A PIEZOELECTRIC BERNOULLI-EULER BEAM THEORY CONSIDERING MODERATELY CONDUCTIVE AND INDUCTIVE ELECTRODES

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1 Proceedings of the 6th International Conference on Mechanics and Materials in Design, Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Aores, July 2015 PAPER REF: 5513 A PIEZOELECTRIC BERNOULLI-EULER BEAM THEORY CONSIDERING MODERATELY CONDUCTIVE AND INDUCTIVE ELECTRODES Juergen Schoeftner 1(*), Gerda Buchberger 2 1 Institute of Technical Mechanics, Johannes Keler University, Lin, Austria 2 Institue of Biomedical Mechatronics, Johannes Keler University, Lin, Austria (*) juergen.schoeftner@jku.at ABSTRACT Pieoelectric transducers are commonly used as devices for actuation, for control and for sensing in structural alications. By modifying their local ieoelectric roerties or by varying the width of the ieoelectric devices, the daming caabilities can be further enhanced comared to smart elastic system, which are controlled by ieoelectric atches. Another ossibility to further imrove the daming erformance is by changing the electrical roerties of the electrodes. Most ieoelectric transducers are characteried by the fact that their electrodes are assumed to be ideal, which means that the equiotential area condition is fulfilled at the surface. In our work we develo a theory taking into account moderately conductive and inductive electrodes. In articular we combine this theory with a slender beam. The outcome of our aroach is the Telegrah equation for the electrodes (2 nd order in sace and time), which is couled with the Bernoulli-Euler beam theory (4 th order in sace, 2 nd order in time). Different electrical and mechanical boundary conditions and electrode roerties are considered. Finally, we gave an outlook on the ractical relevance of resistiveinductive electrodes for trajectory control. Keywords: ieoelectricity, Bernoulli-Euler beam theory, electrode modeling, smart structures, electrode modeling. INTRODUCTION Pieoelectric transducers have gained much reutation for influencing and observing the motion of elastic structures. Due to the caability to transfer energy between mechanical and electrical domains, and also vice versa, they are widely acceted and suitable candidates for vibration control (actuation), structural health monitoring (sensing) and energy harvesting. Reviews in ieoelectric devices, ranging from theoretical models to alications, are resented in (Crawley, 1994), (Mason, 1981) and (Chora, 2002). In this contribution we focus on slender elastic beam-tye structures which are equied by ieoelectric transducers with secial electrodes. These electrodes do not fulfill the equiotential area conditions on their surface, i.e. the electric otential varies locally onto the surface. Beam theories considering the direct and the indirect ieoelectric effect are e.g. resented by (Krommer 2001), by (Benjeddou et al, 1997) (for actuation) and by (Schoeftner and Irschik, 2011) (for assive vibration control). All these contributions have in common that ideal electrodes are assumed

2 Symosium_3 Dynamics, Stability and Control in Structural Engineering In our contribution we extend these works by combining the beam theory within the framework of Bernoulli-Euler with moderately conductive and moderately inductive electrodes. We are motivated by the structure deicted in Fig.1 (left): it shows a beam on which ieoelectric atches are glued onto its surface. The electrodes of the atches fulfill the equiotential area condition, but they are connected by a network of resistances and inductances. The question we are interested in is: what haens if these atches became smaller and smaller, as well as the electrical comonents, and what haens with the differential equations if we take the limit x dx? What kind of beam equation we will obtain? In the next section, we will derive in a systematic and logical way two new kind of artial differential equations which are extensions of the well-known Bernoulli-Euler beam equation and an extension of the Telegraher s equation. By means of these couled ieomechanical relations, we intend to get an increased understanding in the dynamics of the structure and to develo new control strategies for the ractical beam configuration like the beam with several ieoelectric atches shown in Fig.1 (left). ELECTRICAL AND MECHANICAL MODELING - FUNDAMENTAL EQUATIONS OF A PIEZOELECTRIC BEAM In this section we will derive two couled artial differential equations for a slender beam whose ieoelectric transducers have non-erfect (i.e. resistive and inductive) electrodes. So electrodes are characteried that the electric otential varies locally. Thus our goal is to find a relation between the structural degree of freedom, which is the vertical dislacement w ( xt, ), 0 and the electrical degree of freedom, which is the voltage V ( xt, ). Fig. 1 - A beam with several ieoelectric atches is the main motiviation (left-to). When taking the limit x dx one finds a ieoelectric beam with so-called resistive-inductive electrodes (right-to). The corresonding electrical model of the ieoelectric transducer including the electrodes is also shown (bottom). Alying Newton s law for a slender beam configuration, one finds

3 Proceedings of the 6th International Conference on Mechanics and Materials in Design, Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Aores, July 2015 M w&& M q (1) w 0, =, where w,, 0 M q are the lateral dislacement, the bending moment and external distributed. Using the constitutive relation of a ieoelectric material σ % % (2) = C11ε e31 E, which relates the axial stress σ to the axial strain ε = w0, and to the electric field or the voltage E = V,, one finds the bending moment by integration with resect to the cross section M = σ d A. (3) A In general, the electric voltage V ( xt, ) cannot be indeendently rescribed along the beam axis. Only when actuating the structure by a voltage source and assuming erfectly conductive electrodes, it is ossible to rescribe the voltage. Nevertheless, since we are also interested to treat active as well as assive systems with only one extended beam theory, where the voltage DOF is an outcome of the deformation of the structure, we have to derive a novel ieoelectric beam model. First, we assume Gauss law of electrodestatics, which reads in its differential form, that the electric dislacement D is constant or The relation between electric dislacement D, = 0. (4) D, axial strain and electric voltage is given by D = e% + % (5) 31ε κ33 E. Using (5), inserting into (4) and integrating with resect to the thickness of the ieoeelectric transducer 2 1, one finds the electric fields as a function of the dislacement and the voltage V e31 E = ( m) w. 0, h + % % κ (6) 33 With the definition of the bending moment (3) and Newton s law (1), one finally finds the equation of motion as ( ) M w&& ( xt, ) + K w ( xt, ) = q ( xt, ) + e% + bv ( xt, ). (7) w 0 m 0, , As it can be observed from (7), the deformation deends on the second derivative of th e electrical voltage. So we need an additional equation which relate the beam dislacement w (, ) 0 xt and the elctrical degree of freedom V ( xt, ), since the electric voltage can not be rescribed in general. For this, we use the blockdiagram of the ieoelectric layer (Fig.1 bottom). The electrical dislacement can be slit u into a mechanical and an electrical term, see (5) D = D + D,mech,elec with D = e% ε, D = % κ E.,mech 31,elec 33 (8)

4 Symosium_3 Dynamics, Stability and Control in Structural Engineering Alying Kirchhoff s current rule which states the sum of incoming and outgoing current at a node is ero, one finds from the model of the ieoelectric transducer, that the leakage current i ( ) D x between between the uer and the lower electrode is equal to the rate of change of the current on the electrodes i 1 ( x), i 2 ( x ) i ( x+ d x) = i ( x) d i ( x) i ( x) =i ( x) 1 1 D 1, x D, x i ( x+ d x) = i ( x) + d i ( x) i ( x) = i ( x). 2 2 D 2, x D, x (9) Integrating with resect to the width b of the transducers and erforming the timederivative, one finds a relation between the electric dislacement and the leakage current d i ( x) = Db & d x i ( x) = D& ( xb ). (10) D D, x Combining (8), (9) and (10), one obtains ( 2+ 1) i1, x ( x) = id, x ( x) = e% 31 bw& 0, + cv& ( x) 2 ( + ) i x i x e b w cv& x c % κ b , x ( ) = D, x ( ) =% & 31 0, ( ) = 2 h (11) A resistance or inductance also causes a otential dro, if the current is non-ero. Thus alying Kirchhoff s voltage law for the electrodes, one finds that d ϕ = ir + il dt d ϕ = + dt 1, x , x ir 2 2 il 2 2 In (12) the electrode resistance and/or the inductance er unit length are denoted by r 1, l 1 for the electrode at = 1 and by r2, l2 for the electrode at = 2. We further assume that the electrode roerties do not deend on the axial coordinate, we assume them to be constant. Considering that the voltage over the electrodes is defined by V ( xt, ) = ϕ2 ( xt, ) ϕ1 ( xt, ), and subtracting the first equation from the second one in (12), taking the satial derivative and using the exressions for the electrode current i1 ( x), i2 ( x ), one finally finds the extended Telegraher s equation as ( ) ( ) ( 2+ 1) ( ) ( ) V c r1 + r2 V & c l1 + l2 V && = e % 31 b r1 + r2 w0, + l1 + l2 w 0,. 2 & && (10) The couling of the deformation of the beam is due to the exression on the right-hand side. The left-hand side is a artial differential equation for the voltage dro over the electrodes and is the damed wave-equation. Thus the electro-mechanical beam is governed by the forthorder differential equation (6), which is an equation for the beam motion, and (10), which gives the solution for the voltage distribution. With these two artial differential equations the dislacement w (, ) 0 xt and the electrode voltage V ( xt, ) may be determined. (12)

5 Proceedings of the 6th International Conference on Mechanics and Materials in Design, Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Aores, July 2015 NUMERICAL RESULTS: A CLAMPED-HINGED PIEZOELECTRIC BEAM A figure of the clamed-hinged ieoelectric bimorh, which is used in our numerical case 4 study, is given in Fig. 2. The thickness of each ieoelectric layer is h = 5 10 m, thus the total height of the beam is 0.001m. As material for the ieoelectric layers PZT-5A is used. A comlete list of the geometrical and material arameters is rovided in Table 1. Fig. 2 - A clamed-hinged beam is used for the numerical case study. The inner electrodes are assumed to be erfectly conductive, whereas the external electrodes are assumed not to fulfill the equiotential area condition. Table 1 - Numerical values for the bimorh that is used in the case study variable (unit) value variable (unit) value -3-2 ρ ( kgm ) 7750 %C 11( Nm ) 1( m) 0 2( m) l( m) 0.04 ( m) -1-1 ( ) -1-1 c ( AsV m ) κ% AsV m ( Asm ) 0( Nm 1 ) b %e q 25 We are esecially interested in how far such a system can be damed around the first structrual eigenfrequency, if the electrode roerties are otimally choosen. From the literature it is already known that for a beam that is equied with erfect ieoelectric transducers, the otimal resistive end is ( ) 1 R = ωc, see also (Schoeftner and Irschik, 2011). Using this value, the vibration is the lowest around the eigenfreuqnuency. In case of a 2 combination of a resistance and an inductance, the otimal inductance is ( ) 1 L = ω C. The disadvantage of this strategy is that it is quite sensitive to arameter uncertainties. We want to comare these results to those configurations, where the electrode resistance is otimally tuned, i.e. when once the sheet resistance r otimal is otimal, and when the electrode resistance and the electrode inductance er unit length are otimal r, l otimal

6 Symosium_3 Dynamics, Stability and Control in Structural Engineering Fig. 3 - Frequency resonses of the beam around the first eigenfreuqency when various tyes of electrodes are assumed (to: ieoelectric transducers without electrodes r (black), erfect electrodes and short-circuited (dash-dot), erfect electrodes and oen-circuited (dark gray), no ieoelectric effect e % 31= 0 ; middle: erfect electrodes and otimal resistive shunt (light gray), otimal electrode resistance and short-circuited (black), otimal electrode resistance and oen-circuited (gray); bottom: otimal end resistance and inductance (black), inductive shunt (black dash-dot), otimal resistive and inductive electrodes (gray), inductive electrodes (light gray)). The first bending eigenfrequency for the clamed-hinged beam is at f 1280 H. Fig.3 show the harmonic resonses of the beam wˆ 0 (0.5 l ) which is excited by the iecewise constant q for various electrodes. The results for the beam with no electrodes (r ), short- and oen circuited electrodes and when the ieoelectric effect are comletely disregarded are shown in Fig.3-to. We observe that the various electrode condtions yield different eigenfrequencies. This characteristic is a quite well-known result, see also (Krommer, 2001) who derived similar results when develoing a beam theory with ieoelectric transducers and erfect electrodes. If no electrodes are assumed, i.e. r, the eigenfrequency f 1296 H is the highest. Considering erfect electrodes r= 0 which are ket at the same otential

7 Proceedings of the 6th International Conference on Mechanics and Materials in Design, Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Aores, July 2015 (short-circuit), the first mode is at f 1275 H. When considering oen-circuited electrical condition the eak is at f 1281H. If the system is otimally damed by an end or terminal resistance R ( C ) 1 18,140, the largest deflection is m. Otimiing the electrode roerties ω Ω 6 r Ω/m and assuming no terminal R 0Ω, one finds m as the largest deflection, which is slightly higher than if the electrodes are otimally chosen and oen-circuit condition are assumed m. Taking into account inductive roerties, one finds an additional resonance eak. Assuming a resistive-inductive end R 2176 Ω, L 2.23H, one finds that the largest frequency resonse is only wˆ 0 (0.5 l ) = m at f 1235 H and at f 1325 H. Using an 6 otimally tuned electrode with resistive-inductive roerties r Ω/m, l 606.9H/m the deflection is even much less: wˆ 0 (0.5 l ) = m. This is one order of magnitude lower the best results obtained with resistive electrodes, and aroximately the half of the dislacement when using a resistive-inductive shunt. This results demonstrates that the roer choice of the electrode resistance rovide a huge otential for assive vibration control. CONCLUSIONS In this contribution, the governing equations of motion for a ieoelectric slender beam with so-called resistive-inductive electrodes have been resented. The develoed theory can be either used when the voltage is rescribed at a certain location along the beam axis, or if the beam is subjected to an external. The outcome of our derivation is an extended Bernoulli-Euler beam equation which is couled to an extended Telegraher s equation for the ieoelectric transducer and its electrode. Finally, the otential of resistive-inductive electrodes is demonstrated in the context of assive vibration control. It is shown that the daming erformance of a beam configuration where the resistive-inductive electrodes are erfectly tuned, is much more efficient than for a ieoelectric beam with erfect electrodes whose terminal is otimally tuned. ACKNOWLEDGMENTS J. Schoeftner acknowledges suort from the Austrian Science Fund FWF via the roject P N30. REFERENCES [1]-Crawley E F. Intelligent structures for aerosace: a technology overview and assessment. AIAA J, 1994, 32(8), [2]-Mason W P. Pieoelectricity, its history and alications. J Acoust Soc Am, 1981, 70(6), [3]-Chora I. Review of the state of art of smart structures and integrated systems. AIAA J, 2002, 40(10),

8 Symosium_3 Dynamics, Stability and Control in Structural Engineering [4]-Benjeddou A, Trindade MA, Ohayon R. A unified beam finite element model for extension and shear ieoelectric actuation mechanisms. J Intell Mat Sys Struct, 1997, 8, [5]-Krommer M. On the correction of the Bernoulli Euler beam theory for smart ieoelectric beams. Smart Mater Struct, 2001, 10, [6]-Schoeftner J and Irschik H 2011 A comarative study of smart assive ieoelectric structures interacting with electric networks: timoshenko beam theory versus finite element lane stress calculations Smart Mater Struct, 20, (13)