Noise Reduction Design Method and Validation of Unequal-Pitch Blade Fan for Traction Motor

Abstract

The traction motor of trains is an essential component that generates the traction force, but it has a significant influence on the indoor and outdoor acoustic environment during the running condition. To enhance passenger comfort and bolster the environmental performance of trains, regulatory standards and specifications governing train noise have imposed elevated criteria on both tonal noise and the overall A-weighted sound pressure level emanating from traction motors. This paper proposes an aerodynamic noise optimization design method based on aerodynamic interference. The objective function of the optimization is defined as the blade-passing frequency (BPF) noise and the A-weighted overall sound pressure level of the traction motor fan. The simulated annealing algorithm is used to optimize the circumferential distribution of the blades. The optimized unevenly spaced blade configuration of the fan effectively reduces the BPF sound pressure level amplitude by 4.6 dBA and the overall A-weighted sound pressure level by 1.2 dBA. The calculation results agree well with the experimental results. The study reveals that unevenly spaced blades contribute to better noise reduction at lower rotational speeds. However, the effect diminishes at higher speeds. The relationship between noise reduction and blade count is nonlinear, suggesting an optimal count for specific speeds.

1. Introduction

Train interior noise is an important factor in determining passenger comfort. At the same time, the external noise propagation during train operation will also have an impact on the environment around the railroad tracks. In recent years, with the growing interest in passenger comfort and the environmental performance of trains, regulations and norms on train noise continue to be strengthened. Accordingly, research into the assessment and reduction of train noise is also ongoing [1,2,3,4,5].

Train traction motors are important components for generating traction power. The noise generated during its operation has a high impact on the indoor and outdoor sound environments. Noise generated by traction motors includes electromagnetic noise, mechanical noise, and pneumatic noise [6]. Among them is the high-speed, self-fanning, air-cooled motor. The main noise comes from the aerodynamic noise generated by the coaxial fan, including discrete and broadband noise. To realize the requirement for forward and reverse rotation of the traction motor, fans usually have straight blades. However, this blade type causes airflow separation at the blade surface. Low efficiency and high noise, especially tonal noise, are more significant. Therefore, research on reducing the aerodynamic noise of traction motors has been of great interest. Kim J-H et al. [7] successfully established a design method to reduce the noise of railroad traction motors using numerical analysis techniques. By optimizing seven design parameters, such as fan diameter and number of blades, a noise level reduction of about 4 dB (A) was successfully achieved, and the applicability of the selected design parameters, the improved methodology, and the final model were verified. Qu X et al. [8], on the other hand, analyzed the aerodynamic performance and noise of centrifugal cooling fans for traction motors of high-speed trains by numerical and experimental methods. Corresponding performance optimizations were proposed. Rama Krishna S et al. [9] utilized the CFD method and the FW-H equation. The fan noise was reduced by 12.8 dB (A) by adopting noise reduction methods such as wing blade section, flared inlet, composite materials, and reducing the number of blades. The use of unequally spaced blade designs is a common approach to reducing tonal noise. The design idea of the method is to change the spectral characteristics of the noise by phase interference. The objective function is optimized to obtain the best leaf distribution scheme. Studies have been carried out from the perspective of leaf distribution. Mellin R.C. et al. [10] were the first to investigate the theory of unequal-pitch blade noise reduction, which is usually applied to helicopter propellers and low-speed fans [11]. Enwald D. et al. [12] proposed the sinusoidal modulation method (SMM), which is used to modulate the spectral characteristics of an axial fan without inducing dynamic balancing problems or a loss of aerodynamic performance of the fan. Duncan P.E. et al. [13] subsequently further improved this modulation method. Dobrazynski W. et al. [14] performed an optimized design of unequal-pitch blades to reduce the aerodynamic noise of the propeller. Ma J et al. [15] took a forward centrifugal fan as an object and used a combination of numerical qualitative prediction and experimental measurements to analyze the aerodynamic noise of a centrifugal fan with unequal-pitch blades, and the experimental results coincided well with the numerical simulation results. Because the sinusoidal modulation method requires that the number of blades of the fan cannot be a prime, Cattanei A et al. [16] proposed a randomized modulation method (RMM) as an alternative method to eliminate the symmetry present in the distribution of the blades and demonstrated that this method provides better modulation. Anghinolfi D et al. [17] used RMM and the simulated annealing algorithm (SA) to reduce the tonal noise of the propeller. Jiang B et al. [18] investigated the sinusoidal modulation method, stochastic modulation method, and associated constraints for unequal-pitch blades of a forward centrifugal fan to achieve lower pitch noise and good sound frequency distribution. Wu Y et al. [19] developed a tonal noise prediction model based on the Lowson model and CFD simulation results, and the BPF noise of the axial fan optimized using this model was reduced by about 9.1 dB. Peng Z et al. [20] developed a theoretical model to quickly predict and control the tonal noise of an automotive cooling fan, which can be used to effectively and quickly predict and control the tonal noise at an early stage of the automotive cooling fan module design. Cattanei A. et al. [21] further investigated the effect of unequally spaced leaves on sound quality.

Although previous studies have focused on the use of unequally spaced blades to reduce tonal noise, optimization considerations for the A-weighted total sound pressure level have yet to be further explored. Therefore, this paper introduces the A-weighting function as the basis for establishing the objective function of BPF noise and the A-weighted overall sound pressure level of tonal noise of the traction motor and uses the simulated annealing algorithm to carry out the optimization design of the unequal-pitch blades of the fan. By adjusting the distribution of the tonal noise of the motor fan in the frequency spectrum, the noise reduction optimization of the motor BPF noise and the A-weighted overall sound pressure level were successfully achieved.

2. Theoretical Background

2.1. Basic Theory

The tonal noise of a fan originates from the superposition of the acoustic fields generated by each blade, and if the receiver is located in the far field, there is a phase difference in the acoustic waves arriving at the receiver due to the different locations of the circumferential distribution of the blades. Therefore, the principle of phase interference can be utilized to modulate the tonal noise of the fan by changing the relative positions of the blades. If the fan has z blades, the design variables of the optimization model are z. The design variable $\mathit{x}$ is a z-dimensional array, $\mathit{x}=\left\{{\alpha }_1,\cdots ,{\alpha }_i\cdots {\alpha }_z\right\}$, where ${\alpha }_i$ is the circumferential angle of the i-th blade. Due to the different circumferential angles of the blades, there is a phase interference of the sound pressure between the blades, and assuming that the blade load is not affected by the blade pitch, the power spectral density $S_{\mathrm{p}\mathrm{p},\mathrm{b}}(\mathit{x},\omega )$ at the angular frequency of ω for a fan with a blade distribution of $\mathit{x}$ is as follows:

(1)$S_{\mathrm{p}\mathrm{p}}\left(\mathit{x},\omega \right)={\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\omega \right)\right|}^2{S}_{\mathrm{p}\mathrm{p},\mathrm{b}}(\mathit{x},\omega )$

where $F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\omega \right)$ is the interference function, the value of which is jointly determined by the spatial distribution of the blades and the rotational speed:

$F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\omega \right)={\sum }_{i=1}^z\exp (-\mathrm{j}\omega ∆t_i)={\sum }_{i=1}^z\exp (-\mathrm{j}\frac{\omega }{\Omega }{\alpha }_i)$

where j is the imaginary unit of the complex number, $\Omega$ is the rotational angular velocity of the fan, and the corresponding frequency is the fundamental frequency of the fan; $∆t_i$ is the delay time when the i-th blade passes through the position of the first blade. Therefore, the power spectral density can be changed to:

$S_{\mathrm{p}\mathrm{p}}\left(\mathit{x},\omega \right)=S_{\mathrm{p}\mathrm{p}}\left(\mathit{x},n\Omega \right){=S}_{\mathrm{p}\mathrm{p}}\left(\mathit{x},n\right)$

and the superposition of the fan’s tonal noise occurs only at the rotating fundamental frequency and its octave; therefore, only the interference at the frequency of $\omega =n\Omega$ needs to be considered; and the interference function can be changed to:

(2)$F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\omega \right)=F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\Omega \right)=F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)={\sum }_{i=1}^z\exp (-\mathrm{j}n{\alpha }_i)$

For an equally spaced fan with $z$ blades, ${\alpha }_i=2\pi (i-1)/z$. $F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)$ has non-zero values only at the fan blade passage frequency (BPF) and its octave:

(3)$F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)=\left\{\begin{array}{c}z,n=kz\\ 0,n\ne kz\end{array}\right.$

According to the above equation, it can be seen that the noise generated by the equally spaced blade fan is mainly distributed at the fundamental frequency and its octave, which results in the presence of high tonal noise at that location. In contrast, the $F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)$ values of the unequally spaced blade fan are non-zero at most octave frequencies, indicating that the acoustic energy is more dispersed, and thus the interference function actually modulates the noise spectrum.

2.2. Objective Function

A-weighting is a frequency weighting used to measure the loudness of sound, mainly used to evaluate human perception of different frequencies of sound. The A-weighted weight curve weights different frequencies of sound according to the characteristics of human auditory perception, making human auditory perception closer to uniform loudness perception under this weight. The aerodynamic noise of traction motors is measured using A-weighting for sound measurement and noise level evaluation.

The noise at the fan blade passage frequency (BPF) of a self-cooled air-cooled traction motor is very significant, and the total sound pressure level of the motor is significantly higher than that of forced air-cooled and water-cooled motors. In order to optimize the noise level of the motor, this paper focuses on the relationship between the unequally spaced blades and the BPF noise as well as the A-weighted overall sound pressure level of the tonal noise, where the determination of the objective function is the key to the study. Specifically, the tonal noise of the traction motor fan is mainly concentrated in the BPF band, and in order to reduce the tonal noise of the fan, the value of the interference function $F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)$ at the BPF can be minimized, i.e., the value of the objective function $F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(z\right)$ can be minimized. In addition, the idea of noise reduction for unequally spaced blades is to use phase interference to reduce the peaks on the noise spectrum, thus spreading the noise energy at the BPF and its octave to other frequencies with the total energy remaining unchanged. Because the A meter is used to evaluate the noise of the traction motor, it is possible to utilize the property that the A meter attenuates more in the low and high frequency bands to distribute the energy as much as possible in the low and high frequency bands, thus reducing the total A level of the tonal noise. The sound pressure level of tonal noise can be calculated according to the following equation:

(4)${SPL}_n^t=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\sum }_{i=1}^z\left[{\left|\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{j}n{\alpha }_i)\right|}^2\frac{P_{n,\mathrm{b}\mathrm{l}}^2}{P_{\mathrm{r}\mathrm{e}\mathrm{f}}^2}\right]$

where ${SPL}_n^t$ is the sound pressure level at n times the fundamental frequency; $P_{n,\mathrm{b}\mathrm{l}}$ is the RMS value of the sound pressure generated by a single blade; $P_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference value of the sound pressure, $P_{\mathrm{r}\mathrm{e}\mathrm{f}}=2\times {10}^{-5}$ Pa.

Introducing the A-weighting function $w\left(n\right)$ into the above equation, the total A level of the tonal noise is:

(5)${SPLOA}_n^t={\sum }_{n=1}^{N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}}{10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}10}^{{0.1w\left(n\right)·SPL}_n^t}$

Therefore, the objective function in this paper is defined as:

(6)$\left\{\begin{array}{l}{F}_{\mathrm{o}\mathrm{b}\mathrm{j},1}=minimize(\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)\right|),n=z\\ F_{\mathrm{o}\mathrm{b}\mathrm{j},2}=minimize(SPLO{A}_n^t),N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}=3z\end{array}\right.$

where $N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$ is the number of fundamental frequency octaves considered in the model, $F_{\mathrm{o}\mathrm{b}\mathrm{j},1}$ is the objective function to minimize the tonal noise at the BPF, and $F_{\mathrm{o}\mathrm{b}\mathrm{j},2}$ is the objective function to minimize the total A level of the tonal noise in the 3 BPFs.

2.3. Restrictive Condition

Fans need to be designed to meet dynamic balancing requirements while minimizing the loss of aerodynamic performance. Therefore, both the dynamic balance nonlinear equation and the linear inequality for the minimum pitch between blades must be satisfied. Figure 1 shows a schematic diagram of the circumferential distribution of the blades of a centrifugal fan with the number of z blades. The circumferential angle ${\alpha }_i$ of the ith blade needs to satisfy the following relationship:

(7)$0={\alpha }_1<{\alpha }_2\cdots <{\alpha }_{z-1}<{\alpha }_z<2\pi$

The symmetry of the equidistant blades allows their dynamic balance to meet the requirements; however, the unequal-pitch blades change the balance of the fan; therefore, in order to meet the dynamic balance condition, the circumferential angle of the blades needs to satisfy the following equation:

(8)$\left\{\begin{array}{c}{\sum }_{i=1}^z\mathrm{c}\mathrm{o}\mathrm{s} {\alpha }_i=0\\ {\sum }_{i=1}^z\mathrm{s}\mathrm{i}\mathrm{n} {\alpha }_i=0\end{array}\right.$

After satisfying the dynamic balance equations, the degrees of freedom of the blade circumferential distribution become z-3.

The circumferential angle ${\alpha }_i^0$ of the isometric blade is:

(9)${\alpha }_i^0=\frac{2\pi (i-1)}{z}$

The circumferential angle ${\alpha }_i$ of the unequally spaced blades is:

(10)${\alpha }_i={\alpha }_i^0+{\epsilon }_i$

In order not to affect the aerodynamic performance of the fan, it is necessary to ensure that the pitch between the blades is not too small. The constraints on the blade spacing are as follows:

(11)$\left|{\alpha }_{i+1}-{\alpha }_i\right|\ge \frac{2\pi }{z}(1-R_{\alpha }), i=1, z$

where $R_{\alpha }$ is the blade pitch coefficient; the value range is 0–1; when $R_{\alpha }=0$, the fan has equidistant blades; and when $R_{\alpha }=1$, the fan adjacent blades overlap. In order to minimize the loss of aerodynamic performance, it is generally recommended that $R_{\alpha }\le 0.3$ [17].

3. Optimization Design

In this paper, the simulated annealing algorithm is chosen to optimize the design of fan blade pitch. The specific steps are as follows:

• The value of the objective function is computed using the blade distribution ${\mathit{x}}_0$ of the equidistant blade as the initial solution and considering this solution as the current optimal solution ${\mathit{x}}_c$.

• A new solution is obtained by giving a small perturbation to the current optimal solution. The blade angle is varied in the range $U\left[{\mathit{x}}_{x,m}-\rho ,{\mathit{x}}_{x,m}+\rho \right]$, where $\rho =0.1\pi /z$, and the new solution is computed with the value of its objective function and its increase in the value of the objective function ${∆F=F}_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\mathit{x}\right)-F_{\mathrm{o}\mathrm{b}\mathrm{j}}\left({\mathit{x}}_c\right)$.

• If this increase $∆F$ is not greater than zero, accept the new solution as the current optimal solution; if this increase $∆F$ is greater than zero, calculate the acceptance probability $P(\mathit{x},{\mathit{x}}_c,k)=\exp \left(\frac{{F_{\mathrm{o}\mathrm{b}\mathrm{j}}\left({\mathit{x}}_c\right)-F}_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\mathit{x}\right)}{T_k}\right)$, take a random number $r(0<r<1)$, and when $P>r$, accept the current value. Otherwise, it is not accepted. $T_k$ is the cooling control parameter.

(12)$T_k=\left|\frac{F_{\mathrm{o}\mathrm{b}\mathrm{j}}\left({\mathit{x}}_c\right)}{\mathrm{l}\mathrm{n}\left(P_k\right)}\right|\delta$

where $\delta$ is the percentage of new solutions received, and its value is a constant 0.9; the initial value of the relative acceptance probability is $P_k$, $P_0=0.99$, and its value varies with iterations, $P_k=\alpha ·P_{k-1}$, where the reduction factor $\alpha =0.999$. This determines the reduction law for $T_k$.

• 4.. The perturbation and acceptance process is repeated 1000 times under the same control parameter $T_k$, i.e., steps 2 and 3 are performed.

• 5.. Determine whether the temperature reaches the termination temperature level; if it is satisfied, terminate the algorithm; otherwise, reduce the temperature and proceed to the next temperature iteration until the termination condition of the algorithm is satisfied and the algorithm is terminated.

The aerodynamic noise emanating from the traction motor fan primarily falls within the three-times-BPF range. For instance, in the case of a fan with 17 blades and a rotational speed of 4000 revolutions per minute (r/min), the third harmonic ($N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$ = 3 z) corresponds to a frequency of 3400 Hz. This implies that the predominant noise source is concentrated within 3400 Hz. To ascertain the impact of $N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$ on the computational outcomes, a series of 34 computations were conducted. These computations employed $F_{\mathrm{o}\mathrm{b}\mathrm{j}, 2}$ as the objective function, varying the number of blades from 7 to 23 and considering $N_{hram}$ values of 3 z and 5 z, respectively. The findings are depicted in Figure 2, revealing a consistent trend in noise reduction regardless of whether $N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$ is set to 3 z or 5 z. Given that ∆SPL primarily serves to gauge the effectiveness of noise reduction, a qualitative analysis of the noise reduction impact can be performed irrespective of the specific value chosen for $N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$. Consequently, in the interest of computational economy, the latter is set to be equivalent to 3 z.

The fundamental frequency of aerodynamic noise generated by a traction motor fan is contingent upon fan speed, while the BPF is determined by both the fundamental frequency and the number of blades. Consequently, the distribution of acoustic energy in the spectrum is intricately linked to both the speed (Ω) and the number of blades (z). The fan’s frequency is a function of the fundamental frequency and blade count. In Figure 3, with 17 blades, the testwas sampled at 500 r/min intervals across a rotational speed range of 2000–8000 r/min. Subsequently, optimization of $F_{\mathrm{o}\mathrm{b}\mathrm{j},2}$ was performed, yielding the extent of noise reduction at various rotational speeds. Herein, ΔSPL denotes the disparity in the total A-weighted sound level post-optimization between evenly and unevenly spaced blades. The magnitude of this difference quantifies the noise reduction effect achieved through the implementation of unevenly spaced blades. The figure illustrates that the noise reduction effect of unevenly spaced blades varies under different operational conditions and speeds. In the low-speed range, optimized unevenly spaced vanes yield a reduction of up to approximately 2.5 dBA compared to evenly spaced vanes. Conversely, the noise reduction effect is less pronounced in the high-speed range. Notably, as depicted in Figure 4, the relationship between noise reduction and blade count is non-monotonic, showcasing instances where certain blades lead to more substantial noise reduction. Figure 5 provides a graphical representation of the ΔSPL distribution across blade counts ranging from 7 to 23, within the speed range of 2000–8000 r/min. Evidently, the noise reduction effect is more pronounced in the low-speed range. Additionally, as rotational speed escalates, enhanced noise reduction is observed at specific blade counts. Consequently, with regards to aerodynamic noise control for traction motors, the optimal blade count can be tailored to align with the motor’s rated speed, maximum speed, and specific noise reduction requirements.

The simulated annealing algorithm optimization model developed above is used to optimize the design of a certain isometric blade fan with unequal-pitch blades. The number of blades of this fan is 17, the angular speed is 418.7 rad/s (4000 r/min) corresponding to a frequency of 66.7 Hz, and in order to avoid the deterioration of the aerodynamic performance of the fan, $R_{\alpha }=0.3$ is selected as the constraint in this paper. On this basis, it is assumed that the sound pressure level at the BPF and its octave of the isometric fan is ${SPL}_{kn}^t=94 \mathrm{d}\mathrm{B}$. In this paper, the simulated annealing algorithm is used to optimize the two objective functions and the calculation results are shown in Table 1.

Scheme 1 with equidistant blades shows that the value of $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)\right|$ at the BPF and its octave is exactly equal to the number of blades. The total tonal noise A level for this scheme was 98.4 dBA, the highest of all the schemes.

Scheme 2 uses an unequally spaced blade optimization design to minimize the objective function $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(17\right)\right|$, i.e., to minimize the tonal noise at the fan BPF. As can be seen from Figure 5, the value of $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)\right|$ at the BPF is 0, but the value of $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)\right|$ at the nearby frequencies increases, which indicates that the unequally spaced fan disperses the energy at the BPF and its harmonics to the nearby frequencies, and its total A sound level decreases compared with that of the equidistant blades.

Scheme 3 minimizes the A-weighted overall sound pressure level of the tonal noise at the fundamental frequency and its harmonics within the three-times-BPF range., and the A-weighted overall sound pressure level of this scheme is about 1.2 dB lower than that of the equidistant blade scheme, which indicates that the unequal blade design can reduce the A-weighted overall sound pressure level of the tonal noise of the fan.

The computational results of the isometric scheme and the two objective function-seeking optimization schemes are given in Figure 6. The $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(n\right)\right|$ values at BPF and BPF multiples are reduced for both Scheme 2 and Scheme 3, while new peaks appear at other frequency locations. It shows that the acoustic energy at this frequency is reduced and the acoustic energy at other frequencies is increased, and the unequal spacing scheme reduces the peak value of the fan’s tonal noise.

4. Test Validation

In this study, a semi-anechoic chamber was used for the noise test experiment, and the arrangement of the test setup and noise measurement points is shown in Figure 7. The test was conducted in accordance with the national standards of the People’s Republic of China, GB/T10069.1 and GB/T25123.2 standards [22,23]. In this paper, the traction motors for the equally spaced blade fan and the traction motor for the unequally spaced blade fan of Scheme 2 were tested separately with a motor speed of 3600 rpm. The test results are shown in Table 2. The average total A sound level at the five measurement points of the optimized motor was reduced by 1.2 dBA.

5. Analysis of Results

Unequally spaced blades have different leaf spacings and different phase differences between the two neighboring blades, and this harmonic coherence manifests itself in the power spectrum by attenuating the peaks of the equidistant fan at the fundamental frequency to a distribution of smaller values. From Figure 8a, it can be seen that the power spectral densities of both equally spaced blade fan and unequally spaced blade fan motors have significantly higher SPL amplitudes at the BPF than at the other peak points, with similar distribution patterns at the other measurement points, as shown in Figure 8b–e. This indicates that the tonal noise at the BPF accounts for the majority of the aerodynamic noise of the traction motor. The unequally spaced blade scheme, on the other hand, has a significant reduction in the peak sound pressure level at the BPF, and its five-measurement-point average of the sound pressure level amplitude at the BPF is reduced by 4.6 dBA, which is a very obvious noise reduction effect. However, there is a peak in the band near the BPF, suggesting that the unequally spaced blades attenuate the tonal noise at the BPF frequency, distributing the energy more evenly across the frequency range. Figure 9 shows the one-third-octave range of the motor noise for an equally spaced blade fan and an unequally spaced blade fan. Compared to the equally spaced blade fan, the peak sound pressure level of the unequally spaced blade fan in the 1000 to 1200 Hz octave band decreases significantly, allowing the sound energy to be distributed to the nearby frequency bands, reducing the tonal noise, and obtaining a more homogeneous spectral energy distribution. It can also be seen from the figure that the unequally spaced blades selectively distribute energy into the frequency band range where the human ear is not sensitive, thus reducing the A-weighted overall sound pressure level of the traction motor. Based on the calculations of the algorithm above, Scheme 2 reduces the $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(z\right)\right|$ value at the BPF by 13 and the A-weighted overall sound pressure level by 0.7 dBA compared to the equidistant blade scheme, and the sound pressure level magnitude and the A-weighted overall sound pressure level at the BPF in the experimental results are reduced by 4.6 dBA and 1.2 dBA, respectively. The feasibility of the method to reduce the BPF tonal noise and the A-weighted total sound pressure level of the traction motor is shown. This study may provide a valuable reference for engineering practice in related fields.

6. Conclusions

To enhance passenger comfort within the train and mitigate the environmental impact of train operations, this study examines the utilization of the free modulation method in the design of unequally spaced blades for the traction motor fan. The methodology effectively achieves a reduction in both the tonal noise emanating from the traction motor and the A-weighted total sound pressure level. These findings are substantiated through experimental verification of the numerical results. The primary conclusions are outlined below:

In the numerical computations, both the $\left|F_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(z\right)\right|$ value at the BPF experiences a reduction of 13, and the A-weighted overall sound pressure level decreased by 0.7 dBA. Conversely, experimental data demonstrate a respective reduction of 4.6 dBA and 1.2 dBA in the amplitude of the SPL at the BPF and in the A-weighted overall sound level. This amplitude reduction effectively alleviates discomfort attributed to single-frequency noise, while the tonal noise in the A-weighted overall sound level contributes to compliance with increasingly rigorous noise standards.

The outcomes obtained through this algorithm reveal that the efficacy of unequal-pitch blade design in reducing the A-weighted overall sound pressure level is intricately linked to motor speed and the number of fan blades. Notably, the noise reduction effect is most pronounced in the low-speed regime, whereas it becomes less evident at higher speeds. Additionally, the magnitude of noise reduction exhibits a non-monotonic variation with the number of blades. Under constant speed conditions, an optimal blade count exists for achieving maximal noise reduction.

7. Prospects

In this paper, based on the principle of phase interference, and with the blade circumferential distribution as the optimization variable, a simulated annealing algorithm is used to obtain the optimization scheme of unequally spaced blade fans for traction motors, and its validity is verified through experiments. Looking forward to the future research direction, the research on the optimal design method for the tonal noise of the traction motor fan can be further deepened. First, more objective functions, such as sound quality-related parameters, can be considered to be introduced in the optimization process to comprehensively improve the overall performance of the traction motor fan. In addition, CFD simulation methods can be combined to further optimize the geometrical parameters of the unequal-pitch blades, such as blade bending shape, number of blades, etc., in order to explore more potential room for improvement. At the same time, combined with advanced numerical simulation tools and experimental validation, the performance of unequal-pitch blade fans under various operating conditions is thoroughly investigated to ensure their reliability and stability in practical applications. Finally, this optimization method is extended to other fields such as wind turbines and air conditioning systems in order to further expand its application and provide more possibilities for improving energy use efficiency and reducing environmental noise levels. These future works will help deepen the understanding and application of optimized design methods for unequally spaced blade fans and provide strong support for technological advancement and sustainable development in related fields.

Author Contributions

Methodology, W.L., L.Y. and A.H.; formal analysis, L.Y.; writing—original draft preparation, W.L.; writing—review and editing, Y.W., A.H. and G.L. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

$F_{\mathrm{i}\mathrm{n}\mathrm{t}}$	interference function of the rotor	$\mathit{x}$	position vector
$F_{\mathrm{o}\mathrm{b}\mathrm{j}}$	the objective function	$\omega \left(n\right)$	weighting function
$f$	frequency (Hz)	$z$	number of blades
$\mathrm{j}$	imaginary unit	$\alpha$	angular position relative to the first blade (rad)
$k$	integer number	$\mathsf{\delta }$	Dirac pulse
$N_{\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{m}}$	number of harmonics of the rotor angular speed	$\Omega$	angular speed of the rotor (rad/s)
$n$	harmonic order	$\epsilon$	blade angular displacement from the evenly spaced position
$P_n$	power of an acoustic pressure component at the nth harmonic	$\omega$	angular frequency (rad/s)
$P_{\mathrm{r}\mathrm{e}\mathrm{f}}$	acoustic pressure, Pa	$\mathrm{b}\mathrm{l}$	blade
$R_{\alpha }$	maximum displacement parameter	$i$	Blade No.
$SPL$	sound pressure level (dB)	Superscripts
$SPLOA$	the overall A-weighted sound pressure level (dBA)	$t$	tonal
$S_{\mathrm{p}\mathrm{p}}$	power spectrum of sound pressure (Pa2s)	$0$	relative to the evenly spaced configuration

Footnotes

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Figures and Tables

Figure 1. Schematic of blade circumferential distribution. (a) Isometric blade. (b) Unequal-pitch blade.

Figure 2. ∆SPL with respect to the number of blades, z.

Figure 3. Noise reduction at different speeds.

Figure 4. Noise reduction at 4000 r/min with different numbers of blades.

Figure 5. Distribution cloud of noise reduction at different blade numbers and speeds.

Figure 6. Distribution of interference functions at the fundamental frequency and its octave for different objective functions.

Figure 7. Motor noise test. (a) Schematic layout of measurement points. (b) Test setup.

Figure 8. Power spectral density: (a) measurement point 1; (b) measurement point 2; (c) measurement point 3; (d) measurement point 4; (e) measurement point 5.

Figure 8. Power spectral density: (a) measurement point 1; (b) measurement point 2; (c) measurement point 3; (d) measurement point 4; (e) measurement point 5.

Figure 9. One-third-octave plots of motor noise for equally spaced blade fans and unequally spaced blade fans.

Table 1

Optimization results for different objective functions.

No.	${\mathit{F}}_{\mathit{i}\mathit{n}\mathit{t}}\left(\mathit{z}\right)$	${\mathit{S}\mathit{P}\mathit{L}\mathit{O}\mathit{A}}_{\mathit{n}}^{\mathit{t}}$	Blade Circumferential Angle/°
1	17.0	98.4	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$	${\alpha }_7$	${\alpha }_8$	${\alpha }_9$
			0	21.2	42.4	63.5	84.7	105.9	127.1	148.2	169.4
			${\alpha }_{10}$	${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{13}$	${\alpha }_{14}$	${\alpha }_{15}$	${\alpha }_{16}$	${\alpha }_{17}$
			190.6	211.8	232.9	254.1	275.3	296.5	317.6	338.8
2	0	97.7	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$	${\alpha }_7$	${\alpha }_8$	${\alpha }_9$
			0	30.8	49.2	67.3	85.3	103.2	121.1	140.1	163.3
			${\alpha }_{10}$	${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{13}$	${\alpha }_{14}$	${\alpha }_{15}$	${\alpha }_{16}$	${\alpha }_{17}$
			196.5	219.9	238.6	256.8	274.5	292.3	310.5	329.2
3	13.5	97.2	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	${\alpha }_5$	${\alpha }_6$	${\alpha }_7$	${\alpha }_8$	${\alpha }_9$
			0	25.3	44.6	64.5	84.7	105.1	125.5	145.9	166.0
			${\alpha }_{10}$	${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{13}$	${\alpha }_{14}$	${\alpha }_{15}$	${\alpha }_{16}$	${\alpha }_{17}$
			194.0	214.2	234.5	254.9	275.3	295.5	315.4	334.7

Table 2

Noise test results.

	Motor A-Weighted Sound Pressure Level (dBA)
Measurement point	1	2	3	4	5	average value
Isometric blade	99.0	100.2	98.3	100.9	96.5	99.2
Unequal-pitch blade	97.2	99.4	98.6	98.6	95.3	98.0

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