A novel vibration-damping windshield: Dynamic characteristics and benefits for vehicle dynamics performance

Abstract

Conventional windshields passively separate vehicle compartments while offering negligible vibration control. With rising high-speed train demands, suppressing inter-car vibrations becomes critical for improving ride comfort. This study innovatively develops a dual-function vibration-damping windshield that concurrently serves structural and dynamic vibration suppression roles. Employing physically parameterized modeling, the dynamic stiffness and damping characteristics of the vibration-damping windshield were systematically characterized and validated against conventional inter-vehicle dampers, confirming their functional equivalence. A coupled dynamics model was developed through a novel co-simulation framework integrating multibody train dynamics and windshield behavior. The dynamics performance comparison between the traditional and vibration-damping windshields demonstrates that the proposed windshield significantly enhanced secondary suspension isolation. The design effectively attenuates carbody vibrations across all degrees of freedom, while substantially suppressing problematic low-frequency carbody swaying behavior to improve ride stability across operational speeds. Importantly, vibration reduction efficacy proves consistently superior in motor cars relative to trailers. Collectively, this work establishes a paradigm-shifting approach for structural-integrated vibration control, validating the windshield’s capacity to functionally replace dedicated dampers and thereby advance performance integration in next-generation rail vehicles.

Keywords

railway vehicles carbody vibration vibration-damping windshield nonlinear mechanical model dynamic characteristics

1. Introduction

The increase in operating speed of railway vehicles has intensified a series of vibration issues related to the carbody, such as carbody swaying (Ding et al., 2024; Jiang et al., 2024; Sun et al., 2021) and carbody shaking (Chang et al., 2023; Qi et al., 2019), which seriously affect the running quality of trains. Beyond addressing abnormal carbody vibrations at their source, the suppression of carbody vibration is mainly achieved by enhancing the constraints on the carbody and dissipating its vibration energy. Currently, this is mainly achieved by adding constraint elements between the carbody and bogies (Huang et al., 2023; Zhao et al., 2024) or by reinforcing the connections between adjacent car bodies through devices with specific stiffness and damping characteristics to enhance inter-car coupling (Fujimoto and Masayuki, 1996; Fujimoto and Miyamoto, 1996).

The installation of damping devices between cars has been widely recognized as an effective approach to improve the dynamic performance of high-speed EMUs, with inter-vehicle dampers being the most commonly employed solution. Extensive studies from the perspective of train system dynamics have investigated the influence of inter-vehicle suspension parameters, particularly the stiffness and damping characteristics of longitudinal or lateral dampers, on vehicle stability, ride comfort, and wheel-rail interaction. Toshimitsu et al. (2007) and Tanifuji et al. (2008) analyzed the installation of lateral dampers between adjacent car bodies and demonstrated their effectiveness in reducing carbody vibrations in tunnels and improving ride comfort. Wang et al. (2024) optimized the inter-vehicle damper parameters for the CRH1 high-speed train, achieving up to a 20% reduction in lateral acceleration. Zhou et al. (2017) and Sun et al. (2017) reported similar improvements for the CRH380 series EMUs, highlighting the suppression of low-frequency carbody vibrations in the 1–3 Hz range. In addition, Zhao et al. (2024) developed and experimentally validated a novel inter-vehicle longitudinal damper designed to mitigate low-frequency carbody vibrations caused by low wheel-rail equivalent conicity, showing significant enhancements in lateral ride quality.

Apart from mechanical damping devices, aerodynamic components between cars also play an important role in improving dynamic performance. The inter-car windshield, which connects adjacent car bodies, serves primarily to seal the gap between cars, ensuring smooth airflow across the inter-vehicle space during high-speed operation (Ma et al., 2023; Tang et al., 2023). This reduces aerodynamic drag and mitigates pressure fluctuations that could otherwise contribute to carbody vibration and noise. Jin and Chen (2025) conducted a computational fluid dynamics study using steady and unsteady Navier–Stokes equations, showing that the installation of external windshields can reduce overall aerodynamic resistance by approximately 14.46% in open-line conditions and 24.48% in tunnels. Tang et al. (2023) further investigated the fluid-structure interaction between rubber external windshields and unsteady aerodynamic loads, revealing that improper structural parameters can lead to large local deformations and aerodynamic loads. However, existing research on inter-car windshields has focused primarily on aerodynamic drag (Niu et al., 2019), with relatively few studies addressing their potential role in attenuating inter-vehicle vibrations. These findings indicate that aerodynamic sealing devices such as inter-car windshields not only improve energy efficiency but also influence vibration characteristics, offering potential for integrated aerodynamic–vibration control.

In this context, this paper proposes a novel external windshield with inter-car vibration reduction functionality, capable of providing both stiffness and damping for effective vibration suppression between carbodies, distinguishing it from conventional windshields and inter-vehicle dampers. The proposed design integrates vibration control into the structural element itself. To characterize its performance, a refined mechanical model is developed, incorporating sub-models of the rubber air chamber, auxiliary chamber, and throttle orifice to relate internal structural parameters to external mechanical characteristics. The model-derived stiffness and damping are further integrated into vehicle dynamics simulations to evaluate vibration attenuation performance. The paper is organized as follows: Sections 2 and 3 present the working principle, structural features, and mechanical models of the vibration-damping windshield. Section 4 analyzes its dynamic stiffness and damping characteristics under different frequencies and amplitudes, comparing its performance with conventional inter-car dampers. Section 5 develops a coupled simulation model combining the mechanical and vehicle dynamic models, comparing the vibration attenuation effects of configurations with the vibration-damping windshield and the traditional windshield. The results demonstrate that the vibration-damping windshield can effectively replace traditional inter-car dampers in suppressing inter-car vibrations and mitigating carbody abnormal vibrations.

2. Structure of the vibration-damping windshield

The vibration-damping windshield is installed between adjacent train cars, providing a smooth aerodynamic transition while improving vibration isolation performance. As shown in Figure 1(a), it is mounted between the front and tail cars and functions as an external enclosure together with an inner windshield. Compared with traditional windshields, the proposed design retains aerodynamic noise and drag reduction as well as weather protection, while introducing additional stiffness and damping capabilities.

Figure 1.

Structure of the vibration-damping windshield: (a) installation location; (b) internal configuration.

The structure of the vibration-damping windshield is illustrated in Figure 1(b). It mainly consists of a primary air chamber (rubber bellow), auxiliary air chamber (auxiliary tank), quick-release interfaces, and a pneumatic refill pipe. The primary chamber is a rubber bellow filled with compressed air, forming a sealed unit with the auxiliary chamber. A throttle orifice is embedded in the mounting seat between these chambers to regulate airflow. The auxiliary chamber is fixed to the front carbody, while the rear part of the structure connects to the tail car via a detachable interface. Compressed air is supplied from the train’s pneumatic system through the refill pipe, maintaining a stable internal pressure. To enhance damping performance, flexible partitions are arranged inside the rubber bellow to divide the primary chamber into multiple subunits. Each subunit operates as an independent “primary chamber–orifice–auxiliary chamber–refill pipe” module, enabling efficient energy dissipation during vibrations.

The vibration-damping mechanism is illustrated in Figure 2. When the train undergoes yaw or longitudinal vibrations, the relative movement between adjacent car bodies leads to deformation of the rubber bellows. In the case of yaw excitation, one side of the windshield is compressed, increasing the air pressure in the primary chamber. Compressed air then flows through the orifice into the adjacent auxiliary chamber. Simultaneously, the opposite side is stretched, decreasing the pressure in the primary chamber and causing reverse airflow from the auxiliary chamber to the primary chamber. The bidirectional airflow through the orifices introduces significant damping due to the throttling effect. For the longitudinal vibration, both bellows experience simultaneous compression or tension. This causes synchronized airflow across both sides through the orifices, producing opposing damping forces that resist the relative motion between cars. In both cases, the compressed air acts as a medium for energy conversion, transforming vibration energy into thermal energy through throttled airflow, thereby suppressing dynamic responses effectively.

Figure 2.

Operating principle of the vibration-damping windshield: (a) initial position of windshield; (b) gas flow response under vibration, including yaw and longitudinal vibration.

3. Modeling of the vibration-damping windshield

The damping characteristics of the vibration-damping windshield are primarily governed by the airflow through the throttle orifices between the rubber bellows (primary air chambers) and the auxiliary chambers. Therefore, the dynamic model of the windshield is developed by considering each subunit—comprising a rubber air chamber, a throttle orifice, and an auxiliary chamber—as the basic modeling unit.

As illustrated in Figure 3, each subunit consists of a sealed rubber chamber with variable internal pressure and volume, an adjacent auxiliary chamber, and an interconnecting orifice characterized by flow parameters such as orifice diameter d_ori and discharge coefficient c_q. Key thermodynamic variables such as pressure P, temperature T, volume V, and density ρ are defined in both chambers. The interface area A_a between the bellow and the carbody, as well as ambient pressure P_atm, are also considered in the system.

Figure 3.

Schematic diagram of the vibration-damping windshield.

The coupling between the chambers is established by enforcing mass conservation, assuming the mass flow rate into the orifice equals the mass flow rate out of it. Based on this principle, the dynamic response of each subunit can be described by a set of differential equations, and the entire vibration-damping windshield can be modeled by integrating these subunits in parallel. This modular approach allows for precise characterization of the damping behavior under various excitation modes.

3.1. Rubber air chamber model

During the vibration process between two car bodies, the vibration-damping windshield provides stiffness and damping characteristics by compressing the gas in the rubber airbag of the main air chamber. Its internal structural parameters and gas state parameters directly determine the damping force of the windshield. The damping force of the windshield subunit during vehicle operation can be expressed as

F = (P_{a} - P_{atm}) A_{a}

(1)

where P_a is the absolute internal pressure of the main chamber, P_atm is the standard atmospheric pressure; A_a is the effective working area of the main chamber.

The effective working area of the main air chamber is usually related to the vibration amplitude and the pressure difference inside the air chamber, while the influence of air pressure is relatively smaller compared to the vibration amplitude (Facchinetti et al., 2010).This study mainly considers the impact of inter-vehicle vibration amplitude on the effective area. Therefore, the effective area during vibration can be expressed as

A_{a} = A_{a 0} + β x

(2)

where A_a0 is the initial working area, x is the relative vibration displacement, and β is a sensitivity coefficient (obtained via experiment or FEM simulation) representing the rate of change of effective area with respect to displacement.

As for the air pressure P_a in the main air chamber of the windshield, it is based on the ideal gas state equation (Zhou et al., 2023) as

P_{a} = \frac{m_{a} R_{g} T_{a 0}}{V_{a}}

(3)

where m_a is the instantaneous gas mass; R_g is the specific gas constant; T_a0 is the temperature inside the air chamber. The pressure in the main air chamber of the windshield is thus mainly determined by the volume of the main air chamber V_a under the current state and the mass of air inside the air chamber.

The volume change of the chamber is caused by the deflection and the mass change of the gas entering and exiting. Although this variation is relatively small, it exerts a significant impact on the performance of the air spring, and its rate of change can be expressed as

\frac{d V_{a}}{d t} = \frac{d V_{x}}{d t} + \frac{d V_{m}}{d t} = A_{a} \cdot \frac{d x}{d t} + \frac{1}{ρ_{a}} \frac{d m_{a}}{d t}

(4)

where V_x is the volume change caused by the deflection of the chamber; V_m is the volume change caused by the inflow and outflow gas flow; ρ_a is the gas density in the chamber.

For the mass of air in the main chamber, according to the continuity equation, the rate of change of air mass with respect to time equals the mass flow rate through the throttle orifice, and can be expressed as

\frac{d m_{a}}{d t} = q_{o r i}

(5)

where q_ori denotes the mass flow rate through the orifice.

The pressure evolution in the main air chamber is governed by differentiating the state equation, taking into account the variations in both gas mass and volume. Furthermore, the gas density ρ_a may also be described by the polytropic relation:

ρ_{a} = ρ_{a 0} {(\frac{V_{a 0}}{V_{a}})}^{k}

(6)

where k is the polytropic exponent; V_a0 is the initial volume of the main chamber; ρ_a0 is the initial gas density in the main air chamber.

3.2. Auxiliary chamber model

The auxiliary air chamber of the vibration-damping windshield is of a metal structure. Compared with the rubber airbag of the main air chamber, its structure does not deform during the vibration process, thus making the volume of the auxiliary air chamber constant. Since the chamber does not deform, the internal gas pressure P_a is influenced solely by the mass exchange with the primary chamber.

The rate of change of mass in the auxiliary chamber m_b is given by

\frac{d m_{b}}{d t} = - \frac{d m_{a}}{d t} = - q_{o r i}

(7)

By applying the ideal gas law and assuming constant volume, the pressure variation in the auxiliary chamber is expressed as

\frac{d P_{b}}{d t} = \frac{R_{g} T_{b 0}}{V_{b}} \cdot \frac{d m_{b}}{d t}

(8)

3.3. Throttle orifice model

The throttle orifice is a key component for regulating the stiffness and damping characteristics of the vibration-damping windshield. During the operation of the vibration-damping windshield, the air flowing through the orifice is compressible, and its flow characteristics exhibit nonlinearity. The orifice is generally modeled as a thin-walled orifice with an equivalent flow area. When gas flows through the orifice, flow resistance is generated due to the sudden contraction and expansion of the flow cross-section, as well as the viscous friction inside the fluid. This resistance causes a pressure drop across the orifice and converts part of the fluid’s pressure energy into thermal energy (dissipation), thereby generating a damping effect.

Since the air flowing through the throttle valve has a small contact area with the pipe wall and a high flow velocity, the air undergoing state changes has no time to exchange heat with the surrounding environment. Therefore, it can be regarded as an adiabatic process and one-dimensional isentropic flow of an ideal gas. Based on the Bernoulli equation, the gas velocity v at the vena contracta of the orifice can be derived as

v = \sqrt{\frac{2 μ}{μ - 1} \frac{P_{u}}{ρ_{u}} [1 - {(\frac{P_{d}}{P_{u}})}^{\frac{μ - 1}{μ}}]}

(9)

where P_u and ρ_u are the pressure and density upstream of the orifice, P_d is the downstream pressure, and μ is the specific heat ratio of the gas. Based on the ideal gas state equation, the adiabatic equation, and equation (9), the theoretical general mass flow rate q_{mor_1} of the orifice can be obtained as

q_{m o r_1} = A_{o} P_{u} \sqrt{\frac{2 μ}{μ - 1} \frac{1}{R T_{u}} [{(\frac{P_{d}}{P_{u}})}^{\frac{2}{u}} - {(\frac{P_{d}}{P_{u}})}^{\frac{μ + 1}{μ}})]}

(10)

where A_o is the cross-sectional area at the vena contracta of the orifice; T_u is the gas temperature in the upstream air chamber.

When the upstream gas pressure remains constant, the orifice mass flow rate increases with decreasing downstream pressure until reaching saturation at a critical downstream pressure value. Assuming that the gas pressure and temperature in the upstream air chamber of the orifice remain unchanged within an extremely short time, equation (10) is only a function of P_d. By differentiating it and setting the derivative to zero, the pressure ratio at the saturation point can be obtained as

(\frac{P_{d}}{P_{u}}) = {(\frac{2}{μ + 1})}^{\frac{μ}{μ - 1}}

(11)

From equations (10) and (11), the theoretical saturated mass flow rate q_mmax of the gas at the orifice can be derived as

q_{m \max} = A_{o} P_{u} \sqrt{\frac{μ}{R T_{u}} {(\frac{2}{μ + 1})}^{\frac{μ + 1}{μ - 1}}}

(12)

Considering the flow loss when gas passes through the orifice and the saturation characteristic of the orifice’s mass flow rate, the actual mass flow rate q_mor1 of the gas in the orifice is obtained as

q_{m or 1} = {\begin{cases} C_{q} A_{o} P_{u} \sqrt{\frac{μ}{R T_{u}} {(\frac{2}{μ + 1})}^{\frac{μ + 1}{μ - 1}}} \frac{P_{d}}{P_{u}} \leq {(\frac{2}{μ + 1})}^{\frac{μ}{μ - 1}} \\ C_{q} A_{o} P_{u} \sqrt{\frac{2 μ}{μ - 1} \frac{1}{R T_{u}} [{(\frac{P_{d}}{P_{u}})}^{\frac{2}{μ}} - {(\frac{P_{d}}{P_{u}})}^{\frac{μ + 1}{μ}}]} \frac{P_{d}}{P_{u}} > {(\frac{2}{μ + 1})}^{\frac{μ}{μ - 1}} \end{cases}

(13)

where C_q is the discharge coefficient of the orifice.

4. Dynamic characteristics of the vibration-damping windshields

4.1. Dynamic characteristics

To investigate the dynamic performance of the vibration-damping windshield, a detailed analysis of the dynamic characteristics of its subunit is conducted. The vibration isolation effect arises from the compressibility of the enclosed gas and the resistance generated by air flowing through the orifice. To evaluate the stiffness and damping of the subunit under various working conditions, sinusoidal displacement excitations with different amplitudes and frequencies are applied to the rubber end of the main air chamber. The resulting damping force at the fixed auxiliary chamber is recorded, and the force-displacement hysteresis loops are generated. Based on these, the equivalent dynamic stiffness (determined by the slope) and dynamic damping (inferred from the enclosed area) can be calculated based on Docquier et al. (2007).

The typical hysteresis loops of the damping windshield under varying amplitudes and frequencies are illustrated in Figure 4(a). It is seen that, at a fixed amplitude of 1 mm, as frequency increases, both the loop slope and enclosed area grow, indicating increased dynamic stiffness and damping. However, the growth trend gradually flattens at higher frequencies due to limited airflow through the orifice. As the frequency is fixed at 1 Hz, larger amplitudes result in wider loops with higher damping force and energy dissipation. The increase becomes less significant beyond 2 mm, suggesting nonlinear amplitude dependence with saturation behavior. The extracted equivalent stiffness, damping, and energy dissipation across the excitation conditions are summarized in Figure 4(b). In the figure, the equivalent stiffness shows two plateau regions at low and high frequencies, with a sharp rise in the mid-frequency range due to increased air flow resistance. The damping behavior reveals that the damping effect is most prominent at intermediate frequencies, corresponding to the optimal condition for orifice-induced resistance. Finally, energy dissipation follows a similar trend: minimal at very low and high frequencies, and maximized in the mid-frequency region.

Figure 4.

Dynamic characteristics of the vibration-damping windshield: (a) hysteresis loops; (b) dynamic parameters.

These results demonstrate that the dynamic response of the damping windshield is strongly nonlinear and frequency-dependent. The system exhibits low stiffness and damping at low frequencies due to unimpeded airflow; increased performance at mid-frequencies due to optimal air resistance; and reduced effectiveness at high frequencies where air cannot flow through the orifice rapidly enough. This behavior is governed by the airflow characteristics through the throttle orifice and contributes to effective vibration suppression by enhancing damping within the dominant vibration frequency range of the carbody.

4.2. Comparison with inter-vehicle dampers

The primary function of the multifunctional inter-car windshield is to attenuate vibration transmission between adjacent car bodies, aiming to replace conventional inter-vehicle dampers. To evaluate its performance, two key dynamic characteristic parameters of the inter-vehicle damper-equivalent stiffness and equivalent damping-are adopted as benchmarks. To accurately capture the actual dynamic characteristics of the inter-vehicle damper, experimental tests were conducted under various operating conditions. The test conditions were consistent with those used in the simulation of the windshield, that is, the excitation was generated by the relative displacement between the two car ends, with vibration amplitudes ranging from 0.5 mm to 3 mm and frequencies from 0.25 Hz to 10 Hz. The influence laws of frequency and amplitude on equivalent stiffness and equivalent damping were analyzed.

The test results are shown in Figure 5(a). Both equivalent stiffness and equivalent damping of the inter-vehicle damper vary with vibration amplitude and frequency, and their trends are generally consistent with those of the vibration-damping windshield. It can be seen that the equivalent stiffness of the inter-vehicle damper exhibits a significant dependence on both vibration frequency and amplitude. For all tested amplitudes, the stiffness increases rapidly in the low-frequency range (0−4 Hz) and then gradually approaches a saturation value at higher frequencies. Moreover, a smaller vibration amplitude results in a higher equivalent stiffness across the entire frequency range, with the 0.5 mm case exhibiting the largest stiffness and the 3.0 mm case the lowest. The equivalent damping decreases sharply with increasing frequency, especially in the range below 2 Hz, and then tends to stabilize at higher frequencies. The influence of vibration amplitude on damping is more pronounced in the low-frequency region, where larger amplitudes correspond to slightly higher damping values.

Figure 5.

Numerical and experimental results: (a) dynamic characteristics of the inter-car damper; (b) comparison between the inter-car damper and vibration-damping windshield.

Within the primary vibration frequency range of the carbody (0.5–3 Hz), the equivalent stiffness and damping of the windshield must reach levels comparable to those of the inter-vehicle damper to ensure equivalent vibration attenuation performance. Since the vibration amplitudes of the inter-vehicle damper in different working conditions are all below 2 mm, the average maximum amplitude (1 mm) was selected as the reference input for parameter comparison between a single coupler subunit of the windshield and the inter-vehicle damper. The comparison indicates that, at an amplitude of 1 mm, both the equivalent stiffness and damping provided by one coupler subunit are significantly lower than those of the inter-vehicle damper, with the gap in damping being more pronounced. Consequently, simply increasing the number of subunits cannot simultaneously match both parameters to the levels of the inter-vehicle damper. Further comparison results in Figure 5(b) demonstrate that when five subunits are connected in parallel, the combined equivalent stiffness and damping approach those of the existing inter-vehicle damper. This configuration is therefore capable of achieving a comparable vibration attenuation effect in practical applications.

5. Influences of the vibration-damping windshield on vehicle dynamics

Building upon the structural design, parameterized model, and identified dynamic characteristics of the novel vibration-damping windshield, this section quantitatively evaluates its influences on overall vehicle dynamics performance. The primary objective is to conduct a comprehensive comparative analysis between the proposed vibration-damping windshield (DW) and a traditional windshield (TW), employing a validated vehicle dynamics model subjected to representative operating scenarios. Specifically, it seeks to demonstrate how the inherent damping and stiffness properties of the vibration-damping windshield, functioning as an inter-vehicle structural element, contribute to enhanced vehicle dynamics, including improved vibration suppression and stability characteristics.

5.1. Coupled dynamics modeling integrating the vibration-damping windshield

To rigorously evaluate the impact of the novel vibration-damping windshield on the dynamic performance of the train, a comprehensive coupled dynamics model was established. This model integrates the detailed train dynamics with the specific behavior of the windshield acting as an inter-vehicle structural element.

A multibody dynamics model of a three-car train formation (Trailer car 1-Motor car 2-Trailer car 3, T1-M2-T3) was developed using SIMPACK. Each car, whether trailer (T1, T3) or motor car (M2), features a complete representation including 4 wheelsets, 8 axle boxes, 2 bogie frames, and 1 carbody, interconnected by primary and secondary suspension systems. The motor car (M2) additionally incorporates representations of its traction motors and gearbox. Adjacent car bodies are coupled via conventional inter-vehicle connection devices. To incorporate the dynamic characteristics of the proposed vibration-damping windshield, a co-simulation strategy was employed. The SIMPACK train dynamics model was interfaced with a dedicated model of the vibration-damping windshield implemented in MATLAB. Data exchange and synchronized simulation between the SIMPACK (multibody dynamics) and MATLAB (windshield model) environments were facilitated through Simulink, as illustrated schematically in Figure 6(a). This approach allows the complex, potentially frequency-dependent behavior of the windshield to directly influence the forces transmitted between car bodies.

Figure 6.

Multibody dynamic model of the train with the vibration-damping windshield: (a) sketch of the co-simulation model; (b) the wheel/rail profiles and contact relationship.

The wheel tread profile was modeled using the standard LMB10N, while the rail profile employed the standard CHN60. The corresponding geometric profiles of these components and the resulting equivalent conicity curve under their matching configuration are illustrated in Figure 6(b). As for the wheel-rail contact force, the nonlinear Hertzian theory is used to calculate the wheel/rail normal contact force and the FASTSIM algorithm is used for wheel-rail creep forces calculation. Within the vehicle suspension systems, the primary suspension incorporates axle box swing arm nodes, primary steel springs, and primary vertical dampers. The secondary suspension comprises air springs, secondary lateral dampers, anti-roll torsion bars, traction rods, lateral stops, and yaw dampers. Track irregularity excitation was applied using the WG50 spectrum, which can be referred to Track_2 in Ref [1].

5.2. Vehicle dynamics comparison

A comparative analysis of ride performance metrics, including the lateral Sperling index, and vertical Sperling index, was conducted for the trailer cars (TC1, TC3) and the motor car (MC2) under both conventional and vibration-damping windshield configurations. As illustrated in Figure 7, the vibration-damping windshield demonstrates a significant improvement in vehicle lateral ride quality and vertical ride quality compared to the conventional windshield. Notably, these performance gains are particularly pronounced within the 200–300 km/h speed range. Furthermore, analysis of performance trends versus speed reveals a distinct characteristic: at speeds exceeding 200 km/h, the motor car (MC2) exhibits higher values (indicating poorer performance) for all metrics than the trailer cars under the conventional windshield configuration. Conversely, this trend is reversed with the vibration-damping windshield installed, where the motor car shows superior performance improvement relative to the trailers. This reversal clearly indicates that the benefits of the vibration-damping windshield are more pronounced for the motor car compared to the trailer cars, especially at higher operating speeds.

Figure 7.

Comparison of carbody vibration-induced dynamics performance: (a) lateral Sperling index; (b) vertical Sperling index.

Within the scope of the evaluated speed range, the vertical Sperling index consistently remained within the “Excellent” classification limits for both configurations. However, the lateral Sperling index of the motor car (MC2) under the conventional windshield configuration exceeded the threshold limit of 2.5 at 300 km/h. Drawing upon established research experience,¹ this deterioration is likely attributable to abnormal lateral oscillation of the carbody.

To further investigate the differences in carbody vibration responses between the conventional and vibration-damping windshield configurations, Figure 8 presents a comparative analysis of translational and rotational accelerations at 300 km/h. The figure compares the maximum Root Mean Square (RMS) values calculated using a non-overlapping sliding window approach with a 5-second window duration. It can be seen that among translational accelerations, lateral vibration is the most pronounced, followed by vertical vibration, with longitudinal vibration being the least significant. Among rotational accelerations, roll vibration exhibits the highest magnitude, followed by yaw vibration, while pitch vibration is the least significant. Crucially, the vibration-damping windshield demonstrates superior vibration attenuation compared to the conventional windshield across all measured degrees of freedom. This improvement is particularly significant for longitudinal acceleration, lateral acceleration, and roll acceleration.

Figure 8.

Comparison of carbody vibration related to (a) translational accelerations; (b) rotational accelerations.

Consistent with the observations in Figure 7, Figure 8 further confirms that the vibration attenuation effect of the proposed windshield is significantly more pronounced in the motor car (MC2) compared to the trailer cars, particularly for longitudinal and lateral vibrations, achieving substantial reductions of 51.7% and 43.5%, respectively. To elucidate the specific manifestation of this damping effect, the carbody vibration response characteristics under both windshield configurations were comparatively analyzed in both the time and frequency domains, as presented in Figure 9.

Figure 9.

Comparison of motor car vibration in time and frequency domains in the direction of: (a) longitudinal; (b) lateral.

The results reveal the following key findings regarding longitudinal vibration: Under the conventional windshield configuration, the longitudinal acceleration exhibits multiple distinct dominant frequencies (1.6 Hz, 4.8 Hz, 5.4 Hz), the resonance frequencies are mainly associated with the longitudinal mode of the carbody and its coupling with the hunting mode. Upon replacing it with the vibration-damping windshield, these dominant frequencies are effectively suppressed. Indeed, significant attenuation is achieved for vibrations across the entire frequency band below 13 Hz. While a new dominant peak emerges at 14.6 Hz, its amplitude remains relatively low. For lateral vibration under the conventional windshield, the time-domain responses exhibit distinct harmonic characteristics. A dominant resonance-related peak at approximately 1.6 Hz is observed, corresponding to the carbody hunting mode involving coupled yaw and lateral motions. Crucially, this phenomenon is identified as the primary cause of the lateral Sperling index exceeding its limit at 300 km/h. When equipped with the vibration-damping windshield, the dominant frequency of these vibrations (1.6 Hz) remains unchanged. However, the corresponding amplitude is effectively suppressed, reduced to less than half of its original magnitude.

The aforementioned results focused on intrinsic carbody vibrations. To examine its influence on vibration transmission throughout the vehicle system, a comprehensive assessment of the vibration-damping windshield’s impact necessitates. Comparative analysis of vibration transmissibility characteristics under both windshield configurations was carried out, and the results are illustrated in Figure 10. It can be seen that for lateral and vertical vibrations, the primary suspension transmissibility (from axlebox to bogie) exhibits nearly identical behavior irrespective of windshield type. However, significant differences emerge in secondary suspension transmissibility (from bogie to carbody), where the vibration-damping windshield configuration demonstrates markedly superior vibration attenuation compared to the traditional design. Crucially, this enhanced attenuation across the secondary suspension is particularly pronounced in the motor car (MC2), yielding substantially greater improvement than observed in the trailer cars (TC1, TC3).

Figure 10.

Comparison of vibration transmission characteristics in the direction of: (a) lateral; (b) vertical.

6. Conclusions

This study proposes a novel windshield structure designed to attenuate inter-vehicle vibrations. A refined mechanical model of the windshield was developed based on its structural configuration to characterize its dynamic properties. The nonlinear mechanical model was further integrated into vehicle dynamics simulations to evaluate the suppression effect of the vibration-damping windshield on inter-vehicle vibrations. The main conclusions are given as follows:

(1) The designed windshield simultaneously fulfills conventional aerodynamic/structural requirements while providing supplemental inter-vehicle stiffness and damping. This integrated functionality effectively suppresses vibration transmission between coupled vehicles, achieving targeted vibration attenuation without compromising primary windshield performance. Based on the proposed vibration-damping windshield structure and working principle, a mechanical model was established, including a rubber air chamber sub-model, an auxiliary chamber sub-model, and a throttle sub-model.

(2) Dynamic characterization confirms that the windshield exhibits frequency- and amplitude-dependent dynamic stiffness and damping properties that are closely aligned with those of traditional inter-vehicle dampers. The dynamic stiffness increases with vibration frequency, ranging from relatively low values at low frequencies (below 1 Hz) to significantly higher levels at high frequencies (6–9 Hz). Within the primary vibration frequency range of the carbody, the equivalent dynamic stiffness reaches approximately 6 MN/m, which is comparable to that of conventional inter-vehicle dampers. This dynamic equivalence enables the windshield to replicate the core vibration control functions of dedicated dampers, thereby validating its role as a multifunctional structural element.

(3) The implementation of the vibration-damping windshield demonstrably enhances secondary suspension isolation performance, achieving significant vibration attenuation across all carbody degrees of freedom. This efficacy extends to suppressing critical low-frequency carbody swaying behavior, directly improving lateral ride quality. Notably, these vibration control benefits exhibit differential effectiveness across vehicle types, delivering substantially superior performance gains in motor cars compared to trailer units.

(4) While establishing a robust theoretical and simulation foundation, this research remains at the proof-of-concept stage. Subsequent efforts will focus on prototype development, experimental validation under operational conditions, and optimization for manufacturing feasibility.

Footnotes

ORCID iDs

Liangcheng Dai

Jianfeng Sun

Cangpeng Zhao

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Key R&D Program of China (2022YFB4301202 and 2022YFB4301303), the National Natural Science Foundation of China under Grant Nos. 52388102 and 52405139, the Zhejiang Province basic public welfare plan research project under Grant No. LQ24E050013, and Open Project of The National Key Laboratory of Rail Transit Vehicle System under Grant No. RVL2514.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

All data included in this study are available upon request to the corresponding author.*

References

Alonso

Gil-Negrete

Nieto

, et al. (2013) Development of a rubber component model suitable for being implemented in railway dynamic simulation programs. Journal of Sound and Vibration 332(12): 3032–3048. https://doi.org/10.1016/j.jsv.2013.01.002

Chang

Ding

Ling

, et al. (2023) Mechanism of high-speed train carbody shaking due to degradation of wheel–rail contact geometry. International Journal of Reality Therapy 11(3): 289–316. https://doi.org/10.1080/23248378.2022.2077850

Ding

Chang

Ling

, et al. (2024) Mechanism analysis of low-frequency swaying motion of high-speed trains induced by aerodynamic loads. Journal of Vibration and Control 30(13–14): 3141–3153. https://doi.org/10.1177/10775463231190445

Docquier

Fisette

Jeanmart

(2007) Multiphysic modelling of railway vehicles equipped with pneumatic suspensions. Vehicle System Dynamics 45(6): 505–524. https://doi.org/10.1080/00423110601050848

Facchinetti

Mazzola

Alfi

, et al. (2010) Mathematical modelling of the secondary airspring suspension in railway vehicles and its effect on safety and ride comfort. Vehicle System Dynamics 48(1): 429–449. https://doi.org/10.1080/00423114.2010.486036

Fujimoto

Masayuki

(1996) Lateral vibration and its decreasing measure of a Shinkansen train (decrease of train vibration with yaw damper between cars). Vehicle System Dynamics 25(S1): 188–199. https://doi.org/10.1080/00423119608969195

Fujimoto

Miyamoto

(1996) Measures to reduce the lateral vibration of the tail car in a high speed train. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 210(2): 87–93. https://doi.org/10.1243/pime_proc_1996_210_331_02

Huang

Dai

(2023) Suppression of carbody hunting for a locomotive caused by low wheel–rail contact conicity by optimising yaw damper location. Vehicle System Dynamics 62(9): 2237–2259. https://doi.org/10.1080/00423114.2023.2280217

Jiang

Zhang

Ling

, et al. (2024) The effect of yaw damper performance degradation on carbody hunting of high-speed trains. Vehicle System Dynamics 62(12): 3122–3145. https://doi.org/10.1080/00423114.2024.2317397

10.

Jin

Chen

(2025) Investigation into the reduction of aerodynamic drag via external windshields on high-speed trains. Journal of Engineering and Applied Science 72(1): 7.

11.

Yang

Qian

, et al. (2023) Investigation on the fluid-solid coupling effect in high-speed train’s windshield structure. Alexandria Engineering Journal 74: 139–152. https://doi.org/10.1016/j.aej.2023.05.034

12.

Niu

Wang

Zhou

(2019) Effect of the outer windshield schemes on aerodynamic characteristics around the carconnecting parts and train aerodynamic performance. Mechanical Systems and Signal Processing 130: 1–16. https://doi.org/10.1016/j.ymssp.2019.05.001

13.

Dai

Song

, et al. (2019) Shaking analysis of high-speed train’s carbody when cross lines. Journal of Mechanical Science and Technology 33(3): 1055–1064. https://doi.org/10.1007/s12206-019-0205-5

14.

Sun

Zhou

Qin

(2017) Research on influence of the characteristic parameter of inter-vehicle longitudinal damper on dynamic performance of high speed EMUs. Journal of Mechanical Engineering 53(24): 170–176. https://doi.org/10.3901/jme.2017.24.170

15.

Sun

Chi

Jin

, et al. (2021) Experimental and numerical study on carbody hunting of electric locomotive induced by low wheel–rail contact conicity. Vehicle System Dynamics 59(2): 203–223. https://doi.org/10.1080/00423114.2019.1674344

16.

Tang

Xiong

, et al. (2022) Vibration characteristics of outer windshield structures of high-speed trains based on fluid–structure interactions. Nonlinear Dynamics 111(3): 2111–2132. https://doi.org/10.1007/s11071-022-07943-0

17.

Tang

Xiong

, et al. (2023) Dynamic response of outer windshield structure in different schemes under aerodynamic load. Applied Sciences 13(6): 3879. https://doi.org/10.3390/app13063879

18.

Tanifuji

Sakanoue

Kikko

(2008) Modelling of aerodynamic force acting on high speed train in tunnel and a measure to improve the riding comfort utilising restriction between cars. Vehicle System Dynamics 46(sup1): 1065–1075. https://doi.org/10.1080/00423110802158747

19.

Toshimitsu

Tanifuji

Soma

(2007) Potential of improving riding comfort of high-speed train with inter-vehicle lateral damper. Transactions of the Japan Society of Mechanical Engineers - Part C 73(733): 2485–2492. https://doi.org/10.1299/kikaic.73.2485

20.

Wang

Liu

, et al. (2024) Optimisation of an inter-vehicle damper for a high-speed train. Vehicle System Dynamics 62(10): 2703–2728. https://doi.org/10.1080/00423114.2024.2302490

21.

Zhao

Dai

Chi

, et al. (2024) Optimization and design of inter-vehicle longitudinal damper parameters of a high-speed train oriented to carbody shaking caused by low equivalent conicity. Journal of Railway Science and Engineering 21(10): 3956–3968.

22.

Zhao

Zhang

Ling

, et al. (2024) Bio-inspired structure constraints following control for enhancing hunting stability of high-speed trains. IET Control Theory & Applications 18(3): 316–334. https://doi.org/10.1049/cth2.12562

23.

Zhou

Qin

Sun

, et al. (2017) Study on influence of inter-vehicle longitudinal damper on dynamic performance of high-speed EMU. Journal of the China Railway Society 39(6): 20–27.

24.

Zhou

Liu

Carnevale

, et al. (2023) Thermodynamic models of the air spring on the high-speed pantograph. Vehicle System Dynamics 62(2): 463–489. https://doi.org/10.1080/00423114.2023.2178016