#### Abstract

In this paper, a novel fuzzy-based analytic hierarchy process (FAHP) is proposed for identifying noise sources of diesel engine and sorting their contributions. The hierarchy tree consists of 5 levels including the component level, the frequency band level, the test point level, the operating condition level, and the overall noise level. The variational mode decomposition (VMD) is employed to decompose the overall noise into several frequency bands. The factor weights between levels are determined by the partial coherence analysis (PCA) to consider the correlation of component vibration and radiated noise in different frequency bands. A fuzzy consistency matrix is then constructed to order the corresponding factors level by level that can avoid the consistence problem of the traditional AHP. Based on the rig test of a diesel engine, the proposed approach is implemented to weight the noise sources identified. It is shown the overall weighting order of the six major noise sources is (in descending order) the oil pan, the left block, the valve cover, the flywheel housing, and the right block.

#### 1. Introduction

NVH problems are increasingly important in nowadays automobile industry, and reduction in engine noise is one of the most concerns. A precise recognition of the noise sources and their contribution is the first step in interior noise control [1–3].

In the noise source identification of the engine, signal processing-based approach is quite popular these days because of its comprehensive performance in accuracy [4, 5], flexibility, and cost-saving [6, 7]. Lots of method have been proposed, such as the cyclic Wiener filtering [8, 9], the time-frequency analysis [10–12], the blind source separation [13, 14], and the coherent spectrum analysis (CSA) [15]. But it is still very difficult for only using one signal processing method to perfectly handle the complex noise problem. For this reason, the analytic hierarchy process (AHP) [16, 17] has been proved suitable for decision-making of complex systems. It has been widely used in social, political, economic, and technological problems with multiple choices. In the noise identification of the diesel engine, a great amount of factors should be considered, such as frequency, engine speed, load, and component. Besides, correlations between factors (e.g., frequency band and component vibration and vibration between component and radiated noise) have influence on the weighting coefficients of the noise source. In this situation, the AHP could be an effective solution in weighting noise sources of a diesel engine. Liang [18] utilized the cepstrum analysis and the AHP in weighting the noise contribution from engine components. The overall noise was decomposed into five frequency bands, and one operating speed was considered. Zhang et al. [19] proposed a combined AHP approach to analyse the contribution of different engine parts to the overall radiated noise in different engine speeds. The decomposition by ensemble empirical mode decomposition (EEMD) was affected by the modal aliasing problem in intrinsic mode functions (IMFs). And the coherent spectrum analysis (CSA) is limited in the case that multiple inputs are not independent from each other.

On the contrary, some limits have been found for the traditional AHP approach [20, 21]: (1) the consistency of the judgment matrix is very costly to achieve, (2) preferences mismatch the objective priority, and (3) the uncertainty of subjective evaluation reduces the accuracy. To overcome the above shortages, the fuzzy-based AHP (FAHP) [22, 23] is developed to take into account the uncertainty in decision-making. A fuzzy scale is used to express the preference or relative importance of the target property instead of adopting an exact value from pairwise comparison. In addition, the fuzzy consistency matrix [24–27] used in the FAHP can be determined by membership functions (MFs), so the consistency of the judgment matrix is met automatically [28, 29]. It can be seen that the FAHP is able to improve the traditional noise source weighting approach as long as reasonable structures and components of the hierarchy tree is built.

In this paper, a novel approach based on the FAHP is proposed to weight the noise sources of the diesel engine. Based on a rig test, the overall noise is decomposed into several frequency bands using the variational mode decomposition (VMD). Then, the correlation between the vibration of engine component and the overall radiated noise is analysed through partial coherence analysis (PCA). A FAHP is constructed to determine the weights of the component to the overall noise. Finally, the identification results are verified by the near-field noise measurement.

#### 2. Methods

##### 2.1. Variational Mode Decomposition

The VMD algorithm [30] is a self-adaptive signal processing method, which can decompose a mixed signal into a number of IMFs. The discrete subsignals have finite orders of mode with specific sparsity properties for being reconstructed back to the original signal [31]. The constrained variational problem is given as follows:where and are the modal components and the corresponding centre frequencies, is the summation of modes, *k* is the number of modes, and *f* is the input signal. To make the problem unconstraint, it is necessary to bring in the quadratic penalty factor *α* and the Lagrange multiplier *λ*. The quadratic penalty factor is used to improve the high reconstruction accuracy in dealing with the Gaussian noise. The Lagrange multiplier can enforce constraints effectively, which can be obtained as follows:

Using the alternate direction method of multipliers for the extended Lagrange expression through alternating optimization of *μ*_{k}, *ω*_{k}, and *λ*, the optimal solution of the equation can be obtained. The VMD method can be implemented by the following steps:

*Step 1. *Initialize , , and , and set *n* = 0.

*Step 2. *Count *n* = *n* + 1.

*Step 3. *For *k* = 1 : 1 : *k*, obtain

*Step 4. *Judge the convergence:where *c* is the convergence judgment condition. The program stops if the convergence is met at Step 4; otherwise, the iteration circulates from Steps 2 to 4.

A simulation signal *S* *=* *S*_{1} *+* *S*_{2} *+* *S*_{3} *+* *S*_{4} is given by the following equation:Figure 1 shows the original components of *S*_{1} to *S*_{4} and the decomposed IMFs by VMD. It can be seen clearly that the simulation signal *S* is accurately decomposed into four IMFs and the IMFs represent the characters of original components exactly.

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##### 2.2. Partial Coherence Analysis

For a multi-input single-output system with independent inputs, CSA is capable to estimate the relationship between input and output efficiently [32] while it is not the case in this paper. For engine noise, acoustic sources are correlated. It is not very appropriate to use CSA because wrong correlation between the radiated noise and the component vibration may be brought in. To avoid this, the PCA is applied because it can remove the coherence of other inputs from the target. So, the correlations between inputs and outputs can be found out properly as shown in Figure 2, and six input channels represent the main acoustic sources of mechanical noise including the oil pan (*X*_{1}), the left block (*X*_{2}), the right block (*X*_{3}), the gear cover (*X*_{4}), the flywheel housing (*X*_{5}), and the valve cover (*X*_{6}). *L*_{1} to *L*_{6} are the conditioned transfer functions of corresponding acoustic source, where *N*(*t*) is the sensor noise, *Y*(*t*) is the IMFs of the measured acoustic signal.

The PCA can be adopted by the following steps [33]:

*Step 1. *Transform signals from time domain into frequency domain, .

*Step 2. *Calculate the self-power spectra and the cross-power spectra of the transformed signals:where *S*_{xx} and *S*_{yy} represent the self-power spectra of the input and the output respectively, *S*_{xy} represents the cross-power spectrum of the input and output, and and are conjugate complex numbers of *X*_{i}(*f*) and *Y*_{i}(*f*).

*Step 3. *Calculate the conditioned transfer functions and the conditioned power spectra:where *L*_{ij} is the conditioned transfer function, *S*_{ij·r!} is the conditioned cross-power spectrum between the input and output without coherent effects from *X*_{1}, *X*_{2}, … , *X*_{r}, and *S*_{jj·r!} is the conditioned self-power spectrum without coherent effects from other signals.

*Step 4. *Calculate the input partial coherent power spectrum:where is the partial coherent function describing the correlation between *X*_{i}(*t*) and *Y*(*t*) in the whole frequency range, *S*_{ii⋅r!} is the conditioned self-power spectrum, and is the input partial coherent power spectrum (PCPS) that shows the relationship between component vibration and radiated noise.

##### 2.3. Fuzzy AHP

The detailed process of the FAHP can be described as follows:

*Step 1. *Establish the fuzzy judgment matrix **A** from pairwise comparisons. Determine the fuzzy judgment matrix **A** = [*a*_{ij}]_{n×n} according to the relative importance of factors in the same layer. The elements in matrix **A** are in the range of 0.1 to 0.9, where 0.5 represents the two factors are equally important.

*Step 2. *Calculate the fuzzy consistency matrix **R** and the reciprocal matrix **M**. Transform **A** = [*a*_{ij}]_{n×n} into **R** = [*r*_{ij}]_{n×n}. The calculation can be done by the following equation:Then, the matrix **R** can be converted into the reciprocal fuzzy consistency matrix **M** through *m*_{ij} = *r*_{ij}/*r*_{ji}.

*Step 3. *Calculate the initial weighting vector *ω*_{0}. The initial weighting vector can be obtained by the least-squares method:

*Step 4. *Calculate the final weighting vector **V**:(1)Initialize *V*_{0} = by *ω*_{0}(2)For *k* = 1, 2, 3, … , *n*,(3)Check the convergence by the following equation:(4)If the convergence is met, output the final weighting vector as follows:Otherwise, jump back to Step 2.

##### 2.4. Flow Chart

The overall framework is given in Figure 3, and the process can be divided into six steps:

*Step 1. *Decompose the noise signal into a series of IMFs in consecutive frequency bands by VMD.

*Step 2. *Analyse the relationship between component vibration and radiated noise through PCA to obtain the weighting order of components in each frequency band.

*Step 3. *Construct the hierarchy tree and establish the fuzzy consistency matrices. Obtain the contributions of noise sources to the decomposed noise in different frequency bands.

*Step 4. *Calculate the contributions of noise in different frequency bands to different test points according to the A-weighted SPL and construct the combined weight matrix of the test point level.

*Step 5. *Obtain the contributions of noise at different test points to different engine speeds and construct the combined weight matrix of the operating speed level.

*Step 6. *Construct the fuzzy consistency matrix of the objective level according to the A-weighted SPL and the usage rate of engine speeds and obtain the overall weights of noise sources.

#### 3. Measurements

The noise and vibration experiment is carried out on a test rig in a semianechoic chamber with less than 18 dB background SPL, as shown in Figure 4. Acoustic signals are collected using five MA231 acoustic sensors (BSWA Technology Co. Ltd, China) 1 m away from the front, the rear, the left, the right, and the top surfaces of engine. The near-field noise (0.01 m away from the measuring surface) as well as the vibration of some highly concerned components is measured including the oil pan, the block, the flywheel housing, and the valve cover. In the vibration measurement, the triaxial accelerometers (PCB Piezotronics, Inc., US) are arranged at different regions for large component and the mean acceleration is employed. In order to reduce uncertainty error during the test, all the microphones and accelerometers are calibrated before the measurement according to the instructions in the test environment, and all the test cases are recorded for three times separately.

The experiment is carried out at four typical engine speeds of 700 r/min (idle), 1400 r/min (maximum torque), 1800 r/min (most commonly used operating condition of construction machinery with 80% load), and 2200 r/min (rated). The combustion settings of the four typical operating conditions are listed in Table 1. The sampling frequency is 10240 Hz, and the intake noise, the exhaust noise, and the fan noise are removed from the measurement.

#### 4. Results and Discussion

In this section, only the 1800 r/min case is used as an example to show the results due to the huge amount of data, and the analysis procedure of other operating conditions is in the same way as the showcase.

##### 4.1. Noise Decomposition

In order to improve signal decomposition accuracy, the trend term and cumulative error are eliminated from the raw data before the formal analysis. The central frequency of IMF is used as the criteria for choosing the best modal number *K*. Overdecomposition occurs when the difference of the central frequencies of two adjacent IMFs is smaller than 10% of the whole frequency range. Four *K* values are employed in this study including 4, 5, 6, and 7. The central frequencies of the corresponding IMFs are listed in Table 2. For *K* = 7, the central frequency difference of IMF4 and IMF5 is 147 Hz < 10% × whole frequency range (2707–240) = 246.7 Hz. As a result, the *K* = 7 case is considered as overdecomposition so that *K* = 6 is employed in the VMD decomposition. The IMFs obtained from the acoustic signal recorded at the left-side test point at 1800 r/min are given in Figure 5.

The lower and upper limits of the frequency band are determined by time-frequency analysis using wavelet transform. As shown in Figure 6, the six frequency bands are 0–500 Hz, 500–800 Hz, 800–1200 Hz, 1200–1800 Hz, 1800–2500 Hz, and 2500–4000 Hz. In lower frequency ranges, e.g., 0–500 Hz and 500–800 Hz, obvious periodicity can be found for the radiated noise. This periodic character corresponds to the ignitions of cylinder. The six consecutive highlight parts along the time correlate to the ignitions of the six cylinders in a working cycle (720°CA in 0.4 s at 1800 r/min). Most engine parts contribute to the low frequency radiated noise because their lower vibration modes fall in these frequency bands. For higher frequency, the energy distribution still shows periodic characters but more details within every single stroke can be found. From this point of view, each IMF has its own physical meaning, and it can improve the reliability of the following PCA.

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##### 4.2. Partial Coherence Analysis

The radiated noise of the engine contains the vibration information of the engine component, so the contributions of various components to the measured noise can be identified according to the correlation between noise and vibration. Many of the accessories are rigidly connected to the engine block, so the vibration between different noise sources is coupled. It is why the PCA is chosen to recognize the correlation of component vibration and radiated noise. The input partial coherent power spectra (PCPS) of the 1800 r/min case are shown in Figure 7. The coherences of the component to the noise are (1) for 0–500 Hz, gear cover > oil pan > right block > left block > flywheel housing > valve cover; (2) for 500–800 Hz, oil pan > valve cover > gear cover > flywheel housing > right block > left block; (3) for 800–1200 Hz, oil pan > left block > gear cover > valve cover > flywheel housing > right block; (4) for 1200–1800 Hz, oil pan > gear cover > valve cover > left block > flywheel housing > right block; (5) for 1800–2500 Hz, gear cover > oil pan > left block > valve cover > right block > flywheel housing; and (6) for 2500–4000 Hz, gear cover > left block > flywheel housing > oil pan > valve cover > right block.

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It can be seen that the input PCPS of the oil pan is the largest between a wide frequency band of 500–1800 Hz. It is because the lower vibration modes of such a thin-walled component are easy to excite due to the excitation of the body skirt. In other frequency bands, the gear cover shows the largest input PCPS below 500 Hz and above 1800 Hz. The main reasons are the resonances of the component in the low-frequency range and the dynamic meshing force between gear teeth at high frequency. So, the above two components are the top 2 contributors in most cases. For the left block, the amplitude of input PCPS in 1800–4000 Hz is considerable because the fuel injection pump and the air compressor radiate great portion of noise in this high-frequency range.

##### 4.3. Weighting Analysis of Noise Sources

###### 4.3.1. Analytic Hierarchy Tree

As analysed above, for a specific component (noise source), its contribution to the noise varies in different aspects, such as the frequency band, the testing point, and the engine speed. On this basis, the hierarchy tree (Figure 8) for weighting the noise sources consists of five levels from top to bottom: (1) objective level, (2) operating speed level, (3) testing point level, (4) frequency band level, and (5) noise sources level.

###### 4.3.2. Fuzzy Consistency Matrix

The fuzzy consistency matrices for the frequency level are constructed according to the condition coherent power spectra and listed in Table 3. Based on the one-third octave SPL, the fuzzy consistency matrices for the test point level and the operating speed level can be calculated in the same way. Table 4 shows the fuzzy consistency matrix for the left testing point (C1). Table 5 presents the fuzzy consistency matrix for the engine speed of 1800 r/min (B3). For the top level, the fuzzy consistency matrix can be obtained based on the usage proportion of the engine speed and the A-weighted SPL, as shown in Table 6.

###### 4.3.3. Weighting Order of Noise Sources

Based on the consistency matrices of adjacent levels, the combined weight matrices can be obtained. Table 7 gives the combined weight coefficients for the left testing point (C1), and Table 8 is the combined weight coefficients for 1800 r/min (B3). The contributions of the noise source to the radiated noise at different engine speeds are calculated and shown in Figure 9.

The contribution of the component to the overall noise changes with the engine speed. But some patterns still can be found. The oil pan is the most significant contributor regardless of engine speed, followed by the gear cover and the left block. At lower engine speeds, idle, and 1400 r/min, the contributions from gear cover and left block are quite close, while the gear cover contributes more at higher engine speeds (1800 r/min and 2200 r/min). The contributions of the right block, the flywheel, and the valve cover are similar and less important in this case.

###### 4.3.4. Result Validation and Discussion

The overall weighting coefficients of the components to the radiated noise (Table 9) are calculated on the basis of the combined weight matrices considering the surface area effect. The surface area effect here relates to the outer surface of the component and acts by means of area factor. A component with the large outer surface is supposed to make greater contribution to the noise radiation comparing with a smaller one. Without considering the surface area effect, the weighting order of components is oil pan > gear cover > left block > valve cover > flywheel housing > right block. When the surface area is taken into account, the weighting order slightly changes into oil pan > left block > gear cover > valve cover > flywheel housing > right block.

The calculated weighting order of the components to the overall noise is verified using the near-field noise in different operating conditions. The measured near-field noise of components and the computed weighting order under 700 r/min are given in Table 10 as an example. It can be seen from the comparison that the calculated weighting order of components are consistent with the near-field noise measurement. This clearly proves the validity of the proposed approach for noise source weighting of the diesel engine.

According to the overall weighting order, the main objective of the noise control of this diesel engine can be determined. But the detailed strategy still depends on the contributions from different aspects: engine speed and frequency band. Table 11 gives the weight coefficients of the engine component regarding the frequency range of 0–500 Hz at the engine speed of 1800 r/min. The calculated weighting orders of noise sources to the noise in different frequency bands at different speeds are given in Figure 10.

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As discussed above, the contribution of components to the overall noise depends on the frequency band and the engine speed. For the radiated noise below 1800 Hz, the oil pan is almost the greatest contribution for all engine speeds considered. The oil pan even dominates over other components for some frequency bands at some speeds, e.g., 500–800 Hz at 700 r/min, 0–1800 Hz at 1400 r/min, and 500–1800 Hz at 1800 r/min. In the case of higher frequency noise, the left block contributes most at 700 r/min and 1400 r/min, and the gear cover takes the first place at higher speeds except for the case of 2500–4000 Hz at 2200 r/min.

The 1800 r/min case is the only exception that the gear cover breaks the dominance of the oil pan in the frequency range below 1800 Hz. At this speed, the weight of the gear cover is a little bit higher than the oil pan. It may be because the low-order vibration modes of the gear cover are excited up due to the rigid connection with the engine block. For the same reason, the flywheel housing also contributes considerably at idle speed (700 r/min).

The left block plays an important role in the frequency range higher than 1800 Hz. Since the right side of the engine block does not show a similar performance as the left side, the combustion excitation can be excluded from the reasons behind. It is noted that fuel injection pump and air compressor are rigidly mounted on the left block, which can radiate high-frequency noise.

Apart from the left block, the gear cover is even more noticeable in the frequency range of 1800 to 4000 Hz at 1800 r/min and 2200 r/min. At high speeds, the dynamic meshing force of gears becomes significant, and it can directly transmit to the gear cover through the transmission shaft. Therefore, a considerable radiated noise from this component can be found in the high-frequency range, e.g., 1800–2500 Hz and 2500–4000 Hz.

The radiated noise from the valve cover mainly relates to the combustion process and the valve movements. At lower speed, it is known the combustion excitation is more significant than the mechanical excitation. Although the noise due to valve movements is low at low speed, the valve cover still has important effect on the overall noise at 700 r/min and 1400 r/min. As the engine speed increases, the contribution of the valve cover decreases first and then rises again. It is because the high-speed movement of the valve parts starts to take the place.

#### 5. Conclusions

An integrated approach of VMD-PCA-FAHP is proposed for noise source weighting of the diesel engine. Based on the noise and vibration measurement, the approach is implemented on the weighting analysis, and the result is validated using the near-field noise. Conclusions can be drawn as follows:(1)The integrated VMD-PCA-FAHP approach is applicable in noise source identification and weighting analysis of the diesel engine. It can provide reliable weighting results in terms of contribution of engine component to noise.(2)The VMD decomposes the noise signal into several meaningful IMFs, and the frequency bands can be determined by the time-frequency analysis. The correlations between component vibration and noise can be dig out by the PCA without coupled effect from other components.(3)The FAHP eliminates the subjective influence of jury evaluation on the conduction of the consistency matrix. It also simplifies the process in searching the weighting matrix of high consistency.(4)Considering the surface area effect, the overall weighting order of engine components to the radiated noise is oil pan > left block > gear cover > valve cover > flywheel housing > right block.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors gratefully acknowledge the National Key R&D Program of China (2017YFC0211301).