Testing the active minimization of the total radiated sound power

from a vibrating plate Mehran Hajilou 1 Helmut-Schmidt-University / University of the Federal Armed Forces Hamburg Holstenhofweg 85 22043 Hamburg, Germany Delf Sachau 2 Helmut-Schmidt-University / University of the Federal Armed Forces Hamburg Holstenhofweg 85 22043 Hamburg, Germany

ABSTRACT This paper presents an experimental investigation of an Active-Noise-Control (ANC) system for reducing unwanted so-called primary sound by destructive interference with sound generated by so-called secondary sources (loudspeakers). In this work the primary source is a vibrating simply supported plate which is radiating sound into the free field. The loudspeakers are placed in front of the plate. The ANC-system utilizes the measured signals of the primary accelerometers which are placed on the surface of the plate. The number of the primary accelerometers corresponds to the number of the loudspeakers. Also, one secondary accelerometer is fixed onto each loudspeaker cone. Furthermore, a microphone in front of each loudspeaker measures the near field sound pressure. An adaptive feed-forward-controller is used to calculate the optimal control signal for each secondary source. The optimal control is achieved by minimizing the radiated active sound intensity in front of each loudspeaker in order to minimize the total radiated sound power of the primary and secondary sources. The ANC- system is tested in different configurations with one or two loudspeakers in front of the plate. Also, an approach for the system calibration is developed to compensate the phase deviation between the true and the measured sound intensity. 1. INTRODUCTION

The minimization of the radiated sound power from an interfering source with a destructive secondary source is derived analytically in [4], [5] for point source radiators in the free field. Here, the idea is used that the sound power of each point source can be described by the product of its sound flow and the sound pressure at the position of the point sources. By formulating this problem into a quadratic error function depending on the complex sound flow of the secondary sound source, the global minimum of the total radiated sound power can be calculated. One important key finding through this derivation is that a reduction in the radiated sound power from the primary and secondary sources in the free sound field can only be achieved if the distance between the sound sources is less than half

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an acoustic wavelength. The resistances components (real part of the acoustic impedance) of both sound sources are required in order to calculate the minimum of the total sound power. In practical applications with loudspeakers, these quantities are difficult to determine metrologically because the volume velocity and the sound pressure should be measured at the position of each source. This question is addressed in [6].

The expanded theoretical foundations are described in [7], [8] by means of the analytical model of the point sources in the free field depending on the frequency. The minimum of the radiated sound power from multiple interfering primary sources and destructive secondary sources is reached when the radiated sound power from each secondary source becomes zero. The volume velocities of the secondary sources become in phase or out of phase with the volume velocities of the interfering sound sources. The limits of this method are met when there are several sound sources which are distributed in a space with a diffuse sound field and when the distance of half an acoustic wavelength in the free sound field is exceeded.

The undertaken analytical and experimental investigations for this purpose are limited to examples with sinusoidal sound in the weakly dampened rooms with a loudspeaker each for the constant periodic disturbance excitation and anti-noise generation [9], [10], [11], [12]. The secondary loudspeakers are controlled manually so that their radiated sound power is zero or the potential sound energy in the entire room is minimal. Placing a loudspeaker directly in front of an interfering loudspeaker is beneficial due to the high acoustic coupling. Another advantage of this method is that a direct accessibility to the acoustic radiation impedance of the interfering source is not required. The question of implementing the method in an adaptive control system remains unanswered. A formulation that minimizes the total radiated sound power will derive a reference solution. This solution can be used for theoretical preliminary investigations of anti-noise systems as a reference point of the highest possible noise reduction.

In [13] an extended analytical study for minimizing structure borne noise is carried in order to minimize the total radiated sound power of a rectangular simply supported plate in a baffle with the point sources as secondary sources. The achieved minimum of the total radiated sound power depends on the number of the vibrating mode of the plate, the position and number of the secondary sources.

An extended approach to the investigations in [13] is demonstrated in [14], [15] on a rectangular simply supported plate. In these studies, the plate is excited with an incident plane wave or a point force. The radiated sound from the vibrating plate and the flat anti-noise source of a finite surface is formulated by subdividing them into surface elements that vibrates in phase (elemental sources). The distance between the plate and the anti-noise sources is much smaller than the excited acoustic wavelength. By knowing the radiation impedance, the scalar error function for minimizing the total radiated sound power is derived with the mean of normal velocity and the sound pressure of the surface of each elemental source. Numerical calculations show that minimizing the total radiated sound power in the low frequency (< 300 Hz) can be achieved with four anti-noise sources. So far, no problem-related adaptive control system is presented. However, the analytical and numerical investigations show that the minimization of the total radiated sound power with optimal control of some anti-noise sources in front of the vibrating plate is possible. In the literatures, there is an elaborated documentation concerning the appropriate position of anti-noise sources. The best results are therefore achieved when the arrangement of the anti-noise sources corresponds to the number of the antinodes of the vibrating plate. The optimal positions of the anti-noise sources in front of the plate are not verified with the experimental investigations.

The so far missing implementation of an adaptive control in a local anti-noise system to minimize the total radiated sound power is the motivation of works in [1], [2] and [3]. In [1] and [2], an adaptive Simple-Input-Simple-Output (SISO) control system is demonstrated which is utilized to minimize the radiated sound power from a primary loudspeaker with a secondary loudspeaker. This work is based on the findings from [7], [8]. This states that the radiated sound power and the efficiency of the radiated sound from a pair of sources become minimal when the radiated sound from the destructive source is in phase or 180° out of phase with the other source and the sources do not radiated any sound power. The radiated active sound power from the secondary loudspeaker becomes zero.

The adaptive system from [1] and [2] is in [3] further developed for a Multiple-Input-Multiple- Output (MIMO) system utilizing a Filtered-Reference-Least-Mean-Square (FxLMS) algorithm. This control system is tested with two loudspeakers as primary (interference) sources in front of two loudspeakers as secondary (destructive) sources. In this regard, an adaptive feedforward controller is designed. The adaptation of the control signal amplitude of the secondary loudspeakers happens until their sound radiation impedance is purely reactive and thus the total radiated sound powers of the primary and secondary loudspeakers become minimal.

The global reduction of the sound pressure level has been proven in experiment [3]. Using a finite element model, it is shown that minimizing the total radiated sound power with optimal control in a poorly damped room is approximately equivalent to minimizing the total sound energy in a room. Consequently, a global noise reduction can be achieved with the local control in comparison with a control of the sound pressure field over a large number of distributed microphones. The question of transferability to radiated sound from the vibrating modes of a plate remains unanswered. In the experimental investigations which have been done so far, the amplitude gains and the phase of the secondary sources in front of a vibrating plate are set manually. There is no experiment with a plate and an adaptive control that includes the destructive loudspeakers and an unknown acoustic impedance of the noise source. These open questions form the basis of this study.

2. Proposed approach: Total-Active-Sound-Power-Minimization (TASPM)

Figure 1 shows the concept for TASPM. It comprises mainly a primary source, 2 loudspeakers as secondary sources and also a digital controller. The primary source is a simply supported plate which is mounted in an infinite baffle. The plate is excited by a point force. The loudspeakers are positioned in front of the plate.

Figure 1: Schematic of a MIMO system in TASPM In the MIMO system [3], as shown in Figure 1, the controller is fed by the primary accelerations 𝒂 𝑝 = [ 𝑎 𝑝1 𝑎 𝑝2 ] 𝑇 measured by the bonded accelerometers on the surface of the plate, the secondary accelerations 𝒂 𝑠 = [ 𝑎 𝑠1 𝑎 𝑠2 ] 𝑇 measured by bonded accelerometers on the membrane of the loudspeakers and also the sound pressures 𝒑 𝑠 = [ 𝑝 𝑠1 𝑝 𝑠2 ] 𝑇 measured by the microphones in front of the loudspeakers .

The TASPM has been suggested in [1]. The aim of this approach is to minimize the active acoustic energy in an interior. This is achieved by minimizing the sound intensity in front of each secondary source which enables the total radiated sound power of the primary and secondary sources to be reduced. In other words:

𝑰 𝑠 = [ 𝐼 𝑠1

𝐼 𝑠2 ] = [0

0 ] (1)

where 𝐼 𝑠𝑙 with 𝑙= 1,2 are the active sound intensity in front of each secondary source. The radiated sound intensities 𝑰 𝑠 can be extended as below :

𝑰 𝑠 = − 1

∗ (𝑗𝜔)} (2)

2𝜔 Im{diag{𝒑 𝑠 (𝑗𝜔)} 𝒂 𝑠

where diag{ . } is the diagonal matrix of the quantity in brackets, ∗ denotes complex conjugation, Im{. } is the imaginary part of the quantity. The term 𝑗𝜔 for each parameter in Equation (2) means that each parameter is a complex number at the angular frequency 𝜔 . The TASPM approach introduces an optimal factor 𝒦 𝑙 assigned to each secondary source. With real amplification factors defined to be 𝓚= [𝒦 1 𝒦 2 ] 𝑇 , the secondary accelerations become:

𝒂 𝑠 (𝑗𝜔) = diag{𝓚} 𝒂 𝑝 (𝑗𝜔) (3) The primary pressures 𝒑 𝑝𝑠 (𝑗𝜔) are superimposed by the secondary pressures 𝒑 𝑠𝑠 (𝑗𝜔) measured at the position of the microphones:

𝒑 𝑠 (𝑗𝜔) = 𝒑 𝑝𝑠 (𝑗𝜔) + 𝒑 𝑠𝑠 (𝑗𝜔) (4)

The secondary pressures 𝒑 𝑠𝑠 (𝑗𝜔) can be expressed by the transfer matrix 𝑯 𝑝𝑎 (𝑗𝜔) between pressure in front and acceleration of each secondary source 𝒂 𝑠 (𝑗𝜔) :

𝒑 𝑠𝑠 (𝑗𝜔) = 𝑯 𝑝𝑎 (𝑗𝜔) 𝒂 𝑠 (𝑗𝜔) (5)

Substituting Equation (5) with Equation (4) in Equation (2) leads to:

𝑰 𝑠 = − 1

∗ } 𝓚} (6)

2𝜔 Im{ diag{ 𝒑 𝑝𝑠 + 𝑯 𝑝𝑎 diag {𝒂 𝑝 } 𝓚 } diag{ 𝒂 𝑝

Hence, the real amplification factors (optimal factors) 𝓚= [𝒦 1 𝒦 2 ] 𝑇 for the secondary sources can be defined as [3]:

−1

𝑰 𝑠 = 0 →𝓚= −Im { diag{ 𝒂 𝑝

∗ } (7)

∗ } 𝑯 𝑝𝑎 diag{ 𝒂 𝑝 } }

Im{diag{𝒑 𝑝𝑠 } 𝒂 𝑝

The column matrix 𝓚 can be calculated in one step. Finally, by substituting Equation 7 in Equation 3 the optimal solutions of the secondary accelerations can be formulated . 3. Implementation of the proposed TASPM in a digital feed-forward-controller

The TASPM approach requires the primary pressures 𝒑 𝑝𝑠 (𝑗𝜔) which can be calculated for steady noise before control [3]. Also, due to the variation of the primary noise or the acoustic interaction between the primary and secondary sources, the optimal factors 𝓚 require to be updated . From Equation 4, the estimation of the primary pressures 𝒑̂ 𝑝𝑠 can be carried out in the controller:

𝒑̂ 𝑝𝑠 (𝑗𝜔) = 𝒑 𝑠 (𝑗𝜔) − 𝑯 ̂ 𝑝𝑎 (𝑗𝜔) 𝒂 𝑠 (𝑗𝜔) (8)

where 𝑯 ̂ 𝑝𝑎 (𝑗𝜔)

𝑯 ̂ 𝑝𝑎 (𝑗𝜔) = 𝒑 𝑠𝑠 (𝑗𝜔)(𝒂 𝑠 (𝑗𝜔)) −1 (9)

is the estimation of the transfer path between the secondary pressures 𝒑 𝑠𝑠 (𝑗𝜔) and the secondary accelerations 𝒂 𝑠 (𝑗𝜔) which has to be identified before control, see Equation 5. Thus, Equation 7 becomes:

−1

𝓚= −Im { diag{ 𝒂 𝑝

∗ } (10)

∗ } diag{𝝋 𝒊 } 𝑯̂ 𝑝𝑎 diag{ 𝒂 𝑝 } }

Im{diag{ diag{𝝋 𝒊 } 𝒑̂ 𝑝𝑠 } 𝒂 𝑝

where 𝝋 𝒊 is the phase error of the measured sound intensity in front of each loudspeaker in the experimental investigation. How to calibrate the system in order to measure the phase error 𝝋 𝒊 of the measured sound intensity is explained in detail in section 3.1.

An adaptive feedforward controller is used for the system. Figure 2 presents the block diagram of the control scheme. The input signals 𝒂 𝑝 , 𝒂 𝑠 and 𝒑 𝑠 which feed the controllers are converted from

analog to digital format and transformed to frequency domain using a discrete Fast-Fourier- Transform (FFT). The Inverse-Fast-Fourier-Transform (IFFT) transforms the output signals 𝒚 from the frequency domain to time domain.

Figure 2:TASPM system with feedforward control algorithm The secondary path 𝑯 𝑎 (𝑗𝜔) is the relation between the control signals 𝒚(𝑗𝜔) and the secondary accelerations:

𝑯 𝑎 (𝑗𝜔) = 𝒂 𝑠 (𝑗𝜔)(𝒚 (𝑗𝜔)) −1 (11) which has to be identified before control. The reference signals 𝑿(𝑗𝜔) are the measured primary accelerations 𝒂 𝑝 (𝑗𝜔) . As shown in Figure 2, the reference signals are filtered by the estimated secondary path 𝑯 ̂ 𝑎 (𝑗𝜔) as the input of the LMS block. The error signals of the controller are:

𝑚 (𝑗𝜔) −diag{𝓚} 𝒂 𝑝 (𝑗𝜔) ⏟

(12)

𝒆(𝑗𝜔) = 𝒂 𝑠

where 𝒂 𝑠

Equation 3

𝑚 (𝑗𝜔) are the measured secondary accelerations and 𝒂 𝑠 (𝑗𝜔) are the desired secondary accelerations calculated from the TASPM approach. The controller computes the complex coefficients 𝒘(𝑗𝜔) of the adaptive filter from the last available error signals 𝒆(𝑗𝜔) :

𝒘 𝑛+1 (𝑗𝜔) = 𝒘 𝑛 (𝑗𝜔)–𝜇 𝑿 ∗ (𝑗𝜔) 𝑯 ̂ 𝑎

𝐻 (𝑗𝜔) 𝒆(𝑗𝜔) (13)

where the subscript 𝑛 is the iteration step, 𝜇 denotes the step size of control algorithm and 𝐻 is the Hermitian conjugate. The optimal output signals 𝒚 (𝑗𝜔) = [𝑦 1 (𝑗𝜔) 𝑦 2 (𝑗𝜔) ] 𝑇 for each secondary loudspeaker can be expressed as:

𝑦 𝑙 (𝑗𝜔) = 𝑋 𝑙 (𝑗𝜔)𝑤 𝑙 (𝑗𝜔) (14)

3.1. System calibration Utilizing the amplifiers, Analog-to-Digital and Digital-to-Analog converters and cables in experiment lead to deviation of the measured active sound intensities from the true active sound intensities [1], [3]. Moreover, sound pressures 𝒑 𝑠 (𝑗𝜔) and secondary accelerations 𝒂 𝑠 (𝑗𝜔) are measured with sensors that have different designs. As a result, these also have different phase responses. These phase errors do not allow the controller to compute the true optimal factors 𝓚 using the TASPM approach. Therefore, the system requires calibration in order to compensate the occurred phase errors. By calibrating the system, the phase error 𝝋 𝑖 for the measured sound intensity in front of each loudspeaker can be identified.

Figure 3: System calibration Figure 3 shows the block diagram of the system which is used to calibrate the system. The system can be calibrated if there is no phase difference between the measured 𝒑 𝑠 (𝑗𝜔) and 𝒂 𝑠 (𝑗𝜔) . To get this:

𝒑 𝑠 (𝑗𝜔) = 𝒑 𝑠𝑠 (𝑗𝜔) (15)

The amplitudes of 𝒑 𝑠 and 𝒑 𝑠𝑠 cannot be equal, because 𝒑 𝑠 (𝑗𝜔) = 𝒑 𝑝 (𝑗𝜔) + 𝒑 𝑠𝑠 (𝑗𝜔) , see Equation 4. But, the phase of 𝒑 𝑠 (𝑗𝜔) and 𝒑 𝑠𝑠 (𝑗𝜔) must be equal. Substituting Equation 5 in Equation 15 leads to the desired secondary accelerations for system calibration as below:

−1

𝒂 𝑠 (𝑗𝜔) = (𝑯 𝑝𝑎 (𝑗𝜔))

𝒑 𝑠 (𝑗𝜔) (16)

hence, the error signal can be built up as:

−𝒂 𝑝 (𝑗𝜔) (17)

𝒆(𝑗𝜔) = 𝒂 𝑠 (𝑗𝜔) ⏟

Equation 16

in other words, the system is calibrated when the phase of measured sound pressures 𝒑 𝑠 (𝑗𝜔) by microphones and the phase of sound pressure of the loudspeakers 𝒑 𝑠𝑠 (𝑗𝜔) are equal, and also the phase of measured secondary accelerations 𝒂 𝑠 (𝑗𝜔) are 180 ˚ out of phase with the measured phase of primary acceleration 𝒂 𝑝 (𝑗𝜔) . When the controller converges, the phase error of the measured active sound intensity can be calculated as:

𝝋 𝒊 = ∠𝒑 𝑠 (𝑗𝜔) −∠𝒂 𝑠 (𝑗𝜔) (18)

The measured phase errors 𝝋 𝒊 can be then used in Equation 10. 4. Experimental setup

In the experimental setup, a plate with the simply supported boundary condition on all edges is used. Figure 4a represents the rectangular plate with the dimensions (𝑎, 𝑏, ℎ) = (350 mm, 450 mm, 2 mm) . The description of a similar experimental setup can be found in [16]. Restricting all edges of the plate from lateral motion while free rotation is allowed describes the simply supported condition. Figure 4b depicts details of designed plate to meet this condition in the experimental setup. First, 60 holes spaced 6 mm apart are drilled into the edges of the plate. Second, four shims of 0.1 mm thickness are attached to 4 edges of the plate with M1.4 ⨯ 4 screws through these holes. Third, the plate and shims are fastened together to a heavy 30 mm thick aluminum frame. The inner dimensions of the frame correspond to the dimensions of the plate. Then, 26 holes spaced 65 mm apart are tapped into 4 sides of the aluminum frame. M5 ⨯ 20 screws are used to connect the

shims and frame together. Finally, the shims are attached to the 4 faces of the heavy frame using 4 aluminum strips.

Figure 4: a) Plate in the experimental setup b) Schematic of the simply supported plate designed for

the experimental setup c) The attached box to the back side of the plate The simply supported plate is mounted in a finite baffle. The 20 cm thick baffle is made of the Medium-Density-Fiberboard (MDF). To avoid the acoustic interaction between the front and back side of the plate, an asymmetrical hollow MDF box is attached to the back side of the plate, see Figure 4c. In addition, the inside of the box is filled with sound absorbing material.

4.1. Modal analysis It is necessary to evaluate if the aluminum plate satisfies the simply supported boundary condition with experimental tests. In addition, the number and positions of secondary sources in front of the plate are determined by the class of the plate's vibration mode [17]. Performing the modal analysis experimentally provides an alternative solution to these problems. In this regard, the eigenfrequencies and mode shapes of the plate are measured experimentally and are compared with the analytical results obtained in [17].

Figure 5: Shaker and force sensor

In this study, the plate is excited by a soundaktor as shaker, model AS1207, which is made by "Kendrion Kuhnke Automation GmbH". In order to excite more modes in the plate, the shaker is positioned close to a corner of the plate [18]. Therefore, the Excitation-Point (EP) is located in the lower right corner of plate, which is 70 mm away from the edges of the plate, see Figure 4a. In addition, as shown is Figure 5, a force sensor, model B&K 8230-001, measures the force applied between the shaker and the plate. The amplitude of the transfer function between the averaged velocity on the surface of vibrating plate and the applied force is measured using a Polytec Laser Scanning Vibrometer PSV-500 Xtra. The measurements were performed in the frequency range from 50 Hz to 200 Hz.

Gr,

a)

100 10 Frequency (Hz) 200

Figure 6: a) Plate velocity and mode of vibration at b) 67 Hz c) 137 Hz d) 179 Hz Figure 6a shows the measured amplitude of the transmission function between the averaged plate velocity and applied force. The peaks in Figure 6a occur at the plate's eigenfrequencies 67 Hz, 137 Hz and 179 Hz respectively. Figures 6b,c and d show the mode shapes of the plate at these eigenfrequencies respectively.

Table 1 shows the modal analysis results from the experimental investigation and from the analytical results given in [17]. The comparison of the analytical results with the experiment results reveals that the experimental setup proposed in section 4 for the plate agrees well with the simply supported condition at all edges of the plate.

Table 1: Eigenfrequency of the vibrating simply supported plate

Mode Number of antinodes in

Eigenfrequency (Hz) Experimental results Analytical results [17] 1 (1,1) 67 64 2 (1,2) 137 136 3 (2,1) 179 183 4.2. Configurations To test the adaptive controller proposed in section 3, the suggested configurations in [17] were used in the experimental setup. Figures 7a,b and c depict these configurations schematically while Figures 7d,e and f show each configuration in the experimental setup. The distance between the primary and secondary sources is 20 cm. The number of the primary accelerometers, secondary accelerometers and microphones corresponds to the number of the loudspeakers in each configuration. In all configurations, t he EP is located in the lower right corner of plate, but on the back side of the plate.

( 𝒙, 𝒚) directions

Figure 7a, compare with Figure 7d, shows the schematic of the arrangement in configuration I which can be considered as a SISO system due to use of one loudspeaker in front of the plate. In this configuration, one primary accelerometer is glued to the center of the plate.

Figure 7: Schematic of configuration a) I b) II c) III

and experimental setup with configuration d) I e) II f) III The arrangements of the configurations II, III provide the MIMO system by means of using two secondary loudspeakers in front of plate, see Figures 7b and c ( compare with Figures 7e and f respectively ). Table 2 lists the position of each component in each configuration.

Table 2: Position of the components in each configuration [17]

Config. Component Coordinate x/ a y/ b z/cm

Loudspeaker LS 1 2 1 2

20 Microphone Mic 18 Primary accelerometer Acc 0

I

1 st Loudspeaker LS1

20 1 st Microphone Mic1 18 1 st Primary accelerometer Acc1 0 2 nd Loudspeaker LS2 3 4

1 4

1 2

II

20 2 nd Microphone Mic2 18 2 nd Primary accelerometer Acc2 0

1 st Loudspeaker LS1 1 4

20 1 st Microphone Mic1 18 1 st Primary accelerometer Acc1 0 2 nd Loudspeaker LS2 3 4

1 2

III

20 2 nd Microphone Mic2 18 2 nd Primary accelerometer Acc2 0 In the experimental setup, 5 monitor microphones are used in order to measure the total radiated Sound-Pressure-Level (SPL). Figure 8 shows the monitor microphones in the experimental setup. In addition, Table 3 lists the position of the monitor microphones.

Table 3: Position of the monitor

microphones

Coordinate x/cm y/cm z/cm 1 -26 29 14 2 32 -36 71 3 56 24 49 4 14 58 66 5 21 67 19

Monitor

mic.

Figure 8: Monitor microphones

in experimental setup

5. Results

The monitor microphones measure the SPL in two scenarios. In the first scenario, the controller is OFF. In this case, only the plate vibrates and radiates sound. In the second scenario, the controller is ON and feed each loudspeaker with the calculated optimal signal according to the TASPM approach. The SPL at each monitor microphone for both scenarios are listed in Table 4. They can be compared in order to evaluate the efficiency of the adaptive controller.

Table 4: SPL in dB at the monitor microphones without controller (OFF) or controlled (ON)

SPL (dB)

OFF ON Difference

Freq.

(Hz) Config.

Monitor mic. Monitor mic. All Mon. mics.

1 2 3 4 5 1 2 3 4 5 Mean value

I 83.8 81.8 82.7 80.0 82.8 76.2 73.3 75.9 72.2 74.2 7.9 II 83.5 81.3 82.3 79.6 82.3 75.0 71.7 74.6 71.2 73.6 8.6 III 83.5 81.3 82.3 79.6 82.3 73.0 70.9 74.6 70.0 72.8 9.5

67

I 86.5 81.0 80.5 77.5 83.0 76.0 69.8 73.0 68.8 72.5 9.7 II 86.8 81.0 80.7 78.2 83.5 76.0 69.5 73.0 68.8 72.5 10.1 III 86.8 81.0 80.7 78.2 83.5 73.0 68.5 72.5 67.0 71.0 11.6

75

I 85.4 81.0 78.0 76.5 81.7 77.3 72.7 72.5 66.7 71.3 8.4 II 86.3 81.6 78.8 77.6 82.6 75.2 69.8 71.7 64.5 68.0 11.5 III 86.3 81.6 78.8 77.6 82.6 72.0 69.6 72.0 64.8 68.5 12.0 125 II 65.5 81.5 74.4 77.0 87.0 65.3 75.0 73.6 72.5 83.0 3.2 137 II 82.0 76.0 78.3 81.0 89.9 67.0 71.3 71.1 68.0 79.5 10.1 170 III 84.2 75.7 76.2 68.7 73.0 77.6 70.1 70.5 68.0 72.8 3.8 179 III 83.4 67.0 78.0 66.7 73.0 75.6 60.6 71.2 65.3 69.4 5.2 In the configuration I at 𝑓= 67 Hz where the plate vibrates with only one antinode, the active noise reduction is 7.9 dB averagely. At this eigenfrequency and in the configuration II and III, the results are approximately the same when compared to the results in configuration I. Hence, it can be concluded that utilization of one loudspeaker in front of the plate vibrating with one antinode is sufficient [17].

85

The results listed in Table 4 at 𝑓= 75 𝐻𝑧 in all configurations show that the use of the TASPM approach in the adaptive controller leads to a minimization of the SPL on average between 10 and 13

dB. The adaptive system utilized in configuration III shows 2 dB higher SPL reduction at the positions of the monitor microphones 1 and 5 compared to the results obtained in configurations I and II.

At the eigenfrequency of 137 Hz, the plate vibrates with two antinodes in 𝑦 direction, see Figure 6c. At this eigenfrequency and in configuration II, two loudspeakers are placed in front of each antinode of the plate, see Figure 7b. The listed results in Table 4 in this case show that the highest SPL reduction occurs at the position of the monitor microphones 1 ( ≌ 15 dB) and 4 ( ≌ 13 dB). In this case, the average SPL reduction measured by all monitor microphone is 10.06 dB.

At 𝑓= 125 Hz and at the position of the monitor microphone 1, the SPL reduction is 0.2 dB when the controller is on compared with when the controller is off, whereas at 𝑓= 137 Hz the system shows ≌ 15 dB SPL reduction. The reason can be attributed to the main path of the sound transmission energy from the vibrating plate which changes at each frequency. Hence, the SPL minimized by using the adaptive controller at the position of each monitor microphone is therefore different at each frequency.

The third eigenfrequency of the plate is 𝑓= 179 Hz. The configuration III is the optimal arrangement for testing the adaptive controller at this frequency due to positioning each loudspeaker in front of each antinode of plate. At the position of the monitor microphone 1, the highest SPL reduction happens in comparison with the position of the other monitor microphones. This SPL reduction is 7.8 dB. At the position of monitor microphone 4, the SPL reduction is less than 1.5 dB.

It is also interesting to point out that when the plate vibrates (controller is off) with one antinode, namely at 𝑓= 67 Hz, the minimum and maximum SPLs measured by the monitor microphones are 80 and 83.8 dB, which are approximately the same compared to each other. But at higher frequencies, when the plate vibrates with two antinodes, for instance at 𝑓= 137 Hz, the measured minimum and maximum SPLs are 76 and 89.9 dB. It is to be noted that the plate vibrating with two antinodes does not radiate the sound uniformly along different paths, and again it is to be emphasized that the main path of sound transmission energy from the vibrating plate with more antinodes changes at each frequency. These results can be also seen at the third eigenfrequency of the vibrating plate, namely at 𝑓= 179 Hz, when the measured minimum and maximum SPLs are 66.7 to 83.4 dB.

6. CONCLUSIONS

It is no longer necessary to manually adapt the radiated sound from the secondary sources in front of the radiated sound from the vibrating plate. It is concluded that the experimental implementation of the proposed TASPM approach in a feedforward controller is feasible to adaptively calculate the optimal control signal for each loudspeaker in front of the radiated interfering sound from the vibrating plate.

As one would expect, the number of loudspeakers required to reduce the total radiated sound power depends on the class of radiating structural modes of the plate. For the (1,1) mode, one loudspeaker is sufficient to minimize the total radiated sound power. For the (1,2) or (2,1) mode, at least two loudspeakers are required to minimize the total radiated sound power. The positions of the secondary sources in front of the vibrating plate have a significant effect on the sound reduction by the TASPM. The optimal positions of the loudspeakers are in front of the antinodes of the plate where the highest displacements occur. 7. REFERENCES

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