Minimum Curve Radius

To allow HSR operation at 125 mph, just how wide does a curve need to be? This is an elementary calculation of railway engineering, and is determined by safety and passenger comfort. Without going into details, speed can be increased in a curve by banking the track into the turn, like a turning airplane or a freeway exit ramp. The outside rail can be canted or super-elevated a maximum of 7 inches (178 mm) higher than the inside rail. Trains can go even a bit faster than the speed that balances this banking, causing passengers to feel a sideways push to the outside of the curve. The technical term for this is cant deficiency, and under current FRA regulations it is limited to 3 inches. Within those limits (7 inches cant + 3 inches cant deficiency), physics dictates the following curve radii:

Speed | Minimum Radius | (Recommended Radius) |

160 km/h (100 mph) | 1200 m (4000 ft) | 1800 m (5900 ft) |

200 km/h (125 mph) | 1900 m (6300 ft) | 2800 m (9200 ft) |

215 km/h (135 mph) | 2200 m (7300 ft) | 3200 m (10500 ft) |

The recommended radius is preferred, in the absence of trackside constraints such as houses and roads, to keep passengers comfortable and reduce wear and tear on the trains and the track. Wherever curve clearances are constrained (i.e. pretty much anywhere on the peninsula), the minimum radius becomes the quantity of interest.

The Cost of Slowing Down

Slowing down from 125 mph to take a curve, and accelerating back up to 125 mph costs several seconds of travel time, compared to an uninterrupted run at 125 mph. It's just a few seconds, but if every curve eats a few seconds out of the schedule, pretty soon HSR starts losing its "high speed." So exactly how many seconds are too many? Maybe the answer lies in the cost of a second. If you assume:

- HSR annual ridership will be 60M passengers / year (considerably less than the CHSRA's estimate)
- About one third of all HSR passenger trips will include the peninsula segment
- The average passenger (leisure and business) values their time at $12/hour (an approximate value based on time value studies)
- The cost of straightening a curve is amortized over 15 years of operation (the continuing benefit beyond 15 years is free)

Using a typical deceleration / acceleration rate of 0.5 m/s^2, the cost of temporarily slowing down for a typical curve from a cruise speed of 125 mph goes as the square of the speed difference:

Curve Speed (mph) | Time Penalty (s) | Delay Cost |

115 | 3 | $3M |

105 | 7 | $7M |

95 | 13 | $13M |

85 | 21 | $21M |

75 | 31 | $31M |

65 | 43 | $43M |

The square relationship means that it's not necessary to straighten curves all the way up to 125 mph. Arbitrarily setting our threshold of diminishing returns at 5 seconds of penalty, 110 mph curves can be considered "good enough" unless they can be straightened to 125 mph within the existing right of way, essentially for free. The reconstruction of any curve below 110 mph should be weighed against the dollar cost of time lost.

While this author is not versed in the fine art of estimating construction costs, we now have enough information to at least prioritize the worst curves where something should be done, short of deciding which ones are actually cost-effective to rebuild.

Existing Curves on the Peninsula

All major sub-125 mph curves in the Caltrain corridor from San Francisco to San Jose are shown in the chart below. Milepost is plotted along the bottom, and the curve's maximum speed is plotted on the vertical axis. The maximum speed is derived from the curve radius by assuming the aforementioned 10 inches of equivalent cant, except for reverse curves where different constraints apply. (Note, these speeds are not possible today; the maximum cant on Caltrain is 5 inches to accommodate freight trains, and the signaling system allows only 79 mph.)

Click for Larger View. First, there are quite a few curves that interfere with a 125 mph speed limit, as indicated by the blue dotted line.

- Several curves fall above the 110 mph "good enough" threshold, indicated by the green dotted line, although they should still be candidates for realignment if they are easy to fix. Recall these speeds are absolute maximum speeds, with 3 inches of cant deficiency (passenger discomfort).
- Some curves are very tight, but would be impossibly expensive to straighten; an example is the Sierra Point curve, which runs around the base of San Bruno mountain. There are other sharp curves in the San Francisco and San Jose terminal areas that fall into this category.
- One curve will be avoided entirely by HSR: the infamous CEMOF double reverse curve in San Jose, where the most expensive way to avoid a curve is planned, namely a tunnel.

The Top Ten Worst Curves

Here's a Google map, although it is much more accurate and instructive to view the KML file directly in Google Earth.

View Larger Map

#10 (Honorable Mention) CEMOF Double Reverse Curve - Milepost 46.5 - While the CHSRA plans a tunnel under this area, you really have to wonder what Caltrain was thinking when they dropped this turd on the approach to San Jose. That's why it gets an honorable mention.

#9 Belmont - San Carlos Reverse Curve - Milepost 22.4 - While we're adding another two tracks here, the incremental cost of straightening this curve to 125 mph ought to be near zero, since it can probably be done within the existing ROW. Savings: 10 seconds.

#8 San Antonio Curve (see curve detail map) - Milepost 34.3 - Great potential for straightening to 125 mph, again within the existing ROW. Savings: a couple of seconds, but it's free!

#7 Bowers Curve (see curve detail map) - Milepost 41.9 - Already OK for nearly 110 mph, but could use as much flattening as practical because of the proximity of Lawrence curve.

#6 Lawrence Curve (see curve detail map) - Milepost 40.6 - This shallow 100 mph curve can easily be straightened all the way up to 125 mph by purchasing a narrow strip of office parking lot (which Sunnyvale has plans to redevelop anyway). This is low-hanging fruit, well worth the 10 second savings.

#5 Hayward Park Curve (see curve detail map) - Milepost 18.8 - This curve was already straightened in the year 2000 by moving the rails by 20 ft. It might now support 95 mph. Would be better at 110 mph, saving about 10 seconds.

#4 Millbrae Curve (see curve detail map) - Milepost 13.9 - An unfortunate consequence of the last Quentin Kopp extravaganza, the BART airport extension. Challenge: BART tail tracks occupy the inside of this 90 mph curve. BART would have to give up one of three tail tracks to straighten for 100 - 110 mph operation. This ought to be feasible: two of the tail tracks were built in anticipation of a BART extension south of Millbrae, which no longer makes sense. Savings: about 15 seconds.

#3 Palo Alto Station - Milepost 30.1 - Already discussed in Focus on Palo Alto. While the existing curve radii are gentle, the problem at Palo Alto is a double reverse curve, which requires long spiral easements to reverse the curvature and prevents the speeds you might deduce from the radius alone. The southbound track is good for just under 90 mph. Challenge: reconfigure the Alma St. overpass; on the plus side, JPB already owns all the required land. Savings: about 25 seconds. A must-do, regardless of whether Palo Alto becomes an HSR station.

#2 Bayshore Curve (see curve detail map) - Milepost 5.1 - Just north of the Bayshore station at the mouth of Tunnel #4, this curve is a piece of cake to straighten to 125 mph, provided Bayshore station is redone. This will probably happen anyway to make room for the approaches to the planned new tunnel bores on each side of the existing tunnel. The new tunnel bores could even have curved ends. Savings: about 20 seconds. Cost of new platforms: $10M tops. Low hanging fruit, just waiting to be picked!

#1 Worst Curve: San Bruno Curve (see curve detail map) - Milepost 10.9 - previously discussed in the San Bruno article. This curve, currently 65 mph, should be straightened to 110 mph minimum. Savings: a whopping 40 seconds. Challenges: well-advanced plans by Caltrain for a new station, locking in the existing curvature; eminent domain for ~$5M worth of houses on the inside of the curve; six I-380 viaduct pillars would need to be moved. If this curve can be fixed even for $30-40M, JUST DO IT!

The total time saved from straightening these 10 curves is about 2 minutes, not including the savings from straightening the other 110 mph+ curves not listed here. These time savings add up to ~7% of the non-stop travel time between San Francisco and San Jose, expected to be around 30 minutes.

The CHSRA and its engineering contractors should not resign themselves to the existing curvature of the peninsula corridor. A rigorous study of curve remediation should be undertaken before the new track alignments are finalized.

Update - 02 Feb 09

It was brought to my attention that the CHSRA published in its considerable (if un-navigable) body of work a series of run simulations. This is what the pros do, instead of the back-of-the-envelope calculations detailed here. A sample San Jose to San Francisco run is detailed below. The train used in the simulation is a Siemens ICE 3. It does not stop in San Jose in this particular example. Total time from San Jose (running start) to San Francisco is a few seconds short of 30 minutes (1793 seconds, to be precise)

This simulation reveals a couple of interesting assumptions on the part of the CHSRA's analysts:

- Total cant is 12 inches (vs. 10 inches assumed in the calculations above) allowing 10% higher curve speeds. This is not outlandish: 12 inches is practiced today on the NEC.
- The Palo Alto and Bayshore curves are evidently straightened out, with a curved platform at Palo Alto
- None of the other bad curves appear to be straightened, as revealed by the three deep notches in the speed profile at Hayward Park, Millbrae and San Bruno.
- The train's throttle is used heavily, and the regenerative brake will certainly get a good workout. Whether this lead-footed driving style is realistic is open to discussion.

I'm fairly sure the highest superelevation used on US tracks is 5 inches. I'm fairly sure the highest cant deficiency for non-tilting trains is around 6 inches. This is higher than the standard 3 inches, but very reasonable in well-designed trains on well-maintained track (Amfleets on the NEC, in particular). The 3 inches was, at least according to this site, based on a single test in the 1950s on a train car with a soft suspension, and doesn't really mean anything at all. So in total you end up with 11 inches (rather than 10). You also have to account for reverse curves, as you have to leave room for the transition spirals.

ReplyDeleteYes, I accounted for the reverse curve easements at both Palo Alto and Belmont.

ReplyDeleteThe 7 inch figure came from Caltrain's own standards. Chapter 2 of the Design Criteria, section 2.0, claims that max actual superelevation on FRA Class 3 to Class 5 track is 7 inches. I assume class 6 and above would be similar. Is this incorrect?

I once took a TGV ride between Tours and Bordeaux, where the cant deficiency is 7 inches. That was definitely a spill-your-drink kind of ride.

Clem: thanks for the link, there's lots of interesting information in there. So the maximum actual superelevation allowed by the FRA is seven inches, but the actual maximum found on Caltrain is five inches. The three inch maximum unbalanced superelevation is indeed a standard found on just about every railroad in the US, except Amtrak, which got a waiver from the FRA to allow six inches on their tracks (and maybe seven inches for the tilting Acela). One limitation in choosing superelevation values for any given piece of Caltrain track is that you need to account for the fact that it may well be used by express trains running full speed as well as local trains accelerating from or braking to a stop, as would be the case, for example, in Palo Alto.

ReplyDeleteI also love the rationale for their choice of spiral: "Since Caltrain

adopted AutoCAD in the design of track alignment, the clothoid spiral shall be used."

The solution to CEMOF is simply to bulldoze it... while the people responsible for its design and placement (hello, City of San Jose council members and staff!) are inside.

ReplyDeleteRebuilding the damned thing elsewhere (even on the moon!) is

guaranteedto be cheaper than building even one foot of rail tunnel.Seriously.

This all assumes that

cost maximization isn't the goal of CHSRA's contractorswhich of course is a laughably-counter-factual (HSR tunnel to Los Banos! BART tunnels all over San Jose!) premise.You mentioned that the south end of the tunnel bores at Bayshore (tunnel #4) could be curved. The south end of the existing tunnel #4 is already curved -- would you happen to know if the existing curve inside the tunnel is broad enough for 125mph operation, or would they have to rebuild this part of the tunnel?

ReplyDeleteVery interesting discussion, by the way.

Interesting fact about the CEMOF by the way: I was looking at old documents about Caltrain (I think this may have been the early-90s electrification study), and not only do they mention the CEMOF there as a near-future project, but they also talk of it becoming the maintenance facility for the Capitol Corridor. That could have been more efficient than building two separate facilities, one in Oakland and another in San Jose.

ReplyDeleteReading the CAHSR Blog there is a post reply from someone that was at one on the scoping meetings that nothing is going to be done to the San Bruno Curve and its already factered into the time goals

ReplyDeleteNot good. At the scoping meeting I attended, the answer was more non-committal, a sort of "we'll see when we start the engineering".

ReplyDeleteIf they do indeed leave San Bruno curve as is, it would reveal a total lack of concern for end-to-end timings on the HSR system. They can't promise LA in 2:38 and then not even pick this low hanging fruit.

Of course, the cynical truth is that they have no financial incentive to fix these curves... think about it: what's in it for the contractors? Nothing but extra complication. Let's hope they have the professional spine to do it right.

At the Santa Clara HSR scoping meeting I specifically asked about the San Bruno curve. I was told that they have no plans to do any eminent domain to fix the curve. It will remain as a ~70mph curve, and that speed has already been taken into consideration in the state-wide time estimates.

ReplyDeleteThe feeling I got from talking with one of the contractors was they really want to use eminent domain in as few places as possible where they are absolutely required to. They might be playing it too conservatively to get the least amount of resistance possible form homeowners, but that's what their plan is.

Oh, also you can pull up the KML file directly in google maps if you don't feel like using google earth.

ReplyDeletelink

@Peter, my understanding is that eminent domain is not that difficult to carry out if a good case can be made for its public utility (certainly easy for San Bruno curve).

ReplyDeleteI think a bigger issue may be the $10 million already spent by Caltrain to design a new San Bruno station. Caltrain is understandably in love with their design and will resist any changes. It is politically difficult to toss everything and start over, given how much community wrangling went into it. Seizing the half dozen houses on the inside of the curve would just add fuel to the fire.

Nevertheless, it's the right thing to do.

Regarding KML: beware of viewing KML files with Google Maps. GM interprets (decimates and re-samples) the coordinates, resulting in errors of several meters. Google Earth is much more accurate in this respect, since what you put in the KML file is exactly what you get on the screen.

Oh, and not to be judgmental about these houses on Montgomery Ave in San Bruno, but they are surrounded by:

ReplyDelete- railroad tracks to the west

- a vacant lot to the south

- industrial buildings to the east

- a freeway viaduct to the north

In Realtor language, that would probably translate to

"good southern exposure, close to freeway and Caltrain"; however, by any objective measure, it's not the choicest real estate on the peninsula.Clem. Okay, your super engineering math is way beyond me, so I'm going to take a dummy's approach to train math.. and ask the dumb question...

ReplyDeleteI'm looking at your graph, and even though there are many curves where 125mph (or even 110) can be achieved, will it be? Because, in between almost every single 110or higher mph curve, seems to be a far sub 100 mph curve.

So in dummy-wize terms, isn't the train going: ZOOM, slam on the breaks, ZOOM, slam on the breaks, ZOOM slam on the breaks.... All the way from SF to SJ. The question really still is, in realistic operational terms, (worst case scenario here) if they don't fix any of the curves, does the train realistically maintain and Average of 100mph from SF to SJ?

Sorry, I'm sure its in here, but I just can't tell. Thanks for humoring me.

Clem: I'm not saying it makes any sense. Those dozen houses, an empty lot, and a warehouse probably aren't worth nearly enough to justify the current turn slowdown. I'm just repeating what I was told.

ReplyDeleteI'm thinking this would be a good point for an official EIR comment.

1 - once CA can once again find buyers for its bonds at affordable rates, the state legislature has the power to hold prop 1A funds hostage. It should use it to ensure the project-level EIR/EIS is done properly and replace one or more board members if expectations set are not met. Perhaps the Caltrans Division of Rail should be beefed up and charged with quality assurance on the details of the tender documents before they go out of the door. Paying a couple of seasoned HSR railroad engineers from SNCF, JR or RENFE costs nothing compared with change orders once ground is broken.

ReplyDelete- it might make sense for Caltrain and CHSRA to sit down with UPRR to negotiate an arrangement that would permit higher superelevations in the peninsula corridor to avoid a lot of the curve straightening costs. For example, if all UPRR does is collect freight cars and bring them to marshaling yard in Fremont, perhaps they could use a non-compliant Tier 4-compliant locomotive with a lower axle load for just that purpose. Not sure what a loco would cost, though.

Of course, FRA would have to sign off on all this.

- what is so sacred about CEMOF? The addition of HSR and BART warrants an integrated design for the entire Santa Clara-SJ Diridon section. The goal should be to keep HSR at grade. If need be, BART could be kept underground up to and including the maintenance yard. Soil from that site could be used for HSR embankments further up the peninsula and, a standard gauge yard for could be built on top of the one for BART. Air rights above that could be used for office space in the context of the TOD project for Santa Clara station.

The hardest thing here is to force all these separate planning bodies into a single room to hammer out an optimal solution for taxpayers.

- Caltrain may well have spent $10 million and a lot of time on plans for a new San Bruno station, but that was before HSR was approved. C'est la vie. The curve needs to be fixed. Most likely, that just means a minor displacement for the new station so the work would not have to be completely redone.

- I don't see why the Sierra Point curve could not be defused with a viaduct across the little triangular lagoon and cutting across the industrial site via eminent domain. Both Lagoon and Tunnel Road would need an overpass.

Fantastic post, Clem.

ReplyDeleteRegarding maximum cant deficiency:

The standard in the US is 3" (as Clem correctly notes). However, these waivers (and possibly more) have been issued:

New Haven to Boston (NEC): 5" for Amfleet equipment pulled by AEM-7, HHP-8, or F-40

Amtrak Cascades: 6" for Talgo equipment

Washington to Boston (NEC): 7" for Acela Express equipment

I think Amtrak wanted to get 9" for Acela Express, but I don't know if they ever did. They may be limited by the infamous "trainset too wide because of ADA compliant toilet" issue or the infamous "trainset weighs a gazillion tons" issue.

Regardless,

non-tilting Amfleet equipment currently runs at 5" underbalance (cant deficiency)between New Haven and Boston at a max speed of 125 mph. I don't see why a similar waiver couldn't apply to HSR between SJ and SF.If I'm reading the formulas correctly, this would increase max speeds through any given curve by just under 10%. I don't think this would change any of Clem's recommendations, but it does make things a little easier.

BTW, I believe the Millbrae curve is supposed to be at MP 13.9, not MP 23.9.

Anon - Clem's post demonstrates that virtually all curves except Sierra Pt and San Bruno can be realigned for 100+ mph within existing public property. There is no realistic scenario in which a lot of these curves won't be improved (remember, they are already rebuilding/installing most of the track anyway).

ReplyDeleteEven for a 100 mph curve, it will take the train about 0.3 miles to go from 100 to 125 (or vice versa), using Clem's figure of 0.5 m/s^2. That is about 8 times the distance, for example, between the Bayshore Curve and the Sierra Point Curve.

So no, the train will not feel like "ZOOM, slam on brakes, ZOOM, slam on brakes." The Peninsula corridor is actually really straight compared to, say, Amtrak's Shore Line between New Haven and Boston. And for the few sub-110 mph curves that remain, remember that braking doesn't waste much energy on HSR because the train just turns the traction motors in reverse and generates energy that gets fed back into the wires (like a hybrid car, but with much, much more generative capacity).

Rafael: I like your idea of getting Caltrans DOR in on this. They at least have some experience with building rail projects, albeit fairly small and isolated ones mostly done by other railroads. As for Sierra Point, the limitation is not the lagoon, it's the configuration of the mountain and the 101 overpass. Any modifications there will have to involve massive earthworks and the reconstruction of the highway. Given that not too far north you have Bayshore and the tunnels, where I doubt you'd be going faster than 80 anyway, I just don't think it's worth it, at least compared to all the other much easier realignments that can be done elsewhere.

ReplyDeleteOne question Clem left unanswered is whether the 31 minute SF-SJ express is feasible.

ReplyDeleteUnder the assumption that Sierra Pt is 85 mph (5" cant deficiency), San Bruno is 75 mph (5" cant deficiency), and all the "low-hanging fruit" curves are improved as Clem suggests, here is what I see:

Transbay to Diridon = 48.5 miles (current route is 47.5 miles)

Transbay to curve at (current) MP 0.7 = 1.7 miles @ 30 mph = 3.4 minutes

MP 0.7 to MP 1.2 @ 50 mph = 0.6 minutes

MP 1.2 to MP 2 @ 90 mph (acceleration phase) = 0.5 minutes

MP 2 to MP 45 @ 125 mph (cruise phase) = 21 minutes

MP 45 to MP 47 @ 60 mph (deceleration phase) = 2 minutes

MP 47 to MP 47.5 @ 30 mph (terminal approach) = 1 minute

+ 31 second penalty for San Bruno curve and 21 second penalty for Sierra Point curve

Total running time = 3.4 + 0.6 + 0.5 + 21 + 2 + 1 + 0.55 + 0.35 = 29.4 minutes.

Add in a couple minutes of dwell time at Diradon Station, and it looks like 31 minutes is achievable even without the removal for the Sierra Point and San Bruno curves (though it could still be worth it to remove the San Bruno curve)

It's actually less than 47 miles from end of track at 4th and Townsend to San Jose-- 46.75 plus or minus 0.1 would be a fair guess. UP changed the milepost at San Jose.

ReplyDeleteThe Bayshore curve used to be 1 degree, but when Caltrain four-tracked they inserted a short straight for the switches-- so as I recall the curve is now 1 deg 30 min. Similarly, the Lawrence curve is a bit sharper than it was in SP days.

I suppose they'll remove that newish offramp overpass south of the 101 overpass at Sierra Point? They can add two tracks outside the piers, but they'd be kinky.

I just updated the post with a CHSRA run time simulation, which reveals many of the assumptions made about curves on the peninsula. Hat tip to Richard M for bringing my attention to this.

ReplyDeleteSo the run time is the inverse of the area under the curve? Then is is a matter of finding the ratio the area with and without the reductions to find the time and cost savings.

ReplyDeleteAhh, hold the presses. I hadn't noticed the little "32.52 kW/tonne"... their simulated train is

ReplyDeleteridiculouslyover-powered.German ICE 3 - 19.6 kW/tonne

French AGV - 22 kW/tonne

Japanese Series 700 - 20.4 kW/tonne

CHSRA "magic train" - 32.5 kW/tonne

@James - the run time is in the CHSRA document. This is a speed vs. distance plot, so the area under the curve doesn't tell you much about time savings.

I object in the strongest terms to the term "Prius brake"! Electric traction has had regenerative braking for many decades, with the recovered energy being returned back to the grid. If anything, the brake on the Prius should be called the "trolley brake" or something.

ReplyDeleteDoing some back-of-the-envelope calculations using your graph (based on the official graph) here are some numbers I calculated. My calculations are from the start of breaking, through the turn, and to the end of acceleration. I don't take 2-turn combos into account, so my seconds lost are vs an ideal running of 200kph the entire distance (some of which might be part of another turn).

ReplyDeleteHayward park: 178kph avg, 3.2km, 7 sec lost

Millbare: 175kph avg, 3.6km, 9 sec lost

San Bruno: 153kph avg, 6.6km, 35 sec lost

Sierra Point: 161kph avg, 4.9km, 21 sec lost

Can someone double check my math?

Something is weird about the "super-powered" ICE-3. It sure doesn't accelerate like it has 33 kW/t.

ReplyDeleteFor example, in the run time simulation it takes 51 km for the ICE-3 to accelerate from 0 to 330 kph on level track. In real life (1989-1990 test runs), a souped-up TGV accelerated from 0 to 300 kph in 11 km, 0 to 350 kph in 16 km, 0 to 400 kph in 22 km, and 0 to 440 kph in 33 km (and this was on a very slight uphill grade). The TGV did have 43 kW/t, but a mere 30% better power-to-weight ratio shouldn't generate this kind of discrepancy...the TGV made it to 350 kph in less than one-third the distance that it takes the supposedly super-powered ICE-3 to get to 330 kph!

At 32.52 kW/tonne, they're assuming an almost 21 MW of total power. That train is going to need to raise all four pans in order not to fry the pan and wire. I'm not sure how many such beasts the Caltrain electrification system will be able to support simultaneously. I'm guessing somewhere between zero and one per substation.

ReplyDeleteAs for the souped-up TGV, what they usually end up with is using higher supply voltage and generally pushing the traction package to its limits, and removing as many intermediate cars as possible. In fact, the latest test run had two locomotives pulling what amounted to a three-car MU with half its axles powered. I don't think this sort of thing is really representative of anything at all.

Also note that it takes this mythical ICE-3 over 1.5 km to get from 0 to 160 kph. I've been on Amtrak trains with only 13-14 kW/t of power that have gone from 0 to 160 kph in roughly that distance (2 km at most). There is no way that these simulations actually correspond to 33 kW/t of power.

ReplyDeleteWe can use physics to calculate an upper bound on how powerful the simulated CHSRA train could be.

ReplyDeleteWe know that a TGV with 43 kW/tonne can reach 400 km/h in 22 km (this was recorded during the 1989-1990 speed trials before they opened LGV Atlantique). The TGV was 300 tonnes with 12,800 kW of power.

We also know that the simulated CHSRA train reaches ~295 km/h in 22 km. The CHSRA train is 642 tonnes with unknown power (power is claimed at 20,800 kW, but this is clearly wrong).

Assume that there is no friction/drag (we'll return to this later).

Energy = Power*Time.

Kinetic Energy = 0.5*Mass*Velocity^2.

Power = Force*Distance.

Force = Mass*Acceleration.

Acceleration = dVelocity/dTime.

Velocity = dDistance/dTime.

If Acceleration is roughly constant (it's not, but for our calculations it turns out not to make a big difference), and we have:

Velocity = Acceleration*Time

Distance = 0.5*Acceleration*Time^2

Plugging in known quantities, we have:

Distance(Time') = 22 km = 0.5*Acceleration*Time'^2

Velocity(Time') = 400 km/h = Acceleration*Time'

So Acceleration = 400 km/h / Time'.

And 22 km = 0.5*(400 km/h / Time')*Time'^2 = 200 km/h*Time'.

So 0.11 hours = Time'.

Thus we estimate that it takes 0.110 hours for the TGV to reach 400 km/h. In real life, where acceleration declines over time (as speed increases), it should take less time. And in fact, it took the actual TGV 0.104 hours to reach 400 km/h, so we're only off by 5%.

For the simulated CHSRA train, the same calculations imply 22 = 147.5 km/h*Time', so 0.149 hours = Time'. This is assuming acceleration is constant. I can't know exactly how long a CHSRA train would take to reach 295 km/h, but the estimated vs. real TGV numbers above suggest that it definitely shouldn't be less than 0.130 hours. So I'll conservatively assume 0.130 hours.

Finally, we can do comparisons. The TGV must expend a minimum of 0.5*(300 tonnes)*(400 km/h)^2 = 24 million units of energy in 0.104 hours. It thus must have at least 230 million units of power. The CHSRA train must expand a minimum of 0.5*642*295^2 = 28 million units of energy in 0.130 hours. It thus must have at least 215 million units of power.

In a dragless world, the CHRSA train is thus using no more than 93% of the TGV's power, or 0.93*12,800 = 11,900 kW. The actual power/weight ratio of the simulated CHSRA thus does not exceed 18.5 kW/tonne, which is actually less than the current ICE-3's power/weight ratio of 19.3 kW/tonne.

Of course, all of these calculations were done without taking into account drag. However, accounting for drag will only

reducethe amount of power than the CHSRA train needs relative to the TGV. Both trains have similar drag coefficients and similar frontal areas, but the TGV is running much faster than the CHSRA train (400 km/h at exit vs 295 km/h at exit). Since drag is proportional to the square of velocity, the TGV is overcoming roughly 83% more drag than the CHSRA train when it crosses the 22 km mark. Actual energy expenditures are equal to kinetic energy plus energy dissipated as drag. Thus the ratio of actual TGV energy expenditures to actual CHSRA energy expenditures is even higher than I calculated above, which implies that the ratio of the TGV's power to the CHSRA's power must also be higher than I calculated above.Bottom line, this CHSRA train is performing like an HSR train that has 17-18 kW/tonne of power. These run times are achievable with a conventional, off-the-shelf HSR trainset. You definitely don't need 33 kW/tonne to achieve them.

Even more evidence? Siemens claims 0-200 km/h in 148 seconds for the Velaro (i.e. ICE-3). The simulated CHSRA train reaches 200 km/h in roughly 4.7 km, which implies a few seconds under 169 seconds (169 seconds is calculated assuming that acceleration is constant). The performance profile sure looks a lot like a conventional ICE-3.

Even more Siemens data:

ReplyDelete0-100 km/h in 0.6 km

0-200 km/h in 3.9 km

0-300 km/h in 17.9 km

Blowing up the PDF and measuring with a very precise ruler, the simulated CHSRA train does:

0-100 km/h in 0.7 km

0-200 km/h in 4.5 km

0-300 km/h in 23.8 km

So the assumed performance is actually slightly

inferiorto an off-the-shelf ICE-3.Clem, it might be helpful to others to update the post, noting that although the run simulation claims 33 kW/t, the actual acceleration curve that they use is totally consistent with current conventional HSR equipment.

Thanks to mike and Richard M for sharpening their pencils and doing the extra math that I didn't bother with.

ReplyDeleteIt turns out that the acceleration profiles shown in the CHSRA simulation are indeed consistent with the capabilities of a standard ICE-3 high speed train, powered at just under 20 kW/tonne.

We armchair experts now all agree: the 32.5 kW/tonne label on the CHSRA plot is evidently some kind of minor typographical error, and is not indicative of unrealistic train performance assumptions, as I originally assumed.

Post has been expunged accordingly; I'm glad, because this was about curves, not timetables.

Looking closer at the Hayward Park curve, I suspect the problem is not the curvature of the track itself but rather the fact that it's a reverse curve (chicane). I believe this is problematic because it limits the length of the transition curve (i.e. the section between the tangent track and the circular curve).

ReplyDeleteThe bottom line is that it may not be possible to fix this curve without substantial eminent domain takings to the east of the ROW, which may be why the run simulations assume that the curve is limited to 95 mph.

If you go to Hayward Park and take a look, I think you'll agree the spiral lengths aren't a problem there.

ReplyDelete(In SP days Hayward Park was a 2-degree curve; as I recall it's now 1 deg 30 min.)

Did some calculation for San Bruno--

Assuming the total angle of the curve is 23.64 degrees, and assuming the centerline of the track is now a 3 deg 08 min curve (chord definition) with 375-ft spirals...

Then if we want a 1-degree curve with 500-ft spirals we need to shift the centerline inward by 83.1 ft at the midpoint of the curve.

(That total angle is from Google Earth-- I'm guessing it's correct within 0.1 degree or so, but can't promise.)

Next question: how much of a problem are those I-380 pillars. Too lazy to work on that yet.

If you zoom way into the map (or better yet, open the KML in Google Earth), you'll see the six I-380 pillars that cause problems at San Bruno, as well as the outline of the 4-track right of way for a 110 mph (@ 10 inch equivalent cant) curve.

ReplyDeleteOn further reflection-- if you're trying to do 110 mph around a 1-degree curve you might want 800-ft spirals, or thereabouts, and maybe that would be a problem at Hayward Park for a four-track line that can't swing as wide across the right-of-way as a two-track line. At San Bruno, a 1-deg curve with 800-ft spirals is 86.0 ft inside the assumed present centerline.

ReplyDeleteCan we all

ReplyDeletepleaseleave behind the 1800s-era degrees (and feet)?What part of "radius" is so terribly difficult for Olde Tyme PTG/HBTB/PCJPB/PBQD pretend-engineers (and their CAD systems) to understand, exactly?

If we simply got rid of

everybodyconnected to this project whose first language is English (or, shudder, American) we'd be several decades, at least, ahead of where we find ourselves.One "degree" = 1746.4m

Deal with it.

800 foot spirals sound a bit too long. If the runout is 50 mm per second, the train velocity 125 mph (55 m/s), the superelevation is 175 mm, then you only need 650 foot spirals. (sorry for the mixed units)

ReplyDeleteOnly half of that spiral extends beyond the ends of the equivalent circular arc. In other words, you can reverse a curve in a run length of about 750 feet. It's a bit tight, but it just might barely fit at Hayward Park.

The lateral offset due to the spiral is minimal (less than a meter) and does not significantly affect how much land is needed inside the curve.

ReplyDeleteI think Amtrak wanted to get 9" for Acela Express, but I don't know if they ever did. They may be limited by the infamous "trainset too wide because of ADA compliant toilet" issue or the infamous "trainset weighs a gazillion tons" issue.The Northeast Corridor has high level platforms at many stations, so I would hazard a guess that Acela trains are the same width as everything else on the NEC.

From what I've read about tilt restrictions it has something to do with the local track being too close to the express track. Wouldn't want to blow the windows out of the Metro North local as Acela speeds through Cos Cob would you? Or shear the top off both...

Non tilt equipment shares the track, Regional and commuter and in places freight, so there is a limit to the amount of super elevation you can add to the track even if your tilt equipment can take advantage of it.

What is up with that profile from the CHSRA? The parts I'm looking at are:

ReplyDelete(1) The south end. What causes the immense speed restrictions there? Not curves, surely!

(2) The north end. Curves? Tunnels?

All curves. The gray bars show the curve-constrained speed limit, which (if lower than track speed) the train must slow down to meet.

ReplyDeleteThe curve in San Bruno is 60mph, not 65mph, just a note.

ReplyDeleteRegarding the San Bruno curve, is something already being built on the empty lot (ex lumberyard)?

ReplyDelete