Summary

A finite element formulation and an efficient numerical method to solve for the combined dynamic response of a Shinkansen train ( bullet train ), rail, and the railway structure are discussed. Mechanical models for the train composed of many vehicles and for the interaction between wheel and rail are described. A practical finite element program, DIASTARS, has been developed for numerical experiments of the train running at a high speed on the rails of various railway structures. DIASTARS consists of three modules of the preprocessing, the solution, and the postprocessing to evaluate the ride comfort and running safety. The specific capabilities of DIASTARS are described. Numerical examples are demonstrated.

Keywords: FEM, Dynamic interaction between train and railway structure,

Simulation, Computer program

Recently much concern has been focused on developing and enhancing high-speed railway technology for excellent ride comfort, and for satisfying safety and environmental requirements at a high-speed operation. One of the important structural problem for this is to study on the combined dynamic behavior of a high-speed train, rail, and the railway structure. A finite element procedure to analyze the combined dynamic response of a Shinkansen ( bullet train ) vehicle and bridge has been reported by Tanabe et al.(1987), Wakui et al.(1993), and Tanabe et al. (1993).

In this paper a finite element formulation and an efficient numerical method to solve for the combined dynamic response of a Shinkansen train, rail, and the railway structure are discussed. The train which has at most 16 connected vehicles is modeled as an assemblage of the bodies, trucks and wheelsets connected to each other by springs and dampers. The effect of stoppers attached to the springs is included in the model to simulate the impact behavior between components of vehicles running at a high speed under an earthquake. Connectors between vehicles are also modeled by springs and dampers. A practical finite element program, DIASTARS, has been developed for numerical experiments of a Shinkansen train running at a high speed on the rails of various railway structures. The specific capabilities of DIASTARS are described, and numerical examples are demonstrated.

2. Equations of Motion of a Shinkansen Train and the Railway Structure

A Shinkansen train is modeled to have n vehicles connected by springs and dampers at the connectors, and each vehicle has components of a body, two trucks, and four wheelsets connected also by springs and dampers as shown in Fig.1. A spring with the stopper between components is modeled by giving a piecewise linear function of the load-displacement relationship. The body and truck each have five degrees of freedoms of transverse and vertical displacements, roll, pitch, and yaw, and each wheelset has four degrees of freedoms of transverse and vertical displacements, roll, and yaw at the center of gravity. Then a Shinkansen train composed of n vehicles has 31n DOF, and the equation of the motion is written in a matrix form as

(1)

where X^v and F^v are the displacement and load vectors of the train, M^v, D^v, and K^v are the mass, damping, and stiffness matrices, respectively. The symbol (.) denotes the derivative with respect to time t. The railway structure such as a bridge, rail and the foundation is modeled by various finite elements such as beam, shell, solid, mass, damper, and spring elements. Then the equation of motion of the structure is obtained

(2)

where X^b and F^b are the displacement and load vectors, and M^b, D^b, and K^b are the mass, damping and stiffness matrices of the structure, respectively.

3. Interactions between wheel and rail

1) Transverse and longitudinal directions

In the transverse and longitudinal directions between wheel and rail, constitutive equations to describe the relationship between creep forces and slipping rates of the wheel are given. The creep force in the transverse direction Q_c and yaw moment T_c due to the creep force in the longitudinal direction between wheel and rail are given as functions of the slipping rates of wheel in the transverse and longitudinal directions, S_y and S_x , respectively

Q_c = Q_c ( S_y ) (3)

T_c = T_c ( S_x ) (4)

2) Vertical and roll directions

In the vertical and roll directions between wheel and rail, a constraint equation is given to connect motions of the wheel and rail

U_w = U_w ( U_{b ,} I_r, t ) (5)

where U_w is a vector of wheel displacements in the vertical and roll directions, U_b is the nodal displacement vector of the railway structure required to interpolate U_w , and I_r is a vector of rail irregularities in the vertical and roll directions. Note that the constraint equation is time-dependent since wheel moves on the rail at a high speed.

3) Impact of wheel flange on the rail

If the relative displacement between wheel and rail in the transverse direction, d, exceeds a clearance u, the wheel flange impacts on the rail. The impact force Q_f is expressed as a function of d and the transverse displacement of the rail at the contact point y_r ( Wakui et al.(1993) )

Q_f = Q_f( d, y_r) (6)

The impact force plays an important role in the transverse motion of vehicles under an earthquake.

4. Numerical method

The combined dynamic response of the train and a railway structure is obtained by solving equations of motion of the train and the structure in eqs.(1) and (2) subject to constitutive equations (3) and (4), and the constraint equation (5). A modal transformation to displacement vectors of the train and the structure is made to solve the response effectively for practical problems with the train composed of many vehicles and complex railway structures. Thus, equations of motion of the train and the structure are solved in the modal coordinates for each time increment by the Newmark time difference scheme. Iterative calculations are necessary during each time increment until the unbalanced force becomes small enough within a tolerance specified. Adaptive time increment is employed to get the response effectively for large-scale practical problems.

5. DIASTARS

Based on the present method, a finite element program, DIASTARS ( Dynamic Interaction Analysis for Shinkansen Train And Railway Structure), has been developed for the simulation of a Shinkansen train running on the railway structure at a high speed. The railway structure is modeled by various finite elements such as truss, beam, shell, 3D solid, spring, damper, and mass elements. Rail irregularities in the transverse, vertical, and roll directions are given by sinusoidal or spline curves. The simulation of two trains passing each other on a double track of the railway structure is also possible by giving a series of moving load at a certain speed to model the passing of another train. Wind loads, depending on the time and the location where each vehicle passes, can be applied to the bodies of vehicles. Forces and moments equivalent to the wind pressure on the body of vehicle are given at the center of gravity of each body. The simulation of a Shinkansen train running on the railway structure at a high speed under an earthquake is also possible by giving the acceleration wave for an earthquake at the supports of the structure.

DIASTARS consists of three modules of pre-processing, the dynamic response analysis, and post-processing. In the pre-processing module, the geometry of a railway structure is created as an assemblage of curved surfaces in quadrilateral and triangular shapes and curves for the frame and shell structure, and of curved blocks in tetrahedral, pentahedral, and hexahedral shapes for the 3D solid structure. The finite element mesh is generated automatically. The geometry of the train is also created as an assemblage of curved surfaces to display the motion. In the analysis module, the dynamic response of the train and railway structure for various problems including wind and seismic loads is obtained efficiently by the modal method mentioned earlier. In the post-processing module, the evaluation of the running safety and ride comfort is made from the analysis results based on the Japanese standards ( Janeway (1948) and JNR(1963) ), and XY graphs of the various results such as displacements, velocities, accelerations, strains, and stresses at the locations of interest are obtained. The visualization of motions of the train and the railway structure in an animation is also available.

6. Numerical examples

The first example concerns a simulation of a Shinkansen train composed of five vehicles connected running at a speed of 350 km/h on a four-span concrete bridge with length=34m and width=9.8m as shown in Fig.2. The bridge is modeled with beam elements for the frame, and 4-node Mindlin shell elements for the concrete slab. Fig.3 and Fig.4 show vertical displacement histories at the center of the first and second span, respectively. Ten Spike-like responses occur corresponding to ten trucks of the train passing over the point of interest. The maximum displacements at the first and second span are shown to be 0.36mm and 0.34mm, respectively. The reason why the response at the second span is smaller than that of the first one is that the load on the first span causes an uplift effect to the second span, since the beam and slab are continuous from the first to fourth span. Fig.5 shows vertical displacement history of the first wheel in the first vehicle. There are four sharp peaks of the response corresponding to the wheel’s passing over the four spans.

The next example is a numerical experiment of a Shinkansen train composed of 12 vehicles running at a speed of 260km/h on the rail mounted on a cable stayed long-spanned concrete bridge of a new Shinkansen line which connects Tokyo and Nagano, that has been built for the 1998 Nagano Winter Olympics ( Fig.6 ). The bridge is modeled with 377 beam elements. It has totally 1254 DOF including the train. Fig.7 shows the vertical displacement of the second wheelset of the ninth vehicle depicted along the position of the bridge where it passes. The vertical displacement of the wheelset shows the maximum of about 60mm at around the position of two third of the second span. The maximum acceleration of about 1.7m/sec² was obtained at around the time t=6.6 sec. when it got out of the bridge ( a broken line in Fig.7 ) because the bent angle of the bridge becomes maximum at edges of the bridge. Since a modal transformation is used to get the combined response, iterative calculations during each time increment were made effectively. It took about 118 minutes of CPU on a CRAY YMP for the simulation of dynamic phenomena during 8 seconds.

7. Conclusions

A finite element formulation to solve the combined dynamic behavior of a Shinkansen train and the railway structure was discussed. Constitutive and constraint equations were given to describe the interaction and connection between wheel and rail. An efficient numerical method based on a modal transformation method and an adaptive time increment scheme was employed to make iterative calculations during time increments effective in practical problems with many vehicles and complex structures. A practical finite element program, DIASRARS, has been developed for the simulation of a high-speed train running on the railway structure, and the numerical examples were demonstrated. The ride comfort, safety, and the strength of the structure at a high-speed operation can be evaluated using the response results based on the standard and design code for railway structures ( Uetake(1975) and Wakui and Matumoto(1993) ) The numerical approach developed here can be used effectively to design and evaluate new types of high-speed trains and railway structures satisfying the safety, ride comfort and various environmental requirements.

References

Janeway,R.N. (1948), Vehicle vibration limit to fit the passenger, SAE Journal, pp.48-49.

JNR(1963), The standard of ride comfort, JNR Train Velocity Investigation Committee, Report No. 3A-2-1.

Tanabe, M. , Yamada, Y. and Wakui, H. (1987), Modal method for interaction of train and bridge, Computers & Structures, Vol.27, No.1, pp.119-127.

Tanabe, M. , Wakui, H. and Matumoto, N. (1993), The finite element analysis of Dynamic interactions of high-speed Shinkansen, rail, and bridge, ASME Computers in Engineering, Book No. G0813A, pp.17-22.

Uetake, Y.(1975), Ride comfort standard for railway vehicle, J. of Railway Engineering Research, Vol.32, No.5, pp.184-190.

Wakui, H., Matumoto, N. and Tanabe, M.(1993), A study on dynamic interaction analysis for railway vehicles and structures - Mechanical model and practical analysis method, RTRI Report Vol.7, No.4, pp.9-18.

Wakui, H. and Matumoto, N.(1993), Design standards for railway structures with explanatory note 3, Concrete Structure, RTRI Report Vol.34, No.4, pp.229-236.