The Transfer paths’ Contribution Analysis to Floor Vibration of Metro Vehicle

Noise control is one of key issues to improve the ride comfort of metro trains. To find out the excitation source and its transfer path is an important prerequisite for noise control. The sound source identification results found that the significant noise source in metro trains is the structural-borne sound radiated by floor vibration. Based on the OTPA method, this study presents a method considering the amplitude and phase of the excitation to analyze the contribution of the secondary suspension path to the floor vibration. The results show that the energy of the passenger room noise mainly concentrates on the frequency range of 300~800Hz, caused by floor vibration; in the frequency range of 300~800Hz, the vertical direction of the anti-rolling torsion bar area provides the maximum contribution to the floor vibration, followed by the longitudinal direction of the air spring area. On the basis of contribution analysis, a transfer path optimization scheme is proposed, which may provide reference for future metro trains noise control.


Introduction
In recent years, China's metro trains have become an indispensable part of urban public transportation. With the rapid development of the metro car, new challenges have emerged, necessitating further research. Noise control is one of key issues to improve the ride comfort of metro trains. To find out the excitation source and its transfer path is an important prerequisite for noise control.
When the train is running at a speed of 30-80km/h, wheel-rail rolling contact becomes the main source of excitation [1,2] . Fan et al. [3] found that the major source of the interior noise is the structural-borne sound radiated by floor vibration. Zhang et al. [4] found that the bogie area noise and the sound source at the middle of the coach contribute significantly to the interior noise. The most direct and effective method of noise control is to eliminate the excitation source. Furthermore, it is also important to identify the transfer path of the main source of excitation.
The transfer path analysis (TPA) method [5][6][7][8] is an effective method to obtain the contribution of each transfer path. However, it takes a lot of manpower and time to get all the transfer functions. Operational transfer path analysis (OTPA) is an advanced vibration and noise transfer path identification and contribution evaluation method [9] . Klerk et al. [10] elaborated on the theory, the modeling principles and the precautions of the OTPA method and applied the OTPA method to practical engineering. De Sitter [11] applied OTPA to study the NVH problems of the car, and the results showed that no disassembling was required and operational forces also don't have to be eliminated. Keizer et al. [12] used the OTPA method to analyze the contribution of the noise sources of a ship, such as engines, gearboxes, and propellers. Robert [13] applied OTPA to study the vibration and noise problems of a high-speed train bogie, and it was found that OTPA was a faster and often cheaper way than traditional TPA. However, when OTPA is applied in complex mechanical systems, there is a problem of crosstalk between vibration sources [14] . So, Putner [15] pointed out that the reference points of the excitation source should be as close as possible to the excitation source to reduce the effect of crosstalk. Mihkel [16] applied singular value decomposition (SVD) and principal component analysis (PCA) to solve the crosstalk problem of OTPA, which cancelled crosstalk by cutting off small singular values or principal components. Cheng et al. [17] proposed a novel crosstalk cancellation method based on independent component analysis (ICA) to eliminate crosstalk effects between reference signals of operational transfer path analysis (OTPA). Zhang [18] performed crosstalk cancellation pre-processing on experimental data to achieve more accurate data for the transfer path analysis, and the experiment proved that it can obtain more accurate contribution results for each path. Lei [19] used the OTPA method to analyze the transfer paths of the metro vehicle interior noise and found that the major path in the structure-borne paths is the vertical damper on both side of the bogie.
In summary, the OTPA method has been gradually applied in the field of railway transportation to analyze the

OTPA method theory
OTPA is a signal processing method, which uses numerical calculation means to obtain the transfer functions between the input and output signals, to obtain the contribution of the excitation sources or transfer paths to the target point response [20][21][22][23] .
The OTPA method theory is based on the system that is linearized and time-invariant. The input and output of the system can be expressed as: Where k is the number of operating conditions, n, m respectively represents the number of vibration reference points and acoustic reference points, and there is a necessary condition is k>m+n. However, the equation is solved directly, numerical problems will appear and the wrong transfer functions matrix may be obtained.
Therefore, there is needed to solve the problem of solving the equation by singular value decomposition (SVD).
The reference points matrix X can be expressed as: : Where + X is the generalized inverse matrix of the input matrix X.
The transfer functions after SVD as follows: The target point response can be synthesized and the contribution of the excitation sources and transfer paths can be calculated by the input signals and equation (1) after the transfer functions is obtained.

Results of interior sound source identification
A spherical microphone array with 50 channels and 12 cameras (B&K WA1565W004) is used in this sound source identification test, as shown in Figure 1. The software B&K NSI array acoustics post-processing was used for the interior noise source identification analysis, and the method is called spherical harmonics beamforming [24,25] . During the test, all the microphones on the spherical array are applied to collect the sound pressure signal. At    In order to find out whether the evaluation method which only considers the transfer path contribution's amplitude can effectively rank contribution to the target response of the transfer paths. The simplified model is shown in Figure   4. Two different sets of excitations are added to the bottom of the four steel columns. The specific information of the excitations is shown in Table 1 and Table 2. The contribution of four excitations and the target point response can be directly obtained because the excitations and transfer functions are known, and the results are as shown in Figure 6.
Where n is the number of frequencies and Zx is the comprehensive contribution of x path to the target point.

The test of operational transfer path analysis
The basic process of the metro vehicle transfer path contribution analysis based on the OTPA method is shown in Figure 7:   Overall/(m/s 2 ) P a t h 3 P a t h 2 P a t h 1 P a t h 6 P a t h 5 P a t h 4 P a t h 9 P a t h 8 P a t h 7 P a t h 1 2 P a t h 1 1 P a t h 1 0   Comprehensive contribution/(m/s 2 ) P a t h 3 P a t h 2 P a t h 1 P a t h 6 P a t h 5 P a t h 4 P a t h 9 P a t h 8 P a t h 7 P a t h 1 2 P a t h 1 1 P a t h 1 0  Table 4.  Table 4 shows that there is a significant effect on Re duc tio n 80% Re duc tio n 60% Re duc tio n 40% Ini tial Re du cti on 20 % P a th 3 P a th 2 P a th 1 P a th 6 P a th 5 P a th 4 P a th 9 P a th 8 P a th 7 P a th 1 2 P a th 1 1 P a th 1 0

Conclusions
Based on the results described in this paper, the following conclusions can be stated: (1) When the metro vehicles run at 60km/h, the main interior sound source is located at the floor areas. The floor vibration caused by structural excitation provides dominant contributions to the total sound power.  (3) When optimizing the major transfer path, the changes in the contribution of other transfer paths should also be considered. The situation that the target response does not decrease after optimization can be avoid and every optimization for transfer paths will be effective.

Availability of data and materials
All data generated or analyzed during this research process are included in this manuscript.