1. Basic Concepts
1.7 Suspension, Articulation and Curving
It can be seen that a vehicle with a perfect suspension would be stable at all opera-tional speeds, would negotiate curves by minimising the forces acting between wheel and rail and in traversing irregular track would minimise the acceleration lev-els in the car body and the stresses applied to both vehicle and track structures.
These requirements are usually conflicting and compromise, informed by analysis, is required in design.
Not only are the parameters that are associated with wheel-rail contact, both geometrical and frictional, not under the control of the designer or operator, but they are not known exactly and can vary over a wide range. It follows that practical designs must be very robust in relation to such parameters. On the other hand, there is enormous scope for the design of the suspension system in terms of the way in which the wheelsets and car bodies in a train are connected.
It has long been the objective of vehicle design to incorporate wheel and steering arrangements that permit a vehicle to follow a chosen path or a track by a motion which involves pure rolling of the wheels, apart from the necessary transmission of traction forces and the reaction of centrifugal force. In the first place pure rolling might be achieved by a choice of configuration of the wheels and the way in which they are articulated. Such rolling motions may or may not be statically or dynami-cally stable. Additionally the introduction of creep, in response to inertia and sus-pension forces may stabilise or destabilise the system. It follows that there are many possible mechanisms of guidance and stability which can be considered. There have been many attempts to provide an alternative to the railway wheelset and reference will be made to some of these in Chapter 3. However, in using the conventional railway wheelset, there are many ways to improve performance in curves by making a vehicle more flexible in plan view, thus encouraging the axles to take up a more-or-less radial position in curves. It will be shown later that a two-axle vehicle that is capable of radial steering on a uniform curve will be dynamically unstable at all speeds so that the design of a two-axle vehicle requires a compromise between sta-bility and curving. This is the subject of Chapters 4 and 5.
As discussed in Chapters 6 and 7 for a vehicle with three or more axles it is pos-sible to arrange the suspension so that radial steering and dynamic stability are both achieved. One approach is to provide elastic or rigid linkages directly between wheelsets in a vehicle. This can be referred to as self-steering as the vehicle body is not involved. Alternatively, a linkage system can be provided which allows the wheelsets to take up a radial position but provides stabilising elastic restraint from the vehicle body. This is so-called forced steering as it can be considered that the vehicle body imposes a radial position on the wheelsets.
There are many designs in which there is articulation of the vehicle bodies of a vehicle or train. Articulation, in the present context, describes an arrangement in which the relative motion between the vehicle bodies is used to influence the stabil-ity and guidance of the vehicle. In many cases the interaction between the vehicles in a train is minimised by the form of coupling between the vehicles, so that longi-tudinal forces can be transmitted between car bodies, but the coupler is capable of transmitting little or no lateral force or yaw couple. In this case it is a good approxi-mation to treat each vehicle as if it were isolated and the lateral dynamics of each vehicle can be considered to be largely independent of that of the rest of the train. In an articulated vehicle the connections between vehicles form an essential part of the running gear. A common design feature is to link the relative angle between vehicle bodies to yaw of the wheelsets. As is discussed in Chapter 8, such designs improve curving performance and other aspects of vehicle design but can exhibit a wide spectrum of various hunting instabilities.
All the configurations discussed so far have been symmetric fore-and-aft. Un-symmetric configurations make it possible, in principle, to achieve a better com-promise between curving and dynamic stability, at least in one direction of motion.
But additional forms of instability can occur as discussed in Chapter 9.
The equations of motion are fundamental for all configurations of vehicle, and the derivation of these for a wheelset and a simple two-axle vehicle is discussed in Chapter 2.
References
1. Wickens, A.H.: The dynamics of railway vehicles-from Stephenson To Carter.
Proc. I. Mech. E. 212, Part F (1998), pp. 209-217.
Gilchrist, A.O.: The long road to solution of the railway hunting and curving problems. Proc. I. Mech. E. 212, Part F (1998), pp. 219-226.
2. Redtenbacher, F.J.: Die Gesetze des Locomotiv-Baues. Verlag von Friedrich Bassermann, Mannheim, 1855, p. 22.
3. Wickens, A.H.: Dynamics and the advanced passenger train. Speaking of Science 1977, Proceedings of The Royal Institution of Great Britain, 50 (1978), pp. 33-65.
4. Vaughan, A.: Isambard Kingdom Brunel − Engineering Knight Errant. John Murray, London, 1992, p. 102.
5. Dendy Marshall, C.F.A.: History of British Railways Down to the Year 1830. Ox-ford University Press, OxOx-ford, 1938, p. 165.
6. Klingel.: Uber den Lauf der Eisenbahnwagen auf Gerarder Bahn. Organ Fortsch.
Eisenb-wes. 38 (1883), pp. 113-123.
7. Timoshenko, S.P.: A History of the Strength of Materials, Mcgraw-Hill, New York, 1953, p. 348.
8. Carter, F.W.: The electric locomotive. Proc. Inst. Civ. Engs. 221, 1916, pp. 221-252.
9. Pooley, R.A.: Assessment of the critical speeds of various types of four-wheeled vehicles. British Railways Research Department Report E557, 1965.
10. King, B.L.: The measurement of the mode of hunting of a coach fitted with stan-dard double-bolster bogies. British Railways Research Department Report E439, 1963.
11. Gilchrist, A. O., Hobbs, A.E.W., King, B.L. and Washby, V.: The riding of two particular designs of four wheeled vehicle. Proc. I. Mech. E. 180 (1965), pp. 99-113.
12. Matsudaira, T.: Hunting problem of high-speed railway vehicles with special reference to bogie design for the New Tokaido Line. Proc. I. Mech. E. 180 (1965), pp. 58-66.
13. Knothe, K. and Bohm, F.: History of stability of railway and road vehicles. Ve-hicle System Dynamics, 31 (1999), pp. 283-323.
14. Carter, F.W.: The running of locomotives, with reference to their tendency to derail. Inst. Civil Engs, Selected Engineering Paper, No. 81, 1930.
15. Mackenzie, J.: Resistance on railway curves as an element of danger. Proc. Inst.
Civ. Engs. 74 (1883), pp. 1-57.
16. Johnson, K.L.: Effect of spin upon the rolling motion of an elastic sphere on a plane. Trans. A. S. M.E. Ser. E, 80 (1958), pp. 332-338.
17. Heumann, H.: Zur Frage des Radreifen-Umrisses. Organ Fortschr. Eisenb.-wes.
89 (1934), pp. 336-342.
18. Gilchrist, A.O. and Brickle, B.V.: A re-examination of the proneness to derail-ment of a railway wheelset. J. Mech. Eng. Sci. 18 (1976), pp. 131-141.
19. Birmann, F.: Theoretical and experimental solutions of track problems for high speeds. Monthly Bulletin of the International Railway Congress Association 45 (1968), pp. 391-460.
2
Equations of Motion
2.1 Introduction
The basic physical phenomena involved in the dynamics of railway vehicles have been described in Chapter 1. Equations of motion governing the stability and dy-namic response of vehicles will now be derived which encompass the essential fea-tures of the wheel-rail geometry, the frictional forces acting between wheel and rail and the elastic and damping forces generated by the suspension.
As attention will be confined to the dynamics at low frequencies, the wheelset and track are assumed to be rigid apart from local elasticity in the contact patch be-tween wheel and rail, and the contributions of the local deflections near the contact patch to the overall motion of the wheelset are neglected. The wheelset, which is assumed to be axisymmetric about the axle centreline, is considered to be con-strained to run along the track at constant speed. The track is arbitrarily curved in plan view and may be canted.
The kinematics of the wheelset is considered first, and this is followed by a dis-cussion of wheel rail geometry. An evaluation of the creep forces acting between wheel and rail makes it possible to formulate equations of motion of a freely run-ning wheelset. This is followed by the derivation of the equations of motion of a complete two-axle vehicle in which the action of the suspension is taken into ac-count.
2.2 Freedoms and Constraints
The track possesses curvature in a horizontal plane with radius R0, cant or cross-levelφ0, and can be displaced locally through a lateral displacement y0, Figure 2.1.
R0,φ0, and y0 vary with the distance s along the track. The wheelset reference frame Oxyz is attached to the centreline of the undistorted track, Figure 2.1, and moves along the track at the speed of the vehicle V. Thus the irregularities y0 are measured from this centreline. The origin of this set of axes is located at the centre of mass of the wheelset when the wheelset is central on the track. Ox lies along the tangent to the track centreline, Oy lies along the wheelset axle centreline when the wheelset is central and lying in the radial direction on the curve, and Oz is mutually perpendicu-lar. The coordinates X, Y and azimuth Ψ of the origin of the frame Oxyz, Figure 2.1,
with reference to an axis system fixed in the earth are given by
dX/ds = cosΨ (1) dY/ds = sinΨ (2)
dΨ/ds = 1/R0 (3) A second set of axes O*x*y*z* has origin at the centre of mass of the wheelset.
O*y* coincides with the axle centreline, O*x* is perpendicular to O*y* and O*z* is mutually perpendicular.
To specify the orientation of the frame O*x*y*z* it will be convenient to select successive rotations, yaw ψ, and roll φ, about the carried axes Oz*, Ox*. Thus the rotations are taken about the position the axes have taken following the previous rotation. The rotation of the wheelset about the carried axis Oy* is denoted by θ.
The displacements of the wheelset centre of mass O* relative to Oxyz are denoted
Ψ X
Y
y* y
O* O
x* x
ψ
z
z*
y y*
O
G φ
φ0
Figure 2.1 Wheelset axis systems and coordinates.
by the vector x0 with components ux, uy and uz, but as O* is in the plane yOz and both O and O* move forward at constant speed V
x0= [ 0 uy uz ] (4) and the longitudinal position along the track s = Vt. Thus the position and orient-ation of a wheelset can be defined in terms of the six variables, s, the lateral and vertical displacements uy and uzand three rotations, yaw ψ, roll φ, and θ.
As the area of contact between wheel and rail is small compared with the dimen-sions of the track contact between the wheelset and the rails can be considered to take place ordinarily at two points. As will be discussed later, two point contact gives rise to two constraint equations which makes it possible to eliminate two of the above coordinates. It will be convenient to eliminate the vertical displacement and roll angle of the wheelset as independent coordinates, so that they become sim-ply dependent functions of lateral displacement and yaw. As the vehicle speed is constrained to be constant, the system has three degrees of freedom.
2.3 Wheel Rail Geometry
As discussed in Chapter 1 the most important geometrical characteristics of the wheel rail geometry are (a) the variation of rolling radius with lateral displacement as this governs the conicity effect and (b) the variation of the slope at the contact point with lateral displacement as this governs the gravitational stiffness effect. For a typical wheel and rail combination, Figure 2.2, both profiles have curvature which varies continuously across the rail head and wheel tread and are defined by
ζw = f (ηw) ζr = g (ηr) (1)
ζ
rη
rFigure 2.2 Typical wheel and rail profiles relative to the rail coordinates ζr,ηr (mm).
whereηw, ζw are the wheel coordinates and ηr, ζr are the rail coordinates, the pro-files being the same for the right-hand and left-hand wheels, Figure 2.3.
In order to derive equations of motion, the position of the contact points, and the slopes and curvatures at these contact points, as functions of the wheelset lateral displacement and yaw are required. It will be assumed that the cross-sectional ge-ometry of the wheel-rail system does not vary with distance along the track. It fol-lows that the cross-sectional geometry is independent of s and θ. The wheels and rails will be assumed to be rigid in so far as their mutual geometry is concerned.
When the wheelset is in the central position on the track, and is not yawed, the angle made by the contact plane with the horizontal is δ0and the tread circles of the wheels have the same radius r0. When the wheelset is displaced laterally, the angles made between the contact planes and the axle centreline at the new points of contact are δwr
andδwl. Similarly, the radii of the tread circles become rrand rl.The position of the contact points is determined by noting that the wheel and rail contact points must occupy the same position in space, the angles made by the contact planes at the points of contact must be the same for wheel and rail, and the contacting bodies can-not penetrate each other. As the angle of yaw of a wheelset is small in most ordinary circumstances, the two-dimensional case where the influence of yaw is neglected will be considered. Consider the right hand wheels and rails, Figure 2.3. When the
ζr
δl ηl ηr
ζl
δr
central
displaced
uy-y0-r0φ ηwr
B δ0
ζrr
A C
ηrr
ζwr
-uz-lφ
Figure 2.3 Wheel rail geometry (right-hand). A is the origin of the rail axes ζr,ηr, B is the origin of the wheel axes ζw,ηw, so that A and B are coincident when the wheelset is cen-tral. When the wheelset is displaced, contact takes place at C.
wheelset is in the central position the contact point is A. The centreline of the track is displaced laterally by y0from the reference axis from which uy is measured. When the wheelset centre of mass is displaced laterally through a distance uy from the ref-erence axis the wheelset rotates about a longitudinal axis through a small angle φ and so the lateral displacement at the contact point is uy - y0- r0φ, and contact is made at a new point C. If the lateral movement of the contact point on the right hand wheel is ηwrand that on the rail is ηrrthen
uy - y0 - r0φ - ηwr+ηrr= 0 (2) Similarly, considering the vertical movement of the wheelset ζwrat the right hand rail
uz + lφ + ζwr-ζrr= 0 (3) Also, if if δrr denotes the angle between the rail axes and the contact plane φ - δwr + δrr = 0 (4) The three corresponding equations for the left hand wheel are
uy - y0- r0φ + ηwl-ηrl= 0 (5) uz - lφ + ζwl-ζrl= 0 (6) φ + δwl - δrl = 0 (7) In addition, the slopes at the contact points are given by
tan δwr = dζwr/dηwr (8) tan δwl = dζwl/dηwl (9) tan δrr = dζrr/dηrr (10) tan δrl = dζrl/dηrl (11) For profiles specified by (1), equations (1) to (11) can be solved to yield the contact positions and slopes, the vertical displacement and roll angle as functions of uy. Measuring equipment has been developed to measure wheel and rail profiles with the necessary accuracy, [1, 2, 3]. The solution has been implemented in computer programs and typically uses a Newton-Raphson iterative procedure to solve the nonlinear algebraic equations. The first step is to determine the points of contact when the wheelset is central. This then defines a new origin for the wheel rail geo-metric data. Then equations (1) to (11) are solved for the contact points. Once the contact points have been established the various geometrical characteristics can be
determined. Detailed discussions of the analytical aspects of wheel rail geometry have been given by de Pater [4] and Yang [5].
As an example, for the wheelset and rail combination shown in Figure 2.2 the variation of φ, uz and the location of the contact points is shown in Figure 2.4. Fig-ure 2.5 shows the variation of rolling radius, contact angle and the transverse curva-tures of the rail and wheel with lateral displacement. The examples shown refer to worn wheel and rail profiles and in this case yield quite smooth characteristics.
However, in practice, many rail and wheel profile combinations yield characteristics with major discontinuities.
Though, as the wheelset is displaced laterally, the rotation φ and vertical dis-placement uzare small it will be seen later that their derivatives with respect to uy
play an important part in the equations of motion. Expressions for these derivatives will now be derived.
Figure 2.4 Roll angleφ, vertical displacement uzand location of contact points as a function of lateral displacement for the wheel rail combination of Figure 2.2.
Figure 2.5 Variation of rolling radius, contact slope, and radii of curvature with lateral dis-placement for the wheel rail combination of Figure 2.2.
I
δuy δz
δφ δuy
l - rltanδwl l - rrtanδwr
rl rr
rltanδwl rrtanδwr
Figure 2.6 Tilting and vertical displacement of wheelset due to lateral displacement. Wheel-set rotates about instantaneous centre I.
r (mm) r δ r
l l
Rr
(mm)
Rw
(mm)
r r
l l
du
which also follow from the geometry of Figure 2.6. For small displacements from the central position
φy= - σ/l ( 1 - r0δ0/ l ) zy= -εuy/l ( 1 - r0δ0/ l ) (16) where
σ = δ0 ε = (δrr - δrl)l/2uy (17) ε is the parameter that determines the change of inclination of the normal force between wheel and rail as the wheelset is displaced laterally and therefore influences the gravitational stiffness. σ is the roll parameter.
Figure 2.7 shows the variation of the geometrical derivatives φy and zy. Numerical values of the parameters for the wheel rail combination of Figure 2.2 are
Differentiating (3) with respect to uy
r0 = 0.4500 m, δ0 = 0.0493 and l = 0.7452 m.
If uy
* is the lateral displacement of the wheelset in the plane of the original points of contact, then from Figure 2.6
u*y = 2luy/(2l - r0tanδwr- r0tanδwl) (18) because the wheelset is rotating about the point I. For small displacements this be-comes
u*y= u y/( 1 - r0δ0/ l ) (19) From (19) the difference between the lateral displacement of the wheelset at the axle and at the contact points for the profiles of Figure 2.2 is only 3%.
For small displacements equations (1)-(11) have a simple solution. In the vicinity of the point of contact Rw and Rr are the radii of curvature of the wheel tread and rail head respectively. When the wheelset is displaced laterally through a distance uy it can be seen from Figure 2.3 that the lateral displacement of the contact point on the rail is, approximately,
ηrr = Rr(δrr-δ0 ) (20) to the first order in δ0 and δrrThe lateral displacement of the contact point on the wheel is
ηwr = Rw(δwr-δ0 ) (21) Substituting in (2) from (20), (21) and (4) and noting from (16) that φ = φyuy, yields for the right hand side
-10 0 10
-0.4 -0.2 0 0.2 0.4
uy
(mm) z
y-10 0 10
-0.6 -0.4 -0.2
φ
y(1/m)
0uy
(mm)
Figure 2.7 Variation of derivatives zy and φy with lateral displacement for the wheel rail combination of Figure 2.2.
δrr,δrl=δ0± ε0uy/l (22) where, since to first order (1+ r0δ0/ l)-1= (1 - r0δ0/ l)
ε0 = l ( 1 + Rwδ0/l)/ (Rw - Rr)(1 - r0δ0/ l) (23) and
δwr,δwl=δ0± ε0
*
u
y/l (24) whereε0
* = l ( 1 + Rrδ0/l)/ (Rw - Rr)(1 - r0δ0/ l) (25) For example, for the wheel rail combination of Figure 2.2, ε0 = 6.423 and ε0∗ = 6.372. Note that ε0 - ε0∗ = σ. For conical wheels, Rw→ ∞ , ε0∗ = 0 and
ε0 = δ0/( 1 - r0δ0/ l ) (26) For profiled or worn wheels Rwδ0/ l << 1 and Rrδ0/ l << 1. For example, for the wheel rail combination of Figure 2.2, Rwδ0/ l = 0.01795 and Rrδ0/ l = 0.00989.
Hence, in this case
ε = 0 ε0
* = l / ( Rw - Rr)( 1 - r0δ0/ l ) (27)
Reference to Figure 2.3 shows that
ζwr = - Rw (cosδwr- cosδ0 ) = - Rw(δ0
2-δwr
2 )/2 (28) approximately, so that substituting from equations (22) and (24), to first order rr, rl = r0 ± λ0uy (29) whereλ0is the effective conicity
λ0 = δ0Rw( 1 + Rrδ0/l)/ ( Rw - Rr )( 1 - r0δ0/ l ) (30) For example, for the wheel rail combination of Figure 2.2,λ0 = 0.1144. For conical wheels
λ0 = δ0( 1 + Rrδ0/l) /( 1 - r0δ0/ l ) (31)
and for profiled or worn wheels where Rwδ0/ l << 1 and Rrδ0/ l <<1,
λ0 = δ0Rw/ (Rw - Rr)(1 - r0δ0/ l) (32) as first derived by Heumann [6].
For purely coned wheels the vertical displacement of the wheelset is negligible.
Considering therefore the case of profiled or worn wheels so that ε 0 = ε 0*
ζrr = - Rr(cosδrr- cosδ0) =- Rr(δ02
-δrr2
)/2 (33) approximately, so that substituting (28) and (33) into (22) and (24) and then adding (3) and (6) the vertical displacement of the centre of mass is found to be
uz = - ( Rw - Rr) ε0 2uy2
/2l2 (34) consistent with (16). For completeness, to this could be added the small vertical
/2l2 (34) consistent with (16). For completeness, to this could be added the small vertical