IDENTIFICATION AND CONTROL OF ACOUSTIC RADIATION MODES

Transcription

1 IDENTIFICATION AND CONTROL OF ACOUSTIC RADIATION MODES Arthur P. Berkhoff University of Twente, Faulty of Eletrial Engineering, P.O. Box 217, 7 AE Enshede, The Netherlands a.p.berkhoff@el.utwente.nl and TNO TPD, Aoustis Division, PO Box, 26AD Delft, The Netherlands berkhoff@tpd.tno.nl Abstrat A formulation is given of redued-order aousti radiation sensors and redued-order atuators for broadband sound fields. Methods are presented to determine these desriptions from measured data, and their appliation in systems for broadband ative noise ontrol is disussed. One appliation area is the redution of sound radiated from plates with strutural atuators and strutural sensors, using measured or modeled versions of the most effiiently radiating patterns of a vibrating body, the so-alled radiation modes. The seond appliation of the radiation mode theory is in ative noise barriers for the redution of traffi noise. Without speial preautions most of these systems suffer from spillover; a tehnique is given to arrive at good redutions at the error sensors with redued spillover. 1 Introdution In problems in whih the radiated sound has to be redued it is of interest to know the vibration patterns that radiate most effiiently (C.R.Fuller, S.J.Elliott and P.A.Nelson, 1996). In strutural aoustis problems, one approah is to determine the radiation effiieny of eah strutural mode independently (W.T.Baumann, W.R.Saunders and H.H.Robertshaw, 1991). In many ases it may be more eonomi to rewrite the problem in terms of a set of virtual vibrations patterns that radiate sound most effiiently (G.V.Borgiotti, 199; S.J.Elliott and M.E.Johnson, 1993). This approah an be used to arrive at sensing and atuation shemes with redued order. A few pratial implementations have been desribed in the literature (G.P.Gibbs, R.L.Clark, D.E.Cox and J.S.Vipperman, 2; E.J.J.Doppenberg, A.P.Berkhoff and M.v.Overbeek, 21). All these methods, also the pratial implementations, are still based on models of aousti radiation, suh as the sound radiation from baffled plates. One reason is that it is not very easy to obtain the radiation modes in pratie (A.P.Berkhoff, E.Sarajli, B.S.Cazzolato and C.H.Hansen, 21), at least not for monohromati sound radiation. Reently, a modifiation of the radiation mode theory has been presented (A.P.Berkhoff,

2 22), whih allows robust identifiation of the radiation mode shapes by using a broadband formulation. In the present paper a formulation is given in the time domain in terms of operator notation and farfield desriptions. The onditions are given under whih one of the measurement proedures leads to valid results. 2 Formulation of the problem As disussed by many authors, suh as (A.D.Piere, 1989), the frequeny domain sound field of a vibrating body in an aousti fluid an be desribed by the Helmholtz integral equation. Upon solution of the equation for the pressure on the vibrating body, the Helmholtz equation an be used to evaluate the aousti far field. In the time domain the integral equation for points x on the body S is given by ( ) ( ) 1 t v k x,t x x 2 p(x,t)= p x,t x x ρ 4π x x k 4π x x ν kdx, x S (1) with p and v k,k =1, 2, 3 the aousti pressure and aousti partile veloity, respetively, ρ the density of air, the speed of sound, ν k,k =1, 2, 3 the outward-pointing unit vetor normal to the surfae, and where t denotes time. The spatial oordinate is x =(x, y, z). Upon the solution of this equation, the pressure in the aousti fluid in domain D an be omputed with ( ) ( ) t v k x,t x x p x,t x x p(x,t)= ρ 4π x x k 4π x x ν k dx, x D (2) In the far field we an make the approximation: t x x t x + ξ qx q x (3) where we have introdued the observation diretion ξ k ξ k = x k x (4) In addition, the denominators an be approximated by x as x. The expression for the pressure beomes p(x,t)= 1 4π x ( k p x,t x + ξ qx q [ρ t v k ( x,t x ) + + ξ qx q )] ν kdx x D, x () Let us define the farfield radiation patterns f and h by: ( {f,h k } {p, v k }(x,t)= 4π x ξ,t x ) (6)

3 Then, f(ξ,t)= Using we an write f(ξ,t)= ( [ρ t v k x,t+ ξ qx ) ( q k p x,t+ ξ qx )] q ν k dx x D, x (7) k p = ξ k tp, x, (8) ( [ρ t v k x,t+ ξ qx ) q ξ ( k tp x,t+ ξ qx )] q ν k dx x D, x (9) The above expressions show that f(ξ,t) is a funtion of the nomal veloity v(x,t), where v = v k ν k. We define the operator E by: f = Ev = E{v(x,t)}(ξ,t)= dτ K(ξ, x,τ)v(x,t τ)dx (1) An inner produt on S is defined as <v 1,v 2 >= v 1 (x,t)v 2(x,t)dx (11) and an inner produt over the angular setor Ω is defined as [f 1,f 2 ]= f1 (ξ,t)f 2 (ξ,t)dξ (12) The radiated power is defined as W (Ω) = E[f,f] (13) where E is the expeted value operator. We also define the adjoint operator E + by [f,ev] =< E + f,v > (14) Substitution of Eq. (1) in Eq. (14) shows that the adjoint operator E + is given by v = E + f = E + {f(ξ,t)}(x,t)= dτ K (ξ, x,τ)f(ξ,t+ τ)dξ () As ompared to E, E + defines a omplex onjugated, anti-ausal system. The inner produt evaluated from the radiation pattern f an be written as The radiated power beomes W (Ω) = E < E + Ev, v >= τ T [f,f] =< E + Ev, v > (16) K(ξ, x,τ)k (ξ, x,τ )E{v(x,t+ τ τ )v(x,t)} dξdτ dτdx dx (17)

4 If we assume that any frequeny dependene in the normal veloity v(x,t) is inorporated in K and that, therefore, v(x,t) an be onsidered to be temporally white, we an write E{v(x,t+ τ τ )v(x,t)} = R vv (x, x )δ(τ τ ) (18) and the radiated power beomes W (Ω) = K(ξ, x,τ)k (ξ, x,τ)r vv (x, x ) dξdτdx dx (19) whih involves a Hermitian and positive definite operator K + K defined by w = K + K{u}(x) = K(ξ, x,τ)k (ξ, x,τ)u(x )dξdτdx (2) that defines a set of eigenvetors V j on S: K(ξ, x,τ)k (ξ, x,τ)v j (x )dξdτdx or, expressed in operator notation: = λ j V j (x), j=1, 2,...,x S (21) K + KV j = λ j V j, j =1, 2,... (22) whih identifies an orthonormal system of vibration patterns on S: <V i,v j >= δ ij, j =1, 2,... (23) where δ ij is the Kroneker delta. The eigenvalues of Eq. (22) are real and positive. The resulting ontrol system is based on the minimization of the signals ν j =<v,v j >, j =1, 2,... (24) whih an be weighted by a matrix of frequeny dependent filters based on the radiation effiieny of the different V j (A.P.Berkhoff, 2). 3 Radiation modes obtained from measured data The radiation modes as defined in this paper an be obtained one the kernel K is known. With diret measurement of this kernel in the frequeny domain, using soures on or near the radiating surfae (G.H.Koopmann and J.B.Fahnline, 1997), reliable results are often diffiult to ahieve(a.p.berkhoff et al., 21). Alternatively, the kernel an be obtained from broadband input-output data using system identifiation tehniques (A.P.Berkhoff, 22), suh as subspae based tehniques (P.v.Overshee and B.D.Moor, 1996), by applying a suffiient number of persistently exiting vibration patterns to the struture. System identifiation methods are also used in one of the examples. Sometimes it an be more effiient to exploit the underlying properties of the medium. In the following a relationship is given between nearfield quantities and

5 farfield quantities by defining two different states of the medium, state A and state B. This relationship an then be used to obtain the shapes V j in a relatively straightforward manner. The two states A and B are desribed by the following equations (M.D.Verweij, 199) k p A + t C t {ρ A,vk A } = f A k (2) k vk A + tc t {κ A,p A } = q A (26) x D k p B + t C t {ρ B,vk B } = f B k (27) k vk B + t C t {κ B,p B } = q B (28) in whih C t denotes temporal onvolution. For reasons of simpliity, the medium is assumed to be lossless. In a homogeneous medium, in whih ρ A = ρ B and κ A = κ B, we have the field reiproity relation [ Ct {p A,vk B } C t {vk A,p B } ] ν k dx = [ ] C t {f A k,vb k } + C t{p A,q B } Ct {vk A,f B k } C t{q A,p B } dx (29) If we take state A to be soure free and state B as the state with a volume injetion soure q B = q B δ(x x B ), then we obtain [ Ct {p A,vk B } C t {vk A,p B } ] ν k dx = C t {p A (x B )q B }, x B D (3) If S denotes a rigid soure then vk Bν k vanishes on S and we have C t {vk A,p B }ν k dx = C t {p A (x B )q B }, x B D (31) If q B (t) equals a temporal Dira impulse at time t =then the orresponding response p B (x,t) equals the Green s funtion g B (x,t) by definition. State A is taken to be the atual medium state, i.e. vk A = v k and p A = p. Then the following equation holds: C t {v k,g B }ν k dx = p(x B ), x B D (32) This shows that the pressure p is related to the normal veloity v k ν k and the Green s funtion g B = g(x, x B,t) on the radiator S, whih is obtained by measuring the pressure on S in response to an impulsive volume injetion soure in the farfield. Alternatively, if suitable exitation signals are used then the desired transfer funtions an be obtained by system identifiation. If x B is in the farfield with respet to S then the Green s funtion an be written as g(x, x B,t+ xb ) = g(x, ξ,t+ xb ).UseofEq. (6) leads to ( ) f B (ξ,t)=4π x B dτ g x, ξ,t+ xb v(x,t τ)dx (33) whih shows that we have the relation ( ) K(ξ, x) = 4π x B g B x, ξ,t+ xb (34)

6 Hene, provided S is rigid, K(ξ, x) an be obtained from reiproal measurements in the farfield by multipliation of g B with 4π x B and advaning the time argument by x B. 4 Results Simulated and measured radiation modes as desribed above have been ompared. The onfiguration is based on a retangular radiating surfae of 6 m 7 m, onsisting of 6 elemental radiating surfaes. The simulated radiation mode shapes with the highest effiienies for frequenies below 4 Hz are shown in Fig. 1. The orresponding measured radiation mode shapes are shown in Fig. 2. It an be seen that there is agreement between the first three shapes. The order of the seond shape and third shape is interhanged whih is aused by the eigenvalues of these shapes, whih are almost equal. Small hanges then an lead to a hange in the order, whih is a relatively harmless phenomenon. The orresponding redutions of the radiated sound power of a vibrating plate an be found in Fig. 3. This result was obtained by using 16 piezoeletri path atuators and 16 aelerometers. Both the sensors and the atuators were ombined in radiation modal sensors/atuators, leading to a redution of the ontroller dimensionality. The most important radiation modes for frequenies below Hz were used, taken into onsideration the atual exitation of the radiation modes by the strutural modes. The ontroller was based on a 1GHz AMD Athlon proessor with RT Linux, using a sampling frequeny of 2 khz. The inident noise used for this experiment was obtained from measurements reorded in-flight in jet airraft. Sine more physial atuators were used than there were independent atuator driving signals, additional onstraints were used to determine weighting fators for the atuator array. The design of the array was based on the use of a seond set of radiation modes, whih were defined in suh a way that they were the most effiient radiators at high frequenies while being ineffiient radiators at low frequenies. The resulting shapes were used to onstrain the output of the atuator array. The results obtained with this method were found to be better than by using onstraints on ontrol effort (D.R.Morgan, 1991) or onstraints on ontroller oeffiients. An example with appliation to ative noise barriers for redution of traffi noise is shown in Fig. 4. Here, the broadband noise produed by 1 independent white noise soures at z = 2m is being redued at the 2 error mirophones in the far field at z =3m. The sampling frequeny is 1 khz. The sensors at z = m detet the sound radiated from the primary noise soures, whih is then used to drive 2 seondary soures at z =m. The sensor signals are proessed by radiation modal tehniques in order to arrive at a redution of the ontroller dimensionality. The radiation modes were obtained from the transfer funtions between detetion signals and error signals. These transfer funtions were obtained by using 2 identifiation soures at the positions of the primary soures. From the resulting detetion signals and error signals the underlying transfer funtions were found by ausal inversion. The results were obtained with an H 2 -optimal multivariable ausal predition tehnique. For the results of Fig. 4, 2 seondary soures were used. At the error mirophones redutions of approximately 12 db are obtained. However, at other positions inreases an be seen. Better results an be obtained if more seondary soures are used, whih are onstrained by the atuator output at other positions. The result for 2 error sensors, seondary soures and 4 onstraint sensors is shown in Fig.. Still better results are obtained if more seondary soures and error sensors are used. An example for 8 error

7 sensors, 8 seondary soures and 4 onstraint sensors is given in Fig. 6. The redutions at the error mirophones are approximately 18 db. The order of the latter system is still low beause of the use of radiation modal weighting shemes. The tehniques an also be applied to asymmetrial onfigurations. Therefore, the extension to systems in whih redutions are to be obtained over larger angular setors and larger inidene angles is relatively straightforward. (1) (2) y [m] y [m] (3) (4) y [m] y [m] Figure 1: Simulated radiation modes. (1) (2) y [m] y [m] (3) (4) y [m] y [m] Figure 2: Measured radiation modes. Referenes A.D.Piere (1989). Aoustis - An introdution to its physial priniples and appliations, 2 edn, Aoustial Soiety of Ameria, Woodbury, New York. A.P.Berkhoff (2). Sensor sheme design for ative strutural aousti ontrol, J Aoust So Am 18:

8 Redution = 8.9 db 2 Without ontrol Feedbak ontrol 3 db frequeny [Hz] Figure 3: Sound power radiated from an aluminium sandwih panel without ontrol and with ontrol of broadband radiated noise using strutural atuators and strutural sensors leading to a redution of 8.9 db (onstant weighting). A.P.Berkhoff (22). Broadband radiation modes: estimation and ative ontrol, J Aoust So Am. A.P.Berkhoff, E.Sarajli, B.S.Cazzolato and C.H.Hansen (21). Inverse and reiproity methods for experimental determination of radiation modes, in K.M.Li (ed.), Pro. ICSV8, Institute of Aoustis and Vibration. C.R.Fuller, S.J.Elliott and P.A.Nelson (1996). Ative ontrol of vibration, Aademi Press, London. D.R.Morgan (1991). An adaptive modal-based ative ontrol system, J Aoust So Am 89: E.J.J.Doppenberg, A.P.Berkhoff and M.v.Overbeek (21). Smart materials and ative noise and vibration ontrol in vehiles, in G.L.Gissinger (ed.), Pro. IFAC Advanes in automotive ontrol, pp G.H.Koopmann and J.B.Fahnline (1997). Designing for quiet strutures: a sound power minimization approah, Aademi Press, San Diego. G.P.Gibbs, R.L.Clark, D.E.Cox and J.S.Vipperman (2). Radiation modal expansion: Appliation to ative strutural aousti ontrol, J Aoust So Am 17: G.V.Borgiotti (199). The power radiated by a vibrating body in an aousti fluid and its determination from boundary measurements, J Aoust So Am 88: M.D.Verweij (199). Modeling spae-time domain aousti wave fields in media with attenuation: The symboli manipulation approah, J Aoust So Am 97: P.v.Overshee and B.D.Moor (1996). Subspae identifiation for linear systems, Kluwer Aademi Publishers, Boston.

9 3 redution [db] z [m] Figure 4: Ative noise barrier using 2 error sensors (indiated by o at z =3m), 2 seondary soures (indiated by * at z =m). Ten broadband primary noise soures are used (indiated by * at z = 3m)) and detetion sensors (indiated by o at z = m). S.J.Elliott and M.E.Johnson (1993). Radiation modes and the ative ontrol of sound power, J Aoust So Am 94: W.T.Baumann, W.R.Saunders and H.H.Robertshaw (1991). Ative suppression of aousti radiation from impulsively exited strutures, J Aoust So Am 9:

10 3 redution [db] z [m] Figure : As Fig. 4, using seondary soures (indiated by * at z =m) with onstraints on 4 mirophones (indiated by o at 19m x 21m). 3 redution [db] z [m] Figure 6: As Fig. 4, using 8 error sensors (indiated by o at z =3m), 8 seondary soures (indiated by * at z = 2m) and 4 onstraint sensors (indiated by o at 19m x 21m).