International Journal of Automotive Technology, Vol. 13, No. 7, pp. 1047−1056 (2012) DOI 10.1007/s12239−012−0107−3

Copyright © 2012 KSAE/ 068−04 pISSN 1229−9138/ eISSN 1976−3832

DEVELOPMENT OF A NEW ANALYTICAL MODEL FOR A RAILWAY VEHICLE EQUIPPED WITH INDEPENDENTLY ROTATING WHEELS Y. CHO1) and J. KWAK2)* 1)

2)

Virtual Engineering, University of Science and Technology, Daejeon 305-3502, Korea Korea Railroad Research Institute, 360-1 Woram-dong, Uiwang-si, Gyeonggi 437-757, Korea (Received 9 March 2011; Revised 25 March 2012; Accepted 12 August 2012)

ABSTRACT−The urban tram introduced recently has a low-floor structure for the convenience of passengers getting on and off. To adjust the low-floor level and improve performance on curves, most low-floor trams have IRWs (independently rotating wheels) with no central axle between the two wheels. Eliminating the central axle, however, creates several inherent problems, such as insufficient guiding force and excessive wear. To analyze these problems, a new analytical model is described in this paper to describe the dynamic characteristics of IRWs more precisely. This analytical model is developed to consider the effects of longitudinal creep in particular, which have been ignored in conventional analytical models of IRWs. In addition, a running stability analysis based on the newly developed analytical model is conducted to compare the critical speeds of IRW-axle vehicles and rigid-axle vehicles. The dynamic characteristics of an initial disturbance are compared to verify that the analytical model is effective in expressing the dynamic characteristics of IRWs. KEY WORDS : IRW (Independently Rotating Wheel), LRT (Light Rail Transit), Stability analysis, Critical speed, Railway vehicle

Csx : longitudinal damping of the 2nd suspension : lateral damping of the 2nd suspension Csy : longitudinal stiffness of the 2nd suspension Ksx : lateral stiffness of the 2nd suspension Ksy : lateral creep force coefficient f11 : lateral/spin creep force coefficient f12 : spin creep force coefficient f22 : longitudinal creep force coefficient f33 : axle load WA FLx, FLy, FRx and FRy: Kalker’s creep force MLx, MRx, MLy and MRy: Kalker’s creep moment Ft : Flange contact force, Kr: Lateral rail stiffness µ : Friction coefficient , δ : Flange clearance

NOMENCLATURE mw mc Iwx Iwy Iwz Iw1 Itx Itz Icz r0 b1 b2 b3 L1 L2 LC λ Kpx Kpy Kpz Cpx Cpy Cpz

: wheelset mass , mt : Bogie frame mass : car body mass : roll moment of inertia of the wheelset : spin moment of inertia of the wheelset : yaw moment of inertia of the wheelset : roll moment of inertia of the wheel : roll moment of inertia of the bogie frame : yaw moment of inertia of the bogie frame : yaw moment of inertia of the car body : Wheel radius, a : Half of the truck gauge : half of the primary longitudinal spring arm : half of the secondary longitudinal spring arm : half of the secondary vertical damping arm : half of the primary lateral spring arm : half of the primary lateral damping arm : half of the distance between bogie center : wheel conicity : longitudinal stiffness of the 1st suspension : lateral stiffness of the 1st suspension : vertical stiffness of the 1st suspension : longitudinal damping of the 1st suspension : lateral damping of the 1st suspension : vertical damping of the 1st suspension

1. INTRODUCTION In general, low-floor trams may have IRWs (independently rotating wheels) without an axle between wheels for the purpose of achieving a low floor area. When trains equipped with IRWs travel in urban centers with various curved lines, the wheels run independently, reducing wear and noise. In contrast, when running on straight track, a lack of centering force results in less guiding force, which results in a skew phenomenon as well as increased wear and possible derailment. Many studies have been conducted to attempt to overcome the shortcomings of independently rotating wheels. For successful implementation of such research, it is necessary to develop analytical

*Corresponding author. e-mail: [email protected]

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models that more accurately reflect the characteristics of independently rotating wheels. In this study, a new analytical model was developed that considers the effects of longitudinal creep, which have been ignored in conventional analytical models of independently rotating wheels. A running stability analysis based on the newly developed model was conducted to compare the critical speeds of IRW-axle vehicles and rigid-axle vehicles. In addition, the dynamic characteristics of an initial disturbance were compared to verify that the analytical model is effective in expressing the dynamic characteristics and stability of IRWs.

2. BACKGROUND The concept of independent wheel rotation has been studied for a long time. The importance of independent wheel rotation has recently increased as trams combined with low-floor vehicle technology have become more commonly used. Previous studies on independent wheel rotation have thus far focused mainly on control and performance improvements designed to overcome the weaknesses of independent rotation and evaluate the characteristics of IRW-axle vehicles in comparison with rigid -axle vehicles. Liang and Iwnicki investigated the torque control of the rigid-axle model and the IRW-axle model (Liang et al., 2004). Mei and others have conducted research on active wheelset controls and guiding force optimal controls (Mei and Li, 2008; Goodall et al., 2003; Mei et al., 2001). R. V. Dukkipati surveyed previous research findings concerning independently rotating wheels. He introduced an analysis and test of independent rotation in Japan in the 1960s, along with the successful case of Talgo. In his paper, R. V. Dukkipati divided trends in the study of IRWs into three categories: study of the wheel shape for enhancing gravity-restoring forces, study of the addition of guiding equipment, and study of implementation of independent rotation with a torsion clutch or coupler (Dukkipati and Narayana Swamy, 1992). R. M. Goodall conducted a study on a means of improving stability in curves by investigating mechanical vibration of IRWs (Goodall and Hong, 2000). Moritz and Lutz made a comparison study of various guidance control types, and Mei studied the sensors in terms of commercialization (Mei and Goodall, 2003). Studies on IRWs have been actively implemented in Japan. Saitoh and Tanifuji indentified characteristics of IRWs and examined a method to improve their performance by using a scale model and an analytical model (Saitho and Tanifuji, 2002). Obata presented results of research to improve curve performance based on torque control of independent wheels (Obata et al., 2006). Y. Suda, who is renowned for research on steering

equipment using an asymmetric bogie, suggested a way of significantly enhancing performance by installing a trailing axle with IRWs (Suda and Maeshiro, 1998). Miyamoto, Sato and Yasuda conducted an investigation into the characteristics of IRWs that involved verification of an analytical model and scale-model tests for various axles and bogie configurations (Miyamoto and Sato, 2001; Satou and Miyamoto, 2002). J. Perez, T. X. Mei and Goodall analyzed and compared several control algorithms for improving the characteristics of IRWs (Perez et al., 2004). When first introduced, independent rotation was regarded as a means of increasing a vehicle’s stability, because longitudinal creep could be reduced or eliminated, and accordingly, yaw moment and lateral force could be decreased, resulting in a decrease in the relevant lateral activation force. Thus, it was expected that independent rotation would enhance the dynamic characteristics of vehicles by removing the hunting movement that conventional vehicles exhibit. However, because independent rotation does not generate guiding force, a skew phenomenon is caused by disturbances and rail irregularities when a railway vehicle runs on a tangent track, and consequently, zigzag-type vibration, unique to independent vibration, may occur. This dynamic characteristic can result in a significant level of noise and wear with straight-ahead motion. Conventional approaches to research of this subject employ a longitudinal creep-free model from among the analytical models available for a general rigid axle or omit elements generated by the longitudinal factors when drawing an analytical model, with the goal of analyzing the dynamic characteristics of IRWs. However, this type of analytical model has characteristics that are relevant only to the trailing bogie (Liang and Iwnicki, 2007; Mei et al., 2008). Moreover, it is difficult to apply this type of model to analyzing the centering characteristics and driving stability characteristics that are unique to IRWs. Hence, it is necessary to consider the effects of longitudinal creep and relevant spin and yaw characteristics to conduct a stability analysis of vehicles equipped with IRWs and explain their gravitational stiffness. Recent findings by Zaazae and Sugiyama indicate that it is not possible to completely ignore longitudinal creep (Zaazae and Whitten, 2007; Sugiyama et al., 2009). Section 3 of this paper summarizes the development of an analytical model that includes the effects of gravitational stiffness and longitudinal creep, which are ignored in the conventional analytical models. The analytical model given in section 3 was developed using dynamics equations presented by Dukkipati (Garg and Dukkipati, 1984). In section 4 of this paper, a stability analysis, including axle, bogie, and vehicle models, is conducted using the new analytical model, to obtain results for design variables affecting critical speed. These results are compared with

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results obtained with a rigid axle model. The effectiveness of the newly developed analytical model is verified by the results of the comparison.

In compliance with Newton’s law, equation (1) is divided into equilibrium equations with respect to each degree of freedom (DOF), which are given as follows (Lee and Cheng, 2008).

3. ANALYTICAL MODEL DEVELOPMENT

ΣX:mx·· = FLx + FRx + NLx + NRx + FSx

(3)

Development of the model used for analysis begins with dynamics equations for the axle model, as shown in Figures 1 and 2. Based on the axle model, a bogie model was derived, including bogie and primary suspension elements, and a car body model was then derived, including car body and secondary suspension elements. The dynamic equations are derived using the equilibrium equation of force and moment, as shown in Figure 3. Each term is obtained from and substituted for basic equations, such as (1) and (2), and the equations are then simplified and linearized through several assumptions. This process illustrates that the dynamic equations of independent axle models presented previously do not consider the effects of longitudinal creep; therefore, the restoring capability and effects on yaw characteristics of a real independent axle cannot be adequately expressed. However, this paper presents the development of dynamic equations for the wheelset in the independently rotating axle model, considering the effects of the previously ignored longitudinal creep. The newly developed axle model was applied to bogie, half car body and vehicle models. with the results from the new model were compared with the result obtained from the rigid-axle model.

ΣY:my·· = FLy + FRy + NLy + NRy + FSy

(4)

ΣZ:mz·· = FLz + FRz + NLz + NRz + FSz – WA

(5)

·· m ΣF:mr = FL + FR + NL + NR + FS – WAk

(1)

dH ΣM: ------- = RR × ( FR + NR ) + RR × ( FL + NL ) + ML + MR + Ms dt

(2)

v Σφ:Iwxφ·· = Iwy⎛⎝ ----⎞⎠ ψ· + RRy( FRz + NRz ) – RRz ( FRy + NRy ) r0 + RLy ( FLz + NLz ) – RLy( FLz + NLz ) + MLx + MRx + MSx

(6)

Σθ:Iwy θ·· = – RRz FRx – RRx( FRz + NRz ) – RRz ( FRy + NRy ) + RLy ( FLz + NLz ) + MLy + MRy + MSy

(7)

v⎞ · ·· = –I ⎛ --Σψ:Iwz ψ wx - φ + RRx ( FRy + N Ry ) – R Ry FRx ⎝ r0⎠ + RLx ( FLy + NLy ) – RLz FLx + MLz + MRz + MSz

(8)

In this paper, dynamic equations are developed, considering lateral motion and yaw motion of the analytical model designed to evaluate lateral dynamic characteristics of a vehicle moving at a constant velocity on a tangent track. Equations (9) and (10) show the linearized dynamic equations of the model in Figure 1 obtained using equations (4) and (8), and equations (11)–(14) present the dynamics equations of the model in Figure 2 obtained using equations (4), (8), and (7). The linear dynamic equations of the rigid-axle model are as follows: –2f –2f r wλ mwy··1 = ⎛⎝– ------- –2Kpy⎞⎠ y1 + ⎛⎝ ----------11- – ----------11- ----0 λ – 2Cpy⎞⎠ y· 1 a v v a 2f 12 · + 2f11ψ1 – -------- ψ1 v

(9)

2f12- --rλ λvλ ·· 1 = – 2a --Iwz ψ – Iy - --- + 2f12 ----0 ---⎞⎠ y· 1 f y + ⎛ ------r0 33 1 ⎝ v r0 a va + ( – 2f12 + aλw – 2Kpx b21 )ψ1 f33- 2f 22 - – 2Cpxb21⎞ ψ· 1 + ⎛⎝ – 2a2 ---– ------⎠ v v

Figure 1. Rigid -axle model.

(10)

The linear dynamic equations of the IRW-axle model are as follows: –2f –2f r wλ mwy··1 = ⎛⎝– ------- – 2Kpy⎞⎠ y1 + ⎛⎝ ----------11- – ----------11- ----0 λ – 2Cpy⎞⎠ y· 1 a v v a 2f 12 · -ψ + 2f11ψ1 – ------v 1

(11)

2f rλ λ vλ Iwz ψ1 = – 2a ---- f33y1 + ⎛⎝ -------12- – Iy ---- --- + 2f12 ----0 ---⎞⎠ y· 1 r0 r0 a v va

Figure 2. Independently rotating wheel (IRW)-axle model.

+ ( – 2f12 + aλw – 2Kpxb21 )ψ1 f33- 2f + ⎛⎝ – 2a2 ---– -------22- – 2Cpx b21⎞⎠ ψ· 1 v v r r + f33 a ----0 θ· 1L – f33a ----0 θ· 1R v v

(12)

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Figure 3. Schematic of wheelset model. 2

r a Iw1θ·· 1L = f33λy1 + r0f33 --- ψ· 1 – f33 ----0 θ· 1L v v

(13)

2

r a Iw1θ·· 1R = –f33λy1–r0f33 --- ψ· 1 – f33 ----0 θ· 1R v v

(14)

Publications by Dukkipati were consulted in the derivation of the formulas (Garg and Dukkipati, 1984). In particular, the longitudinal creep and the relevant gravitational stiffness effects, which had been ignored in other research, were included to more accurately describe the dynamic motion of IRWs. A wheelset equation of motion was used to derive bogie, half car body, and full car body models. Figure 4 shows a schematic diagram of the dynamic equations developed for the bogie a general railway vehicle. Equations (23)–(32) are the dynamic equations of the train’s bogie with IRWs applied. The dynamic equations were developed using the developed wheelset dynamic equations and the configuration illustrated in Figure 4. In the same manner, an 8-DOF half-car-body model and a 14-DOF full-car-body model with a rigid-axle model and a 12-DOF half-car-body model and 22-DOF full-car-body model were developed.

The only differences between a half-car-body model and a full-car-body model are additional car body lateral and yaw movement equations. Consider a half car body moving on curved tracks with radius R. The equations of motion, including lateral and yaw displacement, are as follows: 2

m V myy··t = Fsyt + ⎛⎝mt + -----c⎞⎠ ⎛⎝ ------ – φ se⎞⎠ g 2 gR

(15)

·· t = M Itz ψ szt

(16) 2

1--- ·· 1 V m y = Fsyc + --- mcg⎛⎝ φ se – ------⎞⎠ 2 c c 2 gR

(17)

·· t = M Icz ψ szc

(18) (19)

(20)

(21)

(22)

Figure 4. General bogie model.

where V is the speed, Fsyt and Mszt are the suspension force and moment of the frame, respectively, and Fsyc and Mszc are the suspension force and moment of the car body, respectively. The dots and double dots indicate differentiation and double differentiation, respectively, with respect to time t. A wheelset model and extended versions of this model for bogie, half-car-body and full-car-body models were developed. The characteristics of a rigid axle and IRW axle for each model were then compared and evaluated, and the effects of IRWs on each model were investigated. A nonlinear analytical model was developed, taking into

DEVELOPMENT OF A NEW ANALYTICAL MODEL FOR A RAILWAY VEHICLE EQUIPPED

account the effects of creep saturation on nonlinear creep characteristics and flange contact characteristics in the linear model, to perform an analysis that addresses nonlinear behavior. A comparative analysis of the nonlinear models developed was conducted for each DOF to review the effects of the nonlinear characteristics (Li et al., 2009). In the case of analysis of wheels and rails with large relative displacements, nonlinear shapes for wheels and rails should have a significant influence on the analysis. Therefore, this must be considered in the analysis of sharp curve motion (Park et al., 2009). An analysis of stability on a tangent track was conducted in this study. The response characteristics of initial disturbances, and thus nonlinear shapes of wheels and rails, are not considered (Ko and Song, 2010). The linear dynamic equations of a bogie with the application of IRWs are as follows:

(32)

As equation (33) shows, creep saturation was introduced by including a saturation coefficient (á) in the equation. As equation (34) shows, a lateral nonlinear characteristic associated with flange contact was modified such that the flange contact force could be selectively applied to the equation, according to clearance.

(33)

(23)

(24)

(34)

Equations (35)–(38) present dynamic equations of axle models with the application of IRWs. The nonlinear dynamic equations of the IRW-axle model are as follows: –2a1f11- –--------------2a1f11- --r-0 wλ λ – 2Cpy⎞⎠ y· 1 mwy1 = ⎛⎝– ------- – 2Kpy⎞⎠ y1 + ⎛⎝ --------------– a v v a 2a1f12- · ψ1 – Ft1 + 2α1f11ψ1 – -----------v

2

r a Iw1θ·· 1L = f33λy1 + r0f33 --- ψ· 1 – f33 ----0 θ· 1L v v r2 a Iw1θ·· 1R = –f33λy1–r0f33 --- ψ· 1 – f33 ----0 θ· 1R v v

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(35)

(25) (26) (36) (27) 2

(28)

r2 a Iw2θ·· 2R = f33λy2 + r0f33 --- ψ· 2 – f33 ----0 θ· 2R v v

(29)

2

r a Iw2θ·· 2R = –f33λy2–r0f33 --- ψ· 2 – f33 ----0 θ· 2R v v

(30) (31)

r a Iw1θ·· 1L = α1f33λy1 + r0α1f33 --- ψ· 1 – α1f33 ----0 θ· 1L v v

(37)

r2 a Iw1θ·· 1R = –α1f33λy1–r0α1f33 --- ψ· 1 – α1f33 ----0 θ· 1R v v

(38)

4. ANALYSIS RESULTS This study compares the effects of IRWs on the critical speed of a railway vehicle. Table 1 shows the parameter values of the railway vehicle. Several previous studies have compared and evaluated the dynamic characteristics of IRWs. However, few studies presenting analyses of the stability of IRW railway vehicles have been published. The predominant view is that with the IRWs, the critical speed should be almost unlimited due to elimination of

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Table 1. Parameter values of railway vehicle. Symbol

Value

Unit

Symbol

Value

Unit

mw

800

kg

L2

1.155

m

mt

3,575

kg

Kpx

1.2e+7

N/m

mc

11,970

kg

Iwx Iwy Iwz

400 42 400

Kpy

4.182e+6

N/m

kg·m

2

Kpz

2.093e+6

N/m

kg·m

2

Cpx

2.77e+4

N·s/m

kg·m

2

Cpy

1.16e+4

N·s/m

2

Cpz

1.16e+4

N·s/m

Iw1

100

kg·m

Itx

1,500

kg·m2

Ksx

1.24e+5

N/m

3,100

kg·m

2

Ksy

1.24e+5

N/m

164,000 kg·m

2

f11

2.212e+6

N

Itz Icz r0

0.33

m

f12

3120

N·m2

a

0.7175

m

f22

16

N

λ

0.05

-

f33

2.536e+6

N

b1

1.01

m

W

5.6e+4

N

b2

1.11

m

µ

0.4

-

b3

1.11

m

Kr

1.617e+7

N/m

L1

1.155

m

δ

0.009

m

longitudinal creep. However, it has been verified by tests that when IRWs are actually used on a vehicle, the critical speed of the vehicle is lower than predicted. Some conventionally developed equations predict very low critical speeds because of their unstable terms. This paper demonstrates that it is possible to conduct a more accurate dynamic analysis of an IRW model by deriving dynamics equations that predict critical speed realistically through a stability analysis. Figures 5 through 12 present the dynamic characteristics for an initial disturbance of 5 mm according to the IRWaxle model and rigid-axle model, including axle, bogie, and car body models, running on a tangent track with an initial velocity of 300 km/h. With respect to lateral axle displacement as a result of

Figure 5. Wheelset displacements (Wheelset model).

Figure 6. Wheelset yaw angles (Wheelset model).

Figure 7. Wheelset displacements (Bogie model).

Figure 8. Wheelset yaw angles (Bogie model).

Figure 9. Bogie displacements (Bogie model).

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Figure 10. Wheelset displacements (Half car body model).

Figure 13. Critical speeds by conicity.

Figure 11. Wheelset yaw angles (Half car body model).

Figure 14. Critical speeds by longitudinal primary suspension stiffness.

Figure 12. Bogie displacements (Half car body model). the initial disturbance, a symmetric motion in the center occurs due to the restoration force in the case of the rigid axle. In the case of IRWs, the initial disturbance gradually decreases due to the gravitational stiffness. However, unlike with rigid-axle model, there is no symmetric motion in the center. The effects of the disturbance tend to decrease gradually with a small amount of vibration. According to the rigid-axle model, after a certain period of time, the disturbance is completely eliminated and the axle is positioned at the center. In the case of the IRW-axle model, the initial disturbance effects are not completely eliminated and the axle tends to lean toward the center. The nonlinear model, including creep saturation and

flange contact characteristics, exhibits greater discrepancies with the IRW-axle model than with the rigid-axle model, which indicates that creep saturation characteristic plays a dominant role in the behavior of IRWs. For the wheelset model in Figures 5 and 6, the rigid axle exhibits the most rapid damping. The IRW axle exhibits slow damping with vibration. Because the wheelset model is not influenced by the suspension force, it displays the effects of gravitational stiffness that are unique to the newly developed independent axle model. A bogie model, a half car body model and a full car body model with application of suspension force exhibit large differences, in terms of responses, between the rigid-axle model and the IRW-axle model. Unlike the wheelset model, the IRW-axle model does not exhibit vibration initiating mainly from the center part, and tends to exhibit gradual damping with very little vibration. With respect to yaw angle displacement, the rigid-axle

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Figure 15. Critical speeds by lateral primary suspension stiffness.

model and the IRW-axle model both tend to exhibit rapid damping and vibration. In fact, due to the additional yaw damping force caused by the difference in rotation speed between the left and right wheels, the IRW-axle model displays less yaw displacement than the rigid-axle model. Figures 8 and 11 represent the wheelset yaw angles of the bogie model and the half-car-body model. Figures 13 to 16 present the results of a stability analysis using the dynamics equations developed. In the stability analysis, the moment when the real number of the eigenvalue obtained from the eigenvalue analysis of the linear equation becomes positive is regarded as a critical unstable point, and the critical speed is thus determined. Based on this, the critical speed was calculated. The characteristics of IRWs were found to be properly described using the new dynamic equations developed in this study, with terms included to reflect the effects of

longitudinal creep and relevant gravitational stiffness on the critical speed. The rigid-axle model and the IRW-axle model both predict very high critical speeds. The bogie and car-body models predict low critical speeds, because secondary mass effects and suspension characteristics affect lateral motion. The car-body model, including additional car body elements and secondary suspension characteristics, tends to predict a lower critical speed. In comparison with the rigid-axle model, the IRW-axle model tends to predict a significantly higher critical speed. The damping coefficient included in the suspension elements has little effect on the critical speed, while longitudinal stiffness and lateral stiffness of the primary suspension exert a large influence on the critical speed. For a low stiffness value, the critical speed is very low. However, when the stiffness value is beyond a certain level, the critical speed does not increase further, and instead reaches a point of saturation. Figure 13 presents the results of an analysis of wheel conicity, which is known to have a large effect on the critical speed. A large critical speed is predicted for a very small area of conicity, but the critical speed dramatically decreases with conicity exceeding 0.1, after signs of wear. The half-car-body model shows negative results in the stability analysis, because the asymmetric configuration of the vehicle leads to increased instability. The maximum value of the critical speed is obtained at a specific value of the suspension stiffness. Accordingly, if an appropriate level of stiffness is set in the design of the suspension, the stability of the vehicle can be ensured. A stability analysis and dynamic analysis of the initial disturbance were conducted using the newly developed IRW-axle model, which incorporates longitudinal creep effects and gravitational stiffness elements. The analysis results demonstrate the validity of the newly developed analytical model. The dynamic equations developed in this study are important findings for improving curve performance and motion characteristics of IRW railway vehicles.

5. CONCLUSION

Figure 16. Critical speeds by longitudinal primary suspension damping coefficient.

A stability analysis was conducted using an improved IRW-axle model that includes the gravitational restoration force. According to the analysis results, the critical speed increases with the use of IRWs, because the differences in rotation speed between the left and right wheels caused by independent rotation result in less longitudinal creep and accordingly less yaw moment and lateral force. For IRWs railway vehicles, there is no restoration force. This could be a drawback. In other words, under tangent motion, a disturbance could invite continuous flange contact, and consequently increase noise and friction.

DEVELOPMENT OF A NEW ANALYTICAL MODEL FOR A RAILWAY VEHICLE EQUIPPED

To overcome this drawback, efforts to improve the restoration force by gravity restoration are being made. As part of such efforts, there have been several studies on the use of an abrasion wheel with large conicity or the use of additional guiding equipment or guide steering equipment. Trams on which IRWs are most widely used are typically 3- to 5-car modules, rather than single-car modules. Tests of trams under operation indicate typical critical speeds of approximately 100 km/h. This critical speed appears to be related to lateral motion being amplified by car body inertia as well as characteristics of connection devices under multi-car module operation. In the future, the improved model developed in this study will be applied to IRW 3-car modules or 5-car modules, and stability analyses will be conducted to make comparisons using the test results.

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