Curving simulation and stability of a creep-controlled wheelset for high speed rail-vehicles

Curving Simulation and Stability of a Creep-Controlled Wheelset for High Speed Rail-Vehicles

Reinhold Meisinger

Mechanical Engineering Department

Georg-Simon-Ohm Fachhochschule Nürnberg

90489 Nürnberg, FRG

[e-mail]

Abstract

Higher travel speeds of rail vehicles will be possible by developing sophisticated top performance bogies having creep-controlled wheelsets. In this case the torque transmission between the right and the left wheel is realized by an actively controlled creep coupling. To investigate hunting stability and curving capability the linear equations of motion are written in state space notation. Simulation results are obtained with realistic system parameters from industry and various controller gains. The advantage of the „creep-controlled wheelset" is discussed by comparison the simulation results with the dynamic behaviour of the special cases „solid-axle wheelset" and „loose wheelset" (independent rotation of the wheels). The stability is also investigated with a root-locus analysis.

Keywords: Rail Vehicles, Dynamics, Vehicles Dynamics Simulation, Creep Control, High Speed Vehicles

Introduction

Under the critical speed, railway wheelsets have a sinusoidal hunting mode oscillation with decreasing amplitude after a small disturbance. Over the critical speed, the amplitudes of the wheelsets are increasing, that means, the sinusoidal oscillation gets unstable. In this case the wheel flange avoids wheel climbing and derailment. By optimization with the aid of computer simulation the critical speed was raising to 500 km/h in the past years (Duffek Kortüm, and Wallrapp, 1977). However there are tradeoffs between hunting stability dynamics and curve negotiation capability. To realize higher critical speeds recently from industry the „creep controlled wheelset" was presented, where the design compromise can be more easily found. There the righthand wheel and the lefthand wheel are connected by an actively controlled electromagnetic creep-coupling. (Guetti, 1990 and Leo, 1984). Though all investigations in this paper are made with the fundamental system of a dicone on cylindrical rails, this system contains most of the basic properties of profiled wheels and rails used in industry. The computer program SAMURAI (Kalker, 1967), is used to simulate the time history of the wheelset during transition of straight track to steady curve with constant vehicle velocity. A root locus diagram with the controller gain as parameter is ploted to indicate the stability of the system. This problem is used at Ohm-Fachhochschule Nürnberg to demonstrate the active control of unstable mechanical systems to the students of automotive engineering.

Mathematical Model

The „creep-controlled wheelset" shown in Fig. 1 is longitudinally, laterally suspended by springs on a frame. The frame is fixed to the inertial system which moves with constant nominal speed v₀ in a steady curve with radius R. The mass and the inertia about z-axis of the whole wheelset are m and J_z. The inertia about y-axis of the half wheelset is J_y. The longitudinal and lateral stiffnesses are c_x and c_y. The right and the left wheel are connected by a coupling which is influenced by the torque M computed by a controller. To simplify the system the distance of the bearings is equal to the distance of the rails (i.e. e = 0), the elevation angle of steady curve is ß<<1 and the rotation angle about x-axis is a <<1.

If the small rotation angle about x-axis and the small translation in z-direction are neglected, the system is described by the degrees of freedom u (translation in y-direction), j (rotation angle about z-axis) and y₁ , y₂ (rotation angles of the two wheels about y-axis).

Neglecting gyroscopic effects the equations of motion for small displacements from steady-state position of the centered wheelset can be written as, cf. Fig.1:

(1)

with Tx1/2 and Ty1/2 as the tangential creep-forces in the rail contact points and r₁ and r₂ as the rolling radii. The influence of a creep-moment is neglected in eq.(1). According to KALKERs linear theory (Kalker, 1967) the creep-forces can be computed with the relative velocities vxr1/2 and vyr1/2 and the constant K as

(2)

This linear relation between creep and force is only valied for small creep-rates, for larger creep this relation is nonlinear.

The relative velocities according to the linear theory result in

(3)

where r_o is the nominal rolling radius and l the conicity which can be written as:

.

If the degrees of freedom y₁ and y₂ are replaced by the relative rotation angle y = y₂- y₁ , the differential equations can be expressed by eq.(1) and eq.(2) as follows:

(4)

For the special case „solid axle wheelset" the system is reduced to two degrees of freedom u and l and the according equations of motion are shown in Gasch and Knothe, (1987).

State Space Notation

For simulation with the computer program SAMURAI (Meisinger, R., Fröschl, 1988) and for the control system design the equations of motion (4) will be written in state space notation.

(5)

There A is the 6*6-system matrix, D the 6*1-input matrix and B die 6*2-disturbance matrix. The state vector x, the input vektor u and the disturbance vektor b are with the according dimensions.

The matrix elements in eq.(5) can be written as:

a41 = - cy /m - gl /e a55 = - 2K e2 /(vo Jz ) d61 = 2/Jy

a42 = 2K /m a56 = K ro e /(vo Jz ) b41 = - g

a44 = - 2K /(mvo ) a61 = 2Kl /Jy b42 = vo2

a51 = - 2Kl e /(ro Jz ) a65 = 2Kero /(vo Jy ) b52 = 2Ke2 /Jz

a52 = - cx e2 /Jz a66 = - Kro2 /(vo Jy ) b62 = - 2Kero /Jy

The measurements are equal to the state vector x.

To investigate different linear control concepts, the state feedback controller

M = - [ k1 k2 k3 k4 k5 k6 ] × x (6)

u = - K x

is introduced, (Guetti, 1990 and Meisinger, 1992). There K is the 1*6-feedback matrix.

Computer Simulation

The control law (6) with the special feedback-matrix K = [ 0 0 0 0 0 k6 ], can be realized very easily with an electromagnetic creep-coupling (Leo, 1984).

To simplify the equations, the sensor- and coupling dynamics are neglected. Further control laws and the influence of the actuator dynamics are investigated in Guetti (1990 ).

For simulation with the computer-program SAMURAI (Meisinger, 1988) the nominal velocity v₀ = 75 m/s (270 km/h) and the following system parameters are assumed:

Mass of wheelset m = 1200 kg

Inertia of wheelset about z-axis Jz = 450 kgm2

Inertia of half wheelset about y-axis Jy = 38,5 kgm2

Longitudinal spring stiffness cx = 1,5× 106 N/m

Lateral spring stiffness cy = 1,5× 106 N/m

Half distance of rails e = 0,75 m

Nominal rolling radius of the wheels r₀ = 0,5 m

Conicity l = 0,2 rad

Constant (Kalker) K = 2× 107 N

Radius of steady curve R = 1000 m

Elevation angle of steady curve ß = 0

To enlarge the lateral displacement of the wheelset in the simulation a zero elevation angle of the steady curve is assumed (i.e. ß = 0).

The following controller gains

k6 = 0 Nms (special case „loose wheelset")

k6 = 75 Nms

k6 = 100 Nms

k6 = ¥ Nms (special case „solid axle wheelset")

are obtained with root-locus control system design analysies, cf. Fig. 6.

Figure 2 till Fig. 5 demonstrate the time history of lateral displacement u and the rolling angle j about z-axis. The transition of straight track to steady curve occurs suddenly (no transition arc). Because of the linearity of the equations limit cycles, as shown in Jaschinski (1990), can not be prognosted.

Figure 2 demonstrates the results of the special case „solid axle wheelset". The curving capability is good (eigenvalue with negative real part, cf. Fig. 6) but the hunting dynamic is unstable (complex eigenvalues with positive real parts, cf. Fig. 6). As shown in Gasch and Knothe (1987), Klingel (1883) and Meisinger (1992) the critical speed for this case is already v_crit = 53,2 m/s (191,5 km/h).

Figure 5 demonstrates the results of the special case „loose wheelset". The hunting dynamic is stable (complex eigenvalues with negative real parts, cf. Fig. 6), but the curving capability is unstable in the whole speed range because KALKERs linear theory (Kalker, 1967) (eigenvalue with positive real part, cf. Fig. 6).

Figure 3 and 4 demonstrate the results of the „creep controlled wheelset". Hunting dynamic and curving capability are stable (eigenvalues with negative real parts, cf. Fig. 6). With a controller gain k₆ = 75 Nms the wheelset lateral displacement is u¥ = 9,8 mm and the steady state rotation about z-axis is j ¥ = 2,0 × 10^-4 rad. The controller gain k₆ = 100 Nms gives according results of u¥ = 7,4 mm and j ¥ = 1,1 × 10^-4 rad.

Conclusions

The linear model of a „creep-controlled wheelset" was presented in state space notation. The aids used in the analysis were computer simulation and root-locus calculations for quasi-statical curve running. Hunting stability and curving capability have been investigated for different feedback of the creep between lefthand wheel and righthand wheel. The results show, that the „solid axle wheelset" has a good curving capability but unstable hunting dynamics for the given vehicle speed of v_o = 75 m/s (270 km/h). The „loose wheelset" has stable hunting dynamics but the curving capability is unstable within the whole speed range. The „creep-controlled wheelset" however has stable hunting dynamics and a good curving capability with small lateral displacement.

Presented at DINAME 97 - 7th International Conference on Dynamic Problems in Mechanics, 3 - 7 March 1997, Angra dos Reis, RJ, Brazil. Technical Editor: Agenor de Toledo Fleury.

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Publication Dates

• Publication in this collection
  20 Nov 2002
• Date of issue
  Sept 1999