High precision localisation of the sources of noise of a bent-axis motor

Abstract

The hydraulic motor will generate noise in operation and it has fluid-structure coupling and compact internal structure. The sound intensity measurement is difficult to distinguish the noise sources of motor. In this paper, a high-precision localization method for noise sources of bent-axis motor is studied to improve the resolution of the sound intensity images to locate noise sources accurately. Firstly, the structural characteristics of the motor are analyzed, and the sound intensity images are obtained through the experiment. Then the high-resolution optimization method based on compressed sensing is designed according to the characteristics of the sound intensity field. Finally, the sound intensity images with high resolution are constructed. The high-resolution images can distinguish the position of the prominent points, and the error is less than 0.4 dB. The identification distance of noise sources can reach 30 mm, and it breaks through the highest resolution of the traditional sound intensity test.

Keywords

Bent-axis motor Noise source location Sound intensity image Compressed sensing High-resolution optimization

Introduction

The bent-axis piston motor has the characteristics of high pressure, heavy load, and large torque, which is an actuator in the hydraulic transmission system. It is widely used in the field of construction machinery. However, the noise of the motor increases with the development of performance, making the vibration and noise problem of the hydraulic system more and more serious. The research on vibration and noise reduction is essential to developing the hydraulic motor.¹

Locating its excitation source is the premise of vibration and noise reduction. Currently, the signals used for diagnosing excitation sources of the rotating machinery include vibration signal, thermal imaging signal .and acoustic signal.² Tang et al.³ and Li et al.⁴ studied the noise of piston pumps and diagnosed the fault of the loose slipper and swashplate vibration as the primary noise sources, respectively. However, the above research is mainly based on the targeted identification of the determined excitation sources. In practical engineering, it is challenging to locate the primary noise sources in a bent-axis motor because the vibration sources of the bent-axis motor are more diversified. The distance of each part is close, and the vibration frequencies of components are the same or have multiplicative relationships.^5,6,7

Noise source location based on the acoustic array is a common and effective method.⁸ Zeng⁹ proposed the microphone array based on the SRC-phat algorithm, and Wang¹⁰ put forward the Michelson fiber interferometer array. Both of them have high practicability in the noise source location. Still, the microphone is expensive, and they are ineffective in distinguishing the noise source of the motor accurately in the hydraulic system with complex and changeable noise. As a vector, the sound intensity signal can effectively isolate external noise interference.¹¹ Thus the noise distribution of the motor can be obtained by the sound intensity test to locate the noise sources. However, there is a near-field error¹² in the sound intensity test, which will be smaller when the double microphones are closer to the sound source. And the Rayleigh criterion¹³ holds that the spatial resolution of sound field imaging is related to the distance from the microphone to the sound source surface. So the conventional sound intensity test does not obtain high-resolution sound intensity images to accurately distinguish the dense and complex excitation sources inside the motor. Therefore, high-resolution processing of sound intensity images of bent-axis motors should be proposed to locate the noise sources.

Candés and Donoho et al.^{14,15,16,17,18} proposed the compressed sensing theory, which pointed out that if the signal is compressible in a fixed domain, it can break through the limitation of the Nyquist sampling theorem and construct the high-resolution images by collecting a small number of samples. Wang et al.¹⁹ used compressed sensing to achieve super-resolution reconstruction of a single image. Ming et al.²⁰ proposed a block Bayesian compressive sensing method based on the equivalent source method to improve the performance of sound source reconstruction with low signal-to-noise ratios. Ning et al.²¹ applied sparse sampling to the microphone array acoustic imaging to realize the noise location of the air compressor. Ding et al.²² proposed a spatially extended sound source reconstruction method based on sparse sampling and achieved high-resolution reconstruction of the sound field.

The above compressive sensing technologies for reconstructing the image and the sound field provide a theoretical basis for improving the resolution of positioning the motor noise source. The sound intensity image of the motor consists of noise signals from each measuring point. It is sparse that fits the premise of compressed sensing. Although the current compressed sensing has a significant effect on constructing high-resolution images, these algorithms may not apply to the sound intensity image of the motor, owing to the composition of the sound intensity image is different from the image obtained by the camera or other instruments. So it is necessary to design the observation matrix, the sparse expression, and the optimization algorithm that matches the characteristics of the sound intensity image.²³

In this study, a high-resolution optimization method of the sound intensity image based on compressed sensing is proposed to locate the noise source of the bent-axis motor. A bent-axis motor is taken as the research object, and the structure feature of the motor is analyzed, and the noise images of the motor are obtained through the sound intensity test. The compressed sensing framework is designed based on the sound intensity test and the motor characteristics. Then the resolution of test images is improved by the designed compressed sensing framework. The result shows that the method breaks through the highest resolution of the conventional sound intensity images and can distinguish the noise source of the bent-axis motor.

Structural feature analysis of bent-axis motor

Figure 1 shows the structure of a bent-axis motor. The structure is compact, and the components are coupled. The working process is as follows: the high-pressure oil output by the hydraulic pump flows into the plunger cavity through the oil inlet and port plate, and the force generated by the high-pressure oil pushes the plunger to rotate the spindle. The collision between the plunger and the cylinder block also causes the cylinder block to rotate. And the mechanical noise, such as spindle eccentricity and bearing vibration, is located at the front end of the shell. The fluid noise caused by flow pulsation and pressure impact is concentrated at the port plate.²⁴ The distances between the excitation sources inside the motor are close. Table 1 shows the distance between the adjacent components.

Figure 1.

Structure of bent-axis motor.

Table 1.

Distance between the adjacent components.

Components	Distance
Bearing and plunger ball	30 mm
Bearing and cylinder block	62 mm
Port plate and the bottom of the plunger cavity	25 mm
Inlet and the outlet of the port plate	38 mm

Table 2 shows the frequency of vibration parts. The frequencies of the vibration parts are the same or have a multiplicative relationship due to the transmission of the inner components associated with the spindle rotating.²⁵ Thus there are many uncorrelated signals in the same frequency under the traditional spectrum analysis method. It is difficult to identify each vibration source accurately.

Table 2.

Theoretical excitation source and vibration frequency.

Vibration source	Vibration frequency
plunger vibration	$f_{1} = \frac{2 z n}{60}$
spherical joint vibration	$f_{2} = \frac{z n}{60}$
distribution shock	$f_{3} = \frac{z n}{60}$
eccentric and unbalanced spindle	$f_{4} = \frac{n}{60}$
bearing vibration	$f_{5} = \frac{n N}{60}$
flow and pressure pulsation	$f_{6} = \frac{2 z n}{60}$

Sound intensity image of bent-axis motor

Preparation of sound intensity test

The sound intensity test is carried out on the bench, as shown in Figure 2, to obtain the noise distribution of the motor. The bench consists of two parts, part (a) is the hydraulic control room, which is responsible for controlling the condition of the hydraulic motor and providing power through the transmission shaft and part (b) is a semi-anechoic room where the sound intensity test is carried out in this room. The acquisition module comes from B&K Company and includes a data collector, a sound intensity probe sensor, and supporting test software.

Figure 2.

Test bench of motor sound intensity.

In this study, the dual-microphone P-P method is used for the sound intensity test, and the dual-microphone probes are opposed to microphones. The line connecting two microphones, A and B, is the direction of sound wave travel, and two sound pressures, P_A(t) and P_B(t), can be measured, while the distance between two microphones is set as d, and the following equation can be obtained:

v (t) = - \frac{1}{ρ_{0} d} \int [p_{B} (t) - p_{A} (t)] d t

(1)

Where v(t) is the acoustic velocity, ρ₀ is the air density.

The sound pressure in the middle of the two microphones is the average value of P_A(t) and P_B(t), and the instantaneous sound intensity in this direction can be obtained as follows:

I_{x} (t) = \frac{1}{2 ρ_{0} d} [p_{A} (t) + p_{B} (t)] \int [p_{A} (t) - p_{B} (t)] d t

(2)

Based on the above analysis, a 12 mm pitch microphone suitable for the frequency range of 125 ∼ 6000 Hz is selected to meet the frequency range of motor noise and improve the test accuracy. Meanwhile, the distance between the microphone and the sound source surface is set as r. The relationship between r and the d should be satisfied $r > 2 d$ ²⁶ to reduce the influence of near-field measurement error.

The discrete method is used to measure the sound intensity of the motor surface in the test. Since the spacing of the U-shaped bends of the sound intensity sensor is 30 mm, the single cell size is set as 30 mm × 30 mm, and the test grid with a length of 210 mm × 300 mm is arranged at a distance of 30 mm in the top view direction and the front view direction according to the actual size of the motor. There are 70 measuring points in both directions. Figure 3 shows the grid distribution in the front and upward directions of the motor, and it completely envelops the outer surface of the motor. It can obtain the complete sound intensity distribution on the outer surface of the motor by measuring on this grid to achieve the highest resolution of the sound intensity test.

Figure 3.

Grid distribution for sound intensity test.

Analysis of test results

The bent-axis motor is tested for sound intensity under two conditions. One is the rotating speed of 500 rpm and the inlet pressure of 5 MPa. The other is the rotating speed of 1000 rpm and the inlet pressure of 10 MPa. As shown in Figures 4 and 5, the sound intensity images of the front direction and the top direction are obtained.

Figure 4.

Sound intensity images under 5 MPa.

Figure 5.

Sound intensity images under 10 MPa.

The sound intensity of the motor is generally low under the condition of 500 rpm and 5 MPa. Figure 4(a) shows that the noise radiation is distributed in the whole part of the motor, and Figure 4(b) shows that the noise generated at the motor spindle and bearing is larger than that of other parts. Therefore, the bearing vibration contributes to the main noise under low pressure and speed. Under the condition of 1000 rpm and 10 MPa, the sound intensity increases obviously. It can see in Figure 5(a) that there are apparent noises at the rotational part and the rear end cover, and Figure 5(b) shows the noise near the port plate is higher than the noise at other positions. Therefore, the prominent noise of the motor is caused by the impact of the rotating body and the flow pulsation at the port plate under high pressure and high speed.

The above results show that the sound intensity image can reflect the noise distribution of the motor. Still, the most prominent points lie in a large area with no apparent distinction from the surroundings. The sound intensity test can distinguish the minimum distance between adjacent sound sources is $x \approx 1.6 r$ , about 48 mm according to the Rayleigh criterion, so it is hard to locate the specific position of the sound sources. Therefore, it needs to optimize the test images and break through the highest resolution of conventional sound intensity tests to locate their noise sources accurately.

High-resolution optimization of sound intensity images

The compressed sensing (CS) is combined with the sound intensity test of the motor to enhance the sound intensity image to a high-resolution image.

Y = Φ X

(3)

where Y is the images of sound intensity test,

Φ

is the observation matrix, X is the high-resolution images.

Due to the sound intensity image signal is not sparse that could not applicable to CS, the sparse matrix $Ψ$ is used to transform the X into sparse.

X = Ψ α

(4)

where

Ψ

is the sparse matrix,

α

is the sparse representation. Thus the CS can be defined as

Y = Φ Ψ α = A α

(5)

where

A = Ψ Φ

is the sensing matrix. The sparse representation

α

and the high-resolution X are obtained using the known sound intensity image y, observation matrix Φ, and sparse matrix Ψ.

The DCT redundant dictionary is constructed as the sparse matrix of the sound intensity image. The DCT transform can realize the conversion of the motor noise signal from the time domain to the frequency domain, which makes the motor noise signal sparse. The redundant dictionary is extended on the orthogonal basis dictionary so that the number of dictionary atoms is larger than the dimension of the image signal. Thus the DCT redundant dictionary is easier to express the signal of sound intensity image sparsely. Based on the sparse matrix Ψ of the DCT redundancy dictionary, the sparse coefficient α is obtained through the formula $α = ψ^{T} x$ and the following formula is satisfied.

{‖ α ‖}_{P} = {(\sum_{i} {| α_{i} |}^{P})}^{1 / P} \leq R \begin{array}{c} 0 < P < 2, R > 0 \end{array}

(6)

As shown in Figure 6, the sound intensity image is obtained by point-by-point measurements on the grid using a sound intensity probe. So the observation matrix with a 0/1 distribution is proposed as

Φ = [\begin{array}{l} Φ_{1} \\ Φ_{2} \\ ⃜ \\ Φ_{M - 1} \\ Φ_{M} \end{array}] = [\begin{array}{l} 0 0 0 0 0 0 1 0 0 0 ⃜ 0 0 0 0 0 0 0 0 0 \\ 0 0 1 0 0 0 0 0 0 0 ⃜ 0 0 0 0 0 0 0 0 0 \\ ⃜ \\ 0 0 0 0 0 0 0 0 1 0 ⃜ 0 0 0 0 0 0 0 0 0 \\ 0 0 0 0 0 0 0 0 0 0 ⃜ 0 0 0 0 1 0 0 0 0 \end{array}]

(7)

Figure 6.

Sound intensity imaging.

It corresponds to that the sound intensity probe can only measure a value at each measuring point, representing a single pixel in the sound intensity image. Firstly, the number of measuring points M is defined according to the number of original signal data N and the relationship of $K < M ≪ N$ . Then an M×1 random number vector θ is defined by selecting M values from 1∼N randomly, and an M × N zero matrix $Φ$ is obtained. Finally, find each row’s positions corresponding to the value in θ and set them to 1.

Also, the correlation between the observation matrix and the sparse matrix is usually used to judge the performance of the observation matrix:

u (Φ, Ψ) = \sqrt{N} \max_{\begin{array}{l} 1 \leq k \leq M \\ 1 \leq j \leq N \end{array}} | 〈 Φ_{k}, Ψ_{j} 〉 |

(8)

where

u (Φ, Ψ)

is the correlation. It can obtain a higher resolution signal through the observation matrix and the sparse matrix when the u is smaller. Figure 7 shows the program flow for calculating the irrelevant between the observation matrix and the sparse matrix. The small irrelevant value is achieved by cyclical calculation, and then the observation matrix and the sparse matrix are obtained, which can improve the signal resolution.

Figure 7.

Judgment standard of observation matrix.

The sensing matrix A is obtained based on the above observation matrix and sparse matrix. The sensing matrix A, the sound intensity image Y and the sparsity K are used as inputs to the Regularized Orthogonal Matching Pursuit (ROMP) for the regularization calculation to improve calculation speed and accuracy. And calculate the least square solution $y = A_{t} α_{t}$ :

{\bar{α}}_{t} = \arg \min_{α_{t}} ‖ y - A_{t} α_{t} ‖ = {(A_{t}^{T} A_{t})}^{- 1} A_{t}^{T} y

(9)

to solve the sparse coefficient

{\bar{α}}_{t}

. Then update the error r_t:

r_{t} = y - A_{t} {\bar{α}}_{t} = y - A_{t} {(A_{t}^{T} A_{t})}^{- 1} A_{t}^{T} y

(10)

The high-resolution sound intensity image is obtained by multiplying the sparse coefficient $\bar{α}$ with the sparse matrix Ψ. It can solve the problem that the accuracy of conventional sound intensity tests is difficult to meet the resolution requirements at the hydraulic component level.

Results and discussion

Figure 8 shows the comparison between test images and high-resolution images in the front and top viewing directions under the condition of 5 MPa. It can be seen in Figure 8(a) that there are three prominent points in the high-resolution image, while the test image only shows a large area of sound intensity with no obvious distinction from the surrounding area, and it is unable to locate the specific position of each prominent point in the image. In Figure 8(b), the noise of the bearing in high-resolution image is more concentrated than in the test image. There is a yellow circular area with a diameter of about 10 mm in the middle area about 50 mm away from the bearing in high-resolution image but not in the test image, which is judged to be plunger impact noise based on the distance of the components mentioned above. The results show that the noise of the motor includes the bearing vibration and the collision between the plunger and the cylinder block, which conforms to the fact that under the condition of low speed and pressure, the mechanical noise generated by the vibration of the rotating parts is the prominent noise while the fluid noise is slight. It also shows that the high-resolution images obtained by this optimization algorithm describe the area of prominent points more accurately than test images.

Figure 8.

Comparison between test image and high-resolution image under 10 MPa.

Figure 9 shows the comparison between test images and high-resolution images in the front and top viewing directions under the condition of 10 MPa. In Figure 9(a), the sound field details in high-resolution image are more than those in the test image. There are four prominent points at the position of the rotating part, and the distance between two adjacent parts is 30 mm at the bearing. However, there is a large sound intensity area in the test image, and the noise intensity points at the bearing are not reflected. The positions of the strong points are roughly the same as the noise under the condition of 5 MPa, which verifies the accuracy of these prominent points. The strong point caused by the spherical joint vibration is located 30 mm below the bearing and the flow noise generated by the inlet and outlet of the port plate is more concentrated. So the high-resolution image can effectively distinguish strong points of noise sources. In Figure 9(b), a small prominent point about 30 mm above the rear end cover is effectively distinguished, which is not reflected in the test image, indicating a noise caused by the collision between the plunger and the cylinder block. The noise of the bent-axis motor mainly comes from the collision of rotating parts and the fluid noise of the port plate under high speed and pressure according to the above analysis.

Figure 9.

Sound intensity images with high resolution under 10 MPa.

Figure 10 and Figure 11 show the error distribution between the high-resolution images and the test images under the conditions of 5 MPa and 10 MPa. The error distribution is relatively flat and has a good fit; the max error is 0.38 dB. It explains that the high-resolution images have improved the resolution while retaining the complete information of the sound intensity images.

Figure 10.

Error distribution between the high-resolution images and the test images under 5 MPa.

Figure 11.

Error distribution between the high-resolution images and the test images under 10 MPa.

The above results show that the optimized images based on the compressed sensing framework of this study have higher resolution than the images obtained by the sound intensity test. The high-resolution images can effectively distinguish the noise sources close to each other. Moreover, the method can be applied to the conditions of 5 MPa and 10 MPa with good results, which means that the process applies to the sound intensity test under different working conditions.

This study combines the compressed sensing with the sound intensity test to form a high-resolution optimization method for the sound intensity image of the bent-axis motor, which improves the resolution of the sound intensity images, solves the problem that the motor noise sources are dense and the sound intensity test cannot accurately identify the noise source location, and provides the basis for the motor fault location judgment.

Conclusion

(1) The noise of bent-axis motor mainly includes mechanical noise such as bearing collision, spherical joint vibration, plunger collision and fluid noise such as flow pulsation and pressure impact.

(2) The compressed sensing framework includes a DCT redundant dictionary, a random observation matrix with a 0/1 distribution and the ROMP optimization algorithm is designed according to the characteristics of sound intensity test of the bent-axis motor.

(3) The max error of the high-resolution images is 0.38 dB and the identification accuracy of adjacent noise sources can reach 30 mm.

(4) The optimized images reflect more details of the sound field than the test images, and achieve high-precision positioning of motor noise sources.

(5) Under low speed and pressure, the noise of the bent-axis motor mainly includes bearing vibration and spherical joint vibration, while the noise including bearing noise, plunger collisions, and pressure impact under high pressure and speed.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported by National Natural Science Foundation of China (Grant No. 52105053), General Project of Natural Science Foundation of Fujian science and Technology Department (Grant No. 2020J01452), Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (Grant No. GZKF-202114), Sub Project of Key R & D Project of Ministry of Science and Technology of the People’s Republic of China: Key Technology of Design and Manufacture of Miniature High Speed on off Valve (Grant No. 2019YFB2005103).

ORCID iD

Chen Xu

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