The vehicle is modeled as a complex system comprising 15 rigid bodies: one car body, two bogie frames, four wheelsets, and four axle boxes (represented by eight distinct components). The system features a total of 50 degrees of freedom (DOFs), with the specific distribution detailed in Table 1. The model explicitly accounts for the vehicle's topological structure, incorporating precise representations of the primary suspension, the secondary suspension (integrating the transverse secondary inertial suspension system, TSISS), and nonlinear wheel–rail contact geometry. This high-fidelity framework effectively captures dynamic responses under operational conditions, serving as a reliable numerical benchmark for the subsequent comparative analysis of suspension configurations.

Figure 1a illustrates the vehicle topology, with corresponding nomenclature detailed in Table 2. A bottom-up modeling approach was employed: wheelsets were established as fundamental subsystems and integrated into the bogie frame to form the bogie assembly. These subsystems were then coupled with the car body to complete the full-vehicle dynamics model, as shown in Fig. 1b. This hierarchical procedure accurately captures the kinematic constraints among components, providing a robust foundation for subsequent dynamic simulations and control algorithm integration.

While the comprehensive multibody model serves as the baseline for numerical simulation, a simplified analytical model was derived to elucidate the vibration attenuation mechanism of the TSISS. This reduced-order model facilitates efficient parametric analysis, particularly regarding frequency domain sensitivity to track irregularities.

Assuming small-amplitude oscillations, the system was linearized and reduced to its essential degrees of freedom (DOFs). The model specifically captures the lateral and roll motions of the car body coupled with the lateral displacements of the front and rear bogies. This configuration effectively isolates the interaction between the car body and the secondary lateral suspension. To ensure validity, all physical parameters – including mass, inertia, stiffness, damping, and inertance – remain consistent with those in the high-fidelity multibody model.

Analogous to springs and dampers, the inerter is a two-terminal mechanical element where the generated force is proportional to the relative acceleration between its terminals. The integration of an inerter can effectively attenuate vibration acceleration amplitudes and extend the oscillation period, thereby theoretically enhancing the running stability of railway vehicles. The constitutive equation of the inerter is expressed as Eq. (1): 1F=bdv1-v2dt, where F is the force applied to the two terminals of the inerter (unit: N), b is the inertance (unit: kg), and v1 and v2 denote the accelerations of the two terminals (unit: m s⁻¹).

Drawing on optimization studies by Wang et al. (2022), several inerter configurations were selected to investigate their impact on sensitive-wavelength distribution under lateral excitations. The inerter models were integrated into the vehicle system by modifying the dynamic stiffness matrices. To accurately simulate series topologies, a virtual mass block was introduced as an intermediate node between the inerter and other suspension elements; this ensures consistency between the numerical simulation and theoretical formulation. The schematics of the conventional secondary lateral suspension and the four proposed inertial configurations are presented in Fig. 2.

In Fig. 2, Fkyi and Fkxi (i=1–4) represent the primary lateral stiffness and primary longitudinal stiffness(unit: kN m-1), respectively; bi (i=1–4) denotes the inertance (unit: kg); and kbi (i=1,2) represents the stiffness of the vibration reduction spring in the suspension systems of Conditions S4 and S5.

The dynamic equations for Conditions S1 through S5 correspond to Eqs. (2)–(6), respectively. 2mcy¨c+ks1(yc-yb)+cs1(y˙c-y˙b)=0,3mcy¨c+b1(y¨c-y¨b)+ks2(yc-yb)+cs2(y˙c-y˙b)=0,4mcy¨c+ks3(yc-yb)+cs3(y˙c-y˙i)=0,cs3(y˙c-y˙i)=b2(y¨i-y¨b)5mcy¨c+cs4(y˙c-y˙b)+ks4(yc-yb)+kb1(yc-yi)=0,kb1(yc-yi)=b3(y¨i-y¨b)6mcy¨c+ks5(yc-yb)+kb2(yc-yi)+cs5(y˙c-y˙i)=0,kb2(yc-yi)+cs5(y˙c-y˙i)=b4(y¨i-y¨b) Here yc denotes the lateral displacement of the car body, and yb denotes the lateral displacement of the bogie frame. Furthermore, yi represents the lateral displacement of the virtual mass block, which is introduced to implement the series connection between the inerter and the damper in the model comprising the car body mc and the bogie frame mb.

Lateral car body acceleration is a critical metric for assessing vehicle running stability (Qi et al., 2025a). To investigate the efficacy of different inerter-based suspension systems, numerical simulations were performed using a measured domestic track irregularity spectrum as the excitation source. The evaluation focused on lateral accelerations of the car body and bogie frame, along with car body yaw and roll accelerations. In these simulations, the inertance was fixed at 500 kg, and the operating speed was set to 250 km h⁻¹. The corresponding response characteristics are illustrated in Fig. 3.

Lateral Car Body Acceleration (Fig. 3a). Trends for Conditions S2 and S4 mirror the baseline Condition S1, with similar peak amplitudes, indicating negligible improvement. In contrast, Conditions S3 and S5 demonstrate significant suppression. Specifically, Condition S5 reduces the peak acceleration to 0.21 m s⁻² – an optimization of 34 % relative to S1. This confirms the inerter system's capability to effectively attenuate vibration through an energy redistribution mechanism.

Lateral Bogie Frame Acceleration (Fig. 3b). While S2 and S4 again remain consistent with S1, Condition S3 exhibits pronounced fluctuations during the initial 4.5 s, showing no clear benefit. Notably, Condition S5 results in a slight deterioration (7 % increase) compared to S1. This indicates a trade-off: the inerter suppresses car body vibration by transferring partial energy to the bogie subsystem. Consequently, dynamic performance optimization alters energy transmission paths, necessitating multi-objective optimization for specific performance indices in engineering practice.

Car body yaw acceleration (Fig. 3c). Conditions S2 and S4 show no significant deviation from S1. Conversely, compared to the baseline, Conditions S3 and S5 achieve substantial reductions of 31 % and 34 %, respectively. The response of S5 fluctuates near 0 m s⁻², indicating that the “parallel damper–spring, series inerter, parallel air spring” topology effectively mitigates yaw motion without inducing resonance in this degree of freedom.

Car body roll acceleration (Fig. 3d). S2 and S4 exhibit trends consistent with S1. However, Conditions S3 and S5 show slight performance degradation relative to S1. This is attributed to the series inerter configuration in these cases, which alters system coupling and slightly compromises the balance of multi-degree-of-freedom vibration suppression.

Comparative analysis of Fig. 3 demonstrates that the introduction of inerter systems induces distinct modulation characteristics in vehicle dynamics. Notably, Condition S5 effectively mitigates lateral car body acceleration, particularly under longwave track irregularities. While this suppression involves transferring vibrational energy to the bogie subsystem – raising lateral frame acceleration to approximately 1.5 m s⁻² – the resulting value remains significantly below the safety threshold of 8 m s⁻² mandated by the GB/T 5599-2019 standard. Consequently, leveraging composite structures to enhance frequency band adaptability presents a promising approach for optimizing both ride quality and running stability.

Considering the weak coupling between vertical and lateral vibrations in railway vehicles (Sun et al., 2024), measured vertical irregularities were incorporated to ensure simulation fidelity without compromising the lateral analysis. Conversely, to systematically isolate the effects of wavelength and amplitude, lateral irregularities were modeled as harmonic excitations. The simulation covered a wavelength range of 5–200 m (5 m increments) with a fixed amplitude of 4 mm, at a constant vehicle speed of 250 km h⁻¹. The peak lateral car body acceleration served as the primary metric for identifying the sensitive-wavelength distribution.

As shown in Fig. 4, S1, S2, and S4 exhibit dominant responses in the short-wavelength regime, peaking around 20 m. Compared to the baseline (S1), S2 and S4 attenuate peak accelerations by 16.1 % and 13.3 %, respectively. In contrast, S3 and S5 shift the sensitive range to long wavelengths (∼ 105 m), with S5 achieving the most significant suppression of 32.2 %. These findings substantiate that integrating inerters into the suspension system significantly enhances lateral dynamic performance.

To investigate the minor local peaks observed in Condition S5's lateral acceleration at wavelengths of 70 and 15 m (Fig. 4), further analysis was conducted, as shown in Fig. 5. Corresponding peaks are also evident in the car body yaw motion (Figs. 5a and 6b). Power spectral density (PSD) analysis (Fig. 5c and d) reveals that under 70 m excitation, the dominant frequency for both lateral and yaw modes coincides at 1.17 Hz. Similarly, under 15 m excitation, the dominant frequencies are highly proximal (5.3 and 5.5 Hz, respectively). This frequency alignment implies that the local peaks in lateral acceleration stem from a resonant coupling between lateral and yaw motions, a phenomenon corroborated by recent studies (Alves et al., 2023; Jiang et al., 2022).

The influence of running speed (250–350 km h⁻¹) on sensitive-wavelength distribution was investigated under harmonic lateral track irregularity excitations (Jiang et al., 2022; Wei and Xu, 2021). As illustrated in Fig. 6, the distribution pattern of sensitive wavelengths remains substantially stable across all suspension configurations, demonstrating insensitivity to speed variations. However, regarding amplitude, the maximum lateral car body accelerations for Conditions S2, S4, and S5 exhibit a distinct upward trend as speed increases.

In Condition S3, a counterintuitive phenomenon occurs under specific excitation wavelengths: maximum lateral car body acceleration decreases as vehicle speed increases. Combined time–frequency analysis (Fig. 7) reveals that this anomaly arises from the interaction between multi-degree-of-freedom modal coupling and phase modulation mechanisms.

Specifically, at 270 km h⁻¹ (Fig. 7a and c), resonant coupling between car body lateral and roll modes near 0.78 Hz amplifies the lateral acceleration. Conversely, at 380 km h⁻¹ (Fig. 7b and d), although the dominant frequencies of roll and yaw motions converge near 1 Hz, their time-domain waveforms exhibit anti-phase characteristics. Consequently, the lateral response is significantly attenuated via destructive interference (or vibration energy cancellation) (Li and Xu, 2022). These findings demonstrate that the sensitive-wavelength distribution of the inerter suspension is governed not merely by the speed–wavelength mapping, but also significantly by complex multi-DOF modal interactions.

The selection of an optimal inertance coefficient is critical for the suspension system's vibration attenuation efficiency. Preliminary frequency domain analysis indicates that an inertance range of 10²–10³ kg is required to effectively shift the car body's dominant lateral frequency away from the sensitive band. Furthermore, based on existing mechanical designs, inertance coefficients between 500 and 2000 kg can be readily accommodated within the spatial constraints of the vehicle's secondary suspension (Chen et al., 2022).

Figure 9 depicts the maximum lateral car body acceleration as a function of track irregularity wavelength and inertance coefficient (b). Cases S2 through S4 exhibit negligible sensitivity to parameter variations, with peak fluctuations remaining below 5 % and showing no discernible trend. Conversely, Case S5 demonstrates a pronounced dependence on inertance. Specifically, increasing b from 500 to 1000 kg reduces the peak acceleration at ∼ 105 m from 0.85 to 0.55 m s⁻² – a significant 35.3 % improvement. This confirms the system's efficacy in suppressing longwave excitations within this specific range. However, further increasing b from 1000 to 2000 kg yields marginal returns (reduction <5%), indicating that vibration attenuation performance has reached saturation. This indicates that the inerter suspension system in Condition S5 plays a positive role in vibration attenuation. An inertance within the range of 1000 kg balances dynamic performance with engineering feasibility, thereby providing a basis for parameter design.

In this study, the FRA algorithm was utilized to synthesize random lateral and vertical track irregularities. With the FPF parameter set to 0.25, the simulation generated lateral irregularities characterized by amplitudes of 4–5 mm across a wavelength range of 5–200 m at a vehicle speed of 250 km h⁻¹. Regarding the identification of sensitive wavelengths, a criterion based on the maximum lateral acceleration power spectrum was adopted: a wavelength is deemed sensitive if it corresponds to a spectral peak that coincides with a significant magnitude in the coherence function (Li, 2022).

Figure 10 depicts the coherence functions and PSD plots relating lateral car body vibration to track irregularities. For Conditions S1, S2, and S4, coherence values exceed 0.8 within the spatial frequency band of 0.04–0.047 m⁻¹ (corresponding to wavelengths of 21.1–25 m), identifying sensitive wavelengths of 21.1, 23.3, and 25.5 m, respectively. In contrast, Conditions S3 and S5 demonstrate significant coherence in the low-frequency range of 0.009–0.0092 m⁻¹ (wavelengths of approximately 108–111 m), with sensitive wavelengths identified at 110.7 and 107.4 m. These distributions align closely with the sensitive-wavelength results previously established in Fig. 4.

Notably, although the coherence values for Conditions S3 and S5 fall below the 0.8 threshold at sensitive wavelengths, this can be attributed to the specific suspension topology (Fig. 2). In these configurations, the series arrangement of the inerter modifies the lateral-force transmission path. Due to the inerter's inherent mechanical characteristics (Liu, 2023), the system not only attenuates the acceleration amplitude but also elongates the vibration period. This introduces a significant phase discrepancy (or lag) between the car body response and the track excitation, consequently suppressing the calculated coherence values at these wavelengths.