345:
and the pointed end is like a small diameter wheel. While the nail or screw will turn around in a full circle (and more) the railroad wheelset behaves differently because as soon at it starts to turn in a curve, the effective diameters change in such a way as to decrease the curvature of the path. Note that "radius" and "curvature" refer to the curvature of the trajectory of the wheelset and not the curvature of the railway since this is perfectly straight track. As the wheelset rolls on, the curvature decreases until the wheels reach the point where their effective diameters are equal and the path is no longer curving. But the trajectory has a slope at this point (it is a straight line which crosses diagonally over the centerline of the track) so that it overshoots the centerline of the track and the effective diameters reverse (the formerly smaller diameter wheel becomes the larger diameter and conversely). This results in the wheelset moving in a curve in the opposite direction. Again it overshoots the centerline and this phenomenon continues indefinitely with the wheelset oscillating from side to side. Note that the wheel
258:
2129:
3615:
313:), is given a push forward on a straight and level track. The wheelset starts coasting and never slows down since there are no forces (except downward forces on the wheelset to make it adhere to the track and not slip). If initially the wheelset is centered on the railroad track then the effective diameters of each wheel are the same and the wheelset rolls down the track in a perfectly straight line forever. But if the wheelset is a little off-center so that the effective diameters (or radii) are different, then the wheelset starts to move in a curve of radius
25:
1129:
213:, have no differential, yet they are mostly not affected by hunting oscillation, as most of their wheels rotate independently from one another. The wheels of the power car, however, can be affected by hunting oscillation, because the wheels of the power car are fixed to the axles in pairs like in conventional bogies. Less conical wheels and bogies equipped with independent wheels that turn independently from each other and are not fixed to an axle in pairs are cheaper than a suitable differential for the bogies of a train.
854:
390:
3591:
certain that the lowest is dictated by the wheelset motion. However, the analysis is instructive because it shows why hunting occurs. As the speed increases, the inertial forces become comparable with the adhesion forces. That is why the critical speed depends on the ratio of the axle load (which determines the adhesion force) to the wheelset mass (which determines the inertial forces).
1124:{\displaystyle {\begin{aligned}{\frac {\operatorname {d} \theta }{\operatorname {d} x}}&={\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}=-\left({\frac {2k}{rd}}\right)y\\{\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} x^{2}}}&=-\left({\frac {2k}{rd}}\right){\frac {\operatorname {d} y}{\operatorname {d} x}}=-\left({\frac {2k}{rd}}\right)\theta \end{aligned}}}
3599:
constraints to the bogie also raises the critical speed. Introducing elastic forces into the equation permits suspension designs which are limited only by the onset of gross slippage, rather than classical hunting. The penalty to be paid for the virtual elimination of hunting is a straight track, with an attendant right-of-way problem and incompatibility with legacy infrastructure.
229:, less-conical wheels and other design changes were used to extend truck design speeds above 225 km/h (140 mph). Advances in wheel and truck design based on research and development efforts in Europe and Japan have extended the speeds of steel wheel systems well beyond those attained by the original
3590:
Two main matters, namely assuming the body as a fixed support and influence of the nonlinear elements in calculation of the hunting speed, are mostly focused in studies. A real railway vehicle has many more degrees of freedom and, consequently, may have more than one critical speed; it is by no means
1647:
The actual adhesion forces arise from the distortion of the tread and rail in the region of contact. There is no gross slippage, just elastic distortion and some local slipping (creep slippage). During normal operation these forces are well within the limiting friction constraint. A complete analysis
344:
To illustrate that a wheelset trajectory follows a curved path, one may place a nail or screw on a flat table top and give it a push. It will roll in a circular curve because the nail or screw is like a wheelset with extremely different diameter wheels. The head is analogous to a large diameter wheel
3586:
torques, which will modify the estimate of the critical speed. Conventionally a railway vehicle has stable motion in low speeds, when it reaches to high speeds stability changes to unstable form. The main purpose of nonlinear analysis of rail vehicle system dynamics is to show the view of analytical
1643:
this expression yields a significant overestimate of the critical speed, but it does illustrate the physical reason why hunting occurs, i.e. the inertial forces become comparable with the adhesion forces above a certain speed. Limiting friction is a poor representation of the adhesion force in this
1133:
The angular deflection also follows a simple harmonic motion, which lags behind the side to side motion by a quarter of a cycle. In many systems which are characterised by harmonic motion involving two different states (in this case the axle yaw deflection and the lateral displacement), the quarter
3594:
Alternatively, below a certain speed, the energy which is extracted from the forward motion is insufficient to replace the energy lost by lowering the axles and the motion damps out; above this speed, the energy extracted is greater than the loss in potential energy and the amplitude builds up.
3464:
to wheel set mass. If the treads were truly conical in shape, the critical speed would be independent of the taper. In practice, wear on the wheel causes the taper to vary across the tread width, so that the value of taper used to determine the potential energy is different from that used to
3598:
The potential energy at maximum axle yaw may be increased by including an elastic constraint on the yaw motion of the axle, so that there is a contribution arising from spring tension. Arranging wheels in bogies to increase the constraint on the yaw motion of wheelsets and applying elastic
1908:
1791:
2954:
1328:
1655:
However, the kinematic analysis assumed that there was no slippage at all at the wheel-rail contact. Now it is clear that there is some creep slippage which makes the calculated sinusoidal trajectory of the wheelset (per
Klingel's formula) not exactly correct.
349:
never makes contact with the rail. In this model, the rails are assumed to always contact the wheel tread along the same line on the rail head which assumes that the rails are knife-edge and only make contact with the wheel tread along a line (of zero width).
1220:
381:
tread speeds up, while the smaller slows down. The wheel set steers around a centre of curvature defined by the intersection of the generator of a cone passing through the points of contact with the wheels on the rails and the axis of the wheel set. Applying
292:
of bodies, the forces acting may then be derived from the kinematics by calculating the accelerations of the components. However, if these forces change the kinematic description (as they do in this case) then the results may only be approximately correct.
3626:
feature independent axles and suspension systems on each rail wheel. When this is combined with the presence of road wheels on the rail it becomes difficult to use the formulae above. Historically, road–rail vehicles have their front wheels set slightly
3266:
580:
682:
2737:
2603:
2124:
2013:
1145:
In practice, below the critical speed, the lag between the two motions is less than a quarter cycle so that the motion is damped out but, above the critical speed, the lag is greater than a quarter cycle so that the motion is amplified.
216:
The problem was first noticed towards the end of the 19th century, when train speeds became high enough to encounter it. Serious efforts to counteract it got underway in the 1930s, giving rise to lengthened trucks and the side-damping
3143:
3376:
147:. The expression came into use in the 19th century and describes how a system "hunts" for equilibrium. The expression is used to describe phenomena in such diverse fields as electronics, aviation, biology, and railway engineering.
3602:
Hunting is a dynamic problem which can be solved, in principle at least, by active feedback control, which may be adapted to the quality of track. However, the introduction of active control raises reliability and safety issues.
1401:
2222:
1728:
3581:
The motion of a wheel set is much more complicated than this analysis would indicate. There are additional restraining forces applied by the vehicle suspension and, at high speed, the wheel set will generate additional
3824:
3039:
845:
2828:
2298:
113:
1494:
122:
1820:
1737:
1639:
3557:
741:
2853:
2848:
859:
3456:
1229:
2469:
3885:
447:
2407:
2351:
1166:
3152:
1538:
794:
507:
3587:
investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles in a tangent track. This study describes the
Bogoliubov method for the analysis.
593:
341:
is the "tracking error", the deviation of the center of the wheelset from the straight line of the railway running down the center of the track (midway between the two rails).
209:
rotating at the same angular rate, although differentials tend to be rare, and conventional trains have their wheels fixed to the axles in pairs instead. Some trains, like the
1664:
In order to get an estimate of the critical speed, we use the fact that the condition for which this kinematic solution is valid corresponds to the case where there is no net
325:
of the center of the wheelset projected vertically on the roadbed in the center of the track. This is a trajectory on the plane of the level earth's surface and plotted on an
2612:
2478:
2022:
1917:
753:
out below a critical speed and the ride is correspondingly more comfortable. The kinematic result ignores the forces causing the motion. These may be analyzed using the
2136:
When the axle yaws, the points of contact move outwards on the treads so that the height of the axle is lowered. The distance between the support points increases to:
1813:
1564:
3709:
f. The action of a machine, instrument, system, etc., that is hunting (see hunt v. 7b); an undesirable oscillation about an equilibrium speed, position, or state.
2411:
This is energy lost from the system, so in order for the motion to continue, an equal amount of energy must be extracted from the forward motion of the wheelset.
3048:
265:
While a qualitative description provides some understanding of the phenomenon, deeper understanding inevitably requires a mathematical analysis of the vehicle
4004:
3275:
1337:
2141:
1684:
193:
forces. At low speed, adhesion dominates but, as the speed increases, the adhesion forces and inertial forces become comparable in magnitude and the
54:
4034:
Serajian, Reza (2013). "Parameters' changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie".
3772:
2963:
801:
2761:
2231:
765:; an early presentation that includes these effects in hunting motion analysis was presented by Carter. See Knothe for a historical overview.
4049:
Serajian, Reza (2011). "Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory".
1142:" of road vehicles, as well as hunting of railway vehicles. The kinematic solution derived above describes the motion at the critical speed.
1134:
cycle lag between the two motions endows the system with the ability to extract energy from the forward motion. This effect is observed in "
366:(the tracking error), the radius of the tread in contact with the rail on one side is reduced, while on the other side it is increased. The
3731:
Tracking error will be zero if the path of the wheels runs absolutely straight along the track and the wheel pair is centered on the track.
3606:
Shortly after the onset of hunting, gross slippage occurs and the wheel flanges impact on the rails, potentially causing damage to both.
1422:
2836:
extracted from the forward motion, and manifesting itself as increased kinetic energy of the wheel set at zero yaw, is equal to the
486:
is the progress along the track. This is sometimes called the tracking error. Provided the direction of motion remains more or less
4019:
Wickens, A. H.; Gilchrist, A. O.; Hobbs, A. E. W. (1969–70). "Suspension Design for High-Performance Two-Axle
Freight Vehicles".
4002:
Wickens, A. H. (1965–66). "The
Dynamics of Railway Vehicles on Straight Track: Fundamental Considerations of Lateral Stability".
3752:. Since the tracking error will turn out to be a sine wave, the points of zero slope are at the points of maximum tracking error
174:
70:
1903:{\displaystyle \omega {\frac {\operatorname {d} \omega }{\operatorname {d} \theta }}=-U^{2}\left({\frac {2k}{rd}}\right)\theta }
1786:{\displaystyle \omega {\frac {\operatorname {d} }{\operatorname {d} \theta }}={\frac {\operatorname {d} }{\operatorname {d} t}}}
1575:
3474:
2949:{\displaystyle {\begin{aligned}2U{\frac {ky}{rd}}&=\omega \\\delta E&={\frac {1}{8}}md^{2}\omega ^{2}\end{aligned}}}
695:
301:
This kinematic description makes a number of simplifying assumptions since it neglects forces. For one, it assumes that the
1323:{\displaystyle {\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} t^{2}}}=-U^{2}\left({\frac {2k}{rd}}\right)\theta }
197:
begins at a critical speed. Above this speed, the motion can be violent, damaging track and wheels and potentially causing
317:(depending on these wheelset radii, etc.; to be derived later on). The problem is to use kinematic reasoning to find the
3385:
284:
causing it, so the analysis begins with a description of the geometry of a wheel set running on a straight track. Since
2419:
4093:
Parameters' changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie
3837:
399:
94:
2360:
2226:(to second order of small quantities). the displacement of the support point out from the centres of the treads is:
1215:{\displaystyle {\frac {\operatorname {d} }{\operatorname {d} t}}=U{\frac {\operatorname {d} }{\operatorname {d} x}}}
2307:
202:
46:
4102:
Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory
3899:
is eliminated by combining the two equations, the resulting equation becomes correct by checking the two cases:
3261:{\displaystyle T={\frac {1}{2}}U^{2}\left(C+{\frac {md^{2}}{4}}\right)\left({\frac {2k}{rd}}\right)\theta ^{2}}
749:
This kinematic analysis implies that trains sway from side to side all the time. In fact, this oscillation is
575:{\displaystyle \left|{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}\right|\approx {\frac {1}{R}}}
3640:
1649:
762:
1507:
771:
677:{\displaystyle {\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}=-\left({\frac {2k}{rd}}\right)y}
2732:{\displaystyle {\frac {1}{4}}m\left(U^{2}-2U^{2}{\frac {ky}{r}}+U^{2}{\frac {k^{2}y^{2}}{r^{2}}}\right)}
2598:{\displaystyle {\frac {1}{4}}m\left(U^{2}+2U^{2}{\frac {ky}{r}}+U^{2}{\frac {k^{2}y^{2}}{r^{2}}}\right)}
2119:{\displaystyle {\frac {1}{2}}C\omega ^{2}=-{\frac {1}{2}}CU^{2}\left({\frac {2k}{rd}}\right)\theta ^{2}}
4124:
3614:
2008:{\displaystyle {\frac {1}{2}}\omega ^{2}=-{\frac {1}{2}}U^{2}\left({\frac {2k}{rd}}\right)\theta ^{2}}
59:
3665:
754:
206:
156:
3744:
for mathematical details. The approximate equality becomes equality only when the tracking error,
3569:. This result is derived in Wickens (1965) from an analysis of the system dynamics using standard
768:
If the motion is substantially parallel with the rails, the angular displacement of the wheel set
1567:
758:
50:
687:
285:
234:
144:
3720:
3623:
1798:
178:
1795:
the angular acceleration equation may be expressed in terms of the angular velocity in yaw,
4119:
3967:
Knothe, K. (2008). "History of wheel/rail contact mechanics: from
Redtenbacher to Kalker".
3939:
3138:{\displaystyle \delta E=-{\frac {1}{8}}U^{2}md^{2}\left({\frac {2k}{rd}}\right)\theta ^{2}}
266:
1549:
8:
3741:
3675:
3570:
487:
35:
3943:
3371:{\displaystyle {\frac {Wkd}{2}}=U^{2}{\frac {2k}{rd}}\left(C+{\frac {md^{2}}{4}}\right)}
3984:
3701:
3631:, which has been found to minimise hunting whilst the vehicle is being driven on-rail.
1135:
302:
112:
39:
471:(which is the slope of tread in the horizontal direction perpendicular to the track).
3988:
1414:
383:
74:
1570:. Gross slipping will occur at a combination of speed and axle deflection given by:
1396:{\displaystyle Fd=C{\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} t^{2}}}}
121:
3976:
3947:
3660:
2837:
2217:{\displaystyle {\frac {d}{cos(\theta )}}=d\left(1+{\frac {1}{2}}\theta ^{2}\right)}
1673:
757:(non-linear) but are somewhat difficult to quantify simply, as they arise from the
367:
182:
140:
63:
474:
The path of the wheel set relative to the straight track is defined by a function
3461:
1723:{\displaystyle \omega ={\frac {\operatorname {d} \theta }{\operatorname {d} t}}}
2753:
1669:
468:
218:
3980:
3895:
is not allowed to be negative (per mathematical definition). But after radius
4113:
3645:
3460:
This is independent of the wheel taper, but depends on the ratio of the axle
750:
4084:
Fundamentals of rail vehicle dynamics : guidance and lateral stability
3952:
3927:
3655:
3650:
1139:
289:
3819:{\displaystyle {\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}
3034:{\displaystyle \omega ^{2}=-U^{2}\left({\frac {2k}{rd}}\right)\theta ^{2}}
840:{\displaystyle \theta ={\frac {\operatorname {d} y}{\operatorname {d} x}}}
761:
of the wheel and rail at the regions of contact. These are the subject of
2823:{\displaystyle \delta E={\frac {1}{2}}m\left({\frac {Uky}{r}}\right)^{2}}
2293:{\displaystyle {\frac {1}{2}}\left(d+{\frac {1}{2}}d\theta ^{2}-d\right)}
358:
The train stays on the track by virtue of the conical shape of the wheel
245:
systems. The speed record for steel-wheeled trains is held by the French
194:
3583:
1154:
585:
495:
456:
322:
318:
238:
225:
198:
491:
371:
273:
210:
1153:
forces, it is necessary to express the distance derivatives as time
1499:
378:
277:
190:
186:
3756:. But the equality is approximately correct provided the slope of
2128:
1489:{\displaystyle F=-CU^{2}\left({\frac {2k}{rd^{2}}}\right)\theta }
1150:
162:
389:
257:
3628:
2833:
1676:
of the system, we should be able to derive the critical speed.
1665:
346:
296:
242:
2832:
The motion will continue at constant amplitude as long as the
16:
Self-oscillation about an equilibrium that is usually unwanted
3618:
Independent rail wheel axles are common on road–rail vehicles
310:
306:
281:
237:
keeps such technology dominant over alternatives such as the
3671:
For general methods dealing with this class of problem, see
3721:
https://www.talgo.com/en/rolling-stock/very-high-speed/350/
3566:
2746:
374:
359:
502:
with respect to distance along the track as approximately
4075:
Railroad vehicle dynamics : a computational approach
246:
161:
A classical hunting oscillation is a swaying motion of a
57:
and tools are available to assist in formatting, such as
4018:
2132:
Outward displacement of points of contact with axle yaw
1634:{\displaystyle \theta U^{2}=\mu W{\frac {rd^{2}}{4Ck}}}
4021:
4005:
3552:{\displaystyle U^{2}={\frac {Ward^{2}}{k(4C+md^{2})}}}
2840:
lost by the lowering of the axle load at maximum yaw.
2355:
The work done by lowering the axle load is therefore:
1668:
exchange with the surroundings, so by considering the
1332:
The inertial moment (ignoring gyroscopic effects) is:
736:{\displaystyle \lambda =2\pi {\sqrt {\frac {rd}{2k}}}}
362:. If a wheelset is displaced to one side by an amount
3840:
3775:
3477:
3465:
calculate the kinetic energy. Denoting the former as
3388:
3278:
3270:
The critical speed is found from the energy balance:
3155:
3051:
2966:
2851:
2764:
2615:
2481:
2422:
2363:
2310:
2234:
2144:
2025:
1920:
1823:
1801:
1740:
1687:
1578:
1552:
1510:
1425:
1340:
1232:
1169:
857:
804:
774:
698:
596:
510:
402:
3576:
370:
is the same for both wheels (they are coupled via a
3830:is positive and conversely. The other equation for
3879:
3818:
3700:
3551:
3451:{\displaystyle U^{2}={\frac {Wrd^{2}}{4C+md^{2}}}}
3450:
3370:
3260:
3137:
3033:
2948:
2822:
2731:
2597:
2463:
2401:
2345:
2292:
2216:
2118:
2007:
1902:
1807:
1785:
1722:
1633:
1558:
1532:
1488:
1395:
1322:
1214:
1123:
839:
788:
735:
676:
574:
441:
269:. Even then, the results may be only approximate.
2464:{\displaystyle V={\frac {U}{r}}\left(r+ky\right)}
4111:
3880:{\displaystyle {\frac {1}{R}}={\frac {2ky}{rd}}}
1224:The angular acceleration of the axle in yaw is:
442:{\displaystyle {\frac {1}{R}}={\frac {2ky}{rd}}}
3707:(2nd ed.). Oxford University Press. 1989.
205:because the action depends on both wheels of a
201:. The problem does not occur on systems with a
3565:is now a shape factor determined by the wheel
2402:{\displaystyle E={\frac {1}{4}}Wkd\theta ^{2}}
1502:force between the wheel and rail is given by:
1157:. This is done using the speed of the vehicle
2346:{\displaystyle h={\frac {1}{4}}kd\theta ^{2}}
588:along the track is governed by the equation:
3928:"On the Stability of Running of Locomotives"
746:known as Klingel's formula (derived in 1883)
297:Assumptions and non-mathematical description
38:, which are uninformative and vulnerable to
463:the wheel radius when running straight and
185:depends. It arises from the interaction of
53:and maintains a consistent citation style.
4001:
3995:
2017:so the kinetic energy due to rotation is:
223:truck. In the development of the Japanese
3960:
3951:
3919:
494:of the path may be related to the second
261:Kinematics of railway wheel coning action
95:Learn how and when to remove this message
4099:
4090:
4048:
4033:
3613:
2127:
1409:is the force acting along the rails and
388:
353:
321:of the wheelset, or more precisely, the
256:
130:Hunting oscillation on railway wheelsets
4081:
4073:Shabana, Ahmed A.; et al. (2008).
4072:
4063:
1648:takes these forces into account, using
337:is the distance along the railroad and
305:is zero. A wheelset (not attached to a
4112:
4036:Journal of Measurements in Engineering
3966:
3925:
2414:The outer wheel velocity is given by:
3695:
3693:
3691:
3609:
3380:Hence the critical speed is given by
252:
249:, at 574.9 km/h (357 mph).
4066:Handbook of railway vehicle dynamics
3043:The translational kinetic energy is
1533:{\displaystyle F=\mu {\frac {W}{2}}}
789:{\displaystyle \left(\theta \right)}
280:of motion, without reference to the
150:
18:
45:Please consider converting them to
13:
3797:
3780:
3688:
1841:
1830:
1771:
1767:
1750:
1746:
1708:
1697:
1374:
1357:
1254:
1237:
1200:
1196:
1176:
1172:
1068:
1057:
997:
980:
916:
899:
876:
865:
825:
814:
618:
601:
536:
519:
14:
4136:
3748:, has zero slope with respect to
3577:Limitation of simplified analysis
1659:
3932:Proceedings of the Royal Society
3891:goes negative, since the radius
177:action on which the directional
120:
111:
23:
4042:
4027:
4012:
3926:Carter, F. W. (July 25, 1928).
386:, we have for the turn radius:
3910:
3763:
3734:
3725:
3714:
3543:
3518:
3469:, the critical speed becomes:
2166:
2160:
49:to ensure the article remains
1:
3742:Curvature#Graph of a function
3681:
3147:The total kinetic energy is:
1161:, which is assumed constant:
393:Calculation of radius of turn
143:, usually unwanted, about an
4082:Wickens, A H (Jan 1, 2003).
3641:Frictional contact mechanics
763:frictional contact mechanics
7:
3634:
276:description deals with the
10:
4141:
2843:Now, from the kinematics:
2607:for the inner wheel it is
154:
4086:. Swets & Zeitlinger.
3981:10.1080/00423110701586469
3703:Oxford English Dictionary
1650:rolling contact mechanics
1149:In order to estimate the
1138:" of aircraft wings and "
233:, while the advantage of
157:Wheelset (rail transport)
3916:Iwnicki, p.7 formula 2.1
4100:Serajian, Reza (2011).
4091:Serajian, Reza (2013).
4064:Iwnicki, Simon (2006).
3969:Vehicle System Dynamics
2473:The kinetic energy is:
2302:the axle load falls by
1808:{\displaystyle \omega }
1568:coefficient of friction
235:backwards compatibility
3953:10.1098/rspa.1928.0220
3881:
3820:
3619:
3553:
3452:
3372:
3262:
3139:
3035:
2950:
2824:
2733:
2599:
2465:
2403:
2347:
2294:
2218:
2133:
2120:
2009:
1904:
1809:
1787:
1724:
1635:
1560:
1534:
1490:
1397:
1324:
1216:
1125:
841:
790:
737:
688:simple harmonic motion
678:
576:
443:
394:
288:relates forces to the
262:
165:vehicle (often called
3882:
3821:
3617:
3554:
3453:
3373:
3263:
3140:
3036:
2951:
2825:
2734:
2600:
2466:
2404:
2348:
2295:
2219:
2131:
2121:
2010:
1905:
1810:
1788:
1725:
1636:
1561:
1546:is the axle load and
1535:
1491:
1398:
1325:
1217:
1126:
842:
791:
738:
679:
577:
444:
392:
354:Mathematical analysis
333:graphical plot where
260:
3838:
3773:
3475:
3386:
3276:
3153:
3049:
2964:
2849:
2762:
2613:
2479:
2420:
2361:
2308:
2232:
2142:
2023:
1918:
1821:
1799:
1738:
1732:Using the operator:
1685:
1576:
1559:{\displaystyle \mu }
1550:
1508:
1423:
1338:
1230:
1167:
855:
802:
772:
696:
594:
584:It follows that the
508:
400:
3944:1928RSPSA.121..585C
3887:does not hold when
3676:Control engineering
3571:control engineering
690:having wavelength:
286:Newton's second law
137:Hunting oscillation
3877:
3816:
3624:road–rail vehicles
3620:
3610:Road–rail vehicles
3549:
3448:
3368:
3258:
3135:
3031:
2946:
2944:
2820:
2729:
2595:
2461:
2399:
2343:
2290:
2214:
2134:
2116:
2005:
1900:
1805:
1783:
1720:
1631:
1556:
1530:
1486:
1417:of the wheel set.
1393:
1320:
1212:
1121:
1119:
837:
786:
759:elastic distortion
733:
674:
572:
490:to the rails, the
439:
395:
303:rolling resistance
263:
253:Kinematic analysis
4125:Rail technologies
3875:
3849:
3826:is negative when
3814:
3547:
3446:
3361:
3328:
3295:
3242:
3213:
3170:
3119:
3072:
3015:
2917:
2880:
2808:
2782:
2722:
2673:
2624:
2588:
2539:
2490:
2437:
2378:
2325:
2264:
2243:
2197:
2170:
2100:
2063:
2034:
1989:
1955:
1929:
1891:
1851:
1781:
1760:
1718:
1629:
1528:
1477:
1415:moment of inertia
1391:
1311:
1271:
1210:
1186:
1108:
1078:
1048:
1014:
963:
933:
886:
835:
731:
730:
665:
635:
570:
553:
437:
411:
384:similar triangles
377:), so the larger
151:Railway wheelsets
105:
104:
97:
55:Several templates
4132:
4105:
4096:
4087:
4078:
4069:
4055:
4054:
4051:Int J Auto Engng
4046:
4040:
4039:
4031:
4025:
4024:
4016:
4010:
4009:
3999:
3993:
3992:
3964:
3958:
3957:
3955:
3938:(788): 585–610.
3923:
3917:
3914:
3908:
3886:
3884:
3883:
3878:
3876:
3874:
3866:
3855:
3850:
3842:
3825:
3823:
3822:
3817:
3815:
3813:
3812:
3811:
3795:
3788:
3787:
3777:
3767:
3761:
3738:
3732:
3729:
3723:
3718:
3712:
3711:
3706:
3697:
3661:Vehicle dynamics
3558:
3556:
3555:
3550:
3548:
3546:
3542:
3541:
3513:
3512:
3511:
3492:
3487:
3486:
3457:
3455:
3454:
3449:
3447:
3445:
3444:
3443:
3421:
3420:
3419:
3403:
3398:
3397:
3377:
3375:
3374:
3369:
3367:
3363:
3362:
3357:
3356:
3355:
3342:
3329:
3327:
3319:
3311:
3309:
3308:
3296:
3291:
3280:
3267:
3265:
3264:
3259:
3257:
3256:
3247:
3243:
3241:
3233:
3225:
3219:
3215:
3214:
3209:
3208:
3207:
3194:
3181:
3180:
3171:
3163:
3144:
3142:
3141:
3136:
3134:
3133:
3124:
3120:
3118:
3110:
3102:
3096:
3095:
3083:
3082:
3073:
3065:
3040:
3038:
3037:
3032:
3030:
3029:
3020:
3016:
3014:
3006:
2998:
2992:
2991:
2976:
2975:
2955:
2953:
2952:
2947:
2945:
2941:
2940:
2931:
2930:
2918:
2910:
2881:
2879:
2871:
2863:
2838:potential energy
2829:
2827:
2826:
2821:
2819:
2818:
2813:
2809:
2804:
2793:
2783:
2775:
2752:The increase in
2749:of both wheels.
2738:
2736:
2735:
2730:
2728:
2724:
2723:
2721:
2720:
2711:
2710:
2709:
2700:
2699:
2689:
2687:
2686:
2674:
2669:
2661:
2659:
2658:
2643:
2642:
2625:
2617:
2604:
2602:
2601:
2596:
2594:
2590:
2589:
2587:
2586:
2577:
2576:
2575:
2566:
2565:
2555:
2553:
2552:
2540:
2535:
2527:
2525:
2524:
2509:
2508:
2491:
2483:
2470:
2468:
2467:
2462:
2460:
2456:
2438:
2430:
2408:
2406:
2405:
2400:
2398:
2397:
2379:
2371:
2352:
2350:
2349:
2344:
2342:
2341:
2326:
2318:
2299:
2297:
2296:
2291:
2289:
2285:
2278:
2277:
2265:
2257:
2244:
2236:
2223:
2221:
2220:
2215:
2213:
2209:
2208:
2207:
2198:
2190:
2171:
2169:
2146:
2125:
2123:
2122:
2117:
2115:
2114:
2105:
2101:
2099:
2091:
2083:
2077:
2076:
2064:
2056:
2048:
2047:
2035:
2027:
2014:
2012:
2011:
2006:
2004:
2003:
1994:
1990:
1988:
1980:
1972:
1966:
1965:
1956:
1948:
1940:
1939:
1930:
1922:
1909:
1907:
1906:
1901:
1896:
1892:
1890:
1882:
1874:
1868:
1867:
1852:
1850:
1839:
1828:
1814:
1812:
1811:
1806:
1792:
1790:
1789:
1784:
1782:
1780:
1766:
1761:
1759:
1745:
1729:
1727:
1726:
1721:
1719:
1717:
1706:
1695:
1674:potential energy
1640:
1638:
1637:
1632:
1630:
1628:
1617:
1616:
1615:
1602:
1591:
1590:
1565:
1563:
1562:
1557:
1539:
1537:
1536:
1531:
1529:
1521:
1495:
1493:
1492:
1487:
1482:
1478:
1476:
1475:
1474:
1461:
1453:
1447:
1446:
1402:
1400:
1399:
1394:
1392:
1390:
1389:
1388:
1372:
1365:
1364:
1354:
1329:
1327:
1326:
1321:
1316:
1312:
1310:
1302:
1294:
1288:
1287:
1272:
1270:
1269:
1268:
1252:
1245:
1244:
1234:
1221:
1219:
1218:
1213:
1211:
1209:
1195:
1187:
1185:
1171:
1130:
1128:
1127:
1122:
1120:
1113:
1109:
1107:
1099:
1091:
1079:
1077:
1066:
1055:
1053:
1049:
1047:
1039:
1031:
1015:
1013:
1012:
1011:
995:
988:
987:
977:
968:
964:
962:
954:
946:
934:
932:
931:
930:
914:
907:
906:
896:
887:
885:
874:
863:
846:
844:
843:
838:
836:
834:
823:
812:
795:
793:
792:
787:
785:
755:concept of creep
745:
742:
740:
739:
734:
732:
729:
721:
713:
712:
683:
681:
680:
675:
670:
666:
664:
656:
648:
636:
634:
633:
632:
616:
609:
608:
598:
581:
579:
578:
573:
571:
563:
558:
554:
552:
551:
550:
534:
527:
526:
516:
448:
446:
445:
440:
438:
436:
428:
417:
412:
404:
368:angular velocity
183:adhesion railway
173:) caused by the
141:self-oscillation
124:
115:
100:
93:
89:
86:
80:
78:
67:
27:
26:
19:
4140:
4139:
4135:
4134:
4133:
4131:
4130:
4129:
4110:
4109:
4108:
4059:
4058:
4047:
4043:
4032:
4028:
4017:
4013:
4000:
3996:
3965:
3961:
3924:
3920:
3915:
3911:
3906:
3902:
3898:
3894:
3890:
3867:
3856:
3854:
3841:
3839:
3836:
3835:
3833:
3829:
3807:
3803:
3796:
3783:
3779:
3778:
3776:
3774:
3771:
3770:
3768:
3764:
3759:
3755:
3751:
3747:
3739:
3735:
3730:
3726:
3719:
3715:
3699:
3698:
3689:
3684:
3637:
3612:
3579:
3564:
3559:
3537:
3533:
3514:
3507:
3503:
3493:
3491:
3482:
3478:
3476:
3473:
3472:
3468:
3458:
3439:
3435:
3422:
3415:
3411:
3404:
3402:
3393:
3389:
3387:
3384:
3383:
3378:
3351:
3347:
3343:
3341:
3334:
3330:
3320:
3312:
3310:
3304:
3300:
3281:
3279:
3277:
3274:
3273:
3268:
3252:
3248:
3234:
3226:
3224:
3220:
3203:
3199:
3195:
3193:
3186:
3182:
3176:
3172:
3162:
3154:
3151:
3150:
3145:
3129:
3125:
3111:
3103:
3101:
3097:
3091:
3087:
3078:
3074:
3064:
3050:
3047:
3046:
3041:
3025:
3021:
3007:
2999:
2997:
2993:
2987:
2983:
2971:
2967:
2965:
2962:
2961:
2956:
2943:
2942:
2936:
2932:
2926:
2922:
2909:
2902:
2893:
2892:
2882:
2872:
2864:
2862:
2852:
2850:
2847:
2846:
2830:
2814:
2794:
2792:
2788:
2787:
2774:
2763:
2760:
2759:
2744:
2739:
2716:
2712:
2705:
2701:
2695:
2691:
2690:
2688:
2682:
2678:
2662:
2660:
2654:
2650:
2638:
2634:
2633:
2629:
2616:
2614:
2611:
2610:
2605:
2582:
2578:
2571:
2567:
2561:
2557:
2556:
2554:
2548:
2544:
2528:
2526:
2520:
2516:
2504:
2500:
2499:
2495:
2482:
2480:
2477:
2476:
2471:
2443:
2439:
2429:
2421:
2418:
2417:
2409:
2393:
2389:
2370:
2362:
2359:
2358:
2353:
2337:
2333:
2317:
2309:
2306:
2305:
2300:
2273:
2269:
2256:
2249:
2245:
2235:
2233:
2230:
2229:
2224:
2203:
2199:
2189:
2182:
2178:
2150:
2145:
2143:
2140:
2139:
2126:
2110:
2106:
2092:
2084:
2082:
2078:
2072:
2068:
2055:
2043:
2039:
2026:
2024:
2021:
2020:
2015:
1999:
1995:
1981:
1973:
1971:
1967:
1961:
1957:
1947:
1935:
1931:
1921:
1919:
1916:
1915:
1910:
1883:
1875:
1873:
1869:
1863:
1859:
1840:
1829:
1827:
1822:
1819:
1818:
1800:
1797:
1796:
1793:
1770:
1765:
1749:
1744:
1739:
1736:
1735:
1730:
1707:
1696:
1694:
1686:
1683:
1682:
1662:
1641:
1618:
1611:
1607:
1603:
1601:
1586:
1582:
1577:
1574:
1573:
1551:
1548:
1547:
1545:
1540:
1520:
1509:
1506:
1505:
1496:
1470:
1466:
1462:
1454:
1452:
1448:
1442:
1438:
1424:
1421:
1420:
1412:
1408:
1403:
1384:
1380:
1373:
1360:
1356:
1355:
1353:
1339:
1336:
1335:
1330:
1303:
1295:
1293:
1289:
1283:
1279:
1264:
1260:
1253:
1240:
1236:
1235:
1233:
1231:
1228:
1227:
1222:
1199:
1194:
1175:
1170:
1168:
1165:
1164:
1160:
1131:
1118:
1117:
1100:
1092:
1090:
1086:
1067:
1056:
1054:
1040:
1032:
1030:
1026:
1016:
1007:
1003:
996:
983:
979:
978:
976:
973:
972:
955:
947:
945:
941:
926:
922:
915:
902:
898:
897:
895:
888:
875:
864:
862:
858:
856:
853:
852:
847:
824:
813:
811:
803:
800:
799:
775:
773:
770:
769:
747:
743:
722:
714:
711:
697:
694:
693:
684:
657:
649:
647:
643:
628:
624:
617:
604:
600:
599:
597:
595:
592:
591:
582:
562:
546:
542:
535:
522:
518:
517:
515:
511:
509:
506:
505:
501:
485:
481:
477:
466:
462:
454:
449:
429:
418:
416:
403:
401:
398:
397:
365:
356:
340:
336:
332:
328:
316:
299:
255:
159:
153:
134:
133:
132:
131:
127:
126:
125:
117:
116:
101:
90:
84:
81:
69:
58:
44:
28:
24:
17:
12:
11:
5:
4138:
4128:
4127:
4122:
4107:
4106:
4097:
4088:
4079:
4070:
4060:
4057:
4056:
4041:
4026:
4011:
3994:
3959:
3918:
3909:
3904:
3900:
3896:
3892:
3888:
3873:
3870:
3865:
3862:
3859:
3853:
3848:
3845:
3831:
3827:
3810:
3806:
3802:
3799:
3794:
3791:
3786:
3782:
3762:
3757:
3753:
3749:
3745:
3733:
3724:
3713:
3686:
3685:
3683:
3680:
3679:
3678:
3669:
3668:
3663:
3658:
3653:
3648:
3643:
3636:
3633:
3611:
3608:
3578:
3575:
3562:
3545:
3540:
3536:
3532:
3529:
3526:
3523:
3520:
3517:
3510:
3506:
3502:
3499:
3496:
3490:
3485:
3481:
3471:
3466:
3442:
3438:
3434:
3431:
3428:
3425:
3418:
3414:
3410:
3407:
3401:
3396:
3392:
3382:
3366:
3360:
3354:
3350:
3346:
3340:
3337:
3333:
3326:
3323:
3318:
3315:
3307:
3303:
3299:
3294:
3290:
3287:
3284:
3272:
3255:
3251:
3246:
3240:
3237:
3232:
3229:
3223:
3218:
3212:
3206:
3202:
3198:
3192:
3189:
3185:
3179:
3175:
3169:
3166:
3161:
3158:
3149:
3132:
3128:
3123:
3117:
3114:
3109:
3106:
3100:
3094:
3090:
3086:
3081:
3077:
3071:
3068:
3063:
3060:
3057:
3054:
3045:
3028:
3024:
3019:
3013:
3010:
3005:
3002:
2996:
2990:
2986:
2982:
2979:
2974:
2970:
2960:
2939:
2935:
2929:
2925:
2921:
2916:
2913:
2908:
2905:
2903:
2901:
2898:
2895:
2894:
2891:
2888:
2885:
2883:
2878:
2875:
2870:
2867:
2861:
2858:
2855:
2854:
2845:
2817:
2812:
2807:
2803:
2800:
2797:
2791:
2786:
2781:
2778:
2773:
2770:
2767:
2758:
2754:kinetic energy
2742:
2727:
2719:
2715:
2708:
2704:
2698:
2694:
2685:
2681:
2677:
2672:
2668:
2665:
2657:
2653:
2649:
2646:
2641:
2637:
2632:
2628:
2623:
2620:
2609:
2593:
2585:
2581:
2574:
2570:
2564:
2560:
2551:
2547:
2543:
2538:
2534:
2531:
2523:
2519:
2515:
2512:
2507:
2503:
2498:
2494:
2489:
2486:
2475:
2459:
2455:
2452:
2449:
2446:
2442:
2436:
2433:
2428:
2425:
2416:
2396:
2392:
2388:
2385:
2382:
2377:
2374:
2369:
2366:
2357:
2340:
2336:
2332:
2329:
2324:
2321:
2316:
2313:
2304:
2288:
2284:
2281:
2276:
2272:
2268:
2263:
2260:
2255:
2252:
2248:
2242:
2239:
2228:
2212:
2206:
2202:
2196:
2193:
2188:
2185:
2181:
2177:
2174:
2168:
2165:
2162:
2159:
2156:
2153:
2149:
2138:
2113:
2109:
2104:
2098:
2095:
2090:
2087:
2081:
2075:
2071:
2067:
2062:
2059:
2054:
2051:
2046:
2042:
2038:
2033:
2030:
2019:
2002:
1998:
1993:
1987:
1984:
1979:
1976:
1970:
1964:
1960:
1954:
1951:
1946:
1943:
1938:
1934:
1928:
1925:
1914:
1899:
1895:
1889:
1886:
1881:
1878:
1872:
1866:
1862:
1858:
1855:
1849:
1846:
1843:
1838:
1835:
1832:
1826:
1817:
1804:
1779:
1776:
1773:
1769:
1764:
1758:
1755:
1752:
1748:
1743:
1734:
1716:
1713:
1710:
1705:
1702:
1699:
1693:
1690:
1681:
1661:
1660:Energy balance
1658:
1627:
1624:
1621:
1614:
1610:
1606:
1600:
1597:
1594:
1589:
1585:
1581:
1572:
1555:
1543:
1527:
1524:
1519:
1516:
1513:
1504:
1485:
1481:
1473:
1469:
1465:
1460:
1457:
1451:
1445:
1441:
1437:
1434:
1431:
1428:
1419:
1410:
1406:
1387:
1383:
1379:
1376:
1371:
1368:
1363:
1359:
1352:
1349:
1346:
1343:
1334:
1319:
1315:
1309:
1306:
1301:
1298:
1292:
1286:
1282:
1278:
1275:
1267:
1263:
1259:
1256:
1251:
1248:
1243:
1239:
1226:
1208:
1205:
1202:
1198:
1193:
1190:
1184:
1181:
1178:
1174:
1163:
1158:
1116:
1112:
1106:
1103:
1098:
1095:
1089:
1085:
1082:
1076:
1073:
1070:
1065:
1062:
1059:
1052:
1046:
1043:
1038:
1035:
1029:
1025:
1022:
1019:
1017:
1010:
1006:
1002:
999:
994:
991:
986:
982:
975:
974:
971:
967:
961:
958:
953:
950:
944:
940:
937:
929:
925:
921:
918:
913:
910:
905:
901:
894:
891:
889:
884:
881:
878:
873:
870:
867:
861:
860:
851:
833:
830:
827:
822:
819:
816:
810:
807:
798:
784:
781:
778:
728:
725:
720:
717:
710:
707:
704:
701:
692:
673:
669:
663:
660:
655:
652:
646:
642:
639:
631:
627:
623:
620:
615:
612:
607:
603:
590:
569:
566:
561:
557:
549:
545:
541:
538:
533:
530:
525:
521:
514:
504:
499:
483:
479:
475:
464:
460:
452:
435:
432:
427:
424:
421:
415:
410:
407:
396:
363:
355:
352:
338:
334:
330:
326:
314:
298:
295:
254:
251:
155:Main article:
152:
149:
129:
128:
119:
118:
110:
109:
108:
107:
106:
103:
102:
47:full citations
31:
29:
22:
15:
9:
6:
4:
3:
2:
4137:
4126:
4123:
4121:
4118:
4117:
4115:
4103:
4098:
4094:
4089:
4085:
4080:
4076:
4071:
4067:
4062:
4061:
4052:
4045:
4037:
4030:
4022:
4015:
4007:
4006:
3998:
3990:
3986:
3982:
3978:
3975:(1–2): 9–26.
3974:
3970:
3963:
3954:
3949:
3945:
3941:
3937:
3933:
3929:
3922:
3913:
3903:negative and
3871:
3868:
3863:
3860:
3857:
3851:
3846:
3843:
3808:
3804:
3800:
3792:
3789:
3784:
3766:
3743:
3737:
3728:
3722:
3717:
3710:
3705:
3704:
3696:
3694:
3692:
3687:
3677:
3674:
3673:
3672:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3646:Rail adhesion
3644:
3642:
3639:
3638:
3632:
3630:
3625:
3616:
3607:
3604:
3600:
3596:
3592:
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3436:
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3408:
3405:
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3390:
3381:
3364:
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3352:
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3338:
3335:
3331:
3324:
3321:
3316:
3313:
3305:
3301:
3297:
3292:
3288:
3285:
3282:
3271:
3253:
3249:
3244:
3238:
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3230:
3227:
3221:
3216:
3210:
3204:
3200:
3196:
3190:
3187:
3183:
3177:
3173:
3167:
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3156:
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3126:
3121:
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3107:
3104:
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3084:
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3069:
3066:
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3055:
3052:
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3026:
3022:
3017:
3011:
3008:
3003:
3000:
2994:
2988:
2984:
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2968:
2959:
2937:
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2927:
2923:
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2911:
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2896:
2889:
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2876:
2873:
2868:
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2859:
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2841:
2839:
2835:
2815:
2810:
2805:
2801:
2798:
2795:
2789:
2784:
2779:
2776:
2771:
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2765:
2757:
2755:
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2748:
2725:
2717:
2713:
2706:
2702:
2696:
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2683:
2679:
2675:
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2666:
2663:
2655:
2651:
2647:
2644:
2639:
2635:
2630:
2626:
2621:
2618:
2608:
2591:
2583:
2579:
2572:
2568:
2562:
2558:
2549:
2545:
2541:
2536:
2532:
2529:
2521:
2517:
2513:
2510:
2505:
2501:
2496:
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2434:
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2423:
2415:
2412:
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2386:
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2372:
2367:
2364:
2356:
2338:
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2327:
2322:
2319:
2314:
2311:
2303:
2286:
2282:
2279:
2274:
2270:
2266:
2261:
2258:
2253:
2250:
2246:
2240:
2237:
2227:
2210:
2204:
2200:
2194:
2191:
2186:
2183:
2179:
2175:
2172:
2163:
2157:
2154:
2151:
2147:
2137:
2130:
2111:
2107:
2102:
2096:
2093:
2088:
2085:
2079:
2073:
2069:
2065:
2060:
2057:
2052:
2049:
2044:
2040:
2036:
2031:
2028:
2018:
2000:
1996:
1991:
1985:
1982:
1977:
1974:
1968:
1962:
1958:
1952:
1949:
1944:
1941:
1936:
1932:
1926:
1923:
1913:
1912:integrating:
1897:
1893:
1887:
1884:
1879:
1876:
1870:
1864:
1860:
1856:
1853:
1847:
1844:
1836:
1833:
1824:
1816:
1802:
1777:
1774:
1762:
1756:
1753:
1741:
1733:
1714:
1711:
1703:
1700:
1691:
1688:
1680:
1677:
1675:
1671:
1667:
1657:
1653:
1651:
1645:
1625:
1622:
1619:
1612:
1608:
1604:
1598:
1595:
1592:
1587:
1583:
1579:
1571:
1569:
1553:
1525:
1522:
1517:
1514:
1511:
1503:
1501:
1483:
1479:
1471:
1467:
1463:
1458:
1455:
1449:
1443:
1439:
1435:
1432:
1429:
1426:
1418:
1416:
1385:
1381:
1377:
1369:
1366:
1361:
1350:
1347:
1344:
1341:
1333:
1317:
1313:
1307:
1304:
1299:
1296:
1290:
1284:
1280:
1276:
1273:
1265:
1261:
1257:
1249:
1246:
1241:
1225:
1206:
1203:
1191:
1188:
1182:
1179:
1162:
1156:
1152:
1147:
1143:
1141:
1137:
1114:
1110:
1104:
1101:
1096:
1093:
1087:
1083:
1080:
1074:
1071:
1063:
1060:
1050:
1044:
1041:
1036:
1033:
1027:
1023:
1020:
1018:
1008:
1004:
1000:
992:
989:
984:
969:
965:
959:
956:
951:
948:
942:
938:
935:
927:
923:
919:
911:
908:
903:
892:
890:
882:
879:
871:
868:
850:
831:
828:
820:
817:
808:
805:
797:
796:is given by:
782:
779:
776:
766:
764:
760:
756:
752:
726:
723:
718:
715:
708:
705:
702:
699:
691:
689:
671:
667:
661:
658:
653:
650:
644:
640:
637:
629:
625:
621:
613:
610:
605:
589:
587:
567:
564:
559:
555:
547:
543:
539:
531:
528:
523:
512:
503:
497:
493:
489:
472:
470:
467:is the tread
458:
455:is the track
433:
430:
425:
422:
419:
413:
408:
405:
391:
387:
385:
380:
376:
373:
369:
361:
351:
348:
342:
324:
320:
312:
308:
304:
294:
291:
287:
283:
279:
275:
270:
268:
259:
250:
248:
244:
240:
236:
232:
228:
227:
222:
221:
214:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
171:bogie hunting
168:
167:truck hunting
164:
158:
148:
146:
142:
138:
123:
114:
99:
96:
88:
76:
75:documentation
72:
65:
64:documentation
61:
56:
52:
48:
43:
41:
37:
32:This article
30:
21:
20:
4104:. CRC Press.
4101:
4095:. CRC Press.
4092:
4083:
4077:. CRC Press.
4074:
4068:. CRC Press.
4065:
4050:
4044:
4035:
4029:
4020:
4014:
4003:
3997:
3972:
3968:
3962:
3935:
3931:
3921:
3912:
3765:
3736:
3727:
3716:
3708:
3702:
3670:
3656:Speed wobble
3651:Rail profile
3621:
3605:
3601:
3597:
3593:
3589:
3580:
3560:
3459:
3379:
3269:
3146:
3042:
2957:
2842:
2831:
2751:
2740:
2606:
2472:
2413:
2410:
2354:
2301:
2225:
2135:
2016:
1911:
1794:
1731:
1678:
1663:
1654:
1646:
1642:
1541:
1498:the maximum
1497:
1404:
1331:
1223:
1148:
1144:
1132:
848:
767:
748:
685:
583:
473:
450:
357:
343:
300:
290:acceleration
271:
264:
230:
224:
220:swing hanger
219:
215:
203:differential
170:
166:
160:
136:
135:
91:
82:
71:Citation bot
33:
4120:Oscillation
1155:derivatives
195:oscillation
189:forces and
145:equilibrium
85:August 2022
4114:Categories
4053:: 186–196.
4038:: 195–206.
3769:Note that
3682:References
3584:gyroscopic
1652:theories.
1500:frictional
686:This is a
586:trajectory
496:derivative
323:trajectory
319:trajectory
239:hovertrain
231:Shinkansen
226:Shinkansen
199:derailment
51:verifiable
3989:109580328
3907:positive.
3801:
3790:
3573:methods.
3250:θ
3127:θ
3062:−
3053:δ
3023:θ
2981:−
2969:ω
2934:ω
2897:δ
2890:ω
2766:δ
2645:−
2391:θ
2335:θ
2280:−
2271:θ
2201:θ
2164:θ
2108:θ
2053:−
2041:ω
1997:θ
1945:−
1933:ω
1898:θ
1857:−
1848:θ
1845:
1837:ω
1834:
1825:ω
1803:ω
1775:
1757:θ
1754:
1742:ω
1712:
1704:θ
1701:
1689:ω
1596:μ
1580:θ
1554:μ
1518:μ
1484:θ
1433:−
1378:
1370:θ
1367:
1318:θ
1277:−
1258:
1250:θ
1247:
1204:
1180:
1115:θ
1084:−
1072:
1061:
1024:−
1001:
993:θ
990:
939:−
920:
909:
880:
872:θ
869:
829:
818:
806:θ
780:θ
709:π
700:λ
641:−
622:
611:
560:≈
540:
529:
492:curvature
482:), where
274:kinematic
211:Talgo 350
179:stability
36:bare URLs
3666:Wheelset
3635:See also
1151:inertial
488:parallel
379:diameter
278:geometry
267:dynamics
207:wheelset
191:inertial
187:adhesion
40:link rot
3940:Bibcode
3760:is low.
2745:is the
1670:kinetic
1566:is the
1413:is the
1136:flutter
849:Hence:
163:railway
4023:: 22–.
4008:: 29–.
3987:
3629:toe-in
3561:where
2834:energy
2741:where
1666:energy
1644:case.
1542:where
1405:where
1140:shimmy
751:damped
744:
451:where
360:treads
347:flange
282:forces
243:maglev
181:of an
175:coning
60:reFill
3985:S2CID
3622:Many
1679:Let:
469:taper
457:gauge
372:rigid
311:truck
307:train
139:is a
34:uses
3740:See
3567:wear
3462:load
2958:but
2756:is:
2747:mass
1672:and
375:axle
241:and
68:and
3977:doi
3948:doi
3936:121
498:of
309:or
247:TGV
169:or
4116::
3983:.
3973:46
3971:.
3946:.
3934:.
3930:.
3834:,
3690:^
1815::
459:,
272:A
3991:.
3979::
3956:.
3950::
3942::
3905:y
3901:y
3897:R
3893:R
3889:y
3872:d
3869:r
3864:y
3861:k
3858:2
3852:=
3847:R
3844:1
3832:R
3828:y
3809:2
3805:x
3798:d
3793:y
3785:2
3781:d
3758:y
3754:y
3750:x
3746:y
3563:a
3544:)
3539:2
3535:d
3531:m
3528:+
3525:C
3522:4
3519:(
3516:k
3509:2
3505:d
3501:r
3498:a
3495:W
3489:=
3484:2
3480:U
3467:a
3441:2
3437:d
3433:m
3430:+
3427:C
3424:4
3417:2
3413:d
3409:r
3406:W
3400:=
3395:2
3391:U
3365:)
3359:4
3353:2
3349:d
3345:m
3339:+
3336:C
3332:(
3325:d
3322:r
3317:k
3314:2
3306:2
3302:U
3298:=
3293:2
3289:d
3286:k
3283:W
3254:2
3245:)
3239:d
3236:r
3231:k
3228:2
3222:(
3217:)
3211:4
3205:2
3201:d
3197:m
3191:+
3188:C
3184:(
3178:2
3174:U
3168:2
3165:1
3160:=
3157:T
3131:2
3122:)
3116:d
3113:r
3108:k
3105:2
3099:(
3093:2
3089:d
3085:m
3080:2
3076:U
3070:8
3067:1
3059:=
3056:E
3027:2
3018:)
3012:d
3009:r
3004:k
3001:2
2995:(
2989:2
2985:U
2978:=
2973:2
2938:2
2928:2
2924:d
2920:m
2915:8
2912:1
2907:=
2900:E
2887:=
2877:d
2874:r
2869:y
2866:k
2860:U
2857:2
2816:2
2811:)
2806:r
2802:y
2799:k
2796:U
2790:(
2785:m
2780:2
2777:1
2772:=
2769:E
2743:m
2726:)
2718:2
2714:r
2707:2
2703:y
2697:2
2693:k
2684:2
2680:U
2676:+
2671:r
2667:y
2664:k
2656:2
2652:U
2648:2
2640:2
2636:U
2631:(
2627:m
2622:4
2619:1
2592:)
2584:2
2580:r
2573:2
2569:y
2563:2
2559:k
2550:2
2546:U
2542:+
2537:r
2533:y
2530:k
2522:2
2518:U
2514:2
2511:+
2506:2
2502:U
2497:(
2493:m
2488:4
2485:1
2458:)
2454:y
2451:k
2448:+
2445:r
2441:(
2435:r
2432:U
2427:=
2424:V
2395:2
2387:d
2384:k
2381:W
2376:4
2373:1
2368:=
2365:E
2339:2
2331:d
2328:k
2323:4
2320:1
2315:=
2312:h
2287:)
2283:d
2275:2
2267:d
2262:2
2259:1
2254:+
2251:d
2247:(
2241:2
2238:1
2211:)
2205:2
2195:2
2192:1
2187:+
2184:1
2180:(
2176:d
2173:=
2167:)
2161:(
2158:s
2155:o
2152:c
2148:d
2112:2
2103:)
2097:d
2094:r
2089:k
2086:2
2080:(
2074:2
2070:U
2066:C
2061:2
2058:1
2050:=
2045:2
2037:C
2032:2
2029:1
2001:2
1992:)
1986:d
1983:r
1978:k
1975:2
1969:(
1963:2
1959:U
1953:2
1950:1
1942:=
1937:2
1927:2
1924:1
1894:)
1888:d
1885:r
1880:k
1877:2
1871:(
1865:2
1861:U
1854:=
1842:d
1831:d
1778:t
1772:d
1768:d
1763:=
1751:d
1747:d
1715:t
1709:d
1698:d
1692:=
1626:k
1623:C
1620:4
1613:2
1609:d
1605:r
1599:W
1593:=
1588:2
1584:U
1544:W
1526:2
1523:W
1515:=
1512:F
1480:)
1472:2
1468:d
1464:r
1459:k
1456:2
1450:(
1444:2
1440:U
1436:C
1430:=
1427:F
1411:C
1407:F
1386:2
1382:t
1375:d
1362:2
1358:d
1351:C
1348:=
1345:d
1342:F
1314:)
1308:d
1305:r
1300:k
1297:2
1291:(
1285:2
1281:U
1274:=
1266:2
1262:t
1255:d
1242:2
1238:d
1207:x
1201:d
1197:d
1192:U
1189:=
1183:t
1177:d
1173:d
1159:U
1111:)
1105:d
1102:r
1097:k
1094:2
1088:(
1081:=
1075:x
1069:d
1064:y
1058:d
1051:)
1045:d
1042:r
1037:k
1034:2
1028:(
1021:=
1009:2
1005:x
998:d
985:2
981:d
970:y
966:)
960:d
957:r
952:k
949:2
943:(
936:=
928:2
924:x
917:d
912:y
904:2
900:d
893:=
883:x
877:d
866:d
832:x
826:d
821:y
815:d
809:=
783:)
777:(
727:k
724:2
719:d
716:r
706:2
703:=
672:y
668:)
662:d
659:r
654:k
651:2
645:(
638:=
630:2
626:x
619:d
614:y
606:2
602:d
568:R
565:1
556:|
548:2
544:x
537:d
532:y
524:2
520:d
513:|
500:y
484:x
480:x
478:(
476:y
465:k
461:r
453:d
434:d
431:r
426:y
423:k
420:2
414:=
409:R
406:1
364:y
339:y
335:x
331:y
329:-
327:x
315:R
98:)
92:(
87:)
83:(
79:.
77:)
73:(
66:)
62:(
42:.
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