Investigation on the time/frequency domain prediction model of radiation noise based on the source map

Abstract

The main exterior noise sources of the high-speed train CRH380AM, such as pantograph and bogie, are studied by using the field test sound imaging data, and the equivalent point source spectrums are obtained here. Then, the prediction model of the exterior train noise is presented by using the moving point sound source radiation model in the time/frequency domain. The correctness and effectiveness of the model are verified by the field test data. The radiation contribution of local noise sources is analyzed by using this model, and the following conclusions are obtained: The largest noise sources are in the bogie areas, which provide the radiation contribution rate of about 80% when the train speed is above 300 km/h. The variation of the maximum radiation noise level can be ignored when the train formation or the pantograph lifting position is changed. The model can evaluate the radiated noise of similar vehicles quickly and accurately, which provides technical reference for acoustic optimization design of high-speed train.

Keywords

pantograph bogie acoustic source imaging radiation noise component model

1. Introduction

The noise is one of the most critical indexes for the operation and internationalization of modern high-speed train (Qi-liang et al. 2022). It is particularly important to incorporate noise suppression technology in the design stage to achieve top-down acoustic top-level optimization design (Zhen et al. 2018). Traditional experimental methods and simulation techniques require high costs and long cycles, which are difficult to meet such requirements (Abdel Gawad et al. 2022; Li et al. 2022). But a good acoustic scheme evaluation technology can respond to the design optimization parameters in a timely manner, and then truly realize the forward loop acoustic optimization design. Therefore, in the current situation, the development of a fast and effective noise prediction model is an effective way to solve such problems (Ivanov et al. 2017).

Similar methods have been widely used in the field of aeroacoustics (Pei-xun et al. 2014; Qiao and Michel, 2000). ANOPP civil aircraft noise prediction software (Zorumski, 1982) provides a semi-analytical model for quickly assessing the airworthiness noise of passenger aircraft, which can more accurately predict the noise level of civil aircraft before airworthiness certification. The software classifies, models, and assembles civil aircraft noise. After decades of development and improvement (experimental data correction), it has become an important reference tool for civil aircraft acoustic design (Weiyang et al. 2008), which greatly saves time and labor costs.

Using the basic model proposed by Guo (2008), DJ Thompson studied the combined predictive frequency domain model of high-speed train aerodynamic noise (Iglesias et al. 2015); furthermore, he carried out the parameterization study of the key areas of pantograph and bogie, and gave the combined spectrum model of pantograph area (Iglesias et al. 2017). This model's prediction deviation is only about 1 dB (Liu et al. 2021). However, this model mostly draws on the results of scaled-down model tests and simulation analysis. The accuracy of describing the noise from the train bottom and side is not enough, and there is a lack of test verification. The realization of the accurate assessment of the whole vehicle level still needs further research. In addition, the bogie area contains the wheel-rail rolling noise and the aerodynamic noise, and it is not possible to accurately separate these two noises (Thompson et al. 2015), which results in the accuracy of the respective modeling being affected to a certain extent. Related research work is also carried out in China, but most of them are based on the basic frequency domain “black box” model. For example, Southwest University uses acoustic array test data to construct an equivalent energy prediction model for a certain type of vehicle acoustic radiation (Deng et al. 2014); Central South University also uses acoustic array data to construct an equivalent frequency spectrum separation model for a certain type of vehicle sound source (Xiao-ming et al. 2017) to discuss the scale characteristics of sound sources in different regions.

Based on the test data of the sound source map of the high-speed train of 200–350 km/h, starting from the bogie, pantograph, and other key areas, this paper studies the sound source energy-velocity variation and the dimensionless spectral characteristics, and then constructs time-frequency fast prediction model of the sound radiation of the train sound source.

2. CRH380AM noise source outside the car

The test high-speed train is a 6-car (full-motor train) formation, which runs back and forth in the test section, and the train passes at a speed of 200–350 km/h. The test adopts the 66-channel standard spoke-type phased acoustic array of Denmark B&K company, the diameter of the array is 4.0 m. The array is arranged on the line side, and the array plane is perpendicular to the track surface and parallel to the track. The center of the array is 19.0 m from the center of the test track, and the center of the array is 3.5 m higher than the track. The beamforming optimization algorithm (NNLS, non-negative least squares method) is used to extract the sound power level map of the sound source. Refer to literature (Central South University, 2015) for details of the test.

When the train passes at a speed of 300 km/h, the sound power level spectrum of the sound source outside the train is shown in Figure 1.

It can be seen from Figure 1 that bogie, pantograph, windshield, and other areas are the main noise source areas.

2.1 The evolution of vehicle speed

It is generally believed that at the speed range of 200–350 km/h, the external noise mainly includes the wheel-rail rolling noise and the aerodynamic noise. The wheel-rail rolling noise is proportional to the 3rd power of the train speed, and the aerodynamic noise is proportional to the 6th power of the train speed (Thompson D J, 2008). Therefore, the sound power level L_IS of the noise source outside the vehicle can be written as follows:

L_{I S} = 10 \cdot \log_{10} (\frac{I}{I_{r e f}}) = 10 \cdot \log_{10} [{(\frac{v_{t r a i n}}{v_{0}})}^{3} (\frac{I_{10}}{I_{r e f}}) + {(\frac{v_{t r a i n}}{v_{0}})}^{6} (\frac{I_{20}}{I_{r e f}})]

(1)

where v₀ is the reference speed, which is taken as 200 km/h in this article; v_train is the train speed corresponding to the total sound power I. I₁₀ and I₂₀ are respectively the sound power of wheel-rail rolling noise source and the sound power of aerodynamic noise source at the reference speed, and I_ref is the reference sound power.

Carry out Taylor expansion on the right side of Equation (1), and keep the first two expansion forms (experimental studies have shown that the second-order combination form can be a good approximation to the actual situation (Yu et al. 2001)), we can get

L_{I S} = A + B \cdot {[\log_{10} (\frac{v_{t r a i n}}{v_{0}})]}^{1} + C \cdot {[\log_{10} (\frac{v_{t r a i n}}{v_{0}})]}^{2}

(2)

where A, B, and C are the fitting coefficients. The total sound power level of the noise source is obtained by fitting the field test data as shown below.

L_{I S, t o t a l} = 128.46 + 6.7238 \cdot {[\log_{10} (\frac{v_{t r a i n}}{v_{0}})]}^{1} + 131.58 \cdot {[\log_{10} (\frac{v_{t r a i n}}{v_{0}})]}^{2}

(3)

The corresponding fitting curve is shown in Figure 2.

In Figure 2, the slopes of the fitted curves at 250, 300, and 350 km/h are 31.6, 52.0, and 69.2, respectively. This reflects the evolution law of wheel-rail rolling noise and aerodynamic noise with vehicle speed: with the increase of vehicle speed, the slope of the curve increases, and the proportion of aerodynamic noise increases gradually.

2.2 Normalized spectrum type

Figure 3(a) is the corresponding equal bandwidth spectrum of the total noise source outside the train. The black line and the red dot line respectively correspond to the test data and the fitted data.

The frequency in Figure 3(a) is dimensionless according to Equation (4) to obtain Figure 3(b).

S_{t r} = f \cdot L / v_{t r a i n}

(4)

where S_tr is the Strouhal number, f is the frequency, and L is the characteristic length (4 m is uniformly used in this paper).

It can be seen from Figure 3 that the noise source spectrum is obviously a combination of broadband and broad peaks. The peak Strouhal numbers are relatively consistent at the different vehicle speeds. This peak Strouhal number is 58.2. As the speed increases, its broad peak characteristics are gradually “overwhelmed” by broadband spectral energy. Therefore, under high-speed conditions, it is manifested as a single broadband feature.

Researchers often use the haystack model to describe this broadband or broad peak feature Iglesias et al. (2017). However, the data is obtained by the principle of beamforming, and the data below 200 Hz is missing (the lower limit of test frequency). Therefore, this paper does not use the haystack model, but uses the cubic polynomial model. For the equal bandwidth broadband spectrum of the total noise source outside the vehicle, the fitting results are as follows:

L_{I S, b r o a d b a n d} (f) = (3 \times 10^{- 10}) f^{3} - (3 \times 10^{- 6}) f^{2} + (4 \times 10^{- 3}) f + δ (v_{t r a i n})

(5)

In the formula, δ is the correction function related to the train speed, as shown in formula (6).

δ (v_{t r a i n}) = 50.595 \cdot [\log_{10} (\frac{v_{t r a i n}}{v_{0}})] + 102.57

(6)

The energy differences L_△dB between the fitted broadband spectrum and the real spectrum at different vehicle speeds are shown in Table 1.

Table 1.

Difference between the fitted spectrum and the real spectrum.

v_trainkm/h	202	233	265	290	311	325
L_△dB dB	3.2	2.0	0.5	0.1	0.1	0.1

It can be seen from Table 1 that L_△dB can be ignored at more than 300 km/h, but can reach 3 dB at less than 250 km/h. This is because equation (5) does not describe the characteristics of discrete spectrum. This paper intends to use the data in Table 1 to fit L_△dB(f_c0, S_tr = 58.2) and add it to equation (5), so that the contribution of this broad peak spectrum can be included. This fitted formula (R² = 0.9809) is as follows:

L_{Δ dB} = 80.56 x^{2} + 32.85 x + 110.26

x = \log_{10} (v_{t r a i n} / v_{0})

(7)

3. Simplified spectrum model

3.1. Pantograph area

In the speed range of 200 ˜ 350 km/h, the pantograph area is dominated by dipole aerodynamic noise. Figure 4 shows the relationship between the sound source power level and the train speed in the pantograph area. In this figure, the abscissa is the logarithmic value of the train speed relative to 200 km/h, and the ordinate is the sound source power level.

It can be seen from Figure 4 that the difference of sound source power level between the raised pantograph and the unraised pantograph is 2–3 dB at 200–350 km/h. The sound source power lever fitted formula (R² = 0.9998, 0.9947) is as follows:

L_{I S, p a n t o g r a p h} = {\begin{cases} 80.56 x^{2} + 32.85 x + 110.26 \\ 113.65 x^{2} + 19.05 x + 108.90 \end{cases} w h e r e \begin{array}{l} raised pantograph \\ unraised pantograph \end{array}

(8)

The normalized frequency band strength β is introduced, which is defined as the logarithm of the ratio of the frequency band energy to the total energy, as shown in equation (9).

β_{i} = 10 \log_{10} (\frac{10^{S P L_{i} / 10}}{\sum_{j = 1}^{N} 10^{S P L_{j} / 10}})

(9)

Equations (4) and (9) are applied to obtain the normalized spectrum pattern of the pantograph area, as shown in Figure 5. In the figure, the box is the test result at different train speeds, and the solid line is the corresponding third-order fitted curve.

The fitting curve formula is described as follows:

β_{p a n t o g r a p h} (s t r) = {\begin{cases} - 1.70 \cdot m^{3} + 9.48 \cdot m^{2} - 17.18 \cdot m + 9.00 \\ - 1.74 \cdot m^{3} + 9.62 \cdot m^{2} - 17.07 \cdot m + 8.58 \end{cases} w h e r e \begin{array}{l} raised pantograph \\ unraised pantograph \end{array}

m = \log_{10} (s t r)

(10)

In the raised pantograph area, the noise source spectrum pattern tends to be stable. In the unraised pantograph area, the interaction is strong between the shedding vortex from the rod and the turbulent boundary layer of the train body, which leads to the strong randomness of the disturbance, so the fitting degree is worse than that in the raised pantograph area. In addition, although there is a certain difference in the sound source energy between them, the difference is not significant in the normalized frequency band strength between them.

3.2. Bogie area

The variation law of sound source power level with train speed in the head car 1-position bogie area and the rear car 2-position bogie area is shown in Figure 6(a), and that in other bogie areas is shown in Figure 6 (b).

It can be seen from Figure 6 that there are obvious differences in the sound source power level of each bogie area. It is relatively minimum around Bogie 6-2, but is relatively maximum around Bogie 1-1. With the increase of train speed, the noise source energy of the middle car bogie is gradually close to that of Bogie 1-1. In the middle car, the sound source energy of each car's 2-position bogie is slightly greater than that of its 1-position bogie. Bogie noise includes the traction noise, the wheel-rail rolling noise, and the aerodynamic noise. Since the bogies of this train are the same type of power bogies, the traction noise of each bogie is basically the same, and the wheel-rail rolling noise is also the same, so the aerodynamic noise is the main reason for their difference. Because the velocity and turbulence of the incoming flow to each bogie are different, their aerodynamic noise is different.

The fitting formula of the sound source power level corresponding to each bogie area is as follows:

L_{I S, b o g i e} = {\begin{cases} 82.25 x^{2} + 20.05 x + 113.19 \\ 158.57 x^{2} - 4.67 x + 114.36 \\ 191.20 x^{2} - 10.54 x + 114.85 \\ 188.15 x^{2} - 9.15 x + 113.11 \end{cases} w h e r e \begin{array}{l} bogie 1 - 1 \\ bogie * - 1 \\ bogie * - 2 \\ bogie 6 - 2 \end{array}

(11)

The normalized spectrum pattern of the bogie area is shown in Figure 7 and Figure 8.

It can be seen from Figures 7 and 8 that the noise source spectrum of the head car 1-position bogie area is obviously different from other bogie areas, and there is relatively more energy proportion at low frequency. The noise source spectrum of the middle car 1-position and 2-position bogies is close. The high-frequency spectrum of the noise source drops faster in the 2-position bogie area of the tail car. Their 3rd order fitting formulas are as follows:

β_{bogie} (s t r) = {\begin{cases} - 2.17 \cdot m^{3} + 9.73 \cdot m^{2} - 14.13 \cdot m + 5.58 \\ - 1.74 \cdot m^{3} + 6.38 \cdot m^{2} - 6.15 \cdot m - 0.35 \\ - 1.51 \cdot m^{3} + 5.34 \cdot m^{2} - 4.71 \cdot m - 0.89 \\ - 2.04 \cdot m^{3} + 7.36 \cdot m^{2} - 7.21 \cdot m + 0.13 \end{cases} w h e r e \begin{array}{l} bogie 1 - 1 \\ bogie * - 1 \\ bogie * - 2 \\ bogie 6 - 2 \end{array}

(12)

3.3. Windshield area

The sound source power level of the 5 windshields varies with the train speed as shown in Figure 9.

It can be seen from Figure 9 that as the train speed increases, the slope gradually becomes larger, indicating that the proportion of aerodynamic noise sources increases. The sound source power level fitting formula is as follows in the windshield area.

L_{I S, w i n d s h i e l d} = 166.38 x^{2} - 5.82 x + 108.88

(13)

The sound source power level is fitted with a cubic polynomial in the windshield area, and the normalized fitting spectrum is shown in Figure 10.

The corresponding fitting formula is as follows:

β_{w i n d s h i e l d} (s t r) = 0.60 \cdot x^{3} - 7.18 \cdot x^{2} + 19.84 \cdot x - 16.78

(14)

4. Combined spectrum model of radiated noise from the train

4.1 Combined sound source and sound radiation physical model

The sound radiation formula (Park et al. 2005) of a point sound source i in a free-field linear motion is as follows:

p_{i} (\overset{⇀}{x}, t) = {[\frac{Q_{i}}{4 π r_{i} | 1 - M_{r_{i}} |}]}_{τ}

(15)

where p_i is the radiation sound pressure from the point sound source i,

\overset{⇀}{x}

is the position vector of the observation point, t is the sound receiving time, Q_i is the sound source intensity,

r_{i}

is the radiation diameter from the point sound source i, and Mr is the component of the Mach number in the direction of the radiation direction r. τ is the acoustic delay time.

When the train is running at a constant speed, the bogie area, the pantograph area, the windshield, and other areas are the most important sources of external radiation noise. These noise sources can be simplified as compact point sources. In addition, these noise sources have obvious broadband characteristics; their sound mechanism has no direct correlation and can be simplified as incoherent sound sources, so they can be directly superimposed.

Therefore, the radiation sound pressure level L_p,i,band(j) of the i-th point source energy in the j-th frequency band is as follows (Park et al. 2005):

L_{p, i, b a n d (j)} (t) = L_{I S, i, b a n d (j)} (τ) + 20 \cdot \log_{10} {[F (θ)]}_{τ} - 20 \cdot \log_{10} {[r_{i} | 1 - M_{r_{i}} |]}_{τ}

(16)

where F(θ) is the sound pressure directivity factor (in this paper, suppose F(θ) = |cos(θ)|, θ is the horizontal directivity deflection angle of the train, and the train running direction is 90°); L_IS,i,band(j) is the sound source power level of the i-th point source energy in the j-th frequency band, and the form is as follows:

L_{I S, i, b a n d (j)} (τ) = L_{I S, i} (x) + β_{i} (s t r)

(17)

The frequency band energy of all point sound sources is superimposed separately. Then, the form of the total radiation sound pressure level from all point sound sources in the j-th band is as follows:

L_{p A, b a n d (j)} (t) = 10 \cdot \log_{10} (\sum_{i = 1}^{n s} 10^{\frac{L_{p, i, b a n d (j)}}{10}}) + L_{g r o u n d} + {j_{p o w e r}}_{A}

(18)

where L_ground is the ground effect correction factor, and the leveled ground is generally 3 dB. Since the ground effect has been considered during the sound source imaging test, there is no need to repeat the calculation. j_powerA is the A-weighted correction value of the frequency band.

Further, calculate the broad peak center frequency f_c0 from the peak dimensionless frequency (Str = 58.2), use Equation (7) to obtain the corresponding frequency band energy correction value L_△dB(f_c0), and use this value to correct the energy of the corresponding frequency band j_c. The form is as follows:

\bar{L_{p A, b a n d (j)}} (t) = {\begin{cases} L_{p A, b a n d (j)} & j \neq j c \\ 10 \cdot \log_{10} (10^{\frac{L_{p A, b a n d (j_{c})}}{10}} + 10^{\frac{L_{Δ d B} (f_{c 0})}{10}}) & j = j c \end{cases}

(19)

Finally, superimpose the energy of all frequency bands to obtain the total radiated sound pressure level of the train. The form is as follows:

L_{p A} (t) = 10 \cdot \log_{10} (\sum_{j = 1}^{n b} 10^{\frac{\bar{L_{p A, b a n d (j)}}}{10}})

(20)

It should be noted that since the sound radiation is directly related to the delay time, the total radiated sound pressure level from the moving sound source at different moments can be further obtained, that is, the time-domain radiation result can be obtained. Therefore, this prediction model can give both time and frequency results at the same time.

4.2. Model verification

Using the combined prediction model of this paper, calculate the average spectrum of train passing noise at the position 19 m away from the track center line and 3.5 m above the track surface, as shown in Figures 11 –13.

It can be seen from Figures 11 –13 that the predicted spectrum shape is basically the same as the measured spectrum. In the high-frequency section, the predicted results coincide with the measurement results, which is more obvious at the higher speed where the broadband noise is dominant.

Table 2 shows the tested and predicted values of the maximum noise level and passing noise level of the train at the position 19 m away from the track center line and 3.5 m above the track surface.

Table 2.

Comparison of noise test results and prediction results at different speeds.

Speed/(km·h⁻¹)	19 m Maximum noise level L_maxA			19 m Passing noise level L_{A, Tp}
Speed/(km·h⁻¹)	Test result	Forecast results	Difference	Test result	Forecast results	Difference
244	97.0	95.9	1.1	95.6	94.9	0.7
290.0	99.9	99.3	0.6	99.0	98.1	0.9
325.0	102.3	102.3	0.0	101.5	101.1	0.4

The comparison results show that they are in good agreement. At 325 km/h, the passing noise level prediction deviation is about 0.4 dBA, and the maximum noise level prediction deviation is negligible. In addition, the predicted value is slightly smaller than the tested value, because the prediction model does not include the noise radiation contribution from the flat surface of the train body. With the increase of train speed, this deviation gradually decreases, indicating that the contribution of noise radiation from the flat surface of the train body decreases.

5. Discussion and application

5.1. Noise radiation contribution

Apply this prediction model to calculate the acoustic radiation contribution of each main noise source area, as shown in Table 3.

Table 3.

Contribution of the main noise sources to the total radiated noise (%).

Speed/(km·h⁻¹)	Bogie (%)	Pantograph (%)	Windshield (%)	Else (%)
250	78.2	8.0	10.3	3.5
300	78.8	9.5	7.9	3.8
350	81.5	9.5	5.6	3.4

It can be seen from Table 3 that the bogie is the most important noise source, which contributes approximately 80% of the total radiated noise (including the wheel-rail rolling noise and the aerodynamic noise); the contribution of the pantograph to the total radiated noise increases slowly with the increase of the speed, which is close to 10% at the speed above 300 km/h; the contribution of the windshield to the total radiated noise decreases rapidly with the increase of the speed, which is less than 6% at 350 km/h; and the contribution of other regions to the total radiation noise is less than 4%. Therefore, in the aeroacoustic optimization design of high-speed trains, the bogie, pantograph, and windshield area should be focused on.

5.2. The impact of train formation on radiated noise

The radiated noise from four formations is predicted by the partitioned sound source model in the prediction model. The four groups are 3-car formation, 4-car formation with 1 pantograph, 6-car formation with 2 pantographs, and 8-car formation with 2 pantographs. Figures 14 –16 show the prediction results of train passing noise.

It can be seen from Figures 14 –16 that the top width of the passing noise curve of different groups is different, but the trend is similar. The maximum noise level of passing noise always appears at the passing time of the head car, and its value increases slightly with the increase of the formation. The corresponding maximum passing noise level is further given in Table 4. Interestingly, the curve of 1s to 2 s in the figure is in good agreement. This is mainly because this time period is the time period when the head car passes through, and the head car is the maximum noise source. The maximum passing noise cannot be significantly increased by adding formations.

Table 4.

Comparison of passing maximum noise level under different formations (dB).

Speed (km/h)	3-car formation	4-car formation	6-car formation	8-car formation
250	94.3	94.4	94.4	94.5
300	98.1	98.2	98.3	98.3
350	102.4	102.5	102.7	102.8

Obviously, different train formations have little effect on the maximum passing noise level. This is due to the uniform distribution of the main noise source areas such as bogies along the length of the train.

5.3. Affected by the position of the pantograph

This prediction model is applied to calculate the passing noise level of 6-car train in different pantograph raising modes (including 2-car pantograph raising, 5-car pantograph lowering and 2-car pantograph lowering, and 5-car pantograph raising) at the position 7.5 m away from the track center line and 3.5 m above the track surface. The results are shown in Figures 17 –19.

According to Figures 17 –19, it can be seen that the rising position of pantograph has a certain influence on the radiated noise when the body passes through, but the influence can be ignored in terms of the total passing noise.

6. Conclusion

Based on the sound source imaging data of CRH380AM high-speed train, this paper studies the noise source characteristics of pantograph, bogie, windshield, and other areas, respectively, and constructs a time/frequency prediction model of radiated noise suitable for CRH380AM running in the speed range of 200–350 km/h. The research and analysis of the model show the following:

(1) The model can more accurately predict the far field radiation noise from CRH380AM under different formations and speed levels. As the speed increases, the prediction accuracy improves.

(2) The bogie area is the main radiation noise contribution area, and the pantograph area is the second-level radiation noise contribution area. When the train speed is above 300 km/h, the bogie area and the pantograph respectively contribute about 80% and 10%.

(3) The noise contribution from other areas such as windshield is relatively small.

(4) Formation changes have a direct impact on the passing noise time history curve, but have a slight impact on the maximum passing noise level (corresponding to the passing of the head car).

(5) The pantograph rising models have a certain influence on the local noise source area, but very small influence on the total radiated noise.

The model proposed in this paper can be applied to the accurate prediction (time domain and frequency domain) of the radiated noise from CRH380AM and other trains with the same structure. It can quickly evaluate the radiated noise performance from one or more components; it can also be used to roughly predict other trains with similar structure. It should be noted that the development and improvement of the model need more data accumulation: On the basis of further integrating the test data of more types of high-speed trains, combined with theoretical analysis and local simulation model research (Xiao-ming et al. 2018; Zhu et al. 2017), refine the local sound source model, improve the accuracy and application scope of the method, and form a comprehensive EMU noise rapid assessment technology. It provides a reference tool for the development of high-speed train shape structure and local equipment acoustic optimization technology.

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

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [52272363], and the Key Laboratory of Aerodynamic Noise Control [ANCL20200302], Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province (High Performance Manufacturing Processes and Service Performance Optimisation).

ORCID iD

Xiao-ming Tan

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