active modal control of flexible structures using ...

ACTIVE MODAL CONTROL OF FLEXIBLE STRUCTURES USING RECTANGULAR PIEZOELECTRIC TRANSDUCERS A. Yousefi-Koma’ and G. Vukovich” ‘Graduate Research Assistant, Carleton University; ASME and SEM member Mechanical and Aerospace Engineering Department, Carleton University Ottawa, Ontario, KlS 586, Canada ** Adj. Professor, Carleton University; Research Scientist, Canadian Space Agency Directory of Space Mechanics, Canadian Space Agency 6767 Route De CAeropolt, St. Hubert, Quebec, J3Y 6Y9, Canada

ABSTRACT. An active modal control method is proposed for flexible structures using rectangular piezoekments. The piezo-sensor equations show that output voltages and currents of sensors are linear combinations of modal amplitudes and their time derivatives respective/y. Consequently modal states can be measured for a full state observation. The piezo-actuator equations indicate that the generalkd (modal) forces generated by the actuators are linear combinations of the input voltages of actuators. A methodology for choosing suitable input control voltages of actuators is presented to control any specific mode separately. Results show that rectangular piezoelements can be used as efficient transducers for controlling vibrational modes of flexible structures especial/y in case where several dynamic modes have to be sensed and controlled. Key Words: Modal Control Nexible Structures, Acfive Control, Pkzoelectric Transducers, Smart Structures.

NOMENCLATURE A B CP

Cl

CV

C”’ c” c’ CQ d3t EP I

K n P 9

: : : : : : : : : : : : : : : :

Dynamic state space matrix Control voltage coefficient matrix Piezoelement capacitance Current coefficient matrix Voltage coefficient matrix Voltage-current coefficient matrix Sensor voltage coefficient Sensor current coefficient Generalized (Modal) force coefficient Pie20 strain constant Elastic modulus of the piezoelement Output current of the piezo-sensor Passive stiffness Number of modes Number of actuators Modal amplitude

Q S

tb tP i WP Xl x2 x Y”’ 40) 0 11

: : : : : : : : : : : : : :

Generalized (Modal) force Number of senscrs Beam thickness Piezoelement thickness Piezo-actuator voltage Output voltage of the piezc-sensor Piezoelement width Start point position of the piezo-transducer End point position of the piezc-transducer Modal state vector Output voltage-current vector mode shape Modal Frequency Structural Damping

1. INTRODUCTION The use of piezoelectric materials as distributed transducers in active control of flexible structures has been studied ever the last ten years (Bailey and Hubbard [l], Plumb et al. [7]), and modal techniques have been used for dynamic modelling of flexible structures bonded with piezoelectric transducers (Chen et al. [2], de Luis and Crawley [3], Ycusefi-Koma et al. IS]). The distributed form of senscr and actuator equations can be transformed to modal form which can in turn be easily cast into state space form for control purposes. Lee et al. [4] have introduced shaped piezoelectric sensors/actuators to sense/ actuate individual modes of a flexible beam. In their study, the one dimensional modal sensor/actuator equations are first derived theoretically and then examined experimentally. Shaped modal piezcfilms are also used by Zhou et al. [lo] to measure the natural frequencies of a beam and a plate. The use of the shaped modal piezoelements introduces two major disadvantages. First, these piezo-transducers should ccver the whole length of the structure in order to properly sense or actuate a specific mode. Secondly, construction and shaping of such piezc-transducers involve scme difficult practical

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problems, some of which are addressed by Lee and Moon [51.

piezo strain constant. x1 and x2 define the boundaries of the piezo-sensor along the beam length.

In this paper, an alternative approach to the shaping of piezoelements is introduced for active control of specific modes of a flexible beam using standard rectangular piezoelectric transducers. Employing this method, there is no need for shaping piezoelements or covering the whole structure by them, and is therefore much simpler and cheaper. It is shown that how each mode can be controlled individually under the in!roduced modal controller using rectangular piezoceramics.

The short circuit current of a piezoelectric sensor, I(t), is defined by [9]:

2. DYNAMICS

The output signals of the piezoelectric sensors for an n degree of freedom model can then be written with respect to the state vector as follows:

r(r) - i cpji(:);

i - I, _,,, n

(3)

i-1

cli is the sensor current coefficient expressed by: (4)

A 500 mm long cantilever stainless beam 40 mm in width and 0.6 mm in thickness is considered as the flexible structure in our study. Rectangular piezoceramic transducers (PSI-5A-S2 [6]) with 63.5 mm in length, 36.1 mm in width, and 0.19 mm in thickness are bonded onto the flexible beam (Fig. 1).

Y

“I

= C”‘x

(5)

where: “1 “I Y -

v, 11 - I 6-25X,

41

;

x- 9” 41 PRznx

03

I

qi (t) and;,(f) represent the ith modal amplitude and its rate

respectively. Vi and Ii are output voltage and current of the jth

In the following sections, dynamic equations of the above smart structure (i.e. flexible beam with bonded piezoelectric sensors and actuators) are presented in the modal domain via modal amplitudes and their time derivatives (YousefiKoma, et al. 191). 2.1 Plezoelectric

piezo-sensor respectively. C”‘, which expresses the relation between states and the output voltages and currents of the piezc-sensors, can be obtained from Eqs. (l)-(4) as:

Sensor

Assume that 2s piezc-sensors are bonded onto the flexible beam (in our study four sensors are used). We use half of them, i.e. s, for measuring voltages (proportional to strains) and the second half for the current measurement (proportional to strain rates). The output voltages of the piezo-sensors, V(t), are linear combinations of the modal states and can be expressed as follows:

cj” and cj’ are voltage and current coefficients which depend on the piezo-sensor parameters given by Eqs. (2) and(4) respectively. If the number of sensors, 25, and the number of states, 2% are equal, Eq. (5) indicates that given the output voltages and currents of piezo-sensors, the modal states (i.e.

where q” is the sensor voltage coefficient given by 191: (2)

modal amplitude, qi (0, and modal amplitude rate, 4, (r)) can be obtained:

where @i(x) is the ith mode shape and b is the thickness of the beam; Wp. Ep. and C, are the width, elastic modulus, and capacitance of the piezoelements respectively; and d31 is the

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3. ACTIVE MODAL CONTROL SYSTEM

which can be used for a full state observation and control. This means one can get each modal amplitude and its time derivative individually. Thus the contribution of each mode to the vibration of the structure can be calculated. In other words each mode can be sensed separately. Although the requirement that there be as many sensors as modes to be controlled may seem unduly restrictive, in actuality in many structures we are interested in controlling only a few (generally the lowest) frequencies. 2.2 Piezoelectric

In this section the system and controller are expressed in state space form. 3.1 State Space Equetlone The dynamic equations of the smart structure can be written in state space form as follows: f(f) - Ax (I) + Bu (I) Y”‘W - C”‘x(t)

Actuator

The total generalized force, Qi~pPi’ro generated by p piezo-

(15)

From Eq. (12), matrix A can be obtained and is given by:

actuators can be obtained by the following equation: Q,wrro _ QpirroI + ,,, + Q;i%;

i- 1,...,n

A- [;:;;

(9)

;;jznx2:he=

Lb [“%” j::,~nm].”

where Qp”T”’ is the ith modal force generated by the j* piezo-actuator and is given by: Qi~iczo,

_ ciQ-~‘noj

and

uj

%

*

(10) B, which contains the related coefficients of the control voltages can be derived from Eqs. (9) and (10):

where ui is the voltage applied to the jth piezo-actuator and cia is the ith modal force factor which depends on the piezcactuator parameters and is given by [9]:

* (1, - [Iu, ; (1)

u(t) is the control voltage vector given by:

The final dynamic equation of the structure, i.e. the flexible beam with bonded piezoelements, can be written in the following modal form [S]:

(18)

uP(t)pxl

3.2 Active Model Controller

In order to study the behavior of the flexible structure under control, a model with only the two first dynamic modes is considered. Two piezo-actuators and four piezo-sensors are used in this study. Eqs. (9) and (10) indicate that the general forces generated by piezc-actuators are linear combination of the input voltages of the actuators, so that:

where T$ is the ith modal structural damping ratio, oi is the natural frequency of the ith mode. Kijmiezos is the total passive stiffness of all piezc-transducers (i.e. sensors and actuators) given by:

(19)

For each transducer Kij is given by the following expression:

If we take the two control voltages to be in the ratio: Q-Pi.ZOl

Kij

(14)

4 5 -s4

where tP is the thickness of the piezo-actuator.

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5

Q -piczo2

(20)

Then Eq. (19) tells us that Q#=O. Therefore applying voltages with this ratio to the actuators affects only the first mode. This means that the first mode can be controlled individually. For the second mode control the first modal force should be zero. i.e. Q,(t)=O, which means:

(21)

[I

1 -muI; A- ; BIG(r) -B 5 -B au, u2 [I [I

(22)

(I is our design parameter for active modal control:

Q-piarol

(Q,(r) - 0 ;

First

mode control)

5 Q-pino,

Cl o-Pisio2

(23) (Q l(t)= 0; Second mode control)

Cl

Fig. 2 shows the active modal control system for the smart structure. LQR method is used to find the controller gains, i.e. K. The system is studied under an initial excitation of: U(r) - 2 s i n (w,t)

c2Wsin

(q)

(24)

This voltage, which includes signals of both the first and the second natural frequencies (01=13 radls and 02=62 rad/s) of the smart structure. is applied to the actuators for only 2 seconds (Fig. 3). Fig. 4 represents the open loop tip displacement of the flexible beam under this excitation and shows the effect of both frequencies. Active modal controller is applied for first and the second

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modes separately. The closed loop response of the system under first mode control (Q,(t) = 0) is shown in Fig. 5. As we see the controller does not damp the vibration caused by the high frequency excitation (i.e. second mode) because the actuators are producing the first generalized force exclusively. The controller is pushing just the first modal amplitude of vibration to zero. Fig. 6 illustrates the effect of this controller on the first modal amplitude. As we see the controller is quickly suppressing the vibration. On the other hand from Fig. 7 one can see that the controller does not affect second modal amplitude, which was expected. The same thing was done for second mode control (Cl,(t) = 0) to show the performance of the controller for damping the second mode exclusively (note that figures 6 and 7 show the responses just after the controller is on, i.e the 2-5 second time interval). Fig. 6 shows the closed loop tip displacement of the flexible beam under this controller. Comparing figures 4 and 6 one can see how the effect of the second mode on the tip displacement has been damped whereas that of the first mode has remained untouched. In Fig. 9 it is shown that the controller has almost no effect on the first modal amplitude. On the other hand the damping of the second modal amplitude of the vibration under this controller, shown in Fig. 10, is quite fast. Finally a comparison between the two controllers, i.e. first mode controller and the second mode controller, is made in figures 11 and 12 (note that figures 9-12 show the responses just after the controller is on, i.e the 2-5 second time interval). As wa see the first mode controller suppress only the effect of excitation of the first mode, and the second mode controller suppress only that of the second mode.

4. CONCLUSION

[lo] Zhou, N.. Sumali, H., Cudney, H., “Experimental Development of Piezofilm Modal sensors and Characterization of Plezofilm Strain Rate Gages”, Proceeding of the 3ih AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, pp.735-743, 1991.

A method for active modal control of flexible structures using rectangular piezo-transducers is presented. Application of a controller on a flexible beam with two piezc-actuators shows that the method can effectively control each mode separately. The use of rectangular piezo-transducers in this study eliminates the need of covering the whole flexible structure by modal shaped piezoelements as well as the technical problems due to shaping them. It is shown that the introduced controller can easily be applied and the results for first and second mode control show excellent performance.

5. REFERENCES [l] Bailey, T., and Hubbard, J. E., “Distributed PiezoelectricPolymer Active Vibration Control of a Cantilever Beam”, Journal of Guidance, Control, and Dynamics, Vol.6. pp. 605 611, 1985. [Z] Chen. S.Y.. Ju MS.. and Tsuei Y.G., “Linear Structure Control by Modal Force Technique”, Proceeding of the 33’h AIA~ASME/ASCE/AHS/ASC structures, structural Dynamics and Materials Conference, pp. 1580-l 588, [3] de Luis, J., and Crawley, E. F., “Experimental Results of Active Control on a Prototype Intelligent Structure”, Proceeding of the 31th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, pp. 234~2350,1989. [4] Lee, C. K.. Chiang, W. W., O’Sullivan, T.C., “Piezoeledric Modal Sensors and Actuators Achieving Critical Active Damping on a Cantilever plate”, Proceeding of the 3dh AIAAlASME/ASCWAHS/ASC Structures, Structural Dynamics and Materials Conference, pp. 20182026,1989. [5] Lee, C.K., and Moon, F.C., “Modal Sensors/Actuators”, ASME Journal of Applied Mechanics, Vol. 57, pp. 434-441, 1990. [6] Piezosystem. I n c . , “ P r o d u c t Catalog&‘, 1 8 6 Massachusetts Ave., Cambridge, MA 02139,1992. [7j Plumb J.M., Hubbard J.E.. and Bailey T., “Nonlinear Control of a Distributed System: Simulation and Experimental Results”, Journal of Dynamic Systems, Measurement, and Contd, Vol. 109, pp. 133-139, 1990. [8] Yousefi-Koma, A., Sasiadek, J. Z., Vukovich, G., “Control of the Flexible Arm Space Based Robot Using Piezoeledric Transducers”, EAC Symposium on Robot Control (SVROCO 94j, Italy, pp. 535-540, 1994. [9] Yousefi-Koma, A., Sasiadek, J. Z.. Vukovich, G., “LQG Control of Flexible Structures Using Piezoelements”, A/AA Guidance, Navigation, and Control Conference, AZ, pp. 893900,1994.

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