Dynamics and topology optimization of piezoelectric fans

Transcription

1 Theral Challenges in Next Generation Electronic Systes, Joshi & Gariella (eds) 00 Millress, Rotterda, ISBN Dynaics and toology otiization of iezoelectric fans Phili Bürann, Arvind Raan & Suresh V. Gariella School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA Keywords: iezoelectric fans, resonators, ortable electronics cooling ABSTRACT: Piezoelectric fans are very low ower, sall, very low noise, solid-state devices that have recently eerged as viable theral anageent solutions for a variety of ortable electronics alications including lato couters, cellular hones and wearable couters. Piezoelectric fans utilize iezoceraic atches bonded onto thin, low frequency flexible blades to drive the fan at its resonance frequency. The resonating, low frequency blade creates a streaing airflow directed at key electronics coonents. The otiization of a iezoelectric fan with two syetrically laced iezoelectric atches is investigated through an analytical Bernoulli-Euler odel as well as a finite eleent (FE) odel of the coosite iezo-bea. The closed for analytical solution is used to deonstrate that different otial iezoceraic-to-blade length ratios and iezoceraic-to-blade thickness ratios exist for axiizing the electroechanical couling factor (EMCF), ti deflection and rotation, and stroke volue rate. Such otiization rocedures rovide sile design guidelines for the develoent of very-low ower, high flow rate iezoelectric fans. INTRODUCTION Piezoelectric resonators are used as actuators (Pan et al. 99, Rivory et al. 99, Tiersten 969, Seeley & Chattoadhyay 99, Seshu & Naganathan 998) and sensors (Lee & Saravanos 996, Aldraihe & Kheir 000, Main et al. 99) for a wide variety of alications including iezoelectric fans, otical bea choers, ultrasonic otors, icroelectroechanical density sensors, and resonating viscoeters. For these alications, little sees to be known about an otial actuatorbea configuration including atch location, actuator atch-to-bea length ratio, and atch-tobea thickness ratio. Moreover, the otiization criterion is alication deendent and can vary between axial electroechanical couling factor (EMCF), otial ode shaes and frequencies, and ti deflections. In light of this, we focus in this article on analytical solutions of the dynaic resonse otiization of a iezoelectric fan odeled as a artially electroded (on both faces) elastic bea near resonance. Several authors (Pan et al. 99, Wang & Wang 000, Khdeir & Aldraihe 00, Brennan et al. 997, Lee & Saravanos 996, Crawley & Anderson 990, Aldraihe & Kheir 000, Seeley & Chattoadhyay 99, Seshu & Naganathan 998, Yang 997) have investigated the odeling and otiization for static deflection/actuation of iezoelectric-elastic coosite structures. Analytical, exeriental and coutational investigations of the dynaic resonse of such structures are resented in Rivory et al. (99), Tiersten (969), Barboni et al. (000), Main et al. Currently at the Deartent of Solid Mechanics, Technical University-Dresden, Gerany Corresonding author, raan@ecn.urdue.edu

2 (99), Lobontiu et al. (999), Shen (995), Pan & Hansen (99), Abraovich (998), Crawley & de Luis (987), and Strabi et al. (995). Rovory et al. (99) and Tiersten (969) resent an analytical dynaical odel. However, the free vibration roble is not considered and thus the electroechanical couling factor is not couted. In Barboni et al. (000) a siilar aroach to a solution for the forced ode shae is derived after the tie-deendent ters have been eliinated. In Main et al. (99) dynaic resonse of a bea that is fully covered by the atch is considered. Lobontiu et al (999) resents a bea that is artially covered by two iezoelectric atches. However, the additional ass and stiffness of the iezoelectric atches are not considered, while the actuation is included as two external oint bending oents alied on the bea. The couling effects between curvature and electric field are neglected. The forcing fro the iezoelectric atches in Shen (995) is odeled as an internal bending oent distribution in the bea. The resulting equations are written in state equation for and the control issues are discussed. However, no couling between iezoelectric atches and bea is taken into account. A lued araeter aroach is followed in Pan & Hansen (99). Mass, daing, stiffness and forcing are lued as araeters in a one degree-of-freedo odel. The forcing ter is included exlicitly and an extensive araetric study is carried out. This is however not based on a continuu odel. The wave resonse in beas and lates with attached iezoelectric atches is considered in Abraovich (998). However, the excitation is included through Dirac delta and Heaviside ste functions ilying the neglect of couling between iezoelectric atch and the substrate. In Crawley & de Luis (987) an infinitely long bea attached to iezoelectric atches is considered and is treated as a wave generator. The forcing is included as external oint bending oents neglecting the couling between the iezoelectric atch and substrate. Strabi et al. (995) resent a static and dynaic odel of a bea with two iezoelectric atch actuators, incororating true couling between the iezoelectric atch and bea. However, otiization issues are not discussed and their aroach requires a riori knowledge of the hase difference between forcing and resonse. Wolf (000) resents a corehensive analysis of otial electroechanical couling factors in the context of a iezoelectric atch attached to an infinitely long bea. A siilar aroach is followed in the resent aer, however a focus is laced on the analysis for a finite length bea. This leads to substantially different analysis techniques and otiization issues. Finite eleent aroaches for the otiization of iezoelectric structures are resented in Benjeddou et al. (999), Chandrashekhara and Varadarajan (997), Silva and Kikuchi (999), Yang (997), and Zhai et al. (000). In Chandrashekhara and Baradarajan (997) a cantilever bea with segented shear iezoelectric actuators is analyzed using sandwich bea finite eleents. In Yang (997) the deterination of the otial thickness of a single iezoelectric actuator is addressed. In Zhai et al. (000) a toology otiization of a geoetrically colex, stack actuator is erfored. Finite eleent based otiization techniques are very exensive coutationally. The focus of the resent aer is to develo a closed-for analytical solution for a resonating, coosite iezobea and utilize this solution to develo guidelines for otial fan configurations. Finite eleent odels of the fan are used to confir the validity of the analytical solution. MODELING AND ANALYSIS Consider the bea shown in Figure. The bea is of total length L, while the iezoelectric atches are bonded erfectly to the bea and are of length L L. The thickness of the bea and the actuators is given as h and h, resectively, where the coordinate x is easured fro the neutral axis of the bea. The widths of the bea and atches are b. The art of the fan fro the cla u to the beginning of the bea will be referred to as region, the section covered by the actuators as region and the section fro the end of the atches to the ti of the bea as region. P.Bürann, A.Raan & S.V.Gariella

3 Figure. Piezoelectric fan geoetry. The fan is driven at resonance by iezoelectric atches that are oerated out-of-hase fro an electronic circuit.. Analytical odel The transverse dislaceents of the neutral axis of the substrate in each region v i (x, t), i,,, are easured with resect to the longitudinal coordinate x along the neutral axis of the substrate. The coosite iezoelectric fan is odeled as a coosite Bernoulli-Euler bea. The equations of otion governing the transverse deflection of the iezoelectric fan can be derived through the use of linear iezoelectric constitutive relations, the silifying Bernoulli-Euler assutions and alication of Hailton s rincile inclusive of strain energy and electroechanical energy ters in the forulation. The resulting equations of otion for the three regions and the corresonding boundary conditions can be shown to be EI v '''' + && sv 0, K v' ''' + && v 0, EI v'' '' + & v s 0 v(0, t) 0, v' (0, t) 0, v (L, t) v(l, t), v' (L, t) v' (L, t), EI v '' (L, t) K v'' (L, t) + a (t) A, EI v ''(L, t) K v''' '(L, t), v(l, t) v(l, t), v'(l, t) v' (L, t), K v' ' (L, t) + a (t) A EI v' ' (L, t), K v'' '(L, t) EI v'' '(L, t), v' '(L, t) 0, v' ' (L, t) 0 () where the ries are derivatives with resect to x s ρs As, EI Es Is, ρs As + ρ A, K Es As + a I + a A () and ρ s, ρ, Es, As, A, Is, I denote resectively the ass density of the bea and the iezoelectric atch, the Young's odulus of the substrate, the areas of cross section of the substrate and atch, and the area oent of inertias of the substrate and atch. The coefficients a,a, a are related to the iezoelectric-structure couling and are given by e - e(h + h ) e(h + h ) Φ(t) ( Φ) a (E + ), a, a (t) -,a 5 8 h h () where,eare e, resectively the Young s odulus of the iezoelectric atch, the roduct of Eand the iezoelectric aterial coefficient d, and the roduct of the relative dielectric constant Dynaics and toology otiization of iezoelectric fans 5

4 K and the free sace eritivity 0. (t) is the otential difference between the to and botto of the electrodes of the iezoelectric atches. It ust be noted here that no exlicit forcing ter aears in equation (). The iezoelectric excitation is only contained in the boundary conditions because a (t) deends on the otential difference (t) (equation ). In addition to the boundary conditions at the cla and the free end of the bea, there are atching conditions at each interface between two regions. The deflection and sloe, and the internal bending oent and the internal shear force are related to each other. Note that there is a tie-deendent ju in bending oent and that the stiffness ter K consists of the stiffness of the underlying substrate E I s s, the added echanical stiffness of the iezoelectric actuators a I and an electroechanical couling ter a A. For free vibrations of the iezoelectric fan, (t) can either be 0 (short-circuited or SC electrodes) or be generated naturally by the bending vibrations (oen circuited or OC electrodes). For forced vibrations, (t) is a rescribed, haronic wavefor with a frequency atching the short-circuited fan frequency. The free vibration eigenvalue roble is solved iecewise in each region by use of exressions for the rescribed otential difference (t) and through searation of variables in equation (): iωt (x, t) (x )e, (x) A cos( x ) + B sin( x ) + C cosh( x ) D sinh( x ) () vi i i i i i i i i + i i i,, and the wavenubers i are given by s, EsI s K It can be shown that the artial differential equations and boundary conditions () are satisfied exactly with the above solution fors. Substitution of (5) into the boundary conditions (), casting the syste in atrix for and setting its deterinant to zero, the eigenvalues of the fan can be couted. For both the SC and OC cases, all unknown coefficients A i,b i,c i, Di are exressed in ters of any one aongst the. This last coefficient is found by noralizing the ode shaes in the standard way with resect to the ass oerator Meirovitch (997). This yields an exact closed for analytical solution for the noralized free vibration ode. When the bea is driven by the iezoelectric actuators the atching conditions () for internal bending oents becoe non-hoogeneous and non-autonoous. To hoogenize the boundary conditions, the following transforations are introduced: v (x, t) w (x, t), v (x, t) w (x, t) + h(x) a (t) A, v(x, t) w (x, t) (6) The function h(x ) is deterined such that the boundary conditions () are satisfied and rendered hoogeneous. The resulting equations for w i (x, t) feature hoogeneous artial differential equations with exlicit forcing ters and can be solved using Galerkin s ethod: w i (x, t) i (x) q(t), i,, (7) where i (x ) are the eigenfunctions for the SC free vibration roble. Substitution of the reviously couted i (x ) into (7) and subsequently into the hoogenized equations for w i (x, t), and taking inner roducts in succession with the i (x ), leads to the following discretized equations near resonance: (5) 6 P.Bürann, A.Raan & S.V.Gariella

5 & q + ω I 5 SC L L L I EI q& + I q I A && a I A a L (x )h(x )dx,i K L L (x ) ''''(x )dx + K '''' (x )h (x )dx, (x ) ''''(x )dx + EI L (x ) ''''(x )dx 0 L L (8) Note that linear viscous daing has been added through the daing ratio ς. Eergence of the short-circuited natural frequency ωsc in equation (8) is a consequence of the use of SC eigenfunctions in Galerkin s ethod. The articular steady state solution of (8) is F( Ω) q (t) F( Ω) sin( Ωt -γ ( Ω)) (I6 I )Vˆ SC, ( Ω) -arctan, Vˆ e(h + h )bv ( SC ) + ζω and V is the alitude of excitation in volts. Table. Material Proerties of a Piezoelectric Fan. E s E ρ s ρ Physical Quantity Young s Modulus of the Bea Young s Modulus of the Actuator Density of the Bea Density of the Piezoelectric Actuator Value 0 *0 6. * N 0 N h Height of the Bea 0.06 h Height of the Piezoelectric Actuator (varied) b Width of Bea and Piezoelectric Actuator 8.89 L Length of Region 0.5 L Length of Region and 7 (varied) L Overall Length of the Structure 7. K Relative Dielectric Constant 800 ε 0 Peritivity of Free Sace C 8.9 *0 V d Piezoelectric Strain Coefficient 0 *0 V ζ Daing Ratio 0.0. Finite Eleent Model To confir the validity of the analytical odel the iezo-bea is odeled in ANSYS 5.7 using -D linear solid brick eleents. The iezoelectric constitutive equations are utilized in their three-diensional lanar isotroic for and the roerties chosen for the coutation are listed in Table. Soe three-diensional iezoelectric constants are not retrievable fro the anufacturer s data and are aroxiated fro tyical handbook values. A regular three-diensional esh was generated using D brick iezoelectric 0-node eleents. Overall 6,008 nodes were used in the coutation. Piezoelectric and elastic bea roerties were assigned to the resective volues. All nodes above and below the iezoelectric atches were assigned a voltage couling corresonding to the SC/OC electrical boundary conditions. kg kg SC (9) Dynaics and toology otiization of iezoelectric fans 7

6 . Coarison of analytical and FE odel For tyical actuator lengths and one value of thickness ( h 0.78 ) the SC and OC natural frequencies of the analytical and FE odel corresonding to the first and second bending odes were coared and found to be in excellent agreeent (Table ). For a fixed length of atch L L 6.5 the ass-noralized first and second SC and OC ode shaes are coared in Figure. The results show excellent agreeent, with the difference between the analytical and FE redictions increasing with increasing ode nuber. Table. A coarison of the natural frequencies redicted by the analytical and the FE odel. The FE odel is used as a reference when couting the error. Mode L 7 L L st SC anal. st SC FEM % Error 5.6 Hz 6.97Hz.05% 9.Hz 9.Hz.6% 6.5Hz 58.85Hz.5% st OC anal. st OC FEM % Error nd SC anal. nd SC FEM % Error nd OC anal. nd OC FEM % Error 5.86Hz 7.55Hz.% 6.86Hz 6.8Hz.9% 66.06Hz 67.9Hz.89% 9.79Hz 95.7Hz 0.5% 96.57Hz Hz.60% 96.58Hz 50.09Hz.65% 6.9Hz 6.Hz.9% 78.6Hz 6.00Hz.6% 06.6Hz 07.60Hz 0.08% (a) (b) Figure. Noralized ode shaes for (a) st bending ode, short-circuited (left), and oen-circuited (right), and (b) nd bending ode, short-circuited (left), and oen-circuited (right). Solid-line: analytical odel, dotted line: FEM odel. 8 P.Bürann, A.Raan & S.V.Gariella

7 OPTIMIZATION. Results & discussion Now two araeters of the syste are allowed to vary: (i) the length of the atch, while L is held fixed, and (ii) the height of the atch. All other geoetric constants are held constant (Table ). L is varied fro to 7 in stes of while h ranges fro 0.7 to 0.5 in increents of Only the first bending ode was considered, it being the ost coon resonant ode used in iezoelectric fans. First the natural frequencies for SC and OC electrodes are calculated. These are used to coute the electroechanical couling factors (EMCF), while for the forced resonse only the SC electrode ode shaes and natural frequencies are sufficient. The EMCF is a easure of the electroechanical energy conversion and is defined as EMCF OC SC OC The variation of EMCF s as a function of the atch-to-bea thickness ratio and atch-to-bea length ratio is lotted in Figure a. The EMCF for the first bending ode of the fan is seen to reach a axiu when the atch thickness is reduced to equal the bea thickness (h/h ). Further an otiu atch-to-bea length ratio exists (aroxiately 0.55) where the EMCF is axiized. This variation is closely connected to the variation with otiization araeters of the fan natural frequencies. Piezoelectric atches add distributed elastic and iezoelectric stiffness as well as ass to the fan. For sall atch lengths, the added stiffness effect doinates and increases the natural frequencies while at longer atch lengths the added ass effect reduces the natural frequencies. Therefore the natural frequencies and consequently EMCF are axial at an otial value of atch-to-bea length ratio. Using the SC ode shaes and the relation (L, t) w (L, t) (L ) q(t), the axiu ti v v (L) φ(l) F( ω0 where 0 deflection can be given as ) is the daed natural frequency of the fan with short-circuited electrodes. The variation of the axiu ti deflection as a function of the atch-to-bea thickness ratio and atch-to-bea length ratio is lotted in Figure b. Equal thickness of the atch and bea are seen to lead to axial ti deflections while the otial atch-to-bea length ratio (aroxiately 0.5) is lesser than that required for otial EMCF. For a fixed daing ratio, the axiu ti alitude is affected by variations in ode shae and odal forcing. Maxiizing the ti rotation of the fan is exected to lead to an increase in the centrifugal forces on the fluid under actuation. For this reason, the variation of the ti rotation defined as v ' (L ) ' (L) F( 0) is also lotted as a function of the atch-to-bea thickness ratio and atch-to-bea length ratio in Figure c. Both ti deflection and ti rotation reach their axia for the sae fan configuration. However, the erforance dro for off-otial configurations is sharer if the otiization criterion is osed in ters of ti rotation as coared to ti deflection. Ti rotation ay also be of interest when considering high ti velocities where vortex shedding can significantly affect the erforance of the iezoelectric fan. Finally, the variation of the stroke volue flow rate defined as V & b F( 0 ) 0 (x)dx as a function of the atch-to-bea thickness ratio and atch-to-bea length ratio is lotted in Figure d. This quantity is a easure of the total volue er second of fluid dislaced by the fan and is exected to correlate well to the axial flow rate of the fan. The stroke volue flow rate is also axiized when the atch thickness equals the blade thickness. However, the otial atch-tobea length ratio (aroxiately 0.6) is slightly greater than that required for axial EMCF. Stroke flow rate is consists of both the natural frequency (EMCF) and the axiu ti deflection exressions. Therefore, its otial configuration in general is different fro otial configurations based on each constituent ter. L 0 Dynaics and toology otiization of iezoelectric fans 9

8 EMCF (a) dis (b) (L-L)/L h/h (L-L)/L h/h rot (c) vol (d) h/h h/h (L-L)/L 0.8 (L-L)/L 0.8 Figure : Variation with atch-to-bea length and thickness ratio of (a) Electroechanical couling factor (EMCF), (b) Ti deflection, (c) Ti rotation, and (d) Volue stroke rate of the resonant fan. CONCLUSIONS A closed-for analytical solution of the dynaic resonse of a resonant, coosite iezoelectric bea is develoed for the otiization of iezoelectric fans. The redicted oen- and shortcircuited frequencies and ode shaes atch excellently the results of finite eleent coutations. The results indicate that the otial values of atch-to-bea ratio and atch-to-bea thickness are different deending on whether the otiization criterion is osed in ters of (i) axial electroechanical couling factor (EMCF), (ii) axial ti deflection and rotation at resonance, and (iii) axial volue stroke rate. Such analytical and FE odels are exected to rovide sile design guidelines for the develoent of low ower, high flow rate iezoelectric fans. REFERENCES Abraovich, H Deflection control of lainated coosite beas with iezoceraic layers - closed for solutions. Coosite Structures : 7-. Aldraihe, O.J. & Kheir, A.A Sart beas with extension and thickness-shear iezoelectric actuators. Sart Materials and Structures 9: -9. Barboni, R., Mannini, A., Fantini, E. & Gaudenzi, P Otial laceent of PZT actuators for the control of bea dynaics. Sart Materials and Structures 9: 0-0. Benjeddou, A., Trinade, M.A. & Ohayon, R New shear actuated sart structure bea finite eleent. AIAA 7 (): Brennan, M.J., Elliot, S.J. & Pinnington, R.J The dynaic couling between iezoceraic actuators and a bea. Journal of the Acoustical Society of Aerica 0 (): 9-9. Chandrashekhara, K. & Varadarajan, S Adative shae control of coosite beas with iezoelectric actuators. Journal of Intelligent Material Systes and Structures 8: -. Crawley, E.F. & Anderson, E.H Detailed odel of iezoceraic actuation of beas. Journal of Intelligent Material Systes and Structures : P.Bürann, A.Raan & S.V.Gariella

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