In the previous instalment, we did exact calculations for a dummy line down a 650-foot entry straight, a 180-degree left-hander, and a 650-foot exit chute. Cornering radii vary from 150 feet to 200 feet, and the track is 100 feet wide all the way around. This dummy line carries constant speed around the entire left-hander. We did those calculations to provide reference times to compare against this month’s more sophisticated calculations, in which we unwind the steering wheel and accelerate at the same time. The baseline times for the dummy line over the whole course, as a function of cornering radius, are in the second-to-last column of the following table:

From this point on, we need only look at the last column. It’s after the apex and down the exit chute where we look for improvement; we actually drive the dummy line up to the apex. Many readers will be screaming that we could try to get on the gas before the apex for even more improvement. Others will be screaming “trail brake!,” that is, ease off the brakes at the same time as winding the steering wheel at turn in (thanks to reader Marc Sibilia for pointing this out to me). We leave those refinements to later articles.

The approach in this article is to find a line by building it up, step-by-step, honouring the traction circle and the sides of the track. This is one of the techniques we can use in computer simulations, so we get to kill two birds with one stone: previewing simulation and analysing a particular driving line. For convenience, we need a Cartesian coordinate system, that is, a square grid. Let’s turn the track around 180 degrees for this purpose, and put the centre of the coordinate system at the centre of the corner. Since the inside edge of the track and the outside edge of the track are concentric semicircles, there is only one identifiable centre of the corner.

We’ll work by measuring the position and heading of the centroid of the car with respect to this new coordinate system. We have a goal of arriving at the point x = 200, y = 650, measured in feet, in the least possible time, with a heading of as close to 90 degrees as we can get it, that is, heading straight down the track. We start at the apex, which measures from x = r₀ sin  , y = r₁ cos  . The following sketch illustrates:

I must note, at this point, if you haven’t already noticed, this instalment of The Physics of Racing is going to be more concentrated and intense than previous instalments. I’m just going to blurt out facts without the usual explanations and walkthroughs. The reasons are (1) that we have a lot to get through in a little space and (2) that we assume that if you’ve been following the series this far, you’ve got the fortitude to work through it. So, let’s get it on!

The initial heading is tangent to the inner edge of the track, that is, perpendicular to the line from the centre of the track’s corner to the apex. Therefore, it has the angle up from the horizontal x axis. We know the starting speed, v0, so we know its components in the x direction and in the y direction: v0_x =  v0 cos  , v0_y = v0 sin  .

We perform the entire manoeuvre whilst never exceeding the limits of the traction circle. We set those limits as 1g cornering and braking and 0.5g accelerating, with smooth transitions all way around, as in the following sketch (the horizontal cap shows a way of accounting for engine limitations with non-smooth transitions, which will allow us to accelerate harder with the wheel still turned but probably scare us in the seat. Also, we note that 0.5g is a plausible, if only approximate, number for acceleration. We leave it to the reader to show that 0.5g in the quarter mile results in a realistic 13-second elapsed time, if at an unrealistic speed of 150 mph):

In each step of the calculation, we keep track of the following information:

• the time, t
• the current position, x(t), y(t), which we check to make sure we’re still on the track (x < 200) and to see whether we’re done (y 650)
• the current velocity, v_x(t), v_y(t), which we use to update the current position: , and likewise for y
• the tangential and radial acceleration, a_t(t), a_r(t), that is, tangential and radial to the bit of racing line at each instant (the instantaneous line), which we check to make sure that we’re not cornering over the limit and that we’re not exceeding the capacity of the engine, i.e., that is inside the traction envelope
• the acceleration in the x and y directions, a_x(t), a_y(t), which we use to update the current velocity: , and likewise for v_y

We drive the whole simulation by feeding on the throttle linearly with time over a time span called k and by simultaneously increasing the instantaneous radius of the driving line over a potentially different time span called k_unwind. Feeding on the throttle allows us to increase the tangential acceleration, a_t at each time step, and unwinding allows us to decrease the radial acceleration, a_r so we can stay within the traction circle. Since we’ll still have centripetal traction available after the throttle is buried full on, we ought to be able to unwind more slowly, enabling us to stay on the track, but use it all up. In other words, we ought to look for solutions wherein k_unwind is larger than k, perhaps by twice.

Let’s look at the first few rows of this simulation in a spreadsheet and delve into the formulas more deeply:

[column 1]: increments by each row; we actually computed with = 0.05 sec and display here every fourth actual row; this is an independent column, meaning that it does not depend on data from any other column.

[column 2]: tangential acceleration,

,

accounting for squeezing on the throttle up to g / 2; depends only on column 1.

[column 3]: maximal radial acceleration,

,

accounting for the traction circle; more precisely, for the upper half of the circle treated as a flattened (oblate) ellipse with height g / 2; depends only on column 2.

[column 4]: radial

,

accounting for unwinding the steering wheel; in steps from the inner parentheses outwards: g(1 - t /  k_unwind) slowly decreases from g as time increases from 0, but, it is never allowed to exceed v² / r, by the min expression, as mandated by the traction circle, and then, never allowed to be negative, by the max expression, because we don’t want to start turning back toward the entry straight; depends on columns 1 and 3.

[column 5]:

;

just for amusement, it’s interesting to calculate the instantaneous radius of a circle we could be driving if we were not accelerating tangentially; depends on columns 4 and 12, but no other columns depend on this.

[column 6]:

,

this just selects out the x components of both the radial and tangential accelerations, but makes sure that we never turn the wheel so much that we start going to the left. Note that the radial acceleration always tries to pull the car to the left, hence the minus sign (centripetal: see part 4 of The Physics of Racing); depends on columns 2, 4, 10, 11, and 12.

[column 7]:

,

selecting the y components, this time always pointing down the track, the way we want to go; depends on columns 2, 4, 10, 11, and 12.

[column 8]:

,

just update the x coordinate by the velocity from the prior time step; depends on columns 8 (the prior row of itself) and 10.

[column 9]:

,

do likewise for the y coordinate; depends on columns 9 (prior row) and 11.

[column 10]:

,

for updating the x component of the velocity (but don’t let it go negative, checking yet again, and, yes, this is a hack); depends on columns 10 (prior row) and 6.

[column 11]:

,

likewise for the y coordinate of the velocity; depends on columns 11 and 7.

[column 12]: finally,

,

depends on columns 10 and 11.

I’ve packed all this in an Excel spreadsheet. The spreadsheet should be in the download package for readers who acquired this document electronically.

Enough talk! Let’s drive! Driving means playing with the values of r, k, and k_unwind, and possibly even , to find the lowest overall time at which columns 8 and 9 show 200 or less and 650 or more, respectively. In general, “playing with” should be a sophisticated process involving hill climbing, genetic search, simulated annealing, and other fancy strategies for finding the very best values. In a computer simulation, we’d do that. However, we can do a reasonable job, for the sake of demonstration, by just tweaking the numbers by hand in the spreadsheet.

I have to admit that as I did so, I got kinaesthetic feelings as if I where actually driving. When I ‘ran off the track,’ that is, picked numbers that gave me x > 200, I gritted my teeth and blushed. When I was still unwinding at the end, I got that panicky feeling of understeer, knowing that I wasn’t going to stay on after the end of the segment, and so on.

The best values I found by hand are shown in the following table at r = 167.5, k = 3.25, and k_unwind = 7.22. That means that we take 3.25 seconds to bury the gas and 7.22 seconds to unwind the wheel. There are solutions with lower segment times, but, since we’re still unwinding long after the segment is done, I reject these solutions as assuming too much about what’s going on after our segment is done. With more track to work with, however, we can find lots more time. In fact, it’s a slightly surprising fact that by taking 9 seconds to unwind at r = 167.5, k = 3.25, we lose hardly any time and stay 15 feet inside the outer edge. There is quite a bit of territory to investigate even in this simple model.

Since the best dummy time, with the widest possible circle, is 16.760, and the best time I found here was 16.466, the improvement by unwinding and accelerating simultaneously is 0.294 seconds. This is very significant. If the exit straight were longer, the improvement would be even more dramatic since it would continue to accumulate time down the straight.

Note that this does not involve changing the entry to the corner other than by slowing down! There is no trail braking or lifting-while-turning or other risk-taking going on at corner entry. There is a very important driving lesson, here: to go faster, it is not necessary to take risks on corner entry. It is, in fact, both safer and faster just to slow down on the entry. The improved exit will follow naturally from the combination of looking far ahead and of being smooth. And that’s not even fair!

There is no guarantee that this is the best possible improvement in the model. I found these numbers by ’seat-of-the-pants’ tweaking. A more systematic or algorithmic search would very likely find better ones. In other words, I was able to find almost three tenths by just driving a better line without trying very hard at all. There is another driving lesson, here: just driving a better line gives better times time without changing the driver’s margin for error, that is, without getting deeper into the g limits of the machine.

For the future, we can start taking more risks to get even more improvement. We can risk accelerating before the apex and we can risk deeper entry by trail braking, that is, easing off the brake and winding up the steering wheel at the same time. These manoeuvres do entail more driver risk since they are new opportunities for loss of car control.

Erratum: in part 17, I wrote “By driving a line just one foot larger than the minimum, one is able to apex more than fifteen degrees later!”. I should have written “…fifteen degrees earlier!” The point was that the tightest line does not apex until the geometric exit of the corner, and that’s way too late. The slip-of-the-pen occurred because one is so accustomed to talking about late apexing as preferable.

By Brian Beckman

You may remember way back in part 5 that we did some simple calculations by hand to show that the classic racing line through a 90-degree right-hander is better than the either the line that hugs the inside or the line that hugs the outside of the corner. ‘Better’ means ‘has lowest time.’ The ‘classic racing line’ was, under the assumptions of that article, the widest possible inscribed line.

In this and the next instalment of The Physics of Racing, we raise the bar. Not only do we calculate the times for all lines through a corner, but we show a new kind of analysis for the exit, accounting for simultaneously accelerating and unwinding the steering wheel after the apex. This kind of analysis requires us to search for the lowest time because we cannot calculate it directly. We apply the approximation of the traction circle-subject of part 7-to stay within the capabilities of the car. We also model a more complex segment than in part 5, including an all-important exit chute where we take advantage of improved corner-exit speed. This style of analysis applies directly to computer simulation that we now have in progress in other continuing threads of The Physics of Racing.

The whole point of this analysis is to back up the old mantra: “slow-in, fast-out.” We will find that the quickest way through the whole segment does not include the fastest line around the corner. Rather, we get the lowest overall time by cornering more slowly so we can get back on the gas earlier. It’s always tempting to corner a little faster, but it frequently does not pay off in the context of the rest of the track.

This analysis is sufficiently long that it will take two instalments of this series. In this, the first instalment, we do exact calculations on a dummy line, which is the actual line we will drive up to the apex, but just a reference line after the apex. In the next instalment, we improve on the dummy line by accelerating and unwinding, predicting the times for a line we would actually drive, but entailing some small inexactitude.

Let’s first describe the track segment. Imagine an entry straight of 650 feet, connected to a 180-degree left-hander with outer radius 200 feet and inner radius 100 feet, connected to an exit chute of 650 feet. In the following sketch, we show the segment twice with different lines. The line on the left contains the widest possible inscribed cornering radius, and therefore the greatest possible cornering speed. The sketch on the right shows the line with the lowest overall time. Although its cornering speed is slower than in the line on the left, it includes a lengthy acceleration and unwinding phase on exit that more than makes up for it.

Line with Fastest Cornering Speed	Line with Lowest Overall Time

Note that both lines begin on the extreme right-hand side of the entry straight. Such will be a feature of every corner we analyse. Lines that begin elsewhere across the entry straight may be valid in scenarios like passing. However, we focus here on lines that are more obvious candidates for lowest times. Also, throughout, we ignore the width of the car, working with the ‘bicycle line’. If we were including the width, w, of the car, we would get the same final results on a track with outer radius of 200 + w / 2 feet and inner radius of 100 - w / 2 feet.

First, we compute exact times where we can on the course: the entry straight, the braking zone, and the corner up to the apex. To have a concrete baseline for comparison, we also do a ’suboptimal’ exit computation-the dummy line-that includes completing the corner without unwinding and then running down the exit chute dead straight somewhere in the middle of the track. In the next instalment of The Physics of Racing, we compare the dummy line to the more sophisticated exit that includes simultaneously accelerating and unwinding to use up the entire width of the track in the exit chute.

Let us enter the segment in the right-hand chute at 100 mph = 146.667 fps (feet per second). We want the total times for a number of different cornering radii between two extremes. The largest extreme is a radius of 200 feet, which is the same as the radius of the outer margin of the track. It should be obvious that it is not possible to drive a circle with a radius greater than 200 feet and still stay on the track. This extreme is depicted in the following sketch:

We take the opportunity, here, to define a number of parameters that will serve throughout. First, let us call the radius of the outer edge of the track r1; this is obviously 200 feet, but, by giving it a symbolic name, we retain the option of changing its numeric value some other time. Likewise, let’s call the radius of the inner circle r0, now 100 feet. Let’s use the symbol r to denote the radius of the inscribed circle we intend to drive. In the extreme case of the widest possible line, r is the same as r1, namely, 200 feet. In the other extreme case, that of the tightest inscribed circle, r is 150 feet, as shown in the following sketch:

We’re now ready to discuss the two remaining parameters you may have noticed: h and (Greek letter alpha). Consider the following figure illustrating the general case:

h indicates the point where we must be done with braking. More precisely, h is the distance of the turn-in point below the geometric start of the corner. Its value, by inspection, is (r - r0) cos . is the angle past the geometric top where the inscribed circle-the driving line-apexes the inner edge of the track. We see two values for the horizontal distance between the centre of the inscribed circle and the centre of the inner edge, and those values are (r - r0) sin and r1 - r. Their equality allows us to solve for :

The following table shows numeric values of h and for a number of inscribed radii (Note that if we varied r0 and r1 we would have a much larger ‘book’ of values to show. For now, we’ll just vary r.):

Inscribed Corner Radius (ft) (deg) h (ft) 150 90.00 0.00 151 73.90 14.14 152 67.38 20.00 153 62.47 24.49 154 58.41 28.28 155 54.90 31.62 160 41.81 44.72 165 32.58 54.77 170 25.38 63.25 175 19.47 70.71 180 14.48 77.46 185 10.16 83.67 190 6.38 89.44 195 3.02 94.87 200 0.00 100.00

There are a couple of interesting things to notice about these numbers. First, they match up with the visually obvious values of h = 0, = 90 and h = 100, = 0 when r = 150, r = 200 respectively. This is a good check that we haven’t made a mistake. Secondly, changes very rapidly with corner radius, and this fact has major ramifications on driving line. By driving a line just one foot larger than the minimum, one is able to apex more than fifteen degrees later!

With these data, we’re now equipped to compute all the times up to the apex and beyond. First, let’s compute the speed in the corner by assuming that our car can corner at 1g = 32.1 ft / s² = v² / r, giving us . We express all speeds in miles per hour, but other lengths in feet. We won’t take the time and space to write out all the conversions explicitly, but just remind ourselves once and for all that there are 22 feet per second for every 15 miles per hour.

Now that we have the maximum cornering speed, we can compute how much braking distance we need to get down to that speed from 100 mph. Let’s assume that our car can brake at 1g also. We know that braking causes us to lose a little velocity for each little increment of time. Precisely, dv / dt = g. However, we need to understand how the velocity changes with distance, not with time. Recall that dx / dt = v, dt = dx / v, so we get dx = vdv / g. Those who remember differential and integral calculus will immediately see that is the required formula for braking distance. In any event, the braking distance goes as the square of the speed, that is, like the kinetic energy, and that’s intuitive. However, there’s a factor of two in the numerator that’s easy to miss (the origin of this factor is in the calculus, where we compute limit expressions like ).

We next subtract the braking distance from the entry straight, and also subtract h, to give us the distance in which we can go at 100 mph, top speed, before the braking zone.

Now, we need the time spent braking, and that’s easy: . All the other times are easy to compute, so here are the times for a variety of cornering lines up to the apices (or apexes for those who aren’t Latin majors):

At first glance, it appears that the widest line is a huge winner, but we must realize that these times include only driving up to the apex, and that is far earlier on the widest line, where = 0. Suppose we continued driving all the way around the corner at constant speed and then accelerated out the exit chute at 0.5g? This is the dummy line. We won’t really drive this line after the apex, but discuss it nonetheless to provide a reference time. It’s very easy to compute and provides a foundational intuition for the more advanced exit computation to follow in the next instalment:

So, we see that, driving the dummy line, the widest line yields the slowest time from the entrance up through the complete semicircle, but the quickest overall time when the exit chute is included. The widest line has lower (better) times than the tightest line in

• the entry straight by about half a second, because h is large and the entry straight is shorter for wider circles
• in the braking zone by about three tenths because the cornering speed is higher and less braking is needed
• and in the exit chute by almost a second, again because is h large and the exit chute is thereby shorter

The widest line has higher (worse) times by about a second in the circle itself because the wider circle is also longer. When the balances are all added up, the widest line is about eight tenths quicker than the tightest line, but it’s all because of the effects of the corner on the straights before and after.

Recall once again that the dummy line is not a line we would actually drive after the apex. But, with that as a framework, we’re in position to introduce the next improvement. Everything we do from here on improves just the post-apex portion of the corner and the exit chute. We will actually drive the dummy line up to the apex. So, from this point on, we need only look at the last column in the table above, where we are shocked to see that there are almost three seconds’ spread from the slowest to the quickest way out. A good deal of this ought to be available for improvement by accelerating and unwinding.

By Brian Beckman

This week, we investigate how the effects of road bumps vary with speed. Everyone has experienced that bumps are more punchy as speed increases. A bump that you barely notice at 50 mph can sting at 100 mph. But what about at 200 mph? Will it just smack a little harder, or will it knock your teeth out or, worse, cause you to lose control? Could a bump be the limiting factor in cornering speed? In an aerodynamic car, could a bump cause a sudden and catastrophic loss of downforce and adhesion? To analyse such things, we need an understanding of the variation of bump violence with speed.

At the expense of a little storytelling, let’s explain how this topic came up. In particular, where is an amateur motorhead going to have to worry about bumps at 200 mph? At autocrosses, speeds are low, by design, to give everyone a safe venue to challenge the limits. If you’re going to spin out, an autocross is the place to do it. Low speed also means, though, that bumps, unless very severe, aren’t dominant. On a road course, speeds are higher, as are the consequences of losing control. But speeds are not higher everywhere, not for extended times, and seldom approach 200 mph. There are two commonplace scenarios with extended time at high speeds: oval courses and open-road racing. High-speed oval racing is a specialized sport not often encountered by amateurs. Since the focus of this series is on grassroots, amateur hijinks, we’ll look at open-road racing.

In Part 11 of this series, we took a scenario for braking from 200 mph from the Silver-State Challenge (SSC) in Nevada. My co-author, Jerry Kuch, and I just ran the 2000 Nevada Open Road Challenge (NORC). This is the May version of the SSC, which is held in September. In all other regards, the NORC and the SCC are the same. For most of the 230 cars entered, these are high-speed, time-speed-distance (TSD) rallies. In each of the sixteen TSD classes, the car running as close as possible to the target speed, over or under, wins. There are TSD classes every five mph from 95 to 170 inclusive, with high and low breakout speeds set by safety concerns. There is also an Unlimited, non-TSD class, in which fastest car wins. This May, the winner of Unlimited averaged 207 mph over a ninety-mile distance and another Unlimited car posted a top speed of 227 mph. Jerry and I ran in the 130-mph class with a top speed of 165 mph.

The SCC and NORC run on a ninety-mile stretch of highway 318 from Lund to Hiko in the Nevada outback, roughly along the shortest path from Twin Falls, ID to Las Vegas. The course runs from north to south, and the road is fabulously stark and beautiful in the unique way of remote desert roads. One is humbled by the realization that if stranded, one would surely perish, probably in a few hours’ time, from heat exhaustion, exposure, and dehydration. It’s great.

Hwy 318 events have been run continuously on since 1988. In 1990 and 1991, Mark Thornton, a fellow autocrosser, built up his 1986 Super Stock corvette into a Nevada car. Mark and I had nearly identical SS ‘vettes, and we often swapped cars at autocrosses. These cars happened to be almost the same as the famous yellow ‘vette that Roger Johnson, of multiple SCCA National Championships, still runs in SS, if I’m not mistaken. I know that Roger has driven my car, and I can’t recall whether he ever drove Mark’s, but I did, many times.

Mark, now deceased, was a bit of a bad boy, and Hwy 318 had just the kind of cachet that appealed to him. The legend goes that the events had been organized by the survivors of the old, illegal ‘cannonball’ runs. Of course, the NORC and SCC are properly sanctioned and completely legal, despite the fact that they use temporarily closed public highways rather than dedicated race courses.

Not content to play in the TSD classes, Mark decided to convert the black car into an Unlimited machine. I was with Mark when he handed his car off to Dick Guldstrand for blank-check suspension work, and I was in the loop when it went to John Lingenfelter for a reliable engine capable of 200 mph. I met up with Mark in Las Vegas to help with the final preparation of the car. I took a few, tyre-warming hops in the car, and, with nearly 600 HP, I can tell you it was seriously fast. Feel free to check out the car’s specs at http://www.angelfire.com/wa/brianbec/foober.htm.

Unfortunately, on race day, the car had an oil fire in the first, six-mile straightaway, due to the headers’ being a bit too close to the oil-filter canister. The required, on-board halon system saved the car and Mark and I saved what residual fun we could putting it back together and trailering it home. Later that year, Mark won a Triathlon of Motorsports hosted by a hotrodding magazine in the car, and, if I’m not mistaken, repeated the feat in ‘92. I have been told the car was featured on the cover of the magazine somewhere in those two years, but I have not checked that myself.

I moved to Washington State and lost touch with Mark, who had a non-motorsports accident and passed away. Mark was not uniformly liked, but even his detractors will grant that he was a truly gifted driver and an engaging, entertaining, complex character. Many, currently active autocrossers will remember him.

By sheer, stupid luck, I stumbled across Mark’s Nevada car for sale in Florida in 1999. This is about as far away from Seattle as one can get, but the kismet was too much to ignore. I had driven this car many times in anger, had crewed it, was friends with its creator. It just had to come home to me, didn’t it? Furthermore, it just HAD to run again in Nevada, didn’t it?

I bought the car and began the complex job of preparing it for NORC. One does not contemplate running 200 mph without giving a car a complete check-up. The energy available for destruction at 200 mph is four times the energy available at 100 mph, and sixteen times that available at 50 mph. Furthermore, the car had had an active, open-track life in the intervening years and it was time to tear it down and check it all out. You do NOT want an engine to seize or a suspension part to break at 100 mph, let alone at 200 mph.

With two months to spare, it became obvious that the car would not be ready in time. Better safe than sorry, I asked the mechanics not to hurry and to make sure the car is done right. The standards for mechanical work on high-speed cars must be significantly higher than it is for road-going and autocross vehicles, for safety. The standards should be comparable to those in aviation. Hurrying is a recognized no-no in aviation, and I applied the same logic to the car work. As I write, I have an ultimate goal of running it in SCC and NORC in ‘01 and ‘02.

I had already committed to run the ‘00 NORC, so I slapped a roll cage in my ‘98 Mallett 435 and went on down. This is another fabulous vehicle, but I hadn’t intended to run it in high-speed events until the last minute. It was quite a hustle to get the required safety gear properly installed in time. In hindsight, I don’t regret the decision. The car really came to life at NORC and I’ve run it in several high-speed events since then.

Our flight plan called for holding speeds up to 165 for minutes at a time. As part of planning, we did a survey and calibration run of the course at legal, highway speeds. On the survey run, we noticed several bumpy spots. Driving over them at 70 mph, they were not frightening. But, we had to figure out what to expect at 165. So, right there in the middle of nowhere, we whipped out some envelopes, turned them over, pulled multicolour pens from our pocket protectors, and started scribbling. Geek racing at its best.

Let us take a moment to review the goals and methods of the “back-of-the-envelope” (BOE) style of analysis introduced in Part 3 of this series. Frequently, one simply needs a ballpark estimate or a trend. These are often much easier to get than are detailed, precise answers. In fact, they are often easy enough that they can be literally scribbled out on the backs of envelopes in the field. And that’s the key point: we needed a rough idea of how the violence of the bumps varies with speed, and we needed it right then and there in the field.

Another benefit of the BOE style is that it can give one a quick plausibility check on numerical data back at the lab. Thoroughgoing engineering analysis usually entails dozens of interlocking equations solved on a computer resulting in tables, plots, and charts. The intuition gets lost in the complexity. It’s sometimes impossible to say, just by looking at a table or chart, whether the results are correct. On the other hand, to get our BOEs, we often make very gross approximations, such as treating the car as a rigid body; or ignoring its track width, that is, treating it as infinitely thin; or ignoring the suspension altogether; or even treating the whole car as a point mass, that is, as if all its mass were concentrated at a single point. Even so, the results are often not wildly off the numerical data, and the discrepancies can usually be explained via non-quantitative arguments. If the BOE and numerical results are wildly different, then some detective work is indicated: one or both of them is probably wrong.

BOE is really a semi-quantitative oracle to the physics. These articles are about the physics of racing as opposed to the engineering of racing. We’re primarily interested in the fundamental, theoretical reasons for the behaviour of racing cars. The trends and ballpark estimates we get from BOEs often do the job. Of course, this doesn’t mean we won’t get into more detailed treatments and computer simulation. It’s just that we will always be focusing on the physics.

All that said, as usual for BOE, we start with a simplistic model we can solve easily. Think of a bump in the road as a pair of matched triangles, one leading and one trailing.

Let the width of each triangle be w and the height be h . Suppose a car approaches the bump with horizontal speed v . To assess the violence of the bump, let’s ask what vertical acceleration the car will experience? If we assume a simplistic model of the car as a rigid body, we get an instantaneous, infinite acceleration right at the instant the car contacts the rising edge. We get further infinite, vertical accelerations at the two other cusps of bump the geometry. However, we know that the tyres and suspension will smooth out these sudden impulses. Calculating the effects of tyre and suspension flex is too time-consuming to do in the field even if we had data and computers on hand. However, we can get a useful approximation by assuming that the acceleration is distributed over the entire bump.

If the bump is shallow (h « w) and the car is fast, then the horizontal speed doesn’t change very much and the car goes up the leading edge of the bump in time t = w / v. In that time, the car goes upward a distance h, thereby acquiring a vertical speed of v_y = h / t = vh / w. Since it acquires that velocity, very roughly, in time t, we can estimate the vertical acceleration to be

Uh oh. BOE says that the severity of a bump goes up as the square of the speed. A bump you can feel at 50 mph is going to be sixteen times worse at 200 mph and will most definitely get your attention. The little whoopdeedoos we were noticing at 70 mph would feel (165/70)² = 5.5 times worse at our planned speed: definitely something to anticipate on-course before we hit them. This BOE also says that the nastiness varies inversely as the width. The wider the bump, the less nasty, linearly. This is plausible.

Now, let’s refine the analysis a little. Conservation of energy dictates that the horizontal speed of the car must change. In our simplified, two-dimensional BOE, the velocity vector, , consists of two components, horizontal speed, v_x, and vertical speed, v_y. These quantities obey the equation

whether on the flat or on the bump, that is, no matter what the inclination of the road. We’ve presupposed, here, that vertical always means “in the direction of Earth’s gravitation.” If we do not change the kinetic energy of the moving car, then ½ mv² stays constant, therefore v² stays constant. On the leading-edge ramp of the bump, remembering trigonometry,

Define, as shorthand, , yielding v_x = vw / r, v_y = vh / r. Using the same approximation as above, we assume that we acquire a vertical velocity of v_y in time t = w / v_x = wr / vw = r / v, for a vertical acceleration of

This still varies as the square of the speed, we just take a little more time to go over the bump. The only difference to the prior formula, v²h / w, is the appearance of h² in the denominator.

Consider the case of a high, narrow bump. This case was not covered by our first BOE, which assumed that h « w. Now, with a high bump, h² » w² and , meaning that the severity of the bump will go down linearly with increasing height. Within the confines of our model, this makes sense, because a higher bump gives the car a greater vertical distance in which to suffer its increased vertical velocity, but this doesn’t seem intuitively correct. A higher bump should be nastier, shouldn’t it?

Furthermore, of course, at constant throttle, the kinetic energy of the car will change because the force of gravitation will attenuate the vertical velocity. So, in our next consultation of the BOE oracle, we must reduce a_y by

The bump is getting less nasty all the time, and it’s obvious that we’re hitting the limitations of this BOE analysis. To expose the limitations even more starkly, consider two more questions: (1) what about the trailing edge? and (2) what about depressions, that is, down-bumps?

As to the trailing edge, a simplistic car-as-rigid-body would just launch ballistically from the top of the bump. Of course, in a real car, tyre elasticity and the suspension would endeavour to keep the tyres on the ground. Short of launching, there would just be weight loss causing rebound of the tyre sidewalls and the suspension springs. Nevertheless, everyone knows that a ballistic projectile assumes a parabolic flight path, so, as long as the parabola off the top of the bump remains vertically above the down-ramp, our car-as-rigid-body is assured of taking to the air. With the simple bump geometry, we can see that a parabolic launch always starts off above the trailing-edge triangle. It intersects the road again either somewhere on the down-ramp or on the following flat bit of road, depending on horizontal speed.

As to a depression – a down-bump as opposed to an up-bump – a car-as-rigid-body will simply have a ballistic phase before suffering an upward acceleration. At this point, I think we’ve reached the point of diminishing returns. Let us first repeat that the BOE style is doing what it’s supposed to do: getting us rough trends and quantities in the field. Primarily, we wanted to find out how bump severity varies with speed, and we’ve got our answer: roughly quadratically. We are seeing some ways in which the model departs from intuition and reality and it’s time to think about how to improve it back at the lab.

The first point to notice is that we drew a pair of triangles for our bump, but used them only to calculate the time to traverse the bump and the height acquired over that time. This is not a proper dynamic analysis, in which we would use Newton’s laws to model the motion of the car up and down the bump. At a glance, one can distinguish a dynamic analysis by the presence mass in the equations. Nowhere did we use the mass of the car in our BOEs above. Dynamic analysis is often too hard to do in the field because it involves integrating differential equations, almost always by computer.

Another problem concerns our simplistic bump geometry. As noted above, strictly speaking, the severity of a bump on a rigid body infinite, no matter what the speed. The reason is that the car acquires its vertical component of velocity instantaneously – in zero time – upon hitting the bump, so the rate of change of the vertical velocity, that is, the vertical acceleration, is infinite at the instant the bump is encountered, then zero on the body of the up-ramp.

Our list-of-things-to-do, should we wish to improve the model, includes the following tasks:

Model the geometry of the bump more carefully, accounting for the fact that the initiation of the up-ramp, no matter how severe, cannot, in fact, be mathematically instantaneous. Draw some sort of little sinusoidal or exponential curves to account for the actual road profile.

Integrate the equations motion of the car over the bump.

Model the car more carefully, accounting for tyre flexion, springs, shocks, suspension geometry, mass distribution, moment of inertia, and all the rest. This will entail designing a suspension.

These improvements put us squarely back in the lab. Ultimately, we will resort to computer simulation. As promised years ago, that is the ultimate goal of this series of articles: to spec out a simulation program. Better late than never, right?

By Brian Beckman