Research on rigid-flexible coupling dynamics modeling for rack vehicle

Research on rigid-flexible coupling dynamics modeling for rack vehicle | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Research on rigid-flexible coupling dynamics modeling for rack vehicle Chenglong Dong, Xingqiao Deng, Jialin Liu, Zhendong Zhang, Chang Gao, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4875774/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In order to address the challenge of quantifying the impact of changes in slope and driving speed on individual components of rack vehicle, which is not feasible through experimental methods in practical engineering. Thus, based on the vehicle-track dynamics theory, this paper establishes a rigid-flexible coupling dynamics model of rack vehicle, which consists of vehicle submodel, track submodel, and gear-rack transmission model. In the model, the bogie frames are linked with the wheelsets through the primary suspensions and linked with the vehicle through the secondary suspensions, respectively, while three-dimensional spring-damper elements are used to represent the stiffness and damping characteristics of the primary and secondary suspension in three directions. Furthermore, regard the connection relationship between rack and sleeper as elastic damping, and the wheel-rail contact relationship is established on the nonlinear Hertz contact theory. Based on this model, the dynamic characteristics of rack vehicle are analyzed under various slopes and driving speeds. Results show that the slope has a substantial effect on the dynamic characteristics of each component of rack vehicle, whereas the driving speed primarily affects the root bending stress of rack and the vibration acceleration of gear. The research results is of great significance to the safety and stability of rack vehicle. rack vehicle finite element analysis dynamic characteristic Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1. Introduction As a form of railway transportation, the rack vehicle demonstrates exceptional climbing capabilities and carrying capacity, rendering it extensively utilized in mountainous regions [ 1 – 2 ]. The stability and safety of rack vehicle during operation are affected by many factors, among which the influence of slope and driving speed is particularly obvious. Numerous studies had been conducted on the vehicle system dynamics of traditional trains [ 3 ]. However, the inclusion of gear-rack system in rack vehicle adds complexity to system dynamic characteristics, making it unsuitable to apply traditional train standards for evaluating the stability and safety of rack vehicle. Furthermore, the existing research literature on the dynamic model of rack vehicle is currently limited, leading to a scarcity of studies on vehicle dynamics characteristics of rack vehicle. Current researches in vehicle system dynamics primarily focuse on traditional trains and high-speed and heavy-haul trains. Zhai et al. [ 4 ] developed a comprehensive 35-degree-of-freedom multi-body model. This model offers a systematic approach to optimizing the design parameters of vehicles and track components, providing a theoretical foundation for future research on the vehicle dynamic characteristics and the interaction between vehicle and track. On this basis, Baeza et al. [ 5 ] simply regard the track as a beam in order to simulate the dynamic interaction between the vehicle and the track. However, using a single factor variable cannot provide an accurate analysis of the vehicle dynamic characteristics. In practical engineering, the dynamic characteristics of vehicle are often influenced by multiple variables. Thus, Sun et al. [ 6 ] utilized Green's function to perform dynamic response analysis on vertical vehicle dynamics model. They investigated the vibration response of both rigid and flexible vehicle bodies on three different track models, and examined the impact of track stiffness and ballast stiffness on trains. Based on the traditional longitudinal dynamics model of train, Cole et al. [ 7 ] proposed a combined simulation approach integrating train longitudinal simulation, locomotive traction control, and rolling stock dynamics control. The study examined the effects of lateral forces and coupler impacts on the lateral and longitudinal dynamics of trains. Huang et al. [ 8 ] utilized a well-established vehicle-track coupling dynamics model to investigate the vertical vibration response of trains on floating slab track under various conditions, including different lengths, thicknesses, and vertical damping. The study also involved optimizing floating slab track's parameters. Numerous studies have been conducted on the dynamics of traditional trains. However, there is a noticeable gap in research literature regarding the dynamics of rack vehicle. In current researches, Chen et al. [ 9 ] established a finite element model for gear and rack to analyze the dynamics of gear meshing forces, vertical forces between wheel and rail, and vehicle acceleration when subjected to track random irregularity. They examined response characteristics and investigated how rack base deflection deformation affects the dynamic response of rack. At the same time, Chen et al. [ 10 ] investigated the vibration response characteristics of gear-rack system in three directions when subjected to track random irregularities. Their findings indicated that the vertical and longitudinal vibration of gear-rack system is less affected by the irregularity excitation, while the lateral vibration is significantly affected. The existing literature also examine the influence of different factors on the dynamic characteristics of rack vehicle, such as the layout mode of traction motor [ 11 ], pier settlement [ 12 ], the gear time-varying mesh stiffness [ 13 ], temperature load, nonlinear dynamic meshing characteristics of the gear-rack system, and the nonlinear dynamic contact characteristics of wheel-rail system [ 14 ], etc. Their research content holds significant value in enhancing the dynamic performance of rack vehicle and advancing the research progress in this field. However, the existing studies did not establish a comprehensive rigid-flexible coupling dynamic model of rack vehicle, or fail to fully consider the connection and coupling relationships among various components of rack vehicle in these established model. In this paper, a rigid-flexible coupling dynamics model of rack vehicle is established. In the model, the interconnection and coupling relationships between each component are comprehensively considered, and the contact relationship between wheel and rail is analyzed based on the nonlinear Hertz contact theory. Furthermore, based on the model, this paper analyzes the impact of various slopes and driving speeds on the dynamic characteristics of individual train components, investigates the working conditions necessary for the safe and stable operation of rack vehicle, and offers theoretical insights for designing and optimizing each part of rack vehicle. 2. Dynamic model of rack vehicle Based on the vehicle-track dynamics theory [ 15 ], a complete rigid-flexible coupling dynamics model of rack vehicle is established, as shown in Fig. 1. The model is consists of vehicle submodel, track submodel [ 16 ], and gear-rack transmission model. The interconnection and coupling relationship between each submodel is clearly express-esed in Fig. 1, and the main parameters of rack vehicle are listed in Table 1 .</p Table 1 Main parameters of rack vehicle. Notation Parameter Units \({M_{\text{c}}}\) Mass of vehicle kg \({M_{\text{t}}}\) Mass of bogie frame kg \({M_{\text{w}}}\) Mass of wheelset kg \({M_{\text{m}}}\) Mass of gear kg \({M_{\text{s}}}\) Mass of sleeper kg \({M_{\text{b}}}\) Mass of ballast kg \({I_{\text{c}}}\) Mass moment of inertia of vehicle \({\text{kg}} \cdot {{\text{m}}^2}\) \({I_{\text{t}}}\) Mass moment of inertia of bogie \({\text{kg}} \cdot {{\text{m}}^2}\) \({I_{\text{w}}}\) Mass moment of inertia of wheelset \({\text{kg}} \cdot {{\text{m}}^2}\) \({I_{\text{m}}}\) Mass moment of inertia of gear \({\text{kg}} \cdot {{\text{m}}^2}\) \({K_{{\text{si}}}}\) Stiffness of secondary suspension in three direction( X , Y , Z ) \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{pi}}}}\) Stiffness of primary suspension in three direction( X , Y , Z ) \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{pv}}}}\) Fastener stiffness of rail in Z direction \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{ph}}}}\) Fastener stiffness of rail in Y direction \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{bv}}}}\) Ballast stiffness in Z direction \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{bh}}}}\) Ballast stiffness in Y direction \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{{\text{fv}}}}\) Subgrade stiffness \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{\text{w}}}\) Ballast shear stiffness \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({K_{\text{r}}}\) Fastener stiffness of rack \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{si}}}}\) Damping of secondary suspension in three direction( X , Y , Z ) \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{pi}}}}\) Damping of primary suspension in three direction( X , Y , Z ) \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{pv}}}}\) Fastener damping of rail in Z direction \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{ph}}}}\) Fastener damping of rail in Y direction \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{bv}}}}\) Ballast damping in Z direction \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{bh}}}}\) Ballast damping in Y direction \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{{\text{fv}}}}\) Subgrade damping \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{\text{w}}}\) Ballast shear damping \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({C_{\text{r}}}\) Fastener damping of rack \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) \({Z_{\text{i}}}\) Displacement in Z direction (vehicle, bogie frame, wheelset, rail, sleeper, ballast) mm 2.1. Vehicle submodel The vehicle submodel is consists of a vehicle, two bogie frames, and four wheelsets. In the submodel, the vehicle is supported on two double-axle bogie frames at each end, and the bogie frames are linked with the vehicle through the secondary suspensions. Moreover, the stiffness and damping characteristics of the secondary suspensions in three directions are completely considered in the modeling process. In dynamic analysis, the vibration characteristics of rack vehicle are significantly affected by whether the structure is rigid or flexible. Rigid components do not generate deformation in the dynamic simulation, resulting in only overall translation and rotation. Furthermore, this model does not account for the influence of structural deformation on the vehicle and bogie frame, thus treating them as discrete rigid bodies [ 17 ]. If the wheelset is considered as a rigid component and directly contact with rail, the vibration and noise generated by the wheel-rail contact will be transmitted to vehicle through the wheel and frame. The large vibration generated will not only impact the vehicle, but also pose a threat to the stable operation of rack vehicle. Therefore, in this paper, both the wheelset and axle are set as elastomeric components [ 18 ], with the aim of mitigating wheel-rail contact forces, minimizing wheel-rail wear, reducing wheel-rail vibration and noise, and enhancing the operational stability of rack vehicle. The bogie frames are linked with the vehicle through the secondary suspensions, which primarily consists of air springs, vertical shock absorber, lateral shock absorber, and anti-roll dampers [ 19 ]. The secondary suspension effectively minimizes the transmis-sion of vibration excitation to vehicle and significantly improves the stability of vehicle operation. The axle and wheelset are secured with tie constraint, while the bogie frames are linked with the axles through the primary suspensions. Three-dimensional spring-damper elements are used to represent the stiffness and damping characteristics of the primary and the secondary suspensions in three directions (vertical, longitudinal and lateral), respectively. The specific parameters are shown in Table 2 . Table 2 The stiffness and damping of model. Parameter Description Vertical( Z ) Longitudinal( X ) Lateral( Y ) Stiffness( \({\text{N}} \cdot {{\text{m}}^{{\text{-1}}}}\) ) Primary suspension 2900000 10056000 12700000 Secondary suspension 205000 900000 900000 Fastener of rail 800000 15056000 15056000 Fastener of rack 15056000 15056000 15056000 Damping( \({\text{N}} \cdot {\text{s}} \cdot {{\text{m}}^{{\text{-1}}}}\) ) Primary suspension 10000 50000 50000 Secondary suspension 36000 50000 72000 Fastener of rail 15000 60000 75000 Fastener of rack 75000 75000 75000 2.2. Track submodel Generally there are two types of track submodels: ballasted track submodel and ballastless track submodel [ 20 ]. The rack vehicle is designed for low speed operation, therefore ballasted track submodel is utilized in this paper. The ballasted track submodel is represented as a flexible model, primarily consisting of rails, rail pads, sleepers, ballast and subgrade [ 21 ]. The model takes into account the vertical vibration displacement of rail, sleeper, and ballast. Furthermore, both the left and right sides of rail are treated as continuous Timoshenko beams, which are discretely supported at rail-sleeper junctions by three layers of springs and dampers, representing the elasticity and damping of rail pad, ballast and subgrade, respectively. As shown in Fig. 1. The wheel-rail contact model plays a crucial role in integrating the vehicle submodel with the track submodel. Furthermore, the accuracy of rigid-flexible coupling dynamics model of rack vehicle is heavily reliant on both the wheel-rail contact model, the primary suspensions, and the secondary suspensions. The impact of the primary and secondary suspensions on the dynamics model has been thoroughly addressed in previous research. The wheel-rail contact force in the wheel-rail contact model consists of the normal contact force and the wheel-rail creep force. Based on the elastic compression deforma-tion at the normal contact point, utilizing the nonlinear Hertz contact theory to calculate the wheel-rail normal contact force [ 22 ]. The Hertz contact force calculation formula can be expressed in Eq. ( 1 ): $$P\left( t \right)={\left[ {\frac{1}{G}\delta Z\left( t \right)} \right]^{3/2}}$$ 1 where G is the wheel-rail contact constant; \(\delta Z\left( t \right)\) is the elastic compression between wheel and rail. $$\delta Z\left( t \right)={Z_{\text{w}}}\left( t \right) - {Z_{\text{r}}}\left( {{x_{\text{w}}},t} \right) - {Z_0}\left( t \right)$$ 2 where \({Z_{\text{W}}}\) is the vertical displacement of wheel; \({Z_{\text{r}}}\) is the vertical displacement at the contact point; \({Z_0}\left( t \right)\) denotes the geometrical irregularities of rail and wheel surfaces. 2.3. Gear-rack transmission model The rack utilized in this paper is Strub system, and the relevant parameters of gear and rack are shown in Table 3 . Regarding the connection method between rack and sleeper, Zhang et al. [ 23 ] fixed the connection between rack and sleeper through fasteners, effectively integrating them into a single unit. Nevertheless, this approach may deviate from engineering practices, leading to significant discrepancies and impacting the analysis results. Thus, in this paper, the Strub system is simulated as a Timoshenko beam supported by continuous elastic discrete points, incorporating elastic damping elements to simulate the behavior of fasteners. The interaction between rack and sleeper involves both stiffness and damping effects. Table 3 Parameters of gear and rack. Parameter Modulus (mm) Number of teeth Materials Width (mm) Pressure angle (°) Gear 100/π 22 18CrNiMo7-6 86 14.036 Rack 24 42CrMo 60 3. Dynamic characteristics of rack vehicle under different working conditions Based on the rigid-flexible coupling dynamics model of rack vehicle, the finite element model of rack vehicle is established, as shown in Fig. 2 . The dynamic characteristics of rack vehicle are investigated under various slopes and driving speeds conditions, encompassing the dynamic response of vehicle submodel, track submodel, gear-rack transmission model, and the contact characteristics of gear-rack contact surfaces. By studying the dynamic characteristics of each component and the root bending stress of rack, a reasonable value for slope and driving speed is proposed to ensure the stability and safety of rack vehicle. 3.1. Dynamic response of rack vehicle under different slopes The variation in train running slope has a significant impact on the stability and safety of rack vehicle. This section aims to investigate the effects of slope changes on various components of rack vehicle, including wheel-rail contact force, root bending stress of rack, and dynamic characteristics of each component, within the range of 60‰-300‰. Figure 3 reveals the vertical contact forces between wheelset and rail under different slopes. In Fig. 3(a), the variation of wheel-rail vertical contact force is illustrates when the slope is 60‰, and the trend of wheel-rail contact force at other slopes is comparable to that of 60‰. It can be seen from the Fig. 3(a) that the vertical contact force of wheel-rail changes periodically during uniform motion. The reason is that during the rotation of wheel, different positions of wheel come into contact with rail, causing the contact force to appear up and down in this cycle. During the time interval of 2.5s-4.5s, the periodic variation in wheel-rail vertical contact force is attributed to the cyclic rotation of wheel. Moreover, due to track random irregularity excitation, weld seams of track, etc. The change trend of wheel-rail vertical contact force exhibits variations at 2.8s and 3.5s. Figure 3(b) shows the average wheel-rail vertical contact force under different slopes. It can be seen from the Fig. that the wheel-rail vertical contact force gradually decreases as the slope increases. When the slope increases from 60‰ to 300‰, the wheel-rail vertical contact force decreases from 71.28kN to 66.35kN. The wheel-rail contact force is reduced by approximately 6.9%. Figure 4 shows the vertical vibration acceleration and displacement of vehicle under different slopes. In order to explore the running stability of rack vehicle, the vibration acceleration of vehicle is shown in Fig. 4(a). It can be seen from the Fig. that the vibration acceleration of vehicle increases with the increase of slope. Due to the vibration direction of vehicle is different from the direction of coordinate axis, the vibration acceleration of vehicle can be either positive or negative, which has little impact on the results. According to the above analysis, the wheel-rail vertical contact force gradually decreases with the increase of slope. It shows that the vertical force exerted by each component on rail gradually decreases. Furthermore, all parameters of the secondary suspension are the same, with the support of the secondary suspension, the vertical displacement of vehicle gradually decreases as slope increases. In Fig. 4(b), the vertical displacement of vehicle under different slopes are shown. When the slope increases from 60‰ to 300‰, the vertical displacement of vehicle decreases by about 20mm. Variations in wheel-rail vertical contact force not only influence the vertical vibration acceleration and vertical displacement of vehicle, but also exert a significant influence on vertical displacement of rail. When suffered to a significant vertical force, the rail will has a substantial deformation, leading to increased wear and potential interference with the contact between wheel and rail. In Fig. 5 , the vertical displace-ment of rail under different slopes are shown. The Fig. illustrates that as the slope increases, the vertical displacement of the rail decreases. Specifically, when the slope increases from 60‰ to 300‰, the maximum vertical displacement of rail decreases from 1.03mm to 0.98mm. The analysis of the wheel-rail vertical contact force reveals that as slope increases, the vertical contact force between wheel and rail decreases gradually, while the longitudinal force increases gradually. This results in an increase in slope resistance that the train must overcome during the climbing process. Nevertheless, the incorporation of gear-rack system can significantly compensate for the traction deficiency. Figure 6 shows the root bending stress maps of rack under different slopes, (a)-(e) correspond to five slopes in 60‰-300‰, respectively. As can be seen from the Figure, when the slope increases from 60‰ to 300‰, the stress map of the gear-rack contact surface gradually becomes larger, indicating that the contact force between gear and rack increases gradually. The variation trend of the maximum root bending stress of rack under different slopes is not significant. Therefore, the value of the maximum root bending stress of rack is presented separately in a bar chart format, as shown in Fig. 7 . It can be observed from the Figure. that there is an increasing trend in the maximum root bending stress of rack with an increase in slope. When the slope increases from 60‰ to 300‰, the maximum root bending stress of rack rising from 79.73MPa to 84.15Mpa, with a increase rate of about 5.5%. 3.2. Dynamic response of rack vehicle under different driving speeds The following section will investigate the varying characteristics of each component of the train at different driving speeds. The train will be operated at constant speeds of 15km/h, 20km/h, 25km/h, 30km/h, and 35km/h respectively on a slope of 120‰. Figure 8 illustrates the variations in wheel-rail vertical contact force at various driving speeds. For instance, at 30km/h, the fluctuation in the wheel-rail vertical contact force is depicted in Fig. 8(a). Similar trends can be observed for the wheel-rail contact force at other driving speeds. The Fig. illustrates the periodic variation of the wheel-rail contact force at a driving speed of 30km/h. The factors influencing this change are comparable to those observed at a 60‰ slope in Section3.1. The average of wheel-rail contact force at different driving speeds is shown in Fig. 8(b). As can be seen from the Fig., when the driving speed of train increases from 15km/h to 35km/h, the change of wheel-rail contact force is not obvious, in which the difference between the minimum value (68.68kN) and the maximum value (69.33kN) is only 0.65kN, which is negligible within the allowed range of error. It can be seen that different driving speeds have little effect on the vertical contact force between wheel and rail. In Fig. 9(a), the vertical vibration acceleration and vertical displacement of vehicle at different driving speeds is shown. It can be seen that the vertical vibration acceleration of vehicle does not change significantly with the driving speed, and the vibration acceleration fluctuates within the range of approximately ± 0.01 at different driving speeds. This is because the driving speed of rack vehicle belongs to the low-speed driving mode, and the dynamics of vehicle is not sensitive to the change of speed. Furthermore, the vertical displacements of vehicle at different driving speeds are also the same, as shown in Fig. 9(b). It can be seen from the Fig. that the vertical displacement of vehicle is basically maintained at about 26mm, and the amplitude of the up and down fluctuations is not obvious. The vertical displacement of rail at different driving speeds is illustrated in Fig. 10 . Due to the varying locations of the probe points within the finite element, the displacement of rail appears staggered at different driving speeds, but its changing trend remains consistent. It can be observed that the impact of different driving speeds on the vertical displacement of rail is not significant, with a range between 1.03mm and 1.05mm. Figure 11 illustrates the root bending stress of rack at various driving speeds, ranging from 15km/h to 35km/h denoted as (a)-(e). The diagram reveals that the bending stress does not follow a consistent pattern across different driving speeds. The changing trend of the maximum root bending stress of rack is shown in Fig. 12 . When the train operates at various speeds, the rise in gear speed results in an escalation of the meshing frequency between gear and rack, consequently amplifying the intricacy of the nonlinear dynamics within gear-rack system [ 24 ]. The contact condition between gear and rack is significantly influenced by the multi-state meshing behavior, subsequently causing fluctuations in the root bending stress of rack, displaying a trend of initially decreasing, then increasing, and finally decreasing. Based on the above analysis, the meshing frequency of gear-rack system varies with the driving speed, so that the vertical vibration acceleration of gear is changed. In Fig. 13 , the root mean square of the vertical vibration acceleration of gear at different driving speeds is shown. It can be seen that the change trend of the vertical vibration acceleration of gear aligns with the maximum bending root stress of rack, providing additional confirmation of the impact of gear-rack meshing frequency on both the maximum root bending stress of rack and gear vibration acceleration. Therefore, it is recommended that the speed of rack vehicle is 25km/h. 4. Conclusion Based on the vehicle-track dynamics theory, a complete rigid-flexible coupling dynamics model of rack vehicle is established in this paper. Subsequently, the dynamic characterist-ics of train under various slopes and driving speeds are investigated. Based on the research results, the conclusions are as follows: (1) The slope and driving speed play crucial roles in determining the dynamic characteristics of rack vehicle. The slope affects all the components of rack vehicle significantly, whereas the driving speed specifically impacts the maximum root bending stress of rack and the vertical vibration acceleration of gear. (2) Under varying slope conditions, the wheel-rail contact force, the displace-ment of vehicle and rail all decrease as slope increases. However, the maximum root bending stress of rack actually increases with slope, and this increase is significantly greater than the decrease seen in parameters like the wheel-rail contact force, the displacement of vehicle, etc. Therefore, it is recommended that the slope of rack vehicle should not exceed 300‰ in order to ensure the safety of train operations. (3) Under varying driving speed conditions, the maximum root bending stress of rack and the vertical vibration acceleration of gear displaying a trend of initially decreasing, then increasing, and finally decreasing. Moreover, when the driving speed of train is 25km/h, the maximum root bending stress of rack and the vertical vibration acceleration of gear are the minimum. Therefore, it is recommended that the driving speed of rack vehicle is 25km/h. Declarations Competing interests The authors declare no competing interests. Author contributions Chenglong Dong: Conceptualization, methodology, software, validation, formal analysis, data curation, writing-original draft preparation. Xingqiao Deng: Conceptualization, writing-review and editing, visualization, methodology, resources, project administration, funding acquisition. Jialin Liu: Methodology, software, validation. Zhendong Zhang: Methodology, validation, data curation. Chang Gao: Investigation, formal analysis. Weiping Liu: Investigation, formal analysis. Shisong Wang: Conceptualization, methodology, supervision, software, validation. Acknowledgements The research was supported by the Sichuan Shudao New System Rail Group CO., LTD. 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Nonlinear Dyn 106(3):2035–2060. 10.1007/s11071-021-06891-5 Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4875774","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":341731185,"identity":"c426476b-77ad-4fce-84ba-c04654612e89","order_by":0,"name":"Chenglong Dong","email":"data:image/png;base64,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","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Chenglong","middleName":"","lastName":"Dong","suffix":""},{"id":341731186,"identity":"3e06b3b0-ecf3-49f7-ac0e-37fd7fc23c34","order_by":1,"name":"Xingqiao Deng","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Xingqiao","middleName":"","lastName":"Deng","suffix":""},{"id":341731187,"identity":"3ecc9e8d-42ff-41c9-9d40-9b88afe515dd","order_by":2,"name":"Jialin Liu","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Jialin","middleName":"","lastName":"Liu","suffix":""},{"id":341731188,"identity":"da88e3de-e235-4816-8d69-d655e71c862c","order_by":3,"name":"Zhendong Zhang","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Zhendong","middleName":"","lastName":"Zhang","suffix":""},{"id":341731189,"identity":"1ad393b0-67a4-4822-9b3f-f6533a181c67","order_by":4,"name":"Chang Gao","email":"","orcid":"","institution":"Sichuan Shudao New Standard Rail Group Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Chang","middleName":"","lastName":"Gao","suffix":""},{"id":341731190,"identity":"c73301e0-6480-4619-be35-bb5ac35ea603","order_by":5,"name":"Weiping Liu","email":"","orcid":"","institution":"Sichuan Shudao New Standard Rail Group Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Weiping","middleName":"","lastName":"Liu","suffix":""},{"id":341731191,"identity":"55c197e2-e587-483b-881c-5eb9501e8f65","order_by":6,"name":"Shisong Wang","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Shisong","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2024-08-07 15:18:57","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4875774/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4875774/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":64457663,"identity":"f4003446-9a15-4692-8b7b-772077ca6216","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":146383,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic model of rack vehicle. (a) Elevation (b) End view\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/7676b9a025599e304c3f8bbe.png"},{"id":64457659,"identity":"ade842ec-fad9-44d6-b548-926e22a8197b","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":458595,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element model of rack vehicle.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/535f979f77d8acf2639b600d.png"},{"id":64458314,"identity":"703600de-c39e-43d6-9739-2a4b54e223b2","added_by":"auto","created_at":"2024-09-13 12:08:42","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":126928,"visible":true,"origin":"","legend":"\u003cp\u003eVertical contact forces. (a) Wheel-rail vertical contact force at 60‰ (b) Average of wheel-rail vertical contact force under different slopes.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/8f942464703012a8f2bf970e.png"},{"id":64458820,"identity":"1a6cc0c2-804b-42f1-81fb-ad166cd533a0","added_by":"auto","created_at":"2024-09-13 12:16:42","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":97764,"visible":true,"origin":"","legend":"\u003cp\u003evertical vibration acceleration and displacement of vehicle under different slopes.(a) Vertical vibration acceleration (b) Vertical displacement.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/7a21ab2b8ff8ba4b607b8e47.png"},{"id":64457660,"identity":"3c6fc37a-e69e-4f6c-a8cb-c850eff3c63c","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":87421,"visible":true,"origin":"","legend":"\u003cp\u003eVertical displacement of rail.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/34edec73501c75624862ee79.png"},{"id":64457661,"identity":"cf01f589-9217-4134-9b83-598f14e45ec7","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":323608,"visible":true,"origin":"","legend":"\u003cp\u003eThe root bending stress of rack under different slopes.(a) 60‰ (b) 120‰ (c)180‰ (d) 240‰ (e) 300‰.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/6df4030a5d4a5ef5708c5fbd.png"},{"id":64457665,"identity":"c3d911e6-1806-43e6-88f4-bca52c77f3d9","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":295440,"visible":true,"origin":"","legend":"\u003cp\u003eThe maximum root bending stress of rack under different slopes.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/7e5a7fc7aaf3f2f72e3cbc22.png"},{"id":64458312,"identity":"df7a5e6b-048d-41f5-9ede-eaa0e51769a3","added_by":"auto","created_at":"2024-09-13 12:08:42","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":167403,"visible":true,"origin":"","legend":"\u003cp\u003eVertical contact forces. (a) Wheel-rail vertical contact force at 30km/h (b) Average of wheel-rail vertical contact force under different driving speeds.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/dd2d63e7f9060a68d30e91c0.png"},{"id":64457668,"identity":"8b55a22f-1d53-47dd-b065-27495c55e2ed","added_by":"auto","created_at":"2024-09-13 12:00:42","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":149762,"visible":true,"origin":"","legend":"\u003cp\u003eVertical vibration acceleration and displacement of vehicle under different driving speeds.(a) Vertical vibration acceleration (b) Vertical displacement.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/180dbe996ef602f5b4a39b32.png"},{"id":64457669,"identity":"8fd33ea6-f223-4a89-9fe6-a21ebd3ead62","added_by":"auto","created_at":"2024-09-13 12:00:43","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":94609,"visible":true,"origin":"","legend":"\u003cp\u003eVertical displacement of rail.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/9eab860269a2f0487403ab85.png"},{"id":64457671,"identity":"f672c8b7-8ee8-4b21-bd12-e4b05b731a17","added_by":"auto","created_at":"2024-09-13 12:00:43","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":308000,"visible":true,"origin":"","legend":"\u003cp\u003eThe root bending stress of rack. (a) 15km/h (b) 20km/h (c) 25km/h (d) 30km/h (e) 35km/h.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/96ff7fd450bf861cd28ae833.png"},{"id":64457670,"identity":"01903dc9-770a-4977-85c9-b891f4caf398","added_by":"auto","created_at":"2024-09-13 12:00:43","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":268056,"visible":true,"origin":"","legend":"\u003cp\u003eThe maximum root bending stress of rack at different driving speeds.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/5250467f8423eaf20a094c0d.png"},{"id":64458821,"identity":"434a2a3d-3217-49a1-a472-e5f937302a4e","added_by":"auto","created_at":"2024-09-13 12:16:42","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":245405,"visible":true,"origin":"","legend":"\u003cp\u003eRoot mean square of gear vertical vibration acceleration.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/debabc0c31e0adac058d0ac5.png"},{"id":66416523,"identity":"4b82f73f-3269-4f20-bf70-8740fe8421ab","added_by":"auto","created_at":"2024-10-11 14:43:54","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3260608,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4875774/v1/8165d58f-1535-4359-8839-f6611a398baa.pdf"}],"financialInterests":"","formattedTitle":"Research on rigid-flexible coupling dynamics modeling for rack vehicle","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAs a form of railway transportation, the rack vehicle demonstrates exceptional climbing capabilities and carrying capacity, rendering it extensively utilized in mountainous regions [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The stability and safety of rack vehicle during operation are affected by many factors, among which the influence of slope and driving speed is particularly obvious. Numerous studies had been conducted on the vehicle system dynamics of traditional trains [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. However, the inclusion of gear-rack system in rack vehicle adds complexity to system dynamic characteristics, making it unsuitable to apply traditional train standards for evaluating the stability and safety of rack vehicle. Furthermore, the existing research literature on the dynamic model of rack vehicle is currently limited, leading to a scarcity of studies on vehicle dynamics characteristics of rack vehicle.\u003c/p\u003e \u003cp\u003eCurrent researches in vehicle system dynamics primarily focuse on traditional trains and high-speed and heavy-haul trains. Zhai et al. [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] developed a comprehensive 35-degree-of-freedom multi-body model. This model offers a systematic approach to optimizing the design parameters of vehicles and track components, providing a theoretical foundation for future research on the vehicle dynamic characteristics and the interaction between vehicle and track. On this basis, Baeza et al. [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] simply regard the track as a beam in order to simulate the dynamic interaction between the vehicle and the track. However, using a single factor variable cannot provide an accurate analysis of the vehicle dynamic characteristics. In practical engineering, the dynamic characteristics of vehicle are often influenced by multiple variables. Thus, Sun et al. [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] utilized Green's function to perform dynamic response analysis on vertical vehicle dynamics model. They investigated the vibration response of both rigid and flexible vehicle bodies on three different track models, and examined the impact of track stiffness and ballast stiffness on trains. Based on the traditional longitudinal dynamics model of train, Cole et al. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] proposed a combined simulation approach integrating train longitudinal simulation, locomotive traction control, and rolling stock dynamics control. The study examined the effects of lateral forces and coupler impacts on the lateral and longitudinal dynamics of trains. Huang et al. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] utilized a well-established vehicle-track coupling dynamics model to investigate the vertical vibration response of trains on floating slab track under various conditions, including different lengths, thicknesses, and vertical damping. The study also involved optimizing floating slab track's parameters.\u003c/p\u003e \u003cp\u003eNumerous studies have been conducted on the dynamics of traditional trains. However, there is a noticeable gap in research literature regarding the dynamics of rack vehicle. In current researches, Chen et al. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] established a finite element model for gear and rack to analyze the dynamics of gear meshing forces, vertical forces between wheel and rail, and vehicle acceleration when subjected to track random irregularity. They examined response characteristics and investigated how rack base deflection deformation affects the dynamic response of rack. At the same time, Chen et al. [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] investigated the vibration response characteristics of gear-rack system in three directions when subjected to track random irregularities. Their findings indicated that the vertical and longitudinal vibration of gear-rack system is less affected by the irregularity excitation, while the lateral vibration is significantly affected. The existing literature also examine the influence of different factors on the dynamic characteristics of rack vehicle, such as the layout mode of traction motor [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], pier settlement [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], the gear time-varying mesh stiffness [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], temperature load, nonlinear dynamic meshing characteristics of the gear-rack system, and the nonlinear dynamic contact characteristics of wheel-rail system [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], etc. Their research content holds significant value in enhancing the dynamic performance of rack vehicle and advancing the research progress in this field. However, the existing studies did not establish a comprehensive rigid-flexible coupling dynamic model of rack vehicle, or fail to fully consider the connection and coupling relationships among various components of rack vehicle in these established model.\u003c/p\u003e \u003cp\u003eIn this paper, a rigid-flexible coupling dynamics model of rack vehicle is established. In the model, the interconnection and coupling relationships between each component are comprehensively considered, and the contact relationship between wheel and rail is analyzed based on the nonlinear Hertz contact theory. Furthermore, based on the model, this paper analyzes the impact of various slopes and driving speeds on the dynamic characteristics of individual train components, investigates the working conditions necessary for the safe and stable operation of rack vehicle, and offers theoretical insights for designing and optimizing each part of rack vehicle.\u003c/p\u003e"},{"header":"2. Dynamic model of rack vehicle","content":"\u003cp\u003eBased on the vehicle-track dynamics theory [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], a complete rigid-flexible coupling dynamics model of rack vehicle is established, as shown in Fig.\u0026nbsp;1. The model is consists of vehicle submodel, track submodel [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], and gear-rack transmission model. The interconnection and coupling relationship between each submodel is clearly express-esed in Fig.\u0026nbsp;1, and the main parameters of rack vehicle are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMain parameters of rack vehicle.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNotation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eUnits\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{c}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of vehicle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{t}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of bogie frame\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{w}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of wheelset\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{m}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of gear\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{s}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of sleeper\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M_{\\text{b}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass of ballast\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ekg\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({I_{\\text{c}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass moment of inertia of vehicle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{kg}} \\cdot {{\\text{m}}^2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({I_{\\text{t}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass moment of inertia of bogie\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{kg}} \\cdot {{\\text{m}}^2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({I_{\\text{w}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass moment of inertia of wheelset\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{kg}} \\cdot {{\\text{m}}^2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({I_{\\text{m}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMass moment of inertia of gear\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{kg}} \\cdot {{\\text{m}}^2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{si}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStiffness of secondary suspension in three direction(\u003cem\u003eX\u003c/em\u003e, \u003cem\u003eY\u003c/em\u003e, \u003cem\u003eZ\u003c/em\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{pi}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStiffness of primary suspension in three direction(\u003cem\u003eX\u003c/em\u003e, \u003cem\u003eY\u003c/em\u003e, \u003cem\u003eZ\u003c/em\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{pv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener stiffness of rail in \u003cem\u003eZ\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{ph}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener stiffness of rail in \u003cem\u003eY\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{bv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast stiffness in \u003cem\u003eZ\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{bh}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast stiffness in \u003cem\u003eY\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{{\\text{fv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSubgrade stiffness\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{\\text{w}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast shear stiffness\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_{\\text{r}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener stiffness of rack\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{si}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDamping of secondary suspension in three direction(\u003cem\u003eX\u003c/em\u003e, \u003cem\u003eY\u003c/em\u003e, \u003cem\u003eZ\u003c/em\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{pi}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDamping of primary suspension in three direction(\u003cem\u003eX\u003c/em\u003e, \u003cem\u003eY\u003c/em\u003e, \u003cem\u003eZ\u003c/em\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{pv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener damping of rail in \u003cem\u003eZ\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{ph}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener damping of rail in \u003cem\u003eY\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{bv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast damping in \u003cem\u003eZ\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{bh}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast damping in \u003cem\u003eY\u003c/em\u003e direction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{{\\text{fv}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSubgrade damping\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{\\text{w}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBallast shear damping\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_{\\text{r}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener damping of rack\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_{\\text{i}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDisplacement in \u003cem\u003eZ\u003c/em\u003e direction (vehicle, bogie frame, wheelset, rail, sleeper, ballast)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003emm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Vehicle submodel\u003c/h2\u003e \u003cp\u003eThe vehicle submodel is consists of a vehicle, two bogie frames, and four wheelsets. In the submodel, the vehicle is supported on two double-axle bogie frames at each end, and the bogie frames are linked with the vehicle through the secondary suspensions. Moreover, the stiffness and damping characteristics of the secondary suspensions in three directions are completely considered in the modeling process.\u003c/p\u003e \u003cp\u003eIn dynamic analysis, the vibration characteristics of rack vehicle are significantly affected by whether the structure is rigid or flexible. Rigid components do not generate deformation in the dynamic simulation, resulting in only overall translation and rotation. Furthermore, this model does not account for the influence of structural deformation on the vehicle and bogie frame, thus treating them as discrete rigid bodies [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. If the wheelset is considered as a rigid component and directly contact with rail, the vibration and noise generated by the wheel-rail contact will be transmitted to vehicle through the wheel and frame. The large vibration generated will not only impact the vehicle, but also pose a threat to the stable operation of rack vehicle. Therefore, in this paper, both the wheelset and axle are set as elastomeric components [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], with the aim of mitigating wheel-rail contact forces, minimizing wheel-rail wear, reducing wheel-rail vibration and noise, and enhancing the operational stability of rack vehicle.\u003c/p\u003e \u003cp\u003eThe bogie frames are linked with the vehicle through the secondary suspensions, which primarily consists of air springs, vertical shock absorber, lateral shock absorber, and anti-roll dampers [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. The secondary suspension effectively minimizes the transmis-sion of vibration excitation to vehicle and significantly improves the stability of vehicle operation. The axle and wheelset are secured with tie constraint, while the bogie frames are linked with the axles through the primary suspensions. Three-dimensional spring-damper elements are used to represent the stiffness and damping characteristics of the primary and the secondary suspensions in three directions (vertical, longitudinal and lateral), respectively. The specific parameters are shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe stiffness and damping of model.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVertical(\u003cem\u003eZ\u003c/em\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eLongitudinal(\u003cem\u003eX\u003c/em\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eLateral(\u003cem\u003eY\u003c/em\u003e)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eStiffness(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePrimary suspension\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2900000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10056000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12700000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSecondary suspension\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e205000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e900000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e900000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener of rail\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e800000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15056000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e15056000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener of rack\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e15056000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15056000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e15056000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eDamping(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{N}} \\cdot {\\text{s}} \\cdot {{\\text{m}}^{{\\text{-1}}}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePrimary suspension\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e50000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e50000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSecondary suspension\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e36000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e50000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e72000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener of rail\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e15000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e60000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e75000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFastener of rack\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e75000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e75000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e75000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Track submodel\u003c/h2\u003e \u003cp\u003eGenerally there are two types of track submodels: ballasted track submodel and ballastless track submodel [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. The rack vehicle is designed for low speed operation, therefore ballasted track submodel is utilized in this paper. The ballasted track submodel is represented as a flexible model, primarily consisting of rails, rail pads, sleepers, ballast and subgrade [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. The model takes into account the vertical vibration displacement of rail, sleeper, and ballast. Furthermore, both the left and right sides of rail are treated as continuous Timoshenko beams, which are discretely supported at rail-sleeper junctions by three layers of springs and dampers, representing the elasticity and damping of rail pad, ballast and subgrade, respectively. As shown in Fig.\u0026nbsp;1.\u003c/p\u003e \u003cp\u003eThe wheel-rail contact model plays a crucial role in integrating the vehicle submodel with the track submodel. Furthermore, the accuracy of rigid-flexible coupling dynamics model of rack vehicle is heavily reliant on both the wheel-rail contact model, the primary suspensions, and the secondary suspensions. The impact of the primary and secondary suspensions on the dynamics model has been thoroughly addressed in previous research.\u003c/p\u003e \u003cp\u003eThe wheel-rail contact force in the wheel-rail contact model consists of the normal contact force and the wheel-rail creep force. Based on the elastic compression deforma-tion at the normal contact point, utilizing the nonlinear Hertz contact theory to calculate the wheel-rail normal contact force [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. The Hertz contact force calculation formula can be expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e):\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$P\\left( t \\right)={\\left[ {\\frac{1}{G}\\delta Z\\left( t \\right)} \\right]^{3/2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eG\u003c/em\u003e is the wheel-rail contact constant; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta Z\\left( t \\right)\\)\u003c/span\u003e\u003c/span\u003e is the elastic compression between wheel and rail.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\delta Z\\left( t \\right)={Z_{\\text{w}}}\\left( t \\right) - {Z_{\\text{r}}}\\left( {{x_{\\text{w}}},t} \\right) - {Z_0}\\left( t \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_{\\text{W}}}\\)\u003c/span\u003e\u003c/span\u003e is the vertical displacement of wheel; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_{\\text{r}}}\\)\u003c/span\u003e\u003c/span\u003eis the vertical displacement at the contact point; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_0}\\left( t \\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the geometrical irregularities of rail and wheel surfaces.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Gear-rack transmission model\u003c/h2\u003e \u003cp\u003eThe rack utilized in this paper is Strub system, and the relevant parameters of gear and rack are shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Regarding the connection method between rack and sleeper, Zhang et al. [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] fixed the connection between rack and sleeper through fasteners, effectively integrating them into a single unit. Nevertheless, this approach may deviate from engineering practices, leading to significant discrepancies and impacting the analysis results. Thus, in this paper, the Strub system is simulated as a Timoshenko beam supported by continuous elastic discrete points, incorporating elastic damping elements to simulate the behavior of fasteners. The interaction between rack and sleeper involves both stiffness and damping effects.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of gear and rack.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eModulus\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNumber of teeth\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMaterials\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eWidth\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ePressure angle\u003c/p\u003e \u003cp\u003e(\u0026deg;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGear\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e100/π\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e18CrNiMo7-6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e14.036\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRack\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e42CrMo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e60\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3. Dynamic characteristics of rack vehicle under different working conditions","content":"\u003cp\u003eBased on the rigid-flexible coupling dynamics model of rack vehicle, the finite element model of rack vehicle is established, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The dynamic characteristics of rack vehicle are investigated under various slopes and driving speeds conditions, encompassing the dynamic response of vehicle submodel, track submodel, gear-rack transmission model, and the contact characteristics of gear-rack contact surfaces. By studying the dynamic characteristics of each component and the root bending stress of rack, a reasonable value for slope and driving speed is proposed to ensure the stability and safety of rack vehicle.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Dynamic response of rack vehicle under different slopes\u003c/h2\u003e \u003cp\u003eThe variation in train running slope has a significant impact on the stability and safety of rack vehicle. This section aims to investigate the effects of slope changes on various components of rack vehicle, including wheel-rail contact force, root bending stress of rack, and dynamic characteristics of each component, within the range of 60\u0026permil;-300\u0026permil;.\u003c/p\u003e \u003cp\u003eFigure 3 reveals the vertical contact forces between wheelset and rail under different slopes. In Fig.\u0026nbsp;3(a), the variation of wheel-rail vertical contact force is illustrates when the slope is 60\u0026permil;, and the trend of wheel-rail contact force at other slopes is comparable to that of 60\u0026permil;. It can be seen from the Fig.\u0026nbsp;3(a) that the vertical contact force of wheel-rail changes periodically during uniform motion. The reason is that during the rotation of wheel, different positions of wheel come into contact with rail, causing the contact force to appear up and down in this cycle. During the time interval of 2.5s-4.5s, the periodic variation in wheel-rail vertical contact force is attributed to the cyclic rotation of wheel. Moreover, due to track random irregularity excitation, weld seams of track, etc. The change trend of wheel-rail vertical contact force exhibits variations at 2.8s and 3.5s.\u003c/p\u003e \u003cp\u003eFigure 3(b) shows the average wheel-rail vertical contact force under different slopes. It can be seen from the Fig. that the wheel-rail vertical contact force gradually decreases as the slope increases. When the slope increases from 60\u0026permil; to 300\u0026permil;, the wheel-rail vertical contact force decreases from 71.28kN to 66.35kN. The wheel-rail contact force is reduced by approximately 6.9%.\u003c/p\u003e\u003cp\u003eFigure 4 shows the vertical vibration acceleration and displacement of vehicle under different slopes. In order to explore the running stability of rack vehicle, the vibration acceleration of vehicle is shown in Fig.\u0026nbsp;4(a). It can be seen from the Fig. that the vibration acceleration of vehicle increases with the increase of slope. Due to the vibration direction of vehicle is different from the direction of coordinate axis, the vibration acceleration of vehicle can be either positive or negative, which has little impact on the results.\u003c/p\u003e \u003cp\u003eAccording to the above analysis, the wheel-rail vertical contact force gradually decreases with the increase of slope. It shows that the vertical force exerted by each component on rail gradually decreases. Furthermore, all parameters of the secondary suspension are the same, with the support of the secondary suspension, the vertical displacement of vehicle gradually decreases as slope increases. In Fig.\u0026nbsp;4(b), the vertical displacement of vehicle under different slopes are shown. When the slope increases from 60\u0026permil; to 300\u0026permil;, the vertical displacement of vehicle decreases by about 20mm.\u003c/p\u003e \u003cp\u003eVariations in wheel-rail vertical contact force not only influence the vertical vibration acceleration and vertical displacement of vehicle, but also exert a significant influence on vertical displacement of rail. When suffered to a significant vertical force, the rail will has a substantial deformation, leading to increased wear and potential interference with the contact between wheel and rail. In Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the vertical displace-ment of rail under different slopes are shown. The Fig. illustrates that as the slope increases, the vertical displacement of the rail decreases. Specifically, when the slope increases from 60\u0026permil; to 300\u0026permil;, the maximum vertical displacement of rail decreases from 1.03mm to 0.98mm.\u003c/p\u003e \u003cp\u003eThe analysis of the wheel-rail vertical contact force reveals that as slope increases, the vertical contact force between wheel and rail decreases gradually, while the longitudinal force increases gradually. This results in an increase in slope resistance that the train must overcome during the climbing process. Nevertheless, the incorporation of gear-rack system can significantly compensate for the traction deficiency.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the root bending stress maps of rack under different slopes, (a)-(e) correspond to five slopes in 60\u0026permil;-300\u0026permil;, respectively. As can be seen from the Figure, when the slope increases from 60\u0026permil; to 300\u0026permil;, the stress map of the gear-rack contact surface gradually becomes larger, indicating that the contact force between gear and rack increases gradually. The variation trend of the maximum root bending stress of rack under different slopes is not significant. Therefore, the value of the maximum root bending stress of rack is presented separately in a bar chart format, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e7\u003c/span\u003e. It can be observed from the Figure. that there is an increasing trend in the maximum root bending stress of rack with an increase in slope. When the slope increases from 60\u0026permil; to 300\u0026permil;, the maximum root bending stress of rack rising from 79.73MPa to 84.15Mpa, with a increase rate of about 5.5%.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Dynamic response of rack vehicle under different driving speeds\u003c/h2\u003e \u003cp\u003eThe following section will investigate the varying characteristics of each component of the train at different driving speeds. The train will be operated at constant speeds of 15km/h, 20km/h, 25km/h, 30km/h, and 35km/h respectively on a slope of 120\u0026permil;.\u003c/p\u003e \u003cp\u003eFigure 8 illustrates the variations in wheel-rail vertical contact force at various driving speeds. For instance, at 30km/h, the fluctuation in the wheel-rail vertical contact force is depicted in Fig.\u0026nbsp;8(a). Similar trends can be observed for the wheel-rail contact force at other driving speeds. The Fig. illustrates the periodic variation of the wheel-rail contact force at a driving speed of 30km/h. The factors influencing this change are comparable to those observed at a 60\u0026permil; slope in Section3.1.\u003c/p\u003e \u003cp\u003eThe average of wheel-rail contact force at different driving speeds is shown in Fig.\u0026nbsp;8(b). As can be seen from the Fig., when the driving speed of train increases from 15km/h to 35km/h, the change of wheel-rail contact force is not obvious, in which the difference between the minimum value (68.68kN) and the maximum value (69.33kN) is only 0.65kN, which is negligible within the allowed range of error. It can be seen that different driving speeds have little effect on the vertical contact force between wheel and rail.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;9(a), the vertical vibration acceleration and vertical displacement of vehicle at different driving speeds is shown. It can be seen that the vertical vibration acceleration of vehicle does not change significantly with the driving speed, and the vibration acceleration fluctuates within the range of approximately\u0026thinsp;\u0026plusmn;\u0026thinsp;0.01 at different driving speeds. This is because the driving speed of rack vehicle belongs to the low-speed driving mode, and the dynamics of vehicle is not sensitive to the change of speed. Furthermore, the vertical displacements of vehicle at different driving speeds are also the same, as shown in Fig.\u0026nbsp;9(b). It can be seen from the Fig. that the vertical displacement of vehicle is basically maintained at about 26mm, and the amplitude of the up and down fluctuations is not obvious.\u003c/p\u003e \u003cp\u003eThe vertical displacement of rail at different driving speeds is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e10\u003c/span\u003e. Due to the varying locations of the probe points within the finite element, the displacement of rail appears staggered at different driving speeds, but its changing trend remains consistent. It can be observed that the impact of different driving speeds on the vertical displacement of rail is not significant, with a range between 1.03mm and 1.05mm.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e11\u003c/span\u003e illustrates the root bending stress of rack at various driving speeds, ranging from 15km/h to 35km/h denoted as (a)-(e). The diagram reveals that the bending stress does not follow a consistent pattern across different driving speeds. The changing trend of the maximum root bending stress of rack is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e12\u003c/span\u003e. When the train operates at various speeds, the rise in gear speed results in an escalation of the meshing frequency between gear and rack, consequently amplifying the intricacy of the nonlinear dynamics within gear-rack system [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. The contact condition between gear and rack is significantly influenced by the multi-state meshing behavior, subsequently causing fluctuations in the root bending stress of rack, displaying a trend of initially decreasing, then increasing, and finally decreasing.\u003c/p\u003e \u003cp\u003eBased on the above analysis, the meshing frequency of gear-rack system varies with the driving speed, so that the vertical vibration acceleration of gear is changed. In Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e13\u003c/span\u003e, the root mean square of the vertical vibration acceleration of gear at different driving speeds is shown. It can be seen that the change trend of the vertical vibration acceleration of gear aligns with the maximum bending root stress of rack, providing additional confirmation of the impact of gear-rack meshing frequency on both the maximum root bending stress of rack and gear vibration acceleration. Therefore, it is recommended that the speed of rack vehicle is 25km/h.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eBased on the vehicle-track dynamics theory, a complete rigid-flexible coupling dynamics model of rack vehicle is established in this paper. Subsequently, the dynamic characterist-ics of train under various slopes and driving speeds are investigated. Based on the research results, the conclusions are as follows:\u003c/p\u003e \u003cp\u003e(1) The slope and driving speed play crucial roles in determining the dynamic characteristics of rack vehicle. The slope affects all the components of rack vehicle significantly, whereas the driving speed specifically impacts the maximum root bending stress of rack and the vertical vibration acceleration of gear.\u003c/p\u003e \u003cp\u003e(2) Under varying slope conditions, the wheel-rail contact force, the displace-ment of vehicle and rail all decrease as slope increases. However, the maximum root bending stress of rack actually increases with slope, and this increase is significantly greater than the decrease seen in parameters like the wheel-rail contact force, the displacement of vehicle, etc. Therefore, it is recommended that the slope of rack vehicle should not exceed 300\u0026permil; in order to ensure the safety of train operations.\u003c/p\u003e \u003cp\u003e(3) Under varying driving speed conditions, the maximum root bending stress of rack and the vertical vibration acceleration of gear displaying a trend of initially decreasing, then increasing, and finally decreasing. Moreover, when the driving speed of train is 25km/h, the maximum root bending stress of rack and the vertical vibration acceleration of gear are the minimum. Therefore, it is recommended that the driving speed of rack vehicle is 25km/h.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003cstrong\u003eCompeting interests\u003c/strong\u003e \u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor contributions\u003c/h2\u003e \u003cp\u003eChenglong Dong: Conceptualization, methodology, software, validation, formal analysis, data curation, writing-original draft preparation. Xingqiao Deng: Conceptualization, writing-review and editing, visualization, methodology, resources, project administration, funding acquisition. Jialin Liu: Methodology, software, validation. Zhendong Zhang: Methodology, validation, data curation. Chang Gao: Investigation, formal analysis. Weiping Liu: Investigation, formal analysis. Shisong Wang: Conceptualization, methodology, supervision, software, validation.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThe research was supported by the Sichuan Shudao New System Rail Group CO., LTD. (DJGC2022111001 and DJGC2022111002), Sichuan Provincial CNC Equipment Ultra-precision Drive and Transmission Engineering Research Center, and Sichuan Provincial New Standard Track Gear Transmission Equipment Engineering Research Center.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eHansen B (2009) Gearing up: the mount washington cog railway. Civ Eng 79(4):36\u0026ndash;39. 10.10 61/ciegag.0000525\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWeber J, Abt S (1911) Rack-railway locomotives of the swiss mountain railways. 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Thus, based on the vehicle-track dynamics theory, this paper establishes a rigid-flexible coupling dynamics model of rack vehicle, which consists of vehicle submodel, track submodel, and gear-rack transmission model. In the model, the bogie frames are linked with the wheelsets through the primary suspensions and linked with the vehicle through the secondary suspensions, respectively, while three-dimensional spring-damper elements are used to represent the stiffness and damping characteristics of the primary and secondary suspension in three directions. Furthermore, regard the connection relationship between rack and sleeper as elastic damping, and the wheel-rail contact relationship is established on the nonlinear Hertz contact theory. Based on this model, the dynamic characteristics of rack vehicle are analyzed under various slopes and driving speeds. Results show that the slope has a substantial effect on the dynamic characteristics of each component of rack vehicle, whereas the driving speed primarily affects the root bending stress of rack and the vibration acceleration of gear. The research results is of great significance to the safety and stability of rack vehicle.\u003c/p\u003e","manuscriptTitle":"Research on rigid-flexible coupling dynamics modeling for rack vehicle","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-13 12:00:37","doi":"10.21203/rs.3.rs-4875774/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"08e09fcc-c9a8-4e1e-8e3c-72e5e99864fd","owner":[],"postedDate":"September 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-10-11T14:35:45+00:00","versionOfRecord":[],"versionCreatedAt":"2024-09-13 12:00:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4875774","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4875774","identity":"rs-4875774","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}