Experimental implementation of the modified independent modal space control method

Journal of Sound and Vibration (1990) 139(l), 133-149

EXPERIMENTAL

IMPLEMENTATION

INDEPENDENT

MODAL

SPACE

A. BAZ AND

OF THE CONTROL

MODIFIED METHOD

S. POH

Mechanical Engineering Department of The Catholic Universitv of America, Washington. D.C. 20064. U.S.A. ( Received 12 January 1989, and in revised,form 18 August 1989)

In this study an experimental realization of a Modified Independent Modal Space Control (MIMSC) method to control the vibration of a flexible cantilevered beam is presented. The method relies in its operation on the use of one piezo-electric actuator to control several vibration modes through a uniquely developed “time sharing” strategy. The effectiveness of the MIMSC method, in damping out the beam vibration, is demonstrated clearly by comparing the results with those obtained by other modal control methods. Two methods are considered; namely, the Independent Modal Space Control (IMSC) method and the Pseudo-Inverse (PI) method. The experimental results obtained emphasize the feasibility of the MIMSC method as a viable alternative for controlling large flexible structures with very small number of actuators.

1. INTRODUCTION Active control of large space structures is recognized as essential to the successful operation of these structures as stable platforms for communications and observations. Recently, several active control systems have been successfully implemented to suppress the vibration of simple structural elements. Among these systems are those relying in their operation on the use of piezo-electric actuators. For example, Forward [l], in 1981, used ceramic piezo-actuators to damp out two closely spaced orthogonal bending modes in a cylindrical fiberglass mast. In 1985 and 1987, Hubbard and co-investigators [2,3] utilized polymeric piezo-actuators ( PVF2) to control the vibration of aluminum beams based on Lyapunov’s second method. Crawley and de Luis [4,5] demonstrated also the effectiveness of using PZT ceramic actuators embedded in glass/epoxy and graphite/epoxy beams in attenuating the vibration of these composite beams. In 1987, Fanson and Caughey utilized pairs of PVF? actuators/sensors to control the vibration of aluminum beams using a positive position feedback algorithm [6]. In all these studies, the piezo-electric actuators have demonstrated their light weight, high force and low power consumption capabilities. All these favorable attributes have rendered this class of actuators to be an attractive candidate for controlling structural vibrations. In this study an alternate approach is considered which is based on the control of vibration of multi-mode flexible system with a small number of actuators. The Modified Independent Modal Space Control (MIMSC) method will be used [7,8]. The method was developed by Baz et al, in 1987 to modify the well known Independent Modal Space Control (IMSC) method of Meirovitch and co-investigators [9-l I]. The MIMSC modifies the IMSC to account for the control spillover between the controlled and the uncontrolled modes. It also incorporates an optimal placement procedure to select the optimal locations of the actuators in the structure. Furthermore, the MIMSC includes a time sharing strategy 133 0022-460X/90/100133+

17 %03.00/O

@ 1990 Academic

Press Limited

134

A.

BAZ

AND

S. POH

to share small number of actuators between larger number of modes. Numerically, the MIMSC has been shown to have superior vibration damping characteristics as compared to the IMSC [7] and the Pseudo-Inverse (PI) method [12]. Experimentally, the comparison between the MIMSC, IMSC and PI is yet to be established. It is therefore the purpose of this study to investigate experimentally the merits of the MIMSC in comparison with the other two modal control methods. 2. MATHEMATICAL

MODELING

OF THE BEAM-ACTUATOR

SYSTEMS

A complete description of the mathematical model of the beam-actuator been given in a previous paper by Baz and Poh [8].

system has

MOMENT 2.1. PIEZO-ELECTRIC Briefly, in Figure 1 is shown a general layout of a finite element model of a flexible beam A, the deflection of which is to be controlled by the piezo-electric actuator C. When an electric field is applied across the actuator, it will expand if the field is aligned with the polarization axis and it will contract if the two are opposed. The expansion and contraction of the actuator relative to the beam develops longitudinal stresses in the beam that bend it in a manner similar to a bimetallic thermostat. The generated bending moment M, is given by [S]

Mf=dbE,~(E,t,t*+E.lf,t~+E*t:+2E3f2f3+E3f:)/2(E*t~+E~tz+E,t,),

(1)

where d is the charge constant of the actuator (m/v), b is the beam width (m), E1,2.3are Young’s moduli of elasticity of the actuator, bonding layer and beam respectively (N/m*), ~i,~,~are the corresponding thicknesses (m), and v is the applied voltage (a list of nomenclature is given in the Appendix). Bonding layer B 7

r

Pieza-electric

actuator

C

Externol moments

External forces Node number N Element number

Figure

2.2.

STIFFNESS

MATRIX

OF

1. General

THE

layout of the beam-actuator

BEAM-ACTUATOR

system.

SYSTEM

actuator moment M, is combined with the external forces and moments vi, Mei, K+, and Mei+, acting on the beam element, shown in Figure 2, to form the following load-deflection characteristics of the element: The

(2)

EXPERIMENTAL

135

MIMSC

Piezo-actuator

Figure

2. Schematic

drawing

of a piezo-actuator

bonded

1

C

to a flexible beam.

Here &Ii is the flexural rigidity of the composite beam (Eilt = ES=1 EjZj ), I r,1,2.3 are. the . area moments of inertia of the composite beam, the actuator, the bonding layer and the beam respectively (m4) and Li is the length of element i. Equation (2) can be rewritten as F, = K,&,

(3)

where hi is the force vector acting on element i (N, Nm), Ki is the stiffness matrix of the composite beam (N/m, Nm/rad) and Si is the nodal deflection vector (m, rad). The overall sti&less matrix K of the beam is obtained by combining the stiffness matrices Ki of the individual beam elements, according to the principle of superposition

[131. 2.3.

MASS

MATRIX

OF THE

BEAM-ACTUATOR

SYSTEM

inertial properties of the composite beam are determined by assuming the mass mi and the inertia Ji of the beam elements to be lumped at the n nodal points [ 131. This gives the following diagonal mass matrix: The

ml

Jl . . .

M=

mi Ji

!4)

... m, Jn

The stiffness and mass matrices K and M are used to define the dynamic equations of the undamped beam-actuator system, Mc++KS=

F,

(5)

where 8’is the nodal acceleration (m/s2, rad/s’). This highly coupled system of equations is transformed to the modal space to reduce it to a form suitable for the application of the Modified Independent Modal Space Control method.

136

A.

3. THE

MODIFIED

BAZ

INDEPENDENT

AND

S.

POH

MODAL

SPACE

CONTROL

METHOD

The dynamic equations of the beam-actuator using the modal transformation

system are put in the modal space by

S=$JiJ,

(6)

where U is the vector of modal co-ordinates and &Jis the modal shape matrix of the flexible system. This transformation reduces the coupled equations of motion (5) to the uncoupled form fi+hU=c#?F=f,

(7)

where A is the system eigenvalues matrix and f is the modal force vector. If C modes are controlled by C actuators, then the modal force vector f can be written as

(8) where fc is the modal force vector (C x 1) generated by the C physical control forces, F, and fU is the modal force vector ((2n - C) x 1) resulting from the spillover of the control forces into the uncontrolled modes. This spillover component can be calculated, in terms of the modal control vector fc, from (9) This component does not equal zero, as assumed in the original IMSC method. Therefore the MIMSC method computes the optimal modal control forcesf, [9-111, which are then used to calculate the modal spillover forces fU. These two modal forces fc and fU are applied to the system, represented by equation (7), to yield the modal displacements u and velocities ri. In turn, u and ti form the proportional and derivative components of the modal control forces, J =

-(gli"iui+g2ilii)lRiv

i=l,...,C,

(10)

where wi is the ith normal mode and Ri is a factor that weighs the importance of minimizing the vibration with respect to the control forces. In equation (lo), g,, and g2, are the modal position and velocity feedback gains obtained by minimizing the performance index J, given by OW[(o;u;+tij)+(Rf’)ldt.

J= I

(11)

Meirovitch and co-workers [9-l l] showed that J attains a minimum when the feedback gains gri and g,, are given by g,i=-w;Ri+J(wiRi)‘+“fR,

and

g~~=J2R~~~(-~~R~+J(~iRi)2+~~Ri)+~fRi.

(12,13)

Equations (9), (lo), (12) and (13) form the basic equations that govern the MIMSC method. The method incorporates also a unique time sharing strategy so that a small number of actuators can be used to control a larger number of modes. In this strategy the system modes are ranked according to their modal energies (wfu:+ rif). If one actuator is to be used, then this actuator will be dedicated, at any instant of time, to control the mode that has the highest modal energy at that instant.

EXPERIMENTAL

137

MIMSC

elastic and mertlol propertles of structure

elastic ond mertml properties of actuator

I

Compute normal modes and modal shope of total structure with actuators

Input of actuators

location

I

ond sensors

I

Compute reduced modal shope matrices corresponding to octuotors and sensors locations

I

Compute the optlmol gems for all modes

I

I

Excite the structure

Monitor

??

I

the physical dlsplocements of the structure

Compute the modal dispiocements

and velocities I

I

Compute the modal energies and ronk them In descending order

I

Compute the modal control forces

I

Compute

thephysical

I

control forces and voltages

I

Send the control voltages to the octuotors

1 Figure

The flowchart

4.

4.1.

EXPERIMENTAL

of the MIMSC

EXPERIMENTAL

3. Flowchart

algorithm

of the MIMSC

algorithm.

is shown

in figure 3.

IMPLEMENTATION

OF

THE

MIMSC

SET-UP

For the purposes of validating the MIMSC method a test structure consisting of a thin rectangular cantilevered beam was constructed from polymethyl methacrylate. The main geometrical and physical properties of the beam are given in Table 1.

138

A. BAZ AND TABLE

S. POH

1

Main geometrical and physical properties of test beam Length

Width

(cm)

(cm)

(cm)

(G$m2)

25.78

3.75

0.075

2.96

Thickness

Density (gm/cm’) 1.31

The beam is controlled by one piezo-electric actuator which is made from G1195ceramic shaped in a bimorph form. The actuator is available commercially (model number R205) from Piezo-Electric Products, Inc., Metuchen, New Jersey, 08840-4015, U.S.A. The main geometrical and physical properties of the actuator are given in Table 2. The experimental beam and the piezo-actuator are arranged as shown in Figure 4. The beam is divided into three finite elements. Bonded to the first element, which is near to the beam fixed end, is the piezo-actuator. Three position sensors are used to monitor the physical displacements of the three nodes in the transverse direction. The sensors are of the non-contacting inductive proximity. These sensors are commercially available (model 576013-190) from Veeder-Root Co., Hartford, Connecticut 06102, U.S.A. The signals of the displacement sensors are sampled by a 386-based microprocessor provided with a commercially available sampling and control board (model DASH-16) from METRABYTE Corp., Taunton, Massachusetts 02780, U.S.A. The board is capable of sampling 16 analog signals at a conversion time of 15 l.~swith a resolution of 12 bits. The board can send also two analog ouputs with a settling time of 30 (IS. The microprocessor uses the three sampled signals to compute the beam angular deflections as well as the linear TABLE

2

Main geometrical and physical properties of the actuator

Length (cm)

Width (cm)

4.85

Thickness (cm)

Charge coefficient (m/V)

Max. voltage

0.1

190 x lo-l2

25

1.375

(V/mil)

E, (GN/m’) 63

Fran computer Piczo-octuotor

Figure 4. Schematic

drawing of the experimental

beam, piezo-actuator

and sensors.

Density (gm/cm3) 7.8

EXPERIMENTAL

139

MIMSC

and angular velocities of the nodes by the backward differentiation algorithm [14]. The 12 computed state variables (y,, 6,). . . , y,, tY3, i,, 8,). . . , &, 6,) are used to calculate the modal co-ordinates of the flexible system, the mode that has the highest modal energy, the corresponding optima1 modal control force A., the physical control force FC and the necessary voltage v to be sent to the piezo-actuator. The implementation of these calculations, i.e. the MIMSC algorithm, is carried out in real time in 3*04ms. 4.2.

MODAL

CHARACTERISTICS

OF THE

BEAM-ACTUATOR

SYSTEM

The modal characteristics of the experimental beam are determined from the theoretical mode1 described in section 2. These theoretical predictions have been validated experimentally by using a random excitation technique [ 151. Briefly, a white noise source (model 1405 from Briiel & Kjaer, Denmark) is used to drive a piezo-electric vibration shaker (mode1 F9/F3 from Wilkinson Research, Bethesda, Maryland 20814, U.S.A.) which is coupled to the beam base of support. The base is designed to oscillate freely along two guide rails which are set parallel to the excitation direction of the shaker. The frequency content of the resulting vibration of the beam tip is shown in Figure 5. A comparison between the theoretical and the experimental values of the first five vibration modes of the beam-actuator system is shown in Table 3. The table gives also the modal damping as calculated from the experimental results using the half power approach [16].

0*5r

Frequency

Figure

5. Frequency

response

(Hz)

of beam to white noise showing

TABLE

first five modes of vibration.

3

Modal characteristics of the beam system

Theoretical mode (Hz) Experimental mode (Hz) Error (%) Modal damping (%)

1 1.62 1.55 4.3 0.038

2 11.0 10.1 8.1 0.022

Mode number 3 64.7 49.5 13.5 0.016

4 87.4 80.5 7.9 0.015

5 157.7 167.5 -6.2 0.010

140

A. BAZ AND

S. POH

4.3. EXPERIMENTAL RESULTS In all the experiments conducted in this study, the beam was excited in its second mode of vibration by applying sinusoidal excitations of 20 volts in magnitude to the piezo-actuator. The excitations were maintained for a period of 0.3 seconds. The beam was either left to vibrate freely (i.e. uncontrolled) or under the action of one modal control algorithm or another. The uncontrolled performance is used as a datum for judging the effectiveness of the different control algorithms. The above excitation form was selected in order to excite modes of vibrations other than the first because if only the first mode is excited then the IMSC and the MIMSC will be exactly the same. Furthermore, excitations of modes higher than the third (i.e., with periods of oscillations co.02 s) would require a faster microprocessor in order to sample at least 10 samples per period to achieve meaningful control. Therefore, the above excitation strategy was devised to demonstrate the differences between the IMSC and MIMSC without exceeding the limits of the present equipment or violating the proper sampling conditions. In Figure 6 is shown a comparison between the resulting time response of the uncontrolled beam and when the beam is controlled by the IMSC or the MIMSC. It is evident that the IMSC, by dedicating the single piezo-actuator to the control of the lowest mode of vibration, is not an effective way of damping out the beam vibration. On the other hand, if the same actuator is utilized efficiently, as in the case of the MIMSC algorithm, it can suppress the vibration much faster than the IMSC. In the MIMSC, the

(a)

i

-o.50LI_L, I

2

3

4

5

Tme is)

Figure 6. Time response

of the beam. (a) Uncontrolled;

(b) with IMSC; (c) with MIMSC.

EXPERIMENTAL

141

MIMSC

actuator is dedicated to the control of the mode that has the highest instantaneous modal energy. This is not necessarily the lowest mode of vibration, as it depends on the nature of the external disturbance as can be seen from Figure 7. The figure emphasizes clearly the fact that the mode that has the highest energy varies with time. As the controller suppresses one mode it excites others, at the same time, by virtue of the spillover effect. Accordingly, the effectiveness of the MIMSC stems from its adaptability to this continuously varying nature of the vibrating system. This effectiveness is demonstrated clearly in Figure 8 in which the total modal energy of the beam when using the MIMSC is shown

l_..LI 0

I

2

3

4

5

Time (s) Figure 7. The time history of the maximum

modal energy mode with MIMSC

controller

(a)

(b)

o-o0

/ I

I.

I

IL.

2

3

4

~.~I~

5

Tfme (s) Figure 8. The total modal energy of the beam. (a) With IMSC;

(b) with MIMSC.

142

A. BAZ AND

S. POH

as compared to that when using the IMSC. The figure indicates that the modal energy increases from zero to a maximum, after O-3 seconds, due to the input excitation energy and then starts to decay under the action of the control algorithm. It is evident that the MIMSC algorithm reduces the modal energy faster than the IMSC method. A better insight into the effectiveness of the MIMSC can be gained by considering the Fast Fourier Transform (FFT) of the beam response. In Figure 9 is shown the frequency content of the response of the uncontrolled beam in comparison to those of the beam when controlled by the IMSC and the MIMSC. These frequency-response characteristics were obtained by sampling the beam tip position signal by a spectrum analyzer and performing on it an FFT analysis. It is indicated in Figure 9(a) that the IMSC is very effective in damping out the vibration of the first mode of vibration. However, it did not produce any significant damping of the second mode. However, the MIMSC produces a significant effect on both the first and second modes of vibrations, as seen from Figure 9(b). This emphasizes the fact that the MIMSC “time shares” effectively the single actuator between the modes. A shift in the frequencies of the controlled closed-loop system relative to those of the uncontrolled open-loop system is shown in Figures 9(a) and 9(b). Such a shift is attributed to the dynamic effect resulting from the addition of the proportional and derivative modal controller. This effect has been reported by many investigators such as, for example, Fanson and Caughey [6].

Frequency (Hz) Figure 9. Frequency response and with MIMSC controller.

of the beam. (a) When uncontrolled

and with IMSC controller;

(b) with IMSC

EXPERIMENTAL

143

MIMSC

It is important, however, to note that the MIMSC uses a larger control effort than the IMSC method as displayed in Figure 10. This is attributed to the fact that the MIMSC starts by controlling the second mode as indicated in Figure 7 for 0 < t < 0.3 s. Accordingly, the control effort is expected to be high because the feedback gains g,; and g2, are proportional to the vibration mode W, as implied by equations (12) and (13). A more comprehensive comparison between the MIMSC and the IMSC can be obtained by considering the displacement index U, and the control effort index U,., defined as ,*

U, =

,*

C S:i,

and

,=”

lJ, = c F2,

(14,151

t=ll

where t* is the time of each experimental run. In Figure 11(a) are shown the displacement and control effort indices of the IMSC method for different values of the weighting parameter R, . As R, increases the control effort decreases and the controller becomes inadequate to damp the beam vibration. This results in higher displacement index. But, as R, decreases the control effort increases and better damping is achieved. Further reduction in R, results in excessively large control forces that tend to excite the system more than damping it. Therefore, the displacement index increases again. Accordingly, there is an optimal R, at which the displacement index attains a minimum. This occurs when R, = 6000 with a minimum value of U, = 0.74. In other words, a reduction of about 15% is obtained as compared to the uncontrolled case in which U, =O-87.

I

I

I

I

I

I

2

3

4

5

Time (s) Figure 10. Time history of the actuator control voltage. (a) With IMSC

controller;

(b) with MlMSC

controller.

144

A.

BAZ

AND

S. POH

o.731._120 Weight

O-26

factor,R,

I 4200

I 2200

200

Weight

factor,

I 6200

I

loo

R,

Figure 11. Effect of the weighting parameter on the displacement and control effort indices. (a) With IMSC controller; (b) with MIMSC controller.

-2

E 8

I -0.3 0.3

I

I

I

I

I

3

4

5

E E a

0.2

i= 0.1

0.0

-0.1

-0.2

Tme

Figure 12. Comparison between theoretical (-_) When uncontrolled; (b) with MIMSC controller.

(5)

and experimental (. . . .) time responses of the beam. (a)

EXPERIMENTAL

145

MIMSC

When the actuator is shared between two modes, i.e., the MIMSC, then R, is set at 6000 and the effect of varying R, on U, and CJ, is shown in Figure 11(b). Adjusting R2 to be 1000 results in a minimum CJ, of 0.31, which is about one third that of the uncontrolled case. In other words, the MIMSC is about 2.5 times more effective than the IMSC. Comparisons between the theoretical and experimental time responses of the uncontrolled and MIMSC-controlled cases are shown in Figures 12(a) and 12(b) respectively. The displayed results show adequate agreements between theory and experiments. ‘The MIMSC also can be compared with two other modal control strategies, as shown in Figure 13, first with MIMSC itself but with a sequential time sharing strategy and second with the Pseudo-Inverse (PI) method. It is seen that the MIMSC with time sharing based on the maximum modal energy ranking is superior to the MIMSC with sequential time sharing. In the latter case the actuator is shared in a sequential manner between the modes without any consideration to the modal energy. Of course, this is easy to implement as it requires a lesser amount of computation than the modal ranking approach. It is evident, however, that it is less effective in damping the vibrations. The beam response when it is controlled by the PI algorithm [17] is also shown in Figure 13. In this algorithm, the modal control of multi-modes is realized by a smaller number of actuators through an approximate statistical realization scheme. This approach results in ineffective damping of the vibrations. Quantitatively, the displacement indices for the sequential MIMSC and the PI algorithms are 0.58 and 0.82 respectively. The corresponding control effort indices are 156 and 0.45 respectively.

Time is)

Figure 13. Time history of the beam vibration. sequential MIMSC; (c) with PI.

(a) With max. modal

energy

ranking

MIMSC;

(b) with

146

A. BAZ

_o.701i.._I 0

AND

S. POH

3

2 Time

Figure 14. Time history IMSC; (c) with MIMSC.

of the beam vibration

for sinusoidal

4

5

(5)

excitation

at 1.6 Hz. (a) Uncontrolled;

(b) with

The performance of the MIMSC algorithm can also be compared with that of the IMSC method when the beam is subjected to sinusoidal excitations. The responses when the excitation frequencies are 1.6 and 11 Hz are shown in Figures 14 and 15 respectively. Each figure displays the uncontrolled response as well as the response with the IMSC and MIMSC controllers. It can be seen that when the beam is excited at its first mode of vibration (i.e., 1.6 Hz), the IMSC and MIMSC produce equal control actions. But, when the excitation frequency coincides with the beam second mode of vibration, the MIMSC out performs the IMSC. In this case, the IMSC does not attenuate the beam vibration at all and the beam vibrates exactly as if it were uncontrolled.

5. CONCLUSIONS

experimental implementation of the MIMSC method to control the vibration of a flexible cantilevered beam with one piezo-electric actuator has been presented. A brief description of the mathematical model simulating the dynamic characteristics of the beam-actuator system was presented, and also a review of the MIMSC method. The method was implemented on a 386-based microprocessor with a cycle time (i.e., sample interval) of 3.04 ms. Comparisons have been carried out between the MIMSC, the IMSC and the PI. The results obtained indicate that the MIMSC with maximum modal energy ranking is more effective in damping the vibration than the IMSC, the PI and the MIMSC with sequential time sharing. The MIMSC requires, however, higher control energy than the IMSC. The least control energy expenditure is required by the PI method. An

EXPERIMENTAL

147

MIMSC

_ .-

2 2

0.3s

E

E

H

P

n a t

o-00

-0.35

-0.70

_o.,ooL+_.~

‘_L--

3

2 Tme

---L

4

.

_+I

(s)

Figure 15. Time history of the beam vibration for sinusoidal IMSC; (c) with MIMSC.

excitation

at 11 Hz.(a) Uncontrolled;

(b) with

The study has demonstrated clearly the potential of the MIMSC method as an effective method for controlling large number of vibration modes with a smaller number of actuators. The feasibility of the concept of “time sharing” has been demonstrated. This feature has important practical implications that make the MIMSC viable means for controlling large space structures in real time.

ACKNOWLEDGMENTS

‘This study was supported by the NASA-Goddard Space Flight Center under grant number NAG 5-749. Special thanks are due to the technical monitor Dr J. Fedor for his invaluable technical inputs.

REFERENCES

18, 11-17. Electronic damping of orthogonal bending modes in a cylindrical mast. 2. T. BAILEY and J. E. HUBBARD, JR 1985 Journal ofGuidance and Conrrol8,605-611. Distributed piezo-electric polymer active vibration control of a cantilever beam. 3. J. M.PLuMP,J.E.HuBBARD,JR~~~T.BAILEY~~S~ AmericnnSocietyofMechanicafEngineers Journal of Dynamic Systems, Measurements and Control 109, 133-139. Nonlinear control of a distributed system: simulation and experimental results. 1. R. L. FORWARD

1981

Journal

of Spacecrafr

148

A.

4. E.

F. CRAWLEY and J.

DE

and Materials

Conference,

piezo-ceramics

as distributed

BAZ

AND

S. POH

LUIS 1985 Proceedings of the 26th Structures, Structural Dynamics Part 2, AIAA-ASME-ASCE, Orlando, Florida, 126-133. Use of actuators in large space structures.

5. E. F. CRAWLEY and J. DE LUIS 1987 American Institute of Aeronautics and Astronautics Journal 25, 1373-1385. Use of piezoelectric actuators as elements of intelligent structures. 6. J. L. FANSON and T. K. CAUGHEY 1987 American Institute of Aeronautics and Astronautics Paper Number 87-0902. Positive position feedback control for large space structures. 7. A. BAZ, S. POH and P. STUDER 1989 Journal of The Institution of Mechanical Engineers, Part C (203, 103-112). Modified independent modal space control method for active control of

flexible systems. 8. A. BAZ and S. POH 1988 Journal of Sound and Vibration 126, 327-343. Performance of an active control system with piezoelectric actuators. 9. L. MEIROVITCH and H. BARUH 1981 Journal of Guidance and Control 4, 157-163. Optimal control of damped flexible gyroscopic systems. 10. L. MEIROVITCH and H. BARUH 1982 Journal of Guidance and Control 5, 59-66. Control of self-adjoint distributed-parameter systems. 11. L. MEIROVITCH, H. BARUH and H. Oz 1983 Journal of Guidance and Control 6, 302-310. Comparison of control techniques for large flexible systems. 12. A. BAZ and S. POH 1987 NASA Technical Report Number N87-25605. Comparison between MIMSC, IMSC and PI in controlling flexible systems. 13. R. T. FENNER 1975 Finite Element Methods for Engineers. London: Macmillan. 14. A. W. AL-KHAFAJI and J. R. TOOLEY 1986 Numerical Methods in Engineering Practice. New York: Holt, Rinehart and Winston. 15. D. J. EWINS 1984 Modal Testing: Theory and Practice. Letchworth, England: Research Studies Press. 16. J. T. BROCH 1980 Mechanical Vibration and Shock Measurements. Denmark: Briiel and Kjaer. 17. R. LINDBERG, JR and R. LONGMAN 1984 Journal of Guidance and Control 7, 215-221. On the number and placement of actuators for independent modal space control.

APPENDIX:

nomenclature

b beam width (m) b charge constant of piezo-actuator (m/V) E 1.2.3 Young’s moduli of actuator, bonding layer and beam respectively (N/m2)

Young’s modulus of ith beam element (N/m2) modal force vector (2n x 1) (N, Nm) controlled and uncontrolled modal force vectors (N, Nm) $,” controlled modal force of the ith mode (N, Nm) ; physical force vector (2n x 1) (N, Nm) Physical force vector (C x 1) (N, Nm) F, ;,,,, area moments of_inertia of actuator, bondinqlayer and beam respectively (m”) area moment of mertta of the ith element (m ) ; performance index mass moment of inertia at the ith node (kgm’) Ji stiffness matrix of ith element (4 x 4) K overall stiffness matrix of beam-actuator system (2n x 2n) K length of ith element (m) Li mass of beam at node i (kg) 2 mass matrix of beam-actuator system (2n x 2n) Mei external moment acting on ith node (Nm) MS piezo-electric actuator moment (Nm) n number of nodal points 0 null matrix, in equation (8), ((2n - C) x (2n - C)) weighting parameter of ith control effort with respect to the amplitude of vibration Ri t,,2,3 thickness of actuator, bonding layer and beam respectively (m) modal co-ordinates of the flexible system u V voltage applied across actuator (V) external forces acting on ith node (N) vi linear translation of ith node (m) Yi Ei

EXPERIMENTAL

Greek symbols deflection 4

6 g 6, A 4 0,

MIMSC

of ith node (m, rad) nodal deflection vector (2n x 1) (m, rad) nodal acceleration vector (2n x 1) (m, rad) angular deflection of ith node (rad) diagonal matrix of eigenvalues of the system (2n x 2n) modal shape matrix of the system eigenvectors (2n x 2n) natural frequency of ith mode (rad/s)

149