Simulation and experimental investigation on dynamic response characteristics of rail connection components

2.1. The composition and working principle

The rail connection components (rail fastening system) for high-speed railways consist of a hierarchically assembled set of mechanical parts, each performing a distinct structural or functional role to secure the steel rail to the sleeper and maintain track geometry, as shown in Fig. 1. At the base, the sleeper serves as the primary foundation, with double-headed bolts embedded in its rail-bearing groove to anchor the entire system and transfer loads. On top of the sleeper, the rail height adjustment shim enables precise vertical tuning of the rail elevation, compensating for construction deviations and operational settlement to ensure rail surface smoothness. Above this shim, the rubber pad acts as a critical elastomeric element, absorbing wheel-rail impact energy, reducing track stiffness, and mitigating vibration transmission. The lateral constraint system is formed by the gauge block seat and gauge block: the seat provides a stable positioning reference, while the gauge block directly abuts the rail base to control track gauge and prevent lateral rail displacement under train-induced horizontal forces. At the top, the serpentine spring, flat washer, and nut constitute the vertical clamping mechanism: tightening the nut compresses the serpentine spring, which in turn exerts a sustained, elastic fastening force on the rail head’s lower jaw, pressing the rail firmly against the rubber pad.

Fig. 1The structure and composition of rail connection components

a) Assembly drawing

b) Exploded view

The working principle of the fastening system relies on the coordinated interaction of these components to achieve stable rail fixation, load transfer, dynamic adaptation, and vibration attenuation. Under static and dynamic train loads, the clamping force from the preloaded serpentine spring secures the rail, while vertical wheel loads are transmitted sequentially from the rail through the rubber pad, height adjustment shim, and finally to the sleeper via the double-headed bolt. The serpentine spring’s inherent elasticity allows it to accommodate rail deformations caused by transient train loads, eliminating stress concentration that would occur with rigid connections. Meanwhile, the rubber pad undergoes controlled compression to dissipate impact energy and reduce vibration, protecting both the track structure and train components from excessive fatigue. The gauge block and seat maintain the track’s lateral geometry, preventing rail shift during curve traversal or braking, while the height adjustment shim ensures long-term track smoothness. Collectively, these mechanisms guarantee the system’s reliability, stability, and durability under the repeated and complex dynamic loads of high-speed railway operation. The rigid lateral constraint system formed by the gauge block and gauge block seat can precisely control the track gauge, resist the centrifugal force generated when trains pass through curves and the lateral force induced by wheel-rail hunting motion, and prevent track gauge widening that would endanger driving safety. The rubber pad absorbs wheel-rail impact energy through its own elastic deformation, achieving vibration and noise reduction, while optimizing the track stiffness matching and reducing damage to the lower foundation caused by wheel-rail interaction forces. The double-headed bolt performs the anchoring function of the entire system, transmitting the vertical wheel load, lateral centrifugal force, longitudinal traction force and braking force borne by the rail connection system. The loads on this rail connection system exhibit significant multi-directional coupling, impact, and alternating fatigue characteristics. During its long-term service, it must simultaneously withstand static wheel loads and dynamic impact loads from trains, lateral centrifugal forces in curved sections, and longitudinal forces generated by train traction and braking.

4.1. Longitudinal force condition

To reveal the deformation and force transmission characteristics of the fastening system under rail longitudinal loads induced by train braking, traction, and other operating conditions, and to evaluate its longitudinal restraint capability and structural reliability, simulations and analyses of the mechanical response of the rail fastening system under longitudinal force conditions were conducted. The load and boundary conditions are shown in Fig. 7(a), where a fixed support is applied to the bottom of the sleeper to completely restrict its displacement and rotation, simulating the rigid support of the lower track bed. A cylindrical support is applied to the bolt shank to limit radial displacement and rotation around the axis, while only allowing tiny axial displacement to restore the assembly state. In terms of loading, a horizontal force of 45 kN is applied along the longitudinal direction of the rail to simulate the longitudinal load transmitted by the train, and an 8 kN preload is applied to the bolt to restore the clamping constraint of the spring on the rail.

The deformation nephogram under the longitudinal force condition is shown in Fig. 7(b). It can be seen that the rail produces directional displacement along the longitudinal direction, with deformation attenuating gradually from the loaded side of the rail head to the rail base. As the main deformation component, the spring reaches a maximum deformation of 9.832 mm, which is concentrated in the curved sections in contact with the bolt and gauge baffle, showing obvious tensile opening and shear deformation characteristics. The deformation distribution presents an asymmetric feature: the deformation of the spring on the loaded side is significantly larger than that on the unloaded side, indicating that when the longitudinal load is transmitted from the rail to the spring, the spring on one side bears the main anti-slip restraint function. Its elastic deformation not only absorbs the energy of the longitudinal load but also maintains the clamping state on the rail. Meanwhile, the deformations of the sleeper and the pad are extremely small, indicating that the fixed support effectively blocks the downward transmission of the load, and the system deformation is mainly dominated by the elastic deformation of the spring. This result verifies the longitudinal restraint mechanism of the fastening system.

The maximum deformation response law of the fastening system under different longitudinal force conditions is shown in Fig. 7(c). It can be seen that as the longitudinal force increases from 25 kN to 65 kN, the maximum deformation of the spring shows an approximately linear growth trend, rising steadily from about 7.0 mm to about 11.9 mm, with a significant positive correlation between the deformation increment and the load increment. The overall slope of the curve is stable without obvious abrupt changes or nonlinear inflection points, indicating that the spring remains in the elastic deformation stage within this load range, without yielding or contact failure, and the coupling relationship between deformation response and longitudinal force is stable. Meanwhile, the growth rate of deformation slows down slightly with increasing load, reflecting the stiffness strengthening effect of the spring under large loads, which is closely related to the geometric nonlinear contact characteristics of the elastic clip and also verifies its deformation bearing capacity under the longitudinal load of the train.

The stress distribution law of the spring under the longitudinal force condition is shown in Fig. 7(d). It can be seen that the maximum equivalent stress reaches 1016.431 MPa, which is concentrated in the curved section of the elastic clip and the contact area with the bolt, forming an obvious high-stress zone, while the stress level in the straight section of the elastic clip is relatively low. The stress distribution presents an asymmetric feature: the stress level of the elastic clip on the loaded side is significantly higher than that on the unloaded side, which echoes the deformation-dominated characteristic of the loaded side in the deformation nephogram, indicating that after the longitudinal load is transmitted through the rail, the anti-slip restraint function is mainly borne by the single-side elastic clip, resulting in local stress concentration.

4.2. Rolling force condition

To obtain the mechanical response of the rail fastening system under conditions such as train curve negotiation and track irregularities, the analysis condition is set as the lateral force load induced by train roll, as shown in Fig. 8(a). Regarding boundary conditions, a fixed support is applied to the bottom of the sleeper to completely restrict its displacement and rotation, simulating the rigid support of the lower track bed. A cylindrical support is applied to the bolt shank to limit radial displacement and rotation about the axis, while only allowing minor axial displacement to restore the actual assembly state of the bolt. In terms of load settings, an 8 kN preload is applied to the bolts on both sides to simulate the initial clamping force of the spring on the rail. Meanwhile, a 30 kN rolling force is applied transversely to the rail to simulate the lateral load caused by train roll, fully reproducing the actual stress state of the fastening system.

Fig. 7Mechanical response analysis under longitudinal force condition

a) Setting of load and boundary conditions

b) Calculation results of deformation

c) Maximum deformation under different longitudinal force conditions

d) Stress distribution of spring

The deformation nephogram under the rolling force condition is shown in Fig. 8(b), which intuitively presents the displacement distribution characteristics of the fastening system. It can be seen that the rail produces a directional lateral displacement toward the load application side, with deformation gradually attenuating from the loaded side of the rail head to the unloaded side. The springs on both sides are the main deformation components, among which the deformation of the spring on the loaded side is significant, with a maximum deformation of 10.583 mm concentrated in the curved sections in contact with the bolt and gauge baffle, showing obvious tensile opening and shear deformation characteristics, while the deformation of the spring on the unloaded side is relatively small. This asymmetric deformation distribution reflects that under the action of the rolling force, the spring on the loaded side bears the main lateral restraint and anti-overturning effect, serving as the main deformation carrier of the system.

The maximum deformations under different rolling force conditions are shown in Fig. 8(c). This curve shows the variation law of the maximum deformations of the left and right springs under different rolling forces. It can be seen that as the rolling force increases from 30 kN to 55 kN, the maximum deformations of the springs on both sides show an approximately linear growth trend. The deformation of the right loaded-side spring increases from about 10.1 mm to about 11.6 mm, and the deformation of the left unloaded-side spring increases from about 9.0 mm to about 10.4 mm. The deformation on the loaded side is always greater than that on the unloaded side, and the difference between the two gradually expands with the increase of the load. The slope of the curve is stable without obvious abrupt changes or nonlinear inflection points, indicating that within this load range, the spring remains in the elastic deformation stage without yielding or contact failure, verifying its stable deformation bearing capacity under the rolling force.

The stress nephogram of the spring under the rolling force condition is shown in Fig. 8(d). The maximum equivalent stress reaches 1238.056 MPa, which is concentrated in the curved sections of the spring in contact with the bolt and rail, forming obvious high-stress zones, while the stress level in the straight sections of the spring is relatively low. The stress distribution presents an asymmetric feature, with the stress level of the spring on the loaded side significantly higher than that on the unloaded side, which echoes the deformation distribution law, indicating that after the rolling force is transmitted through the rail, the lateral restraint function is mainly borne by the spring on the loaded side, resulting in local stress concentration. The high-stress zones completely coincide with the main deformation regions of the spring, indicating that the curved sections are the weak links of the spring under the rolling force condition and potential risk points for fatigue failure, providing a key basis for subsequent fatigue life assessment and structural optimization.

Fig. 8Mechanical response analysis under rolling force condition

a) Setting of load and boundary conditions

b) Calculation results of deformation

c) Maximum deformation under different rolling force

d) Stress distribution of spring

5.1. Modal characteristics analysis

Through finite element modal simulation, the first six mode shapes and natural frequencies are obtained as shown in Fig. 9. It can be seen that the first six natural frequencies of the serpentine spring are concentrated in the range of 500-750 Hz, and the mode shapes are dominated by coupled bending and torsional deformation, with the deformation characteristics of each mode showing a clear progressive pattern. The first mode (506.14 Hz) is a basic symmetric bending mode, with deformation concentrated at both ends of the spring in the form of overall swinging, and the maximum deformation reaches 102.25 mm. In the second mode (558.37 Hz), the bending nodes move toward the middle, with the two ends swinging in opposite directions, enhancing the local bending effect. The third (618.87 Hz) and fourth (664.08 Hz) modes gradually exhibit bending-torsion coupling characteristics, with torsional deformation beginning to appear in the middle section of the spring and the deformation distribution expanding from both ends to the center. The fifth mode (703.35 Hz) presents a multi-node bending mode, with deformation becoming more dispersed. The sixth mode (744.81 Hz) is a strongly coupled mode with significant superposition of bending and torsion, where the deformation amplitude of local sections reaches the maximum among the six modes (116.65 mm), reflecting the complexity of the spring's stiffness distribution under high-order modes. Overall, as the mode order increases, the natural frequency rises monotonically, the mode shapes evolve from simple bending to multi-node and multi-coupled deformation, and the deformation distribution becomes more complex, demonstrating the order-wise evolution law of the dynamic characteristics of the serpentine spring structure.

Fig. 9The first six modal shapes

a) First order (506.14 Hz)

b) Second order (558.37 Hz)

c) Third order (618.87 Hz)

d) Fourth order (664.08 Hz)

e) Fifth order (703.35 Hz)

f) Sixth order (744.81 Hz)

Combining the excitation characteristics of train operation with the modal frequencies of the spring, the resonance probability can be analyzed from two aspects: excitation matching and modal suppression effects [10, 11]. On the one hand, the main excitation sources during train operation include track irregularities, wheel-rail impacts, and component vibration transmission. At operating speeds of 100-350 km/h, the dominant frequency of track irregularity excitation is usually below 300 Hz, which is much lower than the first natural frequency of the spring, resulting in an extremely low probability of direct resonance. At 350 km/h, high-frequency excitations such as wheel-rail scratches and joint impacts may partially overlap with the first six natural frequencies of the spring, theoretically posing a resonance risk under impact excitation. However, such impacts are random broadband excitations, making it difficult to maintain an exact match with a single modal frequency for a long time, and the energy attenuates significantly during transmission, leading to limited actual coupling effects [12, 13]. On the other hand, the modal characteristics of the spring have a natural suppressing effect on resonance. The first six mode shapes are all local bending or torsional deformations, with vibration energy concentrated in local sections rather than large overall swings, resulting in a weak energy amplification effect during resonance. In summary, under normal train operating conditions, the probability of destructive resonance occurring in the serpentine spring is extremely low.

The test scheme for the natural frequency of the serpentine spring is shown in Fig. 10. Based on the modal test setup using the electronic hammer excitation method, a pulse excitation is applied to the fastening system by an excitation hammer, while acceleration sensors collect the dynamic response signals of the structure to obtain frequency response data. Fig. 11 presents the frequency response curve under hammer excitation, where the peak positions correspond to the natural frequencies of the structure. The first six natural frequencies measured experimentally are in good agreement with the results of the finite element modal simulation, and the deviations between the measured and simulated values of each natural frequency are all within the allowable engineering range, with a maximum deviation of only 3.2 %.

5.2. Fatigue life characteristics analysis

During the operation of trains, the serpentine springs of track fastening systems are continuously subjected to alternating displacement loads induced by wheel-rail coupling interactions, which easily leads to fatigue cracking and fracture failures. To accurately evaluate the fatigue life of springs under various load conditions and reveal the underlying fatigue failure mechanisms, finite element fatigue simulations are conducted based on the high-cycle fatigue theory, and indoor fatigue tests are performed correspondingly to verify the accuracy of the numerical model. The numerical simulation adopts the S-N fatigue curve and Miner linear cumulative damage criterion as the core theoretical basis, combined with the composite stress fatigue and stress concentration theories to carry out mechanical mechanism analysis. The S-N curve characterizes the quantitative relationship between stress amplitude and fatigue failure cycles of metallic materials under alternating loads, serving as the fundamental principle for fatigue life calculation of structural components. When the cyclic stress exceeds the fatigue limit of the material, the fatigue cycle number decreases exponentially with the increase of stress amplitude. The bending sections of serpentine springs bear coupled bending and shear stresses under service loads, presenting a typical alternating composite stress state. In the simulation calculation, an equivalent conversion method is adopted to convert the composite stress into equivalent stress amplitude, which is substituted into the S-N curve to solve the fatigue life under single-level load conditions. According to the Miner linear cumulative damage theory, the total fatigue damage of a component under alternating loads is the linear superposition of damage caused by each level of load. Fatigue fracture of the component is judged to occur when the cumulative damage value reaches 1.

Fig. 10Test of natural frequency. Photo by the author on March 10, 2026, at the Mechanics Laboratory in Qingdao

Fig. 11Response results of hammer excitation

Four displacement amplitude working conditions are set for the fatigue simulation, including 1.0 mm, 1.4 mm, 1.7 mm, and 2.0 mm. The indoor fatigue test adopts a fixed displacement load of 1.4 mm, which can effectively validate the simulation results. The loading frequency is consistent with the operating frequency of the test system to reproduce the alternating deformation characteristics of serpentine springs in actual train operation. On this basis, the influence law of different load amplitudes on the fatigue performance of serpentine springs is systematically explored. To investigate the fatigue life characteristics of the serpentine spring efficiently, the fatigue test platform was built, as shown in Fig. 12(a). The test bench realizes closed-loop monitoring of vibration displacement and dynamic response through acceleration sensors and laser displacement sensors, which can accurately reproduce the alternating load conditions borne by the serpentine spring during train operation. Fig. 12(b) shows the fatigue fracture state of the spring after the test. The fracture location is located on the inner side of the curved section of the spring, which completely coincides with the high-stress concentration area predicted by the simulation. This indicates that this position, subjected to long-term alternating bending and shear coupling stresses, has become a weak link for fatigue failure. The fatigue life nephogram shown in Fig. 12(c) reveals that the life distribution of the serpentine spring exhibits significant regional differences. The low-life regions are concentrated on the inner side of the curved section corresponding to the experimental fracture location, with a minimum life of approximately 1.923×10¹⁰ cycles, directly reflecting the influence of stress concentration on fatigue performance. The comparison curve shown in Fig. 12(d) indicates that as the vibration displacement load increases from 1.0 mm to 2.0 mm, the fatigue life of the spring shows a logarithmic linear decay trend. Meanwhile, the test result under the 1.4 mm displacement load is in good agreement with the simulation curve, verifying the accuracy of the fatigue simulation model in predicting the load-life relationship. This also demonstrates that the model can be effectively used for fatigue life assessment of serpentine springs under different load conditions.

Fig. 12Simulation and experimental results of fatigue life analysis. Photo by the author on March 6, 2026, at the Mechanics Laboratory in Qingdao

a) Fatigue testing platform

b) Fatigue fracture state

c) Fatigue life simulation results

d) Comparison between simulation results and test results

6.1. Effects on shock response

To carry out the impact load response analysis and modal analysis of serpentine springs with different cross-sectional dimensions, a parametric design method was adopted in this study, and four comparison schemes (DR_1 to DR_4) were designed with the outer radius and inner hole radius of the spring wire as control variables, as shown in Fig. 13 and Table 2. Among them, the outer radius was kept constant for DR_1 and DR_2, and the wall thickness was varied by adjusting the inner hole radius. For DR_3 and DR_4, the outer radius was also kept unchanged, and different wall thickness gradients were formed by setting different inner hole radii. Meanwhile, a comparison of the overall cross-sectional dimensions was realized between DR_2 and DR_3 through the same inner hole radius but different outer radii. This control variate-based parameter selection method can effectively separate the influences of cross-sectional dimension and wall thickness on structural performance, providing a reliable basis for variable control.

In terms of impact load response, different cross-sectional moments of inertia will directly affect the bending stiffness and stress distribution of the spring, and the gradient of cross-sectional dimensions from DR_1 to DR_4 can clearly demonstrate the influence law of stiffness variation on the impact response. In terms of modal analysis, changes in cross-sectional dimension and wall thickness will alter both the mass distribution and stiffness matrix of the structure simultaneously, and the influence weights of cross-sectional parameters on natural frequencies can be quantitatively identified through the comparison of the four schemes.

Fig. 13Sectional structural parameters

The acceleration response test of the serpentine spring under impact load is intended to reproduce the vertical impact conditions transmitted by the wheel-rail system during train operation, collect the dynamic acceleration signals of the spring, and verify its dynamic performance and resonance risk under service conditions [14]. Meanwhile, a unified test benchmark is provided for comparative schemes with different cross-sectional dimensions, the influence law of structural parameters on dynamic response is quantitatively analyzed, and key experimental evidence is offered for structural optimization, impact resistance improvement, and fatigue life evaluation of the spring.

The serpentine spring serves as the key force-transmitting and elastic vibration-damping component of high-speed railway fastening systems. During train operation, the wheel-rail interaction generates dynamic impact loads characterized by high frequency, transience and large amplitude. Such loads are a major factor leading to stress concentration, local deformation and even fatigue cracking of the spring. Analyzing the time-varying characteristics of acceleration in the impact response of the serpentine spring can fully reflect the transient dynamic behavior of the structure under wheel-rail impact in high-speed operation, and quantitatively evaluate its vibration reduction, energy dissipation and dynamic load-bearing performance. The relevant results can provide theoretical and experimental support for the structural optimization, performance improvement and engineering application of track serpentine springs. In accordance with the test specifications, the impact test conditions are summarized in Table 3. By means of this loading method, the amplitude and loading rate of the impact load can be controlled, thereby providing a unified and controllable load boundary condition for the comparative testing of springs with different cross-sectional dimensions and ensuring variable consistency in subsequent performance analysis.

The application method of the impact load is shown in Fig. 14(a). A vertically loaded servo-driven system is adopted, through which dynamic impact loads are applied along the main stress axis of the spring. This allows the vertical impact conditions transmitted by wheel-rail excitation to the serpentine spring during train operation to be accurately reproduced. The sensor installation scheme is presented in Fig. 14(b).

Fig. 14Test of spring acceleration response under impact load. Photo by the author on March 9, 2026, at the Mechanics Laboratory in Qingdao

a) Direction of impact load

b) Installation of sensors

Acceleration sensors and strain gauges are arranged in critical stress-concentrated regions such as the inner side of the curved section of the spring, enabling the synchronous acquisition of dynamic acceleration and stress responses during the impact process. The proposed loading and sensing scheme is directly oriented to the actual stress state of the spring in service. It not only allows the dynamic response characteristics under impact load to be effectively captured but also provides reliable experimental data support for subsequent analysis of the influence of cross-sectional parameters on the impact resistance of the spring, ensuring the scientificity and pertinence of the test scheme.

The acceleration response results of springs under different structural conditions are presented in Fig. 15. From the perspective of response characteristics, significant differences in acceleration amplitudes are observed among the four springs, which are directly related to their cross-sectional parameters. The lowest acceleration peak and the most gentle overall fluctuation are exhibited by DR_1, demonstrating the optimal dynamic stability. For DR_2, a slight increase in vibration amplitude is caused by the reduced wall thickness and decreased section stiffness. The acceleration responses of DR_3 and DR_4 are significantly amplified, with the highest peak observed in DR_3. This is mainly attributed to the imbalance between stiffness and mass after the increase in cross-sectional dimensions, leading the natural frequency of the structure to approach the impact excitation frequency band and triggering an obvious vibration amplification effect. Through comprehensive comparison, it is found that the dynamic response performance of the spring does not increase linearly with the increase in cross-sectional dimensions, but is determined by the synergistic matching of stiffness and mass. The optimal balance between stiffness and mass is achieved by DR_1 with moderate wall thickness and reasonable section moment of inertia, resulting in the best attenuation effect on impact loads. In contrast, for DR_3 and DR_4 with simply increased outer radius, the shift in natural frequency caused by increased mass aggravates the vibration response instead. It is indicated that wall thickness optimization is more effective in improving impact resistance than blindly increasing cross-sectional dimensions, and special attention should be paid to avoiding the natural frequency falling into the sensitive frequency band of wheel-rail impact excitation in the design.

Fig. 15Acceleration response results of springs under different structural conditions

a) DR_1

b) DR_2

c) DR_3

d) DR_4

6.2. Effects on natural frequency and stiffness

Investigating the influence mechanism of cross-sectional geometric parameters on the modal characteristics of serpentine springs can not only distinguish differences in frequency response, but also screen out optimized schemes that can avoid the dominant wheel-rail excitation band and improve the dynamic stability of the structure. Analyzing the evolution characteristics of spring stiffness under cyclic loading enables us to clarify the high-cycle fatigue mechanical behavior of components with different cross-sections. It also reveals the influence laws of cross-sectional dimensions on work hardening, residual stress accumulation and abnormal stiffness variation, and evaluates the mechanical reliability of each scheme during long-term service.

The variations in natural frequencies of serpentine springs with different cross-sectional dimensions as a function of modal order are presented in Fig. 16. The change law of spring stiffness with the number of loading cycles is illustrated in Fig. 17. Together, these two figures reveal the regulatory mechanisms of cross-sectional parameters on both dynamic and fatigue performances. In terms of natural frequency characteristics, the natural frequencies of the four cross-sectional schemes (DR_1 to DR_4) all exhibit a linear increasing trend with the rise in modal order, yet significant differences are observed among different schemes. The natural frequencies of DR_3 and DR_4 are generally higher than those of DR_1 and DR_2, with DR_2 showing the lowest frequencies. This discrepancy is attributed to the synergistic effect of cross-sectional moment of inertia and mass. The increase in stiffness caused by the enlargement of the outer radius exceeds the influence of mass growth, leading to a significant upward shift in the natural frequencies of DR_3 and DR_4. In contrast, DR_2, due to its larger inner radius and the lowest cross-sectional stiffness, exhibits natural frequencies at the lower limit among the four schemes. From the perspective of engineering applications, the frequency curve of DR_1 lies at an intermediate level, demonstrating superior dynamic stability [15, 16].

Regarding the evolution of stiffness, the spring stiffness of all four schemes shows an upward trend with the increase in the number of cyclic loading cycles, but distinct differences exist in the growth rates. The stiffness increments of DR_3 and DR_4 are notably higher than those of DR_1 and DR_2. Among them, the stiffness of DR_3 increases most drastically, with a growth of over 100 % compared to the initial value after 6×10⁶ cycles. In contrast, the stiffness growth of DR_1 is the most moderate, increasing by only approximately 20 %. Owing to their larger wall thicknesses, DR_3 and DR_4 are prone to work hardening and residual stress accumulation under cyclic loading, resulting in a rapid rise in stiffness. On the other hand, the cross-sectional stress distribution of DR_1 is more uniform, leading to a more stable stiffness evolution during the fatigue process.

A comprehensive evaluation of both dynamic and fatigue performances reveals that the DR_1 scheme exhibits the optimal overall performance in terms of natural frequency distribution and stiffness stability. This indicates that the optimization of cross-sectional dimensions requires simultaneous consideration of stiffness-mass matching and mechanical stability during fatigue. Simply increasing the cross-sectional dimension (e.g., DR_3) can improve the initial stiffness but exacerbates stiffness degradation and resonance risks during fatigue, which is instead detrimental to long-term service performance.

Fig. 16The variations in natural frequencies of serpentine springs

Fig. 17The change law of spring stiffness with the number of loading cycles

Conventional elastic clips, such as Type WJ-7 and SKL series, generally have low-order natural frequencies below 500 Hz, particularly the first-order bending mode that is highly susceptible to resonance. This frequency band overlaps substantially with the dominant wheel-rail excitation range, where vibrations are generated by short-pitch rail corrugation, wheel polygonal wear and track irregularities. When the first-order mode (100-300 Hz) is directly coupled with the excitation frequency of short-pitch rail corrugation, modal resonance takes place, which drastically amplifies the dynamic stress amplitude at critical areas including the clip root. The elevated dynamic stress initiates microcracks in stress concentration zones, and these cracks gradually propagate to form macro fractures. The four innovative DR-series elastic clips proposed in this research adopt cross-sectional configuration optimization (e.g., DR-type hollow structure) and stiffness distribution adjustment. As a result, their first six natural frequencies are entirely shifted to 500-750 Hz. All of their first-order natural frequencies exceed 500 Hz, so the structures stay completely outside the dominant energy band of wheel-rail vibration. The excitation energy above 500 Hz is less than 10 % of that in the dominant band. Even if minor frequency coupling occurs, the dynamic stress amplitude remains far lower than that of conventional structures.