A method for reconstructing continuous wheel-rail forces based on discrete fastener reaction forces

Abstract

This invention provides a method for reconstructing continuous wheel-rail forces based on discrete fastener reaction forces, comprising the following steps: Step 1, establishing a wheel load distribution model; Step 2, establishing an algorithmic framework for the inversion method, by introducing a responsibility region and a local deconvolution strategy, mapping the discrete fastener reaction force FRF to the continuous wheel-rail force WRF. This invention effectively solves the problems of traditional onboard sensors' difficulty in long-term fixed-point monitoring of specific road sections, and the inability of existing ground discrete monitoring points to fully characterize the continuous wheel-rail interaction behavior. By constructing an inversion framework based on Gaussian load distribution using a "responsibility region + local deconvolution" approach, it fully utilizes the load transfer characteristics of the vehicle-track coupling system, achieving high-precision reconstruction of continuous wheel-rail forces using ground discrete fastener reaction force data. This eliminates the need for complex global iterative optimization and can accurately restore the dynamic characteristics of wheel-rail contact even with only a limited number of discrete measurement points.

Description

Technical Field

[0001] This invention relates to a method for reconstructing continuous wheel-rail forces based on discrete fastener reaction forces. Background Technology

[0002] Continuous wheel-rail force (WRF) is the dynamic contact force between the wheel and rail during high-speed train operation, typically exhibiting relatively stable oscillating characteristics. However, anomalies in its amplitude or pattern often indicate potential defects in the track structure, which can even threaten operational safety in severe cases. During long-term operation, track structure deterioration phenomena, such as rail wear and uneven track settlement or deformation, frequently trigger abnormal fluctuations in WRF. Therefore, continuous real-time monitoring and analysis of WRF in specific sections can detect track defects early, reveal their evolutionary characteristics, and provide support for preventative maintenance, thereby preventing defects from escalating into serious damage.

[0003] However, effectively measuring wheel-rail forces has long been a major challenge in railway engineering. The most common method is to install sensors on the train to collect wheel-rail force data during train operation. For example, existing technologies use strain sensors mounted on axle boxes to acquire dynamic responses, enabling real-time measurement and defect detection of wheel-rail forces. Similarly, technologies have developed onboard detection frameworks based on axle box acceleration sensors, capable of identifying wheel-rail force changes caused by polygonal wear of the wheels through vibration signals. These methods can accurately capture wheel-rail force data during train operation. However, a fundamental limitation is that trains carrying instruments do not always operate on specific sections of track, making it difficult to achieve long-term, real-time monitoring of continuous wheel-rail forces in specific track sections. Furthermore, given the dynamic adjustments to train timetables and the complexity of operational scheduling, relying solely on onboard sensors cannot fully meet the needs for real-time assessment of track conditions and early warning of structural defects in specific sections. Summary of the Invention

[0004] Purpose of the invention: The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a method for reconstructing continuous wheel-rail force based on discrete fastener reaction force, comprising the following steps:

[0005] Step 1: Establish a wheel load distribution model;

[0006] Step 2: Establish the algorithmic framework of the inversion method. By introducing the responsibility region and local deconvolution strategy, the discrete fastener reaction force (FRF) is mapped to the continuous wheel-rail force (WRF).

[0007] Step 1 includes: when the wheel acts directly on the fastener system, the fastener only bears part of the wheel-rail force WRF, and the remaining load is distributed to several adjacent fastener systems;

[0008] Calculation of wheel load distribution relationship: Based on the Winkler elastic foundation assumption, a wheel load distribution model is constructed. In the wheel load distribution model, the rail is simplified as an infinitely long beam, continuously supported by a fastening system. According to the Euler-Bernoulli beam bending equation, the longitudinal distribution of the uniformly distributed rail-foundation reaction force P satisfies equations (1) and (2):

[0009] (1),

[0010] (2),

[0011] Where e represents the natural constant, Q is the wheel load, and x represents the wheel position. Let E be the position of the fastener support, E be the elastic modulus of the rail, I be the moment of inertia of the rail cross section, L be the length of the rail, and k be the stiffness coefficient of the continuous elastic foundation.

[0012] Step 1 also includes: integrating the reaction force P between the rail and the track bed within the range of a single sleeper to obtain the frequency response function F:

[0013] (3),

[0014] (4),

[0015] Where d is the integral symbol; It is the support spacing; It is the distance between two adjacent fasteners along the longitudinal direction. Different fastener systems The value varies; for CRTS III type slab track, it is typically 0.63m, while for some (double-block slab track) it is 0.65m. This value directly affects the wheel load distribution factor.

[0016] Step 1 also includes: introducing a wheel load distribution coefficient C to characterize the load proportion borne by a single fastener, as shown in equation (5):

[0017] (5),

[0018] The time history curve of wheel load distribution is fitted using a Gaussian function, and the expression is as follows:

[0019] (6),

[0020] Where A and w are the fitted shape parameters. In this scenario, A can be understood as the peak value of the Gaussian function, which is the maximum load proportion borne by a single fastener system; w controls the width of the Gaussian curve, which represents the attenuation rate of load transfer to adjacent fasteners;

[0021] Step 1 also includes: For high-speed trains with two or more carriages, the wheel load distribution time history curve Q is obtained at any fastener support position by superimposing equations (7) and (8):

[0022] (7),

[0023] (8),

[0024] in, This represents the interaction force between the i-th wheel of the o-th carriage and the rail. represents the wheel weight distribution corresponding to the i-th wheel of the o-th car, n is the number of cars in the train, s is the distance between the centers of two adjacent cars, m is the distance between the centers of two bogies in a single car, and r is the wheelbase in a single bogie. It is the result of superposition.

[0025] Step 1 also includes: each fastener support is responsible for load inversion within a defined area of ​​responsibility, with the area of ​​responsibility extending to both sides from the center of the support. The distance is such that the entire rail is seamlessly covered without overlap;

[0026] For any position g within the area of ​​responsibility, the actual wheel-rail force F(g) is calculated according to the formula... The data is reconstructed, where Y(t) is the measured reaction force of the fastener, C(d) is the corresponding Gaussian load distribution coefficient, d is the distance from position g to the support, and t is the time vector (time series), representing the timestamp corresponding to each sampling point during the data acquisition process.

[0027] Step 2 includes:

[0028] For a given FRF time history dataset The first column of the FRF time history dataset matrix represents time, and the subsequent N columns correspond to the frequency domain response function (FRF) of adjacent and equally spaced fastener supports. Represents the real number space; T is the total number of time sampling points (number of rows in the matrix), representing how many time points of data were recorded in a single acquisition process; N is the total number of fastener supports (number of columns in the matrix), representing how many adjacent fastener supports were monitored simultaneously (number of sensor channels).

[0029] sampling frequency The spatiotemporal mapping step size is calculated from the median difference of the time series. Then according to the formula Determined, where v is the train speed;

[0030] The coordinates of the j-th support Represented as j=1,...,N;

[0031] The half-width h of the responsibility area is defined as ;

[0032] Subsequently, in each support channel, the fastener reaction force FRF value of the j-th fastener support point at time t is calculated. Among them Let Y represent the reaction force data of the j-th column in the dataset matrix Y, i.e., the j-th support, at all time points.

[0033] Locate the first peak and construct the threshold thr according to equation (9):

[0034] (9),

[0035] in This is the reaction force signal channel for the j-th support. The peak prominence coefficient (set to 0.4 in this invention) is used to set the threshold sensitivity for peak detection. It is a median function.

[0036] Step 2 also includes:

[0037] By combining threshold determination and local maximum strategy, the arrival time of each wheel when it passes the support is obtained. If a support fails to detect a peak value, the missing value is filled by geometric extrapolation under uniform motion, as shown in equation (10):

[0038] (10)

[0039] in It is the arrival time when the j-th fastener support detects the passing of the wheel. This is a reference time. It is the coordinate position of the j-th support. This is a reference position;

[0040] Next, construct a continuous spatial grid. To cover the entire track segment, as shown in equation (11):

[0041] (11),

[0042] in These represent the starting and ending coordinates of a continuous spatial grid, respectively. These represent the coordinates of the first support and the Nth (last) support, respectively. It is the distance step length of the spatial grid; It is the spatiotemporal mapping step size, calculated by dividing the train speed v by the sampling frequency f. s Calculated;

[0043] Subsequently, the core inversion is performed by identifying the responsibility support j of each spatial node s, as defined in equation (12):

[0044] (12),

[0045] Where floor is the floor function;

[0046] Calculate the time corresponding to position s according to formula (13). :

[0047] (13)

[0048] in Indicates the center coordinates of the j-th (responsible) support;

[0049] The frequency domain response function at this point is obtained through linear interpolation. :

[0050] (14)

[0051] Where interp represents linear interpolation, used to calculate the time... Obtain the corresponding reaction force value from the discrete data Y;

[0052] Based on the Gaussian load distribution kernel, the continuous wheel-rail force F(s) is reconstructed as:

[0053] (15)

[0054] in Indicates based on the current position s and the support distance The calculated Gaussian load distribution coefficients;

[0055] For a given reference point The timeline is represented as follows:

[0056] (16)

[0057] Where t refers to the time history obtained from the reconstruction;

[0058] Then, a forward reconstruction verification was performed on three representative supports, namely the first, middle, and last supports. At each support, the predicted reaction force was calculated according to equation (17). :

[0059] (17)

[0060] Finally, the results were compared with the measured data. A comparison is made to verify the consistency of the inversion.

[0061] The present invention also provides an electronic device, including a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method.

[0062] The present invention also provides a storage medium storing a computer program or instructions that, when the computer program or instructions are run on a computer, execute the steps of the method described.

[0063] This method proposes to measure the dynamic strain of the track pads during high-speed train operation, convert it into fastener reaction force (FRF), and reconstruct the wheel-rail force (WRF) along the track section by combining wheel load distribution relationships with an inversion framework. This method enables real-time monitoring of continuous wheel-rail forces in specific sections, thereby supporting early detection of deterioration signs and preventative maintenance.

[0064] This invention establishes a complete "theory-simulation-verification" research framework for constructing a continuous WRF from a discrete FRF. First, a Gaussian load distribution framework is systematically proposed, clarifying the core concepts of the inversion process. Then, a vehicle-track coupled dynamic model is developed based on the finite element method, where the calculated FRF serves as the input data for the inversion framework. This method realizes the transformation from FRF to WRF, thereby establishing a mapping relationship between FRF and continuous WRF, and achieving cross-scale inversion from local strain to global load. Finally, typical track degradation scenarios are introduced into the verified model, and the proposed method is applied to perceive and evaluate the corresponding responses. Results show that this method achieves significant progress in overcoming the dependence of traditional WRF monitoring on onboard equipment, providing a novel technical path for online assessment of track structure health.

[0065] Compared with existing technologies, this invention has the following advantages: It effectively solves the problems of traditional vehicle-mounted sensors' difficulty in long-term fixed-point monitoring of specific road sections and the inability of existing ground-based discrete monitoring points to fully characterize continuous wheel-rail interaction behavior. By constructing a "responsibility region + local deconvolution" inversion framework based on Gaussian load distribution (GLDF), it fully utilizes the load transfer characteristics of the vehicle-track coupling system to achieve high-precision reconstruction of continuous wheel-rail force (WRF) using ground-based discrete fastener reaction force (FRF) data. This eliminates the need for complex global iterative optimization and can accurately restore the dynamic characteristics of wheel-rail contact even with only a limited number of discrete measurement points. Its core linear non-iterative spatiotemporal mapping algorithm can adaptively match train speed and measurement point spacing. Combined with the accurate fitting of the Gaussian function to the single fastener load sharing ratio (approximately 36.42%), it fully exploits the spatial correlation between discrete measurement points, ensuring high-fidelity waveform reconstruction (correlation coefficient > 0.9) within the dominant frequency band of 1-100Hz. Furthermore, this method exhibits excellent robustness. Even under extreme conditions where track fasteners suffer severe deterioration (such as reduced stiffness), it can still maintain extremely high inversion accuracy (maximum error less than 10%) and keenly capture abnormal load fluctuations. This enables precise location and early warning of track defects, eliminating the reliance on onboard equipment and requiring only ground sensors to meet the needs of long-term online health monitoring at engineering sites. Attached Figure Description

[0066] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.

[0067] Figure 1 This is a schematic diagram of the test methods for discrete fastener reaction force (FRF) and continuous wheel-rail force (WRF).

[0068] Figure 2 This is a schematic diagram of the orbital structure theory.

[0069] Figure 3 This is a schematic diagram of the wheel load distribution ratio of the fastener system.

[0070] Figure 4 This is a schematic diagram of the Gaussian load distribution model.

[0071] Figure 5 This is a diagram illustrating the principle of the wheel load distribution characteristics based on the Gaussian model and the "responsibility region + local deconvolution" inversion strategy.

[0072] Figure 6 A flowchart illustrating the inversion framework.

[0073] Figure 7 This is a schematic diagram of a vehicle-track coupled dynamics model considering the iron pad.

[0074] Figure 8 The finite element model diagram is a schematic diagram of the inversion accuracy under the condition of damage.

[0075] Figure 9 This is a schematic diagram of a vehicle model.

[0076] Figure 10 It shows the on-site test process and a comparison chart of on-site test data and simulation data.

[0077] Figure 11 This is a schematic diagram of the wheel-rail interaction force results in the first carriage of the simulation model.

[0078] Figure 12 This is a schematic diagram illustrating the effectiveness evaluation results of the method proposed in this invention.

[0079] Figure 13 This is a schematic diagram comparing the continuous WRF reconstructed using a Gaussian load distribution frame with the actual WRF extracted from the model.

[0080] Figure 14 This is a schematic diagram of the actual WRF and the WRF reconstructed based on the Gaussian inversion method. Detailed Implementation

[0081] like Figure 1 As shown, this embodiment of the invention provides a method for reconstructing continuous wheel-rail forces based on discrete fastener reaction forces, including the following steps:

[0082] Step 1: Establish a wheel load distribution model;

[0083] When the wheel acts directly on the fastener system, the fasteners only bear a portion of the wheel load (WRF), while the remaining load is distributed among several adjacent fastener systems. The wheel load distribution relationship is calculated as follows: a wheel load distribution model is constructed based on the Winkler elastic foundation assumption. In this model, the rail is simplified as an infinitely long beam continuously supported by the fastener system, such as... Figure 2 As shown. According to the Euler-Bernoulli beam bending equation, the longitudinal distribution of the uniformly distributed rail-foundation reaction force P satisfies equations (1) and (2):

[0084] (1),

[0085] (2),

[0086] Where Q represents wheel load and x represents wheel position. Here, E represents the position of the fastener support, E is the elastic modulus of the rail, I is the moment of inertia of the rail cross section, and k is the stiffness coefficient of the continuous elastic foundation. Figure 2 As shown. The specific parameter values ​​are listed in Table 1.

[0087] Table 1

[0088]

[0089] The typical value of a type of sleeper is =0.63 meters. Under this condition, the reaction force P between the rail and the track bed within the range of a single sleeper is integrated to obtain the frequency response function F, as shown in equation (3):

[0090] (3),

[0091] (4),

[0092] A load distribution factor C is introduced to characterize the proportion of load borne by a single fastener, as shown in equation (5):

[0093] (5),

[0094] Substituting the parameters in Table 1 into equations (1) to (5), the wheel load distribution ratio at the wheel position is 36.42%. Furthermore, the function curve obtained from equation (5) is extremely similar to the Gaussian function. Therefore, the Gaussian function is used to fit the wheel load distribution time history curve, and the expression is:

[0095] (6),

[0096] The comparison results with theoretical calculations are shown below. Figure 3 .

[0097] For high-speed trains with multiple carriages, the wheel load distribution time history curve Q can be obtained at any fastener support position by superimposing equations (7) and (8), such as Figure 4 As shown in (b), the wheel load distribution time history curves for nine consecutive fastener support points are displayed:

[0098] (7),

[0099] (8),

[0100] in, This represents the interaction force between the i-th wheel of the j-th carriage and the rail. This represents the wheel load distribution corresponding to the i-th wheel of the j-th car, n is the number of cars in the train, s is the distance between the centers of two adjacent cars, m is the distance between the centers of two bogies within a single car, and r is the wheelbase within a single bogie. See the diagram for parameter values. Figure 4 (a) and Table 2.

[0101] Table 2

[0102]

[0103] Figure 4 The core principles of the wheel load distribution and inversion method based on the Gaussian model are demonstrated. Figure 5 Figure (a) shows the Gaussian distribution of a single wheel load along the rail: the fastener system directly beneath the wheel bears approximately 36% of the maximum load, while the transmitted load decreases exponentially with distance, with adjacent fasteners bearing approximately 25%, 9%, and 1% of the load, respectively. This pattern reflects the load transfer mechanism of the rail as a continuous beam. Figure 5 (b) illustrates the inversion algorithm based on the concept of "responsibility region + local deconvolution". Each fastener support is responsible for load inversion within its defined responsibility region, which extends to both sides from the center of the support. The distance is determined to ensure that the entire rail is seamlessly covered without overlap. For any position s within the responsibility area, the actual wheel-rail force F(s) can be calculated using the formula... The data is reconstructed, where Y(t) is the measured reaction force of the fastener, C(d) is the corresponding Gaussian load distribution coefficient, and d is the distance from the location to the support. By introducing peak detection and the established time-space mapping, discrete time-domain measurement data are transformed into continuous spatial load distribution, thereby reconstructing the continuous wheel-rail contact force from finite frequency response function measurements.

[0104] Step 2: Establish the algorithmic framework of the inversion method. By introducing the responsibility region and local deconvolution strategy, the discrete FRF is mapped to the continuous WRF.

[0105] For a given FRF time history dataset (Where the first column represents time, and the subsequent N columns correspond to the FRF of adjacent and equally spaced fastener supports), this process first involves data preprocessing. Sampling frequency The spatiotemporal mapping step size is calculated from the median difference of the time series. Then according to the formula Determined, where v is the train speed. The support spacing is denoted as... The coordinates of each support can be expressed as: j=1,...,N; the half-width of the responsibility region is defined as Subsequently, in each support channel In the middle, locate the first peak and construct the threshold thr according to equation (9):

[0106] (9),

[0107] in ;

[0108] By combining the "threshold determination + local maximum" strategy, the arrival time of each wheel when it passes the support can be obtained. If a peak value cannot be detected for a specific support, the missing value is supplemented by geometric extrapolation under uniform motion, as shown in equation (10):

[0109] (10)

[0110] Next, construct a continuous spatial grid. To cover the entire track segment, as shown in equation (11):

[0111] (11),

[0112] Subsequently, the core inversion is performed by identifying the “responsibility support” j of each spatial node s, as defined in equation (12):

[0113] (12),

[0114] Time corresponding to position s Calculate according to formula (13):

[0115] (13)

[0116] Next, as shown in equation (14), the FRF at this time can be obtained through linear interpolation. :

[0117] (14)

[0118] Based on the Gaussian load distribution kernel, the continuous WRF F(s) can be reconstructed into equation (15), thus establishing the mapping relationship from the discrete FRF to the continuous WRF at position s:

[0119] (15).

[0120] By scanning along s, a complete distribution F(s) covering the entire support region can be obtained. To maintain consistency with subsequent time-domain analysis, this spatial distribution is mapped back to the time axis. For a given reference point... This relationship is shown in equation (16), thus yielding the time history representation F(t):

[0121] (16)

[0122] Then, positive reconstruction verification was performed at three representative supports (the first, middle, and last). At each support, the predicted reaction force was calculated according to Equation (17). Finally, the results were compared with the measured data. A comparison is made to verify the consistency of the inversion.

[0123] (17)

[0124] This framework is developed based on the concept of "region of responsibility + local deconvolution," transforming the estimation of continuous force fields into a problem of local deconvolution at a series of discrete points. Its algorithm is linear and deterministic, requiring no iterative computation, thus ensuring good numerical stability and engineering feasibility. The complete algorithm framework is as follows: Figure 6 As shown.

[0125] Step 3: Based on the finite element method, a vehicle-track coupled dynamic model was established and validated using field test data. In the subsequent description, this model will be used to provide the required frequency response function input for the inversion framework, and validation will be performed using WRF data. Following this, a comparative analysis will be conducted to evaluate the feasibility of the proposed inversion method.

[0126] Step 3.1, Overview of the vehicle-track coupling model;

[0127] A theoretical schematic diagram of the vehicle-track coupled dynamics model including the iron pad is shown below. Figure 7 As shown in the diagram, the vehicle subsystem is represented as a multi-degree-of-freedom, multi-rigid-body system moving at a constant speed along the track structure, consisting of a car body, two bogies, and four wheelsets. Each wheelset is connected to the bogie via a primary suspension, while each bogie is connected to the car body via a secondary suspension. The track subsystem includes rails, a fastening system, track slabs, a self-compacting concrete layer, a base, a roadbed, a foundation, and underlying soil. In the fastening system, rail pads and elastic pads are modeled using spring-damped elements and connected to the rails and track slabs, respectively. and These represent the equivalent stiffness and damping of the rail pad, respectively. and This indicates the equivalent stiffness and damping of the elastic pad.

[0128] Step 3.2, build the model;

[0129] Reference "WAN Z. Study on the mechanism of mud pumping in the roadbed and its remediation of ballastless high-speed railway [D / OL]. Zhejiang University, 2022" points out that in numerical simulations, the additional impact of train formations from the fourth to the tenth carriages on the maximum vertical dynamic response of the vehicle-track-roadbed system has become limited. Therefore, this embodiment selects a four-car train formation as the research object for subsequent analysis. To ensure that multi-car trains have sufficient operating range, 42 track slabs are arranged longitudinally, making the total track length reach 238.07 meters. Figure 8 As shown.

[0130] Figure 9 Figure (a) shows a schematic diagram of the dimensions of the CRH3 high-speed train, which is one of the main train types currently operating in China. In the finite element (FE) model, the length of a single car is uniformly set to 25 meters, the center distance between two bogies within the same car is 17.5 meters, the center distance between two adjacent cars is 7.5 meters, and the wheelbase within the bogie is 2.5 meters. Key parameters are listed in Table 3. Figure 9 Figure (b) shows the established finite element model. In the vehicle subsystem, the axle load is set to 15 tons, and the suspension system is modeled using spring-damping elements. In the track subsystem, the material parameters of each component are shown in Table 4. In particular, the iron pad in the fastener system is modeled using beam elements, a modeling method that has been validated in the literature "WANGM, LI P, LI S, et al. A refined track dynamic model considering the bending properties of iron pad: Proposal and validation[J / OL]. Engineering Failure Analysis, 2024, 165: 108780".

[0131] Table 3

[0132]

[0133] Table 4

[0134]

[0135] 3.3 Wheel-rail interaction

[0136] In the finite element modeling of a vehicle-track coupled system, the spatial wheel-rail contact geometry and contact point locations are first determined. Applying Hertzian nonlinear elastic contact theory, the perpendicular wheel-rail force is expressed as:

[0137] (18)

[0138] Where G is the wheel-rail contact constant ( ), (Unit: m) represents the elastic compression between the wheel and rail. For a wear-type tread wheel, the wheel-rail contact constant is given by equation (19):

[0139] (19)

[0140] Where R represents the wheel radius (m).

[0141] Elastic compression This also includes the static compression of the wheel, which is determined by the displacement of the wheel and rail at the contact point:

[0142] (20)

[0143] in, (Unit: m) represents the displacement of the j-th wheel at time t. (Unit: m) represents the displacement of the track below the j-th wheel at time t. (Unit: meters) represents the displacement irregularity input. It is worth noting that when... When the wheel-rail contact is lost, the wheel-rail force p(t) becomes 0.

[0144] Therefore, the WRF with displacement irregularity input is represented in equation (21):

[0145] (twenty one),

[0146] The wheel-rail contact stress is given by equation (22):

[0147] (twenty two),

[0148] Where S is the stress constant determined by Hertzian theory;

[0149] exist Within the commonly used range of meters, the value of S is determined using equation (23):

[0150] (twenty three),

[0151] In the vehicle model, different speeds are assigned to the center of mass of the car body, bogie, and wheelset along the train running direction to simulate the operating conditions at different train speeds. The wheel-rail interaction is modeled using a penalty function in the tangential direction, with the friction coefficient set to 0.3; while in the normal direction, it is defined according to Hertz's nonlinear elastic contact theory. For wheels with worn flanges, the wheel-rail contact constant G is taken as the corresponding parameter of the worn flange. The finite element software can implement this function in the "Pressure-Closing" module by defining user-defined table relationships. "Pressure" refers to the contact stress borne on the wheel-rail contact interface, while "closing" represents the relative displacement or indentation depth between the two contacting bodies. It is worth noting that closing is not directly defined in Hertz's theory. Hertz's theory mainly focuses on the contact area and stress distribution of two elastic bodies under load, rather than the specific deformation. In finite element analysis, closing is a key parameter used to describe the degree of compressive deformation generated during the contact process. The specific values ​​of "Pressure" and "closing" can be determined according to formulas (18) to (23).

[0152] Step 3.4: Model the track irregularities;

[0153] Track irregularities are modeled using the irregularity spectrum of ballastless track in my country's high-speed railways, with a spatial frequency range of 0.005~0.5m⁻¹ and a corresponding wavelength range of λ of 2~200 m, as shown in equation (24):

[0154] (twenty four),

[0155] The irregular spectrum is represented by a four-segment piecewise fitting, and the spatial frequency range and fitting coefficients of each segment are listed in Table 5.

[0156] Table 5

[0157]

[0158] After obtaining the piecewise fitted spectrum, the spatial distribution characteristics of track irregularities can be obtained by performing an inverse Fourier transform on the power spectral density function. The power spectral density (PSD) represents the track irregularity, where f is the spatial frequency, and A and k are fitting parameters. Track irregularities are achieved by modifying the nodal coordinates of each cross-section of each rail in the input file, thereby matching the lateral and vertical deformations of the rails at different cross-sections with the irregularity amplitude.

[0159] Step 3.5, Model Validation;

[0160] The model was validated using field test data from recent studies. Fiber Bragg grating (FBG) strain sensors were used as the measurement devices in the tests. During the tests, the FBG sensors were attached to the lower surface of the iron pad in the fastening system to capture the strain response under dynamic train loads, such as... Figure 10 As shown in (a), the test was conducted during the joint commissioning phase of an operational railway, and the test section used CRTS III type slab track. A schematic diagram of the on-site measurement layout is shown below. Figure 10 As shown in (b). The test train was a CRH380AJ high-speed comprehensive inspection train with an axle load of 15 tons and an operating speed of 200 km / h. The traffic conditions of the measurement section are as follows. Figure 10 As shown in (c). After the sensor is installed, the signal cable is led through the gap in the bridge to the safety fence outside the bridge and connected to the demodulator for data acquisition. The sampling frequency is set to 1000 Hz, and the specific arrangement is as follows. Figure 10 As shown in (d), the simulated data was compared with the field test data to verify the reliability of the model. Figure 10 Figure (e) shows the time-domain response under train load, where the red curve represents simulated data and the black curve represents field test data. The two curves show a high degree of consistency in their fluctuation trends and their amplitudes are roughly equivalent. The peak value clearly reflects the frequency response function (FRF) caused by the passage of a single wheelset, primarily ranging from 0 to 35 kN. Meanwhile, the troughs correspond to the loading characteristics of two wheelsets acting together within the same bogie. Furthermore, this embodiment utilizes Fast Fourier Transform (FFT) to convert the time-domain data to the frequency domain and compares the amplitude characteristics of the simulated results with those of the field measurements, such as... Figure 10 As shown in (f). The results show that the model successfully captures multiple fundamental frequencies and their harmonic components, and the peak trends at characteristic frequencies such as 2.38 Hz, 4.38 Hz, 6.37 Hz, and 8.76 Hz exhibit a high degree of consistency between the two models. Overall, the comparison with field tests fully demonstrates the accuracy and reliability of the developed simulation model in predicting the dynamic response of vehicle-track coupled systems.

[0161] For example, this embodiment extracts the wheel-rail force (WRF) of the first carriage in the simulation model, and the results are as follows: Figure 11As shown in (a), WS1 to WS4 correspond to the four axle positions of the vehicle from front to rear. It can be observed that as the train passes, the wheel-rail force exhibits irregular oscillating characteristics, with its amplitude mainly fluctuating between 90 and 105 kN. These results are consistent with the trends and values ​​reported in the literature “LIU H, SONG L, XU L, et al. Identification of wheel-rail forces on high-speed railways based on physical model and hybrid recursive neural networks[J / OL]. Engineering Structures, 2025, 338: 120547”, and comply with the relevant requirements in the Chinese industry standard TB 10761-2024 "Technical Specification for Dynamic Acceptance of High-Speed ​​Railway Engineering". Specifically, the force level is close to the reference value of 120 kN and consistently below the maximum permissible limit of 170 kN. Figure 11 Figure (b) illustrates the wheel-rail force distribution of a four-car train (16 wheelsets). The results show that the vertical force distribution among the wheelsets is relatively concentrated, with a median stable at approximately 96 kN, and the main fluctuation range being 90 to 100 kN. The narrow interquartile range (25% to 75%) indicates stable force fluctuations, with no significant overload or underload observed in any wheelset. Overall, the wheel-rail force distribution of this train formation exhibits good uniformity and high data concentration, fully demonstrating the stability of the vehicle-track coupling dynamics model and that the wheel-rail interaction remains within a reasonable range. This further confirms the robustness and reliability of the developed model.

[0162] Step 4, Method Verification;

[0163] First, frequency response function (FRF) data are extracted from the validated numerical model and reconstructed into a WRF using the GLDF method. The results are then compared with the actual WRF to evaluate the effectiveness of the method under normal operating conditions. Next, a typical track degradation scenario is introduced into the model, and the WRF is reconstructed using the same inversion method. Analysis of the reconstruction results further verifies the accuracy and applicability of the method under track degradation conditions.

[0164] Step 4.1, Inversion Results;

[0165] The effectiveness of the method proposed in this invention was evaluated based on a validated numerical model, and the results are as follows: Figure 12 As shown. Figure 12(a) shows the frequency response function (FRF) extracted from the 41 consecutive fastener systems located at the mid-span of the track as a train passes. Meanwhile, Figure 12 Figure (b) presents the reconstructed continuous waveform response function (WRF). From Figure 12 As can be seen in (a), these frequency response functions exhibit obvious wave-like propagation characteristics under train load, with an amplitude range of 0 to 35 kN, which is highly consistent with the field test results reported in Section 3.5. Each peak corresponds to the dynamic response of a single fastener system under wheel load, while the sequential activation of adjacent fasteners accurately reflects the spatial load distribution characteristics described by the inversion framework.

[0166] The time delay between adjacent responses also matches the train's operating speed of 200 km / h. Figure 12 Figure (b) shows the continuous WRF reconstructed using the proposed method, with an amplitude range of 86 to 106 kN, successfully preserving the dynamic oscillatory characteristics of wheel-rail interaction. Notably, no discontinuities were observed at the "responsibility boundary," indicating the good applicability of the responsibility region partitioning and linear interpolation / spatiotemporal mapping strategy. Furthermore, the smoothness of the time-domain curves demonstrates that the algorithm can effectively reconstruct a continuous WRF field from a discrete FRF.

[0167] To quantitatively evaluate the inversion accuracy of the proposed method, Figure 13 The continuous load factor (WRF) reconstructed using a Gaussian load distribution frame (GLDF, red curve) was compared with the actual WRF extracted from the model (black curve). For example... Figure 13 As shown, during the 2.5-second train passage, the two curves exhibited a high degree of consistency, with fluctuations ranging from 85 to 105 kN, consistent with the typical characteristics of WRF in high-speed trains. The reconstructed curve closely matched the actual curve at its peaks and troughs, and the time-domain characteristics of the dynamic response were well preserved, indicating that the method can accurately capture the instantaneous S-change of WRF. Further calculation yielded a correlation coefficient of 0.91228, fully demonstrating a strong correlation between the reconstructed force and the actual force. Although slight numerical deviations occurred at certain moments, the overall trend remained consistent, with the maximum error (blue curve) consistently below 5%, which is within the acceptable range for engineering applications.

[0168] To further verify the accuracy of the proposed method in the frequency domain, this embodiment performs Fast Fourier Transform (FFT) on the time-domain WRF data to conduct spectral analysis. Figure 14 The results of the actual WRF and the WRF reconstructed based on the Gaussian inversion method (FRF-WRF) were compared. Figure 14 (a) shows the amplitude spectrum. Figure 14 (b) shows the power spectral density (PSD). Figure 14 As shown in (a), within the frequency range of 1–100 Hz, the frequency domain characteristics of the reconstructed results are generally consistent with the frequency domain characteristics of the actual force. In the low-frequency band (1–10 Hz), the two curves almost completely overlap, indicating that the inversion method can accurately capture the main frequency components of the WRF. Although there are local differences in the mid-to-high frequency band (10–100 Hz), the positions of significant peaks are still basically consistent, successfully reflecting the key dynamic characteristics of the wheel-rail system. In particular, the significant peak appearing in the 10–30 Hz frequency band corresponds to the typical excitation frequency generated when the wheel passes through the fastener system, and this key frequency component is effectively identified by the inversion method. In contrast, the differences are more obvious in the frequency band above 100 Hz. This is mainly because the Gaussian load distribution model is based on the quasi-static assumption, and its inherent limitations make it difficult to accurately characterize high-frequency dynamics and the complex dynamic behavior of the fastener system under high-frequency excitation—these complex dynamic characteristics exceed the descriptive capabilities of the simplified model. Figure 14 The power spectral density comparison in (b) further confirms this conclusion. Throughout the frequency range, the two curves show similar trends and energy distribution patterns: they almost completely overlap in the low-frequency range (1-10 Hz), and although there are differences in amplitude in the mid-to-high frequency range (10-100 Hz), the distribution patterns are basically consistent; while above 100 Hz, the inversion accuracy decreases, which is consistent with the results of amplitude spectrum analysis. Although there are some differences in the high-frequency range above 100 Hz, this does not affect the practical application value of the proposed inversion method in track health monitoring for two reasons: (1) Track degradation identification and safety assessment usually focus on the dynamic response characteristics in the low-to-mid frequency range (1-50 Hz), in which the high-frequency components contribute negligibly to the condition assessment; (2) From the perspective of energy distribution, the main energy of WRF is concentrated in the low-to-mid frequency range, while the energy content in the high-frequency range is relatively small, and its impact on the overall mechanical behavior is also relatively limited. Therefore, the proposed method has strong engineering applicability and shows great application potential in the fields of track degradation identification and dynamic response analysis.

[0169] This method provides a Gaussian Load Distribution Framework (GLDF) for reconstructing continuous wheel-rail forces (WRF) from discrete fastener reaction forces (FRF). The method establishes a Gaussian load distribution model and innovatively introduces a "responsibility region + local deconvolution" strategy to map the discrete FRF to a continuous WRF. In a case study involving a 4-car trainset with 378 continuous fasteners, the reconstructed WRF ranged from 86 to 106 kN, with a correlation coefficient of 0.91 with the reference value and a maximum error controlled within 5%. It also accurately captured the main spectral components within the 1 to 100 Hz frequency band. Under track deterioration scenarios, GLDF can still continuously detect abnormal overload and underload events, maintaining a correlation coefficient of 0.88, fully demonstrating its robustness in damage identification. Overall, GLDF requires no onboard instruments, provides a linear and stable inversion scheme, and effectively bridges the gap between discrete FRF and continuous WRF, thereby supporting long-term online monitoring and deterioration diagnosis of the track.

[0170] This invention provides a method for reconstructing continuous wheel-rail force based on discrete fastener reaction force. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principle of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A method for reconstructing continuous wheel-rail force based on discrete fastener reaction force, characterized in that, Includes the following steps: Step 1: Establish a wheel load distribution model; Step 2: Establish the algorithmic framework of the inversion method. By introducing the responsibility region and local deconvolution strategy, the discrete fastener reaction force FRF is mapped into the continuous wheel-rail force WRF.

2. The method according to claim 1, characterized in that, Step 1 includes: when the wheel acts directly on the fastener system, the fastener only bears part of the wheel-rail force WRF, and the remaining load is distributed to several adjacent fastener systems; Calculation of wheel load distribution relationship: Based on the Winkler elastic foundation assumption, a wheel load distribution model is constructed. In the wheel load distribution model, the rail is simplified as an infinitely long beam, continuously supported by a fastening system. According to the Euler-Bernoulli beam bending equation, the longitudinal distribution of the uniformly distributed rail-foundation reaction force P satisfies equations (1) and (2): （1）， （2）， Where e represents the natural constant, Q is the wheel load, and x represents the wheel position. Let E be the position of the fastener support, E be the elastic modulus of the rail, I be the moment of inertia of the rail cross section, L be the length of the rail, and k be the stiffness coefficient of the continuous elastic foundation.

3. The method according to claim 2, characterized in that, Step 1 also includes: integrating the reaction force P between the rail and the track bed within the range of a single sleeper to obtain the frequency response function F: （3）， （4）， Where d is the integral symbol; It refers to the support spacing.

4. The method according to claim 3, characterized in that, Step 1 also includes: introducing a wheel load distribution coefficient C to characterize the load proportion borne by a single fastener, as shown in equation (5): （5）， The time history curve of wheel load distribution is fitted using a Gaussian function, and the expression is as follows: （6）， Where A and w are the fitted shape parameters.

5. The method according to claim 4, characterized in that, Step 1 also includes: For high-speed trains with two or more carriages, the wheel load distribution time history curve Q is obtained at any fastener support position by superimposing equations (7) and (8): （7）， （8）， in, This represents the interaction force between the i-th wheel of the o-th carriage and the rail. represents the wheel weight distribution corresponding to the i-th wheel of the o-th car, n is the number of cars in the train, s is the distance between the centers of two adjacent cars, m is the distance between the centers of two bogies in a single car, and r is the wheelbase in a single bogie. It is the result of superposition.

6. The method according to claim 5, characterized in that, Step 1 also includes: each fastener support is responsible for load inversion within a defined area of ​​responsibility, with the area of ​​responsibility extending to both sides from the center of the support. The distance; For any position g within the area of ​​responsibility, the actual wheel-rail force F(g) is calculated according to the formula... Reconstruction is performed, where Y(t) is the measured reaction force of the fastener, C(d) is the corresponding Gaussian load distribution coefficient, d is the distance from position g to the support, and t is the time vector.

7. The method according to claim 6, characterized in that, Step 2 includes: For a given FRF time history dataset The first column of the FRF time history dataset matrix represents time, and the subsequent N columns correspond to the frequency domain response function (FRF) of adjacent and equally spaced fastener supports. Represents the real number space; T is the total number of time sampling points; N is the total number of fastener supports; sampling frequency The spatiotemporal mapping step size is calculated from the median difference of the time series. Then according to the formula Determined, where v is the train speed; The coordinates of the j-th support Represented as j=1,...,N; The half-width h of the responsibility area is defined as ; Subsequently, in each support channel, the fastener reaction force FRF value of the j-th fastener support point at time t is calculated. Among them Let Y represent the reaction force data of the j-th column in the dataset matrix Y, i.e., the j-th support, at all time points. Locate the first peak and construct the threshold thr according to equation (9): （9）， in This is the reaction force signal channel for the j-th support. Peak prominence coefficient; It is a median function.

8. The method according to claim 7, characterized in that, Step 2 also includes: By combining threshold determination and local maximum strategy, the arrival time of each wheel when it passes the support is obtained. If a support fails to detect a peak value, the missing value is filled by geometric extrapolation under uniform motion, as shown in equation (10): （10）， in It is the arrival time when the j-th fastener support detects the passing of the wheel. This is a reference time. It is the coordinate position of the j-th support. This is a reference position; Next, construct a continuous spatial grid. To cover the entire track segment, as shown in equation (11): （11）， in These represent the starting and ending coordinates of a continuous spatial grid, respectively. These represent the coordinates of the first support and the Nth support, respectively. It is the distance step length of the spatial grid; It is the spatiotemporal mapping step size, calculated by dividing the train speed v by the sampling frequency f. s Calculated; Subsequently, the core inversion is performed by identifying the responsibility support j of each spatial node s, as defined in equation (12): （12）， Where floor is the floor function; Calculate the time corresponding to position s according to formula (13). : （13）， in Indicates the center coordinates of the j-th support; The frequency domain response function at this point is obtained through linear interpolation. : （14）， Where interp represents linear interpolation, used to calculate the time... Obtain the corresponding reaction force value from the discrete data Y; Based on the Gaussian load distribution kernel, the continuous wheel-rail force F(s) is reconstructed as: （15）， in Indicates based on the current position s and the support distance The calculated Gaussian load distribution coefficients; For a given reference point The timeline is represented as follows: （16）， Where t refers to the time history obtained from the reconstruction; Then, a forward reconstruction verification was performed on three representative supports, namely the first, middle, and last supports. At each support, the predicted reaction force was calculated according to equation (17). : （17）， Finally, the results were compared with the measured data. A comparison is made to verify the consistency of the inversion.

9. An electronic device, characterized in that, It includes a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method as described in any one of claims 1 to 8.

10. A storage medium, characterized in that, It stores a computer program or instructions that, when run on a computer, perform the steps of the method as described in any one of claims 1 to 8.

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