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Active Structural Acoustic Control. I Plate Systems
8 Active Structural Acoustic Control. I Plate Systems
8.1
Introduction
In numerous industrial applications, structurally radiated noise is a persistent problem which is often poorly alleviated by passive means, particularly at low frequencies. The possibility of active noise control was suggested over 50 years ago (Lueg, 1936) but its implementation is a relatively recent development due to advances in fast microprocessors for digital signal processing. In this chapter we discuss the active control of sound radiation from distributed vibrating structures. Two forms of control sources are generally available. The use of acoustic control sources have been investigated by previous workers (Deffayet and Nelson, 1988 and Fuller et al., 1991). In general it has been shown that when the sound source is complex or distributed over multiple surfaces, many acoustic control sources are required in order to provide global control. An alternative approach as embodied in the research of Fuller and his co-workers (Fuller, 1985a, 1987, 1988), is to use control inputs applied directly to the structure in order to reduce or change the vibration distribution with the objective of reducing the overall sound radiation. This technique has been termed Active Structural Acoustic Control (ASAC), an abbreviation that conforms with the generally accepted terminologies of Active Noise Control (ANC) and Active Vibration Control (AVC). Figure 8.1 shows a genetic arrangement of a distributed elastic system excited by an oscillating disturbance. Sound radiation occurs as a result of the continuity of particle displacement at the interface between the structure and the surrounding compressible medium. The objective is to reduce the sound radiation. Obviously completely reducing the overall structural response with active vibration control would lead to an attenuation of the sound radiation. However, as shown later, various modes of vibration have differing radiation efficiencies and some are better coupled to the radiation field than others. This suggests that in order to reduce sound radiation, only selected modes need to be controlled, rather than the whole response. In addition, if the relative phases and amplitudes of a multi-modal response can be adjusted so that they destructively interfere in terms of radiated sound, then the radiation field may be attenuated with little change in the overall response amplitude of the system. In other words, the controlled system will have an overall lower radiation efficiency than the uncontrolled system. The above observations suggest the general arrangement of ASAC; control inputs are applied to the structure while minimising radiated pressure or pressure related variables.
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ACTIVE CONTROL OF VIBRATION
Sound radiation
I
----
--
---
"-
,,, ,, •
/,
Distr!buted
~ ~
Disturbance
\
-
I
fp"
',
',
Radiation
: +senors
,, ,
.
error
/
elastic system
,
',
\
~ ~ , ~ ' w N ~ ~ ~ s \
"~ w
Normal surface velocity
f
Control forces
Fig. 8.1 Generalarrangement for Active Structural Acoustic Control (ASAC). It should be emphasised that ASAC is not simply a matter of applying AVC. Knyazev and Tartakovskii (1967) have demonstrated that AVC can lead to an increase in radiated sound levels. Conversely as will be seen later, reduction of radiated sound is sometimes accompanied by an increase in structural vibration levels. The advantages of ASAC can be seen to be a marked reduction in the number of control channels and the control power consumed in particular applications. In addition when ASAC is implemented using piezoelectric or other induced strain transducers, a very compact and lightweight control configuration is obtained. These advantages have stimulated research in ASAC by a number of workers such as Vyalyshev et al. (1986), Meirovitch and Thangjitham (1990), Hansen and Snyder (1991), Thi et al. (1991), Naghshineh and Koopman (1991) and Thomas et al. (1990). Before we begin an outline of the application of ASAC to plate systems, it is advantageous to review some of the basic theory of structural acoustics, or how vibrating structures couple to their radiated sound fields. We then apply ASAC to harmonic radiation from plates and finish by considering transient and broadband disturbances and the active control of their associated sound radiation.
8.2
Sound
radiation
by planar
vibrating
surfaces;
the Rayleigh integral
One of the most important problems dealt with by researchers in the field of active control in recent years is that of the active suppression of sound radiation from a vibrating plane surface. The reason for the preoccupation with this topic is that it
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
225
represents the simplest idealisation of a whole class of problems of practical interest involving the radiation from, or transmission through, some form of structure, whether it is the radiation from the hull of a submarine or the transmission of sound through the fuselage of an aircraft. Before describing details of recent research into this problem, it will be useful to review briefly some of the basic techniques involved in calculating the sound field radiated by vibrating surfaces. Here we will give an outline of a subject which is dealt with at length and in considerably more depth by authors such as Junger and Feit (1986) or Fahy (1985). The analysis presented in this section is generally for harmonic motion of one mode of plate vibration. The total radiation consists of a superposition of radiation from individual modes that 'interfere' in the radiation field and this will be described in Section 8.4. The evaluation of the Rayleigh integral is probably the conceptually simplest approach to calculating the sound field radiated by an area of vibrating surface that is surrounded by an otherwise rigid infinite plane (see Fig. 8.2). The Rayleigh integral gives the complex pressure (associated with an e j~' time dependence) at a given field point p(r) in terms of the complex velocities v0(r,) associated with an elemental sources at points r s on the surface S. Thus
I ja)p°fv(rs)e-J~ dS, p(r)= s
(8.2.1)
2erR
where R = I r- r, I and v~(r,) is the component of the complex velocity normal to the surface S, while P0 is the density of the acoustic medium. This equation can be derived
p(r) \ \ \ \ \
Elemental source
~fJf I~
)<.
f f
,. f
f
Y
f \
Fig. 8.2 A rectangular plate in an infinite baffle showing nodal lines, coordinate system and an elemental sound source associated with the plate motion.
226
ACTIVE CONTROL OF VIBRATION
from the more general solution to the wave equation presented by, for example Pierce (1981) or Nelson and Elliott (1992). Essemially the integral evaluates the sum of the fields of a distribution of elemental sources, each having a complex volume velocity rO(r~) dS. As an example of the application of this theoretical approach, consider the sound field radiated by the rectangular plate illustrated in Fig. 8.2. The analysis is assisted by expressing the far-field pressure in the spherical coordinates (r, 0, q~). The classical assumption made in order to evaluate the far-field pressure is that the value of R in the exponential term e-ju~ in equation (8.2.1) is approximated by R .~ r - x sin 0 cos q~- y sin 0 sin q~,
(8.2.2)
where x and y define the coordinate position on the plate and (r, 0, q~) are the coordinates of the field point. This assumption is valid provided R~> a, b, the plate dimensions (see Junger and Feit, 1986, for a full discussion). In addition, the term R in the denominator of equation (8.2.1) can be approximated by R--r, a less stringent approximation being required for this term than for the exponential term which determines the relative phase of the contributions to the pressure at the field point from the different elemental sources. A particular form of out-of-plane vibration of a rectangular plate which, with the above assumptions, leads to an analytically tractable form of equation (8.2.1), is given by ~'(r') = W"~ sin(mZ~X)sin(-~)a
{a0}, 0 x, y, , b "
(8.2.3)
This corresponds to the complex velocity distribution associated with a simply supported plate vibrating in its (m, n)th mode as discussed in Chapter 2. Equation (8.2.1) then reduces to P(r'O'~)=jogp°~Vmne-ikr
Io Iao sin ----~-](mz~X/sin(nZCY)eJ(ax/a+~y/b)dxdy---ff-, (8.2.4)
where a = kasin0cosq~ and fl = kbsinOsincp. This integral has been evaluated by Wallace (1972) who gives the solution
p(r,O, qb)=joopo(Vmne-Jk~ ab [(-a)me-~a-a][(-1)~e-Jt~-l] 2z~r mnz~ 2 -~a'~m~'~---1 (fl/nz02-1 ' '
(8.2.5)
The far-field pressure is clearly a complicated function of the geometry and the modal integers (m, n). In the discussions of this section we will initially restrict attention to a single global measure of the far-field sound; that is the total radiated acoustic power. This can be found by integrating the far-field acoustic intensity over a hemisphere surrounding the plate. This intensity is given by
= 2pocol W,,~12 kab 2p0c0
2
COS
COS
sin
sin
z~3rmn [(a/mzO 2- 1][(fl/nzO z- 1
•
(8.2.6)
This expression is in the form given by Wallace (1972), where cos (a/2) is used when
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
227
m is an odd integer and sin (a/2) is used when m is even. Also, cos (fl/2) is used when n is odd and sin (fl/2) is used when n is even. Note that P0 and Co are respectively the density and sound speed of the medium and k - tO~Co is the acoustic wavenumber. An important feature of this result has been pointed out by Fahy (1985). If one evaluates the maximum value of this intensity, when the wavelength 2 of the sound is such that it exceeds both the 'structural trace wavelengths' of the plate vibration (i.e. such that ka ~ m~ and kb~ n~), it can be shown that the intensity produced by the vibrating plate is never greater than that which would be produced by a single 'cell' of vibration acting alone. (The cells are illustrated in Fig. 8.2; a single cell corresponds to an area bounded by nodal lines shown as dashes.) This is one of the most significant features of plate radiation; it is the interference produced by different areas of the vibrating plate which largely characterises the low frequency radiation of most interest in studies of active control. The destructive interference produced by contributions from neighbouring cells has a profound influence on the radiation characteristics, and as we shall see, it is the contribution from 'uncancelled' cells which dominate the radiation characteristics of a given mode of vibration (Maidanik, 1962). In order to quantify these effects we can evaluate the total power radiated by the (m, n)th mode which is given by integrating the far-field intensity over a hemisphere surrounding the plate. Thus the sound power is given by
0, 0
r sin 0 dO dq~.
(8.2.7)
2p0c0
Wallace (1972) has undertaken this integral numerically using equation (8.2.6) for the acoustic intensity. In order to better compare the results for different mode orders, Wallace defines a modal radiation efficiency given by am, =
H
(l fi,'mn[2)lOocoab
,
(8.2.8)
where (I Wreni2) is the temporal and spatial average modal velocity of the plate, which in this case is simply given by ll;Vm, 12/8. Some examples of the radiation efficiency curves computed by Wallace are shown in Fig. 8.3. Note the widely differing form of the radiation efficiency curves for the different modes at low frequency. These radiation efficiencies are applicable when the plate response is dominated by one mode, i.e. at resonance. Off-resonance, when more than one mode is significant, the radiated power cannot be simply calculated using individual radiation efficiencies. The results shown in Fig. 8.3 are plotted as a function of the dimensionless ratio k/k b, where kb is the structural wavenumber given by +
.
(8.2.9)
~at Thus once k/kb,> 1, the radiation efficiency of all modes becomes unity; this corresponds to the condition that the structural wavelength exceeds the acoustic wavelength and under these conditions there is no appreciable interference between the contributions from neighbouring 'cells'.
228
ACTIVE CONTROL OF VIBRATION
10-I
(m, n)
o>~ 1 0-2 ,-.
.
_
o _
t-
.~
-3
-~ 10
.
_
"o rr
10-4
165
(2, 2)
0.1
1
Non-dimensional wavenumber, k~ kb
Fig. 8.3 Radiation efficiency curves for a number of modes of a square plate (after Wallace, 1972).
The frequency at which the structural wavelength equals the acoustic wavelength in the surrounding medium is known as the critical frequency. In terms of free wavenumber it is thus defined as U = k b = ki,
(8.2.10)
where kI is the flexural wavenumber defined in Chapter 2. For a thin plate the critical frequency f~ is given by (Cremer and Heckl, 1988) 2
fc - ~ CO , 1.8CLh
(8.2.11)
where Co is sound velocity, CL is the longitudinal plate wave phase speed given in Chapter 2 and h is the thickness of the plate. The critical frequency is thus an important characteristic of structural response in terms of sound radiation. Below the critical frequency interference effects between neighboring cells are the dominant factor in determining radiation efficiency. Well above the critical frequency there is no significant interference between the radiation from cells and the radiation efficiency is thus largely independent of modal order. Wallace gives the following expressions for the radiation efficiency of different mode types in the low frequency limit when the acoustic wavelength is much greater than either of the plate structural wavelengths ( o r f ~ f~).
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
229
For m, n both odd,
Omn
'~'
2 5 m 2ner
12
(m~) 2 b + 1
20
(mzO2 b + 1
(ny~) 2
.
(n"~'~)2
•
(8.2.12)
For m odd, n even, Omn
~"
8(ka)(kb)3 f ~ - - 1 3mZn2~ 5
For m, n both even, (~mn
~
15m2n2er 5
( 4)a (
f
1
1- ~ + 1 (met) 2 b
64
(~-~)2
(8.2.13)
•
(8.2.14)
When m is even and n is odd, an identical result to equation (8.2.13) follows with m and n interchanged. Still more instructive forms of these expressions result when ka, kb e 1 (i.e. the acoustic wavelength is much larger than the plate dimensions). In that case, assuming for the moment that b = a, we have the following. For m, n both odd, 32(ka) 2 Om n ~
~
m 2n 2y./~5
(8.2.15)
.
For m odd, n even, 8(ka) 4 ~mn
~
3m2n27g5
•
(8.2.16)
For m, n both even,
2(ka) 6 am,, --
15m2n2er 5
•
(8.2.17)
The dependence of the radiation efficiency of these three mode classes on increasingly high powers of ka shows that the three classes exhibit radiation efficiencies which are respectively characteristic of monopole, dipole and quadrupole type sources. This is shown in Fig. 8.4 which illustrates the 'comer monopole' model of low frequency plate radiation. This is based on the notion of perfect cancellation by neighboring 'half-cells' of vibration, such that the only uncancelled cells appear at the comers of the plate. The relative phase between these effective 'comer monopole sources' then determines whether the new source is of monopole, dipole or quadrupole type (Maidanik, 1962). This model was used by Deffayet and Nelson (1988) to describe the effectiveness of using acoustic secondary sources to control the low frequency radiation of a rectangular plate. It was shown that the field of a monopole type mode could be adequately controlled globally using a single secondary monopole source, whereas a dipole type mode required two correctly oriented secondary sources for global control to be achieved. For control of radiation from quadrupole type modes, four appropriately located and phased secondary acoustic sources were required.
ACTIVECONTROLOFVIBRATION
230 Y +
+
+
+
~X
Y +~,,
=
+
m odd, n odd (monopole)
,~X
m even, n odd (dipole)
Y
y i
,
+
I I I . . . . . . .
I. . . . . . . I I I
~X
+
+ m odd, n even (dipole)
+
I
=
~X
m even, n even (quadrupole)
Fig. 8.4 The comer monopole model of low frequency plate radiation.
8.3
The calculation of radiated sound fields by using wavenumber Fourier transforms
Another approach for the calculation of sound fields radiated by vibrating surfaces involves working with the spatial Fourier transforms of the variables involved. The transforms used in dealing with, for example, radiation from a two-dimensional surface in the x - y plane have the form
F(kx' kY) = I~ I~ f(X' y) eJ(kxX+kyY)dx dy,
(8.3.1)
1 -.i(kxx+ kyy) f(x, y)= (270 2 I?**I?= F(kx, ky) e dkx dky.
(8.3.2)
Thus, as discussed in Section 5.9, the Fourier transforms used, rather than transforming from say time to frequency, transform from the spatial variables x and y to the wavenumber variables kx, ky (and vice-versa). The utility of these expressions can be demonstrated by first applying the transform given by equation (8.3.1) to the Helmholtz equation which governs the form of the complex pressure in a three-dimensional harmonic sound field. This is given by (V 2+ kZ)p(x, y, z) = O,
(8.3.3)
where ~r2 is the Laplacian operator. The transformed equation can thus be written as
~92 ~)2 f-,.r-,, (-~X2+ by 2
~92 2) bz 2
Y, z) e j(kxx+kyy)ax
y=o.
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
231
The derivatives with respect to x and y integrate respectively to -k2p(kx, ky, z) and -k2yP(kx, ky, z) while the derivative with respect to z can be taken outside the integral. The transformed equation then becomes (see, for example, Junger and Feit, 1986) 2
2-k2-ky
2
z) = o.
(8.3.5)
This equation has the solution
P (kx, ky, z) = A e -jk~z,
(8.3.6)
where the wavenumber kz is given by
kz = x/k 2 - k 2 - k 2
(8.3.7)
and A is an arbitrary constant. The solution to the transformed equation thus has the form of a simple plane acoustic wave propagating in the positive z-direction with the wavenumber kz. Note that if (k 2 + k2) > k 2, then kz will be imaginary and equation (8.3.6) will represent an exponentially decaying solution in the positive z direction. (The real part of the exponent in equation (8.3.6) must always be negative in order to satisfy the Sommerfeld radiation condition; see Junger and Feit, 1986, Ch. 5.) The transformation of the complex acoustic pressure by using equation (8.3.1) thus effectively decomposes the field into a sum of plane waves. The formal approach adopted in the solution of radiation problems using this technique is to also transform the boundary conditions. Thus, for example, in the case of acoustic radiation from a plane surface, the linearised equation of conservation of momentum requires that on the surface
jwpofv(x, y) +
i)p(x, y, z)
= 0,
(8.3.8)
bz
where w(x, y) is the complex velocity of the surface in the positive z direction (normal to the x - y plane) and P0 is density of the acoustic medium. The transformed boundary condition is then simply, at z = 0
jwpoW(kx, ky) + ~ P(kx, ky, z) = 0.
(8.3.9)
Thus we can use the general solution of equation (8.3.6) and determine the constant A in terms of the transformed surface velocity distribution W(kx, ky). Substitution of equation (8.3.6) into equation (8.3.9) gives
A = wpoW(kx, ky)/kz
(8.3.10)
and therefore the transformed pressure field is related to the transformed velocity field by
P(k x, ky,
z') =
wpo(V(kx, ky) -~k=z e
kz
.
(8.3.11)
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ACTIVE CONTROL OF VIBRATION
The resulting complex pressure field can then be calculated by an inverse double Fourier transformation of this result. Thus
W( x, y)e p(x, Y, z) = J_oo J_oo kz (2Jr) 2
dkx dky.
(8.3.12)
Integrals of this type can be evaluated relatively easily by using the method of stationary phase. A full account of the use of this technique in solving planar radiation problems is given by Junger and Feit (1986). An important observation given by Junger and Feit is that radiation at a particular angle corresponds directly to a particular vector wavenumber quantity of the planar structural motion. Another approach which has been used more recently is to use the Fast Fourier Transform algorithm for efficient numerical evaluation of the integral transforms. Thus for example, given a complex surface velocity distribution, the discrete Fourier transform corresponding to equation (8.3.1) is evaluated numerically in order to approximate W(k~, ky) at a series of discrete wavenumber values. Equation (8.3.11) is then used to evaluate P(kx, ky, z) at a given value of the coordinate z and the inverse transform is evaluated numerically in order to recover p(x, y, z). This is essentially the approach adopted in generalised acoustic holography (Maynard et al., 1985; Veronesi and Maynard, 1987). In that case it is also sometimes useful to make measurements of surface pressure on a plane (at z = zH say) just above the source. The general solution, equation (8.3.6), to the transformed Helmholtz equation can be used to show that if P(kx, ky, zH) is the measured wavenumber transform at z = zH, then the wavenumber transform at any other plane z is simply given by
P(kx, ky, z) = P(kx, ky, ZH) e -ik:(z - z,).
(8.3.13)
A description of the numerical implications of the use of this technique is presented by Veronesi and Maynard (1987). A further use of the wavenumber transform is that it can be used to give expressions for the total sound power radiated by a vibrating planar surface in terms of acoustical and vibrational variables measured on the plate surface. The sound power radiated by harmonic vibrations of a surface whose complex normal velocity is ~i,(x, y) and on which the complex acoustic pressure is given by p(x, y) is given by FI =
g Re
p(x, y)
_oo
(x, y) dx dy ,
(8.3.14)
which is the integral over the surface of the time averaged normal acoustic intensity. Note that Re denotes the operation of taking the real part and (*) denotes the complex conjugate. Clearly if ~i,(x, y) is non-zero over some finite region of the surface, then the limits on the integral become finite. However, in order to derive expressions for the acoustic power radiated in terms of wavenumber transforms, it is helpful to use Parseval's formula which, in this context, can be expressed as
I?ooI?oop(x, y)Cv* (x, y)dx dy= ~
4:r2
I?o. I?o. P(kx ky)(~/r~(k x, ky) dk x dky.
(8.3.15)
Using this relationship together with equation (8.3.11) to relate P(kx, k,) to IiV(k~, ky)
233
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
shows that the acoustic power output can be expressed in terms of the surface velocity transform as I W(kx, k s) 12 dkx dky . kz
H = Wpo Re 8:r 2
(8.3.16)
We now note that 1/kz is only real for acoustic wavenumbers k = vo/c that satisfy k i> a/k 2 + k~
(8.3.17)
and thus equation (8.3.16) reduces to
n OPo8
l '(kx, ~22
-
ks)I.= dkx dk s,
(8.3.18)
+k~ ~<
where the range of integration is only over those wavenumbers satisfying the inequality given by equation (8.3.17). Alternatively one may write the expression for the acoustic power in terms of the wavenumber transform of the acoustic pressure. Again using equations (8.3.15) and (8.3.11) shows that n =
1
II
2 dxx d k y ,
IP(kx) k y ) 1 2 4 k 2 - k 2 - ky
(8.3.19)
8~2wP° k~+~y -< 2 k2 where again the integration is undertaken only over those wavenumbers for which k2 + k2 ~
¢v(x)={ Wsin(mzex/a)O
0x>"
(8.3.20)
The wavenumber transform is given by W(kx) = WIo sin(merx/a)e '~'xxdx,
(8.3.21)
which results in m -jkxa
-1] W(kx) = (V (mJr/a)[(-1) e k 2 _ (mzr/a) 2
(8.3.22)
This has the modulus squared given by [W(kx) 12 --" [ W I
2
2m~r/a k ~ - (mzr/a) 2
sin2 kx a -2 mJr).
(8.3.23)
The resulting wavenumber spectrum is sketched in Fig. 8.5. Note that since the displacement is uniform in the y direction then the ky spectrum is effectively a Dirac
234
ACTIVE CONTROL O F VIBRATION
Radiating wavenumber components /a) increasing~ ,
-m]z"
-(.z)
a)
a
cO
cO
/>,
m_...~ Wavenumber,kx 8
Fig. 8.5 A typical wavenumber spectrum of a vibrating plate and the identification of radiating wavenumber components (after Fahy, 1985). delta function at the origin. This reduces the condition of equation (8.3.17) to k ~ +kx. The peak in the spectrum is given by kx = m~r/a. However, as we have seen in equation (8.3.18), only those values of I kxl k, the acoustic wavenumber, will contribute to acoustic power radiation (for the one-dimensional case). This range of wavenumbers is shown cross-hatched, for example, in Fig. 8.5. Physically speaking, if one associates with the wavenumber kx, a phase speed of propagation Cx, such that kx = cO/Cx then the condition that ] kx[ >-k amounts to requiring that ] c~ ] ~
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
235
field will occur, since the wavenumber transform will consist of two Dirac delta functions as shown in Fig. 8.6(a), outside the radiation circle. On the other hand, if the same standing wave exists only over a finite region, as for a finite plate, then the wavenumber spectrum will be a result of the convolution of the wavenumber spectrum of the infinite system and that of the 'window' associated with the finite size and shape of the structure. In this case, the wavenumber transform will have the approximate form of Fig. 8.6(b) and now radiation will occur due to the presence of supersonic components. In other words the discontinuities associated with the plate edges have led to a scattering of energy from subsonic to supersonic wavenumber motion. Once again this provides an example of the dual interpretation of structural response in terms of waves or modes. Radiation from higher order modes can be either thought of as resulting from uncancelled modal cells as shown from Fig. 8.2 or due to supersonic wave components. Both of these behaviours, due to the edge scattering, lead
v (kx)l I
I I I I I I I I I I I I I I
-kf
(a)
-k
+k
+kf
Wavenumber, kx
Supersonic region
W(kx)I
(b)
I
I
I I I
I I I
I I I I
I I I I
-kf
-k
+k
+kf
,,
Wavenumber,kx
Fig. 8.6 Schematic illustration of the wavenumber distribution on (a) an infinite onedimensional plate with subsonic bending waves, (b) a finite one-dimensional plate with subsonic bending waves.
236
ACTIVE CONTROL OF VIBRATION
to identical radiation in the spatial domain. Observations such as this, aid in understanding the behaviour of active controllers and sometimes lead to alternative approaches to the control problem. This concept will be briefly expanded upon next.
8.4
Sound power radiation from structures in terms of their multi-modal response
In Section 8.2 we have described the acoustic radiation from individual modes of a vibrating plate. In reality, however, at a given frequency of excitation, a plate will exhibit a response which is a superposition of a number of individual modal responses. Strictly speaking, the response can only be fully described by an infinite series of modes, although in practice a finite series will give a good representation; the number of modes used being dependent on the damping of the plate, the frequency of interest and the convergence of the truncated series. Thus the complex velocity of the plate surface excited at a frequency to can be described by
m=M n=N
~jcot
¢v(x, y) = Z Z (Vm~Pmn(X,y)~ ,
(8.4.1)
m=O, n=O
where Wm~ is the complex amplitude of the (m, n)th mode and ~'m~(X,y) is the mode shape function which describes the modal space dependence. Clearly in the case of the simply supported plate described in Section 8.2, ~mn(X,Y) can be expanded into separable functions such that lffmn(X, y ) = lpm(X)lpn(y), where ~m(X) and ~p~(y) are respectively given by sin(mJrx/a) and sin(nzry/b) and a and b are the plate dimensions in the x and y directions, respectively. Equation (8.4.1) can also be written in the form W(x, y ) = vgT~3(X, y),
(8.4.2)
where the vectors ,iv and lp(x, y) are defined by (8.4.3a,b) "/vT= [W01W10W~... WMN], ~pT(x, y)= [~Po(X)~Pl(y) ~P~(X)~Po(y) ~Pl(X)~Pl(Y)... ~PM(X)~pN(Y)]. An expression for the power output due to the plate response at a given frequency can be found by evaluating equation (8.3.18) which expresses the sound power output of a planar vibrating surface in terms of the wavenumber transform of the surface velocity field. Thus if we define the wavenumber transformed vector ~P(kx,ky) by
~p(kx, ky) = I~_~I?o.lp(x, y)e j(kxx+kyy) dx dy,
(8.4.4)
then the modulus squared of the wavenumber-transformed velocity field can be written as
[ W(kx, ky)[ 2= [,~¢T~)(kx, ky)12=~cH~p*(kx, ky)~T(kx, ky)*W.
(8.4.5)
Substitution of this result into equation (8.3.18) then shows that the expression for the sound power output of the plate can be written as II = ~¢HM~¢,
(8.4.6)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
237
where the matrix M is given by
M=
~op0 Re {I °°
8yg2
I;-
~P*
T (k"m' k'-')IP(kx~' kY)dkx dky}. a/k - kx - ky 2
-oo
2
(8.4.7)
2
The matrix M thus has diagonal terms which quantify the 'self' radiation resistances of the individual modes when acting in isolation, whereas the off-diagonal terms quantify the 'mutual' radiation resistances which arise as a direct result of the interference of the fields of two different modes. Thus if we characterise one mode by the integers (m, n) and another mode by the integers (m', n' ), the corresponding entry in the matrix M is given by
Mmn'm'n'
=("0/902 Rell °° I;oo
8 yg
_oo
lP*m(kx)~;(2Y21P--m'('kx)lPn'(kY)5--;5---75 ] ~fk - kx - ky dkx dky .
(8.4.8)
Using the results of equation (8.3.22) which gives the wavenumber transform of sin(mzcx/a), shows that for a simply supported plate, we can write
~ *m(kx)~m'(kx) =
tam'jr 2 fmm,(kxa), aZ[k~- (mz~/a)Z] [k 2 -(m'yc/a) 2]
(8.4.9)
where the function fmm' (k~a) is given by 2 (1 - cos kxa) 2 (1 + cos kxa) f mm,(kxa) = 2j sinxa - 2 j sin kxa
m even, m' even m odd, m' odd m odd, m' even m even, m' odd
(8.4.10)
Exactly analogous results follow for the product ~p* (ky)~pn,(ky). The expression for the entry in the matrix M due to the interference of the (m, n)th and (m', n' )th modes can thus be written as
ogpo mm'nn'275 8yg2 aZb 2
4
Mmm'nn' =
× Re
"
(8.4.11) This integral can be simplified considerably in the low frequency limit. Thus, if one assumes that kxa~mzr and kyb~ nzc, terms like (k 2- ( m ~ / a ) 2) c a n be replaced by (mzr/a) 2 in the denominator of the integrand. In addition, for small values of k,a, series expansion of the expressions given in equation (8.4.10)shows that, to leading order,
fmm,(k,:a) =
(kxa) 2...
m even, m even
4 - (kxa)2 ...
m odd, m odd
2j(kxa)...
m odd, m' even
- 2 j (kxa).. •
m even, m' odd
(8.4.12)
238
ACTIVE CONTROLOF VIBRATION
Exactly analagous results are again produced in the low frequency limit (when kyb ,~. 1) for f,,,,.(kyb). This demonstrates that the only modal interactions of significance at low frequencies are due to the interference of the radiated fields of modes for which m and m' are odd and for which n and n' are odd. In other words it is only the mutual interference of monopole type modes that significantly modifies the acoustic power output of a simply supported plate at low frequencies. This conclusion was reached by Thomas (1992) in studying the interaction of plate modes in contributing to sound power radiation by a plate under active control. This interaction can be illustrated by plotting the self-radiation efficiencies of the lower order modes of a plate, which can be written as 0"mn,mn
--
2Mmn,mn ~
(8.4.13)
,
pocoab
together with the mutual radiation efficiencies between these modes, which are defined to be
Mmnm'n' 0"mn,m'n'
"--
'
(8.4.14)
.
pocoab
The variation of these radiation efficiencies with non-dimensional frequency is shown in Fig. 8.7 for a plate with b/a =0.57 (Elliott and Johnson, 1993). At low frequencies, ka ~ 1, the plate modes with a net monopole component have the highest radiation efficiencies, a~.~ and O'31,31, although the mutual radiation term 0"1~,3~is more important than the self-radiation term 0"31,31" At higher frequencies, ka,> 1, the effect of the mutual radiation term decreases since each of the modes radiates independently and with approximately equal efficiency. 101 100
°11,11
10-1 _ >:,
o
¢-.
1
0.2
-
°11,31
(D
i_~
10 -3 -
,-.
10-4 _
o
1 0-5 tr'
°31,31 °21,21
10_ 6 10-7 _ 10-8 10-1
. . . .
I
100
.
.
.
.
.
.
,,I
101
i
|
|
,
,
,
,,
103
Non-dimensional frequency, ka
Fig. 8.7 The radiation efficiencies of selected structural modes on a rectangular plate. The self-radiation efficiency of the (1, 1), (1,2) and (3, 1) modes are designated al~.~, crzl.z~, cr3~.31 and mutual radiation efficiency of the (1, 1) and the (3, 1) modes is designated 0~.3~.
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATESYSTEMS
239
It is possible to define a set of velocity distributions on the plate which do radiate sound independently at any one frequency. Consider the eigenvalue/eigenvector decomposition of the matrix M in equation (8.4.6): M = pT~p,
(8.4.15)
where, since M is symmetric and positive definite, P is a real unitary matrix of eigenvectors and f2 is a diagonal matrix of positive real eigenvalues. The power radiated by the plate, equation (8.4.6), can now be written as II = ";vHPTf2P-;v= bHf~b,
(8.4.16)
where b = Pw is the set of structural mode amplitudes transformed by the eigenvectors of the radiation resistance matrix. Since fl is diagonal, equation (8.4.16) can be written as N
I-[ = ~
(8.4.17)
f ~ . l b . I 2,
n=0
so that the velocity distributions corresponding to this transformed set of velocity distributions radiate sound independently. The existence of this set of velocity distributions has been described by Borgiotti (1990) and Cunefare and Koopman (1991), and is implicit in the work of Baumann et al. (1991). They have been termed radiation modes by Elliott and Johnson (1993), who have plotted out their radiation efficiencies, proportional to the f2,'s, for the plate described above, as shown in Fig. 8.8. At low frequencies, ka ~ 1, the first-order (1) radiation mode is much more efficient than those labelled (2) and (3), and generally the variation of their radiation efficiency with frequency is simpler than that for the radiation efficiency of the structural modes (Fig. 8.7). The velocity distributions corresponding to these theoretical radiation modes are weak functions of the excitation frequencies. At low frequencies, ka ~ 1, however, the velocity distributions are almost independent of excitation frequency, and for the plate considered here are plotted in Fig. 8.9 (Elliott and Johnson, 1993). 101 10 0 I~
10-1
i
I
i
I
i
i
i
i
i
i
i
I
i
i
i
i
i i
i
Radiatiom n o d e ~
10"2
~
0.3
.E_O
10 .4
.(1:1 "0
10-5
rr
10..6
(3) (4)
10 .7 10 .8 10-1
i
10 °
I
I
I
I
I
I II
I
101
i
i J J i ii102
Non-dimensional frequency, ka
Fig. 8.8 The radiation efficiencies of the first six radiation modes of the rectangular plate.
240
ACTIVE CONTROL OF VIBRATION
velocity, ~,
(1)
(2)
(3)
(4)
(5)
(6) Fig. 8.9 The velocity distributions corresponding to the first six radiation modes of the rectangular plate at an excitation frequency corresponding to k a = O. 1. The most efficiently radiating velocity distribution (1) clearly corresponds to the net volume displacement of the plate, as expected. The advantage of the radiation mode approach is that, by using Fig. 8.8, it allows one to quantify the extent to which other velocity distributions are also significantly radiating at any one excitation frequency. It also suggests an efficient method of sensing the velocity distributions of the plate which are most important in radiating sound, as will be discussed in Section 8.8.
8.5
General analysis of Active Structural Acoustic Control (ASAC) for plate systems
In this section we outline the basic analysis of ASAC applied to plate systems. The system studied consists of a simply supported rectangular plate positioned in an infinite baffle as shown in Fig. 8.10. Two different disturbances to the system are considered as
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
Incident pressure,Pi
241
/,,
Iii
// ~ t e d ~
pressure Pr
Y
,J v!
b2f - I - Uniforml_~ pressure
Point liD'-force
I I I I
b
y
Piezoelectricpatch
I I I I I
-
--~Ye
i
a2
x2
(c)
Fig. 8.10 Coordinate system and arrangement of a baffled rectangular plate with inputs (a) on the incident wave side, (b) on the radiated wave side and (c) configurations of the input forces. examples. Firstly, an acoustic plane wave incident on the plate at an oblique angle is taken as the noise input shown in Fig. 8.10(a). Secondly, a forcing function over a small localised area on the plate is introduced that is considered to be representative of a structural disturbance (in the limit this forcing function will be a point force) shown in Fig. 8.10(c). Both of these disturbances will excite the plate into motion resulting in radiation on the transmitted half plane of the radiated field shown in Fig. 8.10(b). As discussed in Chapter 1 and Section 8.2 the form of the disturbance has an important influence on the resultant plate motion and thus the sound field to which it is coupled. When the disturbance is a low frequency plane wave, the input phase distribution is relatively constant over the plate surface and the result is that the plate response is dominated by lower order modes that are often excited well above their resonance
242
ACTIVE CONTROLOF VIBRATION
frequencies. This behaviour leads to an overall vibration pattern with fewer nodal lines and a high radiation efficiency. For a localised structural input, the plate responds with a much richer modal distribution and higher order modes tend to dominate near their resonance frequencies. In this case at higher frequencies (however, still below the plate critical frequency) the overall radiation efficiency is lower. Thus, as demonstrated by McGary (1988), airborne disturbance inputs to structural systems will tend to radiate/ transmit higher levels of sound than structural inputs for the same frequency and total magnitude of load. In the following analysis a general procedure is outlined. The inputs are assumed to be at single frequencies and all systems (structural, acoustic and electrical) are assumed linear so that superposition of response holds. The type, number and location of the control transducers is assumed known. In this section the fluid loading is assumed small (e.g. as in air) hence the plate response can be determined using the inv a c u o equations described in Chapter 2. The following steps are taken in the general ASAC analysis: (1) An expression is derived for the response of the plate to the input or primary disturbance(s). (2) An expression is derived for the coupled radiation to the far-field from the plate due to the disturbance. (3) An expression is derived for the response of the plate to the multiple structural control or secondary inputs. (4) An expression is derived for the far-field radiation from the plate due to the structural control inputs. (5) The total far-field pressure response is found by superimposing the disturbance and control fields. (6) A quadratic cost function is formed that is based on the required observed radiated pressure field variables.t (7) Quadratic optimisation theory is used to find the optimal control inputs that minimise the cost function as outlined in Section 4.6. (The input disturbance(s) is assumed constant and known.) (8) The optimal control inputs are substituted into the relations for the total field response (i.e. far-field pressure, plate out-of-plane displacement etc.) in order to evaluate the control performance. (Note that the reduction in the cost function also provides a measure of the overall control performance in terms of the observed error variables.) Before we carry out the above steps, we first derive the basic system response equations. Note that as outlined by Pan et al. (1992b) the method can also be formulated by using transfer matrices written in terms of input and radiation impedances. In addition, as described in Section 8.8, design techniques for optimally shaping and locating the control actuators and sensors are available. Figure 8.10 shows the arrangement and coordinate system of the baffled, simply supported, rectangular thin plate excited by a harmonic disturbance pressure acting over an area of the plate. To calculate the radiated sound field, a description is required of
t One could also minimise a pressure related variable such as supersonic wavenumbercomponents.
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
243
the plate complex vibration distribution. For the simply supported thin plate, the displacement distribution is given by equation (2.11.2) in modal form as oo
oo
W(X,y, t) "- Z Z wmn sin kmx sin kny e i~t,
(8.5.1)
m=ln=l
where the eigenvalues are given by km
met =
~ ,
a
kn =
nJr b
.
(8.5.2a,b)
The plate modal amplitudes for various forms and distributions of input forcing functions have been calculated by Wang and Fuller (1991) and are given by Wren "-
Pro. ph(ooZm~-co 2)
,
(8.5.3)
where 09 are the natural frequencies, p the plate density and h is the plate thickness. The modal force, Pmn, due to the input disturbance depends upon an exact description of the external load. Gu and Fuller (1993) have studied active control of sound radiation from a plate in the presence of a heavy fluid. However, for the present discussion we limit the analysis to a light fluid loading such as air and thus radiation loading effects are ignored. Modal forces for various input disturbances acting upon an in-vacuo simply supported plate have been derived by Wang and Fuller (1991) as follows. (1) Uniformly distributed pressure. For a uniformly distributed pressure with amplitude, Q, located between coordinates a~, a2 and b~, b2, as shown in Fig. 8.10(c), the modal force is given by mn
Pmd =
4Q
m nx~ 2
(cos kma 1 -
cos
kma2)(cos knbl -
cos
knb2),
(8.5.4)
where the superscript d will hereafter denote the input disturbance. (2) Obliquely incident plane wave. An obliquely incident plane wave as shown in Fig. 8.10(a) can be described by pi(x, y, t) = Pie j°~t-jksin °i
cos q,,-jksin 0, sin ~i.
(8.5.5)
Again assuming that the fluid loading is light we can calculate the total input pressure at the plate surface from the plate blocked pressure. That is, we assume total reflection of incident waves and thus the total input pressure at the plate surface is twice the incident pressure. Using such an approach the modal force for an oblique incident plane wave has been calculated first by Roussos (1985) and later by Wang and Fuller (1991) and is given by Pare.= 8Pilmln where the coupling constants are given by
-j [m = 2
sgn(sin Oi cos q~i)
if (met)2 = [sin 0icos ~ b i ( o ) a / c ) ] 2,
(8.5.6)
244
ACTIVE CONTROL OF VIBRATION
or
mn{ 1 - ( - 1 ) m exp[-j sin Oi cos ~i(toa/c)} I n --
(mn) 2 - [sin Oi cos ~i(wa/c)] 2 if (mn) 2~ [sin Oi cos q~i(toa/c)] 2
(8.5.7a,b)
and -J sgn(sin Oi sin (])i)
rn- 2
if (n~:)2 = [sin Oi sin q~i(~ob/c)] 2, or
L-
nn{ 1 - (-1)n exp[-j sin 0 i sin qbi(wb/c)] } (net) 2- [sin Oi sin ~ i ( t o b / c ) ]
2
if (nn)2~ [sin Oi sin ~i(tob/c)] 2.
(8.5.Sa,b)
(3) Piezoelectric actuator. As discussed in Chapter 5 a rectangular piezoelectric actuator configured for pure bending is considered to consist of two wafer elements located symmetrically on each side of the plate and driven 180 ° out of phase in the d31 mode. The modal force for such a piezoelectric actuator as shown in Fig. 8.10(c) is given by Wang and Fuller (1991) as P C m = 4C°l~Pe mn~2
(kZm+
k2n)(cos k m x 1 - c o s k m x z ) ( C O S
knYl -
cos knY2)
(8.5.9)
where x~, x2, and Yl, Y2 are the coordinates of the actuator edges. The parameter Co is a constant that is function of the piezoelectric actuator/plate properties and geometries specified by C o - E b I K I, where K I is given by equation (5.6.8). The unconstrained strain, epe, of the piezoelectric element is defined by d31V l?,pe -- ~ ,
(8.5.10)
ha where d31 is the piezoelectric transverse strain constant and h a is the piezoelectric element thickness. Most importantly Epe and thus Pm~ is seen to be linearly related to V, the input complex voltage to the actuator. (4) Point force. The modal force associated with excitation by a harmonic point force of amplitude F located at (Xr, Yr) as shown in Fig. 8.10(c) is pc _
4F
ab
sin k mXf sin k,y I,
(8.5.11 )
where the superscript c denotes control force. The sound radiation caused by the vibration due to the above inputs is related to the plate velocity distribution. As outlined in Sections 8.2 and 8.3 the radiated pressure can
ACTIVI~ STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTtIM$
245
be evaluated by using the Rayleigh integral or spatially Fourier transformed variables in conjunction with the method of stationary phase. As demonstrated by Junger and Feit (1986), both mathods yield identical solutions. Roussos (1985) has derived e×pressions for the radiation from a rectangular plate using the Rayleigh integral. The radiated field shown in Fig. 8.10 (b) is given by eo
e@
p(R, 0, q~)- K Z Z
(8.5.12)
Wm, lml, e m,
m-'-I n=-I
where the radiation constant K is defined as
1[
K - =topoab exp ]to t . . . .
2r~R
c
]}
(a cos ~ + b sin ~) . 2c
(8.5.13)
The coupling constants I,, and I, for the radiated field ere identical to equations (8.5.7a,b) and (8.5.8a,b) except that the coordinate angles (0i, 'hi) am replaced by (0, ,)) which define the coordinate of the observation point in the radiated fieldand the sign in the argument of the exponential function is changed to minus. We now have all the necessary components in hand to derive the response of the total radiated field,i.e.the disturbance field (also called primary) plus the control field (also called secondary). For a feedforward control arrangement the disturbance source and the control actuators act simultaneously and am assumed Imrfectly coherent. By superposition the totalcomplex pressure can be written as p, = pp + p,,
(8.5.14)
where the subscripts t, p and s refer to total, disturbance and control pressure respectively. Using the previous relations the total pressure in the far field can be written for the following configurations. Case I. Plane wave disturbance = point control forces u~ p, = P~B + ~ F;Cj,
(8.5.15)
/-=1
where N~ is the total number of control forces, while P~ and Fj are the complex amplitudes of the disturbance wave and control forces respectively. Case I1. Localised structural disturbance = point control forces #,
p, = O.B + Z r;cj,
(s.5.16)
/=-1
where Q is the amplitude of the disturbance pressure acting over the small area specified previously. Case III. Plane wave disturbance -- piezoelectric actuators N, P,
= PIB + ~ VjCj, /=-1
where Vj is the complex control voltage applied to the jth actuator.
(8.5.17)
246
ACTIVE CONTROLOF VIBRATION
Case IV. Localised structural disturbance - piezoelectric actuators
s pt= QB + Z V~Cj, j=l
(8.5.18)
where Q is the amplitude of the disturbance pressure. In the above equations the transfer functions are defined as: plane wave disturbance, B - K
-~
ImI~,
(8.5.19a)
Imln,
(8.5.19b)
localised structural disturbance, B =K
= = y
where the appropriate form of WPmnis calculated using equation (8.5.3) with either equation (8.5.6) or equation (8.5.4) respectively; point control force, =
= -~j
Iml.,
(8.5.20a)
=
__ - ~ j
ImIn'
(8.5.20b)
piezoelectric control actuator, Cj=K
where the appropriate form of Wm~njis calculated using equation (8.5.3) with equation (8.5.11) and equation (8.5.9) respectively. In the above transfer functions, B and C are normalised with respect to their appropriate forcing function amplitudes in order to put the relationships for total response in a form which can more readily be manipulated in terms of control amplitudes. Thus in effect, B, for example, will represent a complex transfer function between the specified input pressure of an obliquely incident wave and its corresponding radiation pressure at some observation angle in the far field. Control of the disturbance field can be achieved by appropriately choosing the vector of point force amplitudes f~= [F~ F~ F~ ...]T or piezoelectric voltages v~= [V~ V~ V~... ]T in order to minimise a chosen cost function. Choice of the cost function is defined by the form of control performance required. The ideal cost function for globalt control of sound radiation is the total acoustic power radiated by the plate since this is the variable that we seek to reduce. In this case the cost function is defined as the integral of intensity flowing through a hemisphere surrounding the plate which is given by J =
1 f
Ip, 12dS
2p0c0 ' s t Here global means through an extended area or volume.
(8.5.21)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
247
and which can be described in polar coordinates as J =
2p0c0 ' 0 J0
IP,
sin 0 dO dq~.
(8.5.22)
If directional control is required then equation (8.5.22) can be modified to only attenuate power over a required directional sector and is thus written J =
1
fo2f¢2 ]p 12R2 sin 0 dO d~b, 2 p o c o ' o, ~,, '
(8.5.23)
where 01, 02, q~l, q~2 define the limits of the sector. Usually it is not possible to design a sensor such that a cost function of the form of equation (8.5.21) can be measured, and in practice microphones are often used as point error sensors. In this case the cost function is written in discrete form as Ne
1
J=
~ [pI(R~, 0 i, ~)i)12,
(8.5.24)
2poco i= 1 where N e is the number of error sensors and (Ri, Oi, ~)i) is the location of the ith error sensor microphone. Equation (8.5.24) can be written in a more accurate discrete form as 1
J =
N~
~ Ipll = A S i,
(8.5.25)
2p0c0 i= 1 where A S i is the projected area associated with the jth sensor on the hemispherical surface. In practice, however equation (8.5.24) provides a cost function which is a reasonable estimate proportional to the total radiated power as long as a suitable geometry of error sensors are chosen. Choice of number and location of error sensors is an important topic of research and will be briefly discussed later. In general, for global control, the number of point error sensors required is equal to the number of modes contributing to the overall sound power radiation. The error sensors, by analogy with time sampling theory (see, for example, Nelson and Elliott, 1992, Ch. 2), should be spaced in sectors of the radiated field defined by regions of 180 ° phase change through nodal lines in the radiation field associated with that mode. For example, Fig. 8.11 shows a hypothetical low frequency, rectangular plate, radiation pattern which is a combination of radiation from the (1, 1) and (2, 1) modes. Positioning a single error sensor at location (a) will only lead to control of the (1, 1) mode since this angle corresponds to a node in the (2, 1) radiation pattern. Positioning two sensors at locations (b) will not lead to global control since the (1, 1) and (2, 1) modes will cancel at the error sensors and reinforce in the other sector. Locations (c) shows a situation where two sensors are positioned in two sectors which are 180 ° out of phase for the (2, 1) mode. When used with two actuators, the only way the controller can attenuate the error signals is by completely attenuating each mode, thus leading to global control. Another important observation regarding the choice of point sensors is that generally the number of sensors should be equal to or greater than the number of control inputs used. If the number of error sensors equals the number of actuators then total
248
ACTIVE CONTROL OF VIBRATION
~mode
1
mode 2 + (a)
0 ensor
(b)
e error sensor
+ (c)
Fig. 8.11 Influence of error sensor position on modal radiation control. attenuation is theoretically achieved at each error sensor since the system is perfectly determined. This situation generally leads to poor attenuations at other positions especially if the number of degrees of freedom in the system exceeds the number of error sensors. If the number of control actuators used is greater than the number of error sensors then the system is underdetermined. Note, however, that Elliott and Rex (1992) have presented a methodology to ensure well-conditioned relations for underdetermined systems by introducing a control effort term into the cost function. It should again be stressed here that in ASAC, although the control action is applied directly to the structure, the cost function is derived from the far-field radiated pressure (or far-field radiated pressure-related variables). Thus, inherent in the definition of the cost function, is the natural structural acoustic coupling that relates the plate vibration to the radiated sound. This arrangement should be contrasted to the more obvious
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
249
approach in which plate vibration is directly observed and minimised. Naturally, completely reducing overall plate vibration will lead to a reduction in sound radiation. However, as will be demonstrated, this latter approach generally requires many more channels of control (i.e. number of sensors and actuators) and a much more subtle and efficient control paradigm can be implemented when far-field pressure is used as an error variable. One is thus directly observing the field variable to be controlled, which in this case is radiated sound. The following analysis for derivation of the vector of optimal control inputs is for the ideal case of minimising total radiated power and thus equation (8.5.21) is used to define the cost function. Although the following derivation is formulated for the above Case IV, it can be written in exactly the same form for the other three cases with appropriate substitution of variables. When the expression for total pressure Pt from equation (8.5.18) is substituted into the cost function definition, equation (8.5.21), it can be demonstrated that the cost function is a scalar which is quadratic in the vector of complex control voltages v, or control force amplitudes, f,. The reader is referred to the Appendix of Nelson and Elliott (1992), as well as Chapter 4 of this text, for a proof and discussion on the nature and minimum value of the quadratic cost function. It can be shown that this cost function will possess a unique minimum which will define the optimal control voltages. The minimisation procedure is based upon the setting the gradient of the cost function J with respect to the control vector v, to zero in order to find the stationary point of the quadratic form. The total complex pressure can be expressed in vector form as p, = hTq + cTv ~,
(8.5.26)
where q=[Q~Q2Q3 ...]v is the vector of complex input disturbances and h = [H~ H2 H3 ...IV is the vector of aforementioned transfer functions associated with those disturbances. For cases in which there is a single input disturbance these vectors reduce to scalars such that q=Q
(8.5.27)
h = H~.
(8.5.28)
and
In equation (8.5.26) the control transfer function vector is defined by
c = [Ci C2... Cu,]"r,
(8.5.29)
while the input control voltages or control forces can also be written in vector form as
v,=
T.
(8.5.30)
Note that a similar vector expression as equations (8.5.26) and (8.5.30) could be used for f,, the vector of complex control force amplitudes. As outlined by Nelson and Elliott (1992) it is convenient to write the cost function using matrices in a Hermitian quadratic form. For a single input disturbance the squared modulus of the total complex pressure can be written IP, [2= p,p,= v~Cv, + v~x + XHV,+
QH,H*~Q*.
(8.5.31)
250
ACTIVE
CONTROL
OF VIBRATION
In equation (8.5.31) the Hermitian matrix C is defined by C = c*c T
(8.5.32)
x = QHlC.
(8.5.33)
and the vector x is given by
The cost function as defined by equation (8.5.22) which is written in terms of total radiated acoustic power can now be specified as J = v HAv, + Vnb + bHv s +
(8.5.34)
C,
where the individual terms are now given by AN, ×U,
=
1 [2~ [,q2 [c.cT]R 2 sin 0 dO dq~ 2poco ' o Jo
1 I2, [,#2 bN'×l= 2p0c0,0 J0
c=
1012
r/o rjo,
(8.5.35)
QHlcR2 sin 0 dO dq~
(8.5.36)
sin 0 d0 d~.
(8.5.37)
2poco '
As discussed in Nelson and Elliott (1992), all the combined terms of equation (8.5.34) are scalars and as long as A is positive definite then there is a unique minimum value of J. A typical element of matrix A is defined by A~=
~. kli~ mnjI kliI mnj1" sin 0 d 0 d q~ RoCo "
k = l I=1 m--1 n--1
i = 1 , N , ; j = 1,N,.
(8.5.38)
A typical element of vector b is oo
oo
oo
1 [2=Ij/2 K1K* Z ~ Z Z QP~tlQP,~,'PktalS£nsR2sin OdO dq), s= 1,Ns.
pOCo '
k = l !=1 m = l n = l
(8.5.39) Finally the single entry of h for a single input disturbance is defined as n 1-
1 f 2=[zr/2K1K~ Z 0 dO poCo ' k=l
Z
Z
P* IP lP* D 2 sin0d0dq~ QpkllQmnl*kll*mnl*~
(8.5.40)
1=1 m = l n = l
where
w;%
QS,,j = - - , V~
wL1
Q~l = ~ . Q
(8.5.41a,b)
Note in the above equations that the quadruple infinite sums in (k, l) and (m, n) result from the multiplication of two double modal series. As discussed in Section 8.4 the cross terms are important and the additional modal indices (k, l) are employed. The
ACTIVE STRUCTURALACOUSTICCONTROL. I PLATE SYSTEMS
251
optimal solution of control strengths to minimise the cost function of equation (8.5.34) can be found using the result derived in the Appendix of Nelson and Elliott (1992) and Chapter 4 which shows that V~o= - A - l b -
(8.5.42)
Equation (8.5.42) thus defines the vector of optimal control voltages V,o for the case of piezoelectric actuators. On obtaining V,o, the minimised far-field radiated pressure can then be calculated using equation (8.5.26). The minimum of the cost function can also be calculated from Jmin = c - bHA-lb.
(8.5.43)
Equation (8.5.43) can be used to calculate the attenuation in total radiated power obtained when the control is invoked as long as enough error sensors are used to provide a reasonable estimate. Note that if there are more control actuators than error sensors then matrix A will be singular and the procedure developed by Elliott and Rex (1992) for underdetermined systems should be used. The reader is also referred to the Appendix of Nelson and Elliott (1992) for discussions on this and other aspects of finding the minimum of quadratic functions. The previous analysis enabling the derivation of the optimal control voltages can readily be applied to the different cases I, II and III outlined above with use of the appropriate variables and using similar methodology. In the next two sections, results from example applications using this analysis will be discussed.
8.6
Active control of sound transmission through a rectangular plate using point force actuators
In this section we study the active control of sound transmission through a rectangular, baffled, simply supported plate using point force actuators. Systems similar to these have been previously investigated by Fuller (1990) and Wang and Fuller (1991). All disturbance frequencies are well below the coincidence frequency of the plate which is fc_~6300 Hz. Table 8.1 presents the specifications of the steel plate, while the corresponding natural frequencies for the simply supported boundary conditions, computed using equation (2.10.3), are given in Table 8.2. The acoustic medium is assumed to be air with P0 = 1.21 kgm -3 and Co = 343 ms -~. To calculate the plate response and the corresponding radiated field it is necessary to truncate the infinite summations. In the following examples, truncating the indices k, l, m and n at a value of five (i.e. 25 modes are considered in the doubly infinite sums) was found to provide close to 0.01% error in the radiated pressure amplitude at the highest frequency considered. This choice of truncation can be seen from Table 8.2 to effectively limit the input disturbance frequency such that f < 1750 Hz. In addition, it Table 8.1
E = 207 x 1 0 9 N m pp = 7870 kg m - 3
Plate specifications. v = 0.292 h = 2 mm
a = 0.38 m b = 0.30 m
252
ACTIVI~CONTROLOF VIBRATION Table 8.2 Naturalfrequencies of the plate (Hz).
m
1
2
3
4
5
1 2 3 4 5
87.71 188.74 357.13 592.88 895.98
249.81 350.85 519.23 754.98 1058.08
519.98 621.02 789.40 1025.15 1328.25
898.22 999.25 1167.64 1403.39 1706.48
1384.53 1485.56 1653.95 1889.69 2192.79
was necessary to calculate the integrals of equations (8.5.35) to (S.5.37) and this was carried out numerically using Simpson's rule. In order to use the above equations for this case, the input control terms are replaced with those for point control forces. The plate response results presented in Sections 8.6 and 8.7 consist of the distribution of plate vibrational amplitude plotted along the ) , - b / 2 horizontal plate mid plane (see Fig. 8.10). The results, presented in decibels (dB), were normalised to the largest amplitude obtained in each figure. Radiation directivity patterns are also presented along the y-b/2 axis at a distance of R - 2 m. Although this observation point is relatively close to the plate, far-field radiation equations were used for simplicity, and thus the results also reflect the behaviour at large distances from the plate. For convenience, angular positions with a negative sign of 0 in the figures correspond to the coordinate ~ - ~ positions. For the following example the input disturbance was assumed to be a plane wave with amplitude P~-1 N m =2 incident at angles of 0~ - 45 ° , ~ - 0 e. Figure 8.12 presents the radiation directivity patterns with and without control when the excitation frequency was set to 186 Hz. Note that negative values of sound pressure level correspond to pressure magnitudes less than the reference pressure of 2 x 10 =s N m =2. From Table 8.2 it is apparent that this frequency is near the resonance frequency of the (2, 1) mode, and correspondingly, the uncontrolled radiation field appears to have the distorted version of the radiation pattern associated with a (2, 1) mode. An examination of the modal contributions confirms that the radiation field is mostly due to the (1, 1), (2, 1) and (3, 1) modes. The slight offset of the node in the radiation field is due to the presence of a monopole type radiator such as the (1, 1) mode. Applying one control force, as shown in the schematic diagram at the top of Fig. 8.12 as a black dot, leads to little reduction in radiated power. Positions of the control forces are shown to scale in the schematic figures. However, it appears that the (1, 1) radiation contribution has been controlled as the residual radiation pattern has now the characteristic dipole shape associated with the (2, 1) mode and this is confirmed by examining the modal contributions. The single, centrally located control force is unable to couple into the (2, 1) mode of the plate as shown in equation (2.11.4). Using two forces leads to a large reduction in radiated power as now the (2, 1) mode is controlled and residual field is now largely due to the (3, 1) mode contribution. On using three point force actuators positioned as shown in the schematics of Fig. 8.12, control is achievable over the (1, 1), (2, 1) and (3, 1) modes, and power reductions of the order of 67 dB are predicted.
ACTIVI~STRUCTURALACOUSTICCONTROL,I PLATI~$YSTISMS Unoontrollti¢l
I Foroti
a
llllllllll=
i1= ,
Forolill
253
3 Forollit
=III=I=I=IIIIUlIIII=I
I
48,7 (dB)
80,8 (¢IB)
I
I
i
01=_48" 0l =- O*
Power rttduotlon
o,e (aa)
5O ....... -46°
0--'0°
""':::,
I
/
/ 0
F
=50
7/
=100
.............
....... ":1:::~ ~
............ I ............. 48'1 .:=;:'"
,.................. ~
(
Illl $
/
",...,' . . . .
~
I
.::;:"......
........................ i)N /
F........"
",~:~2 ... \
[ %1
L( ( ;-; !....]i......... li '2........... (
-50
=50 =I00 -80 Sound prnmum Iovel (d8)
0
0
60
Fig, 8,12 Radiationdirectivity for different numbers of point for, o actuators, f=- 186 Hz,
Curves of the power transmission loss versus frequency for the same situations as above are plotted in Fi$, 8,13, Her~ power transmission loss is defined as (Wang, 1991) Transmission loss-~ 10 log(lli/l=I~),
(8,6,1)
where the incident power is given for an obliquely incident plane wave by the relation
hi--- Ipil~ ab cos 0~
(8.6.2)
2pete
and the radiated power is given by Jo
~,, Ip(r, 0, O)[~ r ~ sin 0 dO dO, o 2pete
(8.6.3)
whore p ( r , 0, 0) is evaluated in the far field usin8 equation (8,$,26) for the cases with and without control. Note that negative values of tr~amission lo~a are a numodoal artifact of the computer calculations; in practice transmission loss cannot be loss than zero, For the primary or uncontrolled case, the transmissiort loss cu~e ha~ a number of dips a~sociatod with the plate modal resontm,os, When one ,ontrol fo~o is u~od, it ,an be soon from Fig, 8,13 that the fall in transmission loss at the (1,1) re~onanco frequency has been eliminated as discussed above, Similarly, use of two control forces eliminates the dips at the (1, 1) and (2, 1) resonance frequencies, The figure shows that attenuation greater than 50 dB can M theoretically obtained over a frequency range of
254
ACTIVE
Uncontrolled
CONTROL
OF VIBRATION
1 Force
2 Forces
3 Forces
.................................
0i==45°~i 0° i 200
I
~'~"1
|
150
~,,,',,,
A
rn "o
o (0
100
m
tO .B
50
.B
.'~..
E
,,.
...........~~:...~ ~
\ :~"'.....
,.,,..~..-..-..-:....-.,
e--
-50
0
I
I
I
I
I
I
I
1O0
200
300
400
500
600
700
800
Frequency (Hz)
Fig. 8.13 Plate transmission loss for different numbers of point force actuators. 0 to 450 Hz with three control forces. Whether this is achieved in practice for broadband, random disturbances depends upon a number of issues such as causality, filter size, etc. as described in Chapter 4. However, these frequency domain results do define the ultimately achievable performance with the limited number of actuators used in the locations specified.
8.7
Active control of structurally radiated sound using multiple piezoelectric actuators; interpretation of behaviour in terms of the spatial wavenumber spectrum
The previous section briefly discussed results of using point force actuators as control inputs. However, there are disadvantages to using point force actuators such as their size and the need for a back reaction support. In this section we discuss a few representative examples of control of structurally radiated noise (i.e. a structural rather than an airborne input) with arrays of independently controlled piezoelectric actuators. The piezoelectric actuators were assumed to be manufactured from ceramic material with typical properties given in Table 5.1 (corresponding to G1195 material). As discussed in Chapter 5 the actuators were configured to produce pure bending in the plate. The system used for the analysis is exactly the same as that presented in the previous section except that in this case the input disturbance is assumed to act over a very small area of 40 x 40 mm approximating a localised structural input. For this case the input
255
ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS
disturbance amplitude was set to Q = 7.9 x 103 N m -~ giving an input force of 12.65 N. In the following figures the prescribed disturbance source and actuator locations and size are shown in schematics at the top of each figure, drawn to scale looking into the plate surface from the radiated field. The black rectangle represents the size and position of the disturbance source while the clear rectangles represent the size and position of the two-dimensional piezoelectric actuators. At this stage no attempt is made to optimally configure the control actuators; their selection is made on an ad hoc basis, linked with a knowledge of the acoustically significant plate modes and their response shapes. For the first case the disturbance frequency is set to 85 Hz. An examination of Table 8.2 reveals that this frequency is close to the resonance frequency of the (1, 1) mode.t Hence this case corresponds to an 'on-resonance' excitation. Figure 8.14 presents the normalised vibration amplitude distribution with and without control for four different configurations of piezoelectric actuators. In Fig. 8.14 the solid line depicts the displacement distribution of the plate only under the influence of the disturbance, and as expected is close to a (1, 1) mode shape. When the various configurations of control (1)
(2)
(3)
(4)
I
I
I
I
rn I1) "O
,m=
-50
..../.--:':~'"~.L.. -'~.-...
E
.:----, :,,
tl:l ¢...
•~
-~00 -
,.,.~,-~... -~
.dI3 >
/
t;:',, '(" #
"
.,
:
',.k/. "-
" ~1
"-,,
&, "':~, . . . . i':_..-:...... - .
"13
I
-150
i"
I,
\
|".f
',
E 0
z
-200
I
I
I
I
I
I
0
0.2
0.4
0.6
0.8
1
x/a
.............
Uncontrolled 1 Piezo
.......................... 2 P i e z o s 3 Piezos .......
4 Piezos
Fig. 8.14 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 85 Hz. 1 Since no damping is included in the model, the disturbance frequency is not set exactly to the resonance frequency.
256
ACTIVE CONTROLOF VIBRATION
actuators are applied, the vibration amplitudes are significantly reduced and the (1, 1) mode is well controlled. However, increasing the number of actuators does not lead to a significant reduction in vibration. (Note the we are using an acoustic cost function in this example, in contrast to the configurations in Chapter 6.) This effect will be discussed below. Figure 8.15 presents the radiation directivities corresponding to the cases shown in Fig. 8.14. Also presented is the total reduction in radiated acoustic power calculated from equation (8.2.7) for each case. As expected the uncontrolled field is uniform and corresponds to the monopole like radiation of the (1, 1) mode. When one control actuator is applied, reductions in radiated power of the order of 60 dB are obtained. Use of two control actuators brings a further 10 dB reduction. Further increasing the number of actuators has little effect on the radiated power until for case (4) when actuators are located off the y = b/2 line (see Fig. 8.10) and a further 30 dB of power reduction is obtained. From these cases, it is apparent that good sound control is achievable with a single actuator and increasing the number of control channels has no significant practical advantage. In general this observation is true for systems on or near resonance of an efficiently radiating structural mode such as the (1, 1) mode considered here. Figure 8.16 presents the wavenumber transform of the plate vibration calculated (1)
(2)
(3)
~5°1 ,,
(4)
I po.0°
i,,.,~° ....
................... <' -50
~ "1
,0o ............
i
0
50
!
1O0
/
/
150
Sound pressure level (dB) Power reduction (riB) Uncontrolled ............. 1 Piezo .......................... 2 Piezos ..... 3 Piezos ....... 4 Piezos
0.0
60.6 70.0 70.0
93.4
Fig. 8.1$ Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 85 Hz.
ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS
(1)
(4)
(3)
(2)
257
100 A
e~
80
"7
6O
o
~
40
o=1==,====== ` ='1'= 1' I'
==================================
(N
~
20
-20 ~ / 0 k
i
I
I
10
2O kx(m'l )
30
.............
40
Uncontrolled
1 .......................... 2 . . . . . 3 ....... 4
Piezo Piezos Piezos Piezos
Fig. 8.16 Wavenumbcr spectrum of plate response for different numbers of piezoelectric ceramic actuators, f = 85 Hz or k = 1.556 m- ~. Note values are only shown for positive kx.
using equation (8.3.1) applied to the plate out-of-plane vibration distribution (see also Wang, 1991). The free wavenumber k of the radiation field is also shown. Although the wavenumber distribution exists in two dimensions for k,. and ky, values are only presented here for the positive k~. components. As discussed in Section 8.3 only supersonic wavenumber components, where (k~ + k~)~/2~ k will radiate to the far field. When control is applied, the wavenumber spectrum of Fig. 8.16 is attenuated across the complete wavenumber range including the supersonic components. A modal decomposition of the response also shows that all of the amplitudes of modes which are well coupled to the sound radiation are reduced. Thus radiation control has been achieved by suppressing the supersonic wavenumber components, and vibration of the mode(s) which are dominantly coupled to the radiated far field by virtue of their high radiation efficiency. We term this mechanism of control modal suppression (Fuller et al. , 1991). The disturbance frequency is now increased to 128 Hz which is an off-resonance frequency located between the (1, 1) and (3, 1) modes. Figures 8.17 and 8.18 present the plate displacement distributions and corresponding radiation directivity patterns. In this case it is apparent that increasing the number of control actuators leads to a realistic increase in control performance. At this off-resonance frequency the modal density is
258
ACTIVE CONTROLOF VIBRATION
higher; therefore more modes are now contributing significantly to the cost function, thus requiting multiple actuators for high reduction in sound power. The displacement distributions of Fig. 8.17 are also interesting. Using one control actuator leads to very little overall attenuation of the plate vibration; in fact, it is increased in some locations. As the number of actuators is increased, there is also seen to be a lack of significant change in the overall amplitude of plate motion. However, examining Fig. 8.17 it can be seen that the vibration distribution under multiple control action is quite complex with many nodal lines across the plate surface. A calculation of the overall plate radiation efficiency demonstrates that it is considerably lower when control is applied (Wang et al., 1991). Thus in this case the sound attenuation shown in Fig. 8.18 is not achieved by reducing the overall plate vibration, rather the controlled response of the plate exhibits a lower overall radiation efficiency. Figure 8.19 gives the corresponding positive wavenumber distributions computed along the x axis. In this case it can be seen that when control is applied, the supersonic wavenumber components (where kx <~k) are attenuated while the subsonic, non-radiating components are increased, except for case (3). In this off-resonance case, a multiplicity of modes are contributing significantly to the radiation field. The controller modifies the dynamics of the system so that the uncontrolled modes of significance (in terms of radiation) now have differing phases
(1)
(2)
I
I
0
.."
"0
/
.X
: ..~.-.~-"~..
..?,
-40
(4)
I
./"...-S. ........ -",:,.- .......
.. . . . . . . . . . .
133
(3)
"..
,...-
-, :./......
.
E t.--
.£ ~
-80
;
"0
•
..
._-,
I
~N
\1
\
'
li!I
!i~
~ -~2o
o Z
I
I
I
I
I
0
0.2
0.4
0.6
0.8
.............
1
1
x/a Uncontrolled ............. 1 Piezo ..........................2 Piezos ..... 3 Piezos ....... 4 Piezos
Fig. 8.17 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 128 Hz.
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS (1)
(2)
,4-1 I
100
..........
(3)
259
(4)
I
I
-- 0 °
"',,.. ........_45 °
,,,,'""" 45 ° .......
rn
-~ >
50-
"'"x
, ,. ,,, , ,,.
o/ / ////"''"
i,... co
co
t,..
Q. "0
e-
O-
o
-90 -5010 0
90 °
50
0
-50
0
50
100
Sound pressure level (dB) Power reduction (dB) Uncontrolled ............. 1 Piezo ..........................2 Piezos 3 Piezos 4 Piezos
0.0 18.9 30.1 30.1 51.4
Fig. 8.18 Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 128 Hz.
and amplitudes which interact to cancel the overall radiation in the far field. The closed loop structural response now has lower overall radiation efficiency even though its overall vibrational amplitude has not been attenuated (in some cases it may also increase!). A modal decomposition of the response shows that the modal amplitudes of the dominant sound radiators are not significantly changed. This second mechanism of control, where the overall plate vibration amplitude is not significantly attenuated or sometimes increased while supersonic wavenumber components, and the associated sound radiation are reduced, we term control by modal restructuring (Fuller et al., 1991). An alternative point of view of modal restructuring proposed and analysed by Burdisso and Fuller (1992a) is that the controlled structural response has new eigenvalues and eigenfunctions which have lower radiation efficiency. Burdisso and Fuller (1992a) also demonstrate that these controlled eigenfunctions are nonvolumetric. The above work has shown that global control of planar far-field sound radiation always corresponds to a fall in the supersonic components of the plate structural wavenumber spectrum. Furthermore, as discussed by Junger and Feit (1986), the radiation pressure from a plate at a particular angle can be shown to be directly proportional to a particular plate wavenumber component. This suggests that an
260
ACTIV]~ CONTROL OF
VIBRATION
(~)
(a)
(4)
+'4:=1:: ii :--1: llllllllllllll lllllllllllll
60
~i
~,
.lilt
40 ("
~,~,~.,,=,,=~,=~,=~...... Ill/
~=
"="= == =,.,,,,=
A
tiig~qa,
\
I
__O"i o
k
I
I
I
lo
2O
30
40
kxlm =1) Urtoontrollocl ............. 1 Pie=to ..........................a Plozoe . . . . . 3 Plozoo ....... 4 Piozoo
Fig. 8.19 Wavenumber spectrum of plato response for diffaront numbors of piezoelectric ceramic actuators, f-~ 1:28Hz or k -=2.345 m =t, Note values are only shown for positive k~:. alternative control formulation for minimising sound radiation would b~ to e×press the cost function in the wavenumber domain and suppress discrete (or bands of) wavonumbors, Fuller and Burdisso (1992) have analytically formulated a wavonumber control technique that suppresses radiation towards particular anglos. Clark and Fuller (1992a) have o×porimontally and theoretically demonstrated a wavenumber domain controller that is designed to minimiso the band of wavonumbors enclosed within the supersonic circle. This approach shows much potential. However, its practical implementation is dependent upon development of realistic, time domain, wavenumbor structural sensors that work over a broad frequency range. Work of this nature has been carried out by Maillard and Fuller (1994). Finally, Fig. 8.20 presents the radiated power with and without control over a range of frequencies from 0 < 800 Hz, Peaks in the uncontrolled power are associated with modal resonances. It is apparent that good control is achieved with a single actuator for very low frequencies, where f-~ 150 Hz. However, in order to provide good broadband power reduction it is necessm~ to use an array of appropriately positioned actuators. Oood attenuation is obtained over a frequency range of 0 ~f~650 with four actuators with the exception of the peak near 330 Hz, Table 8,2 indicates that this range encompasses the resonance frequencies of 13 structural modes and thus the result demonstrates the efficiency of ASAC.
261
ACTIVE STRUCTURAL ACOUSTIC CONTROL, IPLATE SYSTEMS
(~,)
(~)
(?)
(4)
180
~=
~=0
6O
o I I
=60
I
I
I
I
I
I
I
1O0
200
300
400
BOO
600
700
800
Fmqu#noy (Hz) Unoontroll~d ............. t Plezo ..........................2 Pl~zom
..... .......
Fig, 8,20
8.8
3 Pl~zo~ 4 Pl~zo~
Radiated ~ound power for different numbor~ of piezoelectric ceramic actuator~,
The use of plezoele~ric distributed structural error sensors in ASAC
In many applicationsof ASAC the use of microphone error sensors is impracticable.In this case it is desirable to configure error sensors which provide estimates of far-field pressure when located on, or very close to, the radiatingstructure.Various desisns hays been suggested in order to implemem this concepL Baumann ct el. (1991) have simulated the use of radiationfilterswith good success to modify the structuralstates of systems m radiationstatesin feedback implememations of ASAC. Clark and Fuller (1992b) have used a model reference implementation of the feedforward LMS all~orithmin which the vibrations of the structureare driven to a reference vector (i,e. not zero) corresponding to minimum acoustic power radiation. Fuller and Burdisso (1992) have also cortsidereda paradigm based upon a wavenumber domain approach in which Sul~rsonic structural wavenumber componems am comrolled, loading to minimisation of radiated pressure at selected anglos, In thi~ ~oction we di~cu~ the u~e of rectangular piezoelectric distributed ~tructural sensors as appropriate error sensors in ASAC. In particular we are interested in implementing structural error sensors for control of sound radiation from two-
262
ACTIVE CONTROL OF VIBRATION
dimensional plates. The use of distributed strain actuators and sensors integrated directly on or into the structure in conjunction with a 'learning' type adaptive controller implies that this configuration falls into a class of systems which are part of a rapidly expanding, related field known as adaptive, smart or intelligent structures. The reader is referred to the review paper by Fuller et al. (1992) for more information on the use of adaptive structures for controlling sound radiation. Recalling equation (5.7.4) which represents the charge output of a two-dimensional element of piezoelectric material bonded to the surface of the plate, we must simply define the area of application, F(x, y), to obtain the sensor electrical response as a function of the plate response. In the case of a rectangular sensor element this function is defined as F(x, y) = [ H ( x - Xel)- H ( x - x g ) ] [ H ( y - ye) _ H ( y - y~)],
(8.8.1)
where H(-) is the Heaviside unit step function of the spatial coordinates. Substituting this expression into equation (5.7.4) and integrating over the area of the sensor yields the charge output of the rectangular sensor given by
qe(t)
= (h + She) Mm~l = Nn~lWren[e31 ma nb + e32 × (COSkny e2 - c o s k , yl) e ~_jot,
(COS
kmx2- c o s kmx~) (8.8.2)
where (x~,y~) are the coordinates of the lower left comer of the rectangular piezoelectric element (viewed from the front of the plate), (x~, y~) are the coordinates of the upper fight comer of the sensor and h e is the sensor thickness. An examination of equation (8.8.2) reveals that the sensor charge output is composed of summation over the modal contribution. The degree to which a given mode contributes to the total sensor output is proportional to modal amplitude Wm, and is also related to modal order (m, n) as well as the sensor size and position. Thus different sensor shapes and positions can be used to selectively observe modes or combinations of modes. In order to modify the quadratic optimisation procedure, the output of the sensor is used as an error signal. Thus the control cost function is modified to be Ne g =
Z Iv; I
(8.8.3)
i=1
where it is assumed that voltage output of the sensor Ve is proportional to the charge generated by the s e n s o r qe and N e is the number of independent sensors used. As a preliminary example, we study the use of two distributed PVDF structural sensors shown approximately to scale in Fig. 8.21. The shape of these sensors is based upon the observation that, in general, odd-odd plate modes (e.g. (1, 1), (3, 1), etc.) are more efficient radiators than the even-even modes (e.g. (2, 2), (4, 4) etc.) at frequencies well below the critical frequency of the plate (see Section 8.2). Substituting the form of the sensor shapes of Fig. 8.21 in equation (8.8.2) reveals that the sensor denoted PVDF2 will only observe the odd modes in the x direction, while PVDF1 will only observe the odd modes in the y-direction. Thus if the outputs of both sensors are minimised then the structural motion of the efficiently radiating modes should be minimised (rather than minimising the total response) and this should lead to a fall in radiated sound. Figure
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
263
Plate
J
::::::::
iiiiiiii
iiii iiiii
piezo 1
~iliii
~
iiii
/ PVDF2
300 mm NJNI --- Disturbance
!i!iiiiii :i:i:i$
PVDFI
iiiii!iii iiiil/
Piezo 2
iiiii --------~ X
380 mm
P,
Fig. 8.21 Simply supported rectangular plate with piezoelectric ceramic actuators and piezoelectric PVDF sensors.
8.22 shows a typical analytical result for a f r e q u e n c y o f 349 Hz for the plate system considered in the previous sections. In this e x a m p l e the plate is excited by a point force located at x = 240 m m and y = 130 m m and two piezoelectric actuators were used for control, as shown in Fig. 8.21 (see Fig. 8.10 for coordinate specifications). The piezoelectric sensors were assumed to be m a n u f a c t u r e d f r o m polyvinylidene fluoride ( P V D F ) material with typical material properties given in Table 8.3. A p p r o x i m a t e dimensions and locations o f the sensors are s h o w n to scale in Fig. 8.21. Also s h o w n in Fig. 8.22 are the controlled results w h e n three error microphones were used that were located at mid plane (i.e. y = b/2) radiation angles o f 0 = +45 °, 0 ° and - 4 5 ° and at 0
0o
/
//
/ /
,,",-" ...... , ",, ,, .... ° , -,
,,'" /
k\
\
"'""
X
"
I
-90 ° 80
90 ° 60
40
20
0
20
40
60
80
Radiated sound pressure level (dB)
Fig. 8.22 Radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).
264
ACTIVE CONTROL OF VIBRATION Table 8.3 Typical piezoelectric polymer sensor properties (PVDF). E, = 2 x 109 N m -2 p,= 1.78 x 103 kgm "3 h,. = 0.16 mm
e3~= 65.3 x I0 "3 C m e32= 38.7 x I0 -3 C m e36=0
-2 -2
R = 2 m. The results show that the PVDF structural error sensors provide of the order of 10-15 dB sound reduction. It should be noted here that simply minimising the vibration at two points often leads to sound radiation increases, as demonstrated in the experiments of Metcalf et al. (1992). The control performance can be seen to be not as good as using microphones and this suggests that the P V D F sensors were not correctly placed to properly observe the structuralmotion associated with radiation. Various strategies have been developed for designing P V D F structural sensors in order to correctlyweight the structuralmodes for sound radiationcontrol (see the work of Clark et al., 1992a). The work by Wang et al. (1991) and Clark and Fuller (1992c) has demonstrated that for higher modal densities of response, optimising the shape and location of the control transducersis of the same order of importance as increasingthe number of channels of controlin terms of obtaining global reduction.In one particular example examined by Clark and Fuller (I992c) for a SISO system with relativelyhigh modal density,the optimal actuatorlocationwas found to be near the comer of the plate where itcould effectivelycouple into multiple structuralmodes. The optimal sensor was a long narrow strip located at the bottom edge of the plate where it could observe structuralmotion associatedwith the edge radiationphenomena discussed in Section 8.2. The resultsof both Wang et al. (199I) and Clark and Fuller (I992c) suggest that for more complex, realisticsystems the actuator and sensors will have to be optimally located to obtain reasonable performance with a low number of transducers.The work of Clark et al. (1992a) also demonstrates that extreme care must be taken in accurately manufacturing and positioningthe distributedsensor shape. Section 8.4 has demonstrated how the sound power radiation of a plate can be described in terms of a set of velocitydistributionson the structurewhose sound power radiation is independent of the amplitudes of the other velocity distributions.These velocity distributionsare termed 'radiationmodes' and are obtained using an eigenvalue decomposition as described in Section 8.4. The form of these radiationmodes suggest an approach for designing distributedstructuralsensors which are shaped (as described in Chapter 5) to respond only to the velocity distributionscorresponding to these radiation modes. Such an approach is only strictlyvalid in the low frequency region ka,t I since at higher frequencies the shapes of the radiation modes depend more strongly on the excitation frequency. Nevertheless both Snyder et al. (1993) and Johnson and Elliott (1993) have developed and successfully tested distributedP V D F sensors designed to observe these radiationmodes for the active control of structurally radiated sound. As noted in Section 8.4, by far the most important radiation mode at low frequencies corresponds to the net volume displacement of the plate (see mode (I) of Fig. 8.9). Differentiatingthe signal from the sensor which detected this distribution would give the net volume velocity of the surface, and a P V D F distributed sensor which measures the volume velocity was originallysuggested for A S A C by Rex and Elliott (1992). The shape of the resultantsensor is seen to be almost identicalto that obtained by the design procedures of Clark et al. (I992a).
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
265
Burdisso and Fuller have pursued a different but related approach to the design problem for distributed structural sensors and actuators in ASAC. Burdisso and Fuller (1992a,b) demonstrated that feedforward-controlled radiation from a structure can be described in terms of new eigenvalues and eigenfunctions for the controlled system. Using this formulation Burdisso and Fuller (1994b) developed an eigenassignment design technique by which the actuator and sensor shapes can be designed so as to cause the closed loop system to behave in a controllable manner. In order to minimise sound radiation from a structure excited by multiple frequencies, Burdisso and Fuller (1993, 1994a) used the eigenassignment procedure to design actuator and sensor locations and shapes so that controlled structural modes were radiators of low acoustic power (and thus had a very low radiation efficiency at frequencies well below the plate critical frequency) and the closed loop resonances (i.e. at the closed loop eigenfrequencies) were detuned from the excitation frequencies. The controlled structural mode shapes were shown to be independent of frequency. The procedure enables simultaneous design of the sensor and actuator shapes and position. Analysis and experiments with this procedure demonstrated high, global attenuation of structurally radiated sound over a wide band of frequencies.
8.9 An example of the implementation of feedforward ASAC In this section we briefly discuss a typical arrangement of ASAC in order to illustrate how such systems are implemented in practice. Figure 8.23 shows a schematic layout of the experimental fig and associated control system. The structural system consists of a baffled simply supported plate located in an anechoic chamber. The disturbance to the plate is provided by a point force actuator mounted on the plate through a force transducer and driven by a steady state single frequency. Control inputs are achieved by two piezoelectric ceramic actuators bonded to the plate in the required positions (actuator in this case implies two symmetrically located piezoelectric wafer elements driven 180° out of phase, as discussed in Chapter 5). Two different sensor arrangements are employed; either three microphone error sensors located at (R, 0, ¢~) coordinates of (2m, +45 °, 0°), (2m,-45 °, 180°) and (2m, 0 °, 0 °) or two PVDF strip sensors attached to one side of the plate are used. Figure 8.21 shows the locations and relative size of the piezoelectric actuators and sensors. The configuration corresponds to the analysis of the Section 8.8. Note that for graphical convenience we use 0 = ±45 ° to indicate ¢~= 0, 180°. A signal generator was used to create the harmonic disturbance and the signal was amplified to drive the shaker. The same signal was passed through two adaptive filters to provide the control signals to the piezoelectric actuators. The coefficients of the adaptive filters were updated at each time step using the multi-channel version of the filtered-x LMS algorithm outlined in Chapter 4, in order to drive the error signals to a minimum. Note that, as discussed in Chapter 4, the reference signal has to be prefiltered with estimates of the transfer functions (G~, G~2 etc.) from each actuator to each sensor. These transfer functions are measured at the frequency of interest before carrying out the experiments. As the tests are for a single frequency of excitation only two coefficients are needed in the adaptive and fixed FIR filters. The instantaneous values of the error signals are also required in the update equation to compute the instantaneous gradient estimate. These variables are used to adapt the filter coefficients in the filtered-x LMS update equation presented in Section 4.7.
266
ACTIVE CONTROL OF VIBRATION ) Reference input Disturbance ~__ Simply supported plate shaker 1 - ~ ~ ' - "
1~ ~
PZTactuat°r II
/
r Adaptive filterA
l error sensors I .
.
.
.
.
.
[~apt,ve,,,ter Fixed filters •.-,"~' Gll I ~~1
~i ,MS h/ q algorithmP= I P--q G21 ~ i G22 I - ~ ~-~ G2z
,~11[
LMS E I 1algorithm IZ i r~-
Fixed filters
Fig. 8.23
Schematic layout of controller and test rig for an ASAC experiment.
In order to evaluate the control performance, the radiated field was measured with a movable microphone traverse centred on the plate and with a radius of 2m (the radius being limited by the dimensions of the anechoic chamber). The output of the traverse thus provides radiation directivity plots in the b/2 mid plane of the plate. The magnitude of the disturbance force was also measured in order to provide an absolute comparison between theory and experiment.
0
0o
/')
-9oo 80
i, 60
40
X\
"
, 20
\
/, 0
";,i ,,X 9oo 20
40
60
80
Radiated sound pressure level (dB)
Fig. 8.24 Measured radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).
267
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
Figure 8.24 shows experimental results for a frequency of 349 Hz. When the results are compared to the theoretical predictions of Fig. 8.22 (with a disturbance amplitude set to that measured in the experiment) reasonable agreement is apparent in terms of shape, absolute values and trends for the different types of sensor. This brief example is intended to introduce the reader to the practical implementation of ASAC. Further details of such experiments can be found in the work of Clark and Fuller (1992a). Smith et al. (1993) have also extended the above application to control of broadband plate radiation using integrated piezoelectric actuators and sensors.
8.10
Feedback control of sound radiation from a vibrating baffled piston
Hitherto in this chapter, attention has been restricted to feedforward control of sound radiation from structures where there is a reference signal available which characterises the primary disturbance. Here we begin a discussion of the potential for feedback control of sound radiation from structures where the excitation is neither deterministic nor can be detected at some time well before it excites the structure (a turbulent boundary layer for example). An important contribution to this field of study is that made by Baumann et al. (1991) who introduced the concept of a radiation filter which enables the evaluation of the sound power radiated from a vibrating surface directly from the properties of the surface vibration. The concept is most easily explained with reference to a simple SDOF model of a vibrating piston in an infinite baffle that is illustrated in Fig. 8.25. This model has been used by Thomas and Nelson (1994a) in order to investigate the potential for feedback control of stiff lightweight structures when used as a secondary 'trim' in passenger aircraft. Such stiff light panels can be designed in order to have elastic modes with a relatively high natural frequency such that the control problem is considerably simplified. The use of this type of structure has already shown some benefits in feedforward control of low frequency sound transmission (Thomas et al., 1993a; Cameal and Fuller, 1993) and shows promise when feedback control is applied (Thomas and Nelson, 1994b). Using the model illustrated in Fig. 8.25, the equation of motion of the rigid circular piston can be written as
Mff~(t) + Cvg(t) + Kw(t) = fp(t) + fs(t),
(8.10.1)
where M is the mass of the piston, C and K are the damping and stiffness of the suspension and fp(t) and fs(t) are the 'primary' disturbance and 'secondary' control force respectively. This equation can be written in matrix form as [ ~"= ('!( t]) [-C/M1
-K~M ] (,)LW(,) r ~i, ]+[1/oM]fs(,)+ [l/oM]fp(,) ,
(8.10.2,
which is thus in the state space form (see Section 3.6) ~(t) = Ax(t) + Bu(t) + Av(t).
(8.10.3)
The equation of motion is thus 'forced' by a vector of external inputs v(t) which, in this case, is simply the scalar primary force input fp (t). If it is assumed that this input is simply Gaussian white noise, then we can again apply LQG regulator theory (outlined
268
ACTIVI~CONTROLOF VIBRATION
Random oxoltmtlon, v (t)
Radiation fllt,r ~
Control foroe, u (t )
g(t)
Diaplaooment, w (t)
Fig, 8,25 A circular, baffled piston mountod on a spring and a damper, The radiation filter has an input given by the piston velocity and an output whoso moan ~quarod output is the acoustic power radiated, in Section 7,10) and find the control input that, for o×ample, minimizes a quadratic cost function such as that specifi¢d in ~quation (7,10,4), As an e×ample, Thomas and N~l~on (1994a) cho~o to minimize tho cost function y-- ~[¢~(t) + a~(t)], (s,~o,4) which is thu~ tho ~um of tho moan squared piston voloeity and tho moan ~quamd ~eondary for¢o woightod by tho factor a, Thi~ factor ponali~ the control 'effort' u~od, Tho r~ult~ (Fig, 8,26), which chow the apaetral dandify of the piston volocity when control is appliod, iUuatrat~ the dop~nd~ne~ on th~ ehoic~ of the w~ighting factor a, Clearly, th~ lower the value of a, the higher the value of the control gain (~e Seetion~ 7,10 and 7,11), Minimization of moan ~quaro velocity, however, does not imply minimization of acoustic power radiated*, A direct measure of the acoustic power can b~ d~dueod from a knowledge of th~ piston velocity by pa~ing thi~ signal through a radiation filter who~o timo avoragod ~quar¢d output i~ o×ttctly equal to the acoustic power radiated, Defining the output of the filt~r to b~ ~(t), the acoustic power can b~ written as
l~ = E[~(t)],
(s, ~o,~)
If the filter i~ a~sumod to havo a frequency r¢~pon~o function density of the acoustic power radiatod i~ given by
s.cw)--- s..(,o)-- I
I
G(jto),
then th~ ~p¢ctral
(s.ao.6)
t Minimizationof pistonaccelorationwouldof cour~oprovid~~ b~tter~pproxim~tionto the minimizationof ~ou~fl~ poworr~di~ted,but ~h~u~eof th~ volo~ity~ign~lher~help~explainth~ ~on~optof thQr~diationfiltor,
ACTIVE STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTI~MS
269
=10 open loop 1 xlO =3
=16 t
=20
~
=25
1 xlO =6
=30
1 xlO 4
~=35 =40
1 xlO =8
=45 0
0,2
0,4
0,8
0,8
1
1,2
1,4
1.8
1,8
2
Fill, 8.26 Optimal reduction in piston velocity for values of a between 10 =-~and 10 =~, Results show the ratio of the spectral density of the piston velocity to the spectral density of the input disturbance force. where S,o~(co) is the spectral density of the piston velocity signal. The spectral density of the acoustic power is a real, even and positive function of frequency. This implies that we can find an approximation to S,,(~o) that is in the form of a rational spectrum. That is, we can write
N(coa) S~(co) - DCwa) ,
(8.10.7)
where N(co a) is a polynomial of order q in ¢o~ and D(co 2) is a polynomial of order p in ma. A simple example of a rational spectrum is given by S.(co)
1
-
~ ,
(8.10.8)
o)4+a
where a must be real since S, (co) is real. Now note that if we assume the piston velocity signal to be white noise of unit spectral density (i.e. S,~,~(¢a)= 1) then S,(w)---I G(jco)I a, the modulus squared of the radiation filter. We now write S,(co) as the product of two factors such that
S,,(o)) = O(jo))O* (jo))-- O(joa)O(-jo)).
(8.10.9)
Thus, for example, we could express equation (8.10.8) in the form I
I
S~(oa),, (b+jco) (b-jm)
I
I
(b*+ja~) (b*-jm)
(810.101
where b is the complex number given by b= (a/4) ~14+j(a/4) ~14.The spectral factor G(jco) is simply l/[(b+jco)(b* +jco)] which thus specifies the frequency response function of the radiation filter. This also implies that we could express the transfer
270
ACTIVE CONTROLOF VIBRATION
function of the radiation filter as the more general Laplace transform G(s)= 1/[(b + s) (b* + s)] where the Laplace variable s = cr +jw. It turns out that this type of decomposition, known as spectral factorisation, is generally applicable to rational spectra since the power spectrum must be a real, even, non-negative function of o9. This subject is dealt with in more detail by Van Trees (1968) who points out that the pole-zero plot of G(s) G(-s) must have the following properties. (1) Symmetry about the jw axis (otherwise S~(w) would not be real). (2) Symmetry about the cr axis (otherwise S,(w) would not also be even). (3) Any zeros on the jw axis occur in pairs (otherwise S~(w) would be negative for some value of co). (4) There are no poles on the j to axis. As a direct consequence of these properties we can, in general, find G(s) such that all its poles and zeros are in the left half of the s plane (or <0), while G(-s) has all its poles and zeros in the fight half of the s plane (or>0). Since G(s) has all its poles and zeros in the left half of the s plane, it is guaranteed to be a stable, causal filter. Furthermore, it is minimum phase and will have a stable inverse (see Nelson and Elliott, 1992, Chapter 3). Thomas and Nelson (1994a) used a least squares method to adjust the coefficients of a rational Laplace transform with sixth-order numerator and denominator in order to match the well known variation with frequency of the acoustic radiation resistance of a rigid circular piston. The radiation resistance curve (which exhibits the frequency dependence of S~(w) when the piston velocity is a white noise signal) is shown in Fig. 8.27. The pole-zero plot of S~(s)= G(s) G(-s) shown in Fig. 8.28 provides a fit to this curve to within 0.01% up to ka = 2.5, where a is the piston radius and k = tO~Co. Note that the symmetry of the pole-zero plot complies with conditions (1) and (2) given above. Taking the left half plane poles and zeros shown in Fig. 8.26 to define G(s) one can use one of several techniques to find a state space realisation of the radiation filter with this transfer function (see, for example, Maciejowski, 1989). Such a realisation can be written as il(t) = Asrl(t) + B sw(t),
(8.10.11)
z(t) = Cg11(t ) + Dg'/v(t).
(8.10.12)
Thus rl(t) is the vector describing the 'states' of the radiation filter while z ( t ) = z(t), the scalar output of the radiation filter and "/v(t)= W(t) is its input. This realisation can then be combined with the state space model of the piston dynamics given by equation (8.10.3) such that
[A 0][x,t,] ["0] [o]
il(t ) = Bs),
Ag
n(t)
z ( t ) = [Dg¥ where ¥= [1
+
u(t) +
x(t)] ' Cg] ~l(t)
v(t),
(8.10.13)
(8.10.14)
0]. This state space model has an output z(t) whose time-averaged
271
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
1.2
I
I
I
I
I
I
I
I
I
I 0.4
I
I
I
I
I
I
I
0.2
0.6
0.8
1 ka
1.2
1.4
1.6
1.8
0.8 0 tO O
Q. 0.6
a:
0.4 0.2 00
Fig. 8.27
2
The radiation resistance of a plane circular baffled piston of radius a. I
I
I
I
I
I
o
I
I
I
I 3
! 4
o
21
#0
-
X
-1 -2
--
-3--
-4
o
--
-5 -5
I -4
I -3
I -2
o I -1
I 0
I 1
I 2
a a/c O
Fig. 8.28 The pole-zero diagram for the rational Laplace transform approximation found for the radiation resistance of a plane circular baffled piston. Poles are represented by x and zeros by O. Note that there are a pair of zeros placed close to the origin. squared value is equal to the acoustic power radiated. Therefore the optimal control that minimises the cost function
J= E[zT(t)z(t) + auX(t)u(t)]
(8.10.15)
will minimise acoustic power radiated plus a factor a times the mean square effort used in the control. Thomas and Nelson (1994a) have computed the solution to this problem and the results are illustrated in Fig. 8.29. These clearly show the difference between the
272
ACTIVE CONTROL OF VIBRATION
130
I
I
~2ol
I
loop
110 Vibration control
~100
Radiation control
,o "0
8o
7O 8o
5O
40 30
0
0.5
1.O /ca
1.5
2.0
Fig. 8.29 Soundpower radiated by a plane circular piston without feedback (open loop), with optimal feedback minimising the squared velocity of the piston (vibration control) and with optimal feedback minimising sound power radiated (radiation control). The mean squared velocity of the primary excitation is 0.083 m s "2. results produced by choosing the control that simply minimises vibration and those produced by choosing the control that minimises acoustic power radiation. The results shown in Fig. 8.29 are for the same value of time-averaged control power used in the two cases (which used two different values of a). The frequency weighting provided by the radiation filter's representation of the radiation resistance curve is clearly evident.
8.11
Feedback control of sound radiation from distributed elastic structures
In this section we discuss the active control of sound radiated from a structure exhibiting multi-modal response using feedback control, following the approach of Baumarm et al. (1991) introduced in the previous section for an SDOF system. It was shown in Section 8.4 that the acoustic power radiated by a harmonically excited structure, whose velocity distribution is described by the sum of contributions from N structural modes, can be written in frequency domain form as (8.11.1) l-IC~) = ~'"C~)MC~)~(~), where ~ ( ~ ) is the vector of N complex structural mode velocity amplitudes, and M(o~) is a matrix of self and mutual radiation resistances (which is real and symmetric) while superscript H denotes the Hermitian operator. We can factorise M (~) into the form G" (~)G (~), so that the radiated power can also be written as (8.11.2)
273
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
Defining the vector of transformed modal amplitudes G(co),~(to) to be z(to), equation (8.11.2) can be written as FI (oj) -- ZH(CO)Z(0~).
(8.11.3)
The acoustic power radiation due to any one element of z(to) is thus independent of that due to each of the other elements of z (to) (Borgiotti, 1990; Cunefare, 199 I). This is in contrast to the power radiation due to each structural mode amplitude, which is generally dependent on the amplitudes of many other structural modes, as accounted for by the off-diagonal mutual radiation terms in M(~). This procedure has been discussed in more detail in Section 8.4. If the structure is subject to a transient excitation, the total radiated acoustic energy can then be written as E = [" ",VH(~)M(w) W(co) dto ,/ 0
~
(8 11.4) '
where . ( t o ) is the Fourier transform of the vector of the waveforms of the structural mode amplitudes ",V(t). The evolution of these time histories can be described in state space form by defining a state vector x(t) (as described in Section 3.6) which includes the velocities ~(t), and, for example, the displacements w(t) associated with each of these velocities, which then obey the state space equation :¢(t) = Ax(t) + Bu(t),
(8.11.5)
where A is the state matrix and u(t) is the vector of inputs to the system due to the secondary forces. The radiated energy equation (8.11.4) can also be written, using equation (8.10.3) as E
= I"0
ZH(~0)Z(0J)do~P
(8.11.6)
where at each frequency we again have z(m) - G (m),,V(m)
(8.11.7)
and G(to) can now be identified as the matrix of frequency response of the 'radiation filters' which operate on the structural mode amplitudes to give each of the elements in z(to). Baumann et al. (1991) demonstrate that these 'radiation filters' can be chosen to be causal, and thus realisable (to arbitrary accuracy) in state space form. The procedure to be adopted first requires finding a rational Laplace transform approximation to the elements of M(to). The algorithm provided by Francis (1986) can be used to carry out the necessary spectral factorisation. Note, however, that the algorithm can only be used provided that the rational Laplace transform approximation to M(to) is also positive definite. The state equations of this array of radiation filters can be written as t'(t) = Acr(t ) + Bcx(t ),
(8.11.8)
z(t) = Ccr(t) + Dcx(t ),
(8.11.9)
in which r(t) is the vector of state variables of the radiation filters, and x(t) is the vector of state variables for the structure, as defined above. These two sets of state
274
ACTIVE CONTROL OF VIBRATION
variables can be combined to give an overall state variable model for the radiating system -
[A 0][x]+[.] Ba
Aa
r
0
u,
(8.11.10)
with a corresponding output equation z=[Dc
CG][X].
(8.11.11)
Now that the complete system, including sound radiation, has been cast in state space form, the standard tools of 'modem' control theory can be used to calculate the optimal LQG feedback gains, as described in Section 3.10. The cost function which must be minimised is of the form Jrad = Io [zT(t)Z(t) + auT(t)u(t)] dt,
(8.11.12)
where the first term is the total sound energy radiated by the structure expressed as an integral over time, which must be equal to the frequency domain form, equation (8.11.6), by Parseval's theorem. The second term in equation (8.11.12) must be introduced to make the equations soluble, and may be physically interpreted as being proportional to the control effort, i.e., the sum of the squared inputs to the secondary sources. The optimal set of full state feedback gains which minimises this cost function is the solution of a steady state Riccati equation (see Section 3.10), and the resulting feedback control law can be expressed in the form u(t) = K ~ x ( t ) ,
(8.11.13)
where x = [x TrT] T is the full state vector, and K rad is the optimal feedback gain matrix. Baumann et al. (1991) present some numerical results of the effect of such optimal feedback control of sound radiation on a beam excited by a short duration pulse. The 1 m x 0.125 m steel beam was assumed undamped, clamped at both ends and supported in an infinite baffle. For simplicity it was assumed that the structure and its sound radiation could be accurately modelled by using the first three structural modes only. A single secondary actuator was located half way along the beam, and driven by each of the full state variables via the feedback gain matrix defined by equation (8.11.13). The resulting velocities of the three structural modes are shown in Fig. 8.30(a) (Baumann et al., 1991, Fig. 4). It can be seen that the first (volumetric) mode, mode 1 and the third mode, mode 3, are most strongly controlled using this feedback strategy. This is because the second mode, mode 2, acts as an acoustic dipole at frequencies close to its natural frequency and is thus not an efficient sound radiator. Controlling this mode will thus have little influence on the total acoustic radiated energy. In contrast to this radiation control strategy, the optimal feedback problem can also be solved to suppress the mechanical vibration of the beam. This can be achieved by using the LQG theory described above to minimise a cost function of the form Io [xT(t)X(t) + a2uT(t)U(t)] dt,
(8.11.14)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
275
0.025 O.02
(a)
0.015 ~,
,, :':
0.01!
/I
"~ o.oo5i
i~/: !
~o
i~.
0 i -0.005
'..\~
! ! i /i
i
.--.
, i
..
"..v
" ""--- ' " -
' '....
: :
"~:............ v
"
/
:~ -o.ol i 'i~ -0.015 -
0 •02 ] /
-0.025 /
i
0.02 ~
.
0.015 t [
i~/(~i
i
i
.
i
.
t
.
t
.
~ Mode 2 ..... Mode 3 t i
I
/ / J
. (b)
o0 ii ii i i i
>"
!t i i :
i I. i/!
o ~ ii £,
-ooo5
i i
.~ :. :~!
i i
'i
!'" !
:
.z i
. :: "
~i
i
i ii'ii
il
i
~..
!,~i
"~
!/i
i i
ii
!
-0.015 ~ 1 / ii ! :'
V
-001:
:-
~,:
i
"
:i
~!
ii
:.
.".,
"~ ---
l
I 0.1
""'.
"
"--
""
- -
-0.02 / 0
~--
..
"'~
Mode 1
~
Mode 2 ..... Mode 3
t 0.2
i 0.3
i 0.4
t 0.5
i 0.6
i 0.7
i 0.8
t 0.9
1
Time (sec)
Fig. 8.30 Velocities of the three structural modes of the beam modelled by Baumann et al. (1991) to demonstrate state feedback control of (a) sound radiation and (b) structural vibration. which results in a feedback law of the form u(t) = KV~bx(t).
(8.11.15)
In the results presented by Baumann et al. (1991), the weighting on the effort term in equation (8.11.14), a z, was adjusted to make the total energy used by the controller
276
ACTIVI~ CONTROL OF VIBRATION 70
~
I
I
I
I
00
=
80
=
40
=
3° /
.10 0
~,
....
i 50
.....
i 100
....
:7
i 160
-,.,, .......
J 200
,
260
Frequenoy (Hz) Fig. 8.31 Totalradiatedpower without statefeedback (====) and with statefeedback (= ==) two control forces placed at x - 0.1 m, y -- 0.1 m and x - 0,25 m, y - 0.125 m with a - 10=°. Results presented from Thomas and Nelson (1993) for the predicted reduction in sound power radiated from an aluminium plate excited by a turbulent boundary layer, The plate was simply supported, measured 0.5 x 0.25 m and was 1 mm thick. A structural damping factor of 0.01 was assumed.
equal to that used in the simulation above. Figure 8.30(b) shows the velocities of the three modes modelled in the computer simulation when implementing feedback control of vibration (equation (8.11.15)). In this case, all three structural modes have been controlled to approximately the same degree, with the result that the third mode, in particular, rings for considerably longer than in Fig. 8.30(a). Baumann et al. (1991) state that the total acoustic energy radiated when using feedback control to suppress vibrations (Fig. 8.30(b)) was 38% greater in these simulations than when feedback control was used explicitlyto suppress radiation (Fig. 8.30(a)). A similar formulation has also been used to analyse the feedback control of sound radiation from a structure excited by random disturbances (Baumann et al., 1992). Thomas and Nelson (1993) have also used Baumann's theory to examine the feasibility of providing active control of the sound power radiated from a simply supported flexible plate excited by a turbulent boundary layer. An example of the results they derived is shown in Fig. 8.31 which demonstrates the reduction of sound power radiated that can be in principle achieved with optimal feedback control for an aluminium panel typical of those used in aircraft fuselage construction. The results presented are for a specificchoice of 'effortweighting' a in the cost function used; the reductions produced were found to be crucially dependent on this choice and thus the control gains used, More detailsare presented by Thomas and Nelson (1995).
8.1
Introduction
In numerous industrial applications, structurally radiated noise is a persistent problem which is often poorly alleviated by passive means, particularly at low frequencies. The possibility of active noise control was suggested over 50 years ago (Lueg, 1936) but its implementation is a relatively recent development due to advances in fast microprocessors for digital signal processing. In this chapter we discuss the active control of sound radiation from distributed vibrating structures. Two forms of control sources are generally available. The use of acoustic control sources have been investigated by previous workers (Deffayet and Nelson, 1988 and Fuller et al., 1991). In general it has been shown that when the sound source is complex or distributed over multiple surfaces, many acoustic control sources are required in order to provide global control. An alternative approach as embodied in the research of Fuller and his co-workers (Fuller, 1985a, 1987, 1988), is to use control inputs applied directly to the structure in order to reduce or change the vibration distribution with the objective of reducing the overall sound radiation. This technique has been termed Active Structural Acoustic Control (ASAC), an abbreviation that conforms with the generally accepted terminologies of Active Noise Control (ANC) and Active Vibration Control (AVC). Figure 8.1 shows a genetic arrangement of a distributed elastic system excited by an oscillating disturbance. Sound radiation occurs as a result of the continuity of particle displacement at the interface between the structure and the surrounding compressible medium. The objective is to reduce the sound radiation. Obviously completely reducing the overall structural response with active vibration control would lead to an attenuation of the sound radiation. However, as shown later, various modes of vibration have differing radiation efficiencies and some are better coupled to the radiation field than others. This suggests that in order to reduce sound radiation, only selected modes need to be controlled, rather than the whole response. In addition, if the relative phases and amplitudes of a multi-modal response can be adjusted so that they destructively interfere in terms of radiated sound, then the radiation field may be attenuated with little change in the overall response amplitude of the system. In other words, the controlled system will have an overall lower radiation efficiency than the uncontrolled system. The above observations suggest the general arrangement of ASAC; control inputs are applied to the structure while minimising radiated pressure or pressure related variables.
224
ACTIVE CONTROL OF VIBRATION
Sound radiation
I
----
--
---
"-
,,, ,, •
/,
Distr!buted
~ ~
Disturbance
\
-
I
fp"
',
',
Radiation
: +senors
,, ,
.
error
/
elastic system
,
',
\
~ ~ , ~ ' w N ~ ~ ~ s \
"~ w
Normal surface velocity
f
Control forces
Fig. 8.1 Generalarrangement for Active Structural Acoustic Control (ASAC). It should be emphasised that ASAC is not simply a matter of applying AVC. Knyazev and Tartakovskii (1967) have demonstrated that AVC can lead to an increase in radiated sound levels. Conversely as will be seen later, reduction of radiated sound is sometimes accompanied by an increase in structural vibration levels. The advantages of ASAC can be seen to be a marked reduction in the number of control channels and the control power consumed in particular applications. In addition when ASAC is implemented using piezoelectric or other induced strain transducers, a very compact and lightweight control configuration is obtained. These advantages have stimulated research in ASAC by a number of workers such as Vyalyshev et al. (1986), Meirovitch and Thangjitham (1990), Hansen and Snyder (1991), Thi et al. (1991), Naghshineh and Koopman (1991) and Thomas et al. (1990). Before we begin an outline of the application of ASAC to plate systems, it is advantageous to review some of the basic theory of structural acoustics, or how vibrating structures couple to their radiated sound fields. We then apply ASAC to harmonic radiation from plates and finish by considering transient and broadband disturbances and the active control of their associated sound radiation.
8.2
Sound
radiation
by planar
vibrating
surfaces;
the Rayleigh integral
One of the most important problems dealt with by researchers in the field of active control in recent years is that of the active suppression of sound radiation from a vibrating plane surface. The reason for the preoccupation with this topic is that it
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
225
represents the simplest idealisation of a whole class of problems of practical interest involving the radiation from, or transmission through, some form of structure, whether it is the radiation from the hull of a submarine or the transmission of sound through the fuselage of an aircraft. Before describing details of recent research into this problem, it will be useful to review briefly some of the basic techniques involved in calculating the sound field radiated by vibrating surfaces. Here we will give an outline of a subject which is dealt with at length and in considerably more depth by authors such as Junger and Feit (1986) or Fahy (1985). The analysis presented in this section is generally for harmonic motion of one mode of plate vibration. The total radiation consists of a superposition of radiation from individual modes that 'interfere' in the radiation field and this will be described in Section 8.4. The evaluation of the Rayleigh integral is probably the conceptually simplest approach to calculating the sound field radiated by an area of vibrating surface that is surrounded by an otherwise rigid infinite plane (see Fig. 8.2). The Rayleigh integral gives the complex pressure (associated with an e j~' time dependence) at a given field point p(r) in terms of the complex velocities v0(r,) associated with an elemental sources at points r s on the surface S. Thus
I ja)p°fv(rs)e-J~ dS, p(r)= s
(8.2.1)
2erR
where R = I r- r, I and v~(r,) is the component of the complex velocity normal to the surface S, while P0 is the density of the acoustic medium. This equation can be derived
p(r) \ \ \ \ \
Elemental source
~fJf I~
)<.
f f
,. f
f
Y
f \
Fig. 8.2 A rectangular plate in an infinite baffle showing nodal lines, coordinate system and an elemental sound source associated with the plate motion.
226
ACTIVE CONTROL OF VIBRATION
from the more general solution to the wave equation presented by, for example Pierce (1981) or Nelson and Elliott (1992). Essemially the integral evaluates the sum of the fields of a distribution of elemental sources, each having a complex volume velocity rO(r~) dS. As an example of the application of this theoretical approach, consider the sound field radiated by the rectangular plate illustrated in Fig. 8.2. The analysis is assisted by expressing the far-field pressure in the spherical coordinates (r, 0, q~). The classical assumption made in order to evaluate the far-field pressure is that the value of R in the exponential term e-ju~ in equation (8.2.1) is approximated by R .~ r - x sin 0 cos q~- y sin 0 sin q~,
(8.2.2)
where x and y define the coordinate position on the plate and (r, 0, q~) are the coordinates of the field point. This assumption is valid provided R~> a, b, the plate dimensions (see Junger and Feit, 1986, for a full discussion). In addition, the term R in the denominator of equation (8.2.1) can be approximated by R--r, a less stringent approximation being required for this term than for the exponential term which determines the relative phase of the contributions to the pressure at the field point from the different elemental sources. A particular form of out-of-plane vibration of a rectangular plate which, with the above assumptions, leads to an analytically tractable form of equation (8.2.1), is given by ~'(r') = W"~ sin(mZ~X)sin(-~)a
{a0}, 0 x, y, , b "
(8.2.3)
This corresponds to the complex velocity distribution associated with a simply supported plate vibrating in its (m, n)th mode as discussed in Chapter 2. Equation (8.2.1) then reduces to P(r'O'~)=jogp°~Vmne-ikr
Io Iao sin ----~-](mz~X/sin(nZCY)eJ(ax/a+~y/b)dxdy---ff-, (8.2.4)
where a = kasin0cosq~ and fl = kbsinOsincp. This integral has been evaluated by Wallace (1972) who gives the solution
p(r,O, qb)=joopo(Vmne-Jk~ ab [(-a)me-~a-a][(-1)~e-Jt~-l] 2z~r mnz~ 2 -~a'~m~'~---1 (fl/nz02-1 ' '
(8.2.5)
The far-field pressure is clearly a complicated function of the geometry and the modal integers (m, n). In the discussions of this section we will initially restrict attention to a single global measure of the far-field sound; that is the total radiated acoustic power. This can be found by integrating the far-field acoustic intensity over a hemisphere surrounding the plate. This intensity is given by
= 2pocol W,,~12 kab 2p0c0
2
COS
COS
sin
sin
z~3rmn [(a/mzO 2- 1][(fl/nzO z- 1
•
(8.2.6)
This expression is in the form given by Wallace (1972), where cos (a/2) is used when
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
227
m is an odd integer and sin (a/2) is used when m is even. Also, cos (fl/2) is used when n is odd and sin (fl/2) is used when n is even. Note that P0 and Co are respectively the density and sound speed of the medium and k - tO~Co is the acoustic wavenumber. An important feature of this result has been pointed out by Fahy (1985). If one evaluates the maximum value of this intensity, when the wavelength 2 of the sound is such that it exceeds both the 'structural trace wavelengths' of the plate vibration (i.e. such that ka ~ m~ and kb~ n~), it can be shown that the intensity produced by the vibrating plate is never greater than that which would be produced by a single 'cell' of vibration acting alone. (The cells are illustrated in Fig. 8.2; a single cell corresponds to an area bounded by nodal lines shown as dashes.) This is one of the most significant features of plate radiation; it is the interference produced by different areas of the vibrating plate which largely characterises the low frequency radiation of most interest in studies of active control. The destructive interference produced by contributions from neighbouring cells has a profound influence on the radiation characteristics, and as we shall see, it is the contribution from 'uncancelled' cells which dominate the radiation characteristics of a given mode of vibration (Maidanik, 1962). In order to quantify these effects we can evaluate the total power radiated by the (m, n)th mode which is given by integrating the far-field intensity over a hemisphere surrounding the plate. Thus the sound power is given by
0, 0
r sin 0 dO dq~.
(8.2.7)
2p0c0
Wallace (1972) has undertaken this integral numerically using equation (8.2.6) for the acoustic intensity. In order to better compare the results for different mode orders, Wallace defines a modal radiation efficiency given by am, =
H
(l fi,'mn[2)lOocoab
,
(8.2.8)
where (I Wreni2) is the temporal and spatial average modal velocity of the plate, which in this case is simply given by ll;Vm, 12/8. Some examples of the radiation efficiency curves computed by Wallace are shown in Fig. 8.3. Note the widely differing form of the radiation efficiency curves for the different modes at low frequency. These radiation efficiencies are applicable when the plate response is dominated by one mode, i.e. at resonance. Off-resonance, when more than one mode is significant, the radiated power cannot be simply calculated using individual radiation efficiencies. The results shown in Fig. 8.3 are plotted as a function of the dimensionless ratio k/k b, where kb is the structural wavenumber given by +
.
(8.2.9)
~at Thus once k/kb,> 1, the radiation efficiency of all modes becomes unity; this corresponds to the condition that the structural wavelength exceeds the acoustic wavelength and under these conditions there is no appreciable interference between the contributions from neighbouring 'cells'.
228
ACTIVE CONTROL OF VIBRATION
10-I
(m, n)
o>~ 1 0-2 ,-.
.
_
o _
t-
.~
-3
-~ 10
.
_
"o rr
10-4
165
(2, 2)
0.1
1
Non-dimensional wavenumber, k~ kb
Fig. 8.3 Radiation efficiency curves for a number of modes of a square plate (after Wallace, 1972).
The frequency at which the structural wavelength equals the acoustic wavelength in the surrounding medium is known as the critical frequency. In terms of free wavenumber it is thus defined as U = k b = ki,
(8.2.10)
where kI is the flexural wavenumber defined in Chapter 2. For a thin plate the critical frequency f~ is given by (Cremer and Heckl, 1988) 2
fc - ~ CO , 1.8CLh
(8.2.11)
where Co is sound velocity, CL is the longitudinal plate wave phase speed given in Chapter 2 and h is the thickness of the plate. The critical frequency is thus an important characteristic of structural response in terms of sound radiation. Below the critical frequency interference effects between neighboring cells are the dominant factor in determining radiation efficiency. Well above the critical frequency there is no significant interference between the radiation from cells and the radiation efficiency is thus largely independent of modal order. Wallace gives the following expressions for the radiation efficiency of different mode types in the low frequency limit when the acoustic wavelength is much greater than either of the plate structural wavelengths ( o r f ~ f~).
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
229
For m, n both odd,
Omn
'~'
2 5 m 2ner
12
(m~) 2 b + 1
20
(mzO2 b + 1
(ny~) 2
.
(n"~'~)2
•
(8.2.12)
For m odd, n even, Omn
~"
8(ka)(kb)3 f ~ - - 1 3mZn2~ 5
For m, n both even, (~mn
~
15m2n2er 5
( 4)a (
f
1
1- ~ + 1 (met) 2 b
64
(~-~)2
(8.2.13)
•
(8.2.14)
When m is even and n is odd, an identical result to equation (8.2.13) follows with m and n interchanged. Still more instructive forms of these expressions result when ka, kb e 1 (i.e. the acoustic wavelength is much larger than the plate dimensions). In that case, assuming for the moment that b = a, we have the following. For m, n both odd, 32(ka) 2 Om n ~
~
m 2n 2y./~5
(8.2.15)
.
For m odd, n even, 8(ka) 4 ~mn
~
3m2n27g5
•
(8.2.16)
For m, n both even,
2(ka) 6 am,, --
15m2n2er 5
•
(8.2.17)
The dependence of the radiation efficiency of these three mode classes on increasingly high powers of ka shows that the three classes exhibit radiation efficiencies which are respectively characteristic of monopole, dipole and quadrupole type sources. This is shown in Fig. 8.4 which illustrates the 'comer monopole' model of low frequency plate radiation. This is based on the notion of perfect cancellation by neighboring 'half-cells' of vibration, such that the only uncancelled cells appear at the comers of the plate. The relative phase between these effective 'comer monopole sources' then determines whether the new source is of monopole, dipole or quadrupole type (Maidanik, 1962). This model was used by Deffayet and Nelson (1988) to describe the effectiveness of using acoustic secondary sources to control the low frequency radiation of a rectangular plate. It was shown that the field of a monopole type mode could be adequately controlled globally using a single secondary monopole source, whereas a dipole type mode required two correctly oriented secondary sources for global control to be achieved. For control of radiation from quadrupole type modes, four appropriately located and phased secondary acoustic sources were required.
ACTIVECONTROLOFVIBRATION
230 Y +
+
+
+
~X
Y +~,,
=
+
m odd, n odd (monopole)
,~X
m even, n odd (dipole)
Y
y i
,
+
I I I . . . . . . .
I. . . . . . . I I I
~X
+
+ m odd, n even (dipole)
+
I
=
~X
m even, n even (quadrupole)
Fig. 8.4 The comer monopole model of low frequency plate radiation.
8.3
The calculation of radiated sound fields by using wavenumber Fourier transforms
Another approach for the calculation of sound fields radiated by vibrating surfaces involves working with the spatial Fourier transforms of the variables involved. The transforms used in dealing with, for example, radiation from a two-dimensional surface in the x - y plane have the form
F(kx' kY) = I~ I~ f(X' y) eJ(kxX+kyY)dx dy,
(8.3.1)
1 -.i(kxx+ kyy) f(x, y)= (270 2 I?**I?= F(kx, ky) e dkx dky.
(8.3.2)
Thus, as discussed in Section 5.9, the Fourier transforms used, rather than transforming from say time to frequency, transform from the spatial variables x and y to the wavenumber variables kx, ky (and vice-versa). The utility of these expressions can be demonstrated by first applying the transform given by equation (8.3.1) to the Helmholtz equation which governs the form of the complex pressure in a three-dimensional harmonic sound field. This is given by (V 2+ kZ)p(x, y, z) = O,
(8.3.3)
where ~r2 is the Laplacian operator. The transformed equation can thus be written as
~92 ~)2 f-,.r-,, (-~X2+ by 2
~92 2) bz 2
Y, z) e j(kxx+kyy)ax
y=o.
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
231
The derivatives with respect to x and y integrate respectively to -k2p(kx, ky, z) and -k2yP(kx, ky, z) while the derivative with respect to z can be taken outside the integral. The transformed equation then becomes (see, for example, Junger and Feit, 1986) 2
2-k2-ky
2
z) = o.
(8.3.5)
This equation has the solution
P (kx, ky, z) = A e -jk~z,
(8.3.6)
where the wavenumber kz is given by
kz = x/k 2 - k 2 - k 2
(8.3.7)
and A is an arbitrary constant. The solution to the transformed equation thus has the form of a simple plane acoustic wave propagating in the positive z-direction with the wavenumber kz. Note that if (k 2 + k2) > k 2, then kz will be imaginary and equation (8.3.6) will represent an exponentially decaying solution in the positive z direction. (The real part of the exponent in equation (8.3.6) must always be negative in order to satisfy the Sommerfeld radiation condition; see Junger and Feit, 1986, Ch. 5.) The transformation of the complex acoustic pressure by using equation (8.3.1) thus effectively decomposes the field into a sum of plane waves. The formal approach adopted in the solution of radiation problems using this technique is to also transform the boundary conditions. Thus, for example, in the case of acoustic radiation from a plane surface, the linearised equation of conservation of momentum requires that on the surface
jwpofv(x, y) +
i)p(x, y, z)
= 0,
(8.3.8)
bz
where w(x, y) is the complex velocity of the surface in the positive z direction (normal to the x - y plane) and P0 is density of the acoustic medium. The transformed boundary condition is then simply, at z = 0
jwpoW(kx, ky) + ~ P(kx, ky, z) = 0.
(8.3.9)
Thus we can use the general solution of equation (8.3.6) and determine the constant A in terms of the transformed surface velocity distribution W(kx, ky). Substitution of equation (8.3.6) into equation (8.3.9) gives
A = wpoW(kx, ky)/kz
(8.3.10)
and therefore the transformed pressure field is related to the transformed velocity field by
P(k x, ky,
z') =
wpo(V(kx, ky) -~k=z e
kz
.
(8.3.11)
232
ACTIVE CONTROL OF VIBRATION
The resulting complex pressure field can then be calculated by an inverse double Fourier transformation of this result. Thus
W( x, y)e p(x, Y, z) = J_oo J_oo kz (2Jr) 2
dkx dky.
(8.3.12)
Integrals of this type can be evaluated relatively easily by using the method of stationary phase. A full account of the use of this technique in solving planar radiation problems is given by Junger and Feit (1986). An important observation given by Junger and Feit is that radiation at a particular angle corresponds directly to a particular vector wavenumber quantity of the planar structural motion. Another approach which has been used more recently is to use the Fast Fourier Transform algorithm for efficient numerical evaluation of the integral transforms. Thus for example, given a complex surface velocity distribution, the discrete Fourier transform corresponding to equation (8.3.1) is evaluated numerically in order to approximate W(k~, ky) at a series of discrete wavenumber values. Equation (8.3.11) is then used to evaluate P(kx, ky, z) at a given value of the coordinate z and the inverse transform is evaluated numerically in order to recover p(x, y, z). This is essentially the approach adopted in generalised acoustic holography (Maynard et al., 1985; Veronesi and Maynard, 1987). In that case it is also sometimes useful to make measurements of surface pressure on a plane (at z = zH say) just above the source. The general solution, equation (8.3.6), to the transformed Helmholtz equation can be used to show that if P(kx, ky, zH) is the measured wavenumber transform at z = zH, then the wavenumber transform at any other plane z is simply given by
P(kx, ky, z) = P(kx, ky, ZH) e -ik:(z - z,).
(8.3.13)
A description of the numerical implications of the use of this technique is presented by Veronesi and Maynard (1987). A further use of the wavenumber transform is that it can be used to give expressions for the total sound power radiated by a vibrating planar surface in terms of acoustical and vibrational variables measured on the plate surface. The sound power radiated by harmonic vibrations of a surface whose complex normal velocity is ~i,(x, y) and on which the complex acoustic pressure is given by p(x, y) is given by FI =
g Re
p(x, y)
_oo
(x, y) dx dy ,
(8.3.14)
which is the integral over the surface of the time averaged normal acoustic intensity. Note that Re denotes the operation of taking the real part and (*) denotes the complex conjugate. Clearly if ~i,(x, y) is non-zero over some finite region of the surface, then the limits on the integral become finite. However, in order to derive expressions for the acoustic power radiated in terms of wavenumber transforms, it is helpful to use Parseval's formula which, in this context, can be expressed as
I?ooI?oop(x, y)Cv* (x, y)dx dy= ~
4:r2
I?o. I?o. P(kx ky)(~/r~(k x, ky) dk x dky.
(8.3.15)
Using this relationship together with equation (8.3.11) to relate P(kx, k,) to IiV(k~, ky)
233
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
shows that the acoustic power output can be expressed in terms of the surface velocity transform as I W(kx, k s) 12 dkx dky . kz
H = Wpo Re 8:r 2
(8.3.16)
We now note that 1/kz is only real for acoustic wavenumbers k = vo/c that satisfy k i> a/k 2 + k~
(8.3.17)
and thus equation (8.3.16) reduces to
n OPo8
l '(kx, ~22
-
ks)I.= dkx dk s,
(8.3.18)
+k~ ~<
where the range of integration is only over those wavenumbers satisfying the inequality given by equation (8.3.17). Alternatively one may write the expression for the acoustic power in terms of the wavenumber transform of the acoustic pressure. Again using equations (8.3.15) and (8.3.11) shows that n =
1
II
2 dxx d k y ,
IP(kx) k y ) 1 2 4 k 2 - k 2 - ky
(8.3.19)
8~2wP° k~+~y -< 2 k2 where again the integration is undertaken only over those wavenumbers for which k2 + k2 ~
¢v(x)={ Wsin(mzex/a)O
0
(8.3.20)
The wavenumber transform is given by W(kx) = WIo sin(merx/a)e '~'xxdx,
(8.3.21)
which results in m -jkxa
-1] W(kx) = (V (mJr/a)[(-1) e k 2 _ (mzr/a) 2
(8.3.22)
This has the modulus squared given by [W(kx) 12 --" [ W I
2
2m~r/a k ~ - (mzr/a) 2
sin2 kx a -2 mJr).
(8.3.23)
The resulting wavenumber spectrum is sketched in Fig. 8.5. Note that since the displacement is uniform in the y direction then the ky spectrum is effectively a Dirac
234
ACTIVE CONTROL O F VIBRATION
Radiating wavenumber components /a) increasing~ ,
-m]z"
-(.z)
a)
a
cO
cO
/>,
m_...~ Wavenumber,kx 8
Fig. 8.5 A typical wavenumber spectrum of a vibrating plate and the identification of radiating wavenumber components (after Fahy, 1985). delta function at the origin. This reduces the condition of equation (8.3.17) to k ~ +kx. The peak in the spectrum is given by kx = m~r/a. However, as we have seen in equation (8.3.18), only those values of I kxl k, the acoustic wavenumber, will contribute to acoustic power radiation (for the one-dimensional case). This range of wavenumbers is shown cross-hatched, for example, in Fig. 8.5. Physically speaking, if one associates with the wavenumber kx, a phase speed of propagation Cx, such that kx = cO/Cx then the condition that ] kx[ >-k amounts to requiring that ] c~ ] ~
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
235
field will occur, since the wavenumber transform will consist of two Dirac delta functions as shown in Fig. 8.6(a), outside the radiation circle. On the other hand, if the same standing wave exists only over a finite region, as for a finite plate, then the wavenumber spectrum will be a result of the convolution of the wavenumber spectrum of the infinite system and that of the 'window' associated with the finite size and shape of the structure. In this case, the wavenumber transform will have the approximate form of Fig. 8.6(b) and now radiation will occur due to the presence of supersonic components. In other words the discontinuities associated with the plate edges have led to a scattering of energy from subsonic to supersonic wavenumber motion. Once again this provides an example of the dual interpretation of structural response in terms of waves or modes. Radiation from higher order modes can be either thought of as resulting from uncancelled modal cells as shown from Fig. 8.2 or due to supersonic wave components. Both of these behaviours, due to the edge scattering, lead
v (kx)l I
I I I I I I I I I I I I I I
-kf
(a)
-k
+k
+kf
Wavenumber, kx
Supersonic region
W(kx)I
(b)
I
I
I I I
I I I
I I I I
I I I I
-kf
-k
+k
+kf
,,
Wavenumber,kx
Fig. 8.6 Schematic illustration of the wavenumber distribution on (a) an infinite onedimensional plate with subsonic bending waves, (b) a finite one-dimensional plate with subsonic bending waves.
236
ACTIVE CONTROL OF VIBRATION
to identical radiation in the spatial domain. Observations such as this, aid in understanding the behaviour of active controllers and sometimes lead to alternative approaches to the control problem. This concept will be briefly expanded upon next.
8.4
Sound power radiation from structures in terms of their multi-modal response
In Section 8.2 we have described the acoustic radiation from individual modes of a vibrating plate. In reality, however, at a given frequency of excitation, a plate will exhibit a response which is a superposition of a number of individual modal responses. Strictly speaking, the response can only be fully described by an infinite series of modes, although in practice a finite series will give a good representation; the number of modes used being dependent on the damping of the plate, the frequency of interest and the convergence of the truncated series. Thus the complex velocity of the plate surface excited at a frequency to can be described by
m=M n=N
~jcot
¢v(x, y) = Z Z (Vm~Pmn(X,y)~ ,
(8.4.1)
m=O, n=O
where Wm~ is the complex amplitude of the (m, n)th mode and ~'m~(X,y) is the mode shape function which describes the modal space dependence. Clearly in the case of the simply supported plate described in Section 8.2, ~mn(X,Y) can be expanded into separable functions such that lffmn(X, y ) = lpm(X)lpn(y), where ~m(X) and ~p~(y) are respectively given by sin(mJrx/a) and sin(nzry/b) and a and b are the plate dimensions in the x and y directions, respectively. Equation (8.4.1) can also be written in the form W(x, y ) = vgT~3(X, y),
(8.4.2)
where the vectors ,iv and lp(x, y) are defined by (8.4.3a,b) "/vT= [W01W10W~... WMN], ~pT(x, y)= [~Po(X)~Pl(y) ~P~(X)~Po(y) ~Pl(X)~Pl(Y)... ~PM(X)~pN(Y)]. An expression for the power output due to the plate response at a given frequency can be found by evaluating equation (8.3.18) which expresses the sound power output of a planar vibrating surface in terms of the wavenumber transform of the surface velocity field. Thus if we define the wavenumber transformed vector ~P(kx,ky) by
~p(kx, ky) = I~_~I?o.lp(x, y)e j(kxx+kyy) dx dy,
(8.4.4)
then the modulus squared of the wavenumber-transformed velocity field can be written as
[ W(kx, ky)[ 2= [,~¢T~)(kx, ky)12=~cH~p*(kx, ky)~T(kx, ky)*W.
(8.4.5)
Substitution of this result into equation (8.3.18) then shows that the expression for the sound power output of the plate can be written as II = ~¢HM~¢,
(8.4.6)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
237
where the matrix M is given by
M=
~op0 Re {I °°
8yg2
I;-
~P*
T (k"m' k'-')IP(kx~' kY)dkx dky}. a/k - kx - ky 2
-oo
2
(8.4.7)
2
The matrix M thus has diagonal terms which quantify the 'self' radiation resistances of the individual modes when acting in isolation, whereas the off-diagonal terms quantify the 'mutual' radiation resistances which arise as a direct result of the interference of the fields of two different modes. Thus if we characterise one mode by the integers (m, n) and another mode by the integers (m', n' ), the corresponding entry in the matrix M is given by
Mmn'm'n'
=("0/902 Rell °° I;oo
8 yg
_oo
lP*m(kx)~;(2Y21P--m'('kx)lPn'(kY)5--;5---75 ] ~fk - kx - ky dkx dky .
(8.4.8)
Using the results of equation (8.3.22) which gives the wavenumber transform of sin(mzcx/a), shows that for a simply supported plate, we can write
~ *m(kx)~m'(kx) =
tam'jr 2 fmm,(kxa), aZ[k~- (mz~/a)Z] [k 2 -(m'yc/a) 2]
(8.4.9)
where the function fmm' (k~a) is given by 2 (1 - cos kxa) 2 (1 + cos kxa) f mm,(kxa) = 2j sinxa - 2 j sin kxa
m even, m' even m odd, m' odd m odd, m' even m even, m' odd
(8.4.10)
Exactly analogous results follow for the product ~p* (ky)~pn,(ky). The expression for the entry in the matrix M due to the interference of the (m, n)th and (m', n' )th modes can thus be written as
ogpo mm'nn'275 8yg2 aZb 2
4
Mmm'nn' =
× Re
"
(8.4.11) This integral can be simplified considerably in the low frequency limit. Thus, if one assumes that kxa~mzr and kyb~ nzc, terms like (k 2- ( m ~ / a ) 2) c a n be replaced by (mzr/a) 2 in the denominator of the integrand. In addition, for small values of k,a, series expansion of the expressions given in equation (8.4.10)shows that, to leading order,
fmm,(k,:a) =
(kxa) 2...
m even, m even
4 - (kxa)2 ...
m odd, m odd
2j(kxa)...
m odd, m' even
- 2 j (kxa).. •
m even, m' odd
(8.4.12)
238
ACTIVE CONTROLOF VIBRATION
Exactly analagous results are again produced in the low frequency limit (when kyb ,~. 1) for f,,,,.(kyb). This demonstrates that the only modal interactions of significance at low frequencies are due to the interference of the radiated fields of modes for which m and m' are odd and for which n and n' are odd. In other words it is only the mutual interference of monopole type modes that significantly modifies the acoustic power output of a simply supported plate at low frequencies. This conclusion was reached by Thomas (1992) in studying the interaction of plate modes in contributing to sound power radiation by a plate under active control. This interaction can be illustrated by plotting the self-radiation efficiencies of the lower order modes of a plate, which can be written as 0"mn,mn
--
2Mmn,mn ~
(8.4.13)
,
pocoab
together with the mutual radiation efficiencies between these modes, which are defined to be
Mmnm'n' 0"mn,m'n'
"--
'
(8.4.14)
.
pocoab
The variation of these radiation efficiencies with non-dimensional frequency is shown in Fig. 8.7 for a plate with b/a =0.57 (Elliott and Johnson, 1993). At low frequencies, ka ~ 1, the plate modes with a net monopole component have the highest radiation efficiencies, a~.~ and O'31,31, although the mutual radiation term 0"1~,3~is more important than the self-radiation term 0"31,31" At higher frequencies, ka,> 1, the effect of the mutual radiation term decreases since each of the modes radiates independently and with approximately equal efficiency. 101 100
°11,11
10-1 _ >:,
o
¢-.
1
0.2
-
°11,31
(D
i_~
10 -3 -
,-.
10-4 _
o
1 0-5 tr'
°31,31 °21,21
10_ 6 10-7 _ 10-8 10-1
. . . .
I
100
.
.
.
.
.
.
,,I
101
i
|
|
,
,
,
,,
103
Non-dimensional frequency, ka
Fig. 8.7 The radiation efficiencies of selected structural modes on a rectangular plate. The self-radiation efficiency of the (1, 1), (1,2) and (3, 1) modes are designated al~.~, crzl.z~, cr3~.31 and mutual radiation efficiency of the (1, 1) and the (3, 1) modes is designated 0~.3~.
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATESYSTEMS
239
It is possible to define a set of velocity distributions on the plate which do radiate sound independently at any one frequency. Consider the eigenvalue/eigenvector decomposition of the matrix M in equation (8.4.6): M = pT~p,
(8.4.15)
where, since M is symmetric and positive definite, P is a real unitary matrix of eigenvectors and f2 is a diagonal matrix of positive real eigenvalues. The power radiated by the plate, equation (8.4.6), can now be written as II = ";vHPTf2P-;v= bHf~b,
(8.4.16)
where b = Pw is the set of structural mode amplitudes transformed by the eigenvectors of the radiation resistance matrix. Since fl is diagonal, equation (8.4.16) can be written as N
I-[ = ~
(8.4.17)
f ~ . l b . I 2,
n=0
so that the velocity distributions corresponding to this transformed set of velocity distributions radiate sound independently. The existence of this set of velocity distributions has been described by Borgiotti (1990) and Cunefare and Koopman (1991), and is implicit in the work of Baumann et al. (1991). They have been termed radiation modes by Elliott and Johnson (1993), who have plotted out their radiation efficiencies, proportional to the f2,'s, for the plate described above, as shown in Fig. 8.8. At low frequencies, ka ~ 1, the first-order (1) radiation mode is much more efficient than those labelled (2) and (3), and generally the variation of their radiation efficiency with frequency is simpler than that for the radiation efficiency of the structural modes (Fig. 8.7). The velocity distributions corresponding to these theoretical radiation modes are weak functions of the excitation frequencies. At low frequencies, ka ~ 1, however, the velocity distributions are almost independent of excitation frequency, and for the plate considered here are plotted in Fig. 8.9 (Elliott and Johnson, 1993). 101 10 0 I~
10-1
i
I
i
I
i
i
i
i
i
i
i
I
i
i
i
i
i i
i
Radiatiom n o d e ~
10"2
~
0.3
.E_O
10 .4
.(1:1 "0
10-5
rr
10..6
(3) (4)
10 .7 10 .8 10-1
i
10 °
I
I
I
I
I
I II
I
101
i
i J J i ii102
Non-dimensional frequency, ka
Fig. 8.8 The radiation efficiencies of the first six radiation modes of the rectangular plate.
240
ACTIVE CONTROL OF VIBRATION
velocity, ~,
(1)
(2)
(3)
(4)
(5)
(6) Fig. 8.9 The velocity distributions corresponding to the first six radiation modes of the rectangular plate at an excitation frequency corresponding to k a = O. 1. The most efficiently radiating velocity distribution (1) clearly corresponds to the net volume displacement of the plate, as expected. The advantage of the radiation mode approach is that, by using Fig. 8.8, it allows one to quantify the extent to which other velocity distributions are also significantly radiating at any one excitation frequency. It also suggests an efficient method of sensing the velocity distributions of the plate which are most important in radiating sound, as will be discussed in Section 8.8.
8.5
General analysis of Active Structural Acoustic Control (ASAC) for plate systems
In this section we outline the basic analysis of ASAC applied to plate systems. The system studied consists of a simply supported rectangular plate positioned in an infinite baffle as shown in Fig. 8.10. Two different disturbances to the system are considered as
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS
Incident pressure,Pi
241
/,,
Iii
// ~ t e d ~
pressure Pr
Y
,J v!
b2f - I - Uniforml_~ pressure
Point liD'-force
I I I I
b
y
Piezoelectricpatch
I I I I I
-
--~Ye
i
a2
x2
(c)
Fig. 8.10 Coordinate system and arrangement of a baffled rectangular plate with inputs (a) on the incident wave side, (b) on the radiated wave side and (c) configurations of the input forces. examples. Firstly, an acoustic plane wave incident on the plate at an oblique angle is taken as the noise input shown in Fig. 8.10(a). Secondly, a forcing function over a small localised area on the plate is introduced that is considered to be representative of a structural disturbance (in the limit this forcing function will be a point force) shown in Fig. 8.10(c). Both of these disturbances will excite the plate into motion resulting in radiation on the transmitted half plane of the radiated field shown in Fig. 8.10(b). As discussed in Chapter 1 and Section 8.2 the form of the disturbance has an important influence on the resultant plate motion and thus the sound field to which it is coupled. When the disturbance is a low frequency plane wave, the input phase distribution is relatively constant over the plate surface and the result is that the plate response is dominated by lower order modes that are often excited well above their resonance
242
ACTIVE CONTROLOF VIBRATION
frequencies. This behaviour leads to an overall vibration pattern with fewer nodal lines and a high radiation efficiency. For a localised structural input, the plate responds with a much richer modal distribution and higher order modes tend to dominate near their resonance frequencies. In this case at higher frequencies (however, still below the plate critical frequency) the overall radiation efficiency is lower. Thus, as demonstrated by McGary (1988), airborne disturbance inputs to structural systems will tend to radiate/ transmit higher levels of sound than structural inputs for the same frequency and total magnitude of load. In the following analysis a general procedure is outlined. The inputs are assumed to be at single frequencies and all systems (structural, acoustic and electrical) are assumed linear so that superposition of response holds. The type, number and location of the control transducers is assumed known. In this section the fluid loading is assumed small (e.g. as in air) hence the plate response can be determined using the inv a c u o equations described in Chapter 2. The following steps are taken in the general ASAC analysis: (1) An expression is derived for the response of the plate to the input or primary disturbance(s). (2) An expression is derived for the coupled radiation to the far-field from the plate due to the disturbance. (3) An expression is derived for the response of the plate to the multiple structural control or secondary inputs. (4) An expression is derived for the far-field radiation from the plate due to the structural control inputs. (5) The total far-field pressure response is found by superimposing the disturbance and control fields. (6) A quadratic cost function is formed that is based on the required observed radiated pressure field variables.t (7) Quadratic optimisation theory is used to find the optimal control inputs that minimise the cost function as outlined in Section 4.6. (The input disturbance(s) is assumed constant and known.) (8) The optimal control inputs are substituted into the relations for the total field response (i.e. far-field pressure, plate out-of-plane displacement etc.) in order to evaluate the control performance. (Note that the reduction in the cost function also provides a measure of the overall control performance in terms of the observed error variables.) Before we carry out the above steps, we first derive the basic system response equations. Note that as outlined by Pan et al. (1992b) the method can also be formulated by using transfer matrices written in terms of input and radiation impedances. In addition, as described in Section 8.8, design techniques for optimally shaping and locating the control actuators and sensors are available. Figure 8.10 shows the arrangement and coordinate system of the baffled, simply supported, rectangular thin plate excited by a harmonic disturbance pressure acting over an area of the plate. To calculate the radiated sound field, a description is required of
t One could also minimise a pressure related variable such as supersonic wavenumbercomponents.
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
243
the plate complex vibration distribution. For the simply supported thin plate, the displacement distribution is given by equation (2.11.2) in modal form as oo
oo
W(X,y, t) "- Z Z wmn sin kmx sin kny e i~t,
(8.5.1)
m=ln=l
where the eigenvalues are given by km
met =
~ ,
a
kn =
nJr b
.
(8.5.2a,b)
The plate modal amplitudes for various forms and distributions of input forcing functions have been calculated by Wang and Fuller (1991) and are given by Wren "-
Pro. ph(ooZm~-co 2)
,
(8.5.3)
where 09 are the natural frequencies, p the plate density and h is the plate thickness. The modal force, Pmn, due to the input disturbance depends upon an exact description of the external load. Gu and Fuller (1993) have studied active control of sound radiation from a plate in the presence of a heavy fluid. However, for the present discussion we limit the analysis to a light fluid loading such as air and thus radiation loading effects are ignored. Modal forces for various input disturbances acting upon an in-vacuo simply supported plate have been derived by Wang and Fuller (1991) as follows. (1) Uniformly distributed pressure. For a uniformly distributed pressure with amplitude, Q, located between coordinates a~, a2 and b~, b2, as shown in Fig. 8.10(c), the modal force is given by mn
Pmd =
4Q
m nx~ 2
(cos kma 1 -
cos
kma2)(cos knbl -
cos
knb2),
(8.5.4)
where the superscript d will hereafter denote the input disturbance. (2) Obliquely incident plane wave. An obliquely incident plane wave as shown in Fig. 8.10(a) can be described by pi(x, y, t) = Pie j°~t-jksin °i
cos q,,-jksin 0, sin ~i.
(8.5.5)
Again assuming that the fluid loading is light we can calculate the total input pressure at the plate surface from the plate blocked pressure. That is, we assume total reflection of incident waves and thus the total input pressure at the plate surface is twice the incident pressure. Using such an approach the modal force for an oblique incident plane wave has been calculated first by Roussos (1985) and later by Wang and Fuller (1991) and is given by Pare.= 8Pilmln where the coupling constants are given by
-j [m = 2
sgn(sin Oi cos q~i)
if (met)2 = [sin 0icos ~ b i ( o ) a / c ) ] 2,
(8.5.6)
244
ACTIVE CONTROL OF VIBRATION
or
mn{ 1 - ( - 1 ) m exp[-j sin Oi cos ~i(toa/c)} I n --
(mn) 2 - [sin Oi cos ~i(wa/c)] 2 if (mn) 2~ [sin Oi cos q~i(toa/c)] 2
(8.5.7a,b)
and -J sgn(sin Oi sin (])i)
rn- 2
if (n~:)2 = [sin Oi sin q~i(~ob/c)] 2, or
L-
nn{ 1 - (-1)n exp[-j sin 0 i sin qbi(wb/c)] } (net) 2- [sin Oi sin ~ i ( t o b / c ) ]
2
if (nn)2~ [sin Oi sin ~i(tob/c)] 2.
(8.5.Sa,b)
(3) Piezoelectric actuator. As discussed in Chapter 5 a rectangular piezoelectric actuator configured for pure bending is considered to consist of two wafer elements located symmetrically on each side of the plate and driven 180 ° out of phase in the d31 mode. The modal force for such a piezoelectric actuator as shown in Fig. 8.10(c) is given by Wang and Fuller (1991) as P C m = 4C°l~Pe mn~2
(kZm+
k2n)(cos k m x 1 - c o s k m x z ) ( C O S
knYl -
cos knY2)
(8.5.9)
where x~, x2, and Yl, Y2 are the coordinates of the actuator edges. The parameter Co is a constant that is function of the piezoelectric actuator/plate properties and geometries specified by C o - E b I K I, where K I is given by equation (5.6.8). The unconstrained strain, epe, of the piezoelectric element is defined by d31V l?,pe -- ~ ,
(8.5.10)
ha where d31 is the piezoelectric transverse strain constant and h a is the piezoelectric element thickness. Most importantly Epe and thus Pm~ is seen to be linearly related to V, the input complex voltage to the actuator. (4) Point force. The modal force associated with excitation by a harmonic point force of amplitude F located at (Xr, Yr) as shown in Fig. 8.10(c) is pc _
4F
ab
sin k mXf sin k,y I,
(8.5.11 )
where the superscript c denotes control force. The sound radiation caused by the vibration due to the above inputs is related to the plate velocity distribution. As outlined in Sections 8.2 and 8.3 the radiated pressure can
ACTIVI~ STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTtIM$
245
be evaluated by using the Rayleigh integral or spatially Fourier transformed variables in conjunction with the method of stationary phase. As demonstrated by Junger and Feit (1986), both mathods yield identical solutions. Roussos (1985) has derived e×pressions for the radiation from a rectangular plate using the Rayleigh integral. The radiated field shown in Fig. 8.10 (b) is given by eo
e@
p(R, 0, q~)- K Z Z
(8.5.12)
Wm, lml, e m,
m-'-I n=-I
where the radiation constant K is defined as
1[
K - =topoab exp ]to t . . . .
2r~R
c
]}
(a cos ~ + b sin ~) . 2c
(8.5.13)
The coupling constants I,, and I, for the radiated field ere identical to equations (8.5.7a,b) and (8.5.8a,b) except that the coordinate angles (0i, 'hi) am replaced by (0, ,)) which define the coordinate of the observation point in the radiated fieldand the sign in the argument of the exponential function is changed to minus. We now have all the necessary components in hand to derive the response of the total radiated field,i.e.the disturbance field (also called primary) plus the control field (also called secondary). For a feedforward control arrangement the disturbance source and the control actuators act simultaneously and am assumed Imrfectly coherent. By superposition the totalcomplex pressure can be written as p, = pp + p,,
(8.5.14)
where the subscripts t, p and s refer to total, disturbance and control pressure respectively. Using the previous relations the total pressure in the far field can be written for the following configurations. Case I. Plane wave disturbance = point control forces u~ p, = P~B + ~ F;Cj,
(8.5.15)
/-=1
where N~ is the total number of control forces, while P~ and Fj are the complex amplitudes of the disturbance wave and control forces respectively. Case I1. Localised structural disturbance = point control forces #,
p, = O.B + Z r;cj,
(s.5.16)
/=-1
where Q is the amplitude of the disturbance pressure acting over the small area specified previously. Case III. Plane wave disturbance -- piezoelectric actuators N, P,
= PIB + ~ VjCj, /=-1
where Vj is the complex control voltage applied to the jth actuator.
(8.5.17)
246
ACTIVE CONTROLOF VIBRATION
Case IV. Localised structural disturbance - piezoelectric actuators
s pt= QB + Z V~Cj, j=l
(8.5.18)
where Q is the amplitude of the disturbance pressure. In the above equations the transfer functions are defined as: plane wave disturbance, B - K
-~
ImI~,
(8.5.19a)
Imln,
(8.5.19b)
localised structural disturbance, B =K
= = y
where the appropriate form of WPmnis calculated using equation (8.5.3) with either equation (8.5.6) or equation (8.5.4) respectively; point control force, =
= -~j
Iml.,
(8.5.20a)
=
__ - ~ j
ImIn'
(8.5.20b)
piezoelectric control actuator, Cj=K
where the appropriate form of Wm~njis calculated using equation (8.5.3) with equation (8.5.11) and equation (8.5.9) respectively. In the above transfer functions, B and C are normalised with respect to their appropriate forcing function amplitudes in order to put the relationships for total response in a form which can more readily be manipulated in terms of control amplitudes. Thus in effect, B, for example, will represent a complex transfer function between the specified input pressure of an obliquely incident wave and its corresponding radiation pressure at some observation angle in the far field. Control of the disturbance field can be achieved by appropriately choosing the vector of point force amplitudes f~= [F~ F~ F~ ...]T or piezoelectric voltages v~= [V~ V~ V~... ]T in order to minimise a chosen cost function. Choice of the cost function is defined by the form of control performance required. The ideal cost function for globalt control of sound radiation is the total acoustic power radiated by the plate since this is the variable that we seek to reduce. In this case the cost function is defined as the integral of intensity flowing through a hemisphere surrounding the plate which is given by J =
1 f
Ip, 12dS
2p0c0 ' s t Here global means through an extended area or volume.
(8.5.21)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
247
and which can be described in polar coordinates as J =
2p0c0 ' 0 J0
IP,
sin 0 dO dq~.
(8.5.22)
If directional control is required then equation (8.5.22) can be modified to only attenuate power over a required directional sector and is thus written J =
1
fo2f¢2 ]p 12R2 sin 0 dO d~b, 2 p o c o ' o, ~,, '
(8.5.23)
where 01, 02, q~l, q~2 define the limits of the sector. Usually it is not possible to design a sensor such that a cost function of the form of equation (8.5.21) can be measured, and in practice microphones are often used as point error sensors. In this case the cost function is written in discrete form as Ne
1
J=
~ [pI(R~, 0 i, ~)i)12,
(8.5.24)
2poco i= 1 where N e is the number of error sensors and (Ri, Oi, ~)i) is the location of the ith error sensor microphone. Equation (8.5.24) can be written in a more accurate discrete form as 1
J =
N~
~ Ipll = A S i,
(8.5.25)
2p0c0 i= 1 where A S i is the projected area associated with the jth sensor on the hemispherical surface. In practice, however equation (8.5.24) provides a cost function which is a reasonable estimate proportional to the total radiated power as long as a suitable geometry of error sensors are chosen. Choice of number and location of error sensors is an important topic of research and will be briefly discussed later. In general, for global control, the number of point error sensors required is equal to the number of modes contributing to the overall sound power radiation. The error sensors, by analogy with time sampling theory (see, for example, Nelson and Elliott, 1992, Ch. 2), should be spaced in sectors of the radiated field defined by regions of 180 ° phase change through nodal lines in the radiation field associated with that mode. For example, Fig. 8.11 shows a hypothetical low frequency, rectangular plate, radiation pattern which is a combination of radiation from the (1, 1) and (2, 1) modes. Positioning a single error sensor at location (a) will only lead to control of the (1, 1) mode since this angle corresponds to a node in the (2, 1) radiation pattern. Positioning two sensors at locations (b) will not lead to global control since the (1, 1) and (2, 1) modes will cancel at the error sensors and reinforce in the other sector. Locations (c) shows a situation where two sensors are positioned in two sectors which are 180 ° out of phase for the (2, 1) mode. When used with two actuators, the only way the controller can attenuate the error signals is by completely attenuating each mode, thus leading to global control. Another important observation regarding the choice of point sensors is that generally the number of sensors should be equal to or greater than the number of control inputs used. If the number of error sensors equals the number of actuators then total
248
ACTIVE CONTROL OF VIBRATION
~mode
1
mode 2 + (a)
0 ensor
(b)
e error sensor
+ (c)
Fig. 8.11 Influence of error sensor position on modal radiation control. attenuation is theoretically achieved at each error sensor since the system is perfectly determined. This situation generally leads to poor attenuations at other positions especially if the number of degrees of freedom in the system exceeds the number of error sensors. If the number of control actuators used is greater than the number of error sensors then the system is underdetermined. Note, however, that Elliott and Rex (1992) have presented a methodology to ensure well-conditioned relations for underdetermined systems by introducing a control effort term into the cost function. It should again be stressed here that in ASAC, although the control action is applied directly to the structure, the cost function is derived from the far-field radiated pressure (or far-field radiated pressure-related variables). Thus, inherent in the definition of the cost function, is the natural structural acoustic coupling that relates the plate vibration to the radiated sound. This arrangement should be contrasted to the more obvious
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
249
approach in which plate vibration is directly observed and minimised. Naturally, completely reducing overall plate vibration will lead to a reduction in sound radiation. However, as will be demonstrated, this latter approach generally requires many more channels of control (i.e. number of sensors and actuators) and a much more subtle and efficient control paradigm can be implemented when far-field pressure is used as an error variable. One is thus directly observing the field variable to be controlled, which in this case is radiated sound. The following analysis for derivation of the vector of optimal control inputs is for the ideal case of minimising total radiated power and thus equation (8.5.21) is used to define the cost function. Although the following derivation is formulated for the above Case IV, it can be written in exactly the same form for the other three cases with appropriate substitution of variables. When the expression for total pressure Pt from equation (8.5.18) is substituted into the cost function definition, equation (8.5.21), it can be demonstrated that the cost function is a scalar which is quadratic in the vector of complex control voltages v, or control force amplitudes, f,. The reader is referred to the Appendix of Nelson and Elliott (1992), as well as Chapter 4 of this text, for a proof and discussion on the nature and minimum value of the quadratic cost function. It can be shown that this cost function will possess a unique minimum which will define the optimal control voltages. The minimisation procedure is based upon the setting the gradient of the cost function J with respect to the control vector v, to zero in order to find the stationary point of the quadratic form. The total complex pressure can be expressed in vector form as p, = hTq + cTv ~,
(8.5.26)
where q=[Q~Q2Q3 ...]v is the vector of complex input disturbances and h = [H~ H2 H3 ...IV is the vector of aforementioned transfer functions associated with those disturbances. For cases in which there is a single input disturbance these vectors reduce to scalars such that q=Q
(8.5.27)
h = H~.
(8.5.28)
and
In equation (8.5.26) the control transfer function vector is defined by
c = [Ci C2... Cu,]"r,
(8.5.29)
while the input control voltages or control forces can also be written in vector form as
v,=
T.
(8.5.30)
Note that a similar vector expression as equations (8.5.26) and (8.5.30) could be used for f,, the vector of complex control force amplitudes. As outlined by Nelson and Elliott (1992) it is convenient to write the cost function using matrices in a Hermitian quadratic form. For a single input disturbance the squared modulus of the total complex pressure can be written IP, [2= p,p,= v~Cv, + v~x + XHV,+
QH,H*~Q*.
(8.5.31)
250
ACTIVE
CONTROL
OF VIBRATION
In equation (8.5.31) the Hermitian matrix C is defined by C = c*c T
(8.5.32)
x = QHlC.
(8.5.33)
and the vector x is given by
The cost function as defined by equation (8.5.22) which is written in terms of total radiated acoustic power can now be specified as J = v HAv, + Vnb + bHv s +
(8.5.34)
C,
where the individual terms are now given by AN, ×U,
=
1 [2~ [,q2 [c.cT]R 2 sin 0 dO dq~ 2poco ' o Jo
1 I2, [,#2 bN'×l= 2p0c0,0 J0
c=
1012
r/o rjo,
(8.5.35)
QHlcR2 sin 0 dO dq~
(8.5.36)
sin 0 d0 d~.
(8.5.37)
2poco '
As discussed in Nelson and Elliott (1992), all the combined terms of equation (8.5.34) are scalars and as long as A is positive definite then there is a unique minimum value of J. A typical element of matrix A is defined by A~=
~. kli~ mnjI kliI mnj1" sin 0 d 0 d q~ RoCo "
k = l I=1 m--1 n--1
i = 1 , N , ; j = 1,N,.
(8.5.38)
A typical element of vector b is oo
oo
oo
1 [2=Ij/2 K1K* Z ~ Z Z QP~tlQP,~,'PktalS£nsR2sin OdO dq), s= 1,Ns.
pOCo '
k = l !=1 m = l n = l
(8.5.39) Finally the single entry of h for a single input disturbance is defined as n 1-
1 f 2=[zr/2K1K~ Z 0 dO poCo ' k=l
Z
Z
P* IP lP* D 2 sin0d0dq~ QpkllQmnl*kll*mnl*~
(8.5.40)
1=1 m = l n = l
where
w;%
QS,,j = - - , V~
wL1
Q~l = ~ . Q
(8.5.41a,b)
Note in the above equations that the quadruple infinite sums in (k, l) and (m, n) result from the multiplication of two double modal series. As discussed in Section 8.4 the cross terms are important and the additional modal indices (k, l) are employed. The
ACTIVE STRUCTURALACOUSTICCONTROL. I PLATE SYSTEMS
251
optimal solution of control strengths to minimise the cost function of equation (8.5.34) can be found using the result derived in the Appendix of Nelson and Elliott (1992) and Chapter 4 which shows that V~o= - A - l b -
(8.5.42)
Equation (8.5.42) thus defines the vector of optimal control voltages V,o for the case of piezoelectric actuators. On obtaining V,o, the minimised far-field radiated pressure can then be calculated using equation (8.5.26). The minimum of the cost function can also be calculated from Jmin = c - bHA-lb.
(8.5.43)
Equation (8.5.43) can be used to calculate the attenuation in total radiated power obtained when the control is invoked as long as enough error sensors are used to provide a reasonable estimate. Note that if there are more control actuators than error sensors then matrix A will be singular and the procedure developed by Elliott and Rex (1992) for underdetermined systems should be used. The reader is also referred to the Appendix of Nelson and Elliott (1992) for discussions on this and other aspects of finding the minimum of quadratic functions. The previous analysis enabling the derivation of the optimal control voltages can readily be applied to the different cases I, II and III outlined above with use of the appropriate variables and using similar methodology. In the next two sections, results from example applications using this analysis will be discussed.
8.6
Active control of sound transmission through a rectangular plate using point force actuators
In this section we study the active control of sound transmission through a rectangular, baffled, simply supported plate using point force actuators. Systems similar to these have been previously investigated by Fuller (1990) and Wang and Fuller (1991). All disturbance frequencies are well below the coincidence frequency of the plate which is fc_~6300 Hz. Table 8.1 presents the specifications of the steel plate, while the corresponding natural frequencies for the simply supported boundary conditions, computed using equation (2.10.3), are given in Table 8.2. The acoustic medium is assumed to be air with P0 = 1.21 kgm -3 and Co = 343 ms -~. To calculate the plate response and the corresponding radiated field it is necessary to truncate the infinite summations. In the following examples, truncating the indices k, l, m and n at a value of five (i.e. 25 modes are considered in the doubly infinite sums) was found to provide close to 0.01% error in the radiated pressure amplitude at the highest frequency considered. This choice of truncation can be seen from Table 8.2 to effectively limit the input disturbance frequency such that f < 1750 Hz. In addition, it Table 8.1
E = 207 x 1 0 9 N m pp = 7870 kg m - 3
Plate specifications. v = 0.292 h = 2 mm
a = 0.38 m b = 0.30 m
252
ACTIVI~CONTROLOF VIBRATION Table 8.2 Naturalfrequencies of the plate (Hz).
m
1
2
3
4
5
1 2 3 4 5
87.71 188.74 357.13 592.88 895.98
249.81 350.85 519.23 754.98 1058.08
519.98 621.02 789.40 1025.15 1328.25
898.22 999.25 1167.64 1403.39 1706.48
1384.53 1485.56 1653.95 1889.69 2192.79
was necessary to calculate the integrals of equations (8.5.35) to (S.5.37) and this was carried out numerically using Simpson's rule. In order to use the above equations for this case, the input control terms are replaced with those for point control forces. The plate response results presented in Sections 8.6 and 8.7 consist of the distribution of plate vibrational amplitude plotted along the ) , - b / 2 horizontal plate mid plane (see Fig. 8.10). The results, presented in decibels (dB), were normalised to the largest amplitude obtained in each figure. Radiation directivity patterns are also presented along the y-b/2 axis at a distance of R - 2 m. Although this observation point is relatively close to the plate, far-field radiation equations were used for simplicity, and thus the results also reflect the behaviour at large distances from the plate. For convenience, angular positions with a negative sign of 0 in the figures correspond to the coordinate ~ - ~ positions. For the following example the input disturbance was assumed to be a plane wave with amplitude P~-1 N m =2 incident at angles of 0~ - 45 ° , ~ - 0 e. Figure 8.12 presents the radiation directivity patterns with and without control when the excitation frequency was set to 186 Hz. Note that negative values of sound pressure level correspond to pressure magnitudes less than the reference pressure of 2 x 10 =s N m =2. From Table 8.2 it is apparent that this frequency is near the resonance frequency of the (2, 1) mode, and correspondingly, the uncontrolled radiation field appears to have the distorted version of the radiation pattern associated with a (2, 1) mode. An examination of the modal contributions confirms that the radiation field is mostly due to the (1, 1), (2, 1) and (3, 1) modes. The slight offset of the node in the radiation field is due to the presence of a monopole type radiator such as the (1, 1) mode. Applying one control force, as shown in the schematic diagram at the top of Fig. 8.12 as a black dot, leads to little reduction in radiated power. Positions of the control forces are shown to scale in the schematic figures. However, it appears that the (1, 1) radiation contribution has been controlled as the residual radiation pattern has now the characteristic dipole shape associated with the (2, 1) mode and this is confirmed by examining the modal contributions. The single, centrally located control force is unable to couple into the (2, 1) mode of the plate as shown in equation (2.11.4). Using two forces leads to a large reduction in radiated power as now the (2, 1) mode is controlled and residual field is now largely due to the (3, 1) mode contribution. On using three point force actuators positioned as shown in the schematics of Fig. 8.12, control is achievable over the (1, 1), (2, 1) and (3, 1) modes, and power reductions of the order of 67 dB are predicted.
ACTIVI~STRUCTURALACOUSTICCONTROL,I PLATI~$YSTISMS Unoontrollti¢l
I Foroti
a
llllllllll=
i1= ,
Forolill
253
3 Forollit
=III=I=I=IIIIUlIIII=I
I
48,7 (dB)
80,8 (¢IB)
I
I
i
01=_48" 0l =- O*
Power rttduotlon
o,e (aa)
5O ....... -46°
0--'0°
""':::,
I
/
/ 0
F
=50
7/
=100
.............
....... ":1:::~ ~
............ I ............. 48'1 .:=;:'"
,.................. ~
(
Illl $
/
",...,' . . . .
~
I
.::;:"......
........................ i)N /
F........"
",~:~2 ... \
[ %1
L( ( ;-; !....]i......... li '2........... (
-50
=50 =I00 -80 Sound prnmum Iovel (d8)
0
0
60
Fig, 8,12 Radiationdirectivity for different numbers of point for, o actuators, f=- 186 Hz,
Curves of the power transmission loss versus frequency for the same situations as above are plotted in Fi$, 8,13, Her~ power transmission loss is defined as (Wang, 1991) Transmission loss-~ 10 log(lli/l=I~),
(8,6,1)
where the incident power is given for an obliquely incident plane wave by the relation
hi--- Ipil~ ab cos 0~
(8.6.2)
2pete
and the radiated power is given by Jo
~,, Ip(r, 0, O)[~ r ~ sin 0 dO dO, o 2pete
(8.6.3)
whore p ( r , 0, 0) is evaluated in the far field usin8 equation (8,$,26) for the cases with and without control. Note that negative values of tr~amission lo~a are a numodoal artifact of the computer calculations; in practice transmission loss cannot be loss than zero, For the primary or uncontrolled case, the transmissiort loss cu~e ha~ a number of dips a~sociatod with the plate modal resontm,os, When one ,ontrol fo~o is u~od, it ,an be soon from Fig, 8,13 that the fall in transmission loss at the (1,1) re~onanco frequency has been eliminated as discussed above, Similarly, use of two control forces eliminates the dips at the (1, 1) and (2, 1) resonance frequencies, The figure shows that attenuation greater than 50 dB can M theoretically obtained over a frequency range of
254
ACTIVE
Uncontrolled
CONTROL
OF VIBRATION
1 Force
2 Forces
3 Forces
.................................
0i==45°~i 0° i 200
I
~'~"1
|
150
~,,,',,,
A
rn "o
o (0
100
m
tO .B
50
.B
.'~..
E
,,.
...........~~:...~ ~
\ :~"'.....
,.,,..~..-..-..-:....-.,
e--
-50
0
I
I
I
I
I
I
I
1O0
200
300
400
500
600
700
800
Frequency (Hz)
Fig. 8.13 Plate transmission loss for different numbers of point force actuators. 0 to 450 Hz with three control forces. Whether this is achieved in practice for broadband, random disturbances depends upon a number of issues such as causality, filter size, etc. as described in Chapter 4. However, these frequency domain results do define the ultimately achievable performance with the limited number of actuators used in the locations specified.
8.7
Active control of structurally radiated sound using multiple piezoelectric actuators; interpretation of behaviour in terms of the spatial wavenumber spectrum
The previous section briefly discussed results of using point force actuators as control inputs. However, there are disadvantages to using point force actuators such as their size and the need for a back reaction support. In this section we discuss a few representative examples of control of structurally radiated noise (i.e. a structural rather than an airborne input) with arrays of independently controlled piezoelectric actuators. The piezoelectric actuators were assumed to be manufactured from ceramic material with typical properties given in Table 5.1 (corresponding to G1195 material). As discussed in Chapter 5 the actuators were configured to produce pure bending in the plate. The system used for the analysis is exactly the same as that presented in the previous section except that in this case the input disturbance is assumed to act over a very small area of 40 x 40 mm approximating a localised structural input. For this case the input
255
ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS
disturbance amplitude was set to Q = 7.9 x 103 N m -~ giving an input force of 12.65 N. In the following figures the prescribed disturbance source and actuator locations and size are shown in schematics at the top of each figure, drawn to scale looking into the plate surface from the radiated field. The black rectangle represents the size and position of the disturbance source while the clear rectangles represent the size and position of the two-dimensional piezoelectric actuators. At this stage no attempt is made to optimally configure the control actuators; their selection is made on an ad hoc basis, linked with a knowledge of the acoustically significant plate modes and their response shapes. For the first case the disturbance frequency is set to 85 Hz. An examination of Table 8.2 reveals that this frequency is close to the resonance frequency of the (1, 1) mode.t Hence this case corresponds to an 'on-resonance' excitation. Figure 8.14 presents the normalised vibration amplitude distribution with and without control for four different configurations of piezoelectric actuators. In Fig. 8.14 the solid line depicts the displacement distribution of the plate only under the influence of the disturbance, and as expected is close to a (1, 1) mode shape. When the various configurations of control (1)
(2)
(3)
(4)
I
I
I
I
rn I1) "O
,m=
-50
..../.--:':~'"~.L.. -'~.-...
E
.:----, :,,
tl:l ¢...
•~
-~00 -
,.,.~,-~... -~
.dI3 >
/
t;:',, '(" #
"
.,
:
',.k/. "-
" ~1
"-,,
&, "':~, . . . . i':_..-:...... - .
"13
I
-150
i"
I,
\
|".f
',
E 0
z
-200
I
I
I
I
I
I
0
0.2
0.4
0.6
0.8
1
x/a
.............
Uncontrolled 1 Piezo
.......................... 2 P i e z o s 3 Piezos .......
4 Piezos
Fig. 8.14 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 85 Hz. 1 Since no damping is included in the model, the disturbance frequency is not set exactly to the resonance frequency.
256
ACTIVE CONTROLOF VIBRATION
actuators are applied, the vibration amplitudes are significantly reduced and the (1, 1) mode is well controlled. However, increasing the number of actuators does not lead to a significant reduction in vibration. (Note the we are using an acoustic cost function in this example, in contrast to the configurations in Chapter 6.) This effect will be discussed below. Figure 8.15 presents the radiation directivities corresponding to the cases shown in Fig. 8.14. Also presented is the total reduction in radiated acoustic power calculated from equation (8.2.7) for each case. As expected the uncontrolled field is uniform and corresponds to the monopole like radiation of the (1, 1) mode. When one control actuator is applied, reductions in radiated power of the order of 60 dB are obtained. Use of two control actuators brings a further 10 dB reduction. Further increasing the number of actuators has little effect on the radiated power until for case (4) when actuators are located off the y = b/2 line (see Fig. 8.10) and a further 30 dB of power reduction is obtained. From these cases, it is apparent that good sound control is achievable with a single actuator and increasing the number of control channels has no significant practical advantage. In general this observation is true for systems on or near resonance of an efficiently radiating structural mode such as the (1, 1) mode considered here. Figure 8.16 presents the wavenumber transform of the plate vibration calculated (1)
(2)
(3)
~5°1 ,,
(4)
I po.0°
i,,.,~° ....
................... <' -50
~ "1
,0o ............
i
0
50
!
1O0
/
/
150
Sound pressure level (dB) Power reduction (riB) Uncontrolled ............. 1 Piezo .......................... 2 Piezos ..... 3 Piezos ....... 4 Piezos
0.0
60.6 70.0 70.0
93.4
Fig. 8.1$ Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 85 Hz.
ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS
(1)
(4)
(3)
(2)
257
100 A
e~
80
"7
6O
o
~
40
o=1==,====== ` ='1'= 1' I'
==================================
(N
~
20
-20 ~ / 0 k
i
I
I
10
2O kx(m'l )
30
.............
40
Uncontrolled
1 .......................... 2 . . . . . 3 ....... 4
Piezo Piezos Piezos Piezos
Fig. 8.16 Wavenumbcr spectrum of plate response for different numbers of piezoelectric ceramic actuators, f = 85 Hz or k = 1.556 m- ~. Note values are only shown for positive kx.
using equation (8.3.1) applied to the plate out-of-plane vibration distribution (see also Wang, 1991). The free wavenumber k of the radiation field is also shown. Although the wavenumber distribution exists in two dimensions for k,. and ky, values are only presented here for the positive k~. components. As discussed in Section 8.3 only supersonic wavenumber components, where (k~ + k~)~/2~ k will radiate to the far field. When control is applied, the wavenumber spectrum of Fig. 8.16 is attenuated across the complete wavenumber range including the supersonic components. A modal decomposition of the response also shows that all of the amplitudes of modes which are well coupled to the sound radiation are reduced. Thus radiation control has been achieved by suppressing the supersonic wavenumber components, and vibration of the mode(s) which are dominantly coupled to the radiated far field by virtue of their high radiation efficiency. We term this mechanism of control modal suppression (Fuller et al. , 1991). The disturbance frequency is now increased to 128 Hz which is an off-resonance frequency located between the (1, 1) and (3, 1) modes. Figures 8.17 and 8.18 present the plate displacement distributions and corresponding radiation directivity patterns. In this case it is apparent that increasing the number of control actuators leads to a realistic increase in control performance. At this off-resonance frequency the modal density is
258
ACTIVE CONTROLOF VIBRATION
higher; therefore more modes are now contributing significantly to the cost function, thus requiting multiple actuators for high reduction in sound power. The displacement distributions of Fig. 8.17 are also interesting. Using one control actuator leads to very little overall attenuation of the plate vibration; in fact, it is increased in some locations. As the number of actuators is increased, there is also seen to be a lack of significant change in the overall amplitude of plate motion. However, examining Fig. 8.17 it can be seen that the vibration distribution under multiple control action is quite complex with many nodal lines across the plate surface. A calculation of the overall plate radiation efficiency demonstrates that it is considerably lower when control is applied (Wang et al., 1991). Thus in this case the sound attenuation shown in Fig. 8.18 is not achieved by reducing the overall plate vibration, rather the controlled response of the plate exhibits a lower overall radiation efficiency. Figure 8.19 gives the corresponding positive wavenumber distributions computed along the x axis. In this case it can be seen that when control is applied, the supersonic wavenumber components (where kx <~k) are attenuated while the subsonic, non-radiating components are increased, except for case (3). In this off-resonance case, a multiplicity of modes are contributing significantly to the radiation field. The controller modifies the dynamics of the system so that the uncontrolled modes of significance (in terms of radiation) now have differing phases
(1)
(2)
I
I
0
.."
"0
/
.X
: ..~.-.~-"~..
..?,
-40
(4)
I
./"...-S. ........ -",:,.- .......
.. . . . . . . . . . .
133
(3)
"..
,...-
-, :./......
.
E t.--
.£ ~
-80
;
"0
•
..
._-,
I
~N
\1
\
'
li!I
!i~
~ -~2o
o Z
I
I
I
I
I
0
0.2
0.4
0.6
0.8
.............
1
1
x/a Uncontrolled ............. 1 Piezo ..........................2 Piezos ..... 3 Piezos ....... 4 Piezos
Fig. 8.17 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 128 Hz.
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS (1)
(2)
,4-1 I
100
..........
(3)
259
(4)
I
I
-- 0 °
"',,.. ........_45 °
,,,,'""" 45 ° .......
rn
-~ >
50-
"'"x
, ,. ,,, , ,,.
o/ / ////"''"
i,... co
co
t,..
Q. "0
e-
O-
o
-90 -5010 0
90 °
50
0
-50
0
50
100
Sound pressure level (dB) Power reduction (dB) Uncontrolled ............. 1 Piezo ..........................2 Piezos 3 Piezos 4 Piezos
0.0 18.9 30.1 30.1 51.4
Fig. 8.18 Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 128 Hz.
and amplitudes which interact to cancel the overall radiation in the far field. The closed loop structural response now has lower overall radiation efficiency even though its overall vibrational amplitude has not been attenuated (in some cases it may also increase!). A modal decomposition of the response shows that the modal amplitudes of the dominant sound radiators are not significantly changed. This second mechanism of control, where the overall plate vibration amplitude is not significantly attenuated or sometimes increased while supersonic wavenumber components, and the associated sound radiation are reduced, we term control by modal restructuring (Fuller et al., 1991). An alternative point of view of modal restructuring proposed and analysed by Burdisso and Fuller (1992a) is that the controlled structural response has new eigenvalues and eigenfunctions which have lower radiation efficiency. Burdisso and Fuller (1992a) also demonstrate that these controlled eigenfunctions are nonvolumetric. The above work has shown that global control of planar far-field sound radiation always corresponds to a fall in the supersonic components of the plate structural wavenumber spectrum. Furthermore, as discussed by Junger and Feit (1986), the radiation pressure from a plate at a particular angle can be shown to be directly proportional to a particular plate wavenumber component. This suggests that an
260
ACTIV]~ CONTROL OF
VIBRATION
(~)
(a)
(4)
+'4:=1:: ii :--1: llllllllllllll lllllllllllll
60
~i
~,
.lilt
40 ("
~,~,~.,,=,,=~,=~,=~...... Ill/
~=
"="= == =,.,,,,=
A
tiig~qa,
\
I
__O"i o
k
I
I
I
lo
2O
30
40
kxlm =1) Urtoontrollocl ............. 1 Pie=to ..........................a Plozoe . . . . . 3 Plozoo ....... 4 Piozoo
Fig. 8.19 Wavenumber spectrum of plato response for diffaront numbors of piezoelectric ceramic actuators, f-~ 1:28Hz or k -=2.345 m =t, Note values are only shown for positive k~:. alternative control formulation for minimising sound radiation would b~ to e×press the cost function in the wavenumber domain and suppress discrete (or bands of) wavonumbors, Fuller and Burdisso (1992) have analytically formulated a wavonumber control technique that suppresses radiation towards particular anglos. Clark and Fuller (1992a) have o×porimontally and theoretically demonstrated a wavenumber domain controller that is designed to minimiso the band of wavonumbors enclosed within the supersonic circle. This approach shows much potential. However, its practical implementation is dependent upon development of realistic, time domain, wavenumbor structural sensors that work over a broad frequency range. Work of this nature has been carried out by Maillard and Fuller (1994). Finally, Fig. 8.20 presents the radiated power with and without control over a range of frequencies from 0 < 800 Hz, Peaks in the uncontrolled power are associated with modal resonances. It is apparent that good control is achieved with a single actuator for very low frequencies, where f-~ 150 Hz. However, in order to provide good broadband power reduction it is necessm~ to use an array of appropriately positioned actuators. Oood attenuation is obtained over a frequency range of 0 ~f~650 with four actuators with the exception of the peak near 330 Hz, Table 8,2 indicates that this range encompasses the resonance frequencies of 13 structural modes and thus the result demonstrates the efficiency of ASAC.
261
ACTIVE STRUCTURAL ACOUSTIC CONTROL, IPLATE SYSTEMS
(~,)
(~)
(?)
(4)
180
~=
~=0
6O
o I I
=60
I
I
I
I
I
I
I
1O0
200
300
400
BOO
600
700
800
Fmqu#noy (Hz) Unoontroll~d ............. t Plezo ..........................2 Pl~zom
..... .......
Fig, 8,20
8.8
3 Pl~zo~ 4 Pl~zo~
Radiated ~ound power for different numbor~ of piezoelectric ceramic actuator~,
The use of plezoele~ric distributed structural error sensors in ASAC
In many applicationsof ASAC the use of microphone error sensors is impracticable.In this case it is desirable to configure error sensors which provide estimates of far-field pressure when located on, or very close to, the radiatingstructure.Various desisns hays been suggested in order to implemem this concepL Baumann ct el. (1991) have simulated the use of radiationfilterswith good success to modify the structuralstates of systems m radiationstatesin feedback implememations of ASAC. Clark and Fuller (1992b) have used a model reference implementation of the feedforward LMS all~orithmin which the vibrations of the structureare driven to a reference vector (i,e. not zero) corresponding to minimum acoustic power radiation. Fuller and Burdisso (1992) have also cortsidereda paradigm based upon a wavenumber domain approach in which Sul~rsonic structural wavenumber componems am comrolled, loading to minimisation of radiated pressure at selected anglos, In thi~ ~oction we di~cu~ the u~e of rectangular piezoelectric distributed ~tructural sensors as appropriate error sensors in ASAC. In particular we are interested in implementing structural error sensors for control of sound radiation from two-
262
ACTIVE CONTROL OF VIBRATION
dimensional plates. The use of distributed strain actuators and sensors integrated directly on or into the structure in conjunction with a 'learning' type adaptive controller implies that this configuration falls into a class of systems which are part of a rapidly expanding, related field known as adaptive, smart or intelligent structures. The reader is referred to the review paper by Fuller et al. (1992) for more information on the use of adaptive structures for controlling sound radiation. Recalling equation (5.7.4) which represents the charge output of a two-dimensional element of piezoelectric material bonded to the surface of the plate, we must simply define the area of application, F(x, y), to obtain the sensor electrical response as a function of the plate response. In the case of a rectangular sensor element this function is defined as F(x, y) = [ H ( x - Xel)- H ( x - x g ) ] [ H ( y - ye) _ H ( y - y~)],
(8.8.1)
where H(-) is the Heaviside unit step function of the spatial coordinates. Substituting this expression into equation (5.7.4) and integrating over the area of the sensor yields the charge output of the rectangular sensor given by
qe(t)
= (h + She) Mm~l = Nn~lWren[e31 ma nb + e32 × (COSkny e2 - c o s k , yl) e ~_jot,
(COS
kmx2- c o s kmx~) (8.8.2)
where (x~,y~) are the coordinates of the lower left comer of the rectangular piezoelectric element (viewed from the front of the plate), (x~, y~) are the coordinates of the upper fight comer of the sensor and h e is the sensor thickness. An examination of equation (8.8.2) reveals that the sensor charge output is composed of summation over the modal contribution. The degree to which a given mode contributes to the total sensor output is proportional to modal amplitude Wm, and is also related to modal order (m, n) as well as the sensor size and position. Thus different sensor shapes and positions can be used to selectively observe modes or combinations of modes. In order to modify the quadratic optimisation procedure, the output of the sensor is used as an error signal. Thus the control cost function is modified to be Ne g =
Z Iv; I
(8.8.3)
i=1
where it is assumed that voltage output of the sensor Ve is proportional to the charge generated by the s e n s o r qe and N e is the number of independent sensors used. As a preliminary example, we study the use of two distributed PVDF structural sensors shown approximately to scale in Fig. 8.21. The shape of these sensors is based upon the observation that, in general, odd-odd plate modes (e.g. (1, 1), (3, 1), etc.) are more efficient radiators than the even-even modes (e.g. (2, 2), (4, 4) etc.) at frequencies well below the critical frequency of the plate (see Section 8.2). Substituting the form of the sensor shapes of Fig. 8.21 in equation (8.8.2) reveals that the sensor denoted PVDF2 will only observe the odd modes in the x direction, while PVDF1 will only observe the odd modes in the y-direction. Thus if the outputs of both sensors are minimised then the structural motion of the efficiently radiating modes should be minimised (rather than minimising the total response) and this should lead to a fall in radiated sound. Figure
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
263
Plate
J
::::::::
iiiiiiii
iiii iiiii
piezo 1
~iliii
~
iiii
/ PVDF2
300 mm NJNI --- Disturbance
!i!iiiiii :i:i:i$
PVDFI
iiiii!iii iiiil/
Piezo 2
iiiii --------~ X
380 mm
P,
Fig. 8.21 Simply supported rectangular plate with piezoelectric ceramic actuators and piezoelectric PVDF sensors.
8.22 shows a typical analytical result for a f r e q u e n c y o f 349 Hz for the plate system considered in the previous sections. In this e x a m p l e the plate is excited by a point force located at x = 240 m m and y = 130 m m and two piezoelectric actuators were used for control, as shown in Fig. 8.21 (see Fig. 8.10 for coordinate specifications). The piezoelectric sensors were assumed to be m a n u f a c t u r e d f r o m polyvinylidene fluoride ( P V D F ) material with typical material properties given in Table 8.3. A p p r o x i m a t e dimensions and locations o f the sensors are s h o w n to scale in Fig. 8.21. Also s h o w n in Fig. 8.22 are the controlled results w h e n three error microphones were used that were located at mid plane (i.e. y = b/2) radiation angles o f 0 = +45 °, 0 ° and - 4 5 ° and at 0
0o
/
//
/ /
,,",-" ...... , ",, ,, .... ° , -,
,,'" /
k\
\
"'""
X
"
I
-90 ° 80
90 ° 60
40
20
0
20
40
60
80
Radiated sound pressure level (dB)
Fig. 8.22 Radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).
264
ACTIVE CONTROL OF VIBRATION Table 8.3 Typical piezoelectric polymer sensor properties (PVDF). E, = 2 x 109 N m -2 p,= 1.78 x 103 kgm "3 h,. = 0.16 mm
e3~= 65.3 x I0 "3 C m e32= 38.7 x I0 -3 C m e36=0
-2 -2
R = 2 m. The results show that the PVDF structural error sensors provide of the order of 10-15 dB sound reduction. It should be noted here that simply minimising the vibration at two points often leads to sound radiation increases, as demonstrated in the experiments of Metcalf et al. (1992). The control performance can be seen to be not as good as using microphones and this suggests that the P V D F sensors were not correctly placed to properly observe the structuralmotion associated with radiation. Various strategies have been developed for designing P V D F structural sensors in order to correctlyweight the structuralmodes for sound radiationcontrol (see the work of Clark et al., 1992a). The work by Wang et al. (1991) and Clark and Fuller (1992c) has demonstrated that for higher modal densities of response, optimising the shape and location of the control transducersis of the same order of importance as increasingthe number of channels of controlin terms of obtaining global reduction.In one particular example examined by Clark and Fuller (I992c) for a SISO system with relativelyhigh modal density,the optimal actuatorlocationwas found to be near the comer of the plate where itcould effectivelycouple into multiple structuralmodes. The optimal sensor was a long narrow strip located at the bottom edge of the plate where it could observe structuralmotion associatedwith the edge radiationphenomena discussed in Section 8.2. The resultsof both Wang et al. (199I) and Clark and Fuller (I992c) suggest that for more complex, realisticsystems the actuator and sensors will have to be optimally located to obtain reasonable performance with a low number of transducers.The work of Clark et al. (1992a) also demonstrates that extreme care must be taken in accurately manufacturing and positioningthe distributedsensor shape. Section 8.4 has demonstrated how the sound power radiation of a plate can be described in terms of a set of velocitydistributionson the structurewhose sound power radiation is independent of the amplitudes of the other velocity distributions.These velocity distributionsare termed 'radiationmodes' and are obtained using an eigenvalue decomposition as described in Section 8.4. The form of these radiationmodes suggest an approach for designing distributedstructuralsensors which are shaped (as described in Chapter 5) to respond only to the velocity distributionscorresponding to these radiation modes. Such an approach is only strictlyvalid in the low frequency region ka,t I since at higher frequencies the shapes of the radiation modes depend more strongly on the excitation frequency. Nevertheless both Snyder et al. (1993) and Johnson and Elliott (1993) have developed and successfully tested distributedP V D F sensors designed to observe these radiationmodes for the active control of structurally radiated sound. As noted in Section 8.4, by far the most important radiation mode at low frequencies corresponds to the net volume displacement of the plate (see mode (I) of Fig. 8.9). Differentiatingthe signal from the sensor which detected this distribution would give the net volume velocity of the surface, and a P V D F distributed sensor which measures the volume velocity was originallysuggested for A S A C by Rex and Elliott (1992). The shape of the resultantsensor is seen to be almost identicalto that obtained by the design procedures of Clark et al. (I992a).
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
265
Burdisso and Fuller have pursued a different but related approach to the design problem for distributed structural sensors and actuators in ASAC. Burdisso and Fuller (1992a,b) demonstrated that feedforward-controlled radiation from a structure can be described in terms of new eigenvalues and eigenfunctions for the controlled system. Using this formulation Burdisso and Fuller (1994b) developed an eigenassignment design technique by which the actuator and sensor shapes can be designed so as to cause the closed loop system to behave in a controllable manner. In order to minimise sound radiation from a structure excited by multiple frequencies, Burdisso and Fuller (1993, 1994a) used the eigenassignment procedure to design actuator and sensor locations and shapes so that controlled structural modes were radiators of low acoustic power (and thus had a very low radiation efficiency at frequencies well below the plate critical frequency) and the closed loop resonances (i.e. at the closed loop eigenfrequencies) were detuned from the excitation frequencies. The controlled structural mode shapes were shown to be independent of frequency. The procedure enables simultaneous design of the sensor and actuator shapes and position. Analysis and experiments with this procedure demonstrated high, global attenuation of structurally radiated sound over a wide band of frequencies.
8.9 An example of the implementation of feedforward ASAC In this section we briefly discuss a typical arrangement of ASAC in order to illustrate how such systems are implemented in practice. Figure 8.23 shows a schematic layout of the experimental fig and associated control system. The structural system consists of a baffled simply supported plate located in an anechoic chamber. The disturbance to the plate is provided by a point force actuator mounted on the plate through a force transducer and driven by a steady state single frequency. Control inputs are achieved by two piezoelectric ceramic actuators bonded to the plate in the required positions (actuator in this case implies two symmetrically located piezoelectric wafer elements driven 180° out of phase, as discussed in Chapter 5). Two different sensor arrangements are employed; either three microphone error sensors located at (R, 0, ¢~) coordinates of (2m, +45 °, 0°), (2m,-45 °, 180°) and (2m, 0 °, 0 °) or two PVDF strip sensors attached to one side of the plate are used. Figure 8.21 shows the locations and relative size of the piezoelectric actuators and sensors. The configuration corresponds to the analysis of the Section 8.8. Note that for graphical convenience we use 0 = ±45 ° to indicate ¢~= 0, 180°. A signal generator was used to create the harmonic disturbance and the signal was amplified to drive the shaker. The same signal was passed through two adaptive filters to provide the control signals to the piezoelectric actuators. The coefficients of the adaptive filters were updated at each time step using the multi-channel version of the filtered-x LMS algorithm outlined in Chapter 4, in order to drive the error signals to a minimum. Note that, as discussed in Chapter 4, the reference signal has to be prefiltered with estimates of the transfer functions (G~, G~2 etc.) from each actuator to each sensor. These transfer functions are measured at the frequency of interest before carrying out the experiments. As the tests are for a single frequency of excitation only two coefficients are needed in the adaptive and fixed FIR filters. The instantaneous values of the error signals are also required in the update equation to compute the instantaneous gradient estimate. These variables are used to adapt the filter coefficients in the filtered-x LMS update equation presented in Section 4.7.
266
ACTIVE CONTROL OF VIBRATION ) Reference input Disturbance ~__ Simply supported plate shaker 1 - ~ ~ ' - "
1~ ~
PZTactuat°r II
/
r Adaptive filterA
l error sensors I .
.
.
.
.
.
[~apt,ve,,,ter Fixed filters •.-,"~' Gll I ~~1
~i ,MS h/ q algorithmP= I P--q G21 ~ i G22 I - ~ ~-~ G2z
,~11[
LMS E I 1algorithm IZ i r~-
Fixed filters
Fig. 8.23
Schematic layout of controller and test rig for an ASAC experiment.
In order to evaluate the control performance, the radiated field was measured with a movable microphone traverse centred on the plate and with a radius of 2m (the radius being limited by the dimensions of the anechoic chamber). The output of the traverse thus provides radiation directivity plots in the b/2 mid plane of the plate. The magnitude of the disturbance force was also measured in order to provide an absolute comparison between theory and experiment.
0
0o
/')
-9oo 80
i, 60
40
X\
"
, 20
\
/, 0
";,i ,,X 9oo 20
40
60
80
Radiated sound pressure level (dB)
Fig. 8.24 Measured radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).
267
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
Figure 8.24 shows experimental results for a frequency of 349 Hz. When the results are compared to the theoretical predictions of Fig. 8.22 (with a disturbance amplitude set to that measured in the experiment) reasonable agreement is apparent in terms of shape, absolute values and trends for the different types of sensor. This brief example is intended to introduce the reader to the practical implementation of ASAC. Further details of such experiments can be found in the work of Clark and Fuller (1992a). Smith et al. (1993) have also extended the above application to control of broadband plate radiation using integrated piezoelectric actuators and sensors.
8.10
Feedback control of sound radiation from a vibrating baffled piston
Hitherto in this chapter, attention has been restricted to feedforward control of sound radiation from structures where there is a reference signal available which characterises the primary disturbance. Here we begin a discussion of the potential for feedback control of sound radiation from structures where the excitation is neither deterministic nor can be detected at some time well before it excites the structure (a turbulent boundary layer for example). An important contribution to this field of study is that made by Baumann et al. (1991) who introduced the concept of a radiation filter which enables the evaluation of the sound power radiated from a vibrating surface directly from the properties of the surface vibration. The concept is most easily explained with reference to a simple SDOF model of a vibrating piston in an infinite baffle that is illustrated in Fig. 8.25. This model has been used by Thomas and Nelson (1994a) in order to investigate the potential for feedback control of stiff lightweight structures when used as a secondary 'trim' in passenger aircraft. Such stiff light panels can be designed in order to have elastic modes with a relatively high natural frequency such that the control problem is considerably simplified. The use of this type of structure has already shown some benefits in feedforward control of low frequency sound transmission (Thomas et al., 1993a; Cameal and Fuller, 1993) and shows promise when feedback control is applied (Thomas and Nelson, 1994b). Using the model illustrated in Fig. 8.25, the equation of motion of the rigid circular piston can be written as
Mff~(t) + Cvg(t) + Kw(t) = fp(t) + fs(t),
(8.10.1)
where M is the mass of the piston, C and K are the damping and stiffness of the suspension and fp(t) and fs(t) are the 'primary' disturbance and 'secondary' control force respectively. This equation can be written in matrix form as [ ~"= ('!( t]) [-C/M1
-K~M ] (,)LW(,) r ~i, ]+[1/oM]fs(,)+ [l/oM]fp(,) ,
(8.10.2,
which is thus in the state space form (see Section 3.6) ~(t) = Ax(t) + Bu(t) + Av(t).
(8.10.3)
The equation of motion is thus 'forced' by a vector of external inputs v(t) which, in this case, is simply the scalar primary force input fp (t). If it is assumed that this input is simply Gaussian white noise, then we can again apply LQG regulator theory (outlined
268
ACTIVI~CONTROLOF VIBRATION
Random oxoltmtlon, v (t)
Radiation fllt,r ~
Control foroe, u (t )
g(t)
Diaplaooment, w (t)
Fig, 8,25 A circular, baffled piston mountod on a spring and a damper, The radiation filter has an input given by the piston velocity and an output whoso moan ~quarod output is the acoustic power radiated, in Section 7,10) and find the control input that, for o×ample, minimizes a quadratic cost function such as that specifi¢d in ~quation (7,10,4), As an e×ample, Thomas and N~l~on (1994a) cho~o to minimize tho cost function y-- ~[¢~(t) + a~(t)], (s,~o,4) which is thu~ tho ~um of tho moan squared piston voloeity and tho moan ~quamd ~eondary for¢o woightod by tho factor a, Thi~ factor ponali~ the control 'effort' u~od, Tho r~ult~ (Fig, 8,26), which chow the apaetral dandify of the piston volocity when control is appliod, iUuatrat~ the dop~nd~ne~ on th~ ehoic~ of the w~ighting factor a, Clearly, th~ lower the value of a, the higher the value of the control gain (~e Seetion~ 7,10 and 7,11), Minimization of moan ~quaro velocity, however, does not imply minimization of acoustic power radiated*, A direct measure of the acoustic power can b~ d~dueod from a knowledge of th~ piston velocity by pa~ing thi~ signal through a radiation filter who~o timo avoragod ~quar¢d output i~ o×ttctly equal to the acoustic power radiated, Defining the output of the filt~r to b~ ~(t), the acoustic power can b~ written as
l~ = E[~(t)],
(s, ~o,~)
If the filter i~ a~sumod to havo a frequency r¢~pon~o function density of the acoustic power radiatod i~ given by
s.cw)--- s..(,o)-- I
I
G(jto),
then th~ ~p¢ctral
(s.ao.6)
t Minimizationof pistonaccelorationwouldof cour~oprovid~~ b~tter~pproxim~tionto the minimizationof ~ou~fl~ poworr~di~ted,but ~h~u~eof th~ volo~ity~ign~lher~help~explainth~ ~on~optof thQr~diationfiltor,
ACTIVE STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTI~MS
269
=10 open loop 1 xlO =3
=16 t
=20
~
=25
1 xlO =6
=30
1 xlO 4
~=35 =40
1 xlO =8
=45 0
0,2
0,4
0,8
0,8
1
1,2
1,4
1.8
1,8
2
Fill, 8.26 Optimal reduction in piston velocity for values of a between 10 =-~and 10 =~, Results show the ratio of the spectral density of the piston velocity to the spectral density of the input disturbance force. where S,o~(co) is the spectral density of the piston velocity signal. The spectral density of the acoustic power is a real, even and positive function of frequency. This implies that we can find an approximation to S,,(~o) that is in the form of a rational spectrum. That is, we can write
N(coa) S~(co) - DCwa) ,
(8.10.7)
where N(co a) is a polynomial of order q in ¢o~ and D(co 2) is a polynomial of order p in ma. A simple example of a rational spectrum is given by S.(co)
1
-
~ ,
(8.10.8)
o)4+a
where a must be real since S, (co) is real. Now note that if we assume the piston velocity signal to be white noise of unit spectral density (i.e. S,~,~(¢a)= 1) then S,(w)---I G(jco)I a, the modulus squared of the radiation filter. We now write S,(co) as the product of two factors such that
S,,(o)) = O(jo))O* (jo))-- O(joa)O(-jo)).
(8.10.9)
Thus, for example, we could express equation (8.10.8) in the form I
I
S~(oa),, (b+jco) (b-jm)
I
I
(b*+ja~) (b*-jm)
(810.101
where b is the complex number given by b= (a/4) ~14+j(a/4) ~14.The spectral factor G(jco) is simply l/[(b+jco)(b* +jco)] which thus specifies the frequency response function of the radiation filter. This also implies that we could express the transfer
270
ACTIVE CONTROLOF VIBRATION
function of the radiation filter as the more general Laplace transform G(s)= 1/[(b + s) (b* + s)] where the Laplace variable s = cr +jw. It turns out that this type of decomposition, known as spectral factorisation, is generally applicable to rational spectra since the power spectrum must be a real, even, non-negative function of o9. This subject is dealt with in more detail by Van Trees (1968) who points out that the pole-zero plot of G(s) G(-s) must have the following properties. (1) Symmetry about the jw axis (otherwise S~(w) would not be real). (2) Symmetry about the cr axis (otherwise S,(w) would not also be even). (3) Any zeros on the jw axis occur in pairs (otherwise S~(w) would be negative for some value of co). (4) There are no poles on the j to axis. As a direct consequence of these properties we can, in general, find G(s) such that all its poles and zeros are in the left half of the s plane (or <0), while G(-s) has all its poles and zeros in the fight half of the s plane (or>0). Since G(s) has all its poles and zeros in the left half of the s plane, it is guaranteed to be a stable, causal filter. Furthermore, it is minimum phase and will have a stable inverse (see Nelson and Elliott, 1992, Chapter 3). Thomas and Nelson (1994a) used a least squares method to adjust the coefficients of a rational Laplace transform with sixth-order numerator and denominator in order to match the well known variation with frequency of the acoustic radiation resistance of a rigid circular piston. The radiation resistance curve (which exhibits the frequency dependence of S~(w) when the piston velocity is a white noise signal) is shown in Fig. 8.27. The pole-zero plot of S~(s)= G(s) G(-s) shown in Fig. 8.28 provides a fit to this curve to within 0.01% up to ka = 2.5, where a is the piston radius and k = tO~Co. Note that the symmetry of the pole-zero plot complies with conditions (1) and (2) given above. Taking the left half plane poles and zeros shown in Fig. 8.26 to define G(s) one can use one of several techniques to find a state space realisation of the radiation filter with this transfer function (see, for example, Maciejowski, 1989). Such a realisation can be written as il(t) = Asrl(t) + B sw(t),
(8.10.11)
z(t) = Cg11(t ) + Dg'/v(t).
(8.10.12)
Thus rl(t) is the vector describing the 'states' of the radiation filter while z ( t ) = z(t), the scalar output of the radiation filter and "/v(t)= W(t) is its input. This realisation can then be combined with the state space model of the piston dynamics given by equation (8.10.3) such that
[A 0][x,t,] ["0] [o]
il(t ) = Bs),
Ag
n(t)
z ( t ) = [Dg¥ where ¥= [1
+
u(t) +
x(t)] ' Cg] ~l(t)
v(t),
(8.10.13)
(8.10.14)
0]. This state space model has an output z(t) whose time-averaged
271
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
1.2
I
I
I
I
I
I
I
I
I
I 0.4
I
I
I
I
I
I
I
0.2
0.6
0.8
1 ka
1.2
1.4
1.6
1.8
0.8 0 tO O
Q. 0.6
a:
0.4 0.2 00
Fig. 8.27
2
The radiation resistance of a plane circular baffled piston of radius a. I
I
I
I
I
I
o
I
I
I
I 3
! 4
o
21
#0
-
X
-1 -2
--
-3--
-4
o
--
-5 -5
I -4
I -3
I -2
o I -1
I 0
I 1
I 2
a a/c O
Fig. 8.28 The pole-zero diagram for the rational Laplace transform approximation found for the radiation resistance of a plane circular baffled piston. Poles are represented by x and zeros by O. Note that there are a pair of zeros placed close to the origin. squared value is equal to the acoustic power radiated. Therefore the optimal control that minimises the cost function
J= E[zT(t)z(t) + auX(t)u(t)]
(8.10.15)
will minimise acoustic power radiated plus a factor a times the mean square effort used in the control. Thomas and Nelson (1994a) have computed the solution to this problem and the results are illustrated in Fig. 8.29. These clearly show the difference between the
272
ACTIVE CONTROL OF VIBRATION
130
I
I
~2ol
I
loop
110 Vibration control
~100
Radiation control
,o "0
8o
7O 8o
5O
40 30
0
0.5
1.O /ca
1.5
2.0
Fig. 8.29 Soundpower radiated by a plane circular piston without feedback (open loop), with optimal feedback minimising the squared velocity of the piston (vibration control) and with optimal feedback minimising sound power radiated (radiation control). The mean squared velocity of the primary excitation is 0.083 m s "2. results produced by choosing the control that simply minimises vibration and those produced by choosing the control that minimises acoustic power radiation. The results shown in Fig. 8.29 are for the same value of time-averaged control power used in the two cases (which used two different values of a). The frequency weighting provided by the radiation filter's representation of the radiation resistance curve is clearly evident.
8.11
Feedback control of sound radiation from distributed elastic structures
In this section we discuss the active control of sound radiated from a structure exhibiting multi-modal response using feedback control, following the approach of Baumarm et al. (1991) introduced in the previous section for an SDOF system. It was shown in Section 8.4 that the acoustic power radiated by a harmonically excited structure, whose velocity distribution is described by the sum of contributions from N structural modes, can be written in frequency domain form as (8.11.1) l-IC~) = ~'"C~)MC~)~(~), where ~ ( ~ ) is the vector of N complex structural mode velocity amplitudes, and M(o~) is a matrix of self and mutual radiation resistances (which is real and symmetric) while superscript H denotes the Hermitian operator. We can factorise M (~) into the form G" (~)G (~), so that the radiated power can also be written as (8.11.2)
273
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
Defining the vector of transformed modal amplitudes G(co),~(to) to be z(to), equation (8.11.2) can be written as FI (oj) -- ZH(CO)Z(0~).
(8.11.3)
The acoustic power radiation due to any one element of z(to) is thus independent of that due to each of the other elements of z (to) (Borgiotti, 1990; Cunefare, 199 I). This is in contrast to the power radiation due to each structural mode amplitude, which is generally dependent on the amplitudes of many other structural modes, as accounted for by the off-diagonal mutual radiation terms in M(~). This procedure has been discussed in more detail in Section 8.4. If the structure is subject to a transient excitation, the total radiated acoustic energy can then be written as E = [" ",VH(~)M(w) W(co) dto ,/ 0
~
(8 11.4) '
where . ( t o ) is the Fourier transform of the vector of the waveforms of the structural mode amplitudes ",V(t). The evolution of these time histories can be described in state space form by defining a state vector x(t) (as described in Section 3.6) which includes the velocities ~(t), and, for example, the displacements w(t) associated with each of these velocities, which then obey the state space equation :¢(t) = Ax(t) + Bu(t),
(8.11.5)
where A is the state matrix and u(t) is the vector of inputs to the system due to the secondary forces. The radiated energy equation (8.11.4) can also be written, using equation (8.10.3) as E
= I"0
ZH(~0)Z(0J)do~P
(8.11.6)
where at each frequency we again have z(m) - G (m),,V(m)
(8.11.7)
and G(to) can now be identified as the matrix of frequency response of the 'radiation filters' which operate on the structural mode amplitudes to give each of the elements in z(to). Baumann et al. (1991) demonstrate that these 'radiation filters' can be chosen to be causal, and thus realisable (to arbitrary accuracy) in state space form. The procedure to be adopted first requires finding a rational Laplace transform approximation to the elements of M(to). The algorithm provided by Francis (1986) can be used to carry out the necessary spectral factorisation. Note, however, that the algorithm can only be used provided that the rational Laplace transform approximation to M(to) is also positive definite. The state equations of this array of radiation filters can be written as t'(t) = Acr(t ) + Bcx(t ),
(8.11.8)
z(t) = Ccr(t) + Dcx(t ),
(8.11.9)
in which r(t) is the vector of state variables of the radiation filters, and x(t) is the vector of state variables for the structure, as defined above. These two sets of state
274
ACTIVE CONTROL OF VIBRATION
variables can be combined to give an overall state variable model for the radiating system -
[A 0][x]+[.] Ba
Aa
r
0
u,
(8.11.10)
with a corresponding output equation z=[Dc
CG][X].
(8.11.11)
Now that the complete system, including sound radiation, has been cast in state space form, the standard tools of 'modem' control theory can be used to calculate the optimal LQG feedback gains, as described in Section 3.10. The cost function which must be minimised is of the form Jrad = Io [zT(t)Z(t) + auT(t)u(t)] dt,
(8.11.12)
where the first term is the total sound energy radiated by the structure expressed as an integral over time, which must be equal to the frequency domain form, equation (8.11.6), by Parseval's theorem. The second term in equation (8.11.12) must be introduced to make the equations soluble, and may be physically interpreted as being proportional to the control effort, i.e., the sum of the squared inputs to the secondary sources. The optimal set of full state feedback gains which minimises this cost function is the solution of a steady state Riccati equation (see Section 3.10), and the resulting feedback control law can be expressed in the form u(t) = K ~ x ( t ) ,
(8.11.13)
where x = [x TrT] T is the full state vector, and K rad is the optimal feedback gain matrix. Baumann et al. (1991) present some numerical results of the effect of such optimal feedback control of sound radiation on a beam excited by a short duration pulse. The 1 m x 0.125 m steel beam was assumed undamped, clamped at both ends and supported in an infinite baffle. For simplicity it was assumed that the structure and its sound radiation could be accurately modelled by using the first three structural modes only. A single secondary actuator was located half way along the beam, and driven by each of the full state variables via the feedback gain matrix defined by equation (8.11.13). The resulting velocities of the three structural modes are shown in Fig. 8.30(a) (Baumann et al., 1991, Fig. 4). It can be seen that the first (volumetric) mode, mode 1 and the third mode, mode 3, are most strongly controlled using this feedback strategy. This is because the second mode, mode 2, acts as an acoustic dipole at frequencies close to its natural frequency and is thus not an efficient sound radiator. Controlling this mode will thus have little influence on the total acoustic radiated energy. In contrast to this radiation control strategy, the optimal feedback problem can also be solved to suppress the mechanical vibration of the beam. This can be achieved by using the LQG theory described above to minimise a cost function of the form Io [xT(t)X(t) + a2uT(t)U(t)] dt,
(8.11.14)
ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS
275
0.025 O.02
(a)
0.015 ~,
,, :':
0.01!
/I
"~ o.oo5i
i~/: !
~o
i~.
0 i -0.005
'..\~
! ! i /i
i
.--.
, i
..
"..v
" ""--- ' " -
' '....
: :
"~:............ v
"
/
:~ -o.ol i 'i~ -0.015 -
0 •02 ] /
-0.025 /
i
0.02 ~
.
0.015 t [
i~/(~i
i
i
.
i
.
t
.
t
.
~ Mode 2 ..... Mode 3 t i
I
/ / J
. (b)
o0 ii ii i i i
>"
!t i i :
i I. i/!
o ~ ii £,
-ooo5
i i
.~ :. :~!
i i
'i
!'" !
:
.z i
. :: "
~i
i
i ii'ii
il
i
~..
!,~i
"~
!/i
i i
ii
!
-0.015 ~ 1 / ii ! :'
V
-001:
:-
~,:
i
"
:i
~!
ii
:.
.".,
"~ ---
l
I 0.1
""'.
"
"--
""
- -
-0.02 / 0
~--
..
"'~
Mode 1
~
Mode 2 ..... Mode 3
t 0.2
i 0.3
i 0.4
t 0.5
i 0.6
i 0.7
i 0.8
t 0.9
1
Time (sec)
Fig. 8.30 Velocities of the three structural modes of the beam modelled by Baumann et al. (1991) to demonstrate state feedback control of (a) sound radiation and (b) structural vibration. which results in a feedback law of the form u(t) = KV~bx(t).
(8.11.15)
In the results presented by Baumann et al. (1991), the weighting on the effort term in equation (8.11.14), a z, was adjusted to make the total energy used by the controller
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ACTIVI~ CONTROL OF VIBRATION 70
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Frequenoy (Hz) Fig. 8.31 Totalradiatedpower without statefeedback (====) and with statefeedback (= ==) two control forces placed at x - 0.1 m, y -- 0.1 m and x - 0,25 m, y - 0.125 m with a - 10=°. Results presented from Thomas and Nelson (1993) for the predicted reduction in sound power radiated from an aluminium plate excited by a turbulent boundary layer, The plate was simply supported, measured 0.5 x 0.25 m and was 1 mm thick. A structural damping factor of 0.01 was assumed.
equal to that used in the simulation above. Figure 8.30(b) shows the velocities of the three modes modelled in the computer simulation when implementing feedback control of vibration (equation (8.11.15)). In this case, all three structural modes have been controlled to approximately the same degree, with the result that the third mode, in particular, rings for considerably longer than in Fig. 8.30(a). Baumann et al. (1991) state that the total acoustic energy radiated when using feedback control to suppress vibrations (Fig. 8.30(b)) was 38% greater in these simulations than when feedback control was used explicitlyto suppress radiation (Fig. 8.30(a)). A similar formulation has also been used to analyse the feedback control of sound radiation from a structure excited by random disturbances (Baumann et al., 1992). Thomas and Nelson (1993) have also used Baumann's theory to examine the feasibility of providing active control of the sound power radiated from a simply supported flexible plate excited by a turbulent boundary layer. An example of the results they derived is shown in Fig. 8.31 which demonstrates the reduction of sound power radiated that can be in principle achieved with optimal feedback control for an aluminium panel typical of those used in aircraft fuselage construction. The results presented are for a specificchoice of 'effortweighting' a in the cost function used; the reductions produced were found to be crucially dependent on this choice and thus the control gains used, More detailsare presented by Thomas and Nelson (1995).