GB2256111A - Distributed sensors for active vibration control - Google Patents

Abstract

A transducer, for example, a piezoelectric wire or sheet (Figs. 4, 7) has its response weighted so that its output is proportional to the net volume velocity of a surface across which it extends. The weighting function is preferably of quadratic form. The square of the output from the transducer is thus proportional to the sound power radiated by the surface of the body to which it is secured. Such a transducer is particularly useful in the active control of sound radiation.

Description

DISTRIBUTED SENSORS FOR ACTIVE VIBRATION CONTROL The present invention relates to transducers, in particular, to sensors for use in the active control of vibration and radiated sound.

The concept of active control of noise and vibration, that is to say, the use of a secondary applied force to cancel out or, at least, to reduce the vibration in a body caused by a primary disturbance, for example, heavy machinery or aircraft engines, is well known. The basic scheme of a typical active structural vibration control system is shown in Figure 1.The essential elements of such a control system which are intended to control the vibration of a structure, for example, a beam 10 to which a primary force is applied at point 12, are a reference sensor 14 which, directly or indirectly derives a reference signal related to the primary force that is the source of the unwanted vibration; a control actuator 16 which converts a control signal into a secondary force which acts on the structure at point 18 and an error sensor 20 which measures vibration in the beam at a third point and provides an error signal which is used to further control the operation of the control actuator 16. Typically, the error sensor is positioned so that the control actuator 16 and the point 18 at which it applies a force to the structure 10 lie between the error sensor 20 and the point 12 at which the primary vibrating force is applied to the structure 10.The control actuator 16 is operated by means of a control algorithm which varies the secondary applied force so as to minimise vibration of the structure 10. The control algorithm may be modified by an adaption algorithm operating on the output from the error sensor 20 to further minimise vibration.

In a simple active control system, the reference sensor 14, control actuator 16 and control algorithm can be designed beforehand so that the control force will always control the vibrations optimally without the need for the error sensor 20. However, in general, better vibrational control is obtained by the use of an error sensor and adaption algorithm to modify the control algorithm.

Conventional transducers employed in active vibration control systems are vibration sensors such as accelerometers, capacitance probes and laser Doppler velocimeters; vibration actuators, such as electrodynamic and hydraulic shakers; or vibrational response modifiers such as variable-rate hydraulic dampers. All of these are relatively bulky and heavy and must be mounted externally to the structure whose vibration is to be controlled. To avoid the transducers themselves materially affecting the vibration of the structure to be controlled, separate support structures are frequently required.

Over the last few years, it has become possible to use instead of these large and bulky external transducers, integrated transducers which are mounted directly on the surface of the controlled structure or, in some cases, within its fabric. Such integrated transducers are formed of materials which react to or generate mechanical strains within themselves, for example, piezoelectric, magnetostrictive or electrostrictive materials.

Whilst the remainder of this specification discusses, in particular, transducers used for sensing vibration, it will be appreciated that similar considerations apply to transducers used as actuators.

A piezoelectric material may be used to sense vibration by measuring the charge which builds up when it is subject to strain.

Alternatively, by applying a voltage to the piezoelectric material a strain will be generated in the material itself and in any structure to which it is firmly bonded.

Piezowire is a low-cost robust and easy-to-use material for constructing one-dimensional strain-integrating sensors. It is available from FOCAS Ltd., under the trade name Vibetek cable. It consists of a thin, cylindrical wire of PVDF piezoelectric polymer, with one metal electrode running along its central axis, and the other deposited on its outside, in the manner of a coaxial cable.

It is protected by an outer sheath of another polymer, and is supplied with electrical leads connected to one end. Piezowire is formed by a die extrusion process, which means that very long lengths can be made. It is activated, or "poled", by applying a large DC voltage across its electrodes at a temperature above its Curie temperature, and then cooling it with the voltage still present. When strained lengthwise, a piezowire produces an electric charge on its electrodes, proportional to the overall change in its length. The output charge is usually converted to a voltage signal by a charge amplifier, in the same fashion as other piezoelectric sensors. Piezowire could also be used as a strain-integrating actuator, by driving it with a high voltage.

In general, vibration sensors used in conventional active control systems have been point sensors responding to motion at a single point on the structure to be controlled.

A single, suitably-positioned point motion sensor can be used, or multiple sensors, positioned across the structure, if a more comprehensive picture of the vibration pattern is required.

Other forms of sensor used for active control include microphones.

These measure the radiated sound directly, and thus have the advantage that they can be used with any form of vibrating structure, over a wide frequency range, with little processing of their outputs. However, if the aim is to minimise the total far field radiation from the structure, or the sound level over an extended volume, many microphones will be needed to provide a useful error signal. Also, it is inconvenient, in many situations, to mount microphones in the region where sound is to be controlled, remote from the radiating structure. If vibration sensors mounted directly on the structure can be used to measure its sound radiation, this problem is overcome.

Acoustic intensity measurements, using microphone pairs placed close to the radiating surface, could also be used to determine the total sound power radiated by a structure. But, again, it is often inconvenient to mount microphones in this way, and many microphone pairs, distributed across the surface, would be required to give a good measure of the radiated power, if the surface vibration pattern was at all complicated.

Distributed strain-integrating sensors are an alternative means of measuring the transverse vibrations of a radiating surface and are more convenient than conventional point sensors in many applications, where their ability to be laid flat on the surface of a structure, or to be integrated into its fabric, is advantageous.

The special materials used to make integrated transducers can, in addition to being used to form point sensors also be made relatively large so that the transducer or sensor acts over an area as a distributed transducer.

A fundamental advantage of distributed strain-integrating sensors lies in their very different spatial response to the pattern of vibration on a surface, compared with point sensors. The sound power radiated by a surface depends very much on the pattern of its transverse vibrations, as well as their overall amplitude. One class of materials used frequently for transducers in active vibration control are piezoelectric materials. Two commonly used piezoelectric materials are ceramics of the PZT family and polyvinylidene fluoride (PVDF) and related polymers of PVDF. Both of these classes of materials are usually produced in thin sheets with electrodes, in the form of thin metal films, deposited on the opposite faces of the sheet.Piezoelectric materials can be fabricated in other forms, in particular, in the case of PVDF, into the so-called "piezowire", referred to above, in which a thin cylindrical wire of PVDF has a metal electrode running along its central axis and a second electrode on its outside in a cylindrical layer rather similar to the construction of a coaxial cable. For practical reasons, the wire is usually protected by an outer sheet of inert material. Piezoelectric materials can be formed in relatively large thin sheets and piezowire can be made to any desirable length. Distributed integrated piezoelectric sensors can be formed by firmly attaching a piece of piezoelectric material, with the appropriate electrodes directly to the surface of a vibrating structure, usually with adhesive. Alternatively, the piezoelectric material or piezowire can be embedded in a composite structure.

A point sensor can generally be defined as a sensor of dimensions much smaller than the "wave length" of the spatial variations in the strain being measured. In this case, the sensor effectively measures strain at a point. A distributed sensor, on the other hand, is of similar size to, or larger than, this "wave length".

A one dimensional strain integrating sensor, for example, a piezowire, has an output which is proportional to the integral over its length of the strain parallel to itself. This is equal to the change in the overall length of the piezowire due to applied strains. A two dimensional strain integrating sensor, for example, those employing a thin sheet of piezoelectric material give an output proportional to the change in their overall area, assuming that they are isotropic, that is, that they respond equally to strains in any direction in the plane of the piezoelectric sheet.

If a two-dimensional sensor is not isotropic, its output is related in a more complicated way to the applied strain pattern. If the structure is vibrating, the strain will, of course, be a function of time and, consequently, the output of the sensor will vary with time too.

A single unweighted strain-integrating sensor can give a measure of low-frequency sound radiation from certain simple structures, such as simply-supported beams and panels, because it responds only to the same modes of vibration that are significant in the structure 5 radiation response. However, its usefulness is limited.

In accordance with the invention, there is provided a transducer for use in the active control of viration in or sound radiation from a vibrating body, the active portion of the transducer being of dimensions such that it extends over substantially the whole of the body along at least one dimension thereof, the transducer response being so weighted that the signal output by or the drive signal applied to the transducer is proportional to the net volume velocity of a vibrating body on which it is mounted.

Preferably, the weighting function is of quadratic form.

Quadratically-weighted sensors are ideal at low frequencies because they effectively measure the net volume displacement or volume velocity of any surface with fixed edges and, hence, give a measure of its low frequency sound radiation. This holds irrespective of the shape of the surface and is independent of any modal analysis of the surface motion. For point motion sensors to give a reasonably accurate measure of the net volume displacement of a surface, many such sensors would be required, positioned across the surface.

The invention further provides a method for active control of vibration in or sound radiation from a vibrating body using such a sensor.

Embodiments of the invention will now be described in detail, by way of example, with reference to the drawings, in which: Figure 1 illustrates schematically a basic active control scheme intended to reduce or minimise noise or vibration; Figure 2 shows a thin Euler-Bernoulli beam with a one-dimensional strain-integrating sensor; Figure 3 shows a section of the beam of Figure 2 when deformed; Figure 4 shows a weighted piezoelectric sheet sensor; Figures 5(a) and (b) show a quadratic weighting function and its effective displacement weighting, respectively; Figure 6 shows a quadratic weighting function; Figure 7 shows an irregular planar surface to which a distributed strain-integrating sensor has been affixed; and Figure 8 is a schematic representation of an 'undetermined' active control system utilising a distributed strain-integrating sensor.

As indicated above, integrated sensors measure the spatial integral of the strain applied over their length or area, and so they have been termed "strain-integrating sensors". Where the sensor element is of similar size to, or larger than, the "wavelength" of spatial vibrations, it performs a spatial integration over the varying structural strain distribution, which is quite different to the behaviour of conventional point sensors.

This spatial integration function can profitably be exploited in active control systems, particularly if the strain can be spatially weighted before integration. As mentioned above, it is to be noted that although this description concentrates on strain-integrating sensors, strain-integrating actuators can be constructed along similar principles.

For one-dimensional strain-integrating sensors, such as piezoelectric wires, resistive strain gauges, or optical fibres, the output of the sensor is proportional to the integral over its length of the strain parallel to itself, which equals the change in its overall length due to applied strains. Suppose the unstrained sensor lies along a line from l=li to 1=12, where 1 is the spatial coordinate along the length of the sensor (the sensor need not lie along a straight line), if a strain S(1) is applied to the sensor (S(1) is parallel to the sensor at any point along it), then its output will be proportional to its length change A L, which is given by

Two dimensional strain-integrating sensors also exist; usually they employ thin sheets of piezoelectric material.If they are isotropic (i.e. they respond equally to in-plane strains in any direction), then these sensors give an output proportional to the change in their overall area A A, which is given by

where A is the sensor area, x and y are Cartesian coordinates within the sensor sheet, and Sx and Sy are strains in the x and y directions. If a two-dimensional sensor is not isotropic, its output is related to the applied strain pattern in a more complicated way.

When a strain-integrating sensor is firmly attached to the surface of a structure, or embedded within a structure, it will strain in unison with the structure, so that the variable S in the above Equations will equal the structural strain adjacent to the sensor. If the structure is vibrating, S will, of course, be a function of time, and the sensor output will be too.

Figure 2 shows a thin Euler-Bernoulli beam 30 with a one-dimensional strain-integrating sensor 32 integrated into it at a constant height, h, above its neutral axis 34 (h is measured from the centre of the sensor element). The sensor extends from x=xi to X=X2, where x is the distance along the beam, as shown in Figure 2.

The sensor could lie on the surface of the beam, or within the beam as shown above. A short section of the beam, between x and x+Ax, is deformed as the beam vibrates transversely, as shown in Figure 3.

At some instant, the transverse displacement of the beam 30 is z(x), and its spatial gradient is z'(x). Hence, the beam element is bent through an angle,n8, which, to a first order approximation, is given by d = = z'(x) - z'(x+t x) (3) There is no strain on the beam's neutral axis 34 (this is a condition of Euler-Bernoulli behaviour), but the section of sensor 32 is stretched from length x to ax+h ~o (again, this is a first order approximation).Thus, the strain in the section of sensor 32 S(x) = ht 8 = h z'(x) - z'(x+Ax) Ax Ax (4) As 4 x tends to zero, this Equation becomes S(x) = -hz"(x) (5) where z"(x) is the second spatial derivative of z(x), i.e. the curvature of the beam 30. The output of the sensor 32 is proportional to the overall change in its length, AL, which is given by

This shows that a uniform strain-integrating sensor laid along an Euler-Bernoulli beam at a constant height, h, above the neutral axis will give an output signal which depends only on h and the difference between the beam's gradients at either end of the sensor.Sometimes it may be useful to measure the difference between the gradients at two points on a beam; at other times, a different form of spatial response may be required from the strain-integrating sensor. This can be achieved by making the sensitivity of the sensor vary with distance along the beam, as discussed below.

Strain-integrating sensors can also be used to measure the transverse vibrations of thin plates by laying them on, or inside, the plate. It is assumed that the plate has an unstrained neutral surface within it, analogous to the neutral axis of an Euler-Bernoulli beam. A uniform one-dimensional strain-integrating sensor, at constant height above the neutral surface, will then integrate the plate's curvature along its length as it does on a beam, even if it does not follow a straight line across the plate.

Thus, its output is still proportional to the difference between the gradients at its two ends. An isotropic two-dimensional sensor at height h above the neutral surface will, by similar arguments to the above, integrate the two-dimensional curvature of the plate over its area. Its output is proportional to

where z(x,y) is the plate's transverse displacement, and zxx is its second derivative with respect to x, and similarly for zYY.

Real structures often do not vibrate in the idealised way assumed above; there may well be some finite strain in their neutral surface or axis. For example, a beam whose ends are held a fixed distance apart must experience a lengthening of its neutral axis as it flexes transversely. A strain-integrating sensor on such a beam will also be subject to this overall stretching, and will generate a proportionate output.

In these circumstances, the transverse motion of a beam with fixed ends can be divided into two components; an Euler-Bernoulli part, governed by bending moments, and a "stretched string" part, governed by tension along the whole length of the beam. Both components can produce a net length change in a strain-integrating sensor on the beam, but the stretched string component will be at double the frequency of the Euler-Bernoulli component. This is because tension stretches the whole beam positively both when it flexes positively (upwards) and when it flexes negatively (downwards). Hence, with single frequency vibrations, the Euler-Bernoulli part of the strain-integrating sensor output will be at the vibration frequency, and the stretched string part will appear as a second harmonic, which can be filtered out and ignored, if required.For broadband vibrations, the stretched string part of the beam's motion will cause nonlinearity of the strain-integrating sensor output, though this will, in many cases, be a small effect.

On a thin plate, any output from the strain-integrating sensor due to stretching of the neutral surface will be distinguishable from the output due to Euler-Bernoulli-type flexure, just as on a beam.

If a strain-integrating sensor is not at a uniform height above the neutral surface of the structure whose transverse vibrations it is measuring, then h in the above derivations can be replaced by h(x), or h(x,y) for two-dimensional sensors. Hence, Equation 6 becomes

and Equation 7 becomes

Effectively, the sensitivity of the sensor is spatially weighted by the variation in h, and the results of the above integrations depend very much on the form of this h-variation. It is to be noted that the spatial weighting of the sensor s sensitivity need not necessarily be obtained by varying the height of the sensor above the neutral surface; the intrinsic sensitivity of the sensor material could be varied to give the same effect.

Hence, from now on the variable h is used to represent a general sensitivity-weighting function, and # L, to represent the output of a one-dimensional strain-integrating sensor, rather than the actual length change of the sensor.

Usually, the way in which the sensor output is related to transverse displacement is of more interest than its relationship to curvature. This relationship is somewhat difficult to analyse for two-dimensional sensor weighting, but for a one-dimensional sensor, we can express iLL in Equation 8 in terms of an integral over displacement, z(x), by integrating by parts twice:

Thus, in general, ss L has terms which depend on the displacements and gradients at the ends of the sensor, plus a term which is the integral over the sensor length of the displacement, z(x), weighted by -h"(x). The factor -h"(x) can be thought of as a weighting of the sensor's transverse displacement sensitivity, which happens to equal minus one times the second derivative of the actual sensitivity weighting, h(x). Strictly speaking, Equation 10 is only valid provided h(x) is differentiable twice over the interval x=xi to X=X2. However, if there are discontinuities in h'(x) or h(x), we can represent them as Dirac 6-functions, or derivatives of 6-functions, in h"(x). The way these are interpreted, in terms of transverse displacement sensitivity, is described below in relation to some of the forms of sensitivity weighting that can be useful for active vibration control purposes.

The simplest way to spatially weight the sensitivity of a strain-integrating sensor is to remove lengths, or areas, of the sensor, so it only covers parts of the structure. Then the sensitivity weighting function, h, is zero for the uncovered areas, and the integrals over the covered areas. With piezoelectric sheets, it is sufficient to etch away one of the metal electrodes from part of the sheet, to make h equal zero in that area. If a more graded sensitivity weighting function is required, the height of the sensor above the neutral surface could be varied accordingly.

The intrinsic sensitivity of the sensor material could be varied instead; with piezoelectric sensors this can be done by varying the thickness of the piezoelectric material, and with resistive strain gauges by varying the width of the conducting strip.

A sheet of piezoelectric material fixed to or embedded in the top surface of a beam can be regarded as a one-dimensional strain-integrating sensor because all strains are parallel to the length of the beam. The sensitivity of this type of transducer can be weighted along its length by varying its width, or the width of one of its electrodes. If regions of negative sensitivity (h < 0) are required, the effective polarity of parts of the piezoelectric sheet can be reversed, by reversing the electrical connections to those parts of its two electrodes. One other technique, that could be used to give a graded weighting to the sensitivity of a piezoelectric sheet in one or two dimensions, would be to remove many small patches from one of its electrodes, right across that electrode.Provided the deactivated patches were small compared with the "wavelength" of the measured strain variations, the resultant sensitivity at any part of the sheet would be proportional to the fraction of electrode area still remaining in that part. An example of this technique is shown in Figure 4; a piezoelectric sheet sensor 40 with its sensitivity greatest at its centre, and falling off towards its edges.

Weighted strain-integrating sensors have great potential as error sensors in active vibration control systems, because they can have responses tailored to measure exactly the vibrations the system is designed to minimise, and to ignore any other vibrations.

They can measure directly quantities that would otherwise have to be measured by multiple point sensors, with processing circuitry to combine their outputs. Only one-dimensional weighting of strain-integrating sensors is discussed here; two-dimensional weighting is somewhat more complicated to analyse. Although only sensors are discussed, the responses of strain-integrating actuators can be tailored by spatial sensitivity weighting in much the same way. In active vibration control systems, such actuators could directly control the vibrations that the system is designed to minimise, without exciting any other vibrations.

A form of weighting function that is particularly useful in sound radiation control is the quadratic form illustrated in Figures 5(a) and 5(b): h(x) = b(x-xl)(x2 - x) (11) where b may be chosen to be b=4ho/(x#-xi)2 (12) so that the minimum value of h(x) is ho.

There are two negative 8-functions in -h"(x), at x=xl and X=X2, SO we expect a sensor weighted with the above h(x) to measure minus one times the sum of the displacements at its two ends.

Additionally, -h"(x) is a constant between x=xi and X=X2, SO the sensor should also measure the integral of the displacement over its length. To confirm this interpretation of -h"(x), the h(x) shown in Figure 5(a) is substituted into Equation 8. As the weighting function is differentiable twice over the interval x=xi to X=X2, we can first integrate Equation 8 by parts twice, which gives

but h(xi)=h(x#)=0, and h'(x1)=b(x#-xi), and h'(x2)= -b(xz-xi), and -h"(x) = 2b from x=xi to x=x2.Hence, the Equation above becomes

Thus, the quadratic weighting function above will give a strain-intregrating sensor that measures the integral of the transverse displacement of the structure over its length, minus the point displacements at each end.

If such a sensor was fixed along the full length of a simply-supported beam, it would simply measure the integral of the beam's displacement over its length, since the beam s end displacements are zero. Again, it would only respond to odd natural modes of the beam, because this integral is zero for even modes, so it is useful for the active control of low-frequency sound radiation from a baffled simply-supported beam. In fact, this type of quadratically-weighted strain-integrating sensor is particularly appropriate for measuring low-frequency sound radiation from any baffled transversely vibrating surface. This is because such sensors can be used to measure the integral of transverse displacement over any surface which is fixed at its edges, and, at low frequencies, the sound power radiated by a baffled surface is proportional to the square of that integral.

In the case of a planar structure, parallel to the x,y plane, and vibrating transversely such that there is no net strain in a neutral surface that lies below its outer surface (i.e. the structure is undergoing Euler-Bernoulli-type bending), the transverse displacement of the structure is z(x,y,t). If a onedimensional strain-integrating sensor is laid across it, parallel to the x-direction at y=yl, between the points x=xi and X=X2, at a height h above the neutral surface. The net length change of this sensor due to the transverse vibrations is (from Equation 6)

i.e. the sensor measures the difference between the x-gradients of the structure at the two ends of the sensor.If this sensor was spatially weighted, its output would now be proportional to

As indicated above in relation to a one-dimensional surface, x, L may not now represent the actual net length change of the strain-integrating sensor, but it is still referred to as the net length change, for convenience.

For "quadratic" weighting of the sensor, the weighting function is of the form h(x) = b(x-x#)(x#-x) (17) As indicated above, this is a "dome-shaped" function, zero at x=xi and X=X2, illustrated at Figure 6.

With this quadratic weighting, the net length change of the example sensor on a two-dimensional vibrating surface becomes

The sensor thus measures the integral of the transverse displacement of the planar structure over the length of the sensor, minus the point displacements at each end of the sensor. If the edges of the structure are fixed so that they have no transverse displacement, and the sensor is laid from one edge to another, the sensor will simply measure the integral of transverse displacement over the full width of the structure. Such a strain-integrating sensor could be used to measure the volume displacements of many types of vibrating structure.

It can also be seen that a quadratically-weighted strain-integrating sensor laid along the length of a uniform Euler-Bernoulli beam with fixed ends could measure the beam's volume displacement, if the beam's displacement is invariant across its width. We consider such a beam, of length L in the x-direction and width w in the y-direction, with transverse displacement x(x,t), which is independent of y.From Equation 18 the output of a quadratically-weighted sensor laid along the length of the beam is proportional to

If it is further assumed that the beam is vibrating at a single frequency cd , its motion can be expressed in terms of a "complex displacement", z(x), where z(x,t) = Re(z(x) erXt) (20) and the output of the sensor can be expressed as a "complex length change",##, which is related to aL by = = Renal e > -t) (21) Hence, Equation 19 becomes

so the length change of the strain-integrating sensor is proportional to the beam's volume displacement:: IL = 2b V w (23) It can be shown that, at low frequencies, the total sound power, W, radiated into the far field by a plane, baffled surface vibrating transversely is proportional to the squared magnitude of the surface's complex volume displacement. Hence, the squared magnitude of the complex length change of the above sensor will be a measure of the beam's low-frequency sound radiation. Thus

This result holds for any pattern of displacement, z(x), along the beam, as long as z(0) = #(L) = 0. A quadraticallyweighted strain-integrating sensor mounted along a baffled beam will act as a 11surface microphone" and measure the total sound radiation of the beam at low frequencies.The squared output of such a sensor would thus be an ideal error signal to use in the active control of low-frequency sound radiation from a baffled beam.

Quadratically-weighted strain-integrating sensors could also be used to measure low-frequency sound radiation from baffled surfaces with fixed edges whose transverse displacement varies with both x and y. As shown in Figure 7, the surface 50 would be divided into a number of parallel strips 50(a), (b), (c)..., and the volume displacement of each strip 50(a),(b), (c)..., would be measured separately, and the results added to give the total volume displacement of the surface 50. Each strip 50(a), (b), (c)... is assumed to have little variation in transverse displacement across its width, so its volume displacement can be measured by a quadratically-weighted sensor laid along its length, as for a beam.

The measurements of the sensors 54 on all the strips 50(a), (b), (c)..., are added by physically connecting all the sensors 54 in series, so that they formed one large strain-integrating sensor, whose net length change is the sum of the length changes of each of its sections.

The quadratically-weighted sections of the sensor 54 are connected by insensitive sections, and its output is converted to a voltage signal by a suitable electronic circuitry. As in the case of a single-length sensor on a beam, the squared magnitude of the output of this combined sensor would be a measure of the total low-frequency sound power radiated by the surface 50. The assumption that transverse displacement is invariant across the width of each strip might lead to some inaccuracy, but this could be reduced by using a larger number of thinner strips.

An active control system utilising a distributed strainintegrating sensor is shown in Figure 8. A single quadraticallyweighted distributed sensor 60 is secured to a vibrating panel 62 driven by primary source 66. The output signal e from the sensor 60 is processed by means of a control processor 64 which in turn applies control signals xl and X2 to actuators (not shown) coupled to the panel 62 so as to drive the output of the sensor 60 to zero.

As indicated above, the output of a quadratically-weighted sensor is directly related to the net volume velocity of the vibrating panel and, hence, the square of that output is proportional to the sound power radiated by the panel.

Higher order modes of the' panel whose net volume velocity is zero, will, however, still radiate power to some extent. It is, thus, desirable to minimise the excitation of these modes by driving the volume velocity to zero with the smallest possible signals xi, X2,... to the secondary actuators. This control system is 'underdetermined', as will be any similar control system with fewer sensors than actuators.

The output of the sensor 60 can be written in the form e=d + Cx (25) where d is the contribution to the error sensor from the primary source 66, C is the physical coupling matrix between the actuators and sensor 60 and x is the vector of complex input signals xi, X2,... to the secondary actuators. The control effort J can be written as XHRX (26) where H is the Hermitian transpose of the vector and R is the real part of the acoustic impedance matrix relating the complex normal velocities at a number of locations on the panel to the corresponding complex acoustic pressures at those locations.

If R is I, the control effort will equal the sum of the mean square control signals but it could also be made proportional to the mean square excitation of the higher order modes by the secondary actuators, for example. The control effort is, as suggested above, ideally to be minimised but subject to the constraint e = d + Cx = 0 (27) The exact solution to this problem can be found using a vector of complex Lagrange multipliers X and minimising a cost function of the form J = xHRx+ A,#(d+Cx)+(d+Cx)K# (28) Setting the differentials of J with respect to the real and imaginary parts of the elements of x and 99 to zero, the optimal solution for the secondary actuator signals is obtained as x0#t=#R-lCH [ CR-1CH ] -1d (29) Using the method described by Frost [O.L. Frost, Proc.IEEE 60, pp. 926-935, 19721 of using a steepest descent algorithm and then imposing the constraint (27) at every step, leads to the iterative algorithm x(k+1)= [ I-ptI-CH [ C C ] -lC ] ] x(k)-C [ C C# ] -1e(k) (30) where p is a convergence coefficient and we assume R = I for simplicity. If the transfer matrix C and the initial error e(0) = d are perfectly known, the first iteration of Equation (30) reduces to the optimal solution and the algorithm will converge in one step.

For a non-stationary system, in which e(k) changes with time, the algorithm converges fastest with p=l.

When applied to a physical system in which there is some delay between the application of the new input, x(k), and the error reaching the steady state, e(k), Equation (30) can only be applied over blocks of data, of duration greater than these delays. It is, however, possible to update the input every sample, n, if a modified form of Equation (30) is used in which x(n) is updated by an amount a(x(k+l)-x(k)) where x(k-l) is derived from Equation (30) with p=1.

This leads to the adaptive algorithm x(n+1)=(I-a)x(n)-aCH [ CC ] -l [ e(n)-Cx(n) ] (31) where a is a convergence coefficient. For a single error sensor CCH is a scalar and no matrix inversion is required.

It has been found using computer simulations that use of this algorithm to control the signals applied to the secondary actuators causes more rapid and effective convergence of the mean square control signals than do other known algorithms including Newton's method and conventional least mean squares methods.

Claims (11)

1 A transducer for use in the active control of vibration in or sound radiation from a vibrating body, the active portion of the transducer being of dimensions such that it extends over substantially the whole of the body along at least one dimension thereof, the transducer response being so weighted that the signal output by or the drive signal applied to the transducer is proportional to the net volume velocity of a vibrating body on which it is mounted.

2. A transducer according to claim 1 in which the weighting function is of quadratic form.

3. A transducer according to any preceding claim in which the active portion is of piezoelectric material.

4. A transducer according to claim 3 in which the active portion is of piezoelectric wire.

5. A transducer according to any preceding claim in which the active portion includes a plurality of active segments of weighted response joined to one another by inactive segments of uniform response so that the active segments can be secured to the surface of a vibrating body with the inactive segments projecting beyond the edges thereof.

6. A transducer for use in the active control of vibration in or sound radiation from a vibrating body, the transducer being substantially as hereinbefore described.

7. A method for providing active control of vibration in or sound radiation from a vibrating body, the method comprising securing the active portion of a transducer in accordance with any of claims 1 to 6 to the body so that the transducer extends across substantially the whole of the body along at least one dimension thereof and so that the transducer vibrates therewith, and providing a countervibration in the body so as to minimise the output from the transducer.

8. A method according to claim 7 in which the transducer is a transducer in accordance with claim 5, the active segments of the transducer being secured to the body with the inactive segments thereof projecting beyond the edges of the body.

9. A method according to claim 8 in which the active segments of the transducer are arranged to extend substantially parallel to one another.

10. A method according to any of claims 7 to 9 in which the countervibration is controlled in accordance with the algorithm defined by x(nfl)=(I-a)x(n)-aC [ CC ] -1 [ e(n)-Cx(n) ] and the variables x, n, a, C, H and e are as hereinbefore described.

11. A method for providing active control of vibration in or sound radiation from a vibrating body, the method being substantially as hereinbefore described.