Study on the gear-rack meshing dynamic performance based on a multi-body dynamic model of rack rail vehicle

Abstract

Rack rail vehicles show strong capability of mountain climbing and have become a popular transportation mode for mountain areas in recent years. Therefore, this paper investigated on the dynamic performance of rack rail vehicle through simulation of a multibody dynamic model. Firstly, a multibody dynamic model of rack rail vehicle considering both the wheel/rail interaction and the gear-rack meshing was developed in the SIMPACK software platform. A rack rail with a large slope of 250‰ was taken into consideration in the proposed study. Subsequently, the gear-rack meshing dynamics in terms of the meshing stiffness and the meshing force in both circumferential- and radial- directions were investigated. Moreover, two types of driving modes were compared to illustrate the effects of driving mode on the wheel/rail interaction and the gear-rack meshing. The results show that due to the gear-rack meshing the meshing stiffness and the meshing force are subject to the periodical variations and the gear-rack meshing can significantly affect the wheel/rail interaction, especially in the presence of track irregularities. The driving mode that the gear wheel is installed on the wheelset axle through a hollow shaft is considered as a more practical design for the rack vehicle.

Keywords

Rack rail vehicle meshing dynamics vehicle system dynamics meshing stiffness driving mode

Introduction

Rack railroad is a kind of mountain climbing railroad, which enables the vehicle operating on the mountain with a large slope up to 480‰.¹ The first rack railroad in the world was successfully put into operation as early as July 1869 at Mount Washington, USA, with a maximum slop of 377‰.^2,3 Subsequently, a rack railroad was opened in Europe in 1871 at Mount Rigi, Switzerland.⁴ The rack railroads have experienced a rapid development in the pasted 150 years, and more than 180 rack railroads have been developed in nearly 30 countries, mainly in Switzerland, Germany, Japan, France and Australia.^5,6 Unlike the traditional railway vehicles, the rack rail vehicles need the large traction and braking forces to overcome the gravitational force when the vehicle climes the mountain.^7,8 Therefore, a rack rail is installed at the middle of track to provide enough driving force in the climbing process. The existing investigations show that the meshing force generated by the gear/rack interaction can provide enough driving force for a slope up to 480‰.^9,10

A number of rack railway systems have been developed in the world, as shown in Figure 1. The early rack railway system, built by Blenkinsop, was designed for the steam locomotive for the Middleton railway, in which a 20-tooth cog wheel with a diameter of 914 mm on the left side was used to engage the rack teeth on the outer of the rail.¹¹ The marsh rack and pinion system was developed by Sylvester Marsh for Mount Washington cog railway.¹² The pinion wheels on the locomotives have deep teeth that ensure that at least two teeth are engaged with the rack at all times. The Riggenbach rack system¹³ using a ladder rack was invented by Niklaus Riggenbach which is similar to the Marsh system. The Abt system¹⁴ was devised by Carl Roman Abt, where a new rack was built using solid bars with vertical teeth. Two or three of these bars are mounted centrally between the rails which can match with the cog wheel. The Locher rack system,¹³ invented by Eduard Locher, has gear teeth cut in the sides rather than the top of the rail, engaged by two cog wheels on the locomotive. The Strub rack system was invented by Emil Strub,¹⁵ which consists of a rolled flat-bottom rail with rack teeth machined into the head approximately 100 mm (3.9 inches) apart. Safety jaws fitted to the locomotive can engage with the underside of the head to prevent derailments and serve as a brake.

Figure 1.

Typical rack railway system.

Recent years, the rack rail railway system was considered as an alternative solution for the transportation of mountain area in China.¹⁶ Feng¹⁷ compared the traditional railway system with the rack railroad system in terms of climbing capacity, construction cost and transportation speed and concluded that the rack rail system is the most suitable transportation system for the mountain area in China. Shang et al.¹⁸ summarised the features of rack rail lines and rack vehicles. Li and Liu¹⁹ investigated on the braking and traction system of rack rail vehicle. Niu et al.¹⁶ summarised characteristics of bogie design for the rack rail vehicle. Wang²⁰ took Zhangjiajie-Qixing rack railway line as an objective, and further determined key parameters of rack rail line, including the curvature, transition, maximum slope, vertical curve and length of slope section.

Liu^21,22 studied the dynamic loads of bridges for the rack railroad and its influences on the bridge, and concluded that the slope shows negligible effect on the dynamics of bridge. Cai et al.²³ pointed out that the problems faced by the ballasted rack railroad in the world, and further proposed a design for the rack railroad based on the ballastless track. Zhang et al.²⁴ studied the interaction between the track and simple girder bridge under typical loads, and suggested that the staggered arrangement of rack rail seam and girder seam can effectively avoid excessive variation of rack rail seam. Cai et al.²⁵ established a discrete element model for a large slope ballast bed and concluded that the longitudinal resistance of the bed decays with the increased track slope, and the maximum stress is located at the tooth root of the rack rail. Zhang et al.²⁶ established a numerical model of ballastless rail for rack railroad, and concluded that the longitudinal force and displacement of rail increase with the increased track slope and the force and deformation of rail for the ballastless track are much lower than those of ballasted track. Xie et al.²⁷ investigated on the fastener system of rack railroad.

Considering the characteristics of rack rail vehicle a number of investigations have been made to study the dynamic performance of rack rail vehicle. Liu et al.²⁸ developed a dynamic model for the rack and gear transmission system and further demonstrated the validity through the orthogonal tests. Liu et al.²⁹ conducted a test to measure the vibration from a mountain self-propelled electric monorail transport (MSEMT) considering different rack and gear parameters, and concluded that the pressure angle and the gear tooth ratio can significantly affect the dynamic performance of MSEMT. Chen and Li³⁰ developed a rack vehicle/track coupled dynamic model considering the non-linear meshing behaviours of the gears and rack together with wheel/rail contact. Nakamura et al.³¹ defined an optimal ratio of the traction forces applied by the front and rear gears, to reduce the stress of rack. The results showed that the maximum compressive stress could be reduced by 47% based on the optimal power distribution ratio.

It is known that the dynamic performance is the essential for the operational safety and ride comfort of rack rail vehicles. However, few investigations related to the dynamic performance of rack vehicles have been reported in the open publications. Therefore, this paper developed a multi-body system dynamic model for the rack vehicle to investigate influences of gear meshing on the dynamic performance of rack vehicle. Moreover, the influences of two types of driving modes for rack vehicles were also investigated through the simulation of proposed dynamic model.

Modelling of multi-body dynamic model of rack rail vehicle

Dynamic model of rack rail vehicle

The rack rail vehicle is equipped with several gears on the bogie, which can overcome the insufficient wheel-rail adhesion through the rack and gear meshing. The rack rail vehicle mainly consists of a car body, two bogie frames, two drive gears, four wheels and eight axle boxes. The bogie frame and car body are connected by the secondary suspension (including air springs, lateral and vertical dampers, anti-roll bars and lateral stops), the bogie frame and the axle box are connected by the primary suspension system (including vertical dampers, coil springs and swivel arm positioning), the axle box and wheel are coupled through the roller bearing, and the gear and the wheelset axle are coupled through a hollow axle. The rack rail mainly includes racks, two rails, rail sleepers, ballast-less road bed and rack rail transmission facilities. The rails are connected to the rail sleepers by fasteners, and the racks are connected to the rail sleepers by rack rail fasteners. In the model, the spring-damper parallel force element is used to model the air spring, and the spring-damper serial force element is used to model the hydraulic damper. Figure 2 illustrated the dynamic model for rack rail vehicle, and its topology for the adhesive bogie and the rack rail bogie. In this study, the rack rail vehicle is equipped with an adhesive bogie and a rack bogie. Figure 3 illustrated the nonlinearities of suspension considered in the rack rail vehicle model.

Figure 2.

Dynamic model of rack rail vehicle: (a) multibody dynamic model for rack rail vehicle, (b) wheel/rail adhesive bogie and (c) rack rail bogie.

Figure 3.

Nonlinearities of suspension in the rack rail vehicle: (a) force characteristic of lateral stop and (b) damping force characteristic of hydraulic damper.

In the model, the wheel-rail creep force was calculated using the Kalker nonlinear creep theory, and the wheel-rail contact force was calculated using the Hertz nonlinear contact theory. Figure 4 illustrated a schematic diagram of the wheel-rail contact relationship, in which the profiles for the rail and wheel are CHN50 and LM, respectively.

Figure 4.

Wheel/rail contact relationship for both left and right wheels: (a) contact relationship for left wheel/rail interaction and (b) contact relationship for right wheel/rail interaction.

Gear-rack meshing model

In this study, two types of driving modes for rack rail vehicle were taken into consideration, as shown in Figure 5. For the type A in Figure 5(a), the brake disc, driving gear and the rack gear are installed on a hollow shaft sleeve, and the hollow shaft sleeve can rotate independently with respect to the wheelset axle. Therefore, the torque generated by the traction motor is initially transferred to the driving gear through the gearbox, and then transmitted to rack rail through the rack gear wheel. For the type B in Figure 5(b), the brake disc, driving gear and the rack gear are directly installed on the wheelset axle.

Figure 5.

Two types of driving unit for rack rail vehicle: (a) Type A and (b) Type B.

Considering that the impact arising from the gear meshing could affect the dynamics of rack rail vehicle, the gear meshing dynamics is taken into consideration in this study. The typical gear force element defined in the SIMACK platform is used to simulate the gear and rack meshing for the rack rail vehicle, which enable us to consider the meshing time-depended stiffness and transmission error, as shown in Figure 6. Table 1 lists main parameters used in the gear/rack pair model.

Figure 6.

Gear-rack meshing model.

Table 1.

Main parameters used in the gear rack model.

Parameters	Values	Parameters	Values
Number of gear teeth	24	Damping model	Linear
Modulus (mm)	31.831	Friction model	Simple
Pressure angle (°)	14.0362	Stiffness ratio	0.8
Tooth top height coefficient	1	Young’s modulus (N/m²)	2.1 × 10¹¹
Top clearance coefficient	0.25	Poisson’s ratio	0.3
Displacement coefficient	0.3	Engagement/exit damping (N·s/m)	5000

The characteristics of the gear-rack meshing can be described through the following meshing dynamic equations³²:

\begin{matrix} {\begin{matrix} J \overset{\cdot\cdot}{θ} + r_{b} \sum_{i} [k_{vi} (r_{b} θ - s - e_{i}) + c_{vi} (r_{b} \overset{\cdot}{θ} - \overset{\cdot}{s} - {\overset{\cdot}{e}}_{i})] = T \\ m \overset{\cdot\cdot}{s} - cosa \sum_{i} [k_{vi} (r_{b} θ - s - e_{i}) + c_{vi} (r_{b} \overset{\cdot}{θ} - \overset{\cdot}{s} - {\overset{\cdot}{e}}_{i})] = - G \end{matrix} \end{matrix}

(1)

where, r_b is the radius of base circle for the gear; k_v is the meshing stiffness; e is the tooth profile error; c_v is the meshing damping factor; T is the gear driving torque; G is the drag force on the rack; θ is the gear angle; J is the gear rotational inertia; s and m are the rack displacement and mass respectively; the gear rotation direction is taken clockwise as positive. Based on the above equations, it can be seen that even if the speed and external load of the gear are constant, the change of meshing position can cause the variation of meshing stiffness and further lead to the fluctuation of the motion of gear. Due to the time-varying characteristics of meshing stiffness, the dynamic meshing forces are expected during the gear meshing process, thereby the excitations for the gear-rack transmission system.

Track model for rack rail vehicle

Figure 7 illustrated the typical track model for rack rail vehicle. The rack rail is installed in the middle of track, and the rail and rack rail are discretely supported on the sleeper through rail pads. In the modelling of rack track, the rail pad and fasteners are modelled as the linear spring-damper elements and the rail and each segment of rack rail are modelled as the rigid component. The sleeper spacing is 0.65 m, the length for rack rail segment is taken as 2.4 m.

Figure 7.

Track model for rack rail vehicle.

The rack track is characterised by the large slope and vertical curve. In China, the slope of rack rail could range from 0 to 250 ‰. Therefore, the rack rail vehicle has to operate from the horizontal line to the slope section. Figure 8 shows the geometry of vertical curve considering in the track model. Before the vehicle enters the slope, the vehicle operates on a horizontal line with a length of 100 m. A vertical curve with a length of 100 m serves as the transition section between the horizontal and slope line. The length of the vertical slope is taken as 600 m.

Figure 8.

Typical vertical curve considering in this study.

Regarding to the excitations of rack vehicle, the track irregularities serve as the main resource of excitation. Figure 9 illustrated the vertical and lateral irregularities for both left and right rails. The amplitudes of vertical irregularities are mainly smaller than 4 mm, and 2 mm for the lateral irregularities.

Figure 9.

Vertical and lateral irregularities considered in this study: (a) vertical irregularities and (b) lateral irregularities.

Apart from the track irregularities, the gearing error excitation is another kind of periodic excitation, which includes the tooth profile error and tooth pitch error of the meshed gear tooth profile. The gearing error can be expressed as:

e (t) = R_{b 1} φ_{1} - R_{b 2} φ_{2}

(2)

where, e denotes the gear transmission error; R_b denotes the radius of the base circle of the master and driven gears; - denotes the actual rotation angle of the master and driven gears (rad), respectively. The meshing stiffness excitation is a parametric excitation caused by the periodic change in the number of pairs of teeth involved in the gear mesh transmission process, which is defined as the load required to produce a unit deformation displacement on a pair or several pairs of simultaneously meshed teeth, and can be expressed as³³

K (t) = \sum_{i = 1}^{n} k_{i} = \sum_{i = 1}^{n} \frac{F_{i}}{δ_{pi} + δ_{bi}}

(3)

where, k denotes the gear meshing stiffness (N/m); n denotes the number of gear teeth involved in meshing at the same time; F_i denotes the meshing contact force; $δ_{pi}$ and $δ_{bi}$ denote the elastic deformation of the first pair of gear teeth on the master and slave gears (m), respectively.

Results and discussion

Based on the proposed dynamic model for the rack vehicle, the dynamic behaviours of rack vehicle were studied in following section, including the gear-rack meshing dynamics of rack rail vehicle and the influences of two driving modes.

Gear-rack meshing dynamics of rack rail vehicle

Figure 10 illustrated the time history of meshing stiffness for a given speed of 40 km/h. To illustrate the influences of the gear-rack meshing, the track is considered as a smooth rail. The results show that the meshing stiffness expresses periodic variations with respect to the mean value 3.09 GN/m. In the double-tooth meshing area, the meshing stiffness (3.6 GN/m) is much greater than the single-tooth meshing area (2.5 GN/m). This periodic variation in the system could give rise to an internal excitation for the rack vehicle system. Figure 11 further shows the characteristics of gear-rack meshing in the frequency domain, the result shows that the typical meshing frequency f_c for the gear-rack meshing stiffness is at the frequency of 111.1 Hz. The meshing frequency of gear/rack interaction can be determined as $f_{c} = vz / (π \cdot d)$ , where v is the vehicle speed 40 km/h, z is taken as 21 which is the number of teeth of rack gear, d is the circle diameter of rack gear, 0.668 m. Therefore, the meshing frequency can be theoretically calculated as 111.1 Hz, which is in good agreement with the numerical results. This suggests that the developed model can effectively describe the rack-gear meshing dynamics. Apart from the fundamental meshing frequency, the multiple frequencies, including 2fc, 3fc and 4fc, also can be identified in the frequency domain.

Figure 10.

Meshing stiffness variations of gear/rack interaction, (a) meshing stiffness in time domain, (b) zoom in meshing stiffness and (c) meshing stiffness in the frequency domain.

Figure 11.

Meshing force variations of gear/rack interaction, (a) meshing force in time domain, (b) zoom in meshing force and (c) meshing force in the frequency domain.

Figure 11 illustrated the meshing forces of the rack-gear meshing. Due to the transmission error and the variation of gear/rack meshing stiffness, the circumferential and radial forces of gear meshing also show periodic variations. It can be seen that the circumferential force varies from 25 to 59 kN, and the radial force lies in the range of 6–15 kN. The mean values for the circumferential and radial forces are 40.53 and 10.18 kN, respectively. The radial force F_r can be calculated through the relationship between the circumferential force F_t and the press angle $α$ ,³⁴ $F_{r} = F_{t} \cdot \tan α$ , where the pressure angle is taken as 14.0362°. Besides, the meshing frequency f_c of 111.1 Hz also can be identified in the frequency domain for the circumferential and radial forces of the rack-gear interaction.

It is known that the gear is installed on the wheelset axle and the resulting fluctuations in the meshing forces of gear/rack interaction could further affect the wheel/rail interaction. Figure 12 illustrated the variations of wheel/rail normal forces. The wheel/rail normal force shows periodical fluctuations with respect to the mean value of 44.8 kN, and is predominated by the frequency of 111. 1 Hz corresponding to the meshing frequency of rack/gear interaction. This suggests that the variations of wheel/rail normal forces are mainly induced by the gear/rack interaction. Moreover, the obtained mean wheel load 44.8 kN is less than the static wheel load of 50.1 kN of rack vehicle, which is induced by the radial force of gear wheel when the rack rail vehicle operates in the slope section.

Figure 12.

Wheel/rail normal forces, (a) in time domain and (b) in frequency domain.

Effects of driving modes on the gear-rack meshing dynamic performance

To better understanding the gear-rack meshing dynamics, this section further investigated on the effects of driving modes on the gear/rack meshing dynamic performance, as shown in Figure 5. Two types of driving modes are taken into consideration, in which the gear wheel is mounted on the wheelset axle through a hollow shaft in type A while the gear wheel is directly mounted on the wheelset axle in the type B.

Figure 13 illustrated the gear-rack meshing forces when the vehicle operates at the speed of 20 km/h on the slope of 250‰. The track is considered as a smooth rail. The results show that the meshing stiffness for both types are quite comparable in both the time and frequency domains (in Figure 13(a) and (b)). Whereas, the huge differences are observed for the gear/rack meshing forces in both circumferential and radial directions. The circumferential and radial forces obtained for type A is much higher than those obtained by type B, while smaller wheel/rail normal forces are observed in type A comparing the type B (in Figure 14(a)). This is because that the tangential force is generated by both the wheel/rail interaction and the rack-gear meshing on the slope section for type B. However, for type A, the tangential force is initially supported by the rack-gear meshing alone. Therefore, the type A driving modes gives rise to larger circumferential and radial forces with respect to type B.

Figure 13.

Meshing stiffness and forces for the gear-rack interaction, (a and b) the meshing stiffness in both time and frequency domains, (c) circumferential force and (d) radial force.

Figure 14.

Wheel/rail normal forces for two types of driving mode: (a) wheel/rail normal force and (b) FFT of wheel/rail normal force.

Figure 14(b) further shows the characteristics of wheel/rail normal force in the frequency domain. The wheel/rail normal forces of type B driving mode are predominated at the frequencies of 4.3, 13.5 and 37.4 Hz apart from the gear/rack meshing frequency of 55.5 Hz and the related multiple frequencies. The amplitudes for the dominating frequencies of 4.3, 13.5 and 37.4 Hz are higher than for the impact caused by the gear/rack meshing. This further suggests that the impact caused by the gear/rack meshing can excite the vibration modes of rack rail vehicle for type B driving mode. Whereas, for the type A, the wheel/rail normal force is mainly affected by the gear/rack meshing interaction. This suggests a strong coupling effecting between the wheel/rail interaction and rack-gear meshing for type B with respect to the type A. Because the gear wheel is rigid connected to the wheelset axle for the type B.

In the real operating condition, the rack rail vehicle is expected to be subjected to track irregularities in both vertical and lateral directions, which could affect both the wheel/rail interaction and gear-rack mesh. Therefore, this study further investigated on the effects of track irregularities through applying track irregularities at the wheel/rail interface.

Figure 15 illustrated the meshing dynamics in the presence of track irregularities. The meshing stiffness for type B is subjected to severe fluctuations due to the track irregularities, and in some scenario the meshing stiffness decreases to 0 MN/m, which suggests a discontinuous gear-rack meshing arising from the track irregularities. Moreover, the circumferential force obtained in type B is greater than those observed in type A. When the rack rail vehicle is subjected to the upward track irregularities, the results show increases in both meshing stiffness and radial force, and reduction in the wheel/rail normal force (Figure 16).

Figure 15.

Meshing dynamics in the presence of track irregularities: (a and b) meshing stiffness in time and frequency domains, (c and d) circumferential forces in time and frequency domains, (e and f) radial forces in time and frequency domains, and (g) vertical track irregularities.

Figure 16.

Wheel/rail normal forces in the presence of track irregularities: (a) wheel/rail normal force, (b) zoom in the wheel/rail normal force and (c) wheel/rail normal force in the frequency domain.

This suggests that the upward track irregularities between the rail and the rack in the vertical direction can pose significant influences on both the wheel/rail interaction and the gear-rack meshing. The type A driving mode is thus considered as a more practical driving mode comparing with the type B driving mode.

Conclusions

In this study, a multibody system dynamic model for a rack rail vehicle, considering both wheel/rail interaction and gear-rack meshing, was established. The meshing dynamics, including meshing stiffness and meshing forces in both circumferential and radial directions, were investigated and two driving modes were compared to illustrate their influences on the dynamic performance of rack rail vehicle.

The results showed that the developed model for rack rail vehicle can effectively simulate both the gear-rack meshing and wheel/rail interaction. During the gear/rack meshing process, the double tooth meshing was identified which gives rise to huge fluctuations in the meshing stiffness, thereby the variations in the meshing forces in both circumferential and radial directions. The resulting meshing force variations can further affect the wheel/rail interaction. Type B driving mode shows significant coupling effects between the gear/rack meshing and the wheel/rail interaction, the resulting meshing impact could excite some vibration modes of rack rail vehicle. In addition, the irregularities between the rail and the rack rail can pose significant influence on both the wheel/rail interaction and the gear-rack meshing. It is thus desirable to maintain the relative irregularities between the rail and the rack. The type A driving mode that the gear wheel is mounted on the wheelset axle through a hollow shaft, is recommended for the rack rail vehicle.

The aforementioned investigations invariably show the obvious influences arising from the gear/rack meshing. Whereas, this study is subjected to the limitation of lacking of model validation, because the rack rail line is still under construction in China. Therefore, the validation of the modelling of gear/rack meshing could be considered as a potential area for future research.

Footnotes

Handling Editor: Sharmili Pandian

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author sincerely appreciates the help provided by the research team (Wheel/rail interaction, Vibration and Noise Research Team) and CRRC. In addition, this study has also been supported by the China Association of Science and Technology Young Talent Support Project (No. 2019QNRC001), Sichuan Science and Technology Program (2023NSFC0374, 2022NSFSC0469 and 2024NSFSC0187), the Science and Technology Research Plan of China Railway General Corporation (N2022J009), and Fundamental Research Funds for the Central Universities(2682024CG007).

ORCID iD

Xingwen Wu

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