Experimental study and modeling of rubber joints for railway vehicles using magnetorheological shear stiffening elastomers

With the rapid development of transportation industry, advanced rail vehicle technology receives more attention than ever. The stiffness of the train’s rubber joint at the primary suspension system has a crucial influence on the operation stability and curve-passing performance. When the train is running on straight track at high speed, a high primary longitudinal stiffness in bogie design is required, whereas running on the curve track calls for a soft primary longitudinal stiffness. To solve this critical problem, a new rubber joint based on magnetorheological shear stiffening elastomer (MSSE) was proposed. Its stiffness can be adjusted by not only external magnetic field but also its inherent frequency-dependent property, ensuring the functionality of the rubber joint even when the controller fails. The prototype of the MSSE joint was fabricated and assembled. Stiffness controllability of the MSSE joint was evaluated using an material testing system (MTS) machine, with MTS testing performed under varying displacement amplitude at fixed frequency to investigate the influence of the varying displacement amplitude on the effective stiffness. The results revealed that the stiffness of this MSSE joint can be controlled effectively credited to the rate-dependent SSE and adjustable electromagnetics, exhibiting exceptional fail-safe characteristics. Lastly, a dynamic model was established to describe the dynamic performance of the rubber joint. All the above studies demonstrate the feasibility of the joint to satisfy the conflicting stiffness requirements to achieve high speed stability and curve trafficability simultaneously.

With the rapid development of transportation industry, advanced rail vehicle technology receives more attention than ever. The stiffness of the train's rubber joint at the primary suspension system has a crucial influence on the operation stability and curve-passing performance. When the train is running on straight track at high speed, a high primary longitudinal stiffness in bogie design is required, whereas running on the curve track calls for a soft primary longitudinal stiffness. To solve this critical problem, a new rubber joint based on magnetorheological shear stiffening elastomer (MSSE) was proposed. Its stiffness can be adjusted by not only external magnetic field but also its inherent frequency-dependent property, ensuring the functionality of the rubber joint even when the controller fails. The prototype of the MSSE joint was fabricated and assembled. Stiffness controllability of the MSSE joint was evaluated using an material testing system (MTS) machine, with MTS testing performed under varying displacement amplitude at fixed frequency to investigate the influence of the varying displacement amplitude on the effective stiffness. The results revealed that the stiffness of this MSSE joint can be controlled effectively credited to the rate-dependent SSE and adjustable electromagnetics, exhibiting exceptional fail-safe characteristics. Lastly, a dynamic model was established to describe the dynamic performance of the rubber joint. All the above studies demonstrate the feasibility of the joint to satisfy the conflicting stiffness requirements to achieve high speed stability and curve trafficability simultaneously. * Author to whom any correspondence should be addressed.
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Introduction
As the world's population continues to expand and satisfying the transportation demands becomes increasingly challenging, it is imperative to promote the advancement of railway technology. Increasing the train speed is an effective way to improve the transportation efficiency and thus to flow the passenger capacity. However, the maximum achievable speed of conventional railway vehicles is limited by the consideration of the train's stability and safety during travel. Trains are susceptible to hunting motion, which is generated by the difference between the inside and outside diameters of a wheelset. Although this motion is designed to guide the train to run on straight lines, it can induce instability or even derailment during high-speed operation. To maintain stability and suppress hunting motion, a hard primary longitudinal stiffness is crucial. However, the high stiffness required for stability directly impairs the train's ability to pass through curves. As a result, conflicting requirements for stiffness arise when trains operate on straight and curved tracks.
To address the issue of conflicting stiffness requirement, many solutions, including active suspensions and passive solutions, have been proposed. Active steering bogies have attracted more attention due to their controllability and adaptability, and there have been numerous theoretical and simulation studies on their design [1][2][3][4]. This type of steering control offers high steering controllability and the potential to achieve nearperfect steering. Using actuators to control the lateral motion of the wheelset is another fully active control solution [5][6][7]. This mode of active control offers the advantage of utilising lateral actuator control to prevent hunting oscillation without impeding the natural cornering action of the wheelsets while passing through curves. Despite the excellent performance of active steering bogies, they require expensive and bulky actuation components, and due to its dependence on external power supply, the control system is susceptible to a heightened risk of instability in the event of failure, potentially resulting in derailment.
In contrast to fully active control options, semi-active control methods are generally deemed as more cost-effective and safer, with significantly lower power requirements. Semiactive control operates by modulating material properties through control signals, as opposed to physical actuation, to attain the intended motion [8,9]. Magnetorheological materials are a type of smart materials that exhibits rapid and reversible response to induced magnetic field stimulation [10][11][12]. They have been widely used in vehicle suspension [13][14][15], shock absorption [16,17] and robotics [18,19]. Several rubber joints based on MR materials have been reported [20][21][22].
Our group developed an MR elastomer (MRE) rubber joint and an MR fluid rubber joint, which can increase stiffness by 74.7% and 97% respectively when the control currents apply [23,24]. Fail-safe characteristics is of significance for train's operation because the train should be ensured with continuous operation even in the event of a joint controller failure, without resulting in critical accident. To meet this aim, our team proposed an MRE based rubber joint with fail-safe characteristics to achieve an initial hard primary longitudinal stiffness with no current supply [25]. However, if the joint controller fails, this rubber joint can only provide hard stiffness to ensure highspeed stability and scarifies the curve trafficability of trains, which results in high operation cost of railway vehicles. As a result, it is imperative to develop a new variable stiffness rubber joint that can accommodate both operating environments in the event of a failure.
To address this issue, shear stiffening materials represent a potential solution. These materials exhibit the ability to rapidly increase in storage modulus when subjected to surgent shear frequency [26][27][28]. During high-speed operation, the excitation of frequency of the hunting motion is elevated, thereby leading to a significant increase in the joint stiffness. Conversely, the curved track is associated with a low frequency passing performance, resulting in reduced joint stiffness. For instance, Yang et al reported a rubber joint based on shear thickening fluid (STF), considerably increasing the stiffness of the STF joint from 0.162 to 0.438 MN m −1 as the excitation frequency rising from 0.1 to 3 Hz [29]. However, these rubber joint can only adjust its stiffness passively according to the loading frequency and the stiffness variation range can be improved as well.
Based on the above analysis, it is apparent that the incorporation of the shear stiffening effect into magnetorheological materials can enable the stiffness controllability across diverse tracks, even when the joint controller fails. In this study, a novel rubber joint, constructed using a magnetorheological shear stiffening elastomer (MSSE), is introduced. MSSE is responsible to both external magnetic field and excitation frequency. With the magnetic field increases, the storage modulus of MSSE rises. Furthermore, the material performs high stiffness under high frequency excitation and low stiffness under low frequency excitation even without the control of external magnetic field. Consequently, the MSSE-based joint can serve as a semi-active control technique, wherein the stiffness can be adjusted by inducing the magnetic field to achieve both high-speed stability and curve feasibility. Furthermore, in the event of a supply failure, the joint can still rely on the inherent shear stiffening properties of the MSSE material to achieve the desired stiffness to some extent, to guarantee the high-speed stability and curve passing performance. The structure of the paper is organised as follows: section 2 details the structure, prototype and analysis of the MSSE joint. Section 3 manifests the fabrication of the MSSE material and the characterisation of its properties. The evaluation of the stiffness variation of the MSSE joint is presented in section 4. Section 5 showcases the dynamic model and simulation. Finally, the conclusion for this work is provided in section 6.

The structure and prototype of the MSSE joint
The H-frame design with rotary arm primary suspension is a popular feature in the bogies of high-speed trains. The rubber joints which connect the bogie frame and the rotary arm play a crucial role in determining the primary longitudinal stiffness of the wheelset. To ensure the safety and comfort, the joint stiffness needs to be soft for curved tracks and hard for stability during high-speed operation on straight tracks. To address this adaptive requirement, a new MSSE joint has been developed. The MSSE joint offered a controllable stiffness, making it an ideal solution to facilitate the variations of the primary longitudinal stiffness. Figure 1 illustrates the position of the MSSE joint on the bogie and structure of the joint. The longitudinal position stiffness of the wheelset is determined by the rubber joint that is installed between the bogie frame and the rotary arm. The joint consists of a central shaft, three groups of steel rings, two sets of electromagnetic coils, MSSE layers, and a steel cylinder. To ensure the optimal permeability, the central shaft, steel rings, and cylinder are entirely made of 1010 steel. The central shaft and the outer steel cylinder are the connection and contact components between the MSSE joint and the bogie frame. The MSSE joint comprises three groups of parallel-installed MSSE multilayered structures. To optimise the magnetorheological performance and the reliable structure stiffness, the thickness of the MSSE layers and steel rings were considered. The thickness of each MSSE layer and steel ring were designed as 1.5 and 2.5 mm, respectively. Each group of multilayered structures includes six MSSE layers, which can be viewed as six springs in series. The stiffness of the rubber joint is determined by the elastic modulus of the MSSE when they are not exposed to a magnetic field.
The working principle of the MSSE joint lies in the variable storage modulus of MSSE material with excitation frequency and magnetic field. Specifically, when the power supply is functioning normally, the current applied to the coils generates a magnetic field that alters the joint stiffness. However, in the event of an accidental power supply failure, the passive frequency-dependent control mechanism can serve as a reliable backup option, enabling the joint to traverse different tracks and displaying superior fail-safe characteristics compared to other rubber joints.
For the current control of the MSSE joint, when there is no current applied, the MSSE joint exhibits relatively low stiffness. However, the electromagnetic field can be generated by the coil current. When the electromagnetic coil is energised, the magnetic field is induced, and the magnetic circuit is formed, leading to a boost in magnetic flux density as the current raises. Consequently, the storage modulus of MSSE increases, endowing the structure with higher stiffness. In this case, the MSSE joint performed with hard stiffness when current is applied to energise the electromagnets, meeting the stiffness requirements of the train to maintain stability when moving at high speeds in a straight line. Conversely, when the external power supply is turned off, the magnetic field working on the MSSE layer would be vanished accordingly, thus contributing to favourable curving trafficability with low stiffness characteristics.
For the case of joint controller failure, the shear thickening effect of the MSSE contributes to the dynamic performance of the train. Specifically, the frequency of longitudinal excitation on the rubber joint is very low during curving and the MSSE layer will have a relatively low modulus and exhibit a soft state. Therefore, the MSSE joint behaves with minimum stiffness under this condition. From this perspective, the proposed MSSE joint can satisfy the requirement for soft stiffness on curving trafficability. For straight track scenarios at high speeds, the frequency of the load produced from the hunting motion of the bogie increases significantly and the primary longitudinal stiffness must be high to ensure the train's stability. Fortunately, the MSSE joint can generate high stiffness during high-frequency excitation. The load developed during this operation is transmitted to the MSSE layer through the outside steel cylinder, inducing the MSSE layer to become rigid. The storage modulus of MSSE increases by several orders of magnitude under the simulation of high-frequency excitation. Therefore, the MSSE joint behaves with increased stiffness when the train runs on straight track at high speed. As the rheological property of MSSE is rate-dependent, i.e., frequency-dependent, the stiffness of the rubber joint exhibits frequency-dependent characteristics, catering to the conflicting needs of trains with different rigidities when the train runs on straight and curve tracks.

Magnetic field analysis of the MSSE joint
To optimise the size of the shaft and the cylinder, a magnetic field analysis of the MSSE joint using COMSOL Multiphysics 5.6 software was conducted. The analysis aimed to evaluate whether the magnetic field would saturate the shaft and the cylinder. The electromagnetic coil was set as 1000 turns with the diameter of 0.5 mm. In the software, the constitutive relationship of steel was assigned according to its B-H curve, which was pre-defined in the software. For more accurate analysis results, the B-H curve of MSSE measured in the section 3.2 was defined in the simulation. Through adjusting diameter of the shaft, width of the electromagnetic coil and thickness of the cylinder, the final results were obtained. The section view of visual magnetic field distribution of the MSSE joint was shown in figure 2(a). The unit of the magnetisation is Tesla (T). The maximum magnetic flux density in the shaft was 2.13 T,  indicating that there was no magnetic saturation in the MSSE joint [30]. To ensure better performance of MR effect, the magnetic flux density of the MSSE layer was numerically evaluated. The probes were positioned to measure the magnetic field densities of the inside and outside MSSE layers in the middle position. It can be seen that the magnetic flux density of both layers indicated an increased trend, with value reaching up to 1.29 and 0.80 T, respectively, when a current of 5 A was applied. The distribution of magnetic field strength and value changes results demonstrated that the magnetic field could be controlled effectively.

Fabrication of the MSSE material
The properties of MSSE, especially the magnetorheological effect (MR effect), have great influence on the performance of MSSE joint. It is crucial to choose a suitable matrix and mass ratio to obtain better MR effect of MSSE. Hydroxy terminated polydimethylsiloxane, boric acid and benzoyl peroxide (BPO) were purchased from Sinopharm Chemical Reagent Co. Ltd, Shanghai, China. The vinyl methyl silicone rubber (VMQ 110-2) was provided by Shenzhen Muwei Technology Co. Ltd. All chemical reagents were of analytical purity and used as received without further purification. Firstly, a mixture of boric acid and hydroxy terminated polydimethylsiloxane was stirred with the mass ration of 1:30 and heated in oven for 2 h (figure 3). A few drops of N-octanoic were added during the heating process. Shear stiffening gel (SSG) could be obtained after cooling down to room temperature. The precursor of MSSE was fabricated by mixing SSG, vinyl methyl silicone rubber (VMQ) and carbonyl iron particles (CIPs) with a two-roll miller. The fixed mass ratio of SSG and VMQ was 7:3. Benzoyl peroxide (BPO) was then added as the vulcanised agent. Consequently, the hybrid polymer matrix was molded and cured in a metal mold under 20 MPa at 100 • C for 15 min to produce the MSSE.

Characterisation of the MSSE material
The rheological properties of MSSE were characterised by a commercial rheometer (Physica MCR 302, Anton Paar Co., Austria). The tested samples were molded into cylinders with the thickness of 1 mm and the diameter of 20 mm. Dynamic frequency scan measurements of shear stiffening elastomers with different mass fractions of CIPs were conducted using the rheometer. As shown in figure 4(a),  owing to the rate-dependent characteristic, the SSE exhibited a typical shear stiffening behaviour. Due to particlestrengthening effect, MSSE containing CIPs was stiffer than pure SSE. With the mass fraction of CIPs in the composite increased, both the minimum storage modulus (G min ′ ) and the maximum storage modulus (G max ′ ) of MSSE showed a growing trend, indicating that the incorporation of CIPs strengthened the mechanical properties of SSE. Take the MSSE with the content of 70 wt% CIPs as an example, when the shear frequency was kept at 0.1 Hz, the initial storage modulus (G min ′ ) was low (0.020 MPa), presenting a soft viscosity state. As soon as the shear frequency reached 100 Hz, the maximum storage modulus (G max ′ ) increased by an order of magnitude (0.559 MPa), exhibiting the typical shear stiffening effect. Here, the MSSE with different CIPs content were tested by HyMDC to evaluate the magnetic properties ( figure 4(b)). With the mass fraction of CIPs rising from 0 wt% to 100 wt%, the saturation magnetisation of the materials soared from 0.14 emu g −1 to 207.69 emu g −1 . The variation tendency of saturation magnetisation was coincident with that of the CIPs content, reflecting the stable magnetisation performance of CIPs inside the MSSE. As shown in figure 4(c), the magnetisation test was conducted under the shear frequency of 1 Hz and strain amplitude of 0.1%. The storage modulus of MSSE evidently boosted with the increasing magnetic field, indicating a typical magnetorheological effect of MSSE. By increasing the CIPs content from 0 wt% to 70 wt%, both the initial modulus (G 0 ′ ) and the magnetic induced modulus (∆G ′ ) significantly increased, proving that the more of the mass ratio of the CIPs will enhance the magnetic induced properties. For MSSE with CIPs content of 70 wt%, the magnetorheological effect ( ∆G ′ G ′ 0 ) was 1625%. In conclusion, the rheological property and magnetic property of the MSSE demonstrated that the material can be sensitive to both external magnetic fields and excitation frequency. The frequency-dependent characteristics can be effective without the need for external magnetic field control, exhibiting the potential fail-safe attribute applied on the joint.

Experimental setup
So as to investigate the variable stiffness performance of the MSSE joint after assembly, material testing system (MTS) was utilised. As shown in figure 5, The MSSE joint was fixed by an upper clamp and a lower clamp attached to the top and bottom of the MTS machine, respectively. The upper clamping jaw was connected to a vertical hydraulic displacement cylinder. The upper displacing cylinder can be programmed to displace according to a predefined path. The harmonic excitation with variable displacement frequency and amplitude was applied. The lower clamp was connected to a load cell which measured and recorded the force signals.
The radial stiffness of the MSSE joint plays a crucial role in the hunting stability and curve passing performance of the train. Hence, this work focused on the MSSE joint's varying capacity of radial stiffness. During the test, the MTS machine was programmed to perform periodic displacements, with an original moving velocity of 20 mm min −1 . To evaluate the varying stiffness performance under different displacements, three sets of tests were conducted using displacement of 0.05, 0.1 and 0.15 mm, respectively. For each group of tests, different current levels ranging from 0 to 4 A and different frequency ranging from 0.1 to 3 Hz were applied to obtain the force-displacement loops under various working conditions.

Frequency-dependent testing
For the purpose of investigating how the stiffness of MSSE joint responds to the varying loading frequencies, harmonic signals with different loading frequencies ranging from 0.1 to 3 Hz at a fixed displacement amplitude of 1 mm was used to excite the MTS machine.
The force response and displacement data of the MSSE joint at different loading frequencies were shown in figure 6. It can be seen that the force-displacement relationship exhibited consistent hysteresis loop, and that slopes of the hysteresis loops exhibited obvious increase as the loading frequency grew, indicating the frequency-dependent property of the MSSE joint. In order to analyse the stiffness changing effect quantitatively and intuitively, the effective stiffness of the MSSE joint for each hysteresis loop was calculated by equation: where F max and F min are the maximum and minimum force, respectively. d max and d min are the maximum and minimum displacements for a given force-displacement loop, and k stands for the effective stiffness of the structure responded to the external excitation. Figure 6 depicted that when the displacement amplitude was 0.1 mm and no current was applied to the coil, the MSSE joint had a minimum effective stiffness of 1.61 MN m −1 . As the loading frequency increased from 0.1 to 3 Hz, the MSSE joint's effective stiffness increased steadily due to the ratedependent characteristics of the MSSE layers. At 3 Hz, the effective stiffness reached 4.44 MN m −1 ( figure 6(a)). When a current of 3 A was applied, the MSSE joint's effective stiffness increased from 3.86 to 6.80 MN m −1 across variable loading frequencies ranging from 0.1 to 3 Hz ( figure 6(b)). Due to the fact that the interface between MSSE and steel cannot be bonded completely the same, the performance of the rubber joint is not completely centrally symmetrical. The loop results reflecting the relationship between force and displacement illustrated that the proposed MSSE joint can adjust its radial stiffness evidently attributed to the remarkable rate-dependent property of MSSE layers.

Current-dependent testing
In order to investigate how the stiffness of MSSE joint responds under varying applied currents, different current levels (0 A, 1 A, 2 A, 3 A and 4 A) were applied to obtain the force-displacement loops of the joint under different magnetic fields. As shown in figures 7(a)-(e), under the displacement amplitude of 0.1 mm, the MSSE joint exhibited varying hysteresis loop with different loading frequency from 0.1 to 3 Hz. For example, with the current level increased from 0 to 4 A at 0.1 Hz, the magnetic field induced on the MSSE layer was strengthened, and the MSSE joint manifested a steady boost in effective stiffness with the increment of 1 A, reaching up to 4.40 MN m −1 at 4 A ( figure 7(a)); at a loading frequency of 2 Hz, the effective stiffness of the MSSE joint rose from 4.18 to 7.13 MN m −1 with a variable current input ranging from 0 to 4 A ( figure 7(d)). The results under different loading frequencies indicated that the proposed MSSE joint can adjust the radial stiffness effectively under various frequency conditions. Table 1 presents the effective stiffness values calculated for different current levels and a loading frequency of 1 Hz. The results indicate that, for a displacement amplitude of 0.05 mm, the effective stiffness increased by 83.26% as the current level was increased from 0 to 4 A. Similarly, for a displacement amplitude of 0.1 mm, the effective stiffness increased by 86.36% with the same current level increase. Finally, for a displacement amplitude of 0.15 mm, the effective stiffness increased by 67.50% under the same current level increase. The results suggest that the stiffness of the MSSE joint can be significantly controlled by adjusting the input current, enabling remarkable stiffness adjustments with active control strategy.

Amplitude-dependent testing
To investigate whether the MSSE joint's stiffness varied in response to the varying displacement amplitudes or not, the   performance of the MSSE joint with different displacement amplitudes ranging from 0.1 to 2 mm was explored. The slopes of the hysteresis loops, i.e. the effective stiffness showed slight decrease as the loading amplitude increased. Through plotting figure 8 by putting the frequency-dependent performance and the amplitude-dependence behaviours together and comparing them, it is observed that the variations induced by the increasing amplitude is insignificant compared to what is induced by the varying frequency. Figure 9 manifested the effective stiffness of MSSE calculated under diverse conditions. In figure 9(a), the loading frequency was fixed at 2 Hz, and on the whole the effective stiffness showed a dropping trend as the displacement amplitude increased. This is because, with a constant loading frequency, a lower displacement amplitude resulted in a shorter time for velocity change when the upper clumper changes its direction of movement. This slightly affected the loading frequency, resulting in a relatively higher stiffness. Without energising the coil, the effective stiffness raised from 3.62 to 4.19 MN m −1 , and then from 4.19 to 5.17 MN m −1 as the displacement amplitude was 0.15, 0.1 and 0.05 mm, respectively. With an applied current of 4 A, the effective stiffness increased from 5.78 to 8.76 MN m −1 as the displacement amplitude declined. Similarly, with a fixed applied current of 3 A, the effective stiffness increased as the displacement amplitude decreased from 0.15 to 0.05 mm ( figure 9(b)). At a loading frequency of 0.1 Hz, the effective stiffness expanded from 3.69 to 4.30 MN m −1 . When the loading frequency was 3 Hz, the effective stiffness enhanced from 5.83 to 8.61 MN m −1 as the displacement amplitude diminished.

Modelling of the MSSE joint
For the sake of describing the behaviour of the MSSE joint under various conditions, a dynamic modelling was established. Stiffness testing demonstrated that the proposed MSSE joint exhibited nonlinear and hysteretic response characteristics and a model is essential to further understand and control the performance. The Bouc-Wen model is a widely used approach to describe nonlinearity and hysteretic behaviour, featuring a combination of a linear spring, a viscous dashpot, and a hysteretic component. However, using the classical Bouc-Wen model involve the issue of solving nonlinear ordinary differential equations, and the model may not fully reflect strain stiffening effects when subjected to extreme working conditions, such as high displacement amplitudes and loading frequencies.
Inspired by a model proposed by Berg [31], this study proposed a dynamic model to portray the force-displacement loop. The dynamic model schematic is represented in figure 10.
The proposed MSSE spring model is one-dimensional and the relationship between force and motion is based on a superposition of four forces: where F e , F v , F f1 and F f2 are the elastic, viscous, fractional Maxwell and friction forces, respectively. The relationship between the elastic force F e , and the displacement x, is simply given by where K e is the elastic stiffness constant corresponding to linear elastic behaviour. Only one model parameter K e , is therefore used for the elastic part of the model. The elastic model response is independent of the frequency and the energy loss is zero (no hysteresis). Introducing a viscous force F v , increasing stiffness with increasing frequency as well as rate-dependent hysteresis can be described. The viscous force is here represented by a linear viscous damper with damping c, in series with a strain stiffening term to reflecting the strain stiffening effect when subjected to extreme working conditions [32]. The mathematical expression of this part is shown in equation: where x 1 is the motion displacement of the damper. For a harmonic excitation this part gives an elliptic force-displacement graph.
Fractional-derivative Maxwell model can be seen as adding more sets of viscous dampers with series spring in parallel to the basic sets, and the drop can be pushed towards higher frequencies. More importantly, the present viscous model should in general be conservative due to the underestimated energy loss at higher frequency, and should therefore be satisfactory in most situations. The inclusion of a friction force means that an increased stiffness at small displacement amplitudes as well as rateindependent hysteresis can be considered. This also means that the model becomes non-linear. The friction force F f2 , in the present model depends, like the elastic force, on the displacement x over the element, but also on a reference state (x s , F fs ) in the friction force versus displacement graph. Depending on how x is related to x s , the friction force F f , in the model is expressed as below The three equations correspond to x = x s , x > x s , and x < x s , respectively. The force F fmax above is the maximum friction force whereas the displacement x 2 is the displacement needed to get the friction force F f = F fmax /2, given the reference state (x s , F fs ) = (0, 0). The parameters F fmax and x 2 are input parameters to the friction part of the MSSE spring model. The auxiliary quantity a = F fs /F fmax ranges from −1 to 1.
Based on the experimental data, parameter identifications were conducted using MATLAB (R2020b). It was observed that both loading frequencies and current levels could affect these parameters being studied. For instance, the forcedisplacement loops were estimated for displacement amplitude of 0.05 mm under current levels ranging from 0 to 4 A. The comparison between the experimental data and the simulation data was shown in figure 11, indicating good agreement between the two for different applied currents.
The simulated force generated by the MSSE joint can be calculated by inputting the displacement obtained from experimental data. Figure 12 illustrated the comparison of results  obtained in the frequency range of 0.1 to 3 Hz. Based on the fitting results, it is evident that the generalised model effectively portrayed the response of the MSSE joint under various frequencies. To check the effectiveness of the simulations in capturing the experimental force-displacement loops, the identification error was computed by minimising the cost function of sum squared error. For the sake of quantitatively evaluating the accuracy of proposed model, the identification error was calculated using the following equation: where N exp is the number of experiment samples, f exp is the experimental force, and f sim is the simulated radial force of the MSSE joint. The reduction of the J is continuously iterated by the method of least squares fitting. Through several iterations, the identified error under various applied currents and frequencies was eventually reduced to 0.0346 and 0.0272, respectively. The relative error of the effective stiffness of the MSSE joint between experiment and simulation was calculated as well. With the applied current of 0 A and displacement amplitudes of 0.05 mm, the relative errors of the effective stiffness were 11.2% and 7.8% when the loading frequencies are 1 and 3 Hz, respectively, exhibiting that the proposed model can accurately depict the response of the MSSE joint with acceptable accuracy. The identified error was reduced to a satisfactory level, indicating that the presented model can effectively capture the behaviour of the MSSE joint.

Conclusion
This paper presented a comprehensive study on the design, manufacturing, testing and evaluation of an MSSE-based joint for high-speed trains. The MSSE joint offers adjustable stiffness through inherent rate-dependent characteristics and magnetic fields induced by applied current, which can maintain excellent train stability and curve-passing performance simultaneously with remarkable fail-safe characteristics. The simulation of the magnetic field demonstrated the reasonability and effectiveness of the magnetic circuit. In order to explore the stiffness adjustment of the MSSE joint, stiffness tests were conducted under various conditions using an MTS machine. The results show that the effective stiffness of the joint can reach up to 4.44 MN m −1 at a loading frequency of 3 Hz, exhibiting a good frequency-dependent response capacity. Additionally, when the displacement amplitude was 0.1 mm, the effective stiffness increased from 3.52 to 6.56 MN m −1 as the current level was increased from 0 to 4 A, indicating an effective controllability with the applied current level. Finally, a joint model has been proposed that accurately represents the response of the MSSE joint to external excitation. The proposed model is capable of accurately portraying the behaviour of the MSSE joint in response to a variety of external stimuli. In conclusion, a novel MSSE joint with stiffness variation capability, even when the controller fails, is successfully built and can effectively enhances train stability on straight tracks and improves curve-passing performance on curved tracks.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).