Theoretical and experimental analysis of noise source identification and sound intensity in box-shaped structures

Abstract

For underwater vehicles, it is expected to pinpoint the location of the sources and strip out the critical noise level from a mixed noise data for the purpose of noise reduction. This article presents beamforming technique of noise source identification and sound intensity estimation of two box-shaped sources theoretically and experimentally. Considering the number of the microphones and the best acoustic performances, a spiral array is designed. To imitate the practical device, sound-making equipment is packaged in two box-shaped enclosures. The simulative and experimental results demonstrate that the error of a single line spectrum between the real sound level and the estimation is within 2 dB, there are multiple sound sources on the box-shaped sources, and the experimental error of a broadband spectrum is within 2.5 dB. It is proved that the proposed approach for noise identification and quantification has high precision and can be utilized in practical engineering.

Introduction

In practical engineering, when the underwater vehicle is cruising in the ocean, sound from the functioning equipment will reduce the stealth of the underwater vehicles. For the installed equipment, it is essential to quickly identify the strongest sound source from a complex acoustic environment and estimate whether the highest sound level is as low as required. In this study, the box-shaped source is a specific design to imitate the acoustic radiation of the mechanical equipment, such as the sea water pump, the gear box, the electrical machinery, and so forth.

The radiation sound field of the equipment compartment is a mixture of multiple sources. In this instance, it is challenging to distinguish between radiation energy information and the spatial location of the main noise source information. Near-field acoustic holography,^1,2 machine learning, ³ and beamforming technique ⁴ are practical methods to identify the strongest source location. ⁵ Moreover, with the sound intensity mapping ⁶ and scaling technique, ⁷ the radiation level of the strongest source can be determined. Near-field acoustic holography is usually utilized in the low frequency range with considerable microphones when the radiation sources is comparable large. While beamforming is applicable to the middle and high frequency range.

Up to now, studies on the identification and intensity estimation of the noise sources is mainly conducted with classical simple sources, rarely applied to complex box-shaped sources, which is imitate large practical equipment. Scholars have focused their research on identifying and separating the sources of noise in significant equipment,^1,8,9 such as engines,^10,11 cars,^12,13 machine tool axle systems, ¹⁴ airplanes,^15,16 and transportation environments. ¹⁷ For example, Huang ¹⁸ used continuous wavelet transform and partial coherence analysis to study the contribution of different noise sources in the vehicle. Shu et al. ¹⁹ identified complex engine noise sources based on coherent spectral analysis. For mechanical systems generating periodic shocks, Yao ²⁰ proposed a noise source identification method with continuous wavelet transform to separate the overlapped noises among the diesel engine noise sources, combustion noise, and piston slap noise. The listed publications whether identified the source location or separated the sound from the mixed noise signals with data processing. However, there is little literature reporting the combination of source identification and sound intensity estimation of practical equipment for sound reduction. Mechanical equipment emits noise inside the steering cabin and radiates sound waves into the water through the shell. ²¹ The radiated sound waves will affect the performance of the ship’s built-in sonar, and if detected by local sonar, it will threaten its concealment. ²² Therefore, noise control must be carried out for equipment that exceeds the ocean background noise. For practical large equipment in an engine cabin in this study, they radiate broadband sound with strong line spectra in the middle and high frequency range. Therefore, beamforming technique is suitable to be exploited to identify the source location. It helps the marine engineers to make quick judgement of the strongest radiation source and whether the vehicle’s radiation from is over-required. Moreover, sound and vibration control must be targeted and cannot be blindly guessed by humans, otherwise it will threat the invisibility of the vehicle. Therefore, the combination of these to helps to identify the contribution of different noise sources and pinpoint the precise location. Therefore, it is crucial and significant to identify the noise source location and estimate the sound level of the practical equipment to pave a way to guide the control of the noise, as shown in Figure 1.

Figure 1.

Diagram of underwater vehicles for sound and vibration suppression.

This paper presents the noise identification and sound intensity estimation formulae, conducts simulations of two single line spectrum sources, and carries out an experiment of noise evaluation of two box-shaped enclosures with optimized spiral array. The experiment is conducted in a semi anechoic chamber. The significant contribution in the manuscript can be stated that: (1) using beamforming algorithm to quickly and accurately locate multi box containers with wideband, coherent, and equivalent radiation source positions that vary with frequency, forming a noise source identification method suitable for rapid engineering applications, providing reference for real-time control of equipment status on real ships. (2) Based on the spiral array sound intensity scaling technology, a method of setting the equivalent sound source center for frequency division is proposed to accurately estimate the radiation intensity of multiple box mounted sound sources, providing technical support for real-time monitoring of ship noise levels.

Beamforming algorithm, scaling theorem of sound intensity

Theory of beamforming algorithm

In the actual test, when the box-shaped and other noise sources exist at the same time, the sound field response has both a strong narrow band noise source and a lower broadband continuous spectrum noise source. Due to the different distribution positions of each microphone in the array, there are fixed phase and amplitude differences between the received signals of each array element, and the differences are only related to the specific coordinates of the array element. The principle of the beamforming algorithm is to search the possible areas of the sound source of the box-shaped by dividing the grid points, and then according to the position of each grid point, the phase and amplitude of the received signal of each array element in the microphone array are compensated by multiplying a complex weighting coefficient. Finally, the beamforming space spectrum corresponding to the box-shaped sound source can be obtained by superimposing the output.

The underlying assumptions during the beamforming process can be summarized as: (1) in the source identification process, the radiated source is simplified to multiple point sources. Every source point radiates spherical waves in near-field and far field, while in the noise estimation stage, it is assumed the waves propagates in plane wave forms in free-field condition. (2) Broadband signals are broken down into multiple narrowband signals. When the signal time delay is small, the delay in the exponential in the frequency item is neglected. (3) During the derivation of the received signals of acoustic array, there is no mutual coupling between the elements of each array. Therefore, the gain is neglected. (4) In the beamforming process, it is assumed that reverberation is neglected. (5) In the sound estimation, the energy of the whole plane at specific frequency is limited to the width of the main lobe. All the assumptions are bot applied in the practical engineering. Some assumptions are just limited to the testing methods, while some assumptions are reasonable simplification of the practical engineering.

The beamforming process is shown in Figure 2.

Figure 2.

Process diagram of beamforming algorithm.

The beamforming output of a microphone array can be expressed by equation [1] with matric form.

$y\left(t\right)=\frac{1}{M}{\displaystyle \sum }_{m=1}^M{w}_m^{*}{x}_m\left(t\right)=\frac{1}{M}{\mathit{w}}^H\mathit{x}\left(t\right)$

(1)

where $y\left(t\right)$ is the output of a microphone array, and $M$ is the quantity of the microphones. $*$ represents conjugate symbol while $H$ represents the conjugate transpose of a matric. $w_m$ denotes the received signal of each microphones, with $\mathit{w}={\left[w_1\hspace{0.5em}{w}_2...w_M\right]}^T$ , and $x\left(t\right)$ represents the weighting coefficient, with $\mathit{x}\left(t\right)={\left[x_1\left(t\right)\hspace{0.5em}{x}_2\left(t\right)...x_M\left(t\right)\right]}^T$ .

In the actual noise sources identification, power P is usually used to represent the beam output result.

$P=E\left(y{\left(t\right)}^2\right)=\frac{1}{M^2}{\mathit{W}}^H{R}_{xx}\mathit{W}$

(2)

where ${\mathit{R}}_{xx}$ is the covariance matrix of the microphone array data.

In practice, due to the limited data from the sampling process, the covariance matrix ${\mathit{R}}_{xx}$ of the array data can be used instead.

${\widehat{\mathit{R}}}_{xx}=\frac{1}{N}{\displaystyle \sum }_{n=1}^N\left(\mathit{x}\left(n\right)\mathit{x}{\left(n\right)}^H\right)$

(3)

${\widehat{\mathit{R}}}_{xx}$ is the sample mean of the number of $N$ snapshots, $\mathit{x}\left(n\right)$ is the signal vector of the array at n snapshots.

In actual box mounted noise source identification, the sound signal is often a broadband signal that cannot be compensated for phase and amplitude in the time domain. When identifying the noise source, it is necessary to convert the domain data into frequency domain data through filtering or discrete Fourier transform to obtain the frequency domain model of the broadband array signal. At this point, the frequency domain model of the broadband array signal at each frequency point is structurally identical to the time domain model of the narrowband array signal. Therefore, most narrowband sound source recognition can be applied to various frequency points of broadband sound sources.

Principles of noise source quantization, separation, and intensity scaling methods

The corresponding distribution of a sound source’s radiated acoustic energy in space is known as the beamforming spatial spectrum, and it is used in noise source detection. Consequently, the main lobe region contains the majority of the acoustic energy. In order to ensure that the integration result of the scaled output in the main petal region of the spatial spectrum equals the sound power of the sound source radiated to the hemisphere on the side of the microphone array, the beamforming sound intensity scaling method’s basic idea is to scale the beamforming output result by the constructed scaling coefficients. The physical meaning of the scaled beamforming result is the sound intensity. To derive the sound intensity scaling coefficient, the beamforming sound source identification model shown in Figure 3 is constructed.

Figure 3.

Spiral array noise source identification model.

Assuming that the reference sound pressure at the origin position is $P_e$ , The angular frequency, the wavelength and the speed of sound of the incident sound source are respectively $\omega$ , $\lambda$ , and $c$ . $r_m$ is the position vector of microphone numbered $m\left(m=1,...,M\right)$ . $W\left(K\right)$ is defined as the array mode, which is used to express the beam output result of the array at each search point. The conventional beam output result in the focusing direction $k$ is expressed as

$B^2\left(k,\omega \right)={\left|P_e\right|}^2{\left|W\left(K\right)\right|}^2$

(4)

$\left|W\left(K\right)\right|=\frac{1}{M}{\displaystyle \sum }_{m=1}^M{e}^{jK\times r_m}=\frac{1}{M}{\displaystyle \sum }_{m=1}^M{e}^{j\left(k-k_0\right)\times r_m}$

(5)

where $k_0$ is the wave number vector in the incidence direction of the acoustic source, and $k=\frac{\omega }{c}$ is the wave number. $K$ is the difference between the wave number vector in the focusing direction and the wave number vector in the incidence direction of the acoustic source. For a multi-armed logarithmic spiral array, the array pattern can be approximated as

$\left|W\left({\rm K}\right)\right|\approx \left|W\left(K\right)\right|=2\frac{J_1\left(KD/2\right)}{KD/2}$

(6)

where $D$ is the array aperture. $J_1$ Represents the first class of Bessel functions. $K$ is the mode of the projection of $K$ onto the XOY plane.

The projection of the array beam output onto the XOY plane as in equation [12].

$B\left(\mathit{k},\omega \right)\approx P_e\overline{W}\left(k\hspace{0.17em}\mathit{sin}\hspace{0.17em}\theta \right)$

(7)

where $\theta$ is the angle between the focusing direction and the axial direction of the array. The beam output at each grid point in the main lobe region of the spatial spectrum is described by the distance $R$ from the grid point to the z-axis and its azimuthal angle $\varphi$ . When the main lobe region is narrow, $\theta$ satisfies the assumption of a small angle, therefore

$R=L\hspace{0.17em}\mathit{tan}\hspace{0.17em}\theta \approx L\hspace{0.17em}\mathit{sin}\hspace{0.17em}\theta$

(8)

$B\left(R,\varphi \right)\approx P_0\overline{W}\left(\frac{kR}{L}\right),R\le R_1$

(9)

with $R_1=L\hspace{0.17em}\mathit{sin}\hspace{0.17em}{\theta }_1\approx L\frac{K_1}{k}\approx \frac{2{\xi }_{1}L}{Dk}=1.22\frac{\lambda L}{D}$ being the radius of the main lobe region,

The beam output is transformed using a scaling factor $\alpha$ , such that the integration of the transformed beam output in the main lobe region is equal to the hemispherical radiated sound power from the sources to the side of the array.

$\underset{0}{\overset{R_1}{{\displaystyle \int }}}\underset{0}{\overset{2\pi }{{\displaystyle \int }}}\alpha {\left|B\left(R,\varphi \right)\right|}^{2}RdRd\varphi =2\pi \alpha {\left|P_e\right|}^2\underset{0}{\overset{R_1}{{\displaystyle \int }}}{\overline{W}}^2\left(\frac{kR}{L}\right)RdR=2\pi L^2\frac{{\left|P_e\right|}^2}{\rho c}$

(10)

The scaling factor $\alpha$ can be expressed as

$\alpha =\frac{L^2}{\rho c\underset{0}{\overset{R_1}{{\displaystyle \int }}}{\overline{W}}^2\left(\frac{kR}{L}\right)RdR}$

(11)

Assuming $u=\left(\frac{kR}{L}\right)\left(\frac{D}{2}\right)$ , then the integral in the above equation can be expressed as

$\underset{0}{\overset{R_1}{{\displaystyle \int }}}{\overline{W}}^2\left(\frac{kR}{L}\right)RdR={\underset{0}{\overset{{\xi }_1}{{\displaystyle \int }}}\left[2\frac{J_1\left(u\right)}{u}\right]}^2{\left(\frac{2L}{kD}\right)}^{2}udu={\left(\frac{2L}{kD}\right)}^2\underset{0}{\overset{{\xi }_1}{{\displaystyle \int }}}\left[2\frac{J_1\left(u\right)}{u}\right]udu$

(12)

Therefore, the scaling factor $\alpha$

$\alpha \approx \frac{5.88}{\rho c}{\left(\frac{D}{\lambda }\right)}^2$

(13)

It can be seen from equation [13], the scaling factor $\alpha$ is only related to the array aperture and the source frequency, the factor is determined before performing the array test. Once the array beamforming noise source identification is complete, the scaling factor can be applied to convert the beam output result into an equivalent sound intensity measurement to estimate the radiated sound power of the source.

Spiral array design and numerical simulation

Spiral array design

In noise source localization technology, the distribution of array elements receiving sound source signals has a significant impact on the recognition accuracy of noise source algorithms. Due to the different distances between each microphone in the array and the sound source, the time for the sound waves emitted by the noise source to reach each microphone varies, resulting in different phases and amplitudes of the received signals. The noise source identification algorithm compensates for the phase and amplitude of the received signals from each array unit in a targeted manner, offsetting the signal inconsistency caused by differences in microphone positions. The original intention of designing a noise source array is that the position of the array elements will affect the delay of the received signal, which will affect the final recognition effect. Therefore, this chapter conducts research on array design to obtain an array distribution with low redundancy, easy processing, more balanced weighting, high element utilization, and more accurate noise source positioning.

In terms of the array design, there are multiple general arrays, such as cross slot array, the circular array, the spiral array, Fibonacci array ²¹ etc. The evaluation criteria of its performance is that the narrower main lobe width, and the wider dynamic range, the better performance of the designed array to locate source. Due to the limitations of the actual test environment, the aperture of the identification array is limited to 1.5 m, and the maximum number of microphones that can be utilized is 25. The target frequency range is 500 Hz∼5000 Hz. The distance between the array center and the center of the two box sources is limited to 5 m. Consequently, the author compared the performance of linear array, circle array, and spiral array, shown in Figure 4. It is found out that within 500 Hz∼2000 Hz, the linear array and the circular array show no main lobe width, which means that under the limitation of the actual situation, the linear and circular array cannot identify the source location. While the multi-armed logarithmic spiral array shows good performance, and is exploited and optimized to show narrower main lobe width.

Figure 4.

Identification array diagrams: (a) linear array, (b) circular array, (c) spiral array.

The array element distribution strategy is determined by the equal area distribution method with smaller redundancy, more balanced weighting, and higher array element utilization. The spiral angle is set to 60°. The multi-armed logarithmic spiral array is configured in five different ways concerned of the limited 25 microphones, with a combination of spiral arms and array elements on each arm. The available combinations are 3 × 8, 4 × 6, 5 × 5, 6 × 4, and 8 × 3, the acoustic performance of the array being shown in Figure 5.

Figure 5.

Comparison of performance parameters of noise source identification arrays in different combinations: (a) main lobe width, (b) dynamic range.

From Figure 5, it can be seen that the main lobe width of the five configurations are almost the same, while the dynamic ranges are different, which manifest different sidelobes level. Within the frequency range of 750 Hz∼3000 Hz, the dynamic range of the 5*5 spiral array is highest, implying the lowest sidelobe level, which is easier to identify the location of the main noise source. Therefore, in this study, the 5 × 5 spiral array is chosen to capture the location of the noise sources.

To determine the optimal inner diameter parameters, the acoustic performance of the array with different inner diameters are simulated under the condition of other parameters unchanged. Results are shown in Figure 6.

Figure 6.

Comparison of performance parameters of noise source identification arrays with different inner diameters: (a) main lobe width, (b) dynamic range.

Figure 6 demonstrates that the array with inner diameter of 0.1 m has a comparatively narrow main lobe width and the largest dynamic range. The larger the dynamic range, the lower the sidelobe. This makes it easier to differentiate between the main and secondary sidelobes and avoid confusion between a strong source’s sidelobe and a weak source’s main sidelobe. Based on the simulation results displayed in Figure 4, the array with an inner diameter of 0.1 m is determined to design the array.

Figure 7 shows the final design structure of the multi-arm spiral array. The center of the array is the coordinate axis, and the Z-axis is the vertical array direction.

Figure 7.

Schematic diagram of spiral array design results.

Simulation of identification of single line spectrum

The identification principle of the beamforming algorithm is to search for grid points in the possible areas where the strong narrowband sound source of the box shaped sources may exist, and then multiply the received signal of each element by a complex weighting coefficient for phase and amplitude compensation based on the position of each grid point. Finally, the beamforming spatial spectrum corresponding to the narrowband sound source of the box mounted body is obtained by superimposing and outputting. When the signals of each array element are adjusted to the same phase, the beam output of the array at that grid point is maximized, corresponding to the spatial spectral peak. The grid point corresponding to the peak is the position of the equivalent sound source center of the identified strong narrowband noise source in the container.

In this paper, when the box-shaped sources are identified, it can be decomposed into cases of identification of multiple point sources propagation. Therefore, the theoretical analysis of coherent signal and incoherent signal is just the foundation of the box shaped sources.

With the center of the array serving as the origin, the specific simulation schematic is displayed in Figure 8. Since the distance of the practical equipment is 4 m, in the simulation, the distance of the two sound sources is set as 4 m. The vertical distance between the sound sources and the array is 5 m, which is based on the assumption of plane wave radiation and limited by the geometry of the semi-anechoic chamber. The coordinates of the two sound sources in the search plane are sound source A (‒2 m, 0) and sound source B (2 m, 0).

Figure 8.

Schematic diagram of the simulation model of box noise source identification and intensity estimation.

The simulation utilizes a five-armed logarithmic spiral array with 25 array elements, as shown in Figure 7. The multi-armed spiral array is located in the XOY plane, with the center point of the array serving as the origin of the coordinates. The speed of sound is $c=343$ m/s, $SNR=10$ dB. The frequency of the sound source is $f$ , the air density $\rho$ is taken as 1.293 kg/m³.

The specific simulation process is as follows. Firstly, the sound source transmits signals of different frequency bands respectively to simulate the sound pressure signal received by each array element; secondly, the beam output of the acoustic signal received by the array is performed to get the corresponding spatial spectrum of the beamforming, and the radius of the main lobe and the scaling coefficient is determined according to the information of the test distance, the frequency of the sound source, and the parameters of the array; then, the scaling coefficients are used to convert the results of beam output at the various search points into the Then, the scaling factor is used to convert the beam output at each search point into an equivalent sound intensity quantity, and numerical integration is carried out in the main lobe region to obtain the estimated radiated sound power of the sound source on the array side. Finally, the theoretical value of the sound power is obtained as a reference value according to the parameters of the vibration speed, radius, and medium density sound speed of the point source in combination with the theoretical calculation formula.

After conducting pre-relevant acoustic tests, it was found that, the main sound sources have specific line spectra, which determines the total sound level. The frequencies corresponding to the line spectra are 600 Hz, 1100 Hz, 3500 Hz, and 4700 Hz. Therefore, multiple simulation cases are conducted to test the capability of the designed spiral array. Since the broadband noise identification is composed of a lot of single line spectra, the simulation cases are set with two coherent/incoherent sources premised on the practical engineering.

Simulation case 1

It is supposed that both source A and source B radiate a line spectrum at 800 Hz with different intensities, as plotted in Figure 9.

Figure 9.

Spectral characteristics of radiation from dual sources.

When two sound sources occur simultaneously, the beamforming algorithm processes the data at the 800 Hz analysis frequency with the presented approach and deconvolution algorithm. ²² The spatial spectrum of the noise source identification is shown in Figure 10.

Figure 10.

Spatial spectrum of noise source identification for aperture 1.5 m analyzing frequency 800 Hz with different methods: (a) beamforming algorithm, (b) deconvolution algorithm.

According to Figure 10, both the beamforming theory and the deconvolution algorithm can differentiate the source location accurately between a strong and a weak source, even if the line spectra is coherent. However, it should be noticed that the consumption time of beamforming is just 10 s, while the deconvolution algorithm needs nearly 600 s. Therefore, the beamforming algorithm has a high efficiency for noise identification, especially for broadband cases. When the vehicle is cruising underwater, it is a must to identify the strongest source and diagnosis whether the sound intensity is over the requirement. Therefore, the beamforming algorithm is expected in practical engineering. The sound source A is positioned at (‒2 m, 0.1 m), and the sound source B is located at (2.1 m, 0). After accurately identifying the noise source, the sound power is converted using the spatial spectrum.

Sound power estimation of the two sources at different frequency points are obtained by applying the formula of the radiated sound power of the point source. From Figure 11, it is seen that the estimated values of the two sources are 120.2 dB and 125.3 dB, while the theoretical values are 118.6 dB and 124.6 dB. The estimation is higher than the theoretical values, which can be attributed to that the two main lobe regions are superimposed on the other source of the side of the valve, causing the integration results to be larger. The estimated results of the weak source will be larger, while the influence of the strong source will be smaller. Thus, the sound power can be accurately estimated.

Figure 11.

Estimated power spectrum of radiated noise from dual sources.

Simulation case 2

Line spectrum of 600 Hz radiated by source A and Line spectrum of 1100 Hz radiated by source B are shown in Figure 12.

Figure 12.

Spectral characteristics of radiation from dual sources.

The spatial spectrum of the noise source identification obtained by using the beamforming algorithm for data processing at the analyzed frequencies of 600 Hz and 1100 Hz, respectively, is shown in Figure 13.

Figure 13.

Analysis of the spatial spectrum of noise source identification at frequencies of 600 Hz and 1100 Hz: (a) aperture 1.5 m analyzing frequency 600 Hz, (b) aperture 1.5 m analyzing frequency 1100 Hz.

According to Figure 13, when the line spectra of sources do not overlap, the algorithm can separate the two sources vividly. Based on the accurate identification of the noise source, the spatial spectrum is used to calculate the sound power, as shown in Figure 14.

Figure 14.

Estimated power spectrum of radiated noise from two sources.

From Figure 14, it can be seen that the presented theory can estimate the sound power of two sound sources at different frequencies using the spatial spectrum when their line spectra do not overlap. From Figure 10, the radiated sound power of a point source can be theoretically calculated with values of A and B are 118.6 dB and 124.6 dB, respectively. Comparing the theoretically calculated and estimated sound power, the differences are 0.6 dB and 0.1 dB. Therefore, it can be concluded that presented theory can identify the incoherent sources’ location and estimate the sound power correctly.

In summary, through two simulation cases, it can be seen that the identification and sound intensity estimation approach proposed in this article can accurately obtain the positions of two sound sources. These processes can be realized under the condition of the coherent/incoherent sources. The presented theory can also accurately obtain the sources’ sound power levels. The simulation lays a reliable foundation for experimental verification.

Experiment of noise source separation, and intensity scaling methods

Experiment set-up

The actual environment of the semi-anechoic chamber for this test is 12 m long, 9 m wide, and 7 m high. The geometry of the box-shaped source is length 1.8 m, width 1.4 m and height 2.4 m. The material of the box-shaped source is shown in Table 1. Inside the box, there stands a large voice box, which generates sound within 500 Hz∼10,000 Hz. The sound power level can be modulated according to difference situation requirements. In practical engine cabin, inside the cabin wall, there exists sound absorbing material. Therefore, this experiment conducted in a semi-anechoic chamber is just an analogy of the practical engine cabin. The spiral array is designed in Section 3 shown in Figure 7.

Table 1.

Material parameters for the box-shaped sources.

Name of the material	Density (kg/m3)	Young’s modulus (Pa)	Structural damping	Poisson ratio
Steel	7850	2.1 × 1011	0.005	0.3
Damping layer	1200	3.1 × 1011	0.2	0.49
Foam	200	1 × 107	0.1	0.45

During the test, sensors named AWA144223 with the sensitivity of 54.32 mv/Pa are used. The noise analyzer, Model SCM2E09, is connected to the sensor and the PC for data acquisition and storage. The ST-1509 Bluetooth speakers are placed inside the box to play sound files of different cases for testing purposes. Figure 15 shows the experimental schematic. The target frequency range is 500 Hz∼10,000 Hz. To better capture the signal, the sampling frequency is 52,000 Hz. To simulate the different cases of the actual power equipment, the test cases are constructed according to the following combinations of sound sources, and the dual sources at the same time. The cases are shown in Table 2. Case 1 and 2 is simple cases with single line spectrum. Case 3 is a complex case, and the complex case is a dual-source emitting a broadband noise signal in a broadband frequency regime.

Figure 15.

Schematic diagram of the experimental geometric relationships.

Table 2.

Simple cases line spectrum.

	Line spectrum frequency from source A	Line spectrum frequency from source B
Case 1	1051 Hz	1051 Hz
Case 2	1051 Hz	2629 Hz
Case 3	Broadband spectra

The data processing flow is illustrated in Figure 16, with the array center designated as the origin and the horizontal direction as the X-axis, the grid points are divided in the area where the box and another interfering sound source are located. Using the beamforming algorithm, the amplitude and phase compensation of the noise signal data received by each array element is displayed at each grid to perform the surface acoustic imaging of the noise source of the box-shaped enclosures. The beam output at each search point is converted into the equivalent sound intensity using the scaling factor. The numerical integration is conducted in the main lobe region to get the estimated radiated sound power.

Figure 16.

Schematic diagram of noise source identification process.

Single line spectrum cases

The actual engineering source spectrum is broadband spectra with multiple strong line spectra. Owing to the similarity among the cases, only three cases are displayed to validate applicability of the presented theory in practical engineering. Case 1 is imitating the coherent sources, while Case 2 is the incoherent sources. Case 3 is imitating the practical broadband noise sources. Figures 17 and 18 show the dual-source spatial spectrum from a mixture noise environment for Case 1 and Case 2, respectively.

Figure 17.

Spatial spectrum of source identification for Case 1 with coherent frequencies: (a) source A, (b) source B.

Figure 18.

Spatial spectrum of source identification for Case 2 with incoherent frequencies: (a) source A, (b) source B.

As can be seen in Figures 17 and 18, regardless of whether the analyzed frequency is coherent or not, the two sound sources A and B of the box-shaped can be accurately identified, which effectively solves the problem of coherent/incoherent dual-source location identification. Using the spiral array sound intensity scaling model, based on spatial energy spectrum, the two sound sources beam outputs are integrated into the main lobe region, and the two sound power of the box-shaped are estimated. Table 3 shows the sound power comparison between the estimation and tested individually of the two sound sources of the box-shaped.

Table 3.

Comparison of the scaling results of sound intensity for dual sources of box-shaped under simple cases (dB).

	Case 1	Case 2
Total sound power level of Source A tested individually	73.7	73.7
Estimation of sound power of Source A from the spatial spectrum	74.2	74.8
Total sound power level of Source B tested individually	70.9	68.5
Estimation of sound power of Source B from the spatial spectrum	71.9	69.1

It can be seen from Table 3, the estimated radiated sound power is comparative with that tested individually, with a maximum difference of 2.5 dB. It should be noted that the sources are enclosed by the boxes, strong linear spectra of the sound sources in the sound emission leads to an increase in the cavity modes and the modes of the box itself, and the form of the radiation is more complex, which makes it more difficult for the identification and the quantification of the noise sources. However, the test shows that the presented theory can realize the identification and separation of the box-shaped source successfully under simple cases.

Practical engineering cases

In this study, a practical engineering spectrum of a pump is presented in the box-shaped enclosures corresponding Case 3, shown in Figure 19. The analysis frequency range is set to 10 Hz to 10,000 Hz with the frequency resolution 10 Hz. The conducted results at 4731 Hz of the box-shaped sources are shown in Figure 20.

Figure 19.

The spectra of Case 3: (a) source A, (b) source B, (c) source A + B.

Figure 20.

Spatial spectrum of source identification for Case 3 at 4731 Hz: (a) source A, (b) source B.

Beamforming is performed at all frequency points within the analyzed frequency band, and then sound power estimation is performed based on the beam output results of the sound source at each frequency point, and the estimated radiation intensity of several strong line spectra, which are the most prominent for the complex sound source, is obtained as shown in Table 4. The results of sound power estimation at each frequency point over the entire frequency band range are plotted as shown in Figure 20.

Table 4.

Comparison of the estimated and tested sound power of the strong line spectra of the two box-shaped sources for Case 3 (dB).

	638 Hz	1489 Hz	2629 Hz	4731 Hz	8085 Hz
Total sound power level of Source A tested individually	72.4	82.7	79.4	79.8	73.5
Estimation of sound power of Source A from the spatial spectrum	72.7	83.5	79.8	80.3	75.6
Total sound power level of Source B tested individually	86.4	84.8	87.7	80.1	73.8
Estimation of sound power of Source B from the spatial spectrum	86.7	85.6	87.9	81.4	74.4

As can be seen from Figure 21 and Table 4, the difference between the estimation of the strong line spectra from the combined signal and the ground truth of the tested box-shaped sources alone is no more than 2.5 dB, in the meanwhile, the estimated sound power curves of single sources from the combined signal are consistent with the ground truth of the tested box-shaped sources alone. It is also found that the perturbation of estimated sound power increased with the increase of frequency. Since the two source files are the same, the two main lobe regions are superimposed on the other source of the side of the value, causing the integration results to be larger. The estimated results of the weak source will be larger, while the influence of the strong source will be smaller. Moreover, with frequency increases, the secondary lobe will increase, which will disturb the engineer to determine the main highlight point. The estimated sound power is determined by using a fixed highlight. Therefore, as frequency increases, the highlight will drift, which will result in deviation from the ground value. Both the consistencies prove that the presented approach is capable to separate the broadband noise sources and quantify the noise sources. In the current test results, the error is up to 2.5 dB for both simple and complex cases, effectively solving the problem of quantitatively separating the box-shaped sources under multi-source cases with different operating cases.

Figure 21.

Estimated sound power of two sources of the two box-shaped sources for Case 3: (a) Source A tested alone, (b) source B tested alone, (c) source A estimated from the combined signal, (d) source B estimated from the combined signal.

Discussion

After analyzing the source identification and sound estimation of the box-shaped structures, it is essential to provide explanation of the objectively existing errors between the experiment and the practical engineering. There are multiple potential error sources discussed as follows.

(1) The background noise. It should be noted that in real cabin, even though there exist sound absorbing material inside the cabin wall, its absorbing coefficient cannot be identical as that in the semi-anechoic chamber, leading to a smaller signal to noise ratio and a higher dynamic range. Therefore, the sidelobe will be higher, which misguides to wrong source location.

(2) The multipath reflections. Since the ground in the semi-anechoic chamber is sound hard boundary, it will reflect the sound waves. Therefore, the box-shaped sources are installed away from the ground to eliminate the disturbance.

(3) In practical mechanical cabin, there may exist other equipment to reflect the sound waves. It is better to install half open covers to the microphones to prohibit the reflected waves.

Actually, the beamforming process and noise estimation can be used to identify stereoscopic source location for larger and more complex underwater vehicles by combining multiple spiral arrays to a three-dimensional measurement array. The identifying and estimation time at a single frequency is about 10 s. Such a time is not sufficiently short for swift dynamic noise sources. It’s better to optimize the identification algorithm, along with better processors in the computer if real-time implementation is anticipated.

Conclusion

In this paper, based on the beamforming technique, the site identification of the box-shaped enclosures noise source and the sound intensity scaling method is used to estimate the intensity of the box-shaped enclosures noise source, which effectively solves the problem of quantitative separation of the box-shaped enclosures noise source under the complex test cases. Simulation and experimental results show that both single-linear spectral sound sources and broadband sound sources, the method can not only accurately acoustically image the box-shaped enclosures noise source, but also realize the quantitative separation of the radiated sound power of the box-shaped enclosures noise source, and the maximum result of the intensity estimation of the noise source is not more than 2.5 dB. The test method has the following advantages: (1) Simplify the measurement process, high measurement efficiency, and reliable results. (2) In the presence of multiple sources, whether coherent, incoherent dual sources, or narrowband and broadband sources, the sources can be accurately identified and accurately separated quantitatively in terms of acoustic power, to study the spectral characteristics of the radiation of the sources. The research in this paper provides an effective solution for broadband testing of box-shaped noise sources.