Active control of a circular plate to reduce transient noise transmission

Journal of Sound and Vibration (1995) 183(4), 643–662

ACTIVE CONTROL OF A CIRCULAR PLATE TO REDUCE TRANSIENT NOISE TRANSMISSION J. L.  N Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, South Africa 0002

 B. H. T Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.

 A. K. P Department of Mechanical Engineering, University of California at Berkeley, Berkeley, California 94720, U.S.A. (Received 21 June 1993, and in final form 2 May 1994) An active control approach that reduces transient noise transmission through a plate in a circular duct is presented. Two circular pieces of piezoceramic material are used as actuators to induce a moment in the plate. A model for this configuration is developed and then used in an H2 optimal control procedure to design a dynamic feedforward/feedback controller. In the experiment the forward travelling wave is estimated from the measurements of two microphones. This information is used as the feedforward signal, while the velocity of the center of the plate is measured with a laser vibrometer and used as the feedback signal. The discretized controller is implemented on a digital signal processor (DSP). A transient pressure pulse is used to excite the plate, and reductions of up to 15 dB are observed at a microphone placed downstream of the plate when the controller is active.

1. INTRODUCTION

The experiment reported on in this paper is part of an ongoing research project concerned with the active control of transient noise and vibration in structures. As a first experiment, the active control of a transient acoustic pulse through a circular membrane placed in a duct was investigated [1]. Using polyvinylidene fluoride (PVDF) film actuators, it was possible dynamically to alter the membrane’s tension (parametric control), thereby controlling the system’s response. Due to the fact that the controller could only affect the structure if it was displaced from its equilibrium position, only the effective damping in the system could be increased. No overt control of the structure itself was possible in the absence of an excitation. In order to overcome this limitation, the membrane was replaced with a plate, and the PVDF film actuators replaced with piezoceramic actuators. The applications of transient noise control are quite varied. Transient dynamics, which manifest themselves as noise, are present in structures varying from motor vehicles to buildings. One can think of an automobile’s tire going through a pothole, for example. The consequence of the tire hitting the pavement is an impulse that will excite a transient vibration in the structure. This vibration will travel along the structure and will eventually lead to noise inside the vehicle’s passenger compartment. 643 0022–460X/95/240643 + 20 $08.00/0

7 1995 Academic Press Limited

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Active control of steady state noise has been successfully implemented by making use of adaptive feedforward control schemes. However, these controllers do not have the speed to adapt to rapidly changing acoustic excitations and, in general, need a signal correlated with the sound that is being controlled. The obvious advantage of adaptive controllers is their low modelling requirements but, if one wishes to look at transient dynamics, the accuracy of the model used in the controller design process becomes more important. In this paper, a review of the literature applicable to the problem of active control of structures to reduce noise transmission is presented. A reliable model for the system at hand is developed, which is then used to design a dynamic, feedforward/feedback controller. Simulation as well as experimentally obtained results are presented and discussed. Finally, some conclusions are drawn. 2. LITERATURE SURVEY

The first reference made to active noise control was a patent application by Paul Lueg in 1936 [2]. After the patent was granted in 1938, very little was accomplished in the field due to the constraints of the available electronic circuitry. As the state of the art in electronic components improved, researchers again turned to the problem of active noise control in ducts. Some of the early work was done by Jessel and Mangiante [3, 4]. As interest increased a host of other researchers started to work in the field; for a review of the early years, see references [5] and [6]. The active control of sound only became feasible after the introduction of the digital signal processor (DSP). The most favored control approach was that of an adaptive feedforward controller. This type of controller requires a reference signal that is correlated with the actual noise to be cancelled. An error measurement is then used to adapt the coefficients of the digital filters through which the reference signal is passed. Initially, loudspeakers were used as secondary sources to inject anti-noise. The mechanisms of cancellation are source modification or source loading by the secondary noise sources. For a recent review of this topic, see reference [7]. The first mention of structural control to reduce noise transmission into a cavity was by Fuller and Jones in 1987 [8]. They made use of a single electrodynamic actuator to reduce the sound inside a cylindrical enclosure excited by an exterior monopole source. They reported attenuation of 10–20 dB in the sound pressure level (SPL) measured inside the enclosure. Since the experiment was aimed at noise reduction in aircraft making use of turbopropellers, all the experiments were carried out at monotones corresponding to the blade passing frequencies. The control signal was determined by varying the phase and amplitude of a sinusoid of the same frequency as the excitation until the sound measured inside the enclosure was of the same magnitude as the noise created by the exterior source alone, but with opposite phase. In another significant paper, Fuller et al. [9], studied the reduction of sound transmission through a circular elastic plate by making use of electrodynamic actuators and an adaptive feedforward controller. They used both accelerometers and microphones as error sensors, and it was concluded that minimizing plate motion does not necessarily lead to a reduction in the transmitted sound field. Reductions of up to 28 dB were reported when microphones were used as error sensors. Microphone measurements will include the effect of the coupling between the plate vibration and acoustic field, which explains the better performance. In related work, Pan, Hansen and others [10–13], studied the physical mechanisms by which sound transmission through a rectangular, simply supported plate and into a reverberant cavity can be reduced. Two such mechanisms were observed. In the first, when

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the system response is dominated by the plate modes, the noise is reduced by controlling only those modes of the plate that couple well into the acoustic cavity. On the other hand, if the system response is dominated by the acoustic modes of the cavity, then reduction is obtained by altering the velocity distribution of the plate. The next significant step in the field came when the electrodynamic actuators used for structural control were replaced by piezoelectric materials. The first article on the use of piezoelectric polymers as actuators on continuous dynamic systems was by Bailey and Hubbard [14]. In subsequent work, Crawley and de Luis [15] developed a detailed model to characterize the interaction beteween the piezoelectric material and the structure. Dimitriadis et al. [16] extended this model so that they could investigate the use of piezoelectric actuators to reduce the sound pressure radiated by thin plates. They concluded that these types of actuators show great promise for controlling the vibration in distributed systems and subsequently the control of sound radiation. It was shown that the shape and position of the actuators markedly affects the distribution of the response among the different modes. Fuller et al. [17, 18], reported on an experiment in which a single piezoceramic actuator was attached to a rectangular plate. The plate was excited by a non-contacting electrodynamic actuator. To reduce the sound radiated from the plate due to the monotone excitation by the electrodynamic actuator, the signal to the actuators was varied in amplitude and phase until the best reduction was observed. The tests were performed at frequencies close to the natural modes of the plate. A global reduction of 45 dB was obtained at the two frequencies that were excited. Subsequent work by Clark and Fuller [19] made use of an adaptive feedforward controller to cancel noise radiated from a rectangular plate due to excitation by an electrodynamic actuator, while using piezoceramics as actuators. They concluded that if the plate was excited close to one of its modes there was no marked improvement as the number of actuators was increased. However, in the case of off-resonance excitation, increasing the number of actuators did improve the performance. The same authors also employed piezoelectric material as sensors instead of microphones [20]. The position of the film was chosen in such a way that the measured response was dominated by the plate modes with an even number of nodal lines. In general, these modes are good radiators. The sensors acted, in a sense, as modal filters. Their results indicate that although similar reductions were obtained using the PVDF sensors in lieu of microphones in some cases, there were other cases in which the microphones outperformed the PVDF sensors. They attributed this to the fact that it was not possible to obtain the same level of modal restructuring, redistributing the energy from one mode to another, using the PVDF sensors, and they suggest that the solution might be an optimization of the shape and position of the PVDF sensors. Wang et al. [21] developed an analytical model of noise transmitted through rectangular plates making use of multiple piezoelectric and point force actuators. An optimal control methodology was followed to reduce the far field noise distribution by inputting suitable control signals to the actuators. It was found that point actuators are more effective than piezoceramics but the practicality of the piezo actuators make them more suitable for implementation in actual structural control. Other researchers who have also completed work in the field are Tzou [22, 23], Kim and Jones [24, 25], Akishita et al. [26] and Zhou and Cudney [27]. Kim and Jones investigated the coupling of piezoelectric actuators with the structural substructure. They used a more refined model than previous researchers, which included the effect of the bonding layers. Some results on the optimal thickness for maximum induced bending moment were presented.

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Clark et al. [28] present a model of a simply supported beam driven by multiple piezoceramic actuators. They also present experimental results verifying the accuracy of the model by predicting the modal distribution of certain excitation conditions analytically and then verifying it from the experimental data. Gu and Fuller [29] looked at the control of sound radiation due to subsonic wave scattering from discontinuities on fluid-loaded plates. They showed that steady state feedforward control can lead to large reductions in the radiated sound pressure due to flexural waves interacting with discontinuities. The only reference known to the authors that addresses transient noise reduction is by Baumann et al. [30] on the active suppression of acoustic radiation from impulsively excited structures. Their LQR feedback approach deals more with damping of the ensuing free vibrations than controlling the structure while the excitation is acting on it. In further work [31], they expanded the scope of the controller to control broadband disturbances. Here they made use of the LQG methodology to design a feedback controller to minimize the sound radiated from a clamped–clamped beam. Other researchers who investigated a feedback approach include Batta and Shourezi [32] and Hull et al. [33]. Yang et al. [34, 35] used a robust Ha design technique to obtain a feedback controller for a reverberant enclosure, using speakers as actuators. 3. EXPERIMENTAL SET-UP

In this experiment, we studied the sound transmission through an aluminum plate, placed within a duct. Here, one only has to deal with one-dimensional acoustics, since the acoustic pressure waves having frequencies below the cut-off frequency of the waveguide, 2 kHz in this case, become planar after having travelled a very short distance along the duct. A plate was chosen since it was previously found that feedforward control was only possible if the actuators had the authority to displace the structure before the acoustic wave reached it. This problem is the next step towards a general transient control problem. The set-up for this experiment is shown in Figure 1. A loudspeaker at one end is used to excite a transient wave in the duct. As this wave travels down the duct, it quickly comes to resemble a plane wave and the waveform is measured by two microphones as it passes by. The wave reaches the plate, where some of the energy is then reflected as a pressure wave back towards the loud speaker, while the rest of the energy induces vibrations of the

Figure 1. The experimental set-up.

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T 1 List of equipment Model

Instrument

GenRad 2515 Bru¨el & Kjær 4166 Bru¨el & Kjær 2639T Bru¨el & Kjær 2804 Bru¨el & Kjær 2706 Realistic 8 inch Kepco BOP 1000 M TSI 1835 Texas Instruments TMS320C30 cessor 486 IBM compatible Piezo Electric Products G-1195

Dynamic signal analyzer Microphone Microphone pre-amplifier Power supply Power amplifier Loud speaker Voltage amplifier Laser vibrometer

Frequency Range 0–16 384 Hz 2·6 Hz–10 kHz 2 Hz–200 kHz — 10 Hz–20 kHz 0–800 Hz 0–1 kHz 1 Hz–100 kHz Digital signal pro-

10 kHz sampling rate Personal computer Piezoceramic

— —

plate. These vibrations lead to pressure waves on the downstream side of the plate (transmitted waves). The transmitted waves continue down the duct and are measured by a third microphone. In addition, the velocity of the center of the plate is measured by a non-contacting laser vibrometer. The aim of this experiment was to control the plate in order to reduce the magnitude of the transmitted waves by applying an actuation force to it. The piezoceramic actuators are circular in shape and were bonded symmetrically to both sides of the plate so that when a voltage was applied to the actuators they would induce a moment in the plate along the edge of the actuators. The control algorithm was implemented in a DSP housed in a 486 IBM-compatible personal computer. The control signal was amplified 100 times by the voltage amplifier before it was used to drive the piezoceramic actuators. The input to the controller was taken as the velocity measured by the laser vibrometer and the two upstream microphone measurements. A dynamic signal analyzer was used to capture the transient data for analysis. A complete list of the equipment used for the experiment is given in Table 1. The physical dimensions of the experimental set-up are given in Table 2 and the material properties in Table 3. The measurements from the two upstream microphones were used to distinguish the wave travelling towards the plate from waves going in the opposite direction. This was necessary since microphones in general only measure pressure, which is a scalar quantity, and cannot determine the direction of the pressure wave as it passes by. As the speed of sound in air, ca , is known and the distance between the two microphones, lm , can be T 2 Physical dimensions Plate

Diameter Thickness

104 mm 2·54 mm

Actuators

Diameter

80 mm

Microphones

Number 1 to number 2 Number 2 to plate Plate to number 3

200 mm 575 mm 500 mm

Duct

Diameter Length (loud speaker to plate) Length (plate to end of duct)

104 mm 1·1 m 1·0 m

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T 3 Material properties Aluminum

Young’s modulus Poisson ratio Density

Piezoceramic

Young’s modulus Poisson ratio Density Piezoelectric Constant

45·4 GPa 0·334 2693 kg/m3 60 GPa 0·3 1·4 kg/m2 190 × 10−12 m/V

measured, the following algebraic equation was used to determine the forward travelling wave, pf (t): pf (t) = m1 (t) − m2 (t − lm /ca ) + pf (t − 2lm /ca ), (1) where m1 and m2 are the pressure measurements from the microphones, obtained by multiplying the voltage signals from the microphones by the appropriate calibration factors. Taking the Z-transform of this equation gives Pf (z−1) =

1 z−k M (z−1) − M (z−1), (1 − z−2k) 1 (1 − z−2k) 2

(2)

where k = lm /ca T , T being the sampling period of the estimator. This equation has all its poles on the unit circle and will therefore only be marginally stable. To ensure stability of the estimation, the original estimator was modified to pf (t) = m1 (t) − m2 (t − lm /ca ) + 0·99pf (t − 2lm /ca )

(3)

which now has all its poles on the circle with radius 0·99. This estimator will then be stable, but it will underestimate the amplitude of the forward travelling wave. 4. MODEL

4.1.   The modelling process for the interaction between the piezoelectric elements used as circular actuators and a circular plate is similar to that presented by Dimitriadis et al. [16] for a rectangular plate with rectangular actuators. The initial assumptions are as follows: the piezoceramic material is perfectly bonded to the underlying plate; the thickness of the bonding agent can be ignored; the piezoceramic does not add any mass to the structure; the piezoceramic is effective in the radial direction; the circular plate deforms symmetrically in pure bending. The first two assumptions depend on the type of bonding agent that is used and the care taken during the bonding process. However, it has been shown that ignoring the effects of the bonding agent is reasonable [14]. The accuracy of the third modelling assumption depends on the ratio of the thicknesses and densities of the piezoceramic material and the plate. In the present experiment it was calculated that a piezoceramic actuator contributed only 12% of the total mass of the system. As shown in Figures 2 and 3, the actuators were symmetrically attached to the plate. If a voltage of equal magnitude but opposite polarization is applied to the two actuators, one will expand while the other contracts, due to the piezoelectric effect. This symmetry will lead to a symmetrical loading on the plate by the actuators. Furthermore, the actuators are also axisymmetric, as is the acoustical loading, thus justifying the assumption of symmetrical bending.

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Figure 2. The radial stress distribution in an infinite plate with bonded piezoelectric material.

It has been shown previously [16] that for an unconstrained, two-dimensional actuator polarized in the z-direction, the induced strain, L pe (t), can be written as a function of the applied voltage, V(t): L pe (t) =

d31 V(t), hpe

(4)

where d31 is the piezoelectric strain constant relating the strain in the unconstrained direction to the applied voltage and hpe is the thickness of the piezoceramic material in the z-direction. For a circular actuator, the unconstained direction is the radial direction and hence one can write erpe (r, z, t) = L pe (t) =

d31 V(t), hpe

(5)

where erpe (r, z, t) are the dynamic strains in the piezoceramic. Next, one considers the static loading condition of an infinite circular plate with two piezoelectric elements bonded symmetrically to it. The strains induced in the piezoelectric material will lead to tractions at the interface between the piezoelectric elements and the plate. These tractions will be uniform in magnitude but of opposite sign. Since no extension of the mid-plane of the plate is possible, due to the symmetry of this loading condition, the assumption of pure bending of the plate is justified. The distribution of the normal stress in the radial direction is shown in Figure 2.

Figure 3. A clamped plate with piezoceramic actuators.

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In the case in which there is deformation of the plate, one can express the couple which induces this deformation as mr =

g

hp

srp (z)z dz,

(6)

−hp

where srp (z) is the normal stress in the radial direction due to the symmetric bending of the plate and 2hp is the thickness of the plate. From Chapter III in reference [36], the radii of curvature of symmetrical, pure bending of a circular plate is given in terms of the transverse displacement w(r) by 1 2w 1 =− 2 , 1r Rr

1 1 1w =− , Ru r 1r

(7, 8)

where Rr is the curvature in the r–z plane, and Ru is the curvature in the perpendicular plane. The strain distribution, which is linear through the thickness of the plate, is given by eup = z/Ru .

erp = z/Rr ,

(9, 10)

At the interface between the plate and the piezoelectric element, the stress in the plate is a function of erp and eup at z = hp and can be written as s¯ rp = −

0

1

Ep hp 1 2w np 1w + , 1 − np2 1r 2 r 1r

(11)

where Ep is the Young’s modulus and np is the Poisson ratio of the plate. For the piezoelectric elements, the resulting stress is a result of the superposition of the strain due to bending of the plate and the strain Lpe , induced by the applied voltage: s¯ rpe =

0

1

Epe hp 1 2w np 1w Lpe + , − 1 − npe2 1r 2 r 1r hp

(12)

where Epe is the Young’s modulus and npe is the Poission ratio of the piezoceramic material. In the plate the bending stress is linear in the r-direction and can be written as a function of s¯ rp : srp (z) = s¯ rp (z/hp ),

(13)

for −hp E z E hp . The stress in the piezoelectric elements is also linear: srpe (z) = s¯ rpe − (Epe /Ep )s¯ rp (1 − z/hp ),

(14)

for −(hp + hpe ) E z E −hp and hp E z E hp + hpe . From the equilibrium of moments one has

g

hp

srp (z)z dz +

0

g

hp + h p e

srpe (z)z dz = 0,

(15)

hp

which leads to s¯ rp = −Ks¯ rpe ,

(16)

where K=

3hp hpe (2hp + hpe ) . 2hp3 + (Epe /Ep )(2hpe3 + 3hp hpe2 )

(17)

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651

If one returns to the equation relating the interface stress to the strain at the interface and uses the above obtained expressions, one can write hp

0

1

d 2w np dw P + = L , dr 2 r dr 1 − P pe

(18)

where

0

1

Epe 1 − np2 . Ep 1 − npe2

P=K

(19)

The moment applied to the plate is therefore mr =

g

hp

srp (z)

−hp

z2 dz = C0 Lpe , hp

(20)

0 1

(21)

where C0 = 23 hp2

Ep P . 1 − np2 1 − P

If one now considers a circular plate with radius a, clamped at the boundary with two circular piezoelectric elements of radius b bonded on either side (Figure 3), the moment distribution in the plate can be written as mr = C0 Lpe [h(r) − h(r − b)],

(22)

where h(r) is the Heaviside function. Following the approach in Chapter III in reference [36], it is possible to obtain the equation of motion for the system by defining the moments M r = Mr + mr ,

M u = Mu ,

(23, 24)

where Mr and Mu are the moments in the r and u directions, respectively, due to the bending of the plate. The equations relating the displacement w(r, t) to these moments are

0

M r = −D

1

1 2w np 1w + , 1r 2 r 1r

M u = −D

0

1

1 1w 1 2w + np 2 . r 1r 1r

(25, 26)

The transverse forces acting on the plate are the pressure loading p(r, t), which is symmetric for the plane wave case studied here, and the inertial force −rp w¨ . With these forces, the equation of motion can be determined to be

0

D

1

1 4w 2 1 3w 1 1 2w 1 1w 1 2m 2 1mr + rp w¨ = p + 2 r + , 4 + 3 − 2 2 + 3 1r r 1r r 1r r 1r 1r r 1r

(27)

where D = 2Ep hp3 /3(1 − np2 ). Substituting for mr yields D9 4w + rw¨ = C0 Lpe (d'(r) − d'(r − b) + (2/r)[d(r) − d(r − b)]) + p(r, u, t),

(28)

where 9 = 1 2/1r 2 + (1/r) 1w/1r. Note that d'(r), the first derivative of the delta function, is a distributed dipole force representing the moment applied to the plate by the piezoelectric elements. To decouple the equation of motion into an infinite set of second order, ordinary differential equations, one expands the displacement w(r, t) by making use of only the symmetric normal modes of a clamped circular plate [37], and the modal co-ordinates h0n : a

w(r, t) = s A0n W0n (r)h0n (t), n=1

(29)

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T 4 Natural frequencies of clamped circular plate Experimental 949·0 1975·0 3240·0 3694·0

Analytical 949·0 1945·0 3174·0 3695·0

where W0n = [I0 (b0n a)J0 (b0n r) − J0 (b0 na)I0 (b0n r)].

(30)

in which J0 and I0 are the zeroth order Bessel functions of the first and second kinds, respectively. Multiplying the equation of motion by A0q W0q and integrating over the domain, provided that the modes are mass-normalized, one obtains 2 (t) = u0q (t) + f0q (t), h¨ 0q (t) + v0q

p = 1, 2, . . . , q = 1, 2, . . . ,

(31)

where

gg 2p

u0q (t) = C0 Lpe (t)

0

a

A0q W0q [d'(r) − d'(r − b) + (2/r)d(r) − (2/r)d(r − b)]r dr du

0

(32) and

gg 2p

f0q =

0

a

A0q W0q p(r, t)r dr du.

(33)

0

Figure 4. Frequency response functions between the plane wave and the velocity of the plate. ——, model; – – , experiment.

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653

Since the pressure loading consists of plane waves, the effective loading on the plate can be taken as twice the amplitude of the approaching wave: p(r, u, t) = 2Pf(t).

(34)

Dropping the subscript 0, since both the control action and the forcing are symmetrical: u0q (t) = uq V(t),

(35)

uq = −C0 (d/hpe )2pA0q bb0q [I0 (b0q a)J1 (b0q b) + J0 (b0q a)I1 (b0q )b)]

(36)

f0q = fq f(t),

(37)

fq = 2pPA0q (a/b0q )[I0 (b0q a)J1 (b0q a) + J0 (b0q a)I1 (b0q a)].

(38)

where and and where

4.2.   Since circular piezoceramic actuators, poled so that they are effective in the radial direction, were not readily available, piezoceramic material that was poled to be effective in a single direction was cut and orientated in such a fashion as to approximate circular actuators. To test the validity of this apporach, two experiments were conducted. The first experiment aimed at assessing the model’s validity was an experimental model analysis of the plate. The plate was excited with an impact hammer and the response measured with an accelerometer. The density of the plate was determined by measuring the weight of the plate and dividing by the volume. To obtain the correct value of the Young’s modulus, the natural frequency of the first mode (obtained experimentally) was set equal to the frequency of the analytic model’s first mode. The natural frequencies of the plate, experimental and analytical, are listed in Table 4. One can see that the results are very similar, which leads to the conclusion that the boundary conditions of the plate are well approximated as being clamped. To further validate the model, the frequency response function (FRF) between the plane pressure wave input and the velocity response of the plate was calculated using the model developed in this chapter. This FRF was then compared to the actual experimentally obtained one and the result is shown in Figure 4. 5. CONTROLLER DESIGN

From the development in the preceding section, the following model emerged for the dynamic response of a plate controlled by a piezoceramic actuator, and excited by a plane wave: h¨ q + vq2 hq = fq f(t) + uq V(t), q = 1, 2, . . . , N, (39) where the 0 subscript has been dropped and the model truncated to include only N modes. This model does not include any damping effect. In any real system there is always some form of damping present, and the most common way to model this effect is with a viscous damping model, where one assumes that the damping force is proportional to the velocity. The equations of motion, which now include damping, can be written as h¨ q + 2zq vq h˙ q + vq2 hq = fq f(t) + uq V(t),

q = 1, 2, . . . , N,

where the modal damping coefficients, zq , can be determined experimentally.

(40)

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654

If one now defines the vector x to represent the states of the system: xT = [h1

h2

hN h˙ 1

...

h˙ 2

h˙ N ],

...

(41)

then it is possible to write the equations of motion in the state space form x(t) = Ax(t) + Bu(t) + Ed(t),

(42)

where 0 K 0 G 0 0 G . . . G .. . G A= 0 0 G 2 0 G −v1 −v22 G 0 . G .. . . G . k 0 0 BT = [0

0

...

0 u1

...

0

1

0

...

0

... . . . ...

0 . . . 0

0 . . . 0

1 . . . 0

... . . . ...

0 . . . 1

...

0

−2z1 v1

0

...

0

... . . . ...

0 . . . −vN2

0 . . . 0

−2z2 v2 . . . 0

u2 . . .

uN ],

ET = [0

0

... 0 . . . . . . . . . −2zN vN

...

0

f1

f2

L G G G G, G G G G G l ...

fN ].

Also, u(t) = V(t),

d(t) = f(t).

To complete the state space representation, one has to consider the type of measurements that are available. In this particular case there are two, the velocity of the center of the plate, y(t), and the impinging pressure wave, d(t). Thus: y(t) = Cx(t) + Du(t)

(43)

where C[0

0

...

0

A01

A02

...

A0N ],

D = [0].

One can arrange this system in the notation of MATLAB’s mSynthesis toolbox [38], as shown in Figure 5. In this problem, it makes sense to follow two different control strategies. The first is to make use of the fact that the disturbance, the transient plane wave pulse, can be measured before it reaches the plate. This leads to the obvious conclusion that a feedforward control

Figure 5. The system representation.

  

655

Figure 6. Feedforward and feedback controllers.

scheme can be implemented. On the other hand, since it will be almost impossible to cancel out all the excitation due to the transient wave, there will be some resultant vibration present in the plate. Using the velocity measurement, it can be argued that a feedback controller will further dampen these vibrations. This set-up is shown in Figure 6. Since the control algorithm will be implemented using a single DSP, it is necessary to design the feedforward and feedback controllers simultaneously. This can be done quite easily using the function ‘‘h2syn’’ in mTools. The block diagram for the system is shown in Figure 7. In order to make use of the H2 optimal control design technique as implemented in mTools, one has to augment the system by adding noise terms to the measurements and also define errors to be minimized by the designed controller. Two independent noise terms, n1 and n2 , are added to the measurements so that y˜ = y + bn1 ,

d = d + gn2 .

Since the downstream acoustic pressure will be proportional to the velocity of the plate, the first error is taken as e1 = y. The algorithm to be used for the design requires that the control also enters into the error that is to be minimized. This is equivalent to the requirement that R must be non-singular in the standard LQR design procedure. Thus the second error is defined as e2 = au. This augmented system can be represented as shown in Figure 8. The values a, b and g are design parameters to be chosen by the designer. Changing these values will alter the weighting on the errors and will result in different gains. It is therefore important to check the stability of the plant for each controller. If a candidate controller was found to be stable, the transient response was evaluated by making use of time domain simulations.

Figure 7. A block diagram of the plant with a single two-input, one-output controller.

  

657

The Bode plot for a control obtained using the values a = 3·0, b = 0·1 and g = 0·1 is shown in Figure 9. The step response of the closed loop system is shown in Figure 10. Here, one can see the typical behavior of a non-minimum phase system; i.e., the system’s response is at first in a direction opposite to the applied control.

6. RESULTS

To demonstrate the effectiveness of piezoceramics as control actuators, a steady state experiment was performed in which a sinusoidal disturbance of 1 kHz was injected into the duct via the loudspeaker. The sound transmitted through the plate was measured by microphone number 3, and is shown in Figure 11. The control signal used in this experiment was obtained by adjusting the phase and magnitude of the same sinusoid until the amplitude of the transmitted acoustic pressure was minimized. The minimized signal is shown in the lower part of Figure 11. The sound pressure levels of these two signals were also calculated and are shown in Figure 12. It is clear that a substantial reduction, 20 dB, was obtained at 1 KHz. The actual transient control experiment was performed by exciting a transient pulse with the loudspeaker. The input to the loudspeaker was a step function; however, due to the limited bandwidth of the loudspeaker, the actual pulse response was not a pure step. The time hisotry of the actual plane wave pulse is shown in Figure 13. This graph also shows that the initial pulse, after being reflected off the plate, is reflected off the loudspeaker and will excite the plate for a second time. This behavior can be verified if one calculates the time that it takes the wave to travel from microphone number 1 to the plate, all the way back to the loudspeaker and then back to microphone number 1. The total length of this path is 2·18 m and, using the value, ca = 343·5 ms−1, the time is 6·35 ms, as shown in Figure 13. One can also observe that the excitation is quite large, having a peak value of 98 Pa. In Figure 14, the velocity of the plate, when excited by the transient pressure pulse, is shown for the case in which there is no control action, as well as the case in which the controller is active. Measurements for both cases were made by the laser vibrometer. It is clear that not only is the initial peak reduced in amplitude but the ensuing vibration is

Figure 11. Downstream microphone number 3: time history for 1000 Hz disturbance, (a) without and (b) with control.

658

. .    .

Figure 12. The SPL of microphone number 3 for 1000 Hz disturbance, with and without control. ——, disturbance; – – , error.

also damped out faster for the controlled case than for the uncontrolled case. Comparing Figure 14 with Figure 10, one observes that the actual response did indeed follow that of a non-minimum phase system, where the response was in the opposite direction than the applied control. The transmitted noise as measured by the downstream microphone number 3 is shown in Figure 15. The time span of the graph is short enough so that no reflection from the end of the duct will reach the microphone and one can conclude that the reduction shown will be global in nature. To quantify the reduction obtained with the controller, the power spectral densities of the two signals were computed and are shown in Figure 16 as sound

Figure 13. The time history of the input pulse, f(t).

  

659

Figure 14. The velocity response of the plate, y(t). (a) No control; (b) with control.

pressure levels (SPL). Here one can see that a reduction of up to 15 dB was achieved at some frequencies. At frequencies higher than 1200 Hz, there is no marked reduction in the SPL. This can be explained if one observes in Figure 9 that the gains of the controller start to decrease after 1200 Hz. The increase in noise at frequencies less than 200 Hz is partly due to the high gain of the controller at low frequencies, as well as the influence of exterior noise on the present experimental set-up. 7. CONCLUSIONS

The goal of this investigation was to determine how effective a distributed actuator could be in controlling the sound transmission through a flexible plate. Our findings were that a suitably designed actuator could indeed perform very effectively in reducing acoustic

Figure 15. The time history of downstream microphone number 3 measurement. (a) No control; (b) with control.

660

. .    .

Figure 16. The SPL of downstream microphone number 3 ——, no control; – – , with control.

transmissions. The geometry and placement of the actuator allowed it to couple well with the plate’s first vibration mode, the mode that most strongly influences acoustic plane wave generation. It was also demonstrated that the actuators could profitably be used in a feedforward sense, allowing the structural control to begin just prior to the initial impingement of the incident acoustic excitation. This allowed the actuator to counter the full effect of this incident pulse, rather than having to wait until the initial pulse caused plate deformation (which would then be fed back into the controller). These results represent an initial step in transient acoustic control for complex structures and show that, in a one-dimensional case, it is possible to obtain good control. The response characteristics of the piezo-material and the flexibility inherent in a distributed actuator suggest that the approach shown in this paper may well be applicable to more general problems. Another significant outcome of the research was the development of a general design methodology which included both the feedforward and feedback controllers in a single dynamic state feedback controller. This coupled design procedure is far more effective than splitting the design into two distinct phases (one for feedback and one for feedforward) and ensures that the resulting controller blends the best of each control viewpoint together in a synergistic way. Furthermore, it was seen that the H2 optimal control methodololgy as implemented in mTools can be implemented quite easily in an active noise control framework. The higher dimensionality of a more complex control problem, although certainly more difficult in a quantitative sense, should pose no qualitative difficulties for the control design approach developed herein. A final contribution of this work is the demonstration of how one can experimentally track translating pressure waves by means of two discrete pressure measurements, a very useful capability for those concerned with both observing and controlling such acoustic signals. REFERENCES 1. J. L. V N and B. H. T 1995 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics to appear. Active control of transient noise transmission through a circular membrane.

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