FUZZY CONTROL OF MECHANICAL STRESS FOR FATIGUE TEST IN ELECTRODYNAMIC SHAKER

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1 Page 96 FUZZY CONTROL OF MECHANICAL STRESS FOR FATIGUE TEST IN ELECTRODYNAMIC SHAKER Julane Donadel 1, Herbert Martns Gomes, 1, Federal Unversty of Ro grande do Sul, Graduate Program n Mechancal Engneerng, R. Sarmento Lete, 45, sala 0, , Porto Alegre, RS, Brazl. 1 Federal Insttute of Ro Grande do Sul, Campus Farrouplha, R. São Vcente, 785, Cnqüentenáro, , Farrouplha, RS, Brazl. Abstract. The control of systems wth tradtonal methodologes often faces problems of nonlnearty, dffculty n the mathematcal defnton of the problem and uncertanty of system parameters as well. In ths context, controllers based on fuzzy logc shows up as an alternatve to solve ths type of problem snce these methodologes deal wth both lnear and nonlnear systems. Besdes they may control complex multvarable systems. The man objectve of ths paper s to develop and mplement a control algorthm based on fuzzy logc n order to control the mechancal stress of a fatgue test n a metallc materal usng an electrodynamc shaker. In order to reach the proposed objectves, t s used an electrodynamc shaker, an amplfer, a sgnal condtoner and stran-gages attached to a test specmen to measure the deformaton and therefore the mechancal stress. The system s developed n object-orented language (Aglent VEE), responsble to acqure stress sgnals and to process and send the control sgnals to the shaker. It s developed a routne whch allows the user to set up desred cyclng frequency and the prescrbed mechancal stress for the test. The Aglent VEE software was used jontly wth Matlab software for the fuzzy logc mplementaton. Comparsons are made wth tradtonal PID control at the end of the paper. Data concernng the controllers performance for changes n stress for a step type varaton are presented and the correspondng parameters for the system are obtaned. Tests are performed wth dsturbances on amplfer gan n order to synthetcally smulate non-lneartes of the system. The results show that the proposed metheodology s robust aganst such perturbatons. Besdes, the accuracy and response tme of the mplemented fuzzy control s compatble wth the tradtonal PID control. Fnally t s concluded that the fuzzy controller s easer to mplement and has fewer parameters to be adjusted when compared wth the PID one. Keywords: fuzzy logc control, fatgue test, PID control, electrodynamc shaker. 1. INTRODUCTION Dfferent lterature sources estmate that PID controllers are used between 90 and 95% of ndustral controllers. Ths s due to ts low cost, smplcty of operaton and effcency for most systems. However, the control based on tradtonal methodologes faces problems when dealng wth system non-lneartes, dffcultes on mathematcal defnton of the problem and uncertantes n system s parameters as well. Ths can degrade ts performance or even destablze the control system. In ths sense, control systems that are robust enough to adapt and adjust to these changng characterstcs are hghly desrable and ths has motvated nterest n use of smart controllers. In ths context, fuzzy logc based controllers shows up as an alternatve to solve ths type of problem. The fuzzy controller does not need the rgorous mathematcal model of the process, but the model of actons from the knowledge of a specalst, usng for ths lngustc terms (Shaw and Smões, 1999). Moreover, Fuzzy controllers deal wth both lnear and nonlnear systems and stll control complex multvarable systems, performng tasks of decson makng n varous types of plants. (Al-Odenat, 008, Lee, 1990). Flora (008b) presented a contrbutom to the snusodal acceleraton control of electrodynamc shakers appled to vbraton tests. It s stated that consderng the mportance of sne tests to dentfy crtcal frequences of operaton and to determne mechancal weakness n the specfed performance of specmens, vbraton controllers should be desgned to reproduce the ampltude and the frequency of the reference acceleraton waveform specfcally at the nterface between the shaker and the structure under test. The paper presented the Shaker dynamc model, the dentfcaton of electrcal and mechancal parameters and the controller structure. The mplementaton was made n C language mplemented n a TMS30C6713 floatng-pont DSP. Expermental results had demonstrated that ths soluton s capable of guaranteeng good reference trackng performance at a fxed frequency or at fast sweep rate tests wth resonance-free loads. More recently, Xangjun et al. (010) presented a paper where an electrodynamc shaker s controlled wth a PID Fuzzy strategy. In that paper the characterstcs of a shaker are frst analyzed and then the controller s proposed wth a feedback - feedforward structure. The paper performs a trackng control of the acceleraton waveform. They clam that the obtaned results prove that the effect of ther proposed fuzzy PID controller s far better than the classcal PID controller. Expermental results show that the system output acceleraton tracks the desred sesmc acceleraton waveforms nput satsfactory. The control of an electrodynamc shaker was mplemented for tme doman snusodal acceleraton waveform ampltude control by Rana (011). In that paper, the adopted control system was a FLC (Fuzzy Logc Control). Besdes,

2 Page 97 frequency sweep tests are mplemented for typcal automotve and aerospace testng. The FLC was based on PI structure where the nput values were the error and change n error and the output was the control sgnal ncrement. Ths paper presented the shaker model, the desgn of a suspenson mode compensator, a FLC synthess and expermental mplementaton results. It was demonstrated that ths soluton was capable of guaranteeng good reference trackng. The obtaned parameters show that FLC performs well for bare table and wth rgd load. It was also found that the controller stablty s senstve to the resonances to some extent. In ths context, the man objectve of ths work s to develop and mplement a control algorthm based on fuzzy logc to control the mechancal stress of a fatgue test system n a metallc materal usng an electrodynamc shaker. For ths, t s proposed the use of Aglent VEE Pro software to control the electrodynamc shaker, to analyze and acqure/process/send data. It was developed a routne, where desred oscllaton frequency and stress level cycles are defned. It was used the Aglent software and Matlab for Fuzzy Logc programmng. It s known that there are dfferent types of fuzzy controllers, namely: PI, PD, PI+D, PD+I and PID. (Carvajal et al., 1999). In ths paper, the Fuzzy PD + I controller s nvestgated. Takng advantage of the developed control system, t s possble to perform several tests, such as fatgue, resonance frequences evaluaton and tests that are lnked to frequency x magntude of mechancal stress. Furthermore, the mplementaton of such heurstc controller s smple and drect, wthout the need of rgorous mathematcal modelng of the electrodynamc shaker dynamcs. The features and functons of the used equpment are dscussed n secton. Theores of PID and Fuzzy control are brefly explaned n secton 3. Secton 4 shows the developed work, wth the equatons and assumptons made for the controlled system. The results are presented n secton 5, followed by conclusons n secton 6 and bblographcal references.. MATERIALS AND METHODS.1. Characterstcs of electrodynamc shakers A shaker s any equpment that apples a controlled vbraton (acceleraton, frequency, etc.) to a test specmen (Gomes, 008, Flora, 006, Flora, 008 e Lang, 1997). Fgure 1 shows the structure of a generc electrodynamc shaker. Fgure 1. Parts of a generc electrodynamc shaker. The characterstcs of the shaker used (St. 5000/300, TIRA, Vbraton Test Systems) are n Tab. 1 Table 1. Shaker characterstcs. Frequency range Maxmum Force Mass of movng part Table s dsplacement range Mass of the Specmen Under Tested (Statc Load) Amplfer and shaker maxmum requred power (rms) Internal wndng resstance Blower voltage Blower requred power Blower nomnal rotaton Hz 940 N (300kgf) 8,5 kg +/- 6 mm 60 kg 500 Watts 4 Ω 0 / 380 V 0,8 kw 850 rpm.. Sgnal amplfer characterstcs

3 Page 98 The amplfer s ntended to amplfy the sgnal receved from the sound card and send ths amplfed sgnal to the Shaker. The used amplfer s a WATTSOM model DBK 6000, a professonal sound amplfer and the characterstcs are shown n Tab.. Table. Sgnal amplfer characterstcs. Channels Output power per channel 4 Ω, Voltage(AC)=30V 350W / speaker Amplfer class AB Frequency Response (-3 db) at 8 Ω 15 Hz to 40 khz Total Harmonc Dstorton + Nose (THD + N) at 8 Ω < 0,05% from 0Hz to 1 khz and < 0,1% from 0 Hz to 0 khz. Input Impedance (unbalanced) 0 kω Sgnal/Nose rato (unweghted) > 90 db.3. Sgnal condtoner characterstcs In order to condton stran measurements t s used a Dgtal Stran Indcator model TMDE by TRANSDUTEC Company. The devce data are lsted n Tab. 3. Table 3. Sgnal condtoner characterstcs. Supply Voltage 10 * -40 V (AC) * Hz Exctaton voltage of the Wheatstone brdge 5 *,,5 and 1 V ±% ( DC) Low Pass Flter 10Hz and of 1kHz * Channel settngs ½ brdge * or full Wheatstone Brdge Extensometers 100 Ω to 600Ω (10 Ω * ) Adjustable senstvty ±0,05 to ±50 mv/v (1mV/V * ) Sgnal Calbraton Shunt Resstor 1±% mv/v Frequency Response Hz (-1dB) and 0-10Hz(-1dB) Nomnal Output Voltage ±10V Accuracy class 0,1% *Used confguraton.4 Electrcal extensometers characterstcs The electrcal extensometers (Stran Gauges) used to measure the stran deformaton on the surface of the specmen s an electrcal fol resstor composed of a metal grd over an nsulatng layer of polymer substrate. The used stran gages were PA06-15AA-10-LEN model for steel made specmens and PA13-15AA-10-LEN to those made of alumnum, wth nomnal resstance of 10 Ω, all by Excel Sensores Ltd. Company..5 A/D and D/A boards characterstcs It was used an A/D board model USB-108FS by Measurements Computers company. Ths board s an USB sgnal acquston system of 1 bts, 8 nput channels wth +/-5,0V, maxmum acquston samplng rate of 50 khz, wth programmable gans from 1 to 0. For D/A conversons t was used a computer sound board model Soundmax Integrated Dgtal HD Audo by Analog Devces wth channels, 16 bts, +/- 1,5V output levels, maxmum acquston samplng rate of 48 khz..6 Control System Software The software used for acquston, processng and transmsson of data s the Aglent VEE (Vsual Engneerng Envronment). The system conssts of a descrptve language program block (lke flowcharts) wth unlmted number of blocks. The verson used s (Aglent Vee, 005a, 005b). 3. CONTROLLERS 3.1. Tradtonal PID Controller The control sgnal generated by a tradtonal PID controller s well-known and gven by Eq. (1):

4 Page 99 1 t de( t) u( t) = K [ e( t) + e( ) d T ] p ò t t + = d T dt 0 K de( t) = K e( t) + e( ) d + K T = P( t) + I( t) + D( t) dt t p p ò t t p d T 0 (1) where K p s the proportonal gan, T tme ntegratve tme and T d dervatve tme. And stll, K = K p / T s the ntegratve gan and Kd = K ptd s the dervatve gan. At regular ntervals, the ntegral acton fxes the value of u( t ) addng to ths the value of the devaton. The ntegral acton that s drectly lnked to the accuracy of the system s responsble for zero error n steady-state. The dervatve acton has the effect of reducng the speed of changes n process varable, whch prevents t to rse or reduce n a very quckly way. (Ogata, 004). The mplementaton of the PID controller s done by makng numercal approxmatons of the dervatve and the ntegral that appear n the control law. The obtaned samplng perod (that s software/machne dependent) was approxmatelyd T» 80ms. Equaton () shows the mplementaton of the algorthm where y( t), r( t), u( t ) and e( t ) denote, respectvely, the output of the process, the reference sgnal, the control sgnal and the error at tme t k : P( k) = K e( k) I( k) = I( k- 1) + ( KD T )[ e( k) + e( k-1)]/( T ) D( k) = [( pdt- ) /( pd T+ )] D( k- 1) + [ KT / DT ( pd T + )][ e( k) - e( k-1)] d u( k) = P( k) + I( k) + D( k) e( k- 1) = e( k) D( k- 1) = D( k) I( k- 1) = I( k) () where the symbols P, D and I are the of proportonal, ntegratve and dervatve correcton parts as ndcated prevously, and e( t ) s the error at the nstant t k = kd T, k s the ncrement of the teratve process. 3.. Fuzzy Controller A controller wth fuzzy logc s bascally composed of four elements: a rule base, an nference engne, a fuzzfcaton nterface and a defuzzfcaton nterface, as shown n Fg.. (Passno, 1997). Rule Base r Reference sgnal + - S e(t) Error sgnal Fuzzfcaton Defuzzfcato n u(t) Control sgnal Plant y(t) Output Inference mechansm Fgure. Fuzzy controller nserted n the control loop Besdes the error sgnal (reference sgnal mnus output value), a fuzzy controller may have the dervatve of the error or the ntegral of the error wth nputs smlar to a tradtonal PID controller. (Jantzen, 001; Burns, 001) Fuzzy PD + I controller A Fuzzy PD + I controller t s a fuzzy proportonal-dervatve controller added to the conventonal ntegratve part. The error s gven by the dfference of the reference value and the output of the system, defned by, e( t) = r- y( t) and the dervatve of the error s approxmated by e& ( t) =- y& ( t). Those are the two nputs of the control system. The control s gven by Eq. (3) and Eq. (4).

5 Page 100 t u( t) = uf + Kò e( t ) dt (3) 0 uf = Fuzzy( e, de) (4) where u F s the fuzzy PD control part and K s the ntegratve gan from a tradtonal PID framework. For the Fuzzfcaton task, n ths paper the nputs of the system are gven n terms of three fuzzy sets: Postve (P), Zero (Z) and Negatve (N). Ths assumpton was based on the reported lterature that ndcates as suffcent for the control. The outputs are gven n terms of fve fuzzy sets: Negatve Bg (NB), Negatve Medum (NM), Zero (Z), Postve Medum (PM) and Postve Bg (PB). The unverse of dscourse n ths case s the nterval [-1,1], and s multpled by a gan defned by the user (Jantzen, 001). In ths case, the assumed gans were k e = 1 for the error, kde = 10m for the dervatve of the error and k u = 0.8 for the control sgnal. The membershp functons for error, the dervatve of the error and output are all of trangle shape as shown by Fg. 3. The bottom of the Fg. 3 shows the control surface formed by the dfferent values of the two nputs, e and de and the output fuzzy control, u F. (a) (b) (c) (d) Fgure 3. Membershp functons for two nputs (a) e(t) and (b) de(t), control output (c) u F (t) and (d) fuzzy control surface of the Fuzzy PD-controller. The rule base s used to catch expert knowledge n controllng the system by observng varable nteractons and classfyng through lngustc rules. A rule base for a fuzzy PD control system can be summarzed by a matrx of order two, as shown n Tab. 4, snce the membershp functons were separated nto three fuzzy sets. e N Z P de N NB NM Z Z NM Z PM P Z PM PB Table 4. Fuzzy PD-controller rule base for three fuzzy sets of the nput parameters. The man functon of the fuzzy nference engne s to calculate the total value of control output varable based on the ndvdual contrbuton of each rule n the rule database. It was used the Mamdan method (Mamdan, 1975), whch encompasses the rules by logcal operator OR, and each rule are aggregated by logcal operators AND and THEN that are modeled by the mnmum operator. The defuzzfcaton step translates the state of the output varable fuzzy control sgnal for a non-fuzzy ("crsp") value. In ths case t was uses the center of mass method that can be represented by lngustc terms as: Sum of frst moments of area of output Fuzzy set Crsp control sgnal = Sum of areas of output Fuzzy set (5) 4. EXPERIMENTAL SETUP

6 Page 101 The am s to desgn a controller for the electrodynamc shaker such that the mechancal stress appled to the specmen can be easly controlled and mantaned durng cycles of loadng and unloadng n real tme. Fg. 4 presents a dagram and photos of the tested prototype and the structure of the controlled system. Clamp Stran gage Porch reacton Lattce Moton Sgnal Condtoner for stran gages Board data acquston A/D Trangular ruler Table Stran gage Electrodynamc Shaker Computer Amplfer Sound Board D/A Fgure 4. Scheme of control system and photo of the prototype tested. The specmen has a trangular shape so that the measured deformatons at any pont on ts face are the same, rrespectve of postonng of the stran gage, reducng the uncertantes by bondng errors n stran measurng. The tp of the specmen s pn-joned through steel rollers that react aganst the shaker reacton frame. Part of the clamped specmen s rgdly connected to the vbratng table whch mposes prescrbed dsplacements and thus deformatons and mechancal stresses. Wth ths structure t was bult the control system that can be used to perform fatgue tests. The Aglent VEE sends a sgnal from the PC sound card to the sgnal amplfer whch amplfes and sends t to the shaker. In the PC, a reference electrcal sgnal s generated based on the desred stress, frequency of cyclng and parameters defned by the user. Ths sgnal s sent back to the col, whch makes the movng part (voce col and table) to produce an oscllatory movement whch transmts to the specmen under test producng a deflecton and generatng, consequently, a mechancal stress. The deformaton produced n the specmen s measured by the electrcal stran gages and sent to sgnal condtoner whch passes through the computer data acquston board, so that the receved sgnal s analyzed by the program and corrected for the next cycle. The stran gage s arranged n a half Wheatstone brdge. Snce Young modulus, gage factor, brdge exctng voltage are known, the mechancal stress can be evaluated and compared wth the desred one. The control s based on the rms value (root mean square), because ths value takes nto account the energy contaned n the sgnal. 5. EXPERIMENTAL RESULTS After several tests, t was obtaned the parameters that best ft the system to a cyclc loadng frequency of 50Hz. The desred mechancal stress was 0 kpa and can range from 1 kpa to 30 kpa due to expermental geometry and shaker s table dsplacement range. By tral and error adjustment we obtan the ntegratve gan control parameter K = 0.1. For the PID control system, the tuned other two parameters were K p = 0.1, Kd = 10m and K = 0.. Fgure 5 shows the stress reference value and the measured stress along teratons (tme). Accordng to Fg. 5(a) and 5(b) both fuzzy control system and PID showed good results, achevng the proposed goal..1 x x Mechancal stress (Pa) Mechancal stress (Pa) (a) 1.3 Desred stress Controlled stress Tme (s) Tme (s) (b) Fgure 5. (a) Fuzzy controller PD + I and (b) PID controller. 1.3 Desred stress Controlled stress

7 Page 10 The rse tme, that s defned as the tme elapsed to get the response from 10% to 90% of ts fnal value, was measured and resulted n approxmately 0.6s. The settlng tme, that s the tme requred for the response settles and stay wthn a percentage range around % of the value of steady state s approxmately, was measured and resulted 1s. Overshootng s also relatvely low, not exceedng.5% of the reference value. Fgure 6 shows comparsons between the Fuzzy PD+I controller and tradtonal PID controller for dsturbances on the system. Ths dsturbance was synthetcally obtaned by applyng slght varatons through of the sgnal amplfer (6 db varaton on power amplfer gan). Fgure 6 also shows that the system quckly returns to ts set pont. Moreover, f one compares the same test performed by the PID controller, t can be notced that the Fuzzy controller returns to the desred value more quckly presentng less overshootng..8 x x 104 Mechancal stress (Pa).6.4. Mechancal stress (Pa).4. (a) Desred stress Controlled stress Tme (s) Tme (s) (b) Fgure 6. Dsturbances at: (a) Fuzzy controller PD + I and (b) PID Controller. Desred stress Controlled stress Fgure 7 shows the behavor of the error sgnal, the dervatve of the error and control sgnal (Volts) tself. One can observe that the error and the dervatve go to zero quckly and the control sgnal s stablzed at the desred value Voltage (V) Error (V) -0.1 Error dervatve (V) Control sgnal (V) Tme (s) Fgure 7. Control sgnal, error and dervatve of error. Comparng the fuzzy controller wth conventonal PID, they obtan smlar results. The parameters used for the PID controller were: K p = 0.1, K d = 0. and K = 10m. In ths case, the rse tme was 0.3s and the settlng tme of about 0.6s. These results can be seen n Fg. 8, where a comparson s made for the two types of controllers..1 x Mechancal stress (Pa) Desred stress 1.3 Fuzzy PD+I controlled stress PID controlled stress Tme (s) Fgure 8. Comparson between PID and fuzzy controllers.

8 Page 103 It was also notced that the overshootng of the PID controller dd not exceed.8%, e t was slghtly larger than the Fuzzy controller. Therefore, t s concluded that the two used controllers are effcent snce they presented error n the 3 order of 10 - Pa. Generally speakng, the fuzzy controller presented less overshootng, whch s less mpactng n a mechancal pont of vew for a fatgue test. 6. CONCLUSION In ths paper, t was presented the development of a Fuzzy PD + I and smple PID controllers for managng the actons of an electrodynamc shaker amng to control the stresses generated n a specmen coupled between the rgd structure and oscllatory table. To develop the paper t was carred out several tests for parameter adjustment and tunng. 3 On both developed controllers the system could be controlled wth an error n the order of 10 - Pa. Both algorthms presented good results wth short rse tmes and low settlng tme, wthout sgnfcant overshootng. One observed advantage of the fuzzy controller was the ease n the mplementaton. It presents a smaller number of parameters to be adjusted when compared to the PID controller. In addton, both systems are effectve n dsturbance rejecton, quckly returnng to the set pont. Thus, the developed control systems are vald to perform fatgue tests and nvestgate dfferent metallc specmens. Fnally, we conclude that the results were satsfactory and the goal of stress control n the fatgue test was attaned. 7. ACKNOWLEDGEMENTS The authors wsh to acknowledge CNPq and CAPES (by the program Pró-Engenharas) for the support n ths research. 8. REFERENCES AGILENT VEE PRO 7.5, Aglent VEE Pro User s Gude, Aglent Technologes, Inc., 9th Edton, 66p., May, 005a. AGILENT VEE PRO 7.5, VEE Pro Advanced Programmng Technques, Aglent Technologes, Inc., 9th Edton, 61p., May, 005b. AL-ODIENAT, A. I. and LAWAMA, A. A., 008, The advantages of PID fuzzy controllers over the conventonal types. Amercan Journal of Appled Scences, Vol.5, No.6, pp BURNS, R. S., 001, Advanced Control Engneerng. Ed. Butterworth Henemann, Plymouth, UK, 464 p. CARVAJAL, J., CHEN, G. and OGMEN, H., 000, Fuzzy PID Controller: Desgn, Performance, Evaluaton and Stablty Analyss. Informaton Scences, 13, pp FLORA, L.D. and GRUNDLIN, H.A., 006, Acceleraton Control of an Inverter-fed Electrodynamc Shaker, In Proceedngs of the 37th IEEE Power Electroncs Specalsts Conference, Jeju, Korea, pp FLORA, L. D. and GRUNDLING, H. A., 008a, An Electrodynamc Shaker Tme Doman Snusodal Acceleraton Controller, Insttutons of Engneerng and Technology, Control Theory and Applcatons, Vol., No. 1, pp FLORA, L. D. and GRUNDLING, H. A., 008b, Tme doman snusodal acceleraton controller for an electrodynamc shaker IET Control Theory Appl., V., No. 1, pp GOMES, H.M., GASPARETO, D.S., FERREIRA, F.S. and THOMAS, C.A.K, 008, A Smple Closed-Loop Actve Control of Electrodynamc Shaker by a Acceleraton Power Spectral Densty for Envronmental Vbraton Tests, Expermental Mechancs Sprnger, Vol. 48, pp JANTZEN, J., 007, Foundatons of Fuzzy Control. Ed. Wley, England, 0p. LEE, C. C., 1990, Fuzzy logc n Control systems: fuzzy logc controller-part I, IEEE Transacton on systems, man and cybernetcs, Vol. 0, No., pp MAMDANI, E.H. and ASSILIAN, S., 1975, An Experment n Lngustc Synthess wth a Fuzzy Logc Controller, Internatonal Journal of Man-Machne Studes, Vol. 7, No. 1, pp OGATA, K., 004, Engenhara de Controle Moderno (n Portuguese), Ed. Prentce-Hall do Brasl, Ro de Janero, Brazl, 800 p. PASSINO, K. M. and YURKOVICH, S., 1997, Fuzzy Control, Addson-Wesley Longman Inc., 50p. RANA, K. P. S., 011, Fuzzy control of an electrodynamc shaker for automotve and aerospace vbraton testng, Journal of Expert Systems wth Applcatons, V.38, pp SHAW, I. S. and SIMÕES, M. G., 1999, Controle e Modelagem Fuzzy (n Portuguese). Edtora Edgard Blücher LTDA, São Paulo, Brazl, 165p. XIANGJUN, L., YANPING, G., XINZHONG, C. and ZHANJUN, M., 010, Fuzzy PID control of acceleraton for electrc shaker, Proceedngs of the Internatonal Conference on Measurng Technology and Mechatroncs Automaton.

9 Page RESPONSIBILITY NOTICE The authors are the only responsble for the prnted materal ncluded n ths paper.