal functions by vibrations and noise. Recent technological advancements such as the availability of high–power and low–cost computing, smart materials, and advanced control techniques have led to a growing use of AVC systems. The implication of active control is that desirable performance characteristics can be achieved through flexible and clever strategies, whereby actuators excite the structure based on the structure’s response measured by sensors. Latin American Journal of Solids and Structures 1(2012) 1 – 25 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Umapathy and Bandyopadhyay[20] discussed the vibration control aspects of a smart flexible beam for a Single Input Single Output (SISO) case. Hanagud et al. [11] developed a FE model for an active beam based on EBT and applied optimal output feedback control. Hawang et al.[13] developed a FE model for vibration control of a laminated plate with piezoelectric sensors /and actuators. Crawley et al [9] have presented the analytical and experimental development of piezoelectric actuators as elements of intelligent structures. FE models of a structure containing distributed piezoelectric sensors / actuators can also be seen in [10, 24]. Detailed survey on various control algorithms used in active vibration control studies has been presented by Alkhatib and Golnaragi [4]. A detailed comparative studies of different control algorithms on active vibration control of smart beam has been presented in [14, 21]. Kumar and Narayanan [15] carried out optimal location studies of sensor-actuator pairs using Linear Quadratic Regulator (LQR). Li et al. [16] proposed an optimal design methodology for the placement of piezoelectric actuator/sensor pairs. Molter et al. [17] carried out control design analysis for flexible manipulators using piezoelectric actuators. In their paper GA technique is employed for optimization of placement and size of piezoelectric material for optimal vibration control. Optimal controller design for the location, size and feedback of sensor/actuators have been carried out in references [12, 23]. Chandrashekhara and Vardarajan [8] have presented a FE model of a piezoelectric composite beam using higher – order shear deformation theory. Aldraihem et al. [3] have developed a laminated cantilever beam model using EBT and TBT with piezoelectric layers. Abramovich [1] has presented analytical formulation and closed form solutions of composite beam with piezoelectric actuators using TBT. Narayan and Balamurugan [18] have presented finite element formulation for the active vibration control study of smart beams, plates and shells and the controlled response is obtained using classical and optimal control strategies. In the analyses mentioned above, the controlled response has been obtained based on extension mode actuation. There have been very few studies based on shear mode actuation and sensing for the analysis of active structures. The idea of exploiting the shear mode of creating transverse deflection in beams (sandwiched type) was first suggested by Sun and Zhang [19]. A FE approach was used by Benjeddou et al [6] to model a sandwich beam with shear and extension piezoelectric elements. It was observed that the shear actuator is more efficient in rejecting vibration than the extension actuator for the same control effort. Aldraihem and Khdeir [2] proposed analytical models and exact solutions for beams with shear and extension piezoelectric actuators. The models are based on TBT and HOBT. Exact solutions are obtained by using the state – space approach. Azulay and Abramovich [5] studied the effects of actuator location and number of patches on the actuator’s performance for various configurations of patches and boundary conditions under mechanical and/or electrical loads. Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 47 48 49 50 51 52 53 54 2 POF CONTROL A standard result in control theory says that the poles of a linear time invariant (LTI) controllable system can be arbitrarily assigned by state feedback. If the original system is time invariant and the linear combinations are also constrained to the time invariant, the design problem is to choose an appropriate matrix of feedback gains. The problem of pole assignment by piecewise constant output feedback with infrequent observation was studied by Chammas and Leondes [7] for LTI systems. Consider the system ẋ = Ax(t) + Bu(t), y(t) = Cx(t) 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 72 73 74 75 (1) Where x ∈ Rn , u ∈ Rm , y ∈ Rp , A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n A, B, C are constants matrices and it is assumed that the system (A, B, C) is controllable, observable and stable. Assume that output measurements are available from system at time instants t = kτ = 0, 1, 2,. . . . Now, construct a discrete LTI system from these output measurements at rate 1/τ (sampling interval of τ seconds), the system so obtained is called the τ system and is given by, x(k + 1)τ = Φτ x(kτ ) + Γτ u(kτ ) y(kτ ) = Cx(kτ ) (2) u(t) = Kl y(kt), [kτ + l∆] ≤ t < [kτ + (l + 1)∆], Kl+N = Kl (3) Now, design an output injection gain matrix G such that Eigen values of (Φτ + GC) are inside the unit circle i.e., eig(Φτ + GC)<1. For l = 0, 1, - - - - - - - - N –1, where an output sampling interval τ is divided in to N subintervals of width ∆ = τ /N , the hold function being assumed constant. To see the relationship between the gain sequence {Kl } and closed loop behavior, let {Φ, Γ, C} be a new system and denote the system sampled at rate 1/∆ as the ∆ system and collect the gain matrices Kl in to one matrix. If (Φ, Γ) system is controllable and (Φτ , C) is observable, one can first choose and output injection gain G to place the eigen values of (ΦN + GC) in the desired locations inside the unit circle and then compute the POF gain sequence {Kl } such that, ΓK = G 71 3 (4) andρ(ΦN + GC) < 1 is satisfied, where ρ is spectral radius. Werner and Furuta [22] proposed the performance index so that ΓK = G need not be forced exactly. This constraint is replaced by a penalty function, which makes it possible to enhance the closed loop performance by allowing slight deviations from the original design and at the same time improving behavior. The performance index is, Latin American Journal of Solids and Structures 1(2012) 1 – 25 4 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam J(k) = ∑ [ xTl ∞ l=0 uTl ] [ ∞ x Q 0 T ] [ l ] + ∑ (xkN − x∗kN ) P (xkN − x∗kN ) ul 0 R k=1 (5) 78 Where, R ∈ Rm×n , Q and P ∈ Rn×n , are positive definite and symmetric weight matrices. The first term represents ’averaged’ state and control energy whereas the second term penalizes deviation of G. 79 3 80 3.1 76 77 81 82 83 84 85 86 87 88 FORMULATION Surface Mounted Sensors and Actuators The smart cantilever model is developed using a piezoelectric beam element, which includes sensor and actuator dynamics and a regular beam element based on TBT assumptions. The piezoelectric beam element is used to model the regions where the piezoelectric patch is bonded as sensor/actuator, and rest of the structure is modeled by the regular beam element. The longitudinal axis of the regular beam element (Fig. 1), lies along the X – axis. The element has constant moment of inertia, modulus of elasticity, mass density and length. The element is assumed to have two degree of freedom, a transverse shear force and a bending moment act at each nodal point. θ θ θ θ Figure 1 89 A Regular Beam Element Figure 2 Piezoelectric Beam Elements with Sensors and Actuators The displacement relation in the x, y and z direction can be written as, u (x, y, z, t) = zθ(x, t) = z ( ∂w − β (x)) , ∂x v (x, y, z, t) = 0, w (x, y, z, t) = w (x, t) 90 91 92 (6) Where, w is the time dependent transverse displacement of the centroidal axis, θ is the time dependent rotation of the cross – section about ‘Y – axis’. Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 93 For the static case with no external force acting on the beam, the equation of motion is, ∂ [κGA ( ∂w + θ)] ∂x ∂x ∂θ ) ∂ (EI ∂x 94 95 96 5 − κGA ( ∂x The boundary conditions are given as, At x = 0 w = w1 , θ = −θ1 and At The mass matrix is given by, = 0, ∂w + θ) = 0 ∂x x=L w = w2 , (7) θ = −θ2 [Nw ] [Nw ] ρA 0 ][ ] dx [M ] = ∫ [ ] [ [Nθ ] [Nθ ] 0 ρIyy (8) [M ] = [MρA ] + [M ρI ] (9) L T 0 97 98 99 [MρA ] in equation is associated with translational inertia and [MρI ] is associated with rotary inertia, there expressions are given in the appendix. The stiffness matrix is given by, ∂ EI 0 [Nθ ] [Nθ ] ∂x [K] = ∫ [ ] [ ] × [ ] dx ∂ ∂ 0 κGA [Nw ] [Nθ ] + ∂x [Nw ] [Nθ ] + ∂x L T ∂ ∂x (10) 0 100 Finally we obtain; ⎡ ⎢ ⎢ EI ⎢ ⎢ [K] = 3 (1 + ϕ) L ⎢ ⎢ ⎢ ⎣ 101 102 103 104 105 106 12 6L −12 6L 6L (4 + ϕ) L2 −6L (2 − ϕ) L2 −12 −6L 12 −6L 6L (2 − ϕ) L2 −6L (4 + ϕ) L2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (11) EI ), Here ϕ is the ratio of the beam bending stiffness to the shear stiffness given by,ϕ = L122 ( κGA L is the length of beam element. E is the Young’s modulus of the beam material, G is shear modulus of the beam material, k is shear coefficient which depends on the material definition and cross – sectional geometry, I is the moment of inertia of the beam element, A is the area of cross – section of the beam element and ρ is the mass density of the beam material. The consistent force array is given as, [Nw ] q {F } = ∫ [ ] { } dx. [Nθ ] m L T (12) 0 107 108 109 The piezoelectric element is obtained by sandwiching the regular beam element between two thin piezoelectric layers as shown in figure 2. The element is assumed to have two – structural degree of freedom at each nodal point and an electric degree of freedom. The piezoelectric Latin American Journal of Solids and Structures 1(2012) 1 – 25 6 110 111 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam layers are modeled based on EBT as the effect of shear is negligible and the middle steel layer is modeled based on TBT. The mass and stiffness matrix of piezoelectric layers is given by, ⎡ 156 22lp 54 −13lp ⎤ ⎥ ⎢ ⎢ 2 ρp Ap lp ⎢ 22lp Ep Ip 4lp 13lp −3lp2 ⎥ ⎥ p p ⎥ [K ] = ⎢ [M ] = 13lp 156 −22lp ⎥ 420 ⎢ lp ⎥ ⎢ 54 2 ⎥ ⎢ −13lp −3l2 −22lp 4l ⎦ ⎣ p p 112 113 114 115 116 117 118 119 120 121 123 124 125 126 128 − 12 l2 4 − l6p − l6p 2 p 12 2 lp − l6p ⎤ ⎥ ⎥ 2 ⎥ ⎥ ⎥ − l6p ⎥ ⎥ ⎥ 4 ⎥ ⎦ 6 lp (13) ta is the thickness of actuator, tb is the thickness of beam, c is the width of beam. The mass matrix for the piezoelectric beam element is given by, [M piezo ] = [MρA ] + [MρI ] + [M p ] (14) [K piezo ] = [K] + [K p ] (15) Stiffness matrix [K piezo ] for the piezoelectric beam element, 3.1.1 Piezoelectric Strain Rate Sensors and Actuators The linear piezoelectric coupling between the elastic field and the electric field can be expressed by the direct and converse piezoelectric equations, respectively, ε = sE σ + dEf (16) σ is the stress, ϵ is the strain, Ef is the electric field, e is the permittivity of the medium, SE is the compliance of the medium, d is the piezoelectric constants. Sensor Equation: If the poling is done along the thickness direction of the sensors with the electrodes on the upper and lower surfaces, the electric displacement is given by, Dz = d31 × Ep εx = e31 εx 127 6 lp ρp is the mass density of piezoelectric beam element, Ap is the area of piezoelectric patch = 2 ta c, lp (=L) is the length of the piezoelectric patch. Ep is the modulus elasticity of piezoelectric material, Ip is the moment of inertia of piezoelectric layer w. r. t. the neutral axis of the beam 2 (ta + tb ) 1 3 ] Ip = cta + cta [ 12 2 D = dσ + eT Ef 122 ⎡ 122 ⎢ lp ⎢ 6 ⎢ ⎢ lp ⎢ 12 ⎢ − l2 ⎢ p ⎢ 6 ⎢ l ⎣ p (17) e 31 is the piezoelectric stress / charge constants, Ep is the Young’s modulus of piezoelectric material, ϵx is the strain of the testing structure at a point. The sensor output voltage is, lp V (t) = Gc e31 zc ∫ nTl q̇.dx s 0 Latin American Journal of Solids and Structures 1(2012) 1 – 25 (18) Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 129 130 131 132 133 134 135 7 nTl is the second spatial derivative of the shape function of the flexible beam and as a scalar vector product as, V s (t) = pT q̇ (19) V a (t) = KV s (t) (20) εA = d31 Ef (21) q̇ is the time derivative of the displacement vector, pT is a constant vector. The input voltage to an actuator is V a (t) given by, Actuator Equation: The strain developed by the electric field (Ef ) on the actuator layer is given by, The control force applied by the actuator is, fctrl = Ep d31 cz a ∫ n2 .dx.V (t). (22) lp 136 137 138 139 140 141 z = (ta 2+tb ) , is the distance between the neutral axis of the beam and the piezoelectric layer. Or as a scalar product as, fctrl = h.V a (t) (23) f t = fext + fctrl (24) nT2 is the first spatial derivative of shape function of the flexible beam, hT is a constant vector. If any external forces described by the vector fext are acting then, the total force vector becomes, 142 3.1.2 143 The dynamic equation of motion of the smart structure is finally given by, 144 145 146 Dynamic Equation and State Space Model M . q̈ + K . q = fext + fctrl (25) q = T.g , (26) T is the model matrix containing the eigen vectors representing the desired number of modes of vibration of the cantilever beam, g is the modal coordinate vector. Equation (25) is then transformed in to, ∗ ∗ M∗ .g̈ + C∗ .ġ + K∗ .g = fext + fctrl (27) Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 8 147 148 149 M ∗ is the generalized mass matrix K∗ is the generalized stiffness matrix, C∗ is the generalized damping matrix, f∗ ext is the generalized external force vectors, f∗ ctrl is the generalized control force vectors. The structural modal damping matrix is-: C ∗ = α M ∗ + β K∗ , α and β are constants. The state space model is 150 151 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 152 153 ẋ1 ẋ2 ẋ3 ẋ4 ⎡ x1 ⎤ ⎢ ⎥ ⎢ x ⎥ 0 I ⎢ ⎥ ⎥=[ ]⎢ 2 ∗−1 ∗ ∗−1 ∗ ⎢ x3 ⎥ −M K −M C ⎢ ⎥ ⎢ x4 ⎥ ⎣ ⎦ ⎤ ⎥ ⎥ 0 0 ⎥ ⎥+[ ] u (t) + [ ∗−1 T ] r(t) (29) ∗−1 T ⎥ M T h M T f ⎥ ⎥ ⎦ u (t) is the control input, r (t) is the external input to the system, f is the external force coefficient vector. The sensor equation for the modal state space form is given by; y(t) = [ 0 pT 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 (28) The above system may be represented as, ⎡ ⎢ ⎢ ⎢ T ] ⎢ ⎢ ⎢ ⎢ ⎣ x1 x2 x3 x4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (30) ẋ = A .x (t) + B . u (t) + E . r (t) (31) y (t) = CT x (t) (32) 3.1.3 Validation for Surface Mounted Smart Beam To validate the present formulation and the computer program, a cantilever beam made of steel which is surface bonded with two PZT layers on both side is considered. The elastic modulus, poisons ratio and density of steel and PZT are 200GPa, 0.3 and 7500 Kg/m3 and 139 GPa, 0.3 and 7500 Kg/m3 respectively while the strain and stress constants of PZT are 23×10−12 m/V and 0.216 respectively [8]. The length, width and the thickness of the beam are 500 mm, 30 mm and 2 mm respectively while the thickness of each of the PZT layers is 40 µm. The Voltages at the steel and PZT layers are set to zero. The beam is discretized into 20 elements to obtain converged results. The beam is excited with 0.2×10−3 Ns impulse load acting on the tip of the beam. The closed loop response of the tip displacement is obtained using constant gain negative velocity feedback (CGVF) control with gain Gv =1 and linear quadratic Regulator (LQR) control Q=106 and R=1 and compared with the response obtained under the same condition by the Narayanan and Balamurugan [8]. The control is applied after 0.5 seconds. The present response of the system is very well matched with the published results. Next, the Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 169 170 171 first six open-loop and closed-loop natural frequencies of beam are presented in Table 1 and compared with reference [18]. These frequencies are in good agreement with the published results. Table 1 Open loop natural frequencies (Hz) 6.809 41.64 115.7 226.2 373.6 557.8 172 173 174 175 176 9 3.2 First six natural frequencies of smart steel cantilever beam Closed loop natural frequencies (Hz) (CGVF control) 6.855 43.19 121.6 237.5 357.5 506.3 Closed loop natural frequencies (Hz) (LQR control) 6.834 41.69 115.7 226.3 373.7 557.9 Natural frequencies (HZ) [8] 6.89 43.285 121.225 237.72 393.58 589.63 Figure 3 Closed loop response of smart cantilever steel beam (a) Narayanan and Balamurugan [18](reproduced with permission from Elsevier) and (b) Present obtained with negative CGVF control with Gv =1. Figure 4 Closed loop response of smart cantilever steel beam (a) Narayanan and Balamurugan [18] (reproduced with permission from Elsevier) and (b) Present obtained with LQR control with Q = 106 and R=1. Embedded Shear Sensors And Actuators The piezoelectric element is embedded on discrete locations of the sandwich beam as shown in Figure (5). The smart cantilever beam model is developed using a piezoelectric sandwich beam element, which includes sensor and actuator and a regular sandwiched beam element, which includes foam at the core. A FE model of a piezoelectric sandwich beam is developed using Latin American Journal of Solids and Structures 1(2012) 1 – 25 10 177 178 179 180 181 182 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam laminate beam theory. It consists of three layers. The assumption made is that the middle layer is perfectly glued to the carrying structure and the thickness of adhesive can be neglected and each layer behaves as a Timoshenko beam. The longitudinal axis of the sandwiched beam element lies along the X – axis. The element has constant moment of inertia, modulus of elasticity, mass density and length. The element is assumed to have three degree of freedom, a transverse shear force and a bending moment act at each nodal point. Figure 5 183 A Sandwiched Beam Element The displacement relation of the beam u (x, z ) and wu (x, z ) can be written as, u (x, z) = u0 (x) − zθ (x, t) 184 185 186 187 188 w(x, z) = w0 (x) (33) u0 (x) and w0 (x) are the axial displacements of the point at the mid plane,θ (x) is the bending rotation of the normal to the mid plane. The beam constitutive equation can be written as, ⎡ ∂u0 ⎤ ⎡ E11 ⎤ ⎡ Nx ⎤ ⎡ A11 B11 0 ⎤ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ dx ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ∂θ ⎢ Mx ⎥ = ⎢ B11 D11 ⎥ + ⎢ F11 ⎥ 0 ⎥⎢ (34) ∂x ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎥⎢ ∂w0 ⎥ ⎢ ⎢ Qxz ⎥ ⎢ 0 ⎥ ⎢ ⎥ 0 A55 ⎦ ⎣ θ + ∂x ⎦ ⎣ G55 ⎦ ⎣ ⎦ ⎣ A11 , B11 , D11 and A55 are the extensional, bending and shear stiffness coefficients defined according to the lamination theory, A11 = c ∑ (Q11 )k (zk − zk−1 ) , N k=1 N 2 ), B11 = c ∑ (Q11 )k (zk2 − zk−1 D11 = c 3 k=1 N 3 ) ∑ (Q11 )k (zk3 − zk−1 k=1 N A55 = cκ ∑ (Q55 )k (zk − zk−1 ) , k=1 Latin American Journal of Solids and Structures 1(2012) 1 – 25 (35) Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 189 190 191 192 193 Zk is the distance of the k th layer from the X – axis, N is the number of layers, k is the shear correction factor usually taken equal to 5/6. The boundary conditions are given as, At x = 0 w = w1 , θ = θ1 , u = u1 and At x=L w = w2 , θ = θ 2 , u = u2 After solving, we get, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [u] = [Nu ] ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u1 w1 θ1 u2 w2 θ2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ , [w] = [Nw ] ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎭ w1 θ1 w2 θ2 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ , θ = [Nθ ] ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ w1 θ1 w2 θ2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (36) [Nu ] = [ N1 N2 N3 N4 N5 N6 ] [NW ] = [ N7 N8 N9 N10 ] [Nθ ] = [ N11 N12 N13 N14 ] 195 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [M ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 197 198 199 M11 M21 M31 M41 M51 M61 M12 M22 M32 M42 M52 M62 M13 M23 M33 M43 M53 M63 M14 M24 M34 M44 M54 M64 M15 M25 M35 M45 M55 M65 M16 M26 M36 M46 M56 M66 ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ [K] = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎦ ⎣ K11 K21 K31 K41 K51 K61 K12 K22 K32 K42 K52 K62 K13 K23 K33 K43 K53 K63 K14 K24 K34 K44 K54 K64 K15 K25 K35 K45 K55 K65 K16 K26 K36 K46 K56 K66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (38) Values of the mass and stiffness matrix coefficients are given in the appendix. The material constants Q11 , Q22 , Q12 , Q66 , G13 and G23 for foam, steel and piezoelectric materials are given in table 2. These constants are used to calculate the values of A11 , B11 , D11 and A55 Table 2 Material Constants G12 (MPa) G13 (MPa) G23 (MPa) d31 (m/V) d15 (m/V) Q11 (MPa) Q22 (MPa) Q12 (MPa) Q66 (MPa) Material Properties and Constants Piezoelectric Material 24800 24800 24800 -16.6×10−9 1.34×10−9 68400 68400 12600 12600 Steel 78700 78700 78700 # # 215000 215000 2880 78700 Foam 99.9 99.9 99.9 # # 85.4 85.4 75.8 9.99 Sensor Equation: The charge q(t) accumulated on the piezoelectric electrodes is given 200 201 (37) Values of N1 to N12 be given in the appendix, The symmetric mass and stiffness matrices are given by, 194 196 11 by, Latin American Journal of Solids and Structures 1(2012) 1 – 25 12 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam q (t) = ∬ D3 dA (39) A 202 203 204 205 206 207 208 D3 is the electric displacement in the thickness direction, A is the area of electrodes. The current induced in sensor layer is converted in to the open circuit sensor voltage V s (t) using a signal-conditioning device with a gain of Gc and applied to the actuator with the controller gain Kc . T 6η 2 −l 0 −2 −l ] [q̇] , V s (t) = [p] [q̇] V s (t) = Gc i (t) , V s (t) = e15 c (−12η+l 2) [ 0 [q̇] is the time derivative of the modal coordinate vector, [p] is a constant vector. The input voltage of the actuator is V a (t), given by, T V a (t) = Kc V s (t) 209 210 (40) Actuator Equation: The strain produced in the piezoelectric layer is directly proportional to the electric potential applied to the layer. γxz ∝ Ef 211 212 213 γxz is the shear strain in the piezoelectric layer, Ef is the electric potential applied to the actuator. From the constitutive piezoelectric equation, we get, (41) γxz = d15 Ef 214 The shear force Qxz is given as V a (t) = cGd15 ∫ dz tp tp Qxz (42) 0 215 Or as a scalar product, fctrl = h . V a (t) 216 217 218 (43) Where,fctrl = Qxz and hT is a constant vector. If any external forces described by the vector fctrl are acting, then total force vector becomes, f t = fext + fctrl Latin American Journal of Solids and Structures 1(2012) 1 – 25 (44) Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 219 3.2.1 220 The dynamic equation is, Dynamic Equation and State Space Model: ∗ ∗ M∗ .g̈ + C∗ .ġ + K∗ .g = fext + fctrl (45) C ∗ = α M∗ + β K ∗ , (46) The state space form of the system is obtained as, 221 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ẋ1 ẋ2 ẋ3 ẋ4 ẋ5 ẋ6 ⎤ ⎡ x1 ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ x3 0 I ⎥=[ ⎢ ⎥ ∗−1 ∗ ∗−1 ∗ ] ⎢ −M K −M C ⎥ ⎢ x4 ⎥ ⎢ ⎥ ⎢ x5 ⎥ ⎢ ⎥ ⎢ x6 ⎦ ⎣ (48) ẋ = A .x (t) + B . u (t) + E . r (t) (49) y (t) = CT x (t) (50) The above system can be represented as, 223 224 4 225 4.1 228 229 230 231 232 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ y (t) = [ 0 pT 226 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ 0 0 ⎥+[ ] u(t) + [ ] r(t) (47) ∗−1 T ∗−1 T ⎥ M T h M T f ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ The sensor equation for the modal state space form is given by; 222 227 13 T ] x1 x2 x3 x4 x5 x6 SIMULATION Simulation of Surface Mounted Sensors and Actuators The state space representation of the cantilever beam with the surface mounted sensor / actuator is obtained by using nine regular beam elements and one piezoelectric element as shown figure (6). The dimensions and properties of the flexible beam and piezoelectric sensor / actuator used in the numerical simulation are given in tables 3 and 4 respectively. Two state space models of the smart cantilever beam have been obtained by keeping the AR 8 and 15, the length of the beam is kept constant and the thickness of the beam is varied. Latin American Journal of Solids and Structures 1(2012) 1 – 25 14 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 6 Cantilever Beam with Surface Mounted Sensors and Actuators Table 3 Physical Properties of Steel Beam Total length of beam (mm) Width (mm) Young’s modulus (Pa) Density Kg/m3 ) Constants used in C∗ 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 l b Eb ρb α, β 500 24 193x109 8030 10−3 ,10−4 The POF control technique is used to design a controller to suppress vibration of a cantilever beam. For this purpose three vibration modes are considered. In the first case the AR is taken equal to 15. Configuration specifications of smart beam are as per table 3 and 4. The collocated sensor and actuator are placed near the fixed end. The FE model of the surface mounted cantilever beam is developed in MATLAB using TBT. A sixth order space model of the system is obtained on retaining the first three modes of vibration of the system. The first three natural frequencies calculated are 106.03 Hz,658.33Hz, 1826.38Hz respectively. An impulsive force of 10N is applied for duration of 0.05 sec and the open loop response (OLR) of the system is obtained as shown in figure 7. A controller based on the POF control algorithm has been designed to control the first three modes of vibration of the smart cantilever beam. The sampling interval used is 0.0004 sec. The sampling interval is divided in to 10 subintervals (N=10). The periodic output gain for the system is obtained by using the algorithm given for POF controller, the impulse response of the system with POF gain is shown in figure 8. In the second case the AR is taken equal to 8, and again all other parameters are kept same as that in the first case for which AR is 15. The first three natural frequencies calculated are 197.94Hz, 1215.89Hz and 3311.05Hz respectively. The OLR and CLR (with POF gain) of the system is obtained as shown in figure 9 and 10. Table 4 Properties of Piezoelectric Sensor /Actuator Width (mm) Thickness (mm) Young’s Modulus (Pa) Density(Kg/m3 ) Piezoelectric strain Constant (mV−1 ) Piezoelectric stress Constant (VmN−1 ) Latin American Journal of Solids and Structures 1(2012) 1 – 25 b ta Ep ρp d31 e31 24 0.5 68x109 7700 125x10-12 10.5x10−3 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 7 Figure 9 250 251 252 253 254 255 256 257 258 OLR of Surface Mounted Cantilever Beam with AR = 15 OLR of Surface Mounted Cantilever Beam with AR = 8 Figure 8 Figure 10 15 CLR of Surface Mounted Cantilever Beam with AR = 15 CLR of Surface Mounted Cantilever Beam with AR = 8 The variation of control signal with time is shown in figure 11 (for AR = 15) and 12(for AR = 8). A POF controller is designed for the Timoshenko beam models. Two cases (AR=15,8 ) have been considered. By comparing the OLR and CLR in the first case for AR=15, we observed that there was a 89.43% decrease in settling time for the system after applying POF control, and a change of 89% in settling time was observed for the case with AR=8. Thus, it can be inferred from the simulation results, that a POF controller applied to a smart cantilever model based on TBT is able to satisfactorily control higher modes of vibration of the smart cantilever beam for a wide range of AR. Latin American Journal of Solids and Structures 1(2012) 1 – 25 16 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 11 Control Signal of Surface Mounted Cantilever Beam with AR = 15 259 4.2 260 4.2.1 Single Input Single Output (SISO) System 261 262 263 264 265 266 267 268 269 270 271 272 273 274 276 277 278 279 CLR of Surface Mounted Cantilever Beam with AR = 8 Simulation of Embedded Sensors and Actuators The FE model of smart cantilever beam based on laminate beam theory is developed. Keeping the sensor location fixed and varying the position of the actuator, different state space models of the smart cantilever beam are obtained. A POF controller is designed to control the first three modes of vibration of the smart cantilever beam. Here an attempt has been made to find the optimum actuator position for a single input single output (SISO) system. Three cases have been considered. In the first case FE model of the smart cantilever beam is obtained by dividing the beam into 10 elements. The actuator is placed as the 1st element (at the fixed end) and the sensor is placed as the 8th element as shown in figure 13. The length of beam is 200mm and its cross – section is 10mm × 20mm. The length of piezoelectric patch is 200mm and its cross – section is 6mm × 20mm. The material properties used for the generation of FE model are given in table 2. A ninth order space model of the system is obtained on retaining the first three modes of vibration of the system. The first three natural frequencies (same for all three models) are 44.9 Hz, 82.4 Hz and 131.5 Hz respectively. Figure 13 275 Figure 12 Smart Cantilever Beam with Actuator at 1st Position and Sensor at 8th Position An impulsive force of 10N is applied for duration of 0.05sec and the OLR of the system is obtained as shown in figure 14 A controller based on the POF control algorithm has been designed to control the first three modes of vibration of the smart cantilever beam. The sampling interval used is 0.07 m sec. The sampling interval is divided in to 10 subintervals (N =10). The impulse response or CLR of the system with POF gain is shown in figure 15. Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 14 280 281 OLR of Smart Cantilever Beam with Actuator at 1st Position 283 285 286 287 Smart Cantilever Beam with Actuator at 5th Position and Sensor at 8th Position An impulsive force of 10 N is applied for duration of 0.05 sec and the OLR of the system is obtained as shown in figure 17. The CLR of the system with POF gain is shown in figure 18. Figure 17 284 CLR of Smart Cantilever Beam with Actuator at 1st Position In the second case, the actuator is placed as the 5th element and the sensor is placed as the 8th element as shown in figure 17. Other parameters are kept same. Figure 16 282 Figure 15 17 OLR of Smart Cantilever Beam with Actuator at 5th Position Figure 18 CLR of Smart Cantilever Beam with Actuator at 5th Position In the third case, the actuator is placed as the 10th element (at the free end) and the sensor is placed as the 8th element as shown in figure 19. Other parameters are kept same as that of first case. An impulsive force of 10 N is applied for duration of 0.05 sec and the OLR of the system is Latin American Journal of Solids and Structures 1(2012) 1 – 25 18 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 19 288 289 obtained as shown in figure 20. The impulse response of the system with POF gain is shown in figure 21. Figure 20 293 294 295 Figure 21 CLR of Smart Cantilever Beam with Actuator at 10th Position 24. Figure 22 292 OLR of Smart Cantilever Beam with Actuator at 10th Position The variation of control signal with time for all the three cases are shown in figure 22 to 290 291 Smart Cantilever Beam with Actuator at 1st Position and Sensor at 8th Position Control Signal of Smart Cantilever Beam with Actuator Placed at 1st Position Figure 23 Control Signal of Smart Cantilever Beam with Actuator at 5th Position Here in the present case, the performance of the controller is evaluated for different actuator locations while the position of the sensor is kept constant. It can be inferred from the response characteristics that the actuator locations has negligible effect on the performance of the controller. Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 24 296 297 298 299 300 301 302 303 304 305 4.2.2 307 308 309 310 Control Signal of Smart Cantilever Beam with Actuator at 10th Position Multi – Input Multi – Output (MIMO) Systems Active control of vibration of a smart cantilever beam through smart structure concept for a multivariable system (MIMO) case is considered here. The structure is modeled in space state from using Finite element method by dividing the beam in to 10 FE and placing the sensors at the 6th and 10th positions and the actuators at the 4th and 8th position. Thus giving rise to MIMO with two actuator inputs u 1 and u 2 and two sensors outputs y 1 and y 2 , The POF control technique is used to design a controller to suppress the first three modes of vibration of a smart cantilever beam for a multi variable system. The simulations are carried out in MATLAB. The parameters are kept same as that of the model used for SISO case. The first three natural frequencies calculated are 45.2 Hz, 83.2 Hz and 136.4 Hz respectively. Figure 25 306 19 A MIMO Smart Cantilever Beam with Two Inputs and Two Outputs An impulsive force of 10N is applied for duration of 0.05sec. A controller based on the POF control algorithm has been designed to control the first three modes vibration of the smart cantilever beam for the multivariable case. The CLR (sensor outputs y1 and y2 ) with periodic output feedback gain K for the state space model of the system is shown in figure 27 and 31. Figures 29 and 33 show the variation of the control signal with time. Latin American Journal of Solids and Structures 1(2012) 1 – 25 20 311 312 313 314 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam Figure 26 CLR of SISO System with Sensor at 6th Position Figure 27 Response y1 of MIMO System with POF Figure 28 Control Input of SISO System with Sensor at 6th Position and Actuator at 4th Position Figure 29 Control Input u 1 of MIMO System with POF It can be inferred from the simulation results, that the system’s performance meets the design requirements. It is also observed that the maximum amplitude of the sensor output voltage is less for the multivariable case and the response takes lesser time to settle. Controlling time is considerably reduced with MIMO systems as compared to SISO systems. Latin American Journal of Solids and Structures 1(2012) 1 – 25 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 5 Figure 30 30 CLR of SISO System with Sensor at 10th Position and Actuator at 8th Position Figure 31 Response y2 of MIMO System with POF Figure 32 Control Input of SISO System with Sensor at 10th Position and Actuator at 8th Position Figure 33 Control Input u 2 of MIMO System with POF 21 CONCLUSION: An integrated FE model to analyze the vibration suppression capability of a smart cantilever beam with surface mounted piezoelectric devices based on TBT is developed. In practical situations a large number of modes of vibrations contribute to the structures response. In this work a FE model of a smart cantilever beam have been obtained by varying the AR from 8 to 15, the length of the beam is kept constants and the thickness of the beam is varied. POF control technique is used to design a controller to suppress the vibration of the smart cantilever beam by considering three modes of vibration. Two different cases have been considered (AR=8,15). The simulation results show that the POF controller based on TBT is able to satisfactorily control the first three modes of vibration of the smart cantilever beam for different AR. Surface mounted piezoelectric sensors and actuators are usually placed at the extreme thickness positions of the structure to achieve most effective sensing and actuation. This subjects the sensors / actuators to high longitudinal stresses that might damage the piezoceramic material. Furthermore, surface mounted sensors/ actuators are likely to be damaged by contact with surrounding objects. Embedded shear sensors / actuators can be used to alleviate these problems. A FE model of a smart cantilever beam with embedded Latin American Journal of Solids and Structures 1(2012) 1 – 25 22 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam piezoelectric shear sensors / actuators based on laminate theory is developed. A POF controller is designed to control the vibration of the system. The performance of the controller is evaluated for different actuator locations while the position of sensor is kept constant. It was observed from the simulation results that the location of the actuator has negligible effect on the performance of the controller. A MIMO system with two sensors and two actuators has also been considered. A POF controller has been designed for the MIMO smart structure model to control the vibration of the system by considering the three modes of vibration. The beam with embedded shear sensors / actuators has been divided in to 10 FE with the sensors placed at the 6th and 10th positions and the actuators placed at the 4th and 8th positions. It can be inferred from the simulation results, that when the system is placed with the controller, the system’s performance meets the design requirements. It is observed that the maximum amplitude of the sensor output voltage is less for the multivariable case and the response takes lesser time to settle. APPENDIX 1. Translational mass matrix: ⎡ [70φ2 + 147φ + 78] ⎢ ⎢ ⎢ [35φ2 + 77φ + 44] L ρI ⎢ 4 [MρA ] = ⎢ 2 [35φ2 + 63φ + 27] 210(1 + φ) ⎢ ⎢ ⎢ ⎢ − [35φ2 + 63φ + 26] L ⎣ 4 L 4 L2 + 14φ + 8] 4 [35φ2 + 63φ + 26] L 4 2 − [7φ2 + 14φ + 6] L4 [35φ2 + 77φ + 44] [7φ2 [35φ2 + 63φ + 27] + 63φ + 26] [70φ2 + 147φ + 78] − [35φ2 + 77φ + 44] L 4 L2 + 14φ + 6] 4 − [35φ2 + 77φ + 44] L 4 2 [7φ2 + 14φ + 8] L4 − [35φ2 + 63φ + 26] L 4 [35φ2 L 4 − [7φ2 346 347 2. Rotational Mass matrix ⎡ 36 − (15ϕ − 3) L −36 − (15ϕ − 3) L ⎢ ⎢ 2 2 (10ϕ + 5ϕ + 4) L (15ϕ − 3) L (5ϕ2 − 5ϕ − 1) L2 − (15ϕ − 3) L ⎢ [MρI ] = ⎢ 2 2 ⎢ (10ϕ + 5ϕ + 4) L (15ϕ − 3) L 36 (15ϕ − 3) L ⎢ 2 2 ⎢ (5ϕ − 5ϕ − 1) L − (15ϕ − 3) L (15ϕ − 3) L (10ϕ2 + 5ϕ + 4) L2 ⎣ 348 3. The mass matrix for the sandwich beam element , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [M ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 349 M11 M21 M31 M41 M51 M61 M12 M22 M32 M42 M52 M62 M13 M23 M33 M43 M53 M63 M14 M24 M34 M44 M54 M64 M15 M25 M35 M45 M55 M65 M16 M26 M36 M46 M56 M66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Where, 1 γ L 3 I1 1 1 1 γ L 2 I1 , M = M = − , M14 = M41 = LI1 , M11 = LI1 , M12 = M21 = 13 31 2 2 3 2 (12η−L ) 4 (12η−L ) 6 Latin American Journal of Solids and Structures 1(2012) 1 – 25 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 23 350 γ L I1 1 γ L I1 M15 = M51 = − 12 (12η−L 2 ) , M16 = M61 = − 4 (12η−L2 ) , M24 = M42 = 2 2 1 γ L I1 , 2 (12η−L2 ) M22 = , ⎛ L⎜ 1 ⎝ 35 3 ⎞ ⎟ ⎠ −294I3 ηL2 + 35I2 L3 + 1680I2 η 2 + 13I3 L4 − 420I2 ηL + 42I1 L2 + 42γ 2 I1 L2 (12η−L2 ) ⎛ 11I3 L5 − 10080I2 η 2 + 1260I3 η 2 L − 231I3 L3 η + 126γ 2 I1 L3 ⎞ ⎟ L⎜ ⎠ ⎝ + 1260LI1 η + 840I2 ηL2 + 21I1 L3 1 , 2 2 210 (12η−L ) 4 2 2 2 2 2 3 L(3I3 L −28γ I1 L −28I1 L −84I3 ηL +560I3 η ) , 2 70 (12η−L2 ) 5 2 2 ⎛ 13I3 L + 10080I2 η + 2520I3 η L − 378I3 L3 η + 252γ 2 I1 L3 ⎞ ⎟ L⎜ ⎠ ⎝ − 2520LI1 η − 840I2 ηL2 − 42I1 L3 1 2 420 (12η−L2 ) 2 M23 = M32 = M25 = M52 = M26 = M62 = 351 M55 = ⎛ L⎜ 1 ⎝ 35 13I3 L4 − 35I2 L3 + 42γ 2 I1 L2 + 42I1 L2 − 294I3 ηL2 + 420I2 ηL + 1680I3 η 2 352 ⎛ L⎜ ⎝ 1 210 ⎛ L⎜ ⎝ 1 210 11I3 L 252I3 η 5 , (12η−L2 ) + 10080I2 η 2 + 1260I3 η 2 L − 231I3 L3 η + 126γ 2 I1 L3 ⎞ ⎟ ⎠ + 1260LI1 η − 840I2 ηL2 + 21I1 L3 2 M56 = M65 = M66 = ⎞ ⎟ ⎠ (12η−L2 ) L − 42I3 L η + 10080I1 η 2 + 63γ 2 I1 L4 + 28I1 L4 ⎞ ⎟ − 420I1 ηL2 + 2I3 L6 + 2520I2 η 2 L − 210I2 ηL3 ⎠ 2 2 2 , 4 (12η−L2 ) 2 . 1. The stiffness matrix for the sandwich beam element is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [K] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 353 K11 = AA11 , L K12 K22 K32 K42 K52 K62 K15 K25 K35 K45 K55 K65 K16 K26 K36 K46 K56 K66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 11 = K51 = − AB , L 4 2 2 2 2 2 6 A (D11 L +10A55 L +10γ A11 L −20η D11 L +120 D11 η ) , 2 2 5L (12η−L ) (D11 L4 +10A55 L2 +10γ 2 A11 L2 −20η D11 L2 +120 D11 η 2 ) (12η−L2 ) 1 AL(−D11 L −10B11 γ L −60A55 L−60γ A11 L+120γ B11 η) , K62 = 10 2 (12η−L2 ) ⎛ 2L6 D11 − 30L4 D11 η + 180L2 D11 η 2 − 15L5 γ B11 + 2160A55 η 2 ⎞ ⎟ A⎜ ⎝ ⎠ + 60A55 L4 − 360A55 ηL2 + 180γL3 B11 η + 45γ 2 L4 A11 , 15L(12η−L2 )2 3 K33 = K14 K24 K34 K44 K54 K64 AB11 11 , K13 = K31 = 0, K14 = K41 = − AA , K15 L L 3 2 2 AL(D L −10B γ L +60A L+60η A L+120γ B 11 11 55 11 11 η) 1 , − 10 2 (12η−L2 ) 11 , K25 = K52 = − 65 A K24 = K42 = − AB L L K26 = K13 K23 K33 K43 K53 K63 K12 = K21 = K16 = K61 = 0, K22 = K23 = K32 = K11 K21 K31 K41 K51 K61 2 2 , 2 K34 = K43 = 0, Latin American Journal of Solids and Structures 1(2012) 1 – 25 24 Vivek Rathi et al / Vibration attenuation and shape control of surface mounted, embedded smart beam 354 3 2 2 1 AL(D11 L −10B11 γ L +60A55 L+60γ A11 L+120γ B11 η) 11 , K44 = AA , 2 2 10 L (12η−L ) 11 , K46 = K64 = 0, K54 = AB L ⎛ L6 D11 − 60L4 A55 − 90L4 A11 γ 2 − 60L4 η D11 + 360D11 L2 η ⎞ ⎟ −A⎜ ⎠ ⎝ + 4320A55 η 2 − 720A55 ηL2 K63 = , 30L(12η−L2 )2 4 2 2 2 2 2 ) (D L +10A L +10γ A L −20η D L +120 D η 11 55 11 11 11 6A , 2 5L (12η−L2 ) K35 = K53 = K45 = K36 = K55 = 1 K56 = K65 = − 10 K66 = 355 AL(−D11 L3 −10B11 γ L2 −60A55 L−60γ 2 A11 L+120γ B11 η) , (12η−L2 ) 6 5 ⎛ 2L D11 + 15L B11 γ − 30L4 D11 η + 2160A55 η 2 + 180L2 D11 η 2 ⎞ ⎟ A⎜ ⎝ ⎠ + 60A55 L4 − 360A55 ηL2 − 180γL3 B11 η + 45γ 2 L4 A11 . 15L(12η−L2 )2 2 References 356 357 [1] H. 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