The passenger compartment of high-speed trains is a relatively enclosed environment, where internal vibration and noise often involve multiple excitation sources that propagate through various paths, attenuate within the car, and form a complex acoustic environment after multiple reflections. OTPA is developed based on traditional TPA, and its expression is:

Y(jω) = H(jω) × (jω) (1)

In this equation, Y(jω) represents the output vector at the response point; X(jω) represents the input vector at the excitation point; H(jω) represents the operational transfer function matrix. In the analysis of noise and vibration transfer paths, the input from the excitation source often includes vibration, force, or sound pressure signals. Therefore, the input and output variables in the above equation can be expressed in the following form:

Y = [ a y f y p y ] , X = [ a x f x p x ] (2)

where, a y = ( a y ( 1 ) ⋯ a y ( k ) ) , a x = ( a x ( 1 ) ⋯ a x ( l ) ) are the output/input acceleration vector; f y = ( f y ( 1 ) ⋯ f y ( m ) ) , f x = ( f x ( 1 ) ⋯ f x ( n ) ) are the output/input force vectors; p y = ( p y ( 1 ) ⋯ p y ( o ) ) , p x = ( p x ( 1 ) ⋯ p x ( p ) ) are the output/input sound pressure vectors; k, l, m, n, o and p represent the number of measurement points for each type of physical quantity.

Therefore, in OTPA, multiple physical variables of all excitation sources during one experiment operation can be taken as inputs vector, and the transfer relationship between the source and response point can be established:

[ y ( 1 ) ⋯ y ( n ) ] = [ x ( 1 ) ⋯ x ( m ) ] [ H 11 ⋯ H 1 n ⋮ ⋱ ⋮ H m 1 ⋯ H m n ] (3)

where m and n are respectively the number of input and response measurement variables.

By measuring the physical quantities of the excitation and response points during experiment, it is convenient to construct the input and response sample matrices. Therefore, the transfer function matrix of the system under operation conditions can be easily solved based on Equation (1).

H = X − 1 Y (4)

The H matrix is the matrix that links the measured physical quantities of various input sources under operating conditions to the output response points. Through this matrix, it is possible to quantitatively analyze the contributions of each source/transfer path to the output response points.

The high-speed train which was tested consists of 8 cars, using a distributed power unit with 4 powered cars and 4 trailer cars, with auxiliary equipment located at the bottom of the vehicles. To study the problem of cabin interior noise, it is necessary to first determine the noise level in the target analysis area. This study focuses on the interior noise at the end of the middle car of the train, with measurement points located on the centerline of the longitudinal axis of the passenger compartment, at a height of 1.2 m from the passenger compartment floor. The measured quantity is the A-weighted equivalent continuous sound pressure level, with a measurement time of more than 20 seconds. This experiment was conducted on a concrete ballastless track slab, and on a straight track. A-weighted equivalent continuous sound pressure level, in decibels, given by:

L p A e q , T = 10 ⋅ log ( 1 T ∫ 0 T p A 2 ( t ) p 0 2 ) (5)

where p_A is the A-weighted sound pressure and T is the measurement time interval. dB reference level is 2 × 10⁻⁵ Pa [5].

When the vehicle’s operating speed increased from 350 km/h to 400 km/h, the interior noise level increased by approximately 3 dB, as shown in Figure 1. It can be observed in the figure that within the 1/3 octave band of a center frequency of 50 - 10 kHz, the interior noise increased by 2 - 3 dB, with the main peak frequency range concentrated in the 100 - 315 Hz band. The noise within the 20 - 400 Hz band was analyzed using Fast Fourier Transform (FFT) and its frequency spectrum is shown in Figure 2. As seen in the figure, there was a single-frequency peak noise at 154 Hz when operating at 350 km/h. When the speed increases to 400 km/h, the interior noise of 100 - 200 Hz become prominent, with single-frequency peak noise observed at 176 Hz and 194 Hz, which is related to factors such as the vehicle’s diameter, track sleeper spacing, and motor drive ratio.

The interior noise can be divided into 2 main forms based on the energy transmission path of the excitation source (including vibration source and noise source): airborne sound transmission and structure-borne sound transmission. Airborne sound transmission refers to the path through which the excitation source is transmitted through the air fluid medium, mainly related to the noise source intensity, the sealing performance of the car body, and the sound insulation characteristics of the trim panels. Structure-borne sound transmission refers to the path through which the excitation source is transmitted through the structural solid medium, mainly related to the vibration source intensity, the structural characteristics of the car body, and the vehicle suspension parameters [6]. In this paper, the vehicle was tested under unloaded conditions, with an operating speed of 400 km/h. The noise from the bogie under the vehicle is equivalent to the aerodynamic noise of the sidewalls and roof. Therefore, both airborne sound transmission and structure-borne sound transmission have a significant impact on the interior noise of the car. As a result, both types of noise sources need to be tested.

According to the analysis requirements in Section 2.2, the region above the bogie at the end of the high-speed train is determined as the OTPA target analysis area, as shown in Figure 3. In order to determine the level of the external noise source and the contribution of the trim to the target point, 12 vibration accelerometers are arranged on the rim of the target analysis area. The locations of the measuring points include the passenger compartment floor, water deflector, side wall under the window, window, luggage rack top plate, and middle top plate, as shown in Figure 4. The detailed locations are shown in Table 1.

Regarding the external noise source intensity, we tested the vibration and noise of the bogie area, the side wall noise, and the roof noise. The source intensity of the bogie area includes the vibration and noise of the motor and gearbox equipment, as well as the vibrations in the three Cartesian coordinate directions of the center pin, air spring, anti-yaw vibration absorber, and the axle box, as well as the wheel-rail noise. Please refer to Figure 5 for details.

In order to obtain an accurate and reliable OTPA model, the experimental test conditions include acceleration, constant speed, coasting, and braking conditions. Each condition undergoes 2 - 3 sets of tests. The data obtained under these different conditions are subjected to Crosstalk Cancellation (CTC) interference correction and Principal Component Analysis (PCA) analysis to determine the contribution of each path. Then, the transfer path matrix is obtained through the TPS (Transfer Path Analysis) process to identify the main noise sources, as shown in Figure 6.

The calculation results of the interior noise level are synthesized by the vibration acceleration data of the passenger compartment ceiling, side roof panel, window partition wall panel, window glass, lower wall panel, drip shield, floor, and compared with the interior test results, as shown in Figure 7. The total value of the experiment is 72.8 dB(A), while the synthesized result is 72.3 dB (A), with a calculation error of 0.5 dB. The spectral consistency within the main peak frequency range of 100 - 315 Hz shows a good agreement between calculation and experimental results.

The prominent frequency range of the interior noise is within the 1/3 octave band with a center frequency of 100 - 315 Hz. The total value of the panel contribution indicates that the lower sidewall, floor, and roof panel make significant contributions to this frequency band. Further analysis of the frequency results reveals that the lower sidewall contributes significantly in the 20 - 40 Hz, while the interior floor contributes more prominently in the 100 - 200 Hz band. See Figure 8 for details.

Based on the analysis results in section 3.2, the vibration of the floor contributes significantly to the interior noise within the 100 - 200 Hz frequency band.

Therefore, the vibration of the interior floor V3 is selected as the target response point for OTPA analysis. The main sources of interior floor vibration include:

(1) Vibration and noise from the bogie and traction system are transmitted through the center pin, air suspension, and anti-yaw vibration absorber to the car body, and then transmitted to the interior floor.

(2) Aerodynamic noise from the exterior sidewalls and roof is transmitted through the car body to the interior floor.

Comparing the synthesis results of OTPA with the experimental test results, it can be seen that the main spectral characteristics of the synthesized results match well with the test results. This indicates that the synthesized results can effectively reflect the transfer characteristics of the target vibration point, as shown in Figure 9, dB reference level is 1 μm/s².

In Figure 10, the x direction represents the longitudinal direction of the vehicle, the running direction is positive, and vice versa is negative. The y direction represents the lateral direction of the vehicle, pointing to the left is positive, and vice versa is negative. The z direction represents the vertical direction of the vehicle, pointing to the roof is positive, and vice versa is negative. The detailed locations of each region in Figure 10 are shown in Table 2.

From the results shown in Figure 10 and Figure 11, it is evident that the vibration of the cabin floor is primarily caused by structural vibration, with airborne noise contributing minimally to floor vibration. The contribution of bogie to the vibration of the end cabin floor through dampers is in the following order: center pin > air suspension > anti-yaw absorber.