active vibration control system for 3d mechanical structures

8
138 * Bialystok University of Technology, Faculty of Mechanical Engineering, Department on Automatics and Robotics, ul. Wiejska 45C, 15-351 Bia³ystok; [email protected]; [email protected] MECHANICS AND CONTROL Vol. 30 No. 3 2011 Andrzej P. KOSZEWNIK * , Zdzis³aw GOSIEWSKI * ACTIVE VIBRATION CONTROL SYSTEM FOR 3D MECHANICAL STRUCTURES SUMMARY The active vibration control system for 3D mechanical structures is presented in the paper. The steel space frame- work which consists of 126 steel bars and 44 aluminium joints is a subject of our investigations. It is equipped in two piezoelectric stacks which are a part of chosen vertical bars in a plane X-Z. Bars, joints and piezo-stacks have got glue connections. The mathematical model was decoupled to change Two Input Two Output (TITO) system into two Single Input Single Output (SISO) systems. Such approach allowed us to design simple control laws with help of computer simulation procedure. In the last chapter the investigations on the laboratory stand of the active vibration control system with local controllers PD are described. Experimental results have proved that these PD controllers work good increasing the damping level. Additional damp in the system causes the excellent vibration reduction for the whole mechanical structure. Keywords: active vibration control, piezo-stack actuators, decoupled system, active structure AKTYWNY UK£AD STEROWANIA DRGANIAMI PRZESTRZENNEJ PRÊTOWEJ KONSTRUKCJI MECHANICZNEJ Aktywne sterowanie drganiami przestrzennej konstrukcji prêtowej zosta³o przedstawione w poni¿szym artykule. Obiektem badañ jest przestrzenna konstrukcja prêtowa zawieraj¹ca 126 elementów prêto- wych, 44 aluminiowe wêz³y oraz dwa piezostosy. Wspomniane piezoelektryczne aktuatory stosowe zosta³y wklejone do konstrukcji w wybranych pionowych elementach prêtowych w p³aszczynie X-Z. Globalny model matematyczny obiektu jest uk³adem o dwóch wejciach i dwóch wyjciach (TITO), który na potrzeby projektowania praw stero- wania zosta³ rozprzê¿ony na dwa poduk³ady o jednym wejciu i jednym wyjciu (SISO). Projektowanie praw ste- rowania zosta³o przeprowadzone na drodze badañ symulacyjnych w rodowisku Matlab. W koñcowej czêci arty- ku³u opisano badania eksperymentalne wraz z zaprojektowanymi lokalnymi regulatorami PD. Wyniki tych badañ udowadniaj¹, ¿e wspomniane regulatory PD zwiêkszaj¹ poziom t³umienia. Tym samym dodatkowe t³umieniem które pojawi³o siê w uk³adzie powoduje znakomit¹ redukcjê drgañ ca³ej konstrukcji mechanicznej. S³owa kluczowe: aktywne sterowanie drganiami, piezostos, aktywne sterowanie, rozprzê¿ony uk³ad 1. INTRODUCTION The vibrations control of the 3D bar structures has a great practical interest in a many industrial applications, e.g. modern structures of the huge space vehicles, aircrafts, so- lar batteries, robots, rotors, etc. Two demands are required in design of such structures. The excellent dynamic behav- ior is the first one in order to guarantee the stability of the structure and high precision pointing. The necessity to ob- tain light structures, which allow reduce the cost is another one. However, these two requirements are often contradic- tory. The weak internal damping in light structures hinders the accuracy requirements (Premount 2002). These difficulties can be overcame by applying recently developed advanced materials, for instance, piezoelectric materials. Several researchers have proved that piezoelec- tric elements can effectively counteract the vibrations (Pietrzko and Mao 2009, Szolc and Pochanke 2010, Pietrza- kowski 2004). Of course, the appropriate active damping of these vibrations can be achieved by the optimal location and control of the piezo-elements in the structure. The vibration control of 3D mechanical structure with two piezo-stacks as actuators and two eddy-current probes as sensors is consider in the paper. It leads to two input and two output (TITO) control system. Exactly saying, control law was designed to eliminate the vibrations of the structure free end in the plane parallel to the fixed base (2D control). By proper excitations the open loop system was decoupled into two single input and single output (SISO) subsystems. For the decoupled system local control laws were designed in the computer simulation procedure. Next, the experimen- tal stand was designed with actuators and sensors located in the structure. The searching procedure of quasi-optimal lo- cation was presented in earlier paper (Gosiewski and Ko- szewnik 2010) while identification procedure of open-loop system dynamics will be described in (Koszewnik and Gosiewski 2011). The experimental results confirm the ana- lytical and simulation considerations. 2. MATHEMATICAL MODEL OF 3D STRUCTURE Following steps are proposed for the design of the vibration control systems for mechanical structures: 1. Analytical and creative part of the design. 2. Determination of the optimum location of sensors and actuators within the structure (with the use of the finite element method, FEM). 3. Identification of the mathematical model of the vibra- tion process. 4. Control law design. 5. Experimental verification of the design. 6. Return to the first step, if the results are not satisfactory.

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Page 1: active vibration control system for 3d mechanical structures

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The vibrations control of the 3D bar structures has a greatpractical interest in a many industrial applications, e.g.modern structures of the huge space vehicles, aircrafts, so-lar batteries, robots, rotors, etc. Two demands are requiredin design of such structures. The excellent dynamic behav-ior is the first one in order to guarantee the stability of thestructure and high precision pointing. The necessity to ob-tain light structures, which allow reduce the cost is anotherone. However, these two requirements are often contradic-tory. The weak internal damping in light structures hindersthe accuracy requirements (Premount 2002).

These difficulties can be overcame by applying recentlydeveloped advanced materials, for instance, piezoelectricmaterials. Several researchers have proved that piezoelec-tric elements can effectively counteract the vibrations(Pietrzko and Mao 2009, Szolc and Pochanke 2010, Pietrza-kowski 2004). Of course, the appropriate active damping ofthese vibrations can be achieved by the optimal location andcontrol of the piezo-elements in the structure.

The vibration control of 3D mechanical structure withtwo piezo-stacks as actuators and two eddy-current probesas sensors is consider in the paper. It leads to two input andtwo output (TITO) control system. Exactly saying, controllaw was designed to eliminate the vibrations of the structurefree end in the plane parallel to the fixed base (2D control).

By proper excitations the open loop system was decoupledinto two single input and single output (SISO) subsystems.For the decoupled system local control laws were designedin the computer simulation procedure. Next, the experimen-tal stand was designed with actuators and sensors located inthe structure. The searching procedure of quasi-optimal lo-cation was presented in earlier paper (Gosiewski and Ko-szewnik 2010) while identification procedure of open-loopsystem dynamics will be described in (Koszewnik andGosiewski 2011). The experimental results confirm the ana-lytical and simulation considerations.

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Following steps are proposed for the design of the vibrationcontrol systems for mechanical structures:

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The first three steps was carried out and their results arepresented in (Gosiewski and Koszewnik 2010, Koszewnikand Gosiewski 2011).

The laboratory stand with the controlled structure isshown in Figure 1. The identification procedure was carriedout for the case when actuators were used as input and sen-sors as output for different form of excitations. We werelooking for such excitation form which allows us to decou-ple TITO system into two SISO subsystems, one – to con-

trol vibration in X direction and second one – to controlvibration in Y direction in the sensors plane. It was appearedthat the best excitations are obtained when piezo-actua-tors generate force moments around X and Y axis, re-spectively. For both excitations the vibrations were mea-sured by both sensors directed in X, Y axis, respectively.As a results of identification procedure we have obtainedmodel in the form of matrix transfer function given in equa-tion (1).

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XX XY

YX YY

H ( ) H ( )( )

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35,84 2,104 4 33,18 3,575 5 38,03 1,459 6( ) 0,014163

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s s e s s e s s eH s

s s s s e s s e

+ + + + + +=

+ + + + + +(2)

( )( )( )( )( )( )2 2 2

2 2 2

19,79 2,572 4 20,76 3,163 5 68,07 1,033 6( ) 0,011018

2,692 9687 22,4 3,096 5 116,2 7,793 5YX

s s e s s e s s eH s

s s s s e s s e

+ + + + + +=

+ + + + + +(3)

( )( )( )( )( )( )

2 2 2

2 2 2

15,81 1,815 4 99,42 2,68 5 33,79 1,403 6( ) 0,002691

2,418 9629 108,7 2,494 5 77,76 9,789 5XY

s s e s s e s s eH s

s s s s e s s e

+ + + + + +=

+ + + + + +(3)

( )( )( )( )( )( )

2 2 2

2 2 2

6,165 2,001 4 15,49 3,739 5 59,8 1,151 6( ) 0,003595

4,828 1.073 4 27,74 2,19 5 53,92 8,127 5YY

s s e s s e s s eH s

s s e s s e s s e

+ + + + + +=

+ + + + + +(4)

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were the criteria in the design procedure of control laws.The controller parameters were chosen during computersimulation by the comparison of the time plots (responses tostep, impulse and sinusoidal force excitations) and the Bodecharacteristics. So more the plots for closed-loop systemswere compared with the plots for open-loop systems.

First, the local controller PD was designed for the trans-fer function HYY(s). All investigations proved the best PDcontrol law in plane Y-Z has the following form

( ) 0.05.YPD s s= + Therefore, all presented characteristicsare results of the computer simulations for the closed-loopsystem with such control law. The step answers of the open-loop system and closed-loop system are presented in Figu-re 3. Please note different time scales in both plots. Theclosed-loop system is very fast – its control time is 0.04 swhile transient period in open-loop system reaches 1.2 s.

The same results are presented in the Figure 4 forthe impulse excitation. The transient period of closed-loopsystems was reduced several times in comparison withthe open-loop system. Both systems were also harmoni-cally excited (opposite applied voltage to piezo-stacku = 100sin(2*pi*16.5*t)). To underline differences the exci-tation frequency equals frequency of the first resonance(f1Y = 16.5 Hz – see left plot in Fig. 2). As we can notice inFigure 5 the vibration amplitude of closed-loop system wasreduced almost five times in comparison with the vibrationamplitude of the open-loop system.

All time plots showed significant reduction of the Y-Zvibrations of the structure when we have used the controlsubsystem with PD control law. The dynamic behavior ofthe systems was also check in desired frequency band. Theamplitude-frequency characteristics of both open-loop andclosed-loop systems are shown in Figure 6. We can noticethe strong damping in the case of the closed-loop systemparticularly in the range of lower frequencies. Strongerdamping in full desired frequency range (10–200 Hz) re-quires of a higher-order control law.

In interesting range of frequency 10–200 Hz all elemen-tary transfer functions are of 6th order since they have threeresonance peaks. All elementary transfer functions (theiramplitudes) are presented in Figure 2.

As we can see in Figure 2 both models have differentresonance and anti-resonance frequencies. It is correct,since we analyze structure vibrations in two different planesX-Z and Y-Z. Furthermore, we can notice, that excitation inone direction has much smaller influence on the vibrationsin perpendicular direction. The amplitude of the transferfunction HXY(s) is about 10 dB lower than the amplitudeof the transfer function HYY(s) in considered range of fre-quencies. Similar situation we can note also comparingtransfer functions HXX(s) and HYX(s). In this case theamplitude of the transfer function HYX(s) is about 20 dBlower than amplitude of the transfer function HXX(s). So, inequation (1) we can omit out-of-diagonal transfer functions.In such a way we have obtained the decupled system. Itallows us to design appropriate local control law for transferfunction HXX(s) and HYY(s).

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TITO system may be controlled in a global way. It would beresulted in TITO controller. It is not simple to design suchcontroller and so more the number a microprocessor calcu-lations increase in power two. Because the vibration pro-cess is very fast it is desired to make the control laws assimple as possible.

We have designed the local control laws for subsystemsrepresented by diagonal elements in the matrix transferfunction (Eq. (1)). With help of computer simulations theclosed-loop system with two classical PD controllers wasinvestigated. One controller was used to damp the vibrationin the plane X-Z while second – in plane Y-Z. High level ofdamping and very short transient period of the vibrations

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In the second stage similar PD control law was designedfor the open-loop model described by the transfer functionHXX(s). Also in this case all investigations proved the bestPD control law in plane X-Z has the following form

( ) 0.1 3.2.XPD s s= + Therefore, presented below character-istics result from the computer simulation of the closed-loopsystem with this control law. Step answers of both systemsare shown in Figure 7. The closed-loop system is significant-ly faster in comparison the open-loop system. Transient pe-riod in open-loop system reaches up 0.4 s, while this sameparameter for closed-loop system is 0.08 s. Similar resultsare presented in Figure 8 for the impulse excitation. In thiscase the transient period of closed-loop systems was reducedfour times in comparison with the open-loop system. Addi-tional both systems were also sinusoidal excited results areshown in Figure 9. This time the signal excitation was the

sinusoidal force with frequency of the first resonance in di-rection X (f1X = 15.3 Hz – see right plot in Figure 2, appliedvoltage to the piezo-stack is u = 100sin(2*pi*15.3*t)). Aswe can see in Figure 9 in this case the amplitude of closed-loop system was reduced six times in comparison with thevibration amplitude of the open-loop system.

Results from Figure 7 to Figure 9 show correct acting theclosed-loop system in plane X-Z. However we do not know,how this control law will damp the vibrations the structurein consider frequency range 10–200 Hz. Bode plot in Fi-gure 10 is an answer to the question.

Amplitude characteristics in consider frequency rangeconfirm the appearing of the additional damp in the sub-system. Evidently it is significant in vicinity of the twolowest resonances where the amplitude of closed-loop sub-system decreased about 10 dB.

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In this chapter the experimental results obtained for open-loop system and closed-loop system with local control lawsare discussed. The laboratory stand consists of mechanicalbar structure, two eddy-current sensors, two piezo-stacks,two voltage amplifiers and digital signal processor DSPwhich play the role of the controller. As it was mentionedearlier the local control laws have the following form:

( ) ( )( ) 3.2 0.1X

de tu t e t

dt⎛ ⎞= +⎜ ⎟⎝ ⎠

and

( ) ( )( ) 0.05 .Y

de tu t e t

dt⎛ ⎞= +⎜ ⎟⎝ ⎠

Such control law was implemented at the processor DSP.The block diagram of the whole control system is shown inFigure 11.

First, the structure vibrations were excited by the piezo-stacks with the frequency of the Y-Z plane first resonance in(applied voltage: u = 1.2sin(2*pi*16.5*t) [V]), next, thestructure was exited with the frequency of the X-Z planefirst resonance (applied voltage: u = 1.2sin(2*pi*15.3*t)[V]). The voltage signals from both eddy-current sensorswere recorded and recalculated to obtain the displacementsat the free end plane of the structure. During these experi-ments the piezo-stacks working as a vibration generatorswere switched off after several seconds (after about 15 s)to obtain transient response of open loop-systems (upperplots in Fig. 12 and Fig. 13). In some experiments the piezo-stacks were switch off as a generator and immediately were

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switch on as a actuators of closed-loop systems to show thetransient period of the closed-loop system (lower plots inFig. 12 and Fig. 13). The results of such experiments arepresented in both directions for the excitation in directions X(Fig. 12) and for excitation Y (Fig. 13). Please notice againthe different scales of the amplitudes in presented plots.

Experimental results have proved that local PD con-trollers were designed correctly. In both cases the transientvibrations of the closed-loop system significantly fasterdisappear in comparison with the open-loop system. It wasappeared such decoupled closed-loop system with simplecontrollers is sufficient to have the excellent active dampingof the whole structure vibration.

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In the paper we have described the process design of 2Dcontrol laws for the vibration control of 3D mechanical sys-tem. Results of dynamic process identification have showedthat the whole system is really decoupled for chosen formsof the excitation. Such excitation approach was used in con-trol loops. Ipso facto process of design control laws wasreduced to the design of two local controllers each of themfor one diagonal transfer function in TITO model of the fullsystem. In such solutions the local control laws have simpleform, typical for PD controllers. Control laws parameterswere obtained in computer simulations procedure. Nextdynamic behavior of open-loop system and closed-loopsystem was analyzed with help of standard excitations likestep function, impulse function and sinusoidal function. Alldynamics characteristics show the advantage of the closed-loop system in comparison to the open-loop system. Forexample transient period time was reduced several times.

Next, the experimental stand was designed and obtainedcontrol laws were implemented to processor DSP. Experi-

mental results completely confirmed the computer simula-tions and they show proper operation of the closed-loopsystem with two local PD controllers. The active systemsignificantly increases the damp level in structure vibra-tions. The transient period is much shorter than in case ofthe open-loop system.

All carried out investigations and obtained results haveproved that designed 2D active vibration control systemperfectly control the vibrations of 3D flexible bar structure,especially for the first mode of vibration.

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