IDENTIFICATION OF PIEZOELECTRIC ULTRASONIC TRANSDUCERS FOR MACHINING PROCESSES

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1 Prceedings f the 9 th Internatinal Sympsium n Mechatrnics and its Applicatins (ISMA3), Amman, Jrdan, April 9-, 3 IDENTIFICATION OF PIEZOELECTRIC ULTRASONIC TRANSDUCERS FOR MACHINING PROCESSES Tarek Tutunji, Ashraf Saleem, Mhammad Salah 3, and Naseer Ahmad Department f Mechatrnics Engineering, Philadelphia University, Amman, Jrdan Department f Mechanical Engineering, Taibah University, Al-Madinah Al-Mnawara, Saudi Arabia 3 Department f Mechatrnics Engineering, Hashemite University, Zarqa, Jrdan ABSTRACT Smart r active materials expand r defrm as a respnse t external pressure, temperature, r electrmagnetic field. The mst cmmnly used active material in actuatrs is piezelectric as it relates the electric field t mechanical cmpressin and elngatin. Hwever, mathematical mdeling f such actuatrs is tedius and cmplicated because f the nnlinear behavir. Therefre, there is a need t emply apprpriate identificatin algrithms in rder t capture the mdeling prcess f such actuatrs. In this paper, ultrasnic transducer that is used in ultrasnically assisted machining is adpted as a case study. A cmpnent-based mdel fr the piezelectric is derived and the dynamic respnse is analyzed. Then, tw identificatin techniques, ARX and ANN, are used t btain mdels that mimic the system s dynamic respnse with minimal errr.. INTRODUCTION The direct piezelectric effect is the ability f certain materials t generate electric charge in prprtin t externally applied frce. The inverse piezelectric effect is their ability t expand under an electric field. Due t these phenmena, piezelectric materials can be used as bth actuatrs and sensrs. Applicatins include micrmanipulatin such as suppressin f scillatins, micr-rbt, micr-pump, and micr-gripper []. Piezmtrs usage as actuatrs is based n the cnversin f high frequency mechanical scillatins int cntinuus mtin. Advantages f piezmtrs are large trque, high reslutin, excellent cntrllability, small time cnstant, cmpactness, high efficiency, silent peratin, and n electrmagnetic inductin. Researchers have been very active in the area f identificatin and cntrl f piezelectric material. Ru et al. [] have derived a mathematical mdel t describe the hysteresis effect and tested an adaptive cntrl apprach fr micrpsitining experiment. Juhasz et al. [3] described a methd fr parameter identificatin f embedded piezelectric actuatrs fr the use in a real-time FPGA system. In bth f these papers, the applied vltage had a frequency range in hertz. Other researchers investigated the identificatin f the piezelectric structure prperties [4] and develped mdels fr damage identificatin [5]. Anther interesting applicatin is the Ultrasnic Assisted Tuning (UAT) which is a machining technique that is being used t machine hard-t-cut allys. In this technique, a high frequency vibratin (abut khz) is superimpsed n the mvement f the cutting tl that results in a reductin f the required cutting frce by 6% [6]. In this wrk, cmpnent-based mdel is develped fr piezelectric ultrasnic transducer. This transducer cnverts electric energy int ultrasnic vibratin waves by applying external vltage n the piezelectric stack. The vibratin waves travel thrugh the transducer material in the frm f defrmatin and stress waves. The stress wave is then cncentrated at the cutting tip t intensify the effrt. As mentined earlier, mdeling f such system is cmplicated and therefre system identificatin techniques are used in this wrk in rder t predict an apprpriate mdel fr the system based n recrded input-utput data. The rest f the paper is rganized as fllwing: Sectin presents the cmpnent-based mdel f the system. Sectin 3 prvides tw different identificatin methds. Results are discussed in Sectin 4 and Sectin 5 cncludes the paper.. COMPONENT BASED-MODELING Cmpnent-based mdeling is an apprach where the system is decmpsed int its basic cmpnents which can be develped and tested individually. Therefre, a cmpnent is defined as an entity r subsystem which is designed t perfrm certain functins and tasks and can be used as a building blck in the design r mdeling f a larger system. In the cntext f this study, the ultrasnic transducer is divided int tw basic cmpnents as shwn in Figure : - PZT cmpnent: this cmpnent represents the piezelectric rings which are cnsidered the driving frce fr the actuatr. This cmpnent has three inputs and tw utputs as depicted in Figure. Mass-Spring-Damper (MSD) cmpnent: this cmpnent represents a mass that is cnnected with tw springs and tw dampers as shwn in Figure. This cmpnent has fur inputs and tw utputs. Fr ultrasnic transducer tw MSD cmpnents are required t mdel the cncentratr. x x x x Figure. Prpsed cmpnent-based mdel fr piezelectric ultrasnic transducers x x x L x L ISMA3-

2 Prceedings f the 9 th Internatinal Sympsium n Mechatrnics and its Applicatins (ISMA3), Amman, Jrdan, April 9-, 3 The mathematical expressin fr the PZT can be presented as [7] and T m em em px T q x F Cc Cc () p p x x x x F k x b x k b () where x (), t x (), t and x () t dente the displacement, speed, and acceleratin f the PZT effective tip, respectively, x (), t x () t, x (), t and x () t are the displacement and speed f the MSD and MSD, respectively, qt () is the charge induced within the piezelectric, m p is the mass f the piezelectric material, T em is the elngatin cnstant, C c is the internal capacitance, k p, bp, k, b, are the spring cnstants and damping cefficients f the PZT and MSD, respectively, and F is the perpendicular frce acting n the piezelectric actuatr. The inherent nnlinear hysteresis (i.e., q = H(V h ) ) can be mdeled as prvide by Cleman Hdgdn mdel [8] mdel f the system using Finite Element Methd. This was achieved by finding the resnant frequencies f the system and then used t calculate the stiffness and damping cefficients. 3. SYSTEM IDENTIFICATION System identificatin is the field f apprximating mathematical mdels by measuring experimental input and utput data f the system under test. The identificatin mdels can be parametric r nnparametric, linear r nnlinear. In this paper, tw parametric methds are used: () linear methd that uses Aut Regressive with exgenus Input (ARX), and () nnlinear methd that uses Artificial Neural Netwrks (ANN). The general blck diagram fr identifying systems is prvided in Figure where the bjective is t find the apprpriate mdel t minimize the errr between the system s utput y and the mdels utput ŷ [9]. In Figure, the input is dented by u and the nise is dented by n. H F d (3) where Vh V Vc. The signal Vc () t is the vltage acrss the internal capacitance f the piezelectric and V () t is the cntrl input (vltage applied n the piezelectric cmpnent). The functin f ( V h ) is a subsequently defined signal and the functin dv ( h ) is defined as Vh V h q f V h e V h V h ' e g( ) f ( ) e d, when V h ( t) V h d ( ) (4) Vh Vh q f V h e V h V e h ' g( ) f ( ) e d, when V h ( t) V h where q is the induced charge at t t, V h is the input vltage at t t, δ is a psitive cnstant, f ' () f ( V h ) V h and g () is a subs equently designed signal. T validate the expressins in equatin (4), the functins dv ( ), f ( V ) and h h g ( V h ) have t satisfy certain prperties that can be fund in [7, 8]. As fr the MSD and MSD mdels, they can be expressed as b x x k x x mx b x x k x x (5) m x b x x k x x (6) where m and m are the mass f the cncentratr s parts; b and b are its damping cefficients; k and k are its stiffness cefficients. These parameters are estimated by creating a full Figure. System Identificatin Blck Diagram One methd that is used fr system identificatin is the ARX which cmbines a deterministic mdel with a stchastic disturbance. The ARX mdel are given by A( q) y( k) B( q) u( k) v( k) (7) where A(q) and B(q) are a plynm ial functins, q is the delay, and u(k), y(k), and v(k) are the input, utput, and nise, respectively. The equatin can be re-written as y( k) a y( k )... an y( k n) buk ( )... bmuk ( m) v( k) The ARX blck diagram is shwn in Figure 3 where the stchastic signal v is used t mdel the nise in the real system. This signal is usually mdeled as a white nise. Figure 3. ARX blck diagram By taking the z-transfrm f equatin (7), the transfer functin wuld be Y ( z) b b z... bmz Gz ( ) U( z) az... a z n (8) m n. (9) The effect f the nise is eliminated as the transfer functin is used t mdel the utput with respect t the input nly. As the nise used is white, it has an average f zer. ISMA3-

3 Prceedings f the 9 th Internatinal Sympsium n Mechatrnics and its Applicatins (ISMA3), Amman, Jrdan, April 9-, 3 The ANN can als be used fr identifying dynamic systems. The advantage f the ANN mdels is that they can be used fr highly-nnlinear systems. A general recurrent multilayer netwrk is shwn in Figure 4. This is a multi-input singleutput cnfiguratin. The netwrk accepts the present and past samples f the input u as well as past samples f the utput y as input signals and prvides ne utput, y. The netwrk cntains activatin functins (f and g) and adapts its utput by ptimizing its weight matrices (W and V) t minimize a sum square errr between the riginal and netwrk utputs. The back prpagatin algrithm can be used fr ptimizing the weights. Displacement (m) 5 x x -6 PZT Respnse MSD Respnse 4 3 Displacement (m) Figure 4. Architecture fr recurrent multi-layer neural netwrk Frm Figure 4, it can bee seen that the netwrk utput, y, is prvided by the fllwing equatin n y ( k) g v jnet j j ( k) where net j (k) is the utput frm the hidden nde j at pattern k and is given by net j ( k) f wijui ( k i) wij yi ( k i) i i () () Displacement x MSD Respnse Figure 5. (a) PZT displacement (b) MSD displacement (c) MSD displacement Usually, the activatin functins used are nnlinear, sigmid and hyperblic tangent. such as the 4. SIMULATION RESULTS The cmpnent-based mdel fr the ultrasnic transducer was created in Matlab/Simulink. All equatins and system parameters used were based n the mdeling intrduced in Sectin. A sinusidal input vltage with a frequency f.7khz was applied t the system. The utput displacement f the PZT, MSD, and MSD are shwn in Figure 5. Input-utput patterns frm the cmpnent-based mdel were gathered and used in ARX mdels t identify the system. A pulse with amplitude f 5kV was used and the impulse respnse with 8, data pints was gathered. The sampling time was sec and therefre the time perid was 8sec. Fr all ARX mdels, the least squares estimatin methd was used t find the mdels parameters [9]. A simple ARX (6,) was used t identify the system. This mdel has six parameters fr the A plynmial and tw parameters fr the B plynmial in equatin (7). The results are shwn in Figure 6. The ARX results stack almst-perfectly ver the cmpnent-based mdel data. In rder t have better insight int the results, the residues are shwn in Figure 7. Figure 6. Impulse respnse cmparisn: cmpnentbased mdel vs. ARX(6,) Once the identified ARX mdel was realized, it was applied t test data f sinusidal inputs. The results are shwn in Figure 8. Here, the ARX respnse is given in yellw while the cmpnent-based mdel is given in black. It is clear that the mdel was nt able t fllw the test data and that the errr is t large. This is due t the fact the piezelectric mdel is nnlinear while the ARX mdel is linear. In rder t cmpensate fr the nnlinearity, the mdel rder was increased: a mdel f ARX(4,) was used (i.e., 4 and parameters fr plynmials A and B respectively in equatin (7)). Figure 9 shws the impulse respnse data (ARX mdel ISMA3-3

4 Prceedings f the 9 th Internatinal Sympsium n Mechatrnics and its Applicatins (ISMA3), Amman, Jrdan, April 9-, 3 results are in purple). As predicted, the ARX(4,) was able t track the cmpnent-based mdel data Autcrrelatin f residuals fr utput y x -3 Crss crr fr input u and utput y resids Samples Figure 7. Residual analysis f the ARX(6,) mdel identified mdel was successful in predicting the system s test data behavir. Next, ANN was used t identify the system. Fr all ANN runs, the backprpagatin algrithm (with Levenberg- Marquardt training) was used. Als, the hyperblic tangent was used as the activatin functin fr all neurns and the meansquare-errr was used as the perfrmance functin []. A recurrent multi-layer netwrk with a single hidden layer ( hidden ndes) was simulated. The netwrk had five input signals: u(k), u(k-), y(k-), y(k-), y(k-3); and ne utput, y(k). Here, the sinusidal data was divided int tw parts: training data (first 5, pints) and testing data (last 3, pints). The training results are shwn in Figure. By cmparing the tw figures, it can be seen that the ANN was able t prvide similar utput t the cmpnent-based mdel. In rder t get a better view, nly the first 5 data pints are shwn in Figure 3. x Measured and simulated mdel utput x -3 Figure. Test data cmparisn: cmpnent-based mdel vs. ARX(4,) Figure 8. Test Data Cmparisn: cmpnent-based mdel vs. ARX(6,) Figure. Test Data cmparisn zm-in: cmpnentbased mdel vs. ARX(4,) Figure 9. Impulse respnse cmparisn: cmpnentbased mdel vs. ARX(4,) The ARX(4,) identified mdel was then used t predict the sinusidal test data. The results are shwn in Figure fr the 8, data pints and Figure fr the first, pints. It is clear that, unlike the lwer-rder ARX(6,), this higher-rder Then, the trained netwrk was tested fr the last 3, data pints. In rder t display an apprpriate view, 5 data pints are shwn in Figure 4. It is clear that the ANN predicts the testing data with minimal errr. ISMA3-4

5 Prceedings f the 9 th Internatinal Sympsium n Mechatrnics and its Applicatins (ISMA3), Amman, Jrdan, April 9-, 3 5. CONCLUSIONS This paper presented a cmpnent-based mdel f piezelectric ultrasnic transducers and used tw system identificatin techniques (ARX and ANN) t identify the system. The results shwed that the lw-rder ARX mdel identified the training data nly while a high-rder ARX identified the training and testing data. This is due t the fact that the piezelectric ultrasnic transducer mdel is nnlinear while the ARX is a linear mdel and therefre it requires higher-rder mdels t cmpensate fr the nnlinearities. On the ther hand, the ANN, a nnlinear mdel, identified bth the training and testing sinusidal data. Wrk is underway t test physical piezelectric-based systems and cllect real-time experimental data in the labs. Identified mdels can then help in predicting the material s behavir and in designing apprpriate cntrllers. Figure. Output results fr training data: (a) Cmpnentbased mdel (b) ANN ACKNOWLEDGMENT This research was supprted by the Strategic Technlgies Prgram f the Natinal Plan fr Science, Technlgy and Innvatin in the Kingdm f Saudi Arabia under the prject n. 9-ADV89. REFERENCES Figure 3. ANN vs. cmpnent-based mdels: results fr the training data (zm-in 5 data pints) Figure 4. ANN vs. cmpnent-based mdels: results fr the testing data (zm-in 5 data pints) [] Ramutis Bansevicius and Rymantas Tadas Tlcka, Piezelectric Actuatrs. Mechatrnics Handbk by Bishp.. Sectin.3 Page 575 [] Ru, Chen, Sha, Rng, and Sun. A hysteresis cmpensatin methd f piezelectric actuatr: Mdel, identificatin, and cntrl. Cntrl Engineering Practice 7 (9) 7 4. [3] Juhasz, Mass, and Brvac, Parameter identificatin and hysteresis cmpensatin f embedded piezelectric stack actuatrs. Mechatrnics () [4] Rus, Palma, Perez-Aparici. Experimental design f dynamic mdel-based damage identificatin in piezelectric ceramics. Mechanical Systems and Signal Prcessing 6 () [5] Mntemurr, Nassar, Kutsawa, Beluettar, Vincenti, and Vannucci, Identificatin f electrmechanical prperties f piezelectric structures thrugh evlutinary ptimizatin technique. Internatinal Jurnal f Slids Structures 49 () [6] Ahmed, N., et al., Analysis f frces in ultrasnically assisted turning. Jurnal f Sund and Vibratin, 7. 38(3): p [7] M.H. Salah, M.L. McIntyre, D.M. Dawsn J.R.Wagner, and E.Tatliciglu. Charge feedback-based rbust psitin tracking cntrl fr piezelectric actuatrs. IET Cntrl Thery and Applicatins Vl. 6, Issue 5, pages -4 [8] Cleman, B., Hdgdn, M.: On a class f cnstitutive relatins fr ferrmagnetic hysteresis, Arch. Ratin. Mech. Anal., 987, 99, (4), pp [9] Rlf Isermann and Marc Muchhf. Identificatin f Dynamic Systems: An Intrductin with Applicatins. Springer. [] Simn Haykin. Neural Netwrks and Learning Machines. Prentice Hall 8. ISMA3