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Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam

Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 905160, 8 pages http://dx.doi.org/10.1155/2013/905160 Research Article Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam 1 2 2 2 1 Fabio Botta, Daniele Dini, Christoph Schwingshackl, Luca di Mare, and Giovanni Cerri Dipartimento di Ingegneria, Universitad ` egli StudiRomaTre,Via dellaVasca Navale,79-00146Roma, Italy Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK Correspondence should be addressed to Fabio Botta; fabio.botta@uniroma3.it Received 14 July 2013; Revised 11 October 2013; Accepted 21 October 2013 Academic Editor: Marc om Th as Copyright © 2013 Fabio Botta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Damping of vibrations is oeft n required to improve both the performance and the integrity of engineering structures, for example, gas turbine blades. In this paper, we explore the possibility of using piezoelectric plates to control the multimode vibrations of a cantilever beam. To develop an effective control strategy and optimize the placement of the active piezoelectric elements in terms of vibrations amplitude reduction, a procedure has been developed and a new analytical solution has been proposed. eTh results obtained have been corroborated by comparison with the results from a multiphysics finite elements package (COMSOL), results available in the literature, and experimental investigations carried out by the authors. 1. Introduction for a review). The rfi st study concerned with the optimal position to damp a specified mode has been that of Crawley The vibration control is a problem of great interest in anddeLuis[4]. They found that the actuators should be in regions of higher average strain. Analogous results have many engineering elds fi since it allows avoiding problems connected with the vibrations. Blade vibrations in aircraft been found by other researchers [8, 9]. For a cantilever beam engines, for example, are oen ft induced by interactions Sunar and Rao [10], Demetriou [11], and Bruant et al. [12] found that the closer the piezoelectric actuators are to the betweenblades andfluidand theassociatedfatigue phe- nomena can give rise to catastrophic failures [1–3]. Typically, xfi ed end, the more ecffi ient they are. For a simply supported passive damping systems, such as friction damping, are beam Yang et al. [13] found that, to control one specicfi mode, the optimal position for the piezoelectric plates is within used to increase the blade life. es Th e systems are very effective, but, in contrast to active damping elements, they the regions separated by the vibration nodal lines. In the paper by Barboni et al. [14] the possibility of exciting the are not able to change their characteristics depending on flexural dynamics of an Euler-Bernoulli beam, according to the system response. In the last two decades, the adoption of piezoelectric elements has received considerable attention a single mode, is examined. The results show that, to excite a desired mode, the actuator must be placed between two by many researchers for its potential applicability to differ- ent areas of mechanical, aerospace, aeronautical, and civil consecutive points at which the curvature becomes zero. engineering. These elements have an interesting coupling Aldraihem et al. [15] studied the optimal length and location for different boundary conditions. They investigated beams between electrical and mechanical properties: a deformation appearswhenanelectricfieldisapplied andviceversa [ 4, 5]. with one pair and two pairs of piezoelectric actuators. eTh ir Their effectiveness to damp a particular excited mode or a optimization criterion was based on beam modal cost and controllability index; moreover, they added a penalty term multimode combination strongly depends on their position; in fact the study of their optimal position has received to consider the actuator length (and the cost, weight, and space factors associated to this). Baz and Poh [16]studied the increasing attention. Typically, the aim of these studies is to n fi d the position that minimizes an objective function effects of varying the thickness and material of the bonding or maximizes the degree of modal controllability (see [6, 7] layer as well as the position of the piezoelectric actuators. 2 Advances in Acoustics and Vibration They consider three beam elements to model a cantilever Piezoelectric plates beam and the results show that it is preferable to place the a V(t) actuator in a region of large strain. Subsequently Yang and Lee presented an analytical model for simultaneous optimiza- tion of noncollocated [17]and collocated [18] piezoelectric M(t) M(t) sensor/actuator placement and feedback control gain. The L x results show that this procedure can avoid the instability of the structural control system. Q. Wang and C. M. Wang [19] Figure 1: Reference configuration and action of the PZT plates. studied modal and multimodal vibrations. They propose a new controllability index and illustrated various beam exam- ples with apairofcollocatedpiezoelectric actuators. More So by indicating𝑤 the vertical displacement, the virtual recently, studies about their use in blades of turbomachinery work of the PZT plates can be written as (the virtual quantities have been carried out; however, only few of these concern areoversignedbyatilde) active damping [20–23]. Unfortunately in many real cases the loads applied to the structure excite more than one mode, 󵄨 󵄨 󵄨 󵄨 ̃ ̃ 𝜕 𝑤 𝜕 𝑤 󵄨 󵄨 with different amplitudes, and the excited modes can change 󵄨 󵄨 =𝑀( 󵄨 − 󵄨 ). (3) 󵄨 󵄨 󵄨 󵄨 during service. eTh refore, the implementation of an active 󵄨𝑥=𝑎+(ℎ/2) 󵄨𝑥=𝑎−(ℎ/2) system that is capable of changing the work configuration of the piezoelectric plates can increase considerably their The variables 𝑎 andℎ can vary within the domain: ecffi iency in the damping of the vibrations. In this paper, a new function to nd fi the optimal placement of piezoelectric plates to control the multimode vibrations of the cantilever 0≤𝑎− ≤𝐿 , beam, with different amplitudes of the single modes, is den fi ed. An analytical solution is proposed and the results are 0≤𝑎+ ≤𝐿 , also compared with the FEM simulations and results from the 𝑏 (4) literature with very good agreement. 0≤𝑎≤𝐿 , 0≤ℎ≤𝐿 2. Governing Equations for Piezoelectric Coupled Beam and their value depends on the modes that must be damped [14]. In Figure 1 an Euler-Bernoulli cantilever beam with attached The virtual work of the elastic and inertial forces can be piezoelectric patches is schematically shown; the two PZT written, respectively, as plates are applied in a symmetrical position with respect to the mid plane. Indicating by , ,and the 𝑒 in 𝑎 𝐿 2 2 virtual work of the elastic, inertial, and piezoelectric forces, 𝜕 𝑤 𝜕 𝑤̃ =𝐸 𝐼 ∫ 𝑑𝑥, 𝑒 𝑏 𝑏 2 2 respectively, the principle of the virtual work can be written as (5) (in the following, the PZT plates will be considered perfectly 𝜕 𝑤 bonded to the structure, their mass and inertia negligible =−𝜌𝑆 ∫ 𝑤𝑑̃𝑥. in with respecttothe mass andinertia of thebeam, andtheir 𝜕𝑡 thickness very lower than the thickness of the beam) Using the modal analysis technique, indicating by𝜙 (𝑥) the 𝑖 th flexural modal displacement of the cantilever beam +𝛿𝐿 +𝛿𝐿 =0. and by𝑋 (𝑡) its amplitude, the vertical displacement will be (1) 𝑒 in 𝑎 approximated by Using the pin force model [4] the action of the PZT plates 𝑤 𝑥,𝑡 =∑𝑋 𝑡 𝜙 𝑥 , (6) ( ) () ( ) can be modeled by two flexural moments concentrated at the 𝑖 𝑖 𝑖=1 end of the plates (Figure 1)with where 𝑁 is the number of the considered modes. Substituting the last in (3), (5), considering𝑤=𝜙 ̃ (𝑥) , 𝑀 (𝑡 )= 𝐸 𝑇 Λ(𝑡 ), 𝑎 𝑎 𝑏 andsuccessivelyin(1), the following governing equation is 6+𝜓 obtained: .. Λ 𝑡 = 𝑉 𝑡 , () () (2) (7) 𝑇 M (𝑡 )+ KX(𝑡 )= B(𝑎,ℎ )𝑉 (𝑡 ), 𝐸 𝑇 𝑏 𝑏 where X represents the vector of the amplitudes of the 𝜓= . 𝐸 𝑇 𝑎 𝑎 modes, M and K are, respectively, the mass and the stiffness 𝑐𝑇 𝛿𝐿 𝛿𝐿 𝜕𝑥 𝜕𝑥 𝛿𝐿 𝛿𝐿 𝛿𝐿 𝛿𝐿 𝜕𝑥 𝜕𝑥 𝛿𝐿 Advances in Acoustics and Vibration 3 𝑋 𝑡 () matrix, and B(𝑎,ℎ) is the vector control (the prime denotes the rfi st spatial derivative): −𝑡𝛼/2 2 2 2 2 2 2 𝑒 𝐵 𝑉 cosh((1/2)𝑡 √𝛼 −4𝜔 )√𝛼 −4𝜔 (𝜔 −𝜔 ) 𝑖 𝑗 𝑖 𝑖 𝑖 𝑗 =∑ − 2 2 2 2 2 2 B(𝑎,ℎ ) 𝑗=1 √𝛼 −4𝜔 (𝛼 𝜔 +(𝜔 −𝜔 )) 𝑖 𝑗 𝑖 𝑗 ℎ ℎ 󸀠 󸀠 −𝑡𝛼/2 2 2 ̆ 2 2 =𝑀[𝜙 (𝑎+ )−𝜙 (𝑎− ), 𝑒 𝐵 𝑉 (𝛼 sinh((1/2)𝑡 𝛼 −4𝜔 )(𝜔 +𝜔 )) 1 1 𝑖 𝑗 𝑖 𝑗 2 2 2 2 2 2 2 2 (8) √ 𝛼 −4𝜔 (𝛼 𝜔 +(𝜔 −𝜔 )) ℎ ℎ 𝑖 𝑗 𝑖 𝑗 󸀠 󸀠 𝜙 (𝑎+ )−𝜙 (𝑎− ),..., 2 2 2 2 2 2 𝐵 𝑉 𝛼 sin(𝜔 𝑡)𝜔 𝐵 𝑉 cos(𝑡𝜔 )(𝜔 −𝜔 ) 𝑖 𝑗 𝑗 𝑖 𝑗 𝑗 𝑗 𝑖 𝑗 ℎ ℎ 󸀠 󸀠 + + . 𝜙 (𝑎+ )−𝜙 (𝑎− )] 2 2 𝑁 𝑁 2 2 2 2 2 2 2 2 2 2 𝛼 𝜔 +(𝜔 −𝜔 ) 𝛼 𝜔 +(𝜔 −𝜔 ) 𝑗 𝑖 𝑗 𝑗 𝑖 𝑗 (13) with 𝑀=(𝜓/(6+𝜓))𝐸 𝑇 (𝑑 /𝑇 ). 𝑎 𝑎 𝑏 31 𝑎 It is possible to obtain an approximate form of (13)con- If the Rayleigh damping is considered, (7)becomes sidering that the 𝑖 th term is the dominant one (the𝑖 th term of 𝑉(𝑡) is exciting𝑋 (𝑡)near its resonance). Aeft r some algebraic manipulation, (13) can be simplified to .. M 𝑡 + CX 𝑡 + KX 𝑡 = B 𝑎,ℎ 𝑉 𝑡 (9) X() () () ( ) () 𝐵 𝑉 sin(𝜔 𝑡) 𝑖 𝑖 𝑖 𝑋 (𝑡 )≅ 𝛼𝜔 with 2 2 2 2 (14) −(1/2)𝑡(𝛼+ √𝛼 −4𝜔 ) 𝑡√𝛼 −4𝜔 𝑖 𝑖 𝑒 (−1+𝑒 )𝐵 𝑉 𝑖 𝑖 − . C=𝛼 M+𝛽 K. (10) 2 2 𝛼 √𝛼 −4𝜔 The second term of the right hand side of ( 14)represents Indicating by𝜔 the diagonal matrix of the natural fre- the transient part so that, if this is neglected, the amplitude of quencies of the beam, using the normal modes (M = I, K = the displacement of the free end can be written as 𝜔 ), and assuming𝛽=0 ,(9)becomes 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝐵 (𝑎,ℎ )𝑉 𝜙 (𝐿 ) ⌣ 󵄨 󵄨 󵄨 󵄨 𝑖 𝑖 𝑖 𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑤 𝑎,ℎ =∑ . (15) .. 󵄨 ( )󵄨 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 ̇ 󵄨 󵄨 𝛼𝜔 𝑡 +𝛼 X 𝑡 +𝜔 X 𝑡 = B 𝑎,ℎ 𝑉 𝑡 . (11) 󵄨 󵄨 X() () () ( ) () 󵄨 𝑖 󵄨 𝑖=1 In the following a bimodal excitation is considered, indi- If a single mode excitation is considered, Barboni et al. cating by𝑖 ,𝑖 the indexes of the rfi st and second considered 1 2 [14] found the optimal placement of PZTs. However, in modes respectively, and𝑟≤1 therelativecontributionto some practical applications, for example, gas turbine blades, theexcitationinduced by thesecondmodewithrespect to theloadspectrumcan includevarious modes,eachof the rfi st one (i.e., 𝑟=1 corresponds to the second mode which is characterized by different amplitudes, and a new governing the excitation without any contribution from the strategy needs to be developed. In this paper, the multimode first mode). From ( 12): dampingwillbeobtainedbyapplyingacounterphase load, 𝑉 (𝑡 )=(1−𝑟 )cos(𝜔 𝑡)+𝑟 cos(𝜔 𝑡). (16) 𝑖 𝑖 by PZTs plates, to the external excitation. The effectiveness 1 2 of the piezoelectric elements will be measured by the ampli- Substituting (16)into(15) and introducing tude of the vertical displacement of the free end, so that the most eeff ctive (optimal) position, identiefi d in terms 𝑀 1−𝑟 of 𝑎 and ℎ (Figure 1), will be the location which maximizes 𝑔 𝑟,𝜉 = 𝜙 (𝐿 )𝜙 𝜉 , ( ) () 𝑖 𝑖 𝑏 𝑖 1 1 𝛼 𝜔 this amplitude. eTh refore, assigning a general spectrum to the 1 (17) PZT load (which corresponds to the spectrum of 𝑉(𝑡) ;see 𝑀 𝑟 (2)) 𝑔 (𝑟,𝜉 )= 𝜙 (𝐿 )𝜙 (𝜉 ), 𝑖 𝑖 𝑏 𝑖 2 2 𝛼 𝜔 it is obtained that 𝑉 (𝑡 )= ∑𝑉 cos(𝜔 𝑡 ), (12) 𝑗 𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ⌣ ℎ ℎ 󵄨 󵄨 󵄨 󵄨 𝑗=1 󵄨 󵄨 󵄨 󵄨 󵄨 (𝑟,𝑎,ℎ )󵄨 =󵄨𝑔 (𝑟,𝑎+ )−𝑔 (𝑟,𝑎− )󵄨 𝑖 𝑖 󵄨 󵄨 󵄨 󵄨 1 1 󵄨 󵄨 󵄨 󵄨 2 2 󵄨 󵄨 (18) 󵄨 󵄨 󵄨 󵄨 ℎ ℎ 󵄨 󵄨 where 𝑁𝑠 is the number of excited modes. eTh solution of 󵄨 󵄨 + 𝑔 (𝑟,𝑎+ )−𝑔 (𝑟,𝑎− ) . 󵄨 󵄨 󵄨 𝑖 𝑖 󵄨 2 2 󵄨 󵄨 2 2 (11)becomes 󵄨 󵄨 𝑁𝑠 𝑐𝑇 𝑁𝑠 4 Advances in Acoustics and Vibration Indicating by (𝜉 =𝑎+(ℎ/2),𝜉 =𝑎−(ℎ/2)),respectively, of the PZT elements can be found by solving the following 1 2 the position of the right and the left ends of the piezoelectric system of equations: plates it will be (𝑟,𝜉 ) 𝑖 ,𝑖 2 1 2 󵄨 󵄨 ⌣ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜕𝜉 󵄨 󵄨 𝑤 (𝑟,𝜉 ,𝜉 ) = 𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 ) 󵄨 󵄨 2 󵄨 󵄨 1 2 𝑖 1 𝑖 2 󵄨 󵄨 1 1 󵄨 󵄨 󵄨 󵄨 (19) 󵄨 󵄨 𝜙 (𝐿 ) 𝜙 (𝐿 ) 󵄨 󵄨 𝑀 𝑖 𝑏 𝑖 𝑏 󵄨 󵄨 + 𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 ). 1 2 󵄨 󵄨 𝑖 1 𝑖 2 2 2 = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]=0, 󵄨 󵄨 2 2 𝑖 𝑖 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 1 2 ̂ ̂ If (𝜉 ,𝜉 ) are the coordinates of the absolute maximum 2 1 2 𝜕 𝑓 (𝑟,𝜉 ) 𝑖 ,𝑖 2 1 2 of | (𝑟,𝜉 ,𝜉 )|,itispossibletowrite 1 2 2 𝜕𝜉 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ 󵄨 󵄨 ⌢ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ ̂ ̂ ̂ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑤 (𝑟,𝜉 ,𝜉 )󵄨 =󵄨𝑤 (𝑟, 𝜉 ,𝜉 )󵄨 = 𝑔 (𝑟, 𝜉 )−𝑔 (𝑟, 𝜉 ) 𝜙 (𝐿 ) 𝜙 (𝐿 ) 1 2 1 2 󵄨 𝑖 1 𝑖 2 󵄨 󵄨 󵄨 󵄨 󵄨 𝑀 𝑖 𝑏 𝑖 𝑏 1 1 1 2 󵄨 󵄨 󵄨 󵄨 max 󵄨 󵄨 = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]>0. 𝑖 2 𝑖 2 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 󵄨 󵄨 1 2 󵄨 󵄨 ̂ ̂ 󵄨 󵄨 + 𝑔 (𝑟, 𝜉 )−𝑔 (𝑟, 𝜉 ) . 󵄨 𝑖 1 𝑖 2 󵄨 2 2 (25) 󵄨 󵄨 (20) eTh solution will provide all the local minima and the absolute minimum will then be selected among these. Considering that 𝑔 (𝑟,𝜉) has, for all the considered val- ues of𝑘 ,its positive absolute maximumat 𝜉 =𝐿 ,thisvalue 1 𝑏 will also be the abscissa of the positive absolute maximum 3. Results and Discussions for|𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 )|+|𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 )|. This implies 𝑖 1 𝑖 2 𝑖 1 𝑖 2 1 1 2 2 This section reports analytical results for different bimodal that the PZT plates should always be placed with their right excitations and their comparison with FEM simulations. edges terminating at the tip of the beam independently of the eTh se simulations have been performed by using the fre- excited modes and value of𝑟 , and hence quency module of the structural mechanics module of 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ 󵄨 󵄨 󵄨 󵄨 COMSOL Multiphysics software [ 24]. Specifically, a steel 󵄨 󵄨 󵄨 󵄨 𝑤 (𝑟,𝜉 ,𝜉 ) = 𝑤 (𝑟,𝐿 ,𝜉 ) 󵄨 󵄨 󵄨 󵄨 󵄨 1 2 󵄨 󵄨 𝑏 2 󵄨 󵄨 󵄨 max 󵄨 󵄨 max beam of 30 cm of length has been taken into account; with bimodal control in mind and focusing on combinations of 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (21) = 𝑔 (𝑟,𝐿 )−𝑔 (𝑟, 𝜉 ) 󵄨 󵄨 𝑖 𝑏 𝑖 2 1 1 the rst fi ve fi modes, the wavelength of the highest mode 󵄨 󵄨 has been divided into 50 subintervals of length 3 mm, so 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 + 𝑔 (𝑟,𝐿 )−𝑔 (𝑟, 𝜉 ) . 󵄨 󵄨 𝑖 𝑏 𝑖 2 2 2 that Δ𝑎 = Δℎ = 3 mm and 5000 different combinations 󵄨 󵄨 for 𝑎 and ℎ have been considered. For each of these, the Moreover, since amplitude of the response of the tip, to a periodic load, with the frequency corresponding to one of the rfi st vfi e eigen- 0<𝑔 (𝑟,𝐿 )>𝑔 (𝑟,𝜉 ) ∀𝜉∈[0,𝐿 ),∀.𝑗 (22) frequencies has been calculated. Moreover the amplitude 𝑗 𝑏 𝑗 𝑏 response to a linear combination of the previous loads has been obtained by superposition of the response to two of the (21)becomes eigenfrequencies. eTh optimal position has been chosen to 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ be the one which corresponds to the maximum amplitude. 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ 󵄨 𝑤 𝑤 󵄨 (𝑟,𝜉 ,𝜉 )󵄨 =󵄨 (𝑟,𝐿 ,𝜉 )󵄨 =𝑔 (𝑟,𝐿 ) 1 2 𝑏 2 𝑖 𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 max 󵄨 󵄨 max In Figure 2; 𝑓 (𝑟, 𝜉 ), the position of its absolute minimum, 1,2 2 (23) maximum and the position of the center of the piezoelectric +𝑔 (𝑟,𝐿 )−𝑓 (𝑟, 𝜉 ), 𝑖 𝑏 𝑖 ,𝑖 2 2 1 2 plates, for different 𝑟 ratios, are reported. It is possible to observe that for𝑟=0 and𝑟=1 (that correspond, resp., where to theexcitationofthe rfi st andsecondmodes)the optimal congfi uration of the PZT coincides with that obtained in [ 14]. 𝑓 (𝑟, 𝜉 ) 𝑖 ,𝑖 2 1 2 For𝑟=0 , 𝑓 (0,𝜉 )attains its absolute minimum at the 1,2 2 boundary points 𝜉 =0,sothatthe optimalcongfi uration ̂ ̂ =𝑔 (𝑟, 𝜉 )+𝑔 (𝑟, 𝜉 ) 𝑖 2 𝑖 2 1 2 is with the piezoelectric plates distributed along the entire length of the beam. For increasing values of 𝑟 the position of ̆ 𝜙 (𝐿 ) 𝜙 (𝐿 ) 𝑖 𝑏 𝑖 𝑏 1 󸀠 2 󸀠 ̂ ̂ the absolute minimum does not change until the derivative = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]. 2 2 𝑖 𝑖 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 1 2 of 𝑓 (𝑟, 𝜉 )becomes zero in 𝜉 =0 (25.1): 𝑖 ,𝑖 2 2 1 2 (24) 𝜙 (0)𝜙 (𝐿 ) 1 𝑏 𝑟=̂ ≅0.5. (26) The ordinate 𝜉 of theabsolutemaximum of|𝑤 (𝑟,𝜉 ,𝜉 )| 2 1 2 𝜙 (0)𝜙 (𝐿 )−𝜙 (0)𝜙 (𝐿 )(𝜔 /𝜔 ) 1 1 𝑏 2 2 𝑏 1 2 will coincide with the absolute minimum of the 𝑓 (𝑟,𝜉 ). 𝑖 ,𝑖 2 1 2 This position changes with the considered modes and the For values of 𝑟 above 𝑟 , the optimal configuration grad- modes ratio 𝑟 so that the optimal location of the left edge ually changes to reach the optimal one for the second mode 󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠󸀠 󸀠󸀠󸀠 󸀠󸀠 󸀠󸀠 𝜕𝑓 Advances in Acoustics and Vibration 5 1.0 0.5 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.2 0.4 0.6 0.8 1 −0.1 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 2: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 1,2 2 edge (analytical);×: FEM simulations. 1.0 0.2 0.8 0.6 𝜉 0.1 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.1 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 3: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 2,3 2 edge (analytical);×: FEM simulations. 1.0 0.15 0.8 0.10 0.6 0.05 0.4 0.2 0.4 0.6 0.8 1 −0.05 0.2 −0.10 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 4: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 3,4 2 right edge (analytical);×: FEM simulations. f (r, ) 3, 4 2 f (r, ) 1, 2 2 f (r, ) 2, 3 2 6 Advances in Acoustics and Vibration 1.0 0.5 0.4 0.8 0.3 0.6 0.2 r 0.4 0.1 0.2 0.2 0.4 0.6 0.8 1 −0.1 0 0.2 0.4 0.6 0.8 1 𝜉 𝜉 2 2 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 5: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 1,3 2 edge (analytical);×: FEM simulations. 1.0 0.2 0.8 0.6 0.1 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.1 0 0.2 0.4 0.6 0.8 1 𝜉 𝜉 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 6: 𝑓 (𝑟, 𝜉 );red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 2,5 2 right edge (analytical);×: FEM simulations. at𝑟=1 . eTh FEM results show that there is a very good comparisons with the FEM are in agreement with all the agreement between the numerical simulations and the analyt- cases analyzed. eTh proposed analytical method has been ical results. Different mode combinations are represented in also compared with the experimental investigations reported Figures 3, 4, 5, 6,and 7: the optimal configuration varies from in [25–27]. To test different load spectra and configurations theoptimal valuefor theindividual 𝑖 -excited mode (𝑟=0 ) an experimental apparatus has been designed and built; to that for the mode 𝑖 (𝑟=1 ). For all the considered coupled moreover, a MATLAB code has been developed to control modes, the optimal position changes continuously except for the actuation and the modal contributions. Four piezoelectric thecouple((1)–(3)), where a different type of behaviour is plates have been used in three different configurations. observed (Figure 5), with a sharp transition at 𝑟 ≅ 0.8 .This Experiments have been done with one mode and multimodal is duetothe fact that,when 𝑟 is low, theabsoluteminimum vibrations. A good agreement was shown in all the considered of 𝑓 (𝑟,𝜉 ) coincides with that obtained for the optimal cases. 1,3 2 congfi uration of the rfi st mode ( 𝑟=0 ). When the ratio reaches the value of 𝑟 ≅ 0.46 ,arelative minimumappears, but this develops into an absolute minimum and, therefore, 4. Conclusions coincides with the left end of the piezoelectric plates only when 𝑟 > 0.8 ; for such values the PZT elements should be In this work a new theoretical model for the optimal active only over about half of the length of the beam. The placement of piezoelectric plates to control the multimode f (r, ) 2, 5 2 f (r, ) 1, 3 2 Advances in Acoustics and Vibration 7 1.0 0.15 0.8 0.10 0.6 0.05 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.05 −0.10 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 7: 𝑓 (𝑟, 𝜉 );red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 3,5 2 right edge (analytical);×: FEM simulations. 𝛼 : Damping coefficient vibrations of a cantilever beam is proposed. After a detailed description of the theoretical model, bi-modal excitation 𝜙 (𝑥) : 𝑖 th flexural mode of the cantilever beam is considered. eTh designs of the optimal configurations, 𝜌 : Mass density 𝑋 (𝑡): Amplitude of the 𝑖 th mode for different coupled modes and different relative contribu- tions, are reported. eTh comparison between numerical and 𝜔 : Natural frequency. theoretical results is shown to confirm the validity of the optimization technique. This model can be very useful in References many real situations when the spectrum of the load excites more eigenmodes and has, therefore, implications for the [1] E. Poursaeidi and M. Salavatian, “Fatigue crack growth simula- development of strategies to control multimode vibrations. tion in a generator fan blade,” Engineering Failure Analysis,vol. 16, no. 3, pp. 888–898, 2009. This methodology has been extended to rotating systems [ 28, 29]. Its application to real systems subjected to the centrifugal [2] L. Witek, “Experimental crack propagation and failure analysis loads, for example, gas turbine blades [30], and impulse loads of the first stage compressor blade subjected to vibration,” Engineering Failure Analysis,vol.16, no.7,pp. 2163–2170, 2009. [31] will constitute the object of further contributions. [3] J.KubiakSz, G. UrquizaB,J.Garc´ıa C, and F. 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K:Stiffnessmatrix [7] V. Gupta, M. Sharma, and N. ak Th ur, “Optimization criteria 𝐿 : Beam length 𝑏 for optimal placement of piezoelectric sensors and actuators 𝑀 : Piezoelectric bending moment on a smart structure: a technical review,” Journal of Intelligent M:Massmatrix Material Systems and Structures,vol.21, no.12, pp.1227–1243, 𝑟 : Percentage coupling coefficient 2010. 𝑆 : Cross-section area of the beam [8] K. D. Dhuri and P. Seshu, “Piezo actuator placement and sizing 𝑇 : Piezoelectric thickness for good control eeff ctiveness and minimal change in original 𝑇 :Beamthickness system dynamics,” Smart Materials and Structures,vol.15, no.6, pp.1661–1672,2006. 𝑉 :Voltageappliedtothepiezoelectricplates 𝑤 : Vertical displacement [9] K. R. Kumar and S. Narayanan, “Active vibration control of ̃ beams with optimal placement of piezoelectric sensor/actuator 𝑤 : Virtual vertical displacement f (r, ) 3, 5 2 8 Advances in Acoustics and Vibration pairs,” Smart Materials and Structures,vol.17, no.5,Article ID plates for active vibration control of a cantilever beam,” Interna- 055008, 2008. tional Journal of Engineering and Technology,vol.5,no. 5, 2013. [10] M. Sunar and S. S. Rao, “Distributed modeling and actuator [27] N. Marx, Counter vibration on gas turbine compressor blades via location for piezoelectric control systems,” AIAA Journal,vol. piezoelectric plates [M.S. thesis], Imperial College, London, UK, 34, no. 10, pp. 2209–2211, 1996. 2012. [11] M. A. Demetriou, “Numerical algorithm for the optimal [28] F. Botta, N. Marx,C.W.Schwingshackl,G.Cerri,and D. Dini, placement of actuators and sensors for flexible structures,” in “A wireless vibration control technique for gas turbine blades Proceedings of the American Control Conference,pp. 2290–2294, using piezoelectric plates and contactless energy transfer,” in Chicago, Ill, USA, June 2000. Proceedings of the ASME Turbo Expo,San Antonio,Tex,USA, June 2013. [12] I. Bruant, G. Coffignal, F. Lene, and M. Verge, “A methodology for determination of piezoelectric actuator and sensor location [29] F. Botta, D. Dini, and R. de Lieto Vollaro, “A new function on beam structures,” Journal of Sound and Vibration,vol.243, for the optimal placement of piezoelectric plates to control no. 5, pp. 861–882, 2001. multimode vibrations of a rotating beam,” International Journal of Engineering and Technology,vol.5,no. 5, 2013. [13] Y. Yang, Z. Jin, and C. K. Soh, “Integrated optimal design of vibration control system for smart beams using genetic algo- [30] G. Cerri, M. Gazzino, F. Botta, and C. 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[17] S. M. Yang and Y. J. Lee, “Optimization of non-collocated sensor/actuator location and feedback gain in control systems,” Smart Materials and Structures,vol.2,no. 2, pp.96–102, 1993. [18] S. M. Yang and Y. J. Lee, “Vibration suppression with optimal sensor/actuator location and feedback gain,” Smart Materials and Structures,vol.2,no. 4, pp.232–239,1993. [19] Q. Wang and C. M. Wang, “Optimal placement and size of piezoelectric patches on beams from the controllability perspective,” Smart Materials and Structures,vol.9,no. 4, pp. 558–567, 2000. [20] A. Hohl, M. Neubauer, S. M. Schwarzendahl, L. Panning, and J. Wallaschek, “Active and semiactive vibration damping of turbine blades with piezoceramics,” in Active and Passive Smart Structures and Integrated Systems,vol.7288of Proceedings of SPIE, San Diego, Calif, USA, March 2009. [21] I. Goltz, H. Bhmer, R. Nollau, J. Belz, B. Grueber, and J. R. Seume, “Piezo-electric actuation of rotor blades in an axial compressor with piezoceramics,” in Proceedings of 8th European Conference on Turbomachinery (ETC ’09), Graz, Austria, March [22] A. J. Provenza and C. R. Morrison, “Control of fan blade vibrations using piezo-electric and bi-directional telemetry,” in Proceedings of the ASME Turbo Expo, Vancouver, Canada, June [23] S. M. Lin, I. C. Mao, and J. H. Lin, “Vibration of a rotating smart beam,” AIAA Journal,vol.45, no.2,pp. 382–389, 2007. [24] Comsol Multiphysics User’s Guide Version 4.3. [25] F. Botta, N. Marx, S. Gentili et al., “Optimal placement of piezoelectric plates for active vibration control of gas turbine blades: experimental results,” in Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems,vol. 8345 of Proceedings of SPIE, San Diego, Calif, USA, March 2012. [26] F. Botta, N. Marx,D.Dini,R.deLieto Vollaro, andG.Battista, “Experimental results for optimal placement of piezoelectric International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam

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Copyright © 2013 Fabio Botta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 905160, 8 pages http://dx.doi.org/10.1155/2013/905160 Research Article Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam 1 2 2 2 1 Fabio Botta, Daniele Dini, Christoph Schwingshackl, Luca di Mare, and Giovanni Cerri Dipartimento di Ingegneria, Universitad ` egli StudiRomaTre,Via dellaVasca Navale,79-00146Roma, Italy Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK Correspondence should be addressed to Fabio Botta; fabio.botta@uniroma3.it Received 14 July 2013; Revised 11 October 2013; Accepted 21 October 2013 Academic Editor: Marc om Th as Copyright © 2013 Fabio Botta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Damping of vibrations is oeft n required to improve both the performance and the integrity of engineering structures, for example, gas turbine blades. In this paper, we explore the possibility of using piezoelectric plates to control the multimode vibrations of a cantilever beam. To develop an effective control strategy and optimize the placement of the active piezoelectric elements in terms of vibrations amplitude reduction, a procedure has been developed and a new analytical solution has been proposed. eTh results obtained have been corroborated by comparison with the results from a multiphysics finite elements package (COMSOL), results available in the literature, and experimental investigations carried out by the authors. 1. Introduction for a review). The rfi st study concerned with the optimal position to damp a specified mode has been that of Crawley The vibration control is a problem of great interest in anddeLuis[4]. They found that the actuators should be in regions of higher average strain. Analogous results have many engineering elds fi since it allows avoiding problems connected with the vibrations. Blade vibrations in aircraft been found by other researchers [8, 9]. For a cantilever beam engines, for example, are oen ft induced by interactions Sunar and Rao [10], Demetriou [11], and Bruant et al. [12] found that the closer the piezoelectric actuators are to the betweenblades andfluidand theassociatedfatigue phe- nomena can give rise to catastrophic failures [1–3]. Typically, xfi ed end, the more ecffi ient they are. For a simply supported passive damping systems, such as friction damping, are beam Yang et al. [13] found that, to control one specicfi mode, the optimal position for the piezoelectric plates is within used to increase the blade life. es Th e systems are very effective, but, in contrast to active damping elements, they the regions separated by the vibration nodal lines. In the paper by Barboni et al. [14] the possibility of exciting the are not able to change their characteristics depending on flexural dynamics of an Euler-Bernoulli beam, according to the system response. In the last two decades, the adoption of piezoelectric elements has received considerable attention a single mode, is examined. The results show that, to excite a desired mode, the actuator must be placed between two by many researchers for its potential applicability to differ- ent areas of mechanical, aerospace, aeronautical, and civil consecutive points at which the curvature becomes zero. engineering. These elements have an interesting coupling Aldraihem et al. [15] studied the optimal length and location for different boundary conditions. They investigated beams between electrical and mechanical properties: a deformation appearswhenanelectricfieldisapplied andviceversa [ 4, 5]. with one pair and two pairs of piezoelectric actuators. eTh ir Their effectiveness to damp a particular excited mode or a optimization criterion was based on beam modal cost and controllability index; moreover, they added a penalty term multimode combination strongly depends on their position; in fact the study of their optimal position has received to consider the actuator length (and the cost, weight, and space factors associated to this). Baz and Poh [16]studied the increasing attention. Typically, the aim of these studies is to n fi d the position that minimizes an objective function effects of varying the thickness and material of the bonding or maximizes the degree of modal controllability (see [6, 7] layer as well as the position of the piezoelectric actuators. 2 Advances in Acoustics and Vibration They consider three beam elements to model a cantilever Piezoelectric plates beam and the results show that it is preferable to place the a V(t) actuator in a region of large strain. Subsequently Yang and Lee presented an analytical model for simultaneous optimiza- tion of noncollocated [17]and collocated [18] piezoelectric M(t) M(t) sensor/actuator placement and feedback control gain. The L x results show that this procedure can avoid the instability of the structural control system. Q. Wang and C. M. Wang [19] Figure 1: Reference configuration and action of the PZT plates. studied modal and multimodal vibrations. They propose a new controllability index and illustrated various beam exam- ples with apairofcollocatedpiezoelectric actuators. More So by indicating𝑤 the vertical displacement, the virtual recently, studies about their use in blades of turbomachinery work of the PZT plates can be written as (the virtual quantities have been carried out; however, only few of these concern areoversignedbyatilde) active damping [20–23]. Unfortunately in many real cases the loads applied to the structure excite more than one mode, 󵄨 󵄨 󵄨 󵄨 ̃ ̃ 𝜕 𝑤 𝜕 𝑤 󵄨 󵄨 with different amplitudes, and the excited modes can change 󵄨 󵄨 =𝑀( 󵄨 − 󵄨 ). (3) 󵄨 󵄨 󵄨 󵄨 during service. eTh refore, the implementation of an active 󵄨𝑥=𝑎+(ℎ/2) 󵄨𝑥=𝑎−(ℎ/2) system that is capable of changing the work configuration of the piezoelectric plates can increase considerably their The variables 𝑎 andℎ can vary within the domain: ecffi iency in the damping of the vibrations. In this paper, a new function to nd fi the optimal placement of piezoelectric plates to control the multimode vibrations of the cantilever 0≤𝑎− ≤𝐿 , beam, with different amplitudes of the single modes, is den fi ed. An analytical solution is proposed and the results are 0≤𝑎+ ≤𝐿 , also compared with the FEM simulations and results from the 𝑏 (4) literature with very good agreement. 0≤𝑎≤𝐿 , 0≤ℎ≤𝐿 2. Governing Equations for Piezoelectric Coupled Beam and their value depends on the modes that must be damped [14]. In Figure 1 an Euler-Bernoulli cantilever beam with attached The virtual work of the elastic and inertial forces can be piezoelectric patches is schematically shown; the two PZT written, respectively, as plates are applied in a symmetrical position with respect to the mid plane. Indicating by , ,and the 𝑒 in 𝑎 𝐿 2 2 virtual work of the elastic, inertial, and piezoelectric forces, 𝜕 𝑤 𝜕 𝑤̃ =𝐸 𝐼 ∫ 𝑑𝑥, 𝑒 𝑏 𝑏 2 2 respectively, the principle of the virtual work can be written as (5) (in the following, the PZT plates will be considered perfectly 𝜕 𝑤 bonded to the structure, their mass and inertia negligible =−𝜌𝑆 ∫ 𝑤𝑑̃𝑥. in with respecttothe mass andinertia of thebeam, andtheir 𝜕𝑡 thickness very lower than the thickness of the beam) Using the modal analysis technique, indicating by𝜙 (𝑥) the 𝑖 th flexural modal displacement of the cantilever beam +𝛿𝐿 +𝛿𝐿 =0. and by𝑋 (𝑡) its amplitude, the vertical displacement will be (1) 𝑒 in 𝑎 approximated by Using the pin force model [4] the action of the PZT plates 𝑤 𝑥,𝑡 =∑𝑋 𝑡 𝜙 𝑥 , (6) ( ) () ( ) can be modeled by two flexural moments concentrated at the 𝑖 𝑖 𝑖=1 end of the plates (Figure 1)with where 𝑁 is the number of the considered modes. Substituting the last in (3), (5), considering𝑤=𝜙 ̃ (𝑥) , 𝑀 (𝑡 )= 𝐸 𝑇 Λ(𝑡 ), 𝑎 𝑎 𝑏 andsuccessivelyin(1), the following governing equation is 6+𝜓 obtained: .. Λ 𝑡 = 𝑉 𝑡 , () () (2) (7) 𝑇 M (𝑡 )+ KX(𝑡 )= B(𝑎,ℎ )𝑉 (𝑡 ), 𝐸 𝑇 𝑏 𝑏 where X represents the vector of the amplitudes of the 𝜓= . 𝐸 𝑇 𝑎 𝑎 modes, M and K are, respectively, the mass and the stiffness 𝑐𝑇 𝛿𝐿 𝛿𝐿 𝜕𝑥 𝜕𝑥 𝛿𝐿 𝛿𝐿 𝛿𝐿 𝛿𝐿 𝜕𝑥 𝜕𝑥 𝛿𝐿 Advances in Acoustics and Vibration 3 𝑋 𝑡 () matrix, and B(𝑎,ℎ) is the vector control (the prime denotes the rfi st spatial derivative): −𝑡𝛼/2 2 2 2 2 2 2 𝑒 𝐵 𝑉 cosh((1/2)𝑡 √𝛼 −4𝜔 )√𝛼 −4𝜔 (𝜔 −𝜔 ) 𝑖 𝑗 𝑖 𝑖 𝑖 𝑗 =∑ − 2 2 2 2 2 2 B(𝑎,ℎ ) 𝑗=1 √𝛼 −4𝜔 (𝛼 𝜔 +(𝜔 −𝜔 )) 𝑖 𝑗 𝑖 𝑗 ℎ ℎ 󸀠 󸀠 −𝑡𝛼/2 2 2 ̆ 2 2 =𝑀[𝜙 (𝑎+ )−𝜙 (𝑎− ), 𝑒 𝐵 𝑉 (𝛼 sinh((1/2)𝑡 𝛼 −4𝜔 )(𝜔 +𝜔 )) 1 1 𝑖 𝑗 𝑖 𝑗 2 2 2 2 2 2 2 2 (8) √ 𝛼 −4𝜔 (𝛼 𝜔 +(𝜔 −𝜔 )) ℎ ℎ 𝑖 𝑗 𝑖 𝑗 󸀠 󸀠 𝜙 (𝑎+ )−𝜙 (𝑎− ),..., 2 2 2 2 2 2 𝐵 𝑉 𝛼 sin(𝜔 𝑡)𝜔 𝐵 𝑉 cos(𝑡𝜔 )(𝜔 −𝜔 ) 𝑖 𝑗 𝑗 𝑖 𝑗 𝑗 𝑗 𝑖 𝑗 ℎ ℎ 󸀠 󸀠 + + . 𝜙 (𝑎+ )−𝜙 (𝑎− )] 2 2 𝑁 𝑁 2 2 2 2 2 2 2 2 2 2 𝛼 𝜔 +(𝜔 −𝜔 ) 𝛼 𝜔 +(𝜔 −𝜔 ) 𝑗 𝑖 𝑗 𝑗 𝑖 𝑗 (13) with 𝑀=(𝜓/(6+𝜓))𝐸 𝑇 (𝑑 /𝑇 ). 𝑎 𝑎 𝑏 31 𝑎 It is possible to obtain an approximate form of (13)con- If the Rayleigh damping is considered, (7)becomes sidering that the 𝑖 th term is the dominant one (the𝑖 th term of 𝑉(𝑡) is exciting𝑋 (𝑡)near its resonance). Aeft r some algebraic manipulation, (13) can be simplified to .. M 𝑡 + CX 𝑡 + KX 𝑡 = B 𝑎,ℎ 𝑉 𝑡 (9) X() () () ( ) () 𝐵 𝑉 sin(𝜔 𝑡) 𝑖 𝑖 𝑖 𝑋 (𝑡 )≅ 𝛼𝜔 with 2 2 2 2 (14) −(1/2)𝑡(𝛼+ √𝛼 −4𝜔 ) 𝑡√𝛼 −4𝜔 𝑖 𝑖 𝑒 (−1+𝑒 )𝐵 𝑉 𝑖 𝑖 − . C=𝛼 M+𝛽 K. (10) 2 2 𝛼 √𝛼 −4𝜔 The second term of the right hand side of ( 14)represents Indicating by𝜔 the diagonal matrix of the natural fre- the transient part so that, if this is neglected, the amplitude of quencies of the beam, using the normal modes (M = I, K = the displacement of the free end can be written as 𝜔 ), and assuming𝛽=0 ,(9)becomes 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝐵 (𝑎,ℎ )𝑉 𝜙 (𝐿 ) ⌣ 󵄨 󵄨 󵄨 󵄨 𝑖 𝑖 𝑖 𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑤 𝑎,ℎ =∑ . (15) .. 󵄨 ( )󵄨 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 ̇ 󵄨 󵄨 𝛼𝜔 𝑡 +𝛼 X 𝑡 +𝜔 X 𝑡 = B 𝑎,ℎ 𝑉 𝑡 . (11) 󵄨 󵄨 X() () () ( ) () 󵄨 𝑖 󵄨 𝑖=1 In the following a bimodal excitation is considered, indi- If a single mode excitation is considered, Barboni et al. cating by𝑖 ,𝑖 the indexes of the rfi st and second considered 1 2 [14] found the optimal placement of PZTs. However, in modes respectively, and𝑟≤1 therelativecontributionto some practical applications, for example, gas turbine blades, theexcitationinduced by thesecondmodewithrespect to theloadspectrumcan includevarious modes,eachof the rfi st one (i.e., 𝑟=1 corresponds to the second mode which is characterized by different amplitudes, and a new governing the excitation without any contribution from the strategy needs to be developed. In this paper, the multimode first mode). From ( 12): dampingwillbeobtainedbyapplyingacounterphase load, 𝑉 (𝑡 )=(1−𝑟 )cos(𝜔 𝑡)+𝑟 cos(𝜔 𝑡). (16) 𝑖 𝑖 by PZTs plates, to the external excitation. The effectiveness 1 2 of the piezoelectric elements will be measured by the ampli- Substituting (16)into(15) and introducing tude of the vertical displacement of the free end, so that the most eeff ctive (optimal) position, identiefi d in terms 𝑀 1−𝑟 of 𝑎 and ℎ (Figure 1), will be the location which maximizes 𝑔 𝑟,𝜉 = 𝜙 (𝐿 )𝜙 𝜉 , ( ) () 𝑖 𝑖 𝑏 𝑖 1 1 𝛼 𝜔 this amplitude. eTh refore, assigning a general spectrum to the 1 (17) PZT load (which corresponds to the spectrum of 𝑉(𝑡) ;see 𝑀 𝑟 (2)) 𝑔 (𝑟,𝜉 )= 𝜙 (𝐿 )𝜙 (𝜉 ), 𝑖 𝑖 𝑏 𝑖 2 2 𝛼 𝜔 it is obtained that 𝑉 (𝑡 )= ∑𝑉 cos(𝜔 𝑡 ), (12) 𝑗 𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ⌣ ℎ ℎ 󵄨 󵄨 󵄨 󵄨 𝑗=1 󵄨 󵄨 󵄨 󵄨 󵄨 (𝑟,𝑎,ℎ )󵄨 =󵄨𝑔 (𝑟,𝑎+ )−𝑔 (𝑟,𝑎− )󵄨 𝑖 𝑖 󵄨 󵄨 󵄨 󵄨 1 1 󵄨 󵄨 󵄨 󵄨 2 2 󵄨 󵄨 (18) 󵄨 󵄨 󵄨 󵄨 ℎ ℎ 󵄨 󵄨 where 𝑁𝑠 is the number of excited modes. eTh solution of 󵄨 󵄨 + 𝑔 (𝑟,𝑎+ )−𝑔 (𝑟,𝑎− ) . 󵄨 󵄨 󵄨 𝑖 𝑖 󵄨 2 2 󵄨 󵄨 2 2 (11)becomes 󵄨 󵄨 𝑁𝑠 𝑐𝑇 𝑁𝑠 4 Advances in Acoustics and Vibration Indicating by (𝜉 =𝑎+(ℎ/2),𝜉 =𝑎−(ℎ/2)),respectively, of the PZT elements can be found by solving the following 1 2 the position of the right and the left ends of the piezoelectric system of equations: plates it will be (𝑟,𝜉 ) 𝑖 ,𝑖 2 1 2 󵄨 󵄨 ⌣ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜕𝜉 󵄨 󵄨 𝑤 (𝑟,𝜉 ,𝜉 ) = 𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 ) 󵄨 󵄨 2 󵄨 󵄨 1 2 𝑖 1 𝑖 2 󵄨 󵄨 1 1 󵄨 󵄨 󵄨 󵄨 (19) 󵄨 󵄨 𝜙 (𝐿 ) 𝜙 (𝐿 ) 󵄨 󵄨 𝑀 𝑖 𝑏 𝑖 𝑏 󵄨 󵄨 + 𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 ). 1 2 󵄨 󵄨 𝑖 1 𝑖 2 2 2 = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]=0, 󵄨 󵄨 2 2 𝑖 𝑖 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 1 2 ̂ ̂ If (𝜉 ,𝜉 ) are the coordinates of the absolute maximum 2 1 2 𝜕 𝑓 (𝑟,𝜉 ) 𝑖 ,𝑖 2 1 2 of | (𝑟,𝜉 ,𝜉 )|,itispossibletowrite 1 2 2 𝜕𝜉 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ 󵄨 󵄨 ⌢ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ ̂ ̂ ̂ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑤 (𝑟,𝜉 ,𝜉 )󵄨 =󵄨𝑤 (𝑟, 𝜉 ,𝜉 )󵄨 = 𝑔 (𝑟, 𝜉 )−𝑔 (𝑟, 𝜉 ) 𝜙 (𝐿 ) 𝜙 (𝐿 ) 1 2 1 2 󵄨 𝑖 1 𝑖 2 󵄨 󵄨 󵄨 󵄨 󵄨 𝑀 𝑖 𝑏 𝑖 𝑏 1 1 1 2 󵄨 󵄨 󵄨 󵄨 max 󵄨 󵄨 = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]>0. 𝑖 2 𝑖 2 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 󵄨 󵄨 1 2 󵄨 󵄨 ̂ ̂ 󵄨 󵄨 + 𝑔 (𝑟, 𝜉 )−𝑔 (𝑟, 𝜉 ) . 󵄨 𝑖 1 𝑖 2 󵄨 2 2 (25) 󵄨 󵄨 (20) eTh solution will provide all the local minima and the absolute minimum will then be selected among these. Considering that 𝑔 (𝑟,𝜉) has, for all the considered val- ues of𝑘 ,its positive absolute maximumat 𝜉 =𝐿 ,thisvalue 1 𝑏 will also be the abscissa of the positive absolute maximum 3. Results and Discussions for|𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 )|+|𝑔 (𝑟,𝜉 )−𝑔 (𝑟,𝜉 )|. This implies 𝑖 1 𝑖 2 𝑖 1 𝑖 2 1 1 2 2 This section reports analytical results for different bimodal that the PZT plates should always be placed with their right excitations and their comparison with FEM simulations. edges terminating at the tip of the beam independently of the eTh se simulations have been performed by using the fre- excited modes and value of𝑟 , and hence quency module of the structural mechanics module of 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ 󵄨 󵄨 󵄨 󵄨 COMSOL Multiphysics software [ 24]. Specifically, a steel 󵄨 󵄨 󵄨 󵄨 𝑤 (𝑟,𝜉 ,𝜉 ) = 𝑤 (𝑟,𝐿 ,𝜉 ) 󵄨 󵄨 󵄨 󵄨 󵄨 1 2 󵄨 󵄨 𝑏 2 󵄨 󵄨 󵄨 max 󵄨 󵄨 max beam of 30 cm of length has been taken into account; with bimodal control in mind and focusing on combinations of 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (21) = 𝑔 (𝑟,𝐿 )−𝑔 (𝑟, 𝜉 ) 󵄨 󵄨 𝑖 𝑏 𝑖 2 1 1 the rst fi ve fi modes, the wavelength of the highest mode 󵄨 󵄨 has been divided into 50 subintervals of length 3 mm, so 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 + 𝑔 (𝑟,𝐿 )−𝑔 (𝑟, 𝜉 ) . 󵄨 󵄨 𝑖 𝑏 𝑖 2 2 2 that Δ𝑎 = Δℎ = 3 mm and 5000 different combinations 󵄨 󵄨 for 𝑎 and ℎ have been considered. For each of these, the Moreover, since amplitude of the response of the tip, to a periodic load, with the frequency corresponding to one of the rfi st vfi e eigen- 0<𝑔 (𝑟,𝐿 )>𝑔 (𝑟,𝜉 ) ∀𝜉∈[0,𝐿 ),∀.𝑗 (22) frequencies has been calculated. Moreover the amplitude 𝑗 𝑏 𝑗 𝑏 response to a linear combination of the previous loads has been obtained by superposition of the response to two of the (21)becomes eigenfrequencies. eTh optimal position has been chosen to 󵄨 󵄨 󵄨 󵄨 ⌣ ⌣ be the one which corresponds to the maximum amplitude. 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ 󵄨 𝑤 𝑤 󵄨 (𝑟,𝜉 ,𝜉 )󵄨 =󵄨 (𝑟,𝐿 ,𝜉 )󵄨 =𝑔 (𝑟,𝐿 ) 1 2 𝑏 2 𝑖 𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 max 󵄨 󵄨 max In Figure 2; 𝑓 (𝑟, 𝜉 ), the position of its absolute minimum, 1,2 2 (23) maximum and the position of the center of the piezoelectric +𝑔 (𝑟,𝐿 )−𝑓 (𝑟, 𝜉 ), 𝑖 𝑏 𝑖 ,𝑖 2 2 1 2 plates, for different 𝑟 ratios, are reported. It is possible to observe that for𝑟=0 and𝑟=1 (that correspond, resp., where to theexcitationofthe rfi st andsecondmodes)the optimal congfi uration of the PZT coincides with that obtained in [ 14]. 𝑓 (𝑟, 𝜉 ) 𝑖 ,𝑖 2 1 2 For𝑟=0 , 𝑓 (0,𝜉 )attains its absolute minimum at the 1,2 2 boundary points 𝜉 =0,sothatthe optimalcongfi uration ̂ ̂ =𝑔 (𝑟, 𝜉 )+𝑔 (𝑟, 𝜉 ) 𝑖 2 𝑖 2 1 2 is with the piezoelectric plates distributed along the entire length of the beam. For increasing values of 𝑟 the position of ̆ 𝜙 (𝐿 ) 𝜙 (𝐿 ) 𝑖 𝑏 𝑖 𝑏 1 󸀠 2 󸀠 ̂ ̂ the absolute minimum does not change until the derivative = [(1−𝑟 ) 𝜙 (𝜉 )+𝑟 𝜙 (𝜉 )]. 2 2 𝑖 𝑖 1 2 𝛼 𝜔 𝜔 𝑖 𝑖 1 2 of 𝑓 (𝑟, 𝜉 )becomes zero in 𝜉 =0 (25.1): 𝑖 ,𝑖 2 2 1 2 (24) 𝜙 (0)𝜙 (𝐿 ) 1 𝑏 𝑟=̂ ≅0.5. (26) The ordinate 𝜉 of theabsolutemaximum of|𝑤 (𝑟,𝜉 ,𝜉 )| 2 1 2 𝜙 (0)𝜙 (𝐿 )−𝜙 (0)𝜙 (𝐿 )(𝜔 /𝜔 ) 1 1 𝑏 2 2 𝑏 1 2 will coincide with the absolute minimum of the 𝑓 (𝑟,𝜉 ). 𝑖 ,𝑖 2 1 2 This position changes with the considered modes and the For values of 𝑟 above 𝑟 , the optimal configuration grad- modes ratio 𝑟 so that the optimal location of the left edge ually changes to reach the optimal one for the second mode 󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠󸀠 󸀠󸀠󸀠 󸀠󸀠 󸀠󸀠 𝜕𝑓 Advances in Acoustics and Vibration 5 1.0 0.5 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.2 0.4 0.6 0.8 1 −0.1 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 2: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 1,2 2 edge (analytical);×: FEM simulations. 1.0 0.2 0.8 0.6 𝜉 0.1 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.1 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 3: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 2,3 2 edge (analytical);×: FEM simulations. 1.0 0.15 0.8 0.10 0.6 0.05 0.4 0.2 0.4 0.6 0.8 1 −0.05 0.2 −0.10 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 4: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 3,4 2 right edge (analytical);×: FEM simulations. f (r, ) 3, 4 2 f (r, ) 1, 2 2 f (r, ) 2, 3 2 6 Advances in Acoustics and Vibration 1.0 0.5 0.4 0.8 0.3 0.6 0.2 r 0.4 0.1 0.2 0.2 0.4 0.6 0.8 1 −0.1 0 0.2 0.4 0.6 0.8 1 𝜉 𝜉 2 2 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 5: 𝑓 (𝑟, 𝜉 ); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right 1,3 2 edge (analytical);×: FEM simulations. 1.0 0.2 0.8 0.6 0.1 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.1 0 0.2 0.4 0.6 0.8 1 𝜉 𝜉 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 6: 𝑓 (𝑟, 𝜉 );red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 2,5 2 right edge (analytical);×: FEM simulations. at𝑟=1 . eTh FEM results show that there is a very good comparisons with the FEM are in agreement with all the agreement between the numerical simulations and the analyt- cases analyzed. eTh proposed analytical method has been ical results. Different mode combinations are represented in also compared with the experimental investigations reported Figures 3, 4, 5, 6,and 7: the optimal configuration varies from in [25–27]. To test different load spectra and configurations theoptimal valuefor theindividual 𝑖 -excited mode (𝑟=0 ) an experimental apparatus has been designed and built; to that for the mode 𝑖 (𝑟=1 ). For all the considered coupled moreover, a MATLAB code has been developed to control modes, the optimal position changes continuously except for the actuation and the modal contributions. Four piezoelectric thecouple((1)–(3)), where a different type of behaviour is plates have been used in three different configurations. observed (Figure 5), with a sharp transition at 𝑟 ≅ 0.8 .This Experiments have been done with one mode and multimodal is duetothe fact that,when 𝑟 is low, theabsoluteminimum vibrations. A good agreement was shown in all the considered of 𝑓 (𝑟,𝜉 ) coincides with that obtained for the optimal cases. 1,3 2 congfi uration of the rfi st mode ( 𝑟=0 ). When the ratio reaches the value of 𝑟 ≅ 0.46 ,arelative minimumappears, but this develops into an absolute minimum and, therefore, 4. Conclusions coincides with the left end of the piezoelectric plates only when 𝑟 > 0.8 ; for such values the PZT elements should be In this work a new theoretical model for the optimal active only over about half of the length of the beam. The placement of piezoelectric plates to control the multimode f (r, ) 2, 5 2 f (r, ) 1, 3 2 Advances in Acoustics and Vibration 7 1.0 0.15 0.8 0.10 0.6 0.05 0.4 0.2 0.4 0.6 0.8 1 0.2 −0.05 −0.10 0 0.2 0.4 0.6 0.8 1 r = 0.0 r = 0.6 r = 0.2 r = 0.8 r = 0.4 r = 1.0 Figure 7: 𝑓 (𝑟, 𝜉 );red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s 3,5 2 right edge (analytical);×: FEM simulations. 𝛼 : Damping coefficient vibrations of a cantilever beam is proposed. After a detailed description of the theoretical model, bi-modal excitation 𝜙 (𝑥) : 𝑖 th flexural mode of the cantilever beam is considered. eTh designs of the optimal configurations, 𝜌 : Mass density 𝑋 (𝑡): Amplitude of the 𝑖 th mode for different coupled modes and different relative contribu- tions, are reported. eTh comparison between numerical and 𝜔 : Natural frequency. theoretical results is shown to confirm the validity of the optimization technique. 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