Basics Of Shock Absorber Tuning

Maneuvering Input Frequencies

Your car’s suspension is classified by engineers and dynamacists nerdy types to be a spring-mass-damper vibration isolation system. Most of the automotive world looks at this system and wants to know how to best provide a smooth ride for the occupants. They typically use a thing called the transmissibility factor to evaluate how good the system isolates the passengers from the not perfectly smooth roads which impress forces onto the tires. The basic question is, given a bad bump or pot hole, how much of the force (or displacement or acceleration or jerk) at the tire gets transmitted to the passenger? That’s given by the transmissibility factor and (and this is key) the value of the transmissibility factor varies with the frequency of the input disturbance.

It’s really kind of sad. The engineers long ago figured out how to provide an excellent ride. It turns out that 90% of the answer is the shocks. Unfortunately, by and large, the car companies have not wanted to spend the money to provide better shocks, so lots of cars, even luxury cars, give a pretty poor ride compared to what it could be. (Edit: Of course, good shocks can be very expensive. Examples for my Corvette with approximate retails costs are 1) OEM C6Z06- $600, 2) Koni 3013- $1400, 3) Penske/Ohlins/JRZ, etc- $4,000. How many people want to spend 10% of a car’s cost on just the shocks? Answer: Only us idiot autocrossers.)

Of course, passenger comfort is not worth much if the driver can’t control the car. So a subset of these investigations aims at what it takes to assure controllability. A subset of those investigations is concerned with how to maintain best grip from the tires in the guise of least contact patch force variation which is what we autocrossers are mostly interested in. Well, it turns out that maintaining best grip by minimizing contact patch force variation is not totally different from providing best comfort, which is good for us racers or the racing world would have had to invent all the math themselves!

This transmissibility I mentioned is a key component in these ride quality, car control and grip maintenance investigations and is highly dependent upon the relationship between input force frequency and the natural frequency of the spring-mass-damper system, denoted herein as Fn. The Fn is easily calculated if you know the masses, the spring rates and the damping rates. (The damping rates are how fast you take energy out of the system after you start it moving with an input. That’s what the shocks are for.)

Engineers design most mechanical systems to operate far away from the system Fn. With cars and their suspension systems this is not possible. As we drive the car we are going to have inputs at, above and below the system Fn guaranteed, either from what the driver does or the world does. Let’s explore the input from driving a slalom, that signature autocross maneuver.

A slalom is driven at about 1.1s per cone in my Street class Corvette as detailed in a previous post here. An interesting fact is that the distance between the slalom cones doesn’t matter much. It always takes about 1.1s. A complete cycle from left to right cornering is accomplished in this 1.1s. Therefore, the average excitation frequency input to the suspension by a slalom maneuver is 1/1.1s = 0.9Hz.

In a stiffer (non-Street) car the slalom time could be (will be) less, maybe about 0.9s, also shown in the post linked above. Then the frequency is 1/.9 = 1.1Hz. A little faster.

I have no idea what the slalom time is for something like an FSAE car, but it might be even lower, which means again a slightly higher input frequency.

OK, so what? The what is that we really don’t want the forcing frequency input to the suspension to be near its natural frequency, Fn. Bad things tend to occur if we excite resonance around Fn, especially if the damping at the associated shaft velocity is very low, as it typically is with OEM or cheapish after-market shocks. The least of the bad things that can occur is a loss of grip. The worst is total loss of control of the car.

Any guesses as to what the natural frequency of a typical passenger car is on the road today? 1.0Hz to 1.5Hz.

Houston, we have a problem.

Example: Once I had one blown shock on my Corvette and didn’t know it. I pulled out from the Tee at the end of my street, turning and accelerating quickly onto the busy road, as you must since the sight lines are really bad and the cars come fast up the hill even though they can’t see over the top. I nearly lost control of the car and thought for a moment I was going over the curb. I went back to the garage and found a puddle of shock fluid on the floor. I had nearly lost control of the car on a city street just from accelerating and turning through a single corner with no damping on one shock and lots of damping on the other three. (Though to be fair, they were all probably adjusted full soft at the time.)

Also note that whatever damping is provided by the shocks tends to decrease the dynamic Fn. Not much for small amounts of damping, but significantly for large amounts. As much as 40% according to one paper I read at 80% of critical damping. (We’ll get into critical damping at some point later on.) So, large amounts of damping can do two things simultaneously, 1) make the problem worse by lowering the dynamic Fn closer to the input frequency, and 2) make the problem better by damping any bad effects. What happened when I added lots of damping to my Corvette’s 1.5Hz Fn by installing those high-dollar revalved pimp shocks? Perhaps it went from 1.5Hz down 30% to 1.0Hz, right at the slalom input frequency. Sheesh! Ah, but what’s a lot of damping doing to the grip? Turns out that depends on a lot of things I’m not ready to go into very far but I will say this: for frequencies below Fn x the square root of 2 (1.414 for the non-nerdy types), which would be 1.5 x 1.414 = 2.1Hz on my Corvette more damping generally decreases tire force variation and thus increases grip at the same time it increases control. Finally, something working in our favor! (And a justification for those pimp shocks.)

Guess what happens if you add damping above Fn x the square root of 2? Hint: grip doesn’t improve with more damping for inputs above that limit, which contains basically all bumps and even some of the driver induced effects for a Street-class car.

Increasing Fn numbers from the factory as delivered (i.e. stiffer suspensions) may be one reason why modern sports cars with Fns of ~1.5Hz (or higher) are so much better at autocross as compared to standard cars or older sports cars that were significantly softer. (The ride, however, is generally worse, because stiffer springs are cheap but better shocks are not.) This also goes some way to partially explain why the addition of a much stiffer FSB (Front sway Bar) for a softly sprung car in Street-class autocross is so necessary. The stiff FSB increases the effective roll stiffness during transient events, moving the dynamic Fn higher and a safe distance away from the input frequency. So, the car with the stiff FSB gives us faster response and more control. (We won’t mention understeer here, but you can think it.)

In classes where you can change the springs and get Fn anywhere you want then a super-stiff FSB is no longer necessary to avoid inputs near the suspension resonance nor to provide acceptable front-end response.

Even if Fn is well above the input frequency, the analytical studies say we still need good damping at the input frequency to limit force variation at the tire patch and thus maintain good grip. That’s why significant damping forces at 1Hz are necessary. We only get that in good shocks properly valved, i.e. your car didn’t come from the factory that way. But, we don’t see frequency on our shock F-V plots do we? We see shaft velocity. So, what’s the shaft velocity that corresponds to our 1Hz input in a slalom? It’s not difficult to estimate.

Maneuvering Shock Shaft Velocities

If the shock shaft travels a total of 3in in a 1s slalom then the average velocity is 3/1 = 3in/s. This assumes that the shock shaft moves 1.5” on either side of the static position, which seems reasonable to me for a stockish car. (Note: this is in roll, not heave.)

If the shock shaft travels only a total of 2in in 0.9s, which might represent a non-Street autocross car that’s low and stiff then the average velocity is 2/0.9 = 2.2in/s.

The peak velocities will be higher, of course, since at each end of the stroke the velocity is zero, but the peaks are limited if the shock provides significant damping. Shocks slow down the shaft velocity by providing an opposing force while increasing the rate of weight transfer. If that’s not clear to you, go read this old post.

This is good support for Dennis Grant’s claim in Autocross To Win that driver-induced frequencies are mostly up to 3in/s so you need to valve the shocks with this in mind. Of course, DG provided shock shaft velocity data from three autocrosses in his Street Modified car to support his point.

On the other hand this is poor support for Ross Bentley who claims (in Shocks For Drivers, pg.12)  that driver-induced shock shaft speeds only range up to 1 in/s. Bentley provided no shock shaft velocity data to support his point. I think he might have misunderstood the data he was looking at. For instance, it may very well be that the majority of the time, when looking at a shock velocity histogram, the shocks operate below 1in/s. Especially for a race car on a smooth track. That this is true is easy enough to verify on the internet and I have done so. However, this says nothing about what velocities are the most important during specific maneuvers, such as corner entry and exit or, God forbid, an autocross slalom.

One last thought. Even if we are convinced that we need significant damping at low shaft velocities to maintain good grip when maneuvering we are still faced with another choice: do we provide the damping equally split between bump and rebound, or do we weight it one way or the other? Traditionally almost every analytical study you can find assumes equal damping in both directions. (Makes the math so much easier!) Just as traditionally almost every shock you can find that’s ever been fitted to an automobile since it’s invention provides more rebound than bump, sometimes many times more. Why? In the Shock Absorber Handbook, 2nd Edition, 2006, the author John Dixon states, “[I have] questioned various vehicle dynamicists informally on this point, and received less- than-convincing replies. Generally, there is a belief that there is a simple explanation, but this was not actually forthcoming.” But 2006 was back in the dark ages, man! I’ve found some more current data of interest and will be delving into this question.

Let’s Get Straight About Late Apexing- 5 (and Final)

In part 1 I wrote about three definitions of late apexing. By the time we got to part 4 I think I’d muddied the water a little bit. (That happens when you haven’t written each section before publishing the previous one.) So, let’s get it straight once and for all by slightly revising the three definitions:

Definition 1: An apex location that’s beyond the geometric center of the corner.

I don’t think we need to add anything more.

Definition 2: An apex location that’s beyond the accepted correct apex location.

This definition applies when people think that there’s one and only one apex location for a corner. OK, maybe it changes in the rain, but basically it’s the same for all cars.

Definition 3: An apex location that’s beyond the actual, correct apex location, understanding that the correct apex location will be unique to each particular car and situation.

Not much difference between 2 and 3, but the difference is crucial. Def 2 carries with it the implication that a single correct apex location exists based on the geometry of the situation, taking into account what comes before and after the curvy part of the turn itself. For instance, many have thought for the last few centuries that the length of the straight after a corner has an effect on the proper apex location. This has always been false. Many have also thought that the speed of entry, if limited by the preceding track geometry, has an effect on the proper apex location. This is correct. In fact, we find this ALL THE FREAKING TIME in autocross. Let me illustrate with an example taken from a real autocross event, not a made-up situation. Or, at least, not completely made up. As in all real-life scenarios it will be necessarily more complicated, but there may be multiple interesting lessons here to justify writing all these words about something I don’t even believe in, i.e. the concept of late apexing.

In Figure 1, below, we have the finish section of the East course at Solo Nationals in 2014 named Over The Falls by Jeff Cox, the designer. It was my first Nationals, in someone else’s car and before I knew much of anything about line theory. (What I thought I knew was mostly wrong.)

Figure 1- 2014 Solo Nationals Finish, East Course

First of all, recognize that the entry to the Over The Falls section is very fast. You accelerate hard from the top of the figure downward. Where to brake? Where to transition from turning right to turning left? (This is, by definition, a chicane because you can’t optimize the right turn separately from the left turn.) The key question in my mind was where to apex the left turn toward the finish.

I really had no idea.

Why don’t you look at it and decide how you would approach this section. Below I’ll show you what I actually did. Each run was different! In my confusion I seem to have bracketed the possibilities in my three tries at it. Please do learn from my ignorance.

To complicate matters the Corvette I was driving was set up very differently from my own. It wasn’t as over-tired as mine (so it wasn’t mushy) and had high-dollar double-adjustable shocks valved and set to make it handle like a kart. Never before (or since) have I driven a Corvette that could transition like that one. The downside? Less ultimate grip than my car. It was like driving on a knife-edge and very difficult (for me) to drive without provoking an oversteering slide. The owner, who won his first national championship that week, kindly warned me after our runs on the practice course that I needed to control my overly fast hand speed, but that was not advice I could easily implement on my first-ever Solo Nats runs.

On the first run I started slowing early, somewhere near the location marked BP (braking point) in Figure 2, below. This allowed a safe and fairly rapid turn of the wheel at the transition point, TP. I wrapped tightly around the two inside cones, apexing about in the middle at the big red X. From there, being so slow, I could start adding throttle and accelerate toward the finish. Cone 621 was never the least bit in play.

Figure 2- 2014 Finish, Run 1

Seem reasonable? It felt very slow. I think it was slow.

This method guaranteed that I fell below the limit very early, the speed at the apex was very low and I used mostly just the power of the engine to propel the car toward the finish. The turn I executed had a very small radius, severely limiting the time during which the tires were pushing the car at high lateral acceleration toward the new ideal direction. In short, I made the mistake of not using all of the available “track”.

At least I didn’t slide (too much) or spin at the transition point, which is the only saving grace for this conservative (scared to death) approach on the first run. In fact, this run turned out to be my fastest overall, setting me into 9th place of 28 drivers on day 1, but not because of how I executed this section.

On the next run I made the opposite mistake, as shown in Figure 3.

Figure 3- 2014 Finish, Run 2

I entered much faster, getting on the brakes late and soft because of continued fear of losing control during the yaw transient point, which thus became more separated from the braking point in both time and space. As a result of the soft braking the car carried way deep toward the horizontal wall of cones to the left of cone 621. My apex was somewhere near the big red X, after which I struggled (very slowly) to get around cone 621. I don’t know which was slower in this section, run 1 or run 2. Both felt awful.

On run 3 I think I did it closer to correct. See Figure 4, below.

Figure 4- 2014 Finish, Run 3

This time I got max on the brakes about point BP, even though I was already turning right pretty hard. I transitioned at the correct location, TP, while still hard on the brakes and managed, somehow, to not let the yaw moment create a spin. The apex was about at the big red X and was at a higher speed than both previous runs, but especially higher than run 1. I smoothly added gas from there. I barely made cone 621 which, these days, tells me I was doing it better. This better-optimized exit used all the space (track) available.

Starting from TP the lateral acceleration began building up, maxed out at the apex, but continued to help beyond the apex as the car turned toward the finish. Do you understand that the direction of this lateral acceleration is always tending to aim (left) toward the finish? It’s maximum magnitude is double what the engine could produce in that car. That’s why you want to use “all the available track width” to most efficiently add the engine capability after the apex and to best use the lateral acceleration capability of the tires both before and after the apex to push the car in the new ideal direction which, in this case, is toward the finish.

Unfortunately, the third run was not the fastest overall, though the final section felt much better. We got bit by overheated tires. That September day in Lincoln was unseasonably hot and my overdriving didn’t help the cause. After we both got slower on our second runs we realized what was happening and draped towels soaked in ice-water over the tires. It was too late. The carcasses of the tires were already too warm. In spite of each of us feeling like we drove much better on our third runs we both went slightly slower again. We stood on our first runs. Luckily, it was enough to give my co-driver the lead going into Day 2. He managed to very slightly again beat the fastest Honda S2000’s on the Day 2 course and take the win. This was back in the days of parity between non-Z06 C5 Corvettes and S2000s in B-Street when high-performance 200 Treadwear tires were just getting started. Those tires couldn’t put power down very well when exiting a corner which invalidated much of the Corvette power advantage. (That all changed when the Bridgestone RE71R came out.)

I contend that if the entry to this section had been different then our correct apex location would be different. Imagine that the course design causes us to take the path shown in Figure 5, below, approaching from the side and that the car’s speed is slower than before.

Figure 5- 2014 Finish Alternate Path

Now the transition point, TP, is earlier than the BP, which makes things easy. (We much prefer to stay on the throttle during the transition and then brake as the steering wheel crosses back over center.) So, no problem to now treat this as a standard corner of approximately 160 degrees once we are turning left. (We intentionally created a chicane on entry to get some angle, as we so often do in autocross, because it gets us through the corner faster by modifying the horrible pinched entry that this mythical course designer (moi) intended for us.) We apex somewhere about where the red “X” is located, about the same as in Figure 2, and add power from there to the finish.

My point is that the location of the apex is sometimes highly dependent on the speed of entry. If you accept this then you have to also accept that there is no such thing as a single, correct apex for any particular corner. Just the difference in the power to weight ratio of various cars is clearly enough to affect the best apex location, angle and speed even if only because it would affect the entry speed in a situation like this. Grip level also affects the apex, especially when you bring non-linear aerodynamic downforce into the picture. That also means that if the grip level of the surface changes, such as from a morning to afternoon temperature change, rain shower or any other cause, then you may have a new apex. For every corner on the course!

I hope you now have a clear idea of Late Apexing Definition 3: An apex location that’s beyond the actual, correct apex location, understanding that the correct apex will be unique to the particular car and situation.

Why would you ever take this type of late apex per Def 3, except by mistake?

Remember I said I don’t believe in the concept of late apexing? Let’s get into that now. I previously defined an apex as having three components: a location, an angle, and a speed. How is it useful to think about the concept of a late apex when it refers to only one of the three components, the location? Once we realize that an apex is not simply a location then I don’t think it is.

I’m sorry this has become so complicated, but I really do believe that, in theory at least, we have to determine the location, the angle and the speed for the location of minimum speed in any particular corner. At least I didn’t change that… the location of minimum speed is still the apex! Just figure out how fast you can or ought to be going at that point as well as the angle of the car to optimize the exit and you’re all done. Easy peasy.

One last thing: it’s fairly common in autocross, because of the interconnectedness from feature to feature, that we might enter a corner, say the approximately 160 degree corner shown in Figure 5, well below the correct speed at the apex, that is, the speed and angle at which we can rather quickly ramp up all the engine power. (This is exceedingly rare at race tracks for most cars.) If so, then we may be accelerating all the way through it in order to obtain that correct apex speed. In that case there’s no braking point! If you’re turning the car hard, but never have to slow down between the entry and the exit of a corner, is it still a corner? Not really. Brouillard, in his book The Perfect Corner 2, calls such situations “full-throttle corners” and has a special rule for them. I think that in autocross we encounter many such situations. Maybe we can’t be at full-throttle, but we may be using significant throttle to accelerate and there’s no braking point to be found. If you don’t have to brake there’s no corner in the standard sense and standard rules no longer apply. In that case I recommend paraphrasing a previous NFL team owner: Just get there quick, baby!

Let’s Get Straight About Late Apexing-4

Definition 2 of the Late Apex: Intentionally make your apex slightly later than the optimum apex.

Note that implicit in this definition is that there is a known optimum apex location (maybe considered to be the same for all cars, or maybe not) and the driver chooses, maybe for safety reasons, not to apex at that point, but apex a little bit later. This definition has nothing to do with the geometric center of a corner. The optimum apex might be before the center or after the center, but the “late” apex is later than where the optimum apex is considered to be.

Usually the instructor of a novice student (who won’t know where the optimum apex is located in any case) simply shows the student where this “late” apex is located and the student practices precisely hitting that exact spot.

Let’s assume we have a moderately powerful car that needs to late apex a geometrically perfect 90 degree corner per late apex definition 1 that we discussed earlier. Here’s our old Figure 2, repeated, to show us just such a situation, perhaps with the lateness of the apex slightly exaggerated at B’.

Figure 2 (Repeated from part 2 of this series)

Now imagine John Wannagofast in his 505Hp Zed06 Corvette at his first track day. Johnny’s instructor takes one look as his gold chains, leather pants, loafers with no socks and sticker Hoosiers on the car and knows he’s in trouble. While the line and apex location shown in Figure 2 might be exactly the way the instructor would drive John’s car, what he’s worried about is that John will not be very precise in his driving and may, in a testosterone-fueled frenzy brought on by being forced to give a point-by to a Miata with front fenders in two different colors, enter Turn 3 (let’s call it) too fast and apex too early. This is the driving line shown below in Figure 9 where I’ve drawn an arrow to indicate where, by accident, a novice might apex. Once that has happened there’s almost no way the car, in the hands of said novice, doesn’t leave the paved surface, maybe even spinning as it disappears down an embankment in search of a tire wall.

Figure 9- Turn 3

Here’s what happens instead: the instructor gets in his car and zips out to turn 3 and plops a cone down on the inner edge right at B” where I’ve shown the blue square in Figure 10, below.

Figure 10- Turn 3 With Instructor’s Apex Cone

The instructor now tells John, as they circle the course in the parade laps, that the way to take this corner is to get slowed down nice and early, turn in smartly as you come off the brakes, follow an arc around to the cone and then add throttle. About the 3rd session of the day the instructor may even suggest trying to get on the gas early, even before the cone is reached. Just make sure to get close to the cone and then allow the car to track-out to the left edge as you accelerate down the ensuing straight to be set up for the next corner.

This is the longer-dashed and bolded driving line shown in Figure 10.

Problem fixed.

John is now highly unlikely to go agricultural at turn 3. The instructor has built in a margin of safety while giving John plenty of good stuff to work on. Some people used to think (some still do) that this line, by virtue of allowing the start of acceleration to be earlier than the apex, will reduce the lap time because the average speed in the following straight is faster. They were wrong, but that’s another subject.

Let’s Get Straight about Late Apexing-3

So far what we’ve explored is well-trodden ground. That’s okay, we’re just getting our bearings within the late apex landscape.

Now I want to shift gears and look at what we’ve just done from the autocross standpoint. There may be discoveries to be made. Let’s start with Figure 4, below.

Figure 4

Here I’ve deleted the track edges and substituted a simple gate at the entry at point A, a single cone at the former geometric center apex at B and another 2-cone gate at the former track-out point C. I left in some dotted lines to indicate the former track edges, the crosshair that indicated the former inside edge arc center and the 45 degree line for reference. Now that we are free from the inside track edge the only limitation becomes the cone at B. (Yes, it would normally have a pointer cone on it. Please use your imagination!) So, how to get from A to B to C in the least amount of time and should it, or could it be any different than before?

Certainly our path could be different. For instance, a typical novice mistake is to head at top speed more or less directly toward the cone at B as shown below. We can do this because we can cross over what would have been the inside edge of a typical track corner and change the angle of the car at B.

Figure 5

I used to do exactly this. I still have to fight the urge. I would race toward B and slam on the brakes. I’d still be braking hard way past point B where I would turn hard toward C, then apply lots of throttle to get back up to speed. Man, it feels like your really driving! Unfortunately, as dramatic as it is, it is also a very slow technique. This path would look something like what’s shown below in Figure 6.

Figure 6

Frankly, I see a lot of people who are no longer novices making this mistake. Maybe not as egregiously as I did for much too long, but not getting it completely right either. It can be quite difficult to fully recognize the fast from slow lines on autocross courses, especially courses designed intentionally to give the drivers the ability to hang themselves if they really want to. I continue to struggle with determining the correct trade-off between carrying the most possible speed through a corner and sacrificing a little of that speed to better set up for the next feature.

Why is the line shown in Figure 6 slow? While you initially get to B very quickly, you must over-slow the car in order to negotiate what will be a small radius turn after B, followed by using mostly just the engine to accelerate in the new direction beyond C. What you have failed to do is maximize the use of the much greater lateral acceleration capability of the tires to get turned from the initial direction at A to the new direction beyond C. The tires on my Corvette can push me toward C at 1.2Gs just by turning the steering wheel. Feakin’ 200 tread-wear street tires on a Street-class car produce 1.2Gs! The engine can only push me at about 0.45Gs in 2nd gear. Using the tires’ multi-tasking capability for longer and not braking so much will win every time in any car that has a similar difference between tire capability and engine capability. If this is not clear to you I covered this point in more detail recently in this post.

OK, so we can’t massively use the lack of a track edge prior to B to get faster. Is there a way to use the lack of a track edge after point B to get faster? We could take a path something like what’s shown in figure 7, below. Would that help?

Figure 7

What we’ve done now is to brake more prior to point B so that we can execute a smaller radius arc that ends at B after which we add throttle. The problem is that this will be slower than our original line also. We don’t take full advantage of the track width that’s available which in turn means we used the brakes too much and the lateral ability of the tires too little.

Ah, but some of you are seeing a utility to this line, aren’t you? If we need to be at the location of the arrow on the end of the path shown in Figure 7, at the lower boundary of the gate at C instead of the upper boundary, to be best positioned for the next feature, then we will use exactly a line like this. And some autocross people will call this late apexing. But, it’s not really the same as what we previously drew as a late apex line on a track with edges in Figure 2, is it? In track driving the key component of a late apex is the fact that the position moves around the inner edge of the track and the angle moves with it. In autocross the position of your apex in any feature similar to this is not going to change. It’s going to stay at the cone at B. What will change is the speed and angle at B. So I prefer to not even use the term early or late apex for autocross. Leave those terms for the track environment.

You have to figure out how to drive this feature on the course walk. My method is to stand on the cone at B and look toward the exit at C. I estimate the speed the car will have when arriving at B and decide what the angle of the car needs to be when it passes B, that is, what angle will allow maximum possible throttle-up starting at B and still make the cone at C, or be where I want to be within the gate at C. (The car’s angle at point B is something very easy to recognize, easier than the speed.) This is the definition of optimizing the corner exit. Then I work backwards to determine the line, braking and steering inputs I will need to execute from A to B to arrive at that angle and speed. I don’t call the result either an early apex or a late apex. It’s just the correct apex which really consists of three things: a location, a speed and an angle.

When you drive the course for the first time you probably won’t get it perfect (or anywhere near perfect) so you have to understand how to make the right correction. If you find you can’t deploy all your car’s power after B without hitting the cone at C (equivalent of going off the edge of the race track) then you arrived at too vertical an angle and probably also too fast, i.e. you didn’t slow enough and turn enough before B. So on run 2 you slow a little more and turn a little more between A and B and thus arrive at B at a more horizontal angle and a little slower to allow taking full advantage of all the tire capability and whatever engine power you have after B to reduce time in the corner. This will then optimize the exit as well as the entry specifically for your car (and your car only) within the prevailing conditions. If you’re driving an A-Street C6Z06 Corvette don’t go to your E-Street 1999 Miata buddy and tell him he’s apexing at the wrong angle just because it’s different from you. (Especially if he’s beating you on PAX!) He may be doing it right for his car and it should be different than what you’re doing.

If you arrive at your anticipated point at C but don’t feel you were working the tires to the limit all the way there then you must have arrived at B at too horizontal an angle and too slow. (Full throttle in a straight line does not work the tires to their total capacity even in most four wheel drive cars. Most cars can still turn while accelerating fully.) You correct it by slowing less after A and turning less after A so that you arrive at B faster and more vertical. Then you keep the tires on the limit by continuing to turn (while slowly opening the wheel) while feeding in throttle. In a Miata it may well be full throttle immediately. In a GT350 you’re gonna have to be careful with the right foot!

If you’ve applied the corrections then you will arrive at a line that looks something like the one shown in Figure 8, below. Yes, it’s our old Euler spiral, if done correctly. A decreasing, trail-braked radius from A to B and an increasing radius from B to C. Theoretically, the tires are worked to their limit from A to C with combinations of braking-turning (A to B) and accelerating-turning (B to C.)

Figure 8

This line is similar to, but not exactly the same as the late apex line shown in Figure 2 of the previous post. Previously, for a high-power car, we needed the late apex to optimize the exit. Now, for the exact same car, the apex has moved from the previous B’ back down to B and the angle and speed at B change slightly. This new line will probably cross the previous track edge before getting to B. The angle at B will be more vertical since there is a longer distance from B to C during which to complete more of the turn than from B’.

The higher the power to weight ratio, for the same grip level, the more turning has to take place prior to reaching the cone at B. Therefore, the more horizontal the angle will need to be at B in our example. So, in autocross, we don’t make a “later” apex with a more powerful car, i.e. change the location of the apex by moving it around the inside track edge. Instead, we revise two of the three properties of an apex, the angle and speed at the apex location.

We could drive the exact same late-apex line of Figure 2 and it would be almost the same time, but I think it would be slightly slower because it feeds throttle in later and is very slightly longer. We would be some distance off the cone at B which most experienced autocrossers would recognize as a fault. It would not quite optimize the exit that could now start at B instead of B’. If the cone were in fact at B’, instead of at B, then the line from Figure 2 would be correct.

This is not to say that some randomly placed cone in the middle of a corner is always the apex point. Most assuredly this is often not the case. Some cones in the middle of a corner may not even be on the racing line, much less mark the apex! What if there’s no cone at B at all, similar to the case of a 2-cone turnaround discussed in this post? In that case we have to decide where, out in the vastness of space, we will reach our slowest point and decided what speed and angle to be at. All without a good landmark. Ain’t autocross grand?

Now we are ready for the next installment where we’ll present definition 2 for late apexing, the one that I was taught in order to be able to drive track corners safely as a novice, meaning without leaving the pavement! I don’t yet know if it will have any autocross relevance.

Let’s Get Straight About Late Apexing-2

On to definition 1: In the simplest of terms a late apex means the car touches the inside of the corner somewhere beyond the geometric midpoint of the corner.

The figure below is intended to show a perfect 90 degree corner and a typical path through it. The borders of the corner are intended to be circular with the center point of the inside arc indicated. A 45 degree line is drawn through the corner.

In this perfect representation of a corner (which never actually occurs in nature) the geometric midpoint of the corner is point B where the 45 degree line intersects the inside edge of the track. (Of course, we have no track edges in autocross. We’ll get to that later.)

The dashed driving line shown starts on the outside at A, makes an apex at B and tracks out to C. Most of us know that something like this is the fast way through such a corner. It’s assumed that the car approaches A at high speed and must brake. Let’s not quibble just yet about trail-braking, spiral entry arcs or whether the length of an ensuing straight makes any difference. Keep it simple.

Figure 1

A late apex line is, per definition 1, any line where the car contacts the inside edge of the track beyond point B, as shown below. The new apex is at B’ and it is “later” than B. Simple as that.

Figure 2

So, when the road racing expert says “You want to late-apex almost every corner on this track” I think what he means is probably definition 1. Why you should late-apex the corners is left unsaid (other than it’s faster) and many assumptions are being made when he says that.

Assumption number one is that the expert is talking about a range of similar cars. Most cars that don’t have extensive aerodynamic downforce, but have relatively high power to weight ratios need to late-apex most corners at most every track on the planet. Does the expert know why? No way for me to know, but his statement is basically correct anyway. This is because if such cars apex at B while cornering at the limit and then try to apply full power as they unwind the steering wheel they will go off the track somewhere around point C. In Brouillard’s terms, they are unable to optimize the exit. To stay on the track they are unable to use their excess power to decrease time spent in the corner. A late apex “fixes” this issue.

By the way, I tend to roughly define a high power to weight ratio as about 10 lbs/hp or lower. A stock 1999 Hard S Miata at 2200 lbs and 120 hp does not qualify at 18.33 lbs/hp. My 2000 Corvette at 3100 lbs and 345 hp barely qualifies at 9.0 lbs/hp and it will need a slightly late apex. A Mustang GT350 at 3800lbs and 526hp is at a power to weight ratio of 7.2lbs/hp and needs an even later apex. If all three cars drive to the same geometrically centered apex at B the Miata wins.

For the stated Miata, for example, late-apexing at B’ is not the fast way through this corner. That car wants a nearly circular arc that apexes very near the geometric center. The same for a Formula 1 car. In spite of their huge power, F1 cars have proportionally even more downforce, and therefore cornering power, so they take cornering lines more like the Miata. Watch any F1 race and this will be evident. What Brouillard made really clear is that it’s the ratio of power to grip that determines the proper line through the corner for any particular car.

What happens when the “corner” is defined only by an entry gate, a single cone “apex” and an exit gate, like in the figure below? That’s what we’ll talk about in the next installment.

Figure 3