Synthesis of Vibration Damping Control Systems based on the Identified Model of a Mechanical Construction at Broadband Disturbance Influence

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1 International Journal of Computer Application ( ) olume 14 No., May 16 Synthei of ibration Damping Control Sytem baed on the Identified Model of a Mechanical Contruction at Broadband Diturbance Influence Marygin Olekiy College of Aeropace Engineering, State Key Laboratory of Mechanic and Control of Mechanical Structure, Nanjing Univerity of Aeronautic and Atronautic, No. 9 Yudao Street, Nanjing 116, China. Xing Shen College of Aeropace Engineering, State Key Laboratory of Mechanic and Control of Mechanical Structure, Nanjing Univerity of Aeronautic and Atronautic, No. 9 Yudao Street, Nanjing 116, China. ABSTRACT The changing of certain type of energy into other type in machine and mechanim, the tranformation of movement form and carrying out certain working procee inevitably caue the emergence of variable force and vibration-cauing momentum. ibration control and management are important tak in aeropace field. Thi thei olve the problem of the development of an effective control ytem baed on the active method ued for vibration reduction in a mechanical contruction of a cantilever fixed metal plate broadband diturbance lab device. In the ytem ynthei proce a carry out tructural and parametric identification of the mathematical model of the controlled object uing experimental data. LabIE 1. development environment and FPA hardware technology from National Intrument were ued to create a pectrum analyzer meaurement ytem, which allow olving problem of experimental acquiition and identification of the dynamic propertie of mechanical object. ith the help of MATHLAB environment mathematical apparatu and Optimization and Control Sytem Toolbox package, a mathematical model of the control object, regulator' ynthei wa built; their optimal parameter were earched for, and ytem operation with different object tructure and regulator (PID, LQ) wa modeled. Keyword Control ytem, regulator, PID, LQ, vibration, control object, mathematical model, LabIE, MATHLAB. 1. INTRODUCTION Modern aeropace aircraft are lightweight and are highly durable. Their peed and acceleration requirement are getting higher and higher, which caue vibration increae and the neceity of taking it into account when drafting aircraft blueprint. A deire to reduce the weight of the aircraft mechanical ytem doen't allow for the uage of common damping material only. Today one of the way for vibration reduction i claimed to be the uage of piezoelectric material [3,]. Piezoelectric material (PZT) poe phyical and technological propertie, which are important for the aircraft, that i: mall volume, light weight, mechanical implicity and the eaine of flexible tructure integration. Becaue of direct and revere piezoelectric effect, thoe material are ued a enor and a executive mechanim in tructural vibration control ytem. Today paive, emi-paive and active vibration damping method are in development. [1,].. EXISTIN PRINCIPLES OF IBRATION CONTROL The main principle of the paive method i that the energy from piezoelectric material deformation diipate on hunt circuit RLC chain. Energy diipation proce prevent piezoelectric material deformation and the tructural damping increae [,3,4]. Semi-paive method of vibration damping ue external power ource, the energy from which help amplify the ignal taken from piezomaterial and tranmit it in the anti-phae to the ame pzt patch [,3,4]. Active method ue control ytem baed on feedback principle with a regulating device. Thoe ytem are either built in analogue form with operational amplifier and P, PI, PID control principle at their bai; or in dicrete form with digital cheme and different control principle and regulation criteria [,3]. Thi thei preent building an active ytem of vibration control uing digital controller. 3. ELECTROMECHANICAL SYSTEM AS A CONTROL OBJECT A a control object, thi thei ue a cantilever fixed metal plate the vibration of which need to be reduced with the help of an executive mechanim and a pzt patch enor. The aluminum plate ha dimenion of 3х4х1 mm; it elatic modulu i 7 Pa; it denity i 7 kg/m 3. The data above allow for calculation of it reonance frequencie (9.1, 7, 16, 314, 3 Hz) and ue them a reference point when determining the dynamic propertie of the object. Analytical determination of a complex "mechanical ytem pzt converter" tructure i quite a complicated tak; beide, the preence of inhomogeneity in the mechanical tructure, deviation from the form and the introduction of additional mae pzt enor and actuator dratically change the dynamic of the object. Thuly an important direction in tudying the behavior of a mechanical object i the experimental determination of it dynamic qualitie [11,1]. The tak et in thi thei i to reduce the vibration level of a mechanical ytem, where the vibration i induced by HE- electromechanical exciter uing manipulating influence via a PZT actuator (Fig. 1). 18

2 mount rod PZT actuator Exciter HE- Input channel control Input channel diturbance Laer diplacement enor Beam control vibration from the PZT enor Fig. 1: Allotment of control and diturbance channel of the reearch object. The analyi of the aforementioned electromechanical ytem a the control object allow to allot diturbance and control channel. In the diturbance channel, force f generated by HE- influence the mechanical ytem and lead to it diturbance X 1. In the control channel, the ignal ent to the pzt actuator caue it flex and the flex of the plate X proportionally to the level of voltage applied U(t). The flex of the pzt actuator hould level out the flex of the plate caued by the diturbance. Each of the allotted channel ha it own dynamic characteritic, which need to be defined for the further control ytem ynthei (Fig. ). International Journal of Computer Application ( ) olume 14 No., May 16 pecter of the timulating ignal and reaction AA, BB, are the function, neceary for the frequency characteritic aement, and coherence function. Coherence function how the degree of linear dependence for accounted ignal in the frequency domain. Function can be preented a H 1 ( f ) AB AA, H ( f ) BB BA, AB AA( f ) BB.. MEASUREMENT SYSTEM A pectrum analyzer meaurement ytem wa created with the ue of LabIE 1. development environment and FPA hardware technology from National Intrument, which allow olving problem of experimental acquiition and identification of the dynamic propertie of a mechanical object (fig. 4). f() DISTURBAN C E () X 1 () U () () CONTROL X () X() () A() Fig. : eneral tructural cheme of the object 4. OBJECT DYNAMIC CHARACTERISTICS MEASUREMENT METHOD USED A dual-channel pectrum analyzer wa ued for the definition of the dynamic propertie of the control object. Thi method i baed on the pectral analyi of the timulating ignal and repone auming the linearity of the electromechanic ytem. The reearch ignal ued wa of the "white noie" type. Sytem repone of the acceleration type wa meaured with a pzt enor. The working principle of the analyzer i repreented in fig. 3. А Channel В Channel Temporal function а(t) Temporal function b(t) F F Intant pecter A(f) Intant pecter B(f) Self pecter AA(f) Mutual pecter AВ(f) Self pecter BB(f) Frequency characteritic H 1 (f) Coherence function γ (f) Regitration Proceing Averaging Additional proceing H 1 (f) Re[H 1 (f)] Im[H 1 (f)] Fig. 3: Principal block-cheme of a program dual-channel pectrum analyzer. Signal a (t), b (t) applied to the analyzer' input are changed to the digital form a [nk], b [nk] o a to get the erie of digital data (realization). Recorded realization are tranformed into frequency domain a complex intant pecter A and B uing dicrete Fourier tranform [6,7,8]. Mutual and elf φ (f) Fig. 4: eneral picture and cheme of the meaurement ytem. Automatic meaurement ytem include: 1. A PC equipped with a dual-channel analogue-digital tranformation device baed on PCIe-784R and developed oftware;. Tektronix AF ignal generator (output Ug(t), ); 3. Analogue low pa filter (input - Ug(t),, output - Ufil(t)); 4. HEAS- amplifier (input Ufil(t), ; output Ig(t), A);. HE- exciter (input Ig(t), A; output F(t), N); 6. Mounting connection board; 7. The plate being tudied; 19

3 International Journal of Computer Application ( ) olume 14 No., May Pzt enor (input - acceleration, mm/ ; output - Ud(t), ); 9. ain PZT XE1-A (input - Ufil(t), ; output - u(t), )); 1. SCB-68А interface module; The oftware developed for the meaurement ytem preuppoe an ability to carry out harmonic analyi with the allocation of the main frequency; amplitude calculation, tranfer coefficient calculation and, allotted frequency phae calculation. There i a "Signal Analyi" tab for timulating ignal pectral characteritic Ug(t) and repone. There i a "Sytem Analyi" ain PZT tab for ytem enerator frequency characteritic in FRF, Phae and Coherence. Laer PZT patch 6. EXISTIN PRINCIPLES diplacement OF enor IBRATION CONTROL Modul SCB-68А Frequency characteritic Beam of the ytem on diturbance and PXI ytem control channel were acquired with the help of the meaurement board, the Ug(t) configuration of which are hown Exciter uer in HE- fig.. Double integration enerator wa 1 ued application for the output diplacement ignal. Ug(t) Filter Ucontrol(t) Filter x(t) Ufil(t) Ig(t) F(t) Mechanical Amplifier Exciter ytem tudied HEAS- HE- Ug(t) F(t) Ufil(t) a) Reearch object on diturbance channel Uamp(t) Amplifier PZT HPZT actuator a) Mechanical ytem tudied Reearch object on control channel b) PZT enor PZT enor Fig. : Structural cheme of the meaurement ytem by channel a) diturbance; b) control. 6.1 On diturbance channel Experimentally obtained amplitude frequency repone graph (fig. 6) how multiple pronounced reonance in [1, ] Hz frequency range. In [, 1] Hz frequency range the ytem i on the noie level, which i concurred by the coherence function, which reduce to zero. a1(t) a(t) x1(t) x(t) x(t) 6. On control channel On the graph of the experimentally obtained amplitude frequency repone function (fig.7) on control channel in [1, ] Hz frequency range there are a few reonance with a high peak value. Yet the coherence function tetifie the preence of the noie, which require additional reearch in thi frequency domain. There are multiple pronounced reonance in the frequency range of [, ] Hz. Yet their level i much lower compared to the level in the frequency repone range [1, ] Hz, which allow to not take them into conideration when identifying the model. On frequencie above Hz the ytem i on the noie level. Fig. 7: Frequency characteritic on the control channel. According to the frequency characteritic obtained, tructural and parametric identification of the object wa made on control and diturbance channel. 7. IDENTIFICATION Experimental data wa implified toward imple calculation cheme in form of combined mae, tiffnee, damping element and external force' vector. In order to create a ytem of common differential equation of motion according to the calculation cheme, ditributed mae were replaced with the focued one, fig. 8. С b z(t) M Pz Fig. 6: Experimentally obtained frequency characteritic of the object on diturbance channel Becaue of a big amount of noie in the low frequency region, frequency function were rectified in the [.1,1] Hz range during harmonic excitation. Frequency repone function analyi, how three reonance on 1, 19, 6 Hz frequencie. Thoe reonance will be ued for the further identification of the mechanical ytem. Fig. 8: Mechanical ocillation circuit The contruction of the mathematical model of ocillatory link (fig. 8) wa performed with the approximation of experimentally obtained frequency characteritic of an elatic ytem by nonlinear programming [9,1]. Becaue of the practical inability to identify all the reonance found, their number i limited according to a certain criteria. For example, on the diturbance channel, the tranmiion coefficient value of the ytem reearched on the firt reonance (fig. 6) exceed the tranmiion coefficient value on the fourth reonance and on all the following one more than 4 db. ithin diplacement value it mean that the ocillation on the firt reonance are time higher than thoe on the fourth one at the ame excitement level. According to thi, the frequency range of the ytem tudy wa limited to 1 Hz.

4 U pid pid out, U pid out, Phae (deg) Phae (deg) Phae (deg) Ug(t) ain PZT enerator 7.1 On diturbance channel Conideration of the frequency characteritic Laer PZT patch x(t) a of a Modul et of diplacement SCB-68А vibration link (econd-order differential enor equation) allow for the creation of a tructural cheme a hown in fig. 9. The ue of the multiobjective Beam function (vector) of goal achievement PXI ytem from the Optimization Toolbox Matlab package allowed Ug(t) Exciter F() uer obtaining the etimate of the parameter of the propoed HE- enerator 1 application model. Bode diagram of the model i repreented in fig. 1. f (),34 ( 1 ),33,9 1,1384 ( ) 8.76e e - ( 3 ) 6.38 e X1() Fig. 9: Mathematical model of the ytem on diturbance channel Bode Diagram International Journal of Computer Application ( ) olume 14 No., May PID regulator control ytem Finding PID coefficient for the dicrete control ytem fig. 13 wa carried out by the method of nonlinear programming, taking into account the minimization of the variance of the controlled variable Канал управления from the diturbance influence of limited Effectivene coefficient of the ytem at the regulator Кана возмущения coefficient found (k p =., k i =11, k d =) are repreented in table 1. e () f() E () DISTURBANC X 1 () U () X () X() pid() CONTROL () Fig. 13: Structural cheme of a PID regulator control ytem A() The yntheized PID regulator control cheme provided tranmiion coefficient decreae on the diturbance channel in the frequency range of [1-1, *1 ] rad/ compared to the object without a regulator (fig. 14). The ytem i effective in thi range; diturbance tranmiion coefficient decreae. 1 Bode Diagram From: olt To: x1, mm Fig. 1: Bode diagram on diturbance channel PZT patch Ug(t) ain PZT enerator Laer diplacement 7. On control channel enor Modul SCB-68А Conideration of the frequency characteritic on the control Beam PXI ytem channel allowed for the creation of a tructural cheme a Ug(t) hown Exciter in fig. 11. F() Bode diagram of the model uer acquired i on HE- enerator 1 application fig. 1. U() ( ) 1 1 ( ) 1 Frequency (rad/ec) x(t),1978 ( 1 ), ( ) e ( 3 ) e Fig. 11: Mathematical model of the ytem on control channel From: u, olt To: x, mm Frequency (rad/ec) Fig. 1: Bode diagram on control channel 8. SYNTHESIS OF CONTROL SYSTEMS There were three type of feedback-baed control ytem with variou regulator yntheized for the obtained and approximated dynamic characteritic. The effectivene of the modeled ytem' control wa etimated according to the coefficient equal to the variance of the controlled variable without the control ytem to the variance of thi variable with the control ytem working. X() Frequency (rad/ec) Fig. 14: Comparion of frequency characteritic of diturbance channel with and ytem The reult of. the ytem.4 work modeling.6 at.8 harmonic 6 diturbance on the firt reonance of the object i hown in fig. 1 acceleration, mm/ acceleration, mm/ mm/ acceleration with, mm/ acceleration with, with, mm/ with without control control Time, Fig. 1 1: The reult of the modeling of the control ytem at 1 the diturbance in form of the1 Hz harmonic ignal Time, Time, 1

5 Phae (deg) U U pid pid out, out, U pid out, U reg out, f, f, 8. Synthei of a Linear Quadratic Regulator Synthei of the LQ regulator (fig. 16) wa done in the Control Sytem Toolbox package of the MatLAB ytem [11]. The realization of the LQ regulator on the exiting equipment will be carried out with ADC, DAC and an FPAbaed microcontroller, o the model of the ytem (fig. 16) will be dicrete. The control object acquired i tranferred to the tate pace and ha the following matrix dimenion: A=[1x1], B = [1 x ], C = [ x 1], D = [ x ]. Sytem input f() receive input impact of a white noie type with the zero mean, from which diturbing influence with et characteritic are formed inide the model. The aim i to develop an optimal linear regulator. The average value of the quadratic functional J ( u) E n1 T T y[ R1 y[ u[ R u[. i ued a the optimality criterion. Matrice of the weight coefficient R 1 and R, which determine the input of the original object coordinate y[ into the integral control error, and the ratio of the ytem error and the power of the Канал управления controlling influence u[ were elected during the regulator Кана возмущения ynthei proce with a condition that the controlling ignal i in the et range. U in () Model y Model Plant 1 in CONTROL () X () X() 1 out out U out_reg () A() International Journal of Computer 4 Application 6 8( ) olume 14 No., May 16 acceleration, mm/ acceleration, acceleration, mm/ mm/ with, mm/ acceleration acceleration with, with, mm/ mm/ Fig. 18: The reult of the modeling of the ytem with an LQ regulator at the diturbance from the "white noie" type ignal 8.3 Synthei of a Linear Quadratic Regulator for a Modified Object To reearch a poibility of the control ytem effectivene increae with the white noie diturbance influence, the control object can be changed, and meaurement and hift control, rather than acceleration can be demanded (fig. 19). Канал управления hen yntheizing the regulator, demand the tabilization of Кана возмущения the "hift" parameter on the nd output can be demanded. 3 rd output acceleration A() i left there for control. f() in E () DISTURBANC X 1 () Model y Model Plant 1 out U out_reg() Model RegMIMO U in () 1 in CONTROL () X () out X() K X_e Kalman filter f() in E () DISTURBANC X 1() 3 out A() Fig. 16: Structural cheme of a ytem with an LQ regulator 1 From: f To: Out(1) Model RegMIMO K X_e Kalman filter Frequency (rad/ec) Fig. 19: Structural cheme of a ytem with an LQ regulator for a modified object Amplitude-frequency characteritic on acceleration can be tudied. In the frequency range of [1-1, 1 3 ] rad/ the LQ regulator control ytem provided a decreae of the diturbance channel ignal tranmiion coefficient compared to the ytem without a regulator. The ytem i effective in thi range. Effectivene coefficient are hown in table 1 Fig. 17: Comparion of frequency characteritic of diturbance channel with and ytem The LQ regulator control cheme provided tranmiion coefficient decreae on the diturbance channel in the frequency range of [4*1 1, 1 3 ] rad/ compared to the object without a regulator (fig. 17). The ytem i effective in thi range. Fig. 18 how the reult of the ytem working with the diturbing influence of the "white noie" type. Effectivene coefficient are hown in table 1.

6 Phae (deg) Phae (deg) International Journal of Computer Application ( ) olume 14 No., May Bode Diagram From: f To: Out(1) Bode Diagram Frequency (rad/ec) 9 a) b) Frequency (rad/ec) Fig. : Comparion of frequency characteritic of diturbance channel with and ytem (a acceleration, b diplacement). 8.4 Control ytem comparion Table 1 how effectivene coefficient for the yntheized ytem with the diturbing influence on the firt 1 Hz reonance of the object and with the "white noie" ignal influence. Table 1. Effectivene coefficient for the control ytem Control ytem Effectivene coefficient Harmonic influence hite noie 1 PID 14 LQ type 1 8 3, 3 LQ type.e+3 Comparative analyi of the ytem' work at the modeling tage in equal condition how that the ytem built around the LQ regulator, when meauring and hift controlling, i more effective than the ytem baed on calculation and acceleration control. 9. CONCLUSION In thi article: an electromechanic ytem baed on the cantilever fixed plate lab equipment i tudied a a control object with input and output ignal differentiated; a vibration ditribution channel diturbance channel wa defined; and a control channel a way for control influence - wa created; oftware for the experimental definition of the dynamic propertie of an electromechanic ytem wa developed; and the dynamic propertie were meaured; tructural and parametric identification of the experimental data on control and diturbance channel of the object wa carried out; -dicrete regulator with different object and regulator tructure (PID, LQ) were yntheized; ytem effectivene wa compared acceleration control. Identification of the object tudied and modeling different variant of control ytem for it at diturbing influence allow howing a poibility of creating an effective vibration control ytem. Further development of thi tudy i realization of yntheized digital hardware regulator, their adaptation for the object and uage of the uggeted approach for damping of more complex mechanical tructure, a well a the earch for more effective and robut control algorithm. 1. ACKNOLEDMENTS Thi work wa ponored by Qing Lan Project and a Project funded by the PAPD. The author are alo grateful to the Fundamental Reearch Fund for the Central Univeritie (No. NS131). 11. REFERENCES [1] Ehteham Mutafa Qurehi, Xing Shen, JinJin Chen. 14 ibration Control law via Shunted Piezoelectric Tranducer: A Reivew. Int l J. of Aeronautical & Space Sci, 1 (1). [] C. Richard, D. uyomar, D. Audigier, and H. Baaler, Enhanced emi paive damping uing continuou witching of a piezoelectric device on an inductor, in Proc. SPIE Smart Struct. Mater. Conf., Paive Damping and Iolation, pp [3] Donald, J.L. 7 Engineering analyi of mart material ytem John-iley & Son, Inc.: Hoboken, New Jerey. [4] Srinivaan, A. McFarland, D.M., Smart tructure: analyi and deign. Cambridge Univerity Pre. [] Cunefare, K. A. State-witched aborber for vibration control of point-excited beam," in ASME International Mechanical Engineering Conference and Expoition, Orlando, FL. [6] Box, D., att, D Time erie analyi. Prognoi and Control, Iue 1, Mocow pre, pp [7] Box, D., att, D Time erie analyi. Prognoi and Control, Iue, Mocow pre, pp [8] Bendat, J.S. and Pierol, A Analyi and Meaurement Procedure', iley Intercience, New York [9] rop, D Method of Identification of Sytem. orld, Mocow. [1] Deutch, A.M Identification method of dynamic object, Mocow pre, pp.- 4. [11] Lazarev, Yu. Modelling of the procee and ytem in MatLab. The educational coure. SPb: Publihing group BH, p. 1 [1] Nikiforov, A. S., Budrin, S Propagation and aborption of vibration on hip. Leningrad, «Sudotroenie», IJCA TM : 3