Little Things That Matter: Jerk

When you ride a subway train, and the train decelerates to its station, you feel your body pulled forward, and your muscles tense to adjust, but then when the train reaches a sudden stop, you are suddenly flung backward, since you are no longer decelerating, but your muscles take time to relax and stop fighting a braking that no longer exist. This effect is called jerk, and is defined to be change in acceleration, just as acceleration is change in speed and speed is change in position. Controlling jerk is crucial for a smooth railway ride. Unfortunately, American mainline rail is not good at this, leading to noticeable jolts by passengers even though speed limits on curves and acceleration rates are very conservative.

This is particularly important for speeding up mainline trains around New York and other legacy cities in the US, like Boston. Speeding up the slowest segments is more important than speeding up the fastest ones; my schedules for New York-New Haven trains, cutting trip times from 2:09 to 1:24, save 4 minutes between Grand Central and 59th Street just through avoiding slowdowns in the interlocking. The interlocking is slow because the switches have very conservative speed limits relative to curve radius (that is, lateral acceleration), which in turn is because they are not designed with good lateral jerk control. The good news is that replacing the necessary infrastructure is not so onerous, provided the railroads know what they need to do and avoid running heavy diesel locomotives on delicate infrastructure.

Spirals and jerk

In practice, the worst jerk is usually not forward or backward, except in the last fraction of a second at the end of acceleration. This is because it takes about a second for train motors to rev up, which controls jerk during acceleration. Rather, the worst is sideways, because it is possible to design curves that transition abruptly from straight track, on which there is no lateral acceleration, to curved track, on which there is, in the form of centrifugal force centripetal force.

To reduce jerk, the transition from straight track to a circular arc is done gradually. There are a number of usable transition curve (see Romain Bosquet’s thesis, PDF-p. 36), but the most common by far is called the clothoid, which has the property of having constant change in curvature per unit of arc length – that is, constant jerk. Different countries have different standards for how long the clothoid should be, that is what the maximum lateral jerk is. Per Martin Lindahl’s thesis, the limit in Sweden is 55 mm/s (PDF-p. 30) and that in Germany is 69.44 mm/s (PDF-p. 38), both measured in units of cant deficiency; in SI units, this is 0.367 m/s^3 and 0.463 m/s^3 respectively. In France, the regular limit is 50 mm/s (Bosquet’s thesis, PDF-p. 35), that is 0.333 m/s^2, but it is specifically waived in turnouts.

Track switches are somehow accepted as sites of very high jerk. A presentation about various technical limits in France notes on p. 106 that in switches (“appareils de voie” or “aiguilles” or “aiguillages,” depending on source, just like “switch” vs. “turnout” in English), the jerk can be increased to 100 and even 125 mm/s. On p. 107 it even asserts that in exceptional circumstances, abrupt change in cant deficiency of up to 50 mm on main track and 100 on the diverging direction on a switch is allowed; see also PDF-pp. 13-15 of a pan-European presentation. Abrupt changes are not good for passengers, but will not derail a train.

Turnout design in the advanced world

Second derivative control, that is acceleration and cant deficiency, can be done using calculus and trigonometry tools. Third derivative control, that is clothoids and jerk, requires numerical calculations, but fortunately they are approximated well by pretending the clothoid is half straight line, half circular arc, with the length determined by the maximum jerk. Working from first principles, it’s possible to figure out that at typical turnout needs – e.g. move a train from one track to a parallel track 4 meters away – the clothoid is far longer than the curve itself, and at 50 mm/s jerk and 150 mm cant deficiency it’s not even possible to hit a curve radius of 250 meters.

Turnouts are inherently compromises. The question is just where to compromise. Here, for example, is a French turnout design, in two forms: 0.11 and 0.085. The numbers denoting the tangent of the angle at the frog, and the radius is proportional to the inverse square of the number, thus the speed is proportional to the inverse of the number. The sharper turnout, the 0.11, has a radius of 281 meters, a maximum speed of 50 km/h, and a total length of 26 meters from point to frog (“lead” in US usage), of which the clothoid curve (“point”) takes up 11, to limit jerk to 125 mm/s at a cant deficiency of 100 mm. The 0.085 turnout has a radius of 485 meters, a maximum speed of 65 km/h, a lead of about 38 meters, and a point of about 14.5 meters.

In Germany, turnouts have somewhat independent numbers and radii – some have shorter leads than others. The numbers are the inverse of those of France, so what France calls 0.11, Germany calls 1:9, but at the end of the day, the curve radius is the important part, with a cant deficiency of 100 mm. A higher cant deficiency may be desirable, but lengthening the point requires almost as much space as just increasing the curve radius, so might as well stick with the more comfortable limits.

Turnout design in the United States

American turnouts look similar to French or German ones, at first glance. I’ve seen a number of different designs; here’s one by CSX, on PDF-pp. 22 (#8) and 24 (#10), the numbers being very roughly comparable to German ones and inverses of French ones. CSX’s #10 has a curve radius of 779.39′, or 238 meters, and a lead of 24 meters, both numbers slightly tighter than the French 0.11. The radius is proportional to the square of the number, and so speed is proportional to the number.

However, the cant deficiency is just 50 mm. The point is not always curved; Amtrak’s low-number switches are not, so the change in cant deficiency is abrupt. Judging by what I experience every time I take a train between New York and New Haven, Metro-North’s switches have abrupt change in cant deficiency even on the mainline. The recommended standards by AREMA involve a curved point, but the point is still much shorter than in France (19.5′, or just under 6 meters, on a #12), so a 125 mm/s jerk only gets one up to about 62 mm cant deficiency.

The reason for this is that European turnouts are curved through the frog, whereas American ones are always straight at the frog. Extremely heavy American freight trains do not interact well with curved frogs and long points.

One might ask, why bother with such turnout design on rail segments that never see a heavy freight locomotive or 130-ton freight car? And on segments that do see the odd freight locomotive, like the approaches to Grand Central and Penn Station with the rare dual-mode locomotive, why not kick out anything that doesn’t interact well with advanced track design? Making a handful of passengers transfer would save around 4 minutes of trip time on the last mile into Grand Central alone for everyone else, not to mention time savings farther up the line.


  1. Tom M

    Two small corrections, acceleration is the derivative of velocity not speed, (speed is a scalar magnitude only and has on direction component), and you are describing centripetal force not centrifugal.

  2. Nathanael

    It would be worth figuring out which territory actually hosts freight that would dislike the curved frogs. In NY the main question is where CSX runs on MetroNorth, since NY&A runs very light and very slow on LIRR.

    May be nothing to be done at Spuyten Duyvil, but you are right, the approaches to Penn and GCT are mostly freight-free…

    Have to allow for dual modes from the Empire Line through Penn for Amtrak, but they only go to GCT during emergency diversions. And they only go to some of the Penn tracks, too.

    • Alon Levy

      Honestly, even on the mainlines this should be fine, there’s not a lot of freight on the NEC and if Metro-North et al had any idea how things are done in the rest of the world they’d just decide in advance which tracks get freight (the local ones, not the express ones) and have freight take the straight direction on turnouts with curved frogs.

    • F-Line to Dudley

      This can be an issue with full-high platforms and Plate F or greater freight cars, which have to run on the tracks where the platforms ain’t to avoid strike risk or punitive speed restrictions for safe passage. For example a 60 ft. Plate F boxcar, one of the most common types out there, won’t pass many full-highs that have much of any curve to them. Something as curved as this ( on the MBTA Greenbush Line designed to match up gapless with vestibule-end doors exceeds the turning radius of that freight car’s axles and will clip the edge of the car without fail, and so would many platform permutations of less curvature.

      In other cases with tangent platforms or extreme-slight curvature it is OK to pass a full-high with a “high-and-wide” freight car because the absolute width dimensions will technically fit, but it can only be done in-practice with very severe speed restrictions put in place to minimize the strike risk and associated nuisance repair costs to both the platform owner and the freight car owner. In those cases it’s the car’s suspension swinging momentarily out-of-envelope from the harmonics of the entire freight train, an effect most pronounced when the platform in question is tangent but abuts a nearby curve that some of the freight train would still be rounding as the first cars were passing the platform. The freight may get whacked with a blanket 10 MPH restriction through any full-high platform as a hedge against those kinds of nuisance strikes, which only works when traffic is very low (like overnights) or when high-and-wide dimension loads make rare enough appearances on a given freight train’s manifest that the freight carrier is willing to hold its nose and take the schedule hit only every once in a blue moon. Most prominent current example of such a speedo arrangement is T.F. Green Airport station on the NEC, where Providence & Worcester autorack trains craaawwwwl through the lone passenger platform twice a day amid extremely sparse MBTA traffic while staying out of the way of denser high-speed Amtrak traffic. But that’s only a temporary arrangement, as once RIDOT builds out the complete station with a matching northbound platform and increases schedules to somewhat usable levels P&W will be getting a gauntlet track installed to pass that platform at at least thrice its current speed.

      This isn’t a big deal overall as 48-inch platforms are mostly a Northeastern phenomenon, while Plate F freight routes in the Northeast tend to be judiciously pared down to only the highest-volume routes…few of which have more than incidental overlap with high-ridership commuter rail. But there are some high-profile cases like the near-entirety of the Hudson Line where that is going to influence both platform design (e.g. consistent design standard like outer side local platforms w/inner express tracks when possible) and turnout design so the track switching is kept to tolerable minimums.

      See here for some clearance maps where it comes into play:
      — CSX (by clicking “Plate Restriction” on their clearance map):
      — New York State & PANYNJ region clearance map: 21 – 2008 NYS Rail Clearances.pdf
      — Pan Am Railways (MBTA northside):

      TL;DR…it’s really only a tangible issue on: MNRR/Amtrak Hudson Line, Morris Heights to Hudson; MNRR Port Jervis Line + most of NJT Main & Bergen County Lines; MBTA Worcester Line, west-of-Framingham only; MBTA Franklin Line, Readville to Walpole only; MBTA Lowell Line (incl. future NH extensions); MBTA Haverhill Line/Amtrak Downeaster, Ballardvale to Portland; ConnDOT Hartford Line; and very *isolated* spots on the NEC in Rhode Island and Maryland. Basically, the NY-Albany, Albany-Boston + Hartford/Providence spurs, Boston-Portland, Southern Tier NY, and Port of Baltimore freight lanes boiled down to their barest essential spokes. Nearly all stations en route that still have low-level platforms are solvable for raising to full-high using something from the design bag of tricks…but at increased likelihood of ugly turnout hacks being deployed to get that job done. Put most mission-critical shared lines from that list under scrutiny accordingly for how many (good/bad/indifferent-designed) turnouts are too many for the task.

      • Alon Levy

        But the Hudson Line has high platforms, inc. at two-track stations and at express four-track stations. No ugly turnouts, just freight that stays inside the dynamical clearance envelope.

        • F-Line to Dudley

          Not all Plate F cars foul the clearance envelope as the classification is inclusive of both height-on-axle and length-on-axle dimensions…but only some combinations of the two (like the afforementioned big boxcars) actually add up to a “high-and-wide” hazard. So it also requires parsing the makeup of the given freight lane’s traffic, to further muddy the picture. Poughkeepsie-north there’s some trace amounts of high-and-wides, but the overwhelming majority of carloads going to/from Harlem are the huge outbound trash train that runs every night, inbound perishables to Hunts Point, and interchange loads for NYAR on Long Island and P&W on the NEC who are either width or height restricted. So the carloads are legit Plate F, but effectively zero of them are currently high-and-wide. You don’t see maxi boxcars at Oak Point or Harlem yards because no customers exist on that side of the river who need them, including what few Hudson Line customers get served on the daytime locals. That’s very different from the Albany-Framingham lane where the high-and-wide tonnage is much too big to coexist, and too much of it runs during daytime when speed restrictions hurt the most. The worst they have to pay attention to on the Hudson is that some of the cars on the trash train are so dented from repeated loadings they have to watch their speed around a few platforms at/near non-tangent track, but as a strict nocturnal job it’s the only train out there so that’s no biggie.

          This has the potential to change. The line was vertically cleared as far south as Tarrytown for big autoracks from the ex-GM plant there before GM reversed course and closed the plant. And there’s potential now that CSX has vertically cleared the Oak Point Link to develop new overnight traffic beneficial at getting trucks off the road in the shorter term, as much as the state and CSX want to explore the underutilized graveyard shift for further congestion mitigation. Right now there’s enough passing opportunities in 4-track territory that if the former Track 5 passer at Yonkers were reinstated then Ossining is the only one with a potential speed restriction should those carloads materialize. 2-track territory would be further speed restricted, but if they time their shots from Croton-Harmon to Poughkeepsie freight yards entirely on the overnight they get in/out without breaking a schedule sweat.

          Eminently doable with the platforms. But as per the subject of the piece how many turnouts does that end up adding if the overnight carloads in question become lucrative enough to merit the installs for fluidity, and how well-designed do the turnouts have to be to not degrade ride quality for daytime passengers? The Euros can do this on HSR lines with 5x the freight volumes, but can you trust MNRR’s track dept. as far as you can throw them to design a turnout that doesn’t ride like ass in an M7?

      • adirondacker12800

        Something as curved as this ( on the MBTA Greenbush Line designed to match up gapless with vestibule-end doors exceeds the turning radius of that freight car’s axles and will clip the edge of the car without fail, and so would many platform permutations of less curvature.

        I realize this would cause the fabric of the universe to rend, move the station two blocks to where there is straight track? It seems to be single tracked, double track that short section? NJTransit and Conrail cope on a very busy line at Union and Roselle Park with gauntlet tracks, yet a third solution. Or have service that sucks and sit and traffic because the service sucks.

        • F-Line to Dudley

          That reference example for curvature that’s OK for passenger doors but not for certain boxcars was from a line with zero freight traffic, whose connections to the outside world all have restricted clearances so you couldn’t even get a maxi bozcar within 5 miles of there without clipping the underside of a bridge. Curved platform in that spot is a non-issue now and 50 years from now, and a non-issue if tomorrow you started running 200-car trains of oil tankers or anything else in the great big universe of NOT high-and-wide loads past that thing.

          This is the case for the vast majority of the Northeastern passenger network. It is ONLY on a few select spots on a few high-priority lanes with particular traffic profiles where it’s even enough of an issue to do a risk assessment…let alone actually have to install that turnout or slap that speedo. You just have to figure out where on the sliding scale the various factors converge and have some sort of coherent strategy for where/when/how to address if there’s a kink to smooth out.

          Reading the post for context for a change instead of the opportunity to tee up lazy zingers wouldn’t cause the universe to rend either, you know.

          • adirondacker12800

            if there is no traffic there that is a problem, there’s no problem is there? If there is there are solutions. Or everything can stay just the way it is and everyone can sit in traffic.

  3. adirondacker12800

    Making a handful of passengers transfer would save around 4 minutes of trip time on the last mile into Grand Central alone for everyone else, not to mention time savings farther up the line.

    Saves them two minutes assuming they can make a timed transfer. Or electrify to Poughkeepsie. Electrifying to Montreal and Toronto would be nice. Saratoga Springs and Springfield would be a good first phase.
    New York gets complicated if you want to run a dozen high speed trains an hour through Penn Station and serve Wall Street from the suburbs. Have to decide what a “suburb” is – who is going to be expressing through what and where. All suburban stations get service to the three destinations in Manhattan is too much capacity. Or not enough.

      • adirondacker12800

        The Connecticut diesel branches are mostly shuttles now. Electrify to Saratoga Springs, Poughkeepsie gets something M8-ish and Saratoga Springs gets something Acela-y. Until that happens it saves the diesel passengers two minutes because they “waste” two making the timed transfer. Which many of them already do. 149th and Third would be nice because they could change for the Upper West Side and Wall Street. That depends on how you want to give Staten Island service to Manhattan and the Harlem line service to the West Side and Wall Street. It’s a pity the City didn’t knock the railroads heads together in 1890, this could all be much more rational.

  4. Charles Smith

    Designing curves and turnouts using the clothoid would indeed be a step forward. But, if you’re going to introduce a new set of curves because calculation is easy, shouldn’t you extend the control regime to the fourth derivative, Using forth derivative, you can smoothly, likely linearly, build the jerk from zero to a constant level and take it off similarly. Of course, you should do the same for the propulsion control.

    Yes, you can FEEL the improvement brought by third AND fourth derivative control. And, today it really is not costly to do.the computations on a pc and use numerical controlled tools to produce metal with the necessary cursve. (I experienced the feel doing control system design experiments (for a non-rail system) more than fifty years ago, when doing so was expensive.)

    • Alon Levy

      …how much difference in ride quality comes from fourth derivative control? Bosquet’s thesis has a page comparing different transition curves, and IIRC it claims there’s some vertical acceleration coming from using the clothoid without any fourth derivative control, but it also says that the clothoid is by far the most common transition curve…

      • Richard Mlynarik

        The problem with hideous US AREMA (ie design for freight that derails just infrequently enough not to cause bankruptcy when maintenance is done just well enough) approved turnouts isn’t third or fourth derivatives: it’s zeroth-order!

        The switch blades (point rails, whatever the terminology, the inner rails, the ones that move) are barely tapered to fit against the stock rails (the “outside” rails) with thick blunt ends. The stock rails (the fixed, outer ones) are barely cut away for any sort of close point-rail fit, and can even be kinked or bent (no curve radius, just bent) at the blade contact point. The effect — by design at the diverge point is a kink from straight to a diverge angle approximately half the final diverge angle at the frog.

        In contrast, a modern turnout designed for things aside from coal (and coal = death) trains in Wyoming has precisely-machined blades that taper to almost nothing and fit tightly against a tapered undercuts in the stock rails. Even in non-clothoid (sub-130kmh) turnouts with simple circular curves, the difference in ride quality is dramatic, as is that through the tight-tolerance and better-maintained frogs. (AREMA frogs are a SMASH SMASH … SMASH SMASH nightmare every time.)

        Just look at any photo of non-USian trackwork anywhere, or check out PDFs of turnout parts on the web, or ride any first world passenger train outside the US: one should (faintly!) hear turnouts passing under a train more than one should feel them.

        This isn’t rocket science. Or even third-grade primary school stuff. It’s nearly always nothing more than having a approximately differentiable (continous first derivative, keep those transition curves and clothoids and spirals and stuff for the grown-ups) track geometry.

        We don’t have that in the USA, we don’t want that in the USA, we ignore that in the USA, we design against that in the USA, USA USA. Because: USA.

        • Alon Levy

          Things I definitely shouldn’t be doing: figuring out exact geometry for a turnout that permits abrupt changes in cant deficiency up to 100 mm but also 125 mm/s clothoids.

          (I have clothoid geometry. At a design speed of 65 km/h, i.e. French 0.085 – the numbers in France are literally the inverse of the German numbers – the lateral displacement is 1 centimeter 7.5 meters from the start of the curve, which is within general tolerances on running high-speed track per Lindahl’s thesis, PDF-p. 72.)

  5. patrickjensen2013

    The assertion that European countries have curved frogs in turnouts and America has straight ones is a bit oversimplified. Most European countries do have turnouts with straight frogs, sometimes even offering turnouts with the same crossing angle in both straight- and curve-frogged flavors, the latter being reserved for situations when space is at a premium.

    A straight frog simplifies maintenance, since a frog with straight geometry is agnostic with regards to handedness, whereas a frog where one or both tracks are curved is not.

    Since maintainability and geometric flexibility are often at odds with each other, you can easily deduce whether planners or maintainers determine turnout policy. Straight frogs-only is one end of the spectrum for maximum maintainability, whereas custom-fitted curve turnouts galore suggests low emphasis on maintainability for better geometric flexibility. Finland fits somewhere in the middle, as turnouts with curved frogs are allowed, but curved turnouts are practically banned.

    • Richard Mlynarik

      Most European countries do have turnouts with straight frogs, sometimes even offering turnouts with the same crossing angle in both straight- and curve-frogged flavors, the latter being reserved for situations when space is at a premium.

      Space is always at a premium! The question is whether the operational needs driving always-constrained track layout allow for larger, more expensive and slower straight-frog turnouts, or whether the default curved-frog turnouts with higher diverging speed and for the same diverging angle are preferable.

      Let’s compare the very most standard (German, and increasingly though much of UIC-land, with individual national standard designs converging to varying degrees) simple turnouts (“Einfachweichen”):

      With 190m cuve radius (the tightest ever used now), a 190m radius (40kmh diverging speed) turnout with a curved frog attains a 1:7.5 “angle” (you can do the derivation from this arctangent-ish description if you like) and is 25.9m long.
      In contrast, a straight-frog 190m radius turnout attains shallower 1:9 end inclination, and is 27.1m long.

      At 300m radius for 40kmh diverge, EW-300-1:9 (curved frog) is 33.2m, while EW-300-1:14 (straight) is 37.8.
      500m 60kmh EW-500-1:12 (curved) 41.6m EW-500-1:14 (straight) 44.9m
      760m 80kmh EW-760-1:14 (curved) 54.2m EW-760-1:18.5 (straight) 52.9m

      Straight-frog turnouts end up mostly being used in yards (where EW-190-1:9 is a nice “snap-track” for making ladder tracks) and for some crossovers, in situations where close track spacing and higher desired diverging speed demand a longer section of straight track between the reverse curves of the crossover result in a longer crossing with shallower diverge. eg for 80kmh diverge, 500-1:12 curved-frog turnouts are used when inter-track spacing is over 4.2m, while longer 500-1:14 are needed when track spacing is closer. And then there are situations where a shallow diverging angle is geometrically desirable and a straight-frog turnout fits the bill.

      A straight frog simplifies maintenance, since a frog with straight geometry is agnostic with regards to handedness

      Olde Tyme American Railroader love claiming this stuff, but at any scale above a model train set, the overhead of spare parts handedness is zero, and the maintenance … well, let’s not even talk about North American ride quality, OK?

      … custom-fitted curve turnouts galore suggests low emphasis on maintainability for better geometric flexibility …

      There’s pretty much no project I’ve seen (randomly downloading project documents from either procurement sites or from public outreach information) anywhere in Central Europe (my sample is biased because my Romance language skills and rail vocabulary are worse than my Germanic, and I’m lost outside ISO-8859 alphabets) doesn’t involve curved frog turnouts, and very few end up avoiding turnouts where both routes are curved, despite that fact that standard-design un-bent turnouts are always preferred on geometrical, cost and maintenance grounds. It seems to end up that operational needs within limited available space (and space, again, is always limited, except on “green-field” new construction, and almost never near existing stations) nearly always justify the small cost penalties of curved turnouts.

      As just one example, check out the new (December 2015) Wien Hauptbahnhof: so curvy! Apart from EW-190-1:9 turnouts on yard tracks, I’m not sure there is a single straight-frog turnout in the entire new-build layout. And there are dozens of curved (IBW “Innenbogenweiche” where both tracks curve the same way and ABW where they curve away from each other) turnouts at every standard frog angle (1:9, 1:12, 1:14, 1:18.5) and “straight” radii anything from 300m to 600m (I think — the track layout I downloaded ages ago doesn’t correspond to as-built.)

      Finland may get to “practically ban” curved turnouts because Finland is big and sparsely-populated, but you might also be reading what they claim (something like “exception situations only”, which is what everybody says, with reason) but not seeing in real life just how common the exceptions are. (I’ve no appetite for Finnish Google Maps train station tourism right this moment — just guessing, based on Switzerland, Germany, France, Spain, Sweden, Norway, Britain, etc.)

      To only slightly exaggerate, there might be no crossovers between in Swiss mainline tracks if curved turnouts (not just curved frogs) were forbidden.

      And PS: straight-frog turnouts can’t be bent to form curved turnouts, at all. They’re inflexible!

      • Patrick Jensen

        As a track designer I’m intimately familiar with the dearth of space in turnout geometries, but my point wasn’t to defend existing turnout policy, but to explain it. If an infrastructure owner maintains extremely rigid regulations, they must have reasons that make sense to them, whether or not they make sense to us.

        In the case of Finland, you’d be in for a long tour, as only 8 out of some 9500 turnouts are located on curves. The ban on curved turnouts makes crossovers maddeningly difficult to design, but on the other hand it simplifies maintenance a lot.

  6. Marcel

    If upgrading this one switch would save four minutes on 90% of trains, surely just the operational savings would justify spending millions of dollars on it. Why not go for the cadillac of switches, the swingnose crossing?

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.