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Global active control of harmonic noise in a vibro-acoustic cavity using Modal FxLMS algorithm
Applied Acoustics 150 (2019) 147–161
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Global active control of harmonic noise in a vibro-acoustic cavity using Modal FxLMS algorithm Amrita Puri, Subodh V. Modak ⇑, Kshitij Gupta Department of Mechanical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India
a r t i c l e
i n f o
Article history: Received 8 November 2018 Received in revised form 11 January 2019 Accepted 10 February 2019
Keywords: Active control Noise FxLMS Modal Vibro-acoustic
a b s t r a c t The Modal FxLMS algorithm is recently proposed for reduction of global level of noise in vibro-acoustic cavities. The algorithm minimizes acoustic potential energy expressed in modal domain which allows a choice of acoustic modes whose modal amplitudes are desired to be minimized. The working of the algorithm and its effectiveness has been demonstrated previously through a numerical study and an experimental study is required to test the effectiveness of the algorithm in practice. To meet this objective this paper presents an experimental study of this algorithm for global active noise control in a rectangular box cavity. The algorithm utilizes modal filtered reference signals and modal amplitudes of the selected acoustic modes of the cavity. To identify modal filtered reference signals, modal secondary paths are utilized. In the present study, modal secondary paths are identified using experimentally identified physical secondary paths and acoustic mode shapes. Modal amplitudes of the acoustic modes are identified on the basis of acoustic pressure measurements from eight microphones suitably placed throughout the cavity. The performance of the algorithm is studied under the presence of acoustic and structural disturbances using one or two control loudspeakers. The active control is carried out at harmonic frequencies coinciding with cavity-controlled resonances and panel-controlled resonances. It is found from the experimental results that the Modal FxLMS algorithm is successful in reducing modal amplitudes of the selected acoustic modes. When all the significantly contributing acoustic modes are chosen, a reduction in global level of noise is observed at cavity as well as panel controlled resonances. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Car, aeroplane, helicopter and other transportation equipment are vibro-acoustic cavities encountered in real-life and reduction of interior noise in these cavities is an important objective in their NVH design. Passive and active methods are two approaches which can be used for noise control. Passive solutions are generally not much effective or acceptable for reduction of low frequency noise. On the contrary, the active methods can be effectively used in this frequency range. The problem of global active noise control in a cavity has been addressed in different ways in the literature. One of the approaches is to reduce acoustic potential energy in the cavity. Elliott et al. [1] studied minimisation of acoustic potential energy approximated as sum of squares of acoustic pressures at corners of a rectangular acoustic cavity. Cheer [2] extended this approach for global active noise control in a car cavity. Cheer and Elliott [3,4] implemented ⇑ Corresponding author. E-mail addresses: [email protected] (A. Puri), [email protected] (S.V. Modak), [email protected] (K. Gupta). https://doi.org/10.1016/j.apacoust.2019.02.008 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
feedforward control systems to minimise sum of squares of acoustic pressures to reduce road noise in a car and to reduce generator noise in a luxury yacht, respectively. Yang et al. [5] used correlation based FxLMS algorithm to reduce noise at a suitably placed error microphone to control high speed elevator noise. Another approach for global active noise control is minimisation of sum of acoustic energy densities (Parkins et al. [6]). Lau and Tang [7] presented a comparison of acoustic potential energy control, squared pressure control and energy density based control, using secondary acoustic sources, to minimize sound transmission into a rectangular enclosure. They pointed out that an increase in SPL may occur at some locations even if there is global attenuation in the acoustic potential energy. Koshigoe et al. [8] compared velocity feedback, mean square sound pressure minimisation and sound pressure feedback control schemes to reduce the acoustic potential energy inside a rectangular cavity with one flexible panel. Balachandran et al. [9] presented a comparison of different control schemes to minimise noise inside a cavity when a plane wave acts on the flexible panel of a vibroacoustic cavity. Control schemes studied were minimization of mean square pressure inside the cavity, minimisation of mean
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square velocity of panel, minimisation of acoustic potential energy and minimisation of acoustic potential energy formulated with pressure readings at a fixed number of sensors. It was observed that minimising mean square velocity of the panel may not necessarily reduce the acoustic potential energy inside the cavity and similarly, minimizing the acoustic potential energy may increase the kinetic energy of the panel. Another approach used for global active noise control in vibroacoustic cavities is control of so-called ’radiation modes’. These modes are nothing but orthogonal or independent contributors of acoustic potential energy as opposed to the structural modes (Snyder and Tanaka [10], Johnson and Elliott [11]). Active noise control in a stiffened cylindrical cavity has been carried out by minimising amplitudes of radiation modes (Cazzolato and Hansen [12]). Bagha and Modak [13] performed active noise control in a weakly coupled cavity by reducing the amplitudes of radiation modes estimated through a Kalman filter and a frequency weighting filter. Bagha and Modak [14] presented an experimental study to reduce global level of noise in a rectangular box vibro-acoustic cavity using an experimentally updated finite element model of the cavity used in formulation of the feedback controller. Minimisation of modal amplitudes of the contributing modes for active control has been researched a lot in the field of active vibration control (Morgan [15], Fuller et al. [16]). But not much work is reported on these lines in the area of active noise control. An LQG based independent modal space feedback control strategy has been used to reduce noise in a duct and a rectangular room (Bai and Shieh [17]). Lane et al. [18] presented an experimental study to reduce global level of noise in an aircraft fuselage using a modal based H2 feedback control strategy. Clark [19] studied optimal modal feedforward control for a structural reverberant system and suggested use of an adaptive modal control method based on a time averaged gradient descent algorithm. Cheer [2] and Cheer and Elliott [20] proposed optimal modal feedback control system in an internal model control (IMC) architecture. It is observed that these strategies are in the framework of feedback control or IMC and the feedback controller needs to be redesigned for a new choice of acoustic modes to be controlled. For harmonic noise, feedforward technique is often used for active noise control and therefore, a modal based formulation of this technique is needed. Recently, such a formulation called as ’Modal Filtered-x LMS’ algorithm has been presented by authors [21]. The operation and effectiveness of the algorithm have been demonstrated through a numerical study. This paper presents an experimental study of Modal FxLMS method for global active noise control in a vibro-acoustic cavity. The paper discusses the procedure to experimentally implement the Modal FxLMS method. The global active noise control is performed in a rectangular box cavity with one flexible panel under the presence of acoustic and structural disturbances at cavitycontrolled resonances and panel-controlled resonances using one or two control loudspeakers. The issue of identification of modal secondary paths is addressed. The paper is organised as follows: Section 2 describes theory of Modal FxLMS algorithm. Section 3 presents an experimental study of global active noise control using Modal FxLMS algorithm. The section presents results of active noise control under the presence of acoustic and structural disturbance with one and multiple control sources. It also presents computation of modal secondary using data identified using experimental modal analysis. Section 4 concludes the present work.
Following expression gives instantaneous acoustic potential energy inside a vibro-acoustic cavity [22],
Ep ¼
Z
1 2qa c2
V
ð1Þ
p2 dV 0
Using a finite element model of the cavity, Ep , can be written as,
Ep ¼ pH Vp
ð2Þ
where p represents vector of acoustic pressures at all nodes of the cavity, V is volume matrix and superscript ‘H’ represents Hermitian operator.
V¼
Ma 2qa
ð3Þ
where Ma and qa are acoustic mass matrix and density of air, respectively. For a weakly coupled vibro-acoustic cavity, p can be expressed in terms of modal amplitudes of rigid walled acoustic modes, pm , as,
p ¼ Ua p m
ð4Þ
where Ua is a matrix of mass-normalised rigid walled acoustic modes. Using Eqs. (2) and (4), Ep can be written as,
Ep ¼
1 H p p 2 m m
ð5Þ
In Modal FxLMS algorithm, weights of the adaptive filter are updated as,
wðn þ 1Þ ¼ wðnÞ
l 2
$Ep
ð6Þ
where $ represents gradient with respect to weights of the adaptive filter. Substituting Eq. (5) in Eq. (6) and expanding the resulting expression, we get,
wðn þ 1Þ ¼ wðnÞ
M lX
2
pkm : rpkm
ð7Þ
k¼1
where M is total number of acoustic modes and pkm corresponds to modal amplitude of kth acoustic mode. As explained in [21], pkm can be expressed as, k
k
pkm ¼ hpm qp þ hsm qs k
ð8Þ
k
where hpm and hsm represent kth modal impulse responses corresponding to primary and secondary sources, respectively. qp and qs represent inputs to primary and secondary sources, respectively. The symbol ’’represents convolution operator. The modal impulse response corresponding to primary/secondary noise source represents variation of modal amplitude of a particular mode of the cavity when the primary/secondary source is driven by an impulse function. Modal impulse responses corresponding to primary and secondary sources are called as modal primary paths and modal secondary paths, respectively. Using Eq. (8), rpkm can be written as,
rpkm ¼ rhkpm qp þ rhksm qs
rpkm ¼ r hksm qs
ð9Þ ð10Þ
Eq. (10) can be written as, 2. Modal FxLMS algorithm This section presents a brief theory of Modal FxLMS algorithm.
rpkm ¼ r hksm w xr
where xr is reference signal vector.
ð11Þ
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(2)
(6)
(8)
(4)
(7)
(9)
(1)
(5) (3)
Fig. 1. Experimental set-up for global active noise control. [(1) Rectangular box cavity with one flexible plate, (2) Primary loudspeaker, (3) Power amplifier for the primary loudspeaker, (4) Electrodynamic shaker, (5) Power amplifier for the electrodynamic shaker (6) Function generator, (7) First control loudspeaker, (8) Second control loudspeaker, (9) Microphones (8 Nos.)]
signals can be obtained by linear convolution of vectors of modal secondary paths and reference signal. Therefore, modal secondary paths are required to obtain modal filtered reference signals. As explained in [21], instantaneous values of modal amplitudes of acoustic modes are obtained as,
pm ¼ Uya p
ð14Þ
where p represents a vector of measured nodal acoustic pressures,
pm represents a vector of modal acoustic pressures of first few contributing acoustic modes in the desired frequency range and Uya represents pseudo-inverse of Ua which represents matrix of the chosen first few contributing acoustic modes. Rows of Ua correspond to nodes at which acoustic pressure is measured and columns of Ua correspond to the chosen first few contributing acoustic modes. In [21], two methods of identification of modal secondary paths are discussed. This paper utilises the second method of using acoustic mode shapes. The second method is explained below. Acoustic pressure at kth FE node can be expressed as, k
k
pk ¼ hp qp þ hs qs
ð15Þ
Using Eq. (15), writing acoustic pressures at all the nodes of FE model in a vector form, we get,
Fig. 2. Schematic diagram of the experimental set-up for global active noise control.
Assuming slow variation of weights of the adaptive filter, Eq. (11) can be written as,
rpkm ¼ r w hksm xr ¼ hksm xr ¼ xfr km
ð12Þ
where xfr km is modal filtered reference signal corresponding to kth acoustic mode. Substituting Eq. (12) into Eq. (7), we get,
wðn þ 1Þ ¼ wðnÞ
l 2
M X
pkm :
xfr km
ð13Þ
k¼1
Eq. (13) represents final weight updating equation of the Modal FxLMS algorithm which utilises instantaneous values of modal amplitudes of acoustic modes of the cavity and modal filtered reference signals. As observed from Eq. (12), modal filtered reference
8 9 8 9 8 1 9 > h1 qp > > h1 q > p > > > s s > p > > > > > > > > > > > > > > > > > 2 2 > > > > > hp qp > > > > > > hs qs > p2 > > > > > > > > > > > > > > > > > > > < p3 = < 3 = < 3 = h q h q p p s s ¼ þ > > > > > > : > > > > > > : : > > > > > > > > > > > > > > > > > > > : > > > > > > > > > > > : : > > > > : M > > > ; > > > > > > > : M ; : M ; p hs qs hp qp
ð16Þ
Eq. (16) can be expressed as,
p ¼ Hp : qp þ Hs : qs
ð17Þ
Multiplying Eq. (17) with Ua 1 , we get,
Ua 1 p ¼ Ua 1 Hp : qp þ Ua 1 Hs : qs
ð18Þ
Using pm ¼ Ua 1 p, Eq. (18) can be written as,
pm ¼ Ua 1 Hp : qp þ Ua 1 Hs : qs
ð19Þ
From Eq. (19), we get,
Hpm ¼ Ua 1 Hp
ð20Þ
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Fig. 3. (a) Experimental set-up for modal analysis of the plate [(1) Flexible plate of the cavity, (2) Impact hammer, (3) Accelerometer, (4) Signal conditioner and (5) FFT analyser] and (b) Schematic diagram of the experimental set-up for modal analysis of the plate.
133.4 Hz
70.8 Hz
257.1 Hz
172.8 Hz
317.7 Hz
235.7 Hz
340.6 Hz
Fig. 4. First seven experimental mode shapes of the plate backed by the rectangular cavity.
Hsm ¼ Ua 1 Hs
ð21Þ
where Hpm and Hsm are matrices of primary modal paths and secondary modal paths, respectively. From Eq. (21), it is seen that using information about acoustic modes shapes and physical secondary paths, modal secondary paths can be identified. In the present work, modal secondary paths are obtained using experimentally identified acoustic mode shapes and experimentally identified physical secondary paths.
formed on a 3D rectangular box cavity with one flexible panel. The Subsection 3.1 describes experimental set up used for the study. Subsection 3.2 describes experimental modal analysis of the acoustic cavity to obtain acoustic mode shapes. Subsection 3.3 presents results of the Modal FxLMS algorithm using one control loudspeaker when only acoustic disturbance acts on the vibroacoustic cavity and when both acoustic and structural disturbances act on the vibro-acoustic cavity. Subsection 3.4 presents results of the global active noise control using two control sources.
3. Global active noise control using Modal FxLMS algorithm
3.1. Description of experimental set up
This section presents an experimental study of global active noise control using Modal FxLMS algorithm. The study is per-
Fig. 1 shows the experimental set-up used for active noise control in a rectangular box vibro-acoustic cavity. The dimensions of
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151
Fig. 5. (a) Experimental set-up for modal analysis of the cavity [(1) Loudspeaker, (2) Microphone, (3) Power amplifier for the loudspeaker, (4) Signal conditioner for the microphone and (5) FFT analyser] and (b) Schematic diagram of the experimental set-up for modal analysis of the cavity.
the rectangular cavity are 0.522 m 0.60 m 1.372 m. The five surfaces of the cavity are made of acrylic sheet of thickness 15 mm while the remaining sixth surface of the cavity is made of aluminium of thickness 3 mm and is relatively much flexible than the other surfaces. A schematic diagram of the experimental set-up is shown in Fig. 2. One loudspeaker is placed on the top wall of the cavity which acts as a primary noise source. Two loudspeakers mounted diagonally on the acrylic wall opposite to the aluminium panel are used as control sources. The lower left loudspeaker is referred as first control loudspeaker while the upper loudspeaker is referred as second control loudspeaker. To mimic structural disturbances, a shaker is attached to excite the flexible panel of the cavity. The active noise control is carried out for a harmonic primary noise. The primary loudspeaker and/or shaker are driven by a sinusoidal signal using a function generator. Eight 1/200 microphones are positioned in the cavity to measure acoustic pressure. The locations of these microphones are chosen such that they can observe first five acoustic modes beyond rigid body mode. These five acoustic modes are shown in next subsection. The outputs of the microphones are sent to a dSPACE controller board via signal conditioners. A MATLAB Simulink block diagram of the active noise control using the Modal FxLMS method is made and burnt on the dSPACE controller board. The control outputs from the controller board are used to drive the control sources after being passed through analog low pass filters. 3.2. Experimental modal analysis of the rectangular vibro-acoustic cavity This subsection describes experimental modal analysis of the vibro-acoustic cavity to obtain acoustic mode shapes which are
required to estimate modal amplitudes of the acoustic modes of the cavity. The modal analysis also gives information about panel and cavity controlled resonances. Experimental set-ups used for modal analysis of the plate and the acoustic cavity are shown in Figs. 3 and 5, respectively. Modal analysis of the plate is performed by exciting the plate with an impact hammer at 437 (19 23) nodal points and recording the response of the plate with an accelerometer mounted at a fixed location. The frequency response functions (FRFs) are recorded and analysed to identify natural frequencies and mode shapes of the plate giving information about panel-controlled resonances of the vibro-acoustic cavity. First seven mode shapes of the plate identified in this manner are shown in Fig. 4. In these figures, blue colour shows the minimum displacement (or negative displacement) and red colour shows the maximum displacement (or positive displacement). Modal analysis of the acoustic cavity is performed using a sweep sine signal (20 Hz to 660 Hz in 80 s (sweep rate of 8 Hz/s)) fed to a loudspeaker and recording acoustic response using a microphone (Fig. 5). The interior of the cavity is divided virtually in a 3D mesh of 936 points (8 points in x-direction, 9 points in ydirection and 13 points in z-direction). The output of the microphone and voltage across two terminals of the loudspeaker are sent to a FFT analyser which provides a frequency response function. The procedure is repeated at 936 points, by moving the microphone to the desired test point, to cover the whole cavity. The 936 FRFs are analysed using modal analysis software to obtain mode shapes of the cavity. This gives information of acoustic mode shapes and cavity-controlled resonances of the vibro-acoustic cavity. Total 21 acoustic mode shapes (excluding rigid body mode) are identified in the frequency range from 100 Hz to 640 Hz, with first
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131.4 Hz
250.1 Hz
289.8 Hz
314.8 Hz
350.7 Hz Fig. 6. First five experimental acoustic mode shapes of rectangular cavity of dimensions (0.524 m 0.600 m 1.37 m). Table 1 Experimental coupled natural frequencies of the vibro-acoustic cavity. Natural frequencies Natural frequencies identified from modal identified from modal analysis of cavity (Hz) analysis of plate (Hz) 70.8 131.4 133.4 172.8 235.7 250.1 257.1 289.8 314.8 317.7 340.6 350.7
Significance
1st Panel-controlled resonance 1st Cavity-controlled resonance 2nd Panel-controlled resonance 3rd Panel-controlled resonance 4th Panel-controlled resonance 2nd Cavity-controlled resonance 5th Panel-controlled resonance 3rd Cavity-controlled resonance 4th Cavity-controlled resonance 6th Panel-controlled resonance 7th Panel-controlled resonance 5th Cavity-controlled resonance
five identified acoustic mode shapes shown in Fig. 6. In these figures, colour contour depicts the distribution of the acoustic pres-
sure in a mode. Green colour shows the location where acoustic pressure is zero while red/blue colour shows the locations where acoustic pressure is maximum (positive)/minimum (negative). In addition, the acoustic pressure distribution in a mode is also depicted by a proportionate virtual displacement in vertical direction. It is seen from Fig. 6 that mode at 131 Hz is first longitudinal acoustic mode (0, 0, 1) with a nodal plane passing thorough the centre of the length of the cavity. The mode at 250 Hz is second longitudinal mode (0, 0, 2) with two nodal planes along the length and an anti-nodal plane passing thorough the centre of the length of the cavity. The mode at 289 Hz has variation of acoustic pressure along the height of the cavity with nodal plane passing through the centre of the height of the cavity (0, 1, 0 mode). The mode at 315 Hz is a tangential mode (0, 1, 1) with nodal planes passing through centre of the length of cavity as well as through the centre of the height of the cavity. The mode at 350 Hz has variation of acoustic pressure along the width of the cavity with nodal plane passing through the centre of the width of the cavity (1, 0, 0 mode). Table 1 shows the natural frequencies obtained through experimental modal analysis of the plate and the cavity. The table also shows the significance of these coupled natural frequencies as being either panel-controlled or cavity-controlled resonances. The rigid body acoustic mode (0, 0, 0), which has a very low frequency, also needs to be known for a complete modal description of the acoustic cavity. This mode was experimentally identified by excitation of the cavity at harmonic frequency in the low frequency range. The rigid body acoustic mode is identified on the basis of the fact that acoustic pressure distribution over the test mesh is
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Fig. 7. Location of eight microphones in the rectangular box vibro-acoustic cavity.
Table 2 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes and estimated acoustic potential energy at 131 Hz for different cases of modal control with the proposed Modal FxLMS. Values before control (dB)
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
1 2 3 4 5 6 7 8
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
0 1 2 3 4
Estimated acoustic potential energy
Reduction in dB after control One mode control (1st mode control) Case (a)
Five mode control First 5 mode control) Case (b)
122.8 118.4 107.9 96.9 115.9 120.3 121.6 122.8
27.7 17.2 5.2 5.3 14.8 17.6 24.2 24.3
26.7 16.9 5.1 5.3 15.0 17.8 24.9 24.9
96.6 125.0 106.2 107.5 102.2
7.0 38.7 10.8 11.0 9.3
6.9 33.8 10.5 11.3 9.1
31.2
20.0
20.0
dominated by this mode in the low frequency range. This mode is referred as ’0th’ acoustic mode in this paper.
frequency is chosen as 5000 Hz which is more than 10 times the maximum frequency of interest.
3.3. Active noise control using one control source
3.3.1. Identification of modal amplitudes of acoustic modes This subsection presents details of the estimation of modal amplitudes of acoustic modes which are required in Modal FxLMS algorithm for updating weights of the filter (Eq. (13)). The frequencies 131 Hz and 170 Hz at which active noise control is carried out lie in the range of 50–320 Hz. In this frequency range, the first five acoustic modes (including rigid body acoustic mode) are expected
This section presents results of active noise control at harmonic frequencies using Modal FxLMS algorithm using one control source. The results at frequencies 131 Hz (coinciding with first cavity-controlled resonance) and 170 Hz (near to the third panelcontrolled resonance 173 Hz) are presented in this paper. Sampling
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Fig. 8. Acoustic pressure at location of eight microphones locations at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
to make most of the significant modal contribution to the acoustic pressure. Therefore, modal amplitudes of these acoustic modes need to be estimated from measurement of acoustic pressure, for which eight microphones are chosen. Fig. 7 shows the locations where these eight microphones are placed. The modal amplitudes are estimated from measured acoustic pressures as explained in Section 2 (Eq. (14)). The strategy used to locate these microphones is based on requirement that each of the five acoustic modes is observable to at least some of the microphones. Fig. 7 shows that the microphones 1, 7, and 8 are positioned near the anti-nodal planes of the first acoustic mode (131 Hz) whereas the microphones 3 and 4 are positioned near anti-nodal plane of the second acoustic mode
(250 Hz). Similarly, third acoustic mode (289 Hz) is observable at all the eight microphones since all are away from the nodal plane of this mode. The fourth mode (315 Hz) is observable at all microphones except third and fourth microphones. 3.3.2. Identification of modal secondary paths This subsection presents details of the identification of modal secondary paths which are required to obtain modal filtered reference signals which are required for updating weights of the filter (in Eq. (13) in Modal FxLMS algorithm. The method utilises information of acoustic mode shapes and physical secondary paths impulse responses at all locations of the cavity. Eq. (21) is then used to obtain modal secondary path impulse responses.
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155
Fig. 9. Estimated modal amplitudes of first five acoustic modes at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The symbols Hsm , Ua and Hs are explained in Section 2. The superscript E essentially represents that these variables in this equation correspond to experimental data. Ua E is a matrix of 936 rows and 22 columns of experimentally obtained mass normalised acoustic modes. Hs E is a matrix of physical secondary paths impulse responses at a harmonic frequency at 936 points. The length of each
Fig. 10. Plots of estimated acoustic potential energy at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Experimental modal analysis of the cavity (presented in Section 2) yielded 22 acoustic mode shapes (including 0th acoustic mode) between 50 Hz 640 Hz. Hence, Eq. (21) is modified as, E HEsm ¼ UEy a Hs
ð22Þ
secondary path is 30 samples. Therefore, Hs E is a matrix 936 rows and 30 columns. The physical secondary paths impulse responses are obtained using LMS algorithm for secondary path identification. Using the experimentally obtained physical secondary paths impulse responses and experimentally obtained acoustic modes shapes, modal secondary paths are obtained from Eq. (22). Since, as explained in Section 3.3.1, first five acoustic modes are considered adequate for modal description of the acoustic response in the frequency range 50–320 Hz, only first five modal secondary paths are further used in the present experimental study. 3.3.3. Active noise control under the presence of acoustic disturbance This subsection presents the results of active noise control using the Modal FxLMS algorithm under the presence of acoustic disturbance. A loudspeaker mounted in one corner of the top surface of the cavity (Fig. 1) acts as a source of primary noise or acoustic disturbance.
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3.3.3.1. Results at 131 Hz. At 131 Hz (which is the first cavitycontrolled resonance), active noise control was carried out for two choices of modes to be controlled to test the ability of the proposed Modal FxLMS algorithm to perform reduction of modal amplitudes of the chosen/desired modes. These choices are: (a) reduction of modal amplitude of only one acoustic mode (mode 1, which is nothing but the first cavity-controlled mode) and (b) reduction of modal amplitude of first five acoustic modes (modes 0, 1, 2, 3, 4). Table 2 shows dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control along with reductions in these quantities after control for the cases (a) and (b). It is noted that when only mode 1 is controlled, there is a reduction of 38.7 dB in the modal amplitude of the mode 1 whereas when all the five acoustic modes are controlled, there is a reduction of 33.8 dB in modal amplitude of the mode 1. This is because the algorithm minimises the sum of squares of modal amplitudes of all the modes considered and hence, the reduction in the modal amplitude of the mode 1 is somewhat lesser at the cost of reduction in modal amplitudes of other modes. However, it is seen that the reductions in estimated acoustic potential energy for the two cases are nearly same. Similarly, there are significant reductions in acoustic pressures at microphones locations for both the cases of modal control and these reductions are almost same for the two cases. This shows that the ANC using the proposed method can be carried out by controlling only the dominant acoustic mode with equal effectiveness as when many more modes are controlled. This highlights the usefulness of the proposed Modal-FxLMS method in allowing a choice of modes to be controlled. Figs. 8–10 show plots of acoustic pressures at location of eight microphones, estimated modal amplitudes of first five acoustic modes, estimated acoustic potential energy (taken as sum of squares of estimated modal amplitudes) before and after control of only mode 1, respectively. In these figures, the active control is initiated roughly at time t = 20 s. It is seen from Fig. 8 that there is a significant reduction in acoustic pressures at all the microphones except 4th microphone location which is placed near the nodal plane of the mode 1. It is observed from Fig. 9 that there is a significant reduction in modal amplitude of mode 1 and there is a small increase in modal amplitude of mode 0 (which explains the increase in acoustic pressure at 4th microphone). It is observed from Fig. 10 that there is a significant reduction in acoustic potential energy after control.
3.3.3.2. Results at 170 Hz. Table 3 shows dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control and reductions in these quantities after control for four cases of modal control at 170 Hz. These four cases of modal control are: (a) control of only mode 1, (b) control of modes 1 and 4, (c) control of modes 0 and 1 and (d) control of all five modes. It is seen from Table 3 that when only mode 1 is controlled, there is a reduction of 40.3 dB in modal amplitude of mode 1 (1st acoustic mode). There is a reduction of 11.2 dB in modal amplitude of mode 4 (4th acoustic mode). However, modal amplitudes of modes 0, 2 and 3 (0th, 2nd, 3rd acoustic modes) increase by 5.2 dB, 8.8 dB and 5.9 dB, respectively. When modes 1 and 4 are controlled (case (b)), reduction in modal amplitude of 1st acoustic mode is 25.0 dB (which is less than the reduction of 40.3 dB obtained when only mode 1 was controlled). There is a reduction of 12.9 dB in modal amplitude of 4th acoustic mode (which is more than 11.2 dB obtained when only mode 1 was controlled). The increase in modal amplitudes of modes 0, 2 and 3 is similar to that of when only mode 1 was controlled. This shows that the Modal FxLMS algorithm works well to effectively reduce the chosen modes (modes 1 and 4 in this case). When modes 0 and 1 are controlled (case (c)), reduction in modal amplitude of 1st acoustic mode is 16.3 dB (which is less than the reduction of 40.3 dB obtained when only mode 1 was controlled and the reduction of 25.0 dB when modes 1 and 4 are controlled). The increase in modal amplitude of 0th acoustic mode is less. When modes 0 and 1 are controlled, the increase in modal amplitude of 0th acoustic mode is 4.3 dB as against the increases of 5.1 dB and 5.2 dB for cases (a) and (b). This shows that when modes 0 and 1 are controlled, the increase in modal amplitude of 0th acoustic mode and reduction in 1st acoustic mode are lesser than the previous two cases of modal control (only 1st mode control, and 1st and 4th mode control). This shows that Modal FxLMS algorithm successfully considers the contribution of the chosen acoustic modes (modes 0 and 1 here) and attempts to reduce sum of squares of modal amplitudes of these modes. When all the five acoustic modes are considered (case (d)), the reduction in modal amplitudes of 1st and 4th acoustic modes are 7.6 dB and 8.5 dB which is lower than the corresponding reductions obtained from the previous three cases of modal control. The increases in modal amplitudes of 0th, 2nd and 3rd acoustic modes are 3.4 dB, 5.4 dB and 3.9 dB, which
Table 3 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz for different cases of modal control with the proposed Modal FxLMS method. Reduction in dB after control
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Values before control (dB)
One mode control (1st mode control) Case (a)
Two mode control (1st and 4th modes control) Case (b)
Two mode control (0th and 1st modes control) Case (c)
First five modes control Case (d)
102.8 104.3 100.5 96.1 83.6 94.8 98.1 100.1
22.4 4.3 2.0 5.9 17.0 7.2 8.1 6.3
18.3 4.1 2.0 5.8 16.7 6.8 8.8 7.0
14.0 3.7 1.5 5.0 15.1 4.8 14.5 12.0
7.0 2.3 1.2 3.9 11.6 0.6 13.0 16.5
95.7 105.7 91.9 92.3 98.7
5.2 40.3 8.8 5.9 11.2
5.1 25.0 8.6 5.9 12.9
4.3 16.3 7.6 5.0 12.7
3.4 7.6 5.8 3.9 8.5
13.1
2.2
2.3
3.1
3.1
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157
Fig. 11. Acoustic pressure at location of eight microphones at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
are less than the corresponding increase in the previous three cases of modal control. The amount of reduction and amount of increase in modal amplitudes of different modes for different cases of modal control is also reflected in acoustic pressures at eight microphones. For example, as seen from the results in Table 3, the increase in acoustic pressures at the location of 3rd, 4th and 5th microphones for case (d) is lesser than that of for the cases (a), (b) and (c). It is further seen from Table 3 that when five acoustic modes are controlled, a reduction of 3.1 dB in estimated acoustic potential
energy is obtained, respectively. This reduction is higher than or equal to the reductions in the cases (a), (b) and (c) (2.2 dB, 2.3 dB and 3.1 dB, respectively). Figs. 11–13 show plots of acoustic pressures at location of eight microphones, estimated modal amplitude of first five acoustic modes and estimated acoustic potential energy (taken as sum of squares of estimated modal amplitudes), respectively, before and after control for case (d) (control of first five acoustic modes). It is seen from Fig. 12 that there is a reduction in modal amplitudes of 1st and 4th acoustic modes and an increase in modal amplitudes
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of 0th, 2nd and 3rd acoustic modes after control. The reduction in modal amplitudes of 1st acoustic mode is reflected in reduction of acoustic pressures at the location of 1st, 2nd, 7th and 8th microphones (Fig. 11), which are positioned near the anti-nodal planes of the first acoustic mode, and an increase in modal amplitudes of 0th and 2nd acoustic modes is reflected in acoustic pressure at the location of 3rd and 4th microphones (Fig. 11) which are positioned near nodal plane of 1st acoustic mode and anti-nodal plane of 2nd acoustic mode. It is also seen from Fig. 13 that there is a significant reduction in estimated acoustic potential energy after control of first five acoustic modes. 3.3.4. Active noise control under the presence of both acoustic and structural disturbances A structural disturbance is applied on the flexible plate by using an electrodynamic shaker as shown in Fig. 14. The electrodynamic shaker and primary loudspeaker are driven by two different power amplifiers but connected to a single function generator. Gains of these two power amplifiers are suitably adjusted to realise the desired relative contribution of the two disturbances. 3.3.4.1. Results at 170 Hz. This subsection discusses the performance of Modal FxLMS algorithm for different cases of modal control when both acoustic and structural disturbances at 170 Hz act on the vibro-acoustic cavity.
It is noted that 170 Hz is near to 3rd panel-controlled resonance (173 Hz). The cases of modal control with different choices of acoustic modes are studied: (a) control of only mode 1, (b) control of modes 1 and 4, (c) control of modes 0 and 1, (d) control of modes 0, 1, 2 and 4 and (e) control of all five modes. Table 4 shows a comparison of dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control along with reductions in these quantities after control. A comparison of reduction in acoustic potential energy for the various choices of acoustic modes indicates that as the number of modes that are controlled is increased, then there is an increase in total reduction that is obtained. For all the cases, acoustic potential energy is reduced under the presence of both acoustic and structural disturbances. It is also noted that as the number of acoustic modes in the list of modes under control is increased, then the algorithm tries to reduce the modal amplitude of these modes in such a manner as to give a maximum reduction in acoustic potential energy. 3.4. Active noise control using two control sources This section presents results of active noise control in the 3D vibro-acoustic cavity using two control loudspeakers whose locations are shown in Fig. 1. The results for two cases of modal control at 170 Hz under the presence of both acoustic and structural dis-
Fig. 12. Estimated modal amplitudes of first five acoustic modes at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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source alone, (b) using the second control source alone and (c) using both the first and second control sources.
Fig. 13. Plots of estimated acoustic potential energy at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
turbances are given below. In order to see the effectiveness of the use of two control sources simultaneously, active control is also carried out using either of these two cases alone. Therefore, the three cases of control sources studied are: (a) using the first control
3.4.1. Control of acoustic modes 1 and 3 It is seen from Fig. 6 that mode 1 is a longitudinal mode (0, 0, 1) along the length while mode 3 is a mode along the height (0, 1, 0). Mode 2 is a second longitudinal mode (0, 0, 2) while mode 4 is a tangential mode (0, 1, 1). In view of this, control of modes 1 and 3 is studied as these two modes are in two orthogonal directions and two loudspeakers couple with these two modes differently. Table 5 shows a comparison of reduction in modal amplitudes before and after control with the proposed Modal FxLMS for the abovementioned three cases of control sources. When the first control source is used alone, there is a significant reduction of 23.5 dB in the modal amplitude of the first acoustic mode but a reduction of only 2.3 dB is obtained in the modal amplitude of the third acoustic mode. When the second control source is used alone, there is a significant reduction of 18.5 dB in the modal amplitude of the first acoustic mode while the modal amplitude of the third acoustic mode increases by 1.7 dB. When both the control sources are used, a higher reduction (26.9 dB and 17.3 dB) in the modal
Fig. 14. Application of structural disturbance on the cavity.
Table 4 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz for different cases of modal control with the proposed Modal FxLMS.
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Before control (dB)
Reduction after control with second control source (dB) One mode control (1st mode control) Case (a)
Two mode control (1st and 4th mode control) Case (b)
Two mode control (0th and 1st mode control) Case (c)
Four mode control (0th, 1st, 2nd and 4th mode control) Case (d)
First five mode control Case (e)
99.4 99.6 96.0 90.6 83.6 94.4 94.0 97.1
9.8 4.9 2.2 6.7 12.9 2.5 5.5 6.7
6.7 4.7 2.3 6.9 12.9 2.3 5.0 7.6
8.5 4.0 2.0 6.1 11.3 0.1 9.2 10.8
5.8 3.0 1.7 5.3 9.1 3.9 12.7 18.2
5.9 2.9 1.5 5.0 8.4 5.1 14.0 15.2
90.6 102.0 87.7 87.4 97.5
5.9 26.9 9.2 4.1 4.2
6.0 15.7 9.4 4.7 6.0
5.3 16.5 8.4 3.3 4.3
4.5 9.8 7.5 2.9 4.6
4.3 9.0 7.1 2.4 3.9
9.8
2.6
2.6
3.1
3.2
3.3
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Table 5 Comparison of reduction in estimated modal amplitudes of first five acoustic modes at 170 Hz before and after control of the first and third acoustic modes with the proposed Modal FxLMS for the three cases of control sources. Reduction in dB after control of modes 1 and 3
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
0 1 2 3 4
Before control (dB)
Using first control source alone Case (a)
Using second control source alone Case (b)
Using both the control sources simultaneously Case (c)
90.6 102.0 87.7 87.4 97.5
6.4 23.5 9.1 2.3 1.6
4.6 18.5 7.9 1.7 3.8
6.5 26.9 9.1 17.3 0.9
Table 6 Comparison of reduction in acoustic pressures at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz before and after control of the first five acoustic modes with the proposed Modal FxLMS for the three cases of control sources. Reduction after control of first five acoustic modes (dB)
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Values before control (dB)
Using first control source alone Case (a)
Using second control source alone Case (b)
Using both the control sources simultaneously Case (c)
99.4 99.6 96.0 90.6 83.6 94.4 94.0 97.1
5.7 2.2 0.8 3.7 7.3 6.9 6.1 6.5
5.9 2.9 1.5 5.0 8.4 5.1 14.0 15.2
6.8 3.1 1.2 4.8 8.6 5.0 6.0 10.7
90.6 102.0 87.7 87.4 97.5
3.9 5.9 5.5 1.0 1.2
4.3 9.0 7.1 2.4 3.9
4.3 9.1 6.9 4.0 8.2
9.8
2.4
3.3
3.7
amplitude of both the first and third acoustic modes is obtained. Therefore, it is seen that the Modal FxLMS algorithm is able to reduce the modal amplitudes of the modes chosen for control using multiple control sources. While the use of one control source gives a significant reduction of modal amplitude of one acoustic mode, use of two control sources are able to effectively reduce the modal amplitudes of two acoustic modes. 3.4.2. Control of first five acoustic modes Another experiment is done to actively reduce noise using the two control sources but by controlling first five acoustic modes with the Modal FxLMS method. Table 6 shows a comparison of reduction in acoustic pressures at locations of eight microphones, estimated modal amplitudes and estimated acoustic potential energy at 170 Hz before and after control of five acoustic modes. It is observed from Table 6 that the reduction obtained in acoustic potential energy using two control sources is higher than that with use of only one control source. The algorithm changes the modal amplitudes of the five acoustic modes in such a way as to give a maximum reduction in the acoustic potential energy. It is also seen that acoustic modes 1 and 4 make dominant contribution (102.0 and 97.5 dB, respectively) at the current frequency at which active noise control is being studied. When only either of the control sources is used, modal amplitude of these acoustic modes is reduced but when both the control sources are used simultaneously, a higher reduction in modal amplitude of these acoustic modes is obtained. This shows the effectiveness of the Modal FxLMS algorithm in reducing the acoustic potential energy in the cavity through a reduction of modal amplitudes of multiple targeted modes.
4. Conclusion The paper presents global active noise control in a vibroacoustic cavity using Modal filtered-x least mean square algorithm. The method is evaluated on an experimental 3D rectangular box cavity with a flexible plate made of aluminium. Experimental modal analysis of the acoustic cavity and the flexible plate is carried out to obtain acoustic mode shapes, cavity-controlled resonance frequencies and panel-controlled resonance frequencies. The experimentally obtained acoustic mode shapes are used in implementation of the Modal FxLMS algorithm for on-line extraction of modal amplitudes of acoustic modes. The experimentally identified acoustic mode shapes along with experimentally identified physical secondary paths are used for computation of modal secondary paths. The Modal FxLMS algorithm is implemented on a dSPACE controller board. Eight error microphones suitably located throughout the cavity are used for measurement of acoustic pressures based on which estimation of modal amplitudes of first five acoustic modes and acoustic potential energy is carried out by the algorithm. The performance of the method is evaluated at two harmonic frequencies: at 131 Hz (coinciding with a cavitycontrolled resonance) and at 170 Hz (which is near a panelcontrolled resonance (172.8 Hz)). Active noise control is studied under the presence of only acoustic disturbance and when both acoustic and structural disturbances are present. It is observed from the results with Modal FxLMS algorithm that if one acoustic mode is selected, modal amplitude of that acoustic mode is reduced. As the number of acoustic modes is increased in the list of modes under control, then the method tries to reduce the modal amplitudes of these modes in such a manner as to give a maximum reduction in the acoustic potential energy. These results are in line
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with results of numerical study presented in a previous work by the authors. The results presented in this paper also show that the framework provided by Modal FxLMS algorithm is effective in allowing a choice of modes to be controlled as use of two control sources is seen to give a significant reduction in modal amplitude of the two acoustic modes chosen. The experimental study, therefore, shows that the Modal FxLMS method is effective for global active noise control in a vibro-acoustic cavity. References [1] Elliott SJ, Curtis ARD, Bullmore AJ, Nelson PA. The active minimization of harmonic enclosed sound fields, part iii: experimental verification. J Sound Vibration 1987;117:35–58. [2] Cheer J. Active control of the acoustic environment in an automobile cabin. Univ Southampt; 2012. [3] Cheer J, Elliott SJ. Multichannel control systems for the attenuation of interior road noise in vehicles. Mech Syst Signal Process 2015;60–61:753–69. [4] Cheer J, Elliott SJ. Active noise control of a diesel generator in a luxury yacht. Appl Acoust 2016;105:209–14. [5] Yang In-Hyung, Jeong Jae-Eun, Jeong Un-Chang, Kim Jin-Su, Jae-Eung Oh. Improvement of noise reduction performance for a high-speed elevator using modified active noise control. Appl Acoust 2014;79:58–68. [6] Parkins JW, Sommerfeldt SD, Tichy J. Narrowband and broadband active control in an enclosure using the acoustic energy density. J Acoust Soc Am 2000;108:192–203. [7] Lau SK, Tang SK. Sound fields in a rectangular enclosure under active sound transmission control. J Acoust Soc Am 2001;110:925–38. [8] Koshigoe S, Gillis JT, Falangas ET. A new approach for active control of sound transmission through an elastic plate backed by a rectangular cavity. J Acoust Soc Am 1993;94:900–7.
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[9] Balachandran B, Sampath A, Park J. Active control of interior noise in a threedimensional enclosure active control of interior noise in a three-dimensional enclosure. Smart Mater Struct 1996;5:89–97. [10] Snyder SD, Tanaka N. On feedforward active control of sound and vibration using vibration error signals. J Acoust Soc Am 1993;94:2181–93. [11] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power. J Acoust Soc Am 1993;94:2194–204. [12] Cazzolato BS, Hansen CH. Active control of sound transmission using structural error sensing. J Acoust Soc Am 1998;104:2878–89. [13] Bagha AK, Modak SV. Structural sensing of interior sound for active control of noise in structural-acoustic cavities. J Acoust Soc Am 2015;138:11–21. [14] Bagha AK, Modak SV. Active structural-acoustic control of interior noise in a vibro-acoustic cavity incorporating system identification. J Low Freq Noise Vib Act Control 2017;36:261–76. [15] Morgan DR. An adaptive modal-based active control system. J Acoust Soc Am 1991;89:248–56. [16] Fuller CR, Elliott SJ, Nelson PA. Active control of vibration. Academic Press; 1996. [17] Bai MR, Shieh C. Active noise cancellation by using the linear quadratic Gaussian independent modal space control. J Acoust Soc Am 1995;97:2664–74. [18] Lane SA, Clark RL, Southward SC. Active control of low frequency modes in an aircraft fuselage using spatially weighted arrays. J Vib Acoust 2000;122:227. [19] Clark RL. Adaptive feedforward modal space control. J Acoust Soc Am 1995;98:2639–50. [20] Cheer J, Elliott SJ. The design and performance of feedback controllers for the attenuation of road noise in vehicles. Int J Acoust Vib 2014;19:155–64. [21] Puri A, Modak SV, Gupta K. Modal filtered-x LMS algorithm for global active noise control in a vibro-acoustic cavity. Mech Syst Signal Process 2018;110:540–55. [22] Fahy F, Gardonio P. Sound and structural vibration: radiation, transmission and response. Elsevier/Academic; 2007.
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Global active control of harmonic noise in a vibro-acoustic cavity using Modal FxLMS algorithm Amrita Puri, Subodh V. Modak ⇑, Kshitij Gupta Department of Mechanical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India
a r t i c l e
i n f o
Article history: Received 8 November 2018 Received in revised form 11 January 2019 Accepted 10 February 2019
Keywords: Active control Noise FxLMS Modal Vibro-acoustic
a b s t r a c t The Modal FxLMS algorithm is recently proposed for reduction of global level of noise in vibro-acoustic cavities. The algorithm minimizes acoustic potential energy expressed in modal domain which allows a choice of acoustic modes whose modal amplitudes are desired to be minimized. The working of the algorithm and its effectiveness has been demonstrated previously through a numerical study and an experimental study is required to test the effectiveness of the algorithm in practice. To meet this objective this paper presents an experimental study of this algorithm for global active noise control in a rectangular box cavity. The algorithm utilizes modal filtered reference signals and modal amplitudes of the selected acoustic modes of the cavity. To identify modal filtered reference signals, modal secondary paths are utilized. In the present study, modal secondary paths are identified using experimentally identified physical secondary paths and acoustic mode shapes. Modal amplitudes of the acoustic modes are identified on the basis of acoustic pressure measurements from eight microphones suitably placed throughout the cavity. The performance of the algorithm is studied under the presence of acoustic and structural disturbances using one or two control loudspeakers. The active control is carried out at harmonic frequencies coinciding with cavity-controlled resonances and panel-controlled resonances. It is found from the experimental results that the Modal FxLMS algorithm is successful in reducing modal amplitudes of the selected acoustic modes. When all the significantly contributing acoustic modes are chosen, a reduction in global level of noise is observed at cavity as well as panel controlled resonances. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Car, aeroplane, helicopter and other transportation equipment are vibro-acoustic cavities encountered in real-life and reduction of interior noise in these cavities is an important objective in their NVH design. Passive and active methods are two approaches which can be used for noise control. Passive solutions are generally not much effective or acceptable for reduction of low frequency noise. On the contrary, the active methods can be effectively used in this frequency range. The problem of global active noise control in a cavity has been addressed in different ways in the literature. One of the approaches is to reduce acoustic potential energy in the cavity. Elliott et al. [1] studied minimisation of acoustic potential energy approximated as sum of squares of acoustic pressures at corners of a rectangular acoustic cavity. Cheer [2] extended this approach for global active noise control in a car cavity. Cheer and Elliott [3,4] implemented ⇑ Corresponding author. E-mail addresses: [email protected] (A. Puri), [email protected] (S.V. Modak), [email protected] (K. Gupta). https://doi.org/10.1016/j.apacoust.2019.02.008 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
feedforward control systems to minimise sum of squares of acoustic pressures to reduce road noise in a car and to reduce generator noise in a luxury yacht, respectively. Yang et al. [5] used correlation based FxLMS algorithm to reduce noise at a suitably placed error microphone to control high speed elevator noise. Another approach for global active noise control is minimisation of sum of acoustic energy densities (Parkins et al. [6]). Lau and Tang [7] presented a comparison of acoustic potential energy control, squared pressure control and energy density based control, using secondary acoustic sources, to minimize sound transmission into a rectangular enclosure. They pointed out that an increase in SPL may occur at some locations even if there is global attenuation in the acoustic potential energy. Koshigoe et al. [8] compared velocity feedback, mean square sound pressure minimisation and sound pressure feedback control schemes to reduce the acoustic potential energy inside a rectangular cavity with one flexible panel. Balachandran et al. [9] presented a comparison of different control schemes to minimise noise inside a cavity when a plane wave acts on the flexible panel of a vibroacoustic cavity. Control schemes studied were minimization of mean square pressure inside the cavity, minimisation of mean
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square velocity of panel, minimisation of acoustic potential energy and minimisation of acoustic potential energy formulated with pressure readings at a fixed number of sensors. It was observed that minimising mean square velocity of the panel may not necessarily reduce the acoustic potential energy inside the cavity and similarly, minimizing the acoustic potential energy may increase the kinetic energy of the panel. Another approach used for global active noise control in vibroacoustic cavities is control of so-called ’radiation modes’. These modes are nothing but orthogonal or independent contributors of acoustic potential energy as opposed to the structural modes (Snyder and Tanaka [10], Johnson and Elliott [11]). Active noise control in a stiffened cylindrical cavity has been carried out by minimising amplitudes of radiation modes (Cazzolato and Hansen [12]). Bagha and Modak [13] performed active noise control in a weakly coupled cavity by reducing the amplitudes of radiation modes estimated through a Kalman filter and a frequency weighting filter. Bagha and Modak [14] presented an experimental study to reduce global level of noise in a rectangular box vibro-acoustic cavity using an experimentally updated finite element model of the cavity used in formulation of the feedback controller. Minimisation of modal amplitudes of the contributing modes for active control has been researched a lot in the field of active vibration control (Morgan [15], Fuller et al. [16]). But not much work is reported on these lines in the area of active noise control. An LQG based independent modal space feedback control strategy has been used to reduce noise in a duct and a rectangular room (Bai and Shieh [17]). Lane et al. [18] presented an experimental study to reduce global level of noise in an aircraft fuselage using a modal based H2 feedback control strategy. Clark [19] studied optimal modal feedforward control for a structural reverberant system and suggested use of an adaptive modal control method based on a time averaged gradient descent algorithm. Cheer [2] and Cheer and Elliott [20] proposed optimal modal feedback control system in an internal model control (IMC) architecture. It is observed that these strategies are in the framework of feedback control or IMC and the feedback controller needs to be redesigned for a new choice of acoustic modes to be controlled. For harmonic noise, feedforward technique is often used for active noise control and therefore, a modal based formulation of this technique is needed. Recently, such a formulation called as ’Modal Filtered-x LMS’ algorithm has been presented by authors [21]. The operation and effectiveness of the algorithm have been demonstrated through a numerical study. This paper presents an experimental study of Modal FxLMS method for global active noise control in a vibro-acoustic cavity. The paper discusses the procedure to experimentally implement the Modal FxLMS method. The global active noise control is performed in a rectangular box cavity with one flexible panel under the presence of acoustic and structural disturbances at cavitycontrolled resonances and panel-controlled resonances using one or two control loudspeakers. The issue of identification of modal secondary paths is addressed. The paper is organised as follows: Section 2 describes theory of Modal FxLMS algorithm. Section 3 presents an experimental study of global active noise control using Modal FxLMS algorithm. The section presents results of active noise control under the presence of acoustic and structural disturbance with one and multiple control sources. It also presents computation of modal secondary using data identified using experimental modal analysis. Section 4 concludes the present work.
Following expression gives instantaneous acoustic potential energy inside a vibro-acoustic cavity [22],
Ep ¼
Z
1 2qa c2
V
ð1Þ
p2 dV 0
Using a finite element model of the cavity, Ep , can be written as,
Ep ¼ pH Vp
ð2Þ
where p represents vector of acoustic pressures at all nodes of the cavity, V is volume matrix and superscript ‘H’ represents Hermitian operator.
V¼
Ma 2qa
ð3Þ
where Ma and qa are acoustic mass matrix and density of air, respectively. For a weakly coupled vibro-acoustic cavity, p can be expressed in terms of modal amplitudes of rigid walled acoustic modes, pm , as,
p ¼ Ua p m
ð4Þ
where Ua is a matrix of mass-normalised rigid walled acoustic modes. Using Eqs. (2) and (4), Ep can be written as,
Ep ¼
1 H p p 2 m m
ð5Þ
In Modal FxLMS algorithm, weights of the adaptive filter are updated as,
wðn þ 1Þ ¼ wðnÞ
l 2
$Ep
ð6Þ
where $ represents gradient with respect to weights of the adaptive filter. Substituting Eq. (5) in Eq. (6) and expanding the resulting expression, we get,
wðn þ 1Þ ¼ wðnÞ
M lX
2
pkm : rpkm
ð7Þ
k¼1
where M is total number of acoustic modes and pkm corresponds to modal amplitude of kth acoustic mode. As explained in [21], pkm can be expressed as, k
k
pkm ¼ hpm qp þ hsm qs k
ð8Þ
k
where hpm and hsm represent kth modal impulse responses corresponding to primary and secondary sources, respectively. qp and qs represent inputs to primary and secondary sources, respectively. The symbol ’’represents convolution operator. The modal impulse response corresponding to primary/secondary noise source represents variation of modal amplitude of a particular mode of the cavity when the primary/secondary source is driven by an impulse function. Modal impulse responses corresponding to primary and secondary sources are called as modal primary paths and modal secondary paths, respectively. Using Eq. (8), rpkm can be written as,
rpkm ¼ rhkpm qp þ rhksm qs
rpkm ¼ r hksm qs
ð9Þ ð10Þ
Eq. (10) can be written as, 2. Modal FxLMS algorithm This section presents a brief theory of Modal FxLMS algorithm.
rpkm ¼ r hksm w xr
where xr is reference signal vector.
ð11Þ
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(2)
(6)
(8)
(4)
(7)
(9)
(1)
(5) (3)
Fig. 1. Experimental set-up for global active noise control. [(1) Rectangular box cavity with one flexible plate, (2) Primary loudspeaker, (3) Power amplifier for the primary loudspeaker, (4) Electrodynamic shaker, (5) Power amplifier for the electrodynamic shaker (6) Function generator, (7) First control loudspeaker, (8) Second control loudspeaker, (9) Microphones (8 Nos.)]
signals can be obtained by linear convolution of vectors of modal secondary paths and reference signal. Therefore, modal secondary paths are required to obtain modal filtered reference signals. As explained in [21], instantaneous values of modal amplitudes of acoustic modes are obtained as,
pm ¼ Uya p
ð14Þ
where p represents a vector of measured nodal acoustic pressures,
pm represents a vector of modal acoustic pressures of first few contributing acoustic modes in the desired frequency range and Uya represents pseudo-inverse of Ua which represents matrix of the chosen first few contributing acoustic modes. Rows of Ua correspond to nodes at which acoustic pressure is measured and columns of Ua correspond to the chosen first few contributing acoustic modes. In [21], two methods of identification of modal secondary paths are discussed. This paper utilises the second method of using acoustic mode shapes. The second method is explained below. Acoustic pressure at kth FE node can be expressed as, k
k
pk ¼ hp qp þ hs qs
ð15Þ
Using Eq. (15), writing acoustic pressures at all the nodes of FE model in a vector form, we get,
Fig. 2. Schematic diagram of the experimental set-up for global active noise control.
Assuming slow variation of weights of the adaptive filter, Eq. (11) can be written as,
rpkm ¼ r w hksm xr ¼ hksm xr ¼ xfr km
ð12Þ
where xfr km is modal filtered reference signal corresponding to kth acoustic mode. Substituting Eq. (12) into Eq. (7), we get,
wðn þ 1Þ ¼ wðnÞ
l 2
M X
pkm :
xfr km
ð13Þ
k¼1
Eq. (13) represents final weight updating equation of the Modal FxLMS algorithm which utilises instantaneous values of modal amplitudes of acoustic modes of the cavity and modal filtered reference signals. As observed from Eq. (12), modal filtered reference
8 9 8 9 8 1 9 > h1 qp > > h1 q > p > > > s s > p > > > > > > > > > > > > > > > > > 2 2 > > > > > hp qp > > > > > > hs qs > p2 > > > > > > > > > > > > > > > > > > > < p3 = < 3 = < 3 = h q h q p p s s ¼ þ > > > > > > : > > > > > > : : > > > > > > > > > > > > > > > > > > > : > > > > > > > > > > > : : > > > > : M > > > ; > > > > > > > : M ; : M ; p hs qs hp qp
ð16Þ
Eq. (16) can be expressed as,
p ¼ Hp : qp þ Hs : qs
ð17Þ
Multiplying Eq. (17) with Ua 1 , we get,
Ua 1 p ¼ Ua 1 Hp : qp þ Ua 1 Hs : qs
ð18Þ
Using pm ¼ Ua 1 p, Eq. (18) can be written as,
pm ¼ Ua 1 Hp : qp þ Ua 1 Hs : qs
ð19Þ
From Eq. (19), we get,
Hpm ¼ Ua 1 Hp
ð20Þ
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Fig. 3. (a) Experimental set-up for modal analysis of the plate [(1) Flexible plate of the cavity, (2) Impact hammer, (3) Accelerometer, (4) Signal conditioner and (5) FFT analyser] and (b) Schematic diagram of the experimental set-up for modal analysis of the plate.
133.4 Hz
70.8 Hz
257.1 Hz
172.8 Hz
317.7 Hz
235.7 Hz
340.6 Hz
Fig. 4. First seven experimental mode shapes of the plate backed by the rectangular cavity.
Hsm ¼ Ua 1 Hs
ð21Þ
where Hpm and Hsm are matrices of primary modal paths and secondary modal paths, respectively. From Eq. (21), it is seen that using information about acoustic modes shapes and physical secondary paths, modal secondary paths can be identified. In the present work, modal secondary paths are obtained using experimentally identified acoustic mode shapes and experimentally identified physical secondary paths.
formed on a 3D rectangular box cavity with one flexible panel. The Subsection 3.1 describes experimental set up used for the study. Subsection 3.2 describes experimental modal analysis of the acoustic cavity to obtain acoustic mode shapes. Subsection 3.3 presents results of the Modal FxLMS algorithm using one control loudspeaker when only acoustic disturbance acts on the vibroacoustic cavity and when both acoustic and structural disturbances act on the vibro-acoustic cavity. Subsection 3.4 presents results of the global active noise control using two control sources.
3. Global active noise control using Modal FxLMS algorithm
3.1. Description of experimental set up
This section presents an experimental study of global active noise control using Modal FxLMS algorithm. The study is per-
Fig. 1 shows the experimental set-up used for active noise control in a rectangular box vibro-acoustic cavity. The dimensions of
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Fig. 5. (a) Experimental set-up for modal analysis of the cavity [(1) Loudspeaker, (2) Microphone, (3) Power amplifier for the loudspeaker, (4) Signal conditioner for the microphone and (5) FFT analyser] and (b) Schematic diagram of the experimental set-up for modal analysis of the cavity.
the rectangular cavity are 0.522 m 0.60 m 1.372 m. The five surfaces of the cavity are made of acrylic sheet of thickness 15 mm while the remaining sixth surface of the cavity is made of aluminium of thickness 3 mm and is relatively much flexible than the other surfaces. A schematic diagram of the experimental set-up is shown in Fig. 2. One loudspeaker is placed on the top wall of the cavity which acts as a primary noise source. Two loudspeakers mounted diagonally on the acrylic wall opposite to the aluminium panel are used as control sources. The lower left loudspeaker is referred as first control loudspeaker while the upper loudspeaker is referred as second control loudspeaker. To mimic structural disturbances, a shaker is attached to excite the flexible panel of the cavity. The active noise control is carried out for a harmonic primary noise. The primary loudspeaker and/or shaker are driven by a sinusoidal signal using a function generator. Eight 1/200 microphones are positioned in the cavity to measure acoustic pressure. The locations of these microphones are chosen such that they can observe first five acoustic modes beyond rigid body mode. These five acoustic modes are shown in next subsection. The outputs of the microphones are sent to a dSPACE controller board via signal conditioners. A MATLAB Simulink block diagram of the active noise control using the Modal FxLMS method is made and burnt on the dSPACE controller board. The control outputs from the controller board are used to drive the control sources after being passed through analog low pass filters. 3.2. Experimental modal analysis of the rectangular vibro-acoustic cavity This subsection describes experimental modal analysis of the vibro-acoustic cavity to obtain acoustic mode shapes which are
required to estimate modal amplitudes of the acoustic modes of the cavity. The modal analysis also gives information about panel and cavity controlled resonances. Experimental set-ups used for modal analysis of the plate and the acoustic cavity are shown in Figs. 3 and 5, respectively. Modal analysis of the plate is performed by exciting the plate with an impact hammer at 437 (19 23) nodal points and recording the response of the plate with an accelerometer mounted at a fixed location. The frequency response functions (FRFs) are recorded and analysed to identify natural frequencies and mode shapes of the plate giving information about panel-controlled resonances of the vibro-acoustic cavity. First seven mode shapes of the plate identified in this manner are shown in Fig. 4. In these figures, blue colour shows the minimum displacement (or negative displacement) and red colour shows the maximum displacement (or positive displacement). Modal analysis of the acoustic cavity is performed using a sweep sine signal (20 Hz to 660 Hz in 80 s (sweep rate of 8 Hz/s)) fed to a loudspeaker and recording acoustic response using a microphone (Fig. 5). The interior of the cavity is divided virtually in a 3D mesh of 936 points (8 points in x-direction, 9 points in ydirection and 13 points in z-direction). The output of the microphone and voltage across two terminals of the loudspeaker are sent to a FFT analyser which provides a frequency response function. The procedure is repeated at 936 points, by moving the microphone to the desired test point, to cover the whole cavity. The 936 FRFs are analysed using modal analysis software to obtain mode shapes of the cavity. This gives information of acoustic mode shapes and cavity-controlled resonances of the vibro-acoustic cavity. Total 21 acoustic mode shapes (excluding rigid body mode) are identified in the frequency range from 100 Hz to 640 Hz, with first
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131.4 Hz
250.1 Hz
289.8 Hz
314.8 Hz
350.7 Hz Fig. 6. First five experimental acoustic mode shapes of rectangular cavity of dimensions (0.524 m 0.600 m 1.37 m). Table 1 Experimental coupled natural frequencies of the vibro-acoustic cavity. Natural frequencies Natural frequencies identified from modal identified from modal analysis of cavity (Hz) analysis of plate (Hz) 70.8 131.4 133.4 172.8 235.7 250.1 257.1 289.8 314.8 317.7 340.6 350.7
Significance
1st Panel-controlled resonance 1st Cavity-controlled resonance 2nd Panel-controlled resonance 3rd Panel-controlled resonance 4th Panel-controlled resonance 2nd Cavity-controlled resonance 5th Panel-controlled resonance 3rd Cavity-controlled resonance 4th Cavity-controlled resonance 6th Panel-controlled resonance 7th Panel-controlled resonance 5th Cavity-controlled resonance
five identified acoustic mode shapes shown in Fig. 6. In these figures, colour contour depicts the distribution of the acoustic pres-
sure in a mode. Green colour shows the location where acoustic pressure is zero while red/blue colour shows the locations where acoustic pressure is maximum (positive)/minimum (negative). In addition, the acoustic pressure distribution in a mode is also depicted by a proportionate virtual displacement in vertical direction. It is seen from Fig. 6 that mode at 131 Hz is first longitudinal acoustic mode (0, 0, 1) with a nodal plane passing thorough the centre of the length of the cavity. The mode at 250 Hz is second longitudinal mode (0, 0, 2) with two nodal planes along the length and an anti-nodal plane passing thorough the centre of the length of the cavity. The mode at 289 Hz has variation of acoustic pressure along the height of the cavity with nodal plane passing through the centre of the height of the cavity (0, 1, 0 mode). The mode at 315 Hz is a tangential mode (0, 1, 1) with nodal planes passing through centre of the length of cavity as well as through the centre of the height of the cavity. The mode at 350 Hz has variation of acoustic pressure along the width of the cavity with nodal plane passing through the centre of the width of the cavity (1, 0, 0 mode). Table 1 shows the natural frequencies obtained through experimental modal analysis of the plate and the cavity. The table also shows the significance of these coupled natural frequencies as being either panel-controlled or cavity-controlled resonances. The rigid body acoustic mode (0, 0, 0), which has a very low frequency, also needs to be known for a complete modal description of the acoustic cavity. This mode was experimentally identified by excitation of the cavity at harmonic frequency in the low frequency range. The rigid body acoustic mode is identified on the basis of the fact that acoustic pressure distribution over the test mesh is
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Fig. 7. Location of eight microphones in the rectangular box vibro-acoustic cavity.
Table 2 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes and estimated acoustic potential energy at 131 Hz for different cases of modal control with the proposed Modal FxLMS. Values before control (dB)
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
1 2 3 4 5 6 7 8
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
0 1 2 3 4
Estimated acoustic potential energy
Reduction in dB after control One mode control (1st mode control) Case (a)
Five mode control First 5 mode control) Case (b)
122.8 118.4 107.9 96.9 115.9 120.3 121.6 122.8
27.7 17.2 5.2 5.3 14.8 17.6 24.2 24.3
26.7 16.9 5.1 5.3 15.0 17.8 24.9 24.9
96.6 125.0 106.2 107.5 102.2
7.0 38.7 10.8 11.0 9.3
6.9 33.8 10.5 11.3 9.1
31.2
20.0
20.0
dominated by this mode in the low frequency range. This mode is referred as ’0th’ acoustic mode in this paper.
frequency is chosen as 5000 Hz which is more than 10 times the maximum frequency of interest.
3.3. Active noise control using one control source
3.3.1. Identification of modal amplitudes of acoustic modes This subsection presents details of the estimation of modal amplitudes of acoustic modes which are required in Modal FxLMS algorithm for updating weights of the filter (Eq. (13)). The frequencies 131 Hz and 170 Hz at which active noise control is carried out lie in the range of 50–320 Hz. In this frequency range, the first five acoustic modes (including rigid body acoustic mode) are expected
This section presents results of active noise control at harmonic frequencies using Modal FxLMS algorithm using one control source. The results at frequencies 131 Hz (coinciding with first cavity-controlled resonance) and 170 Hz (near to the third panelcontrolled resonance 173 Hz) are presented in this paper. Sampling
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Fig. 8. Acoustic pressure at location of eight microphones locations at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
to make most of the significant modal contribution to the acoustic pressure. Therefore, modal amplitudes of these acoustic modes need to be estimated from measurement of acoustic pressure, for which eight microphones are chosen. Fig. 7 shows the locations where these eight microphones are placed. The modal amplitudes are estimated from measured acoustic pressures as explained in Section 2 (Eq. (14)). The strategy used to locate these microphones is based on requirement that each of the five acoustic modes is observable to at least some of the microphones. Fig. 7 shows that the microphones 1, 7, and 8 are positioned near the anti-nodal planes of the first acoustic mode (131 Hz) whereas the microphones 3 and 4 are positioned near anti-nodal plane of the second acoustic mode
(250 Hz). Similarly, third acoustic mode (289 Hz) is observable at all the eight microphones since all are away from the nodal plane of this mode. The fourth mode (315 Hz) is observable at all microphones except third and fourth microphones. 3.3.2. Identification of modal secondary paths This subsection presents details of the identification of modal secondary paths which are required to obtain modal filtered reference signals which are required for updating weights of the filter (in Eq. (13) in Modal FxLMS algorithm. The method utilises information of acoustic mode shapes and physical secondary paths impulse responses at all locations of the cavity. Eq. (21) is then used to obtain modal secondary path impulse responses.
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Fig. 9. Estimated modal amplitudes of first five acoustic modes at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The symbols Hsm , Ua and Hs are explained in Section 2. The superscript E essentially represents that these variables in this equation correspond to experimental data. Ua E is a matrix of 936 rows and 22 columns of experimentally obtained mass normalised acoustic modes. Hs E is a matrix of physical secondary paths impulse responses at a harmonic frequency at 936 points. The length of each
Fig. 10. Plots of estimated acoustic potential energy at 131 Hz before and after control of only acoustic mode 1 with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Experimental modal analysis of the cavity (presented in Section 2) yielded 22 acoustic mode shapes (including 0th acoustic mode) between 50 Hz 640 Hz. Hence, Eq. (21) is modified as, E HEsm ¼ UEy a Hs
ð22Þ
secondary path is 30 samples. Therefore, Hs E is a matrix 936 rows and 30 columns. The physical secondary paths impulse responses are obtained using LMS algorithm for secondary path identification. Using the experimentally obtained physical secondary paths impulse responses and experimentally obtained acoustic modes shapes, modal secondary paths are obtained from Eq. (22). Since, as explained in Section 3.3.1, first five acoustic modes are considered adequate for modal description of the acoustic response in the frequency range 50–320 Hz, only first five modal secondary paths are further used in the present experimental study. 3.3.3. Active noise control under the presence of acoustic disturbance This subsection presents the results of active noise control using the Modal FxLMS algorithm under the presence of acoustic disturbance. A loudspeaker mounted in one corner of the top surface of the cavity (Fig. 1) acts as a source of primary noise or acoustic disturbance.
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3.3.3.1. Results at 131 Hz. At 131 Hz (which is the first cavitycontrolled resonance), active noise control was carried out for two choices of modes to be controlled to test the ability of the proposed Modal FxLMS algorithm to perform reduction of modal amplitudes of the chosen/desired modes. These choices are: (a) reduction of modal amplitude of only one acoustic mode (mode 1, which is nothing but the first cavity-controlled mode) and (b) reduction of modal amplitude of first five acoustic modes (modes 0, 1, 2, 3, 4). Table 2 shows dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control along with reductions in these quantities after control for the cases (a) and (b). It is noted that when only mode 1 is controlled, there is a reduction of 38.7 dB in the modal amplitude of the mode 1 whereas when all the five acoustic modes are controlled, there is a reduction of 33.8 dB in modal amplitude of the mode 1. This is because the algorithm minimises the sum of squares of modal amplitudes of all the modes considered and hence, the reduction in the modal amplitude of the mode 1 is somewhat lesser at the cost of reduction in modal amplitudes of other modes. However, it is seen that the reductions in estimated acoustic potential energy for the two cases are nearly same. Similarly, there are significant reductions in acoustic pressures at microphones locations for both the cases of modal control and these reductions are almost same for the two cases. This shows that the ANC using the proposed method can be carried out by controlling only the dominant acoustic mode with equal effectiveness as when many more modes are controlled. This highlights the usefulness of the proposed Modal-FxLMS method in allowing a choice of modes to be controlled. Figs. 8–10 show plots of acoustic pressures at location of eight microphones, estimated modal amplitudes of first five acoustic modes, estimated acoustic potential energy (taken as sum of squares of estimated modal amplitudes) before and after control of only mode 1, respectively. In these figures, the active control is initiated roughly at time t = 20 s. It is seen from Fig. 8 that there is a significant reduction in acoustic pressures at all the microphones except 4th microphone location which is placed near the nodal plane of the mode 1. It is observed from Fig. 9 that there is a significant reduction in modal amplitude of mode 1 and there is a small increase in modal amplitude of mode 0 (which explains the increase in acoustic pressure at 4th microphone). It is observed from Fig. 10 that there is a significant reduction in acoustic potential energy after control.
3.3.3.2. Results at 170 Hz. Table 3 shows dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control and reductions in these quantities after control for four cases of modal control at 170 Hz. These four cases of modal control are: (a) control of only mode 1, (b) control of modes 1 and 4, (c) control of modes 0 and 1 and (d) control of all five modes. It is seen from Table 3 that when only mode 1 is controlled, there is a reduction of 40.3 dB in modal amplitude of mode 1 (1st acoustic mode). There is a reduction of 11.2 dB in modal amplitude of mode 4 (4th acoustic mode). However, modal amplitudes of modes 0, 2 and 3 (0th, 2nd, 3rd acoustic modes) increase by 5.2 dB, 8.8 dB and 5.9 dB, respectively. When modes 1 and 4 are controlled (case (b)), reduction in modal amplitude of 1st acoustic mode is 25.0 dB (which is less than the reduction of 40.3 dB obtained when only mode 1 was controlled). There is a reduction of 12.9 dB in modal amplitude of 4th acoustic mode (which is more than 11.2 dB obtained when only mode 1 was controlled). The increase in modal amplitudes of modes 0, 2 and 3 is similar to that of when only mode 1 was controlled. This shows that the Modal FxLMS algorithm works well to effectively reduce the chosen modes (modes 1 and 4 in this case). When modes 0 and 1 are controlled (case (c)), reduction in modal amplitude of 1st acoustic mode is 16.3 dB (which is less than the reduction of 40.3 dB obtained when only mode 1 was controlled and the reduction of 25.0 dB when modes 1 and 4 are controlled). The increase in modal amplitude of 0th acoustic mode is less. When modes 0 and 1 are controlled, the increase in modal amplitude of 0th acoustic mode is 4.3 dB as against the increases of 5.1 dB and 5.2 dB for cases (a) and (b). This shows that when modes 0 and 1 are controlled, the increase in modal amplitude of 0th acoustic mode and reduction in 1st acoustic mode are lesser than the previous two cases of modal control (only 1st mode control, and 1st and 4th mode control). This shows that Modal FxLMS algorithm successfully considers the contribution of the chosen acoustic modes (modes 0 and 1 here) and attempts to reduce sum of squares of modal amplitudes of these modes. When all the five acoustic modes are considered (case (d)), the reduction in modal amplitudes of 1st and 4th acoustic modes are 7.6 dB and 8.5 dB which is lower than the corresponding reductions obtained from the previous three cases of modal control. The increases in modal amplitudes of 0th, 2nd and 3rd acoustic modes are 3.4 dB, 5.4 dB and 3.9 dB, which
Table 3 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz for different cases of modal control with the proposed Modal FxLMS method. Reduction in dB after control
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Values before control (dB)
One mode control (1st mode control) Case (a)
Two mode control (1st and 4th modes control) Case (b)
Two mode control (0th and 1st modes control) Case (c)
First five modes control Case (d)
102.8 104.3 100.5 96.1 83.6 94.8 98.1 100.1
22.4 4.3 2.0 5.9 17.0 7.2 8.1 6.3
18.3 4.1 2.0 5.8 16.7 6.8 8.8 7.0
14.0 3.7 1.5 5.0 15.1 4.8 14.5 12.0
7.0 2.3 1.2 3.9 11.6 0.6 13.0 16.5
95.7 105.7 91.9 92.3 98.7
5.2 40.3 8.8 5.9 11.2
5.1 25.0 8.6 5.9 12.9
4.3 16.3 7.6 5.0 12.7
3.4 7.6 5.8 3.9 8.5
13.1
2.2
2.3
3.1
3.1
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Fig. 11. Acoustic pressure at location of eight microphones at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
are less than the corresponding increase in the previous three cases of modal control. The amount of reduction and amount of increase in modal amplitudes of different modes for different cases of modal control is also reflected in acoustic pressures at eight microphones. For example, as seen from the results in Table 3, the increase in acoustic pressures at the location of 3rd, 4th and 5th microphones for case (d) is lesser than that of for the cases (a), (b) and (c). It is further seen from Table 3 that when five acoustic modes are controlled, a reduction of 3.1 dB in estimated acoustic potential
energy is obtained, respectively. This reduction is higher than or equal to the reductions in the cases (a), (b) and (c) (2.2 dB, 2.3 dB and 3.1 dB, respectively). Figs. 11–13 show plots of acoustic pressures at location of eight microphones, estimated modal amplitude of first five acoustic modes and estimated acoustic potential energy (taken as sum of squares of estimated modal amplitudes), respectively, before and after control for case (d) (control of first five acoustic modes). It is seen from Fig. 12 that there is a reduction in modal amplitudes of 1st and 4th acoustic modes and an increase in modal amplitudes
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of 0th, 2nd and 3rd acoustic modes after control. The reduction in modal amplitudes of 1st acoustic mode is reflected in reduction of acoustic pressures at the location of 1st, 2nd, 7th and 8th microphones (Fig. 11), which are positioned near the anti-nodal planes of the first acoustic mode, and an increase in modal amplitudes of 0th and 2nd acoustic modes is reflected in acoustic pressure at the location of 3rd and 4th microphones (Fig. 11) which are positioned near nodal plane of 1st acoustic mode and anti-nodal plane of 2nd acoustic mode. It is also seen from Fig. 13 that there is a significant reduction in estimated acoustic potential energy after control of first five acoustic modes. 3.3.4. Active noise control under the presence of both acoustic and structural disturbances A structural disturbance is applied on the flexible plate by using an electrodynamic shaker as shown in Fig. 14. The electrodynamic shaker and primary loudspeaker are driven by two different power amplifiers but connected to a single function generator. Gains of these two power amplifiers are suitably adjusted to realise the desired relative contribution of the two disturbances. 3.3.4.1. Results at 170 Hz. This subsection discusses the performance of Modal FxLMS algorithm for different cases of modal control when both acoustic and structural disturbances at 170 Hz act on the vibro-acoustic cavity.
It is noted that 170 Hz is near to 3rd panel-controlled resonance (173 Hz). The cases of modal control with different choices of acoustic modes are studied: (a) control of only mode 1, (b) control of modes 1 and 4, (c) control of modes 0 and 1, (d) control of modes 0, 1, 2 and 4 and (e) control of all five modes. Table 4 shows a comparison of dB values of acoustic pressures, estimated modal amplitudes and estimated acoustic potential energy before control along with reductions in these quantities after control. A comparison of reduction in acoustic potential energy for the various choices of acoustic modes indicates that as the number of modes that are controlled is increased, then there is an increase in total reduction that is obtained. For all the cases, acoustic potential energy is reduced under the presence of both acoustic and structural disturbances. It is also noted that as the number of acoustic modes in the list of modes under control is increased, then the algorithm tries to reduce the modal amplitude of these modes in such a manner as to give a maximum reduction in acoustic potential energy. 3.4. Active noise control using two control sources This section presents results of active noise control in the 3D vibro-acoustic cavity using two control loudspeakers whose locations are shown in Fig. 1. The results for two cases of modal control at 170 Hz under the presence of both acoustic and structural dis-
Fig. 12. Estimated modal amplitudes of first five acoustic modes at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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source alone, (b) using the second control source alone and (c) using both the first and second control sources.
Fig. 13. Plots of estimated acoustic potential energy at 170 Hz before and after control of first five acoustic modes with the proposed Modal FxLMS (Red: before control and Blue: after control). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
turbances are given below. In order to see the effectiveness of the use of two control sources simultaneously, active control is also carried out using either of these two cases alone. Therefore, the three cases of control sources studied are: (a) using the first control
3.4.1. Control of acoustic modes 1 and 3 It is seen from Fig. 6 that mode 1 is a longitudinal mode (0, 0, 1) along the length while mode 3 is a mode along the height (0, 1, 0). Mode 2 is a second longitudinal mode (0, 0, 2) while mode 4 is a tangential mode (0, 1, 1). In view of this, control of modes 1 and 3 is studied as these two modes are in two orthogonal directions and two loudspeakers couple with these two modes differently. Table 5 shows a comparison of reduction in modal amplitudes before and after control with the proposed Modal FxLMS for the abovementioned three cases of control sources. When the first control source is used alone, there is a significant reduction of 23.5 dB in the modal amplitude of the first acoustic mode but a reduction of only 2.3 dB is obtained in the modal amplitude of the third acoustic mode. When the second control source is used alone, there is a significant reduction of 18.5 dB in the modal amplitude of the first acoustic mode while the modal amplitude of the third acoustic mode increases by 1.7 dB. When both the control sources are used, a higher reduction (26.9 dB and 17.3 dB) in the modal
Fig. 14. Application of structural disturbance on the cavity.
Table 4 Comparison of reduction in acoustic pressure at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz for different cases of modal control with the proposed Modal FxLMS.
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Before control (dB)
Reduction after control with second control source (dB) One mode control (1st mode control) Case (a)
Two mode control (1st and 4th mode control) Case (b)
Two mode control (0th and 1st mode control) Case (c)
Four mode control (0th, 1st, 2nd and 4th mode control) Case (d)
First five mode control Case (e)
99.4 99.6 96.0 90.6 83.6 94.4 94.0 97.1
9.8 4.9 2.2 6.7 12.9 2.5 5.5 6.7
6.7 4.7 2.3 6.9 12.9 2.3 5.0 7.6
8.5 4.0 2.0 6.1 11.3 0.1 9.2 10.8
5.8 3.0 1.7 5.3 9.1 3.9 12.7 18.2
5.9 2.9 1.5 5.0 8.4 5.1 14.0 15.2
90.6 102.0 87.7 87.4 97.5
5.9 26.9 9.2 4.1 4.2
6.0 15.7 9.4 4.7 6.0
5.3 16.5 8.4 3.3 4.3
4.5 9.8 7.5 2.9 4.6
4.3 9.0 7.1 2.4 3.9
9.8
2.6
2.6
3.1
3.2
3.3
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Table 5 Comparison of reduction in estimated modal amplitudes of first five acoustic modes at 170 Hz before and after control of the first and third acoustic modes with the proposed Modal FxLMS for the three cases of control sources. Reduction in dB after control of modes 1 and 3
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
0 1 2 3 4
Before control (dB)
Using first control source alone Case (a)
Using second control source alone Case (b)
Using both the control sources simultaneously Case (c)
90.6 102.0 87.7 87.4 97.5
6.4 23.5 9.1 2.3 1.6
4.6 18.5 7.9 1.7 3.8
6.5 26.9 9.1 17.3 0.9
Table 6 Comparison of reduction in acoustic pressures at locations of eight microphones, estimated modal amplitudes of first five acoustic modes and estimated acoustic potential energy at 170 Hz before and after control of the first five acoustic modes with the proposed Modal FxLMS for the three cases of control sources. Reduction after control of first five acoustic modes (dB)
Acoustic pressure at error microphones
Mic Mic Mic Mic Mic Mic Mic Mic
Modal amplitude of acoustic modes
Mode Mode Mode Mode Mode
Estimated acoustic potential energy
1 2 3 4 5 6 7 8 0 1 2 3 4
Values before control (dB)
Using first control source alone Case (a)
Using second control source alone Case (b)
Using both the control sources simultaneously Case (c)
99.4 99.6 96.0 90.6 83.6 94.4 94.0 97.1
5.7 2.2 0.8 3.7 7.3 6.9 6.1 6.5
5.9 2.9 1.5 5.0 8.4 5.1 14.0 15.2
6.8 3.1 1.2 4.8 8.6 5.0 6.0 10.7
90.6 102.0 87.7 87.4 97.5
3.9 5.9 5.5 1.0 1.2
4.3 9.0 7.1 2.4 3.9
4.3 9.1 6.9 4.0 8.2
9.8
2.4
3.3
3.7
amplitude of both the first and third acoustic modes is obtained. Therefore, it is seen that the Modal FxLMS algorithm is able to reduce the modal amplitudes of the modes chosen for control using multiple control sources. While the use of one control source gives a significant reduction of modal amplitude of one acoustic mode, use of two control sources are able to effectively reduce the modal amplitudes of two acoustic modes. 3.4.2. Control of first five acoustic modes Another experiment is done to actively reduce noise using the two control sources but by controlling first five acoustic modes with the Modal FxLMS method. Table 6 shows a comparison of reduction in acoustic pressures at locations of eight microphones, estimated modal amplitudes and estimated acoustic potential energy at 170 Hz before and after control of five acoustic modes. It is observed from Table 6 that the reduction obtained in acoustic potential energy using two control sources is higher than that with use of only one control source. The algorithm changes the modal amplitudes of the five acoustic modes in such a way as to give a maximum reduction in the acoustic potential energy. It is also seen that acoustic modes 1 and 4 make dominant contribution (102.0 and 97.5 dB, respectively) at the current frequency at which active noise control is being studied. When only either of the control sources is used, modal amplitude of these acoustic modes is reduced but when both the control sources are used simultaneously, a higher reduction in modal amplitude of these acoustic modes is obtained. This shows the effectiveness of the Modal FxLMS algorithm in reducing the acoustic potential energy in the cavity through a reduction of modal amplitudes of multiple targeted modes.
4. Conclusion The paper presents global active noise control in a vibroacoustic cavity using Modal filtered-x least mean square algorithm. The method is evaluated on an experimental 3D rectangular box cavity with a flexible plate made of aluminium. Experimental modal analysis of the acoustic cavity and the flexible plate is carried out to obtain acoustic mode shapes, cavity-controlled resonance frequencies and panel-controlled resonance frequencies. The experimentally obtained acoustic mode shapes are used in implementation of the Modal FxLMS algorithm for on-line extraction of modal amplitudes of acoustic modes. The experimentally identified acoustic mode shapes along with experimentally identified physical secondary paths are used for computation of modal secondary paths. The Modal FxLMS algorithm is implemented on a dSPACE controller board. Eight error microphones suitably located throughout the cavity are used for measurement of acoustic pressures based on which estimation of modal amplitudes of first five acoustic modes and acoustic potential energy is carried out by the algorithm. The performance of the method is evaluated at two harmonic frequencies: at 131 Hz (coinciding with a cavitycontrolled resonance) and at 170 Hz (which is near a panelcontrolled resonance (172.8 Hz)). Active noise control is studied under the presence of only acoustic disturbance and when both acoustic and structural disturbances are present. It is observed from the results with Modal FxLMS algorithm that if one acoustic mode is selected, modal amplitude of that acoustic mode is reduced. As the number of acoustic modes is increased in the list of modes under control, then the method tries to reduce the modal amplitudes of these modes in such a manner as to give a maximum reduction in the acoustic potential energy. These results are in line
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