As always, I like to do a preface before jumping into the article. I feel it gives the article a bit more of a human feel. I'll start by saying this: I don't claim to be part of a race team, or a time-attack winning champion. But I am a nerd -- I love math, physics, and cars. It turns out that putting those three things together can give you a decent recipe for success. While I'm only going to talk about some high-level stuff today, hopefully this article will be both informative, and a guide. I even include my Excel-based suspension calculator for you to follow along.

## Natural Frequency

One of the most basic aspects of a suspension setup is natural frequency. If you recall from physics, if you tie a mass to a spring in a perfect environment (no friction, air resistance, dampening, etc.) it will oscillate at a certain frequency. This frequency is known as the "natural frequency". In a sense, a car is the same way. We have weight, over an axle, and a spring. While a car of course experiences many other components, it turns out that this natural frequency is a big governing factor suspension behavior.

## Choosing a Target Frequency

Most factory sports cars are tuned with a suspension frequency of ~1 Hz. This gives a comfortable ride, and decent performance. Bump that up to 5Hz, and you're looking at a high downforce F1 car. For reference, most low downforce race cars are in the neighborhood of 2Hz. We'll bring this down to real numbers (spring rates) soon enough, but bear with me. For my GTI, I factored in one major thing: tires. For track, I know I'll be running R-compounds in the 235-265 range. Knowing this, I'm not quite a low-downforce race car, but somewhere close to, thus my thoughts for spring rates were initially ~1.8Hz (80% of a racecar, ha!) Spring rates are chosen on a per-axle basis (at least for our level of understanding). We'll need to grab some data to put together what rates we need: per axle weight, wheel base, and a target speed. Lost me on that last one? Well, let's walk through an example.

Download my suspension calculator here: www.dropbox.com/s/re6mgeznu1pn7z6/Suspension%20Frequency%20Analysis.xlsx?dl=0

Take a look at the graph above. On the x axis we have time, in seconds, and on the y axis we have motion. The unit here doesn't really matter, as we are just looking at correlation. The blue line simulates the front axle motion, the orange line simulates the rear axle motion. Each axle hits some "imaginary" bump. The axles experience this bump at different times because of the car's wheelbase and the speed at which we hit the bump--the front wheels will hit the bump first, then the rear. We're going to neglect the higher order effects for now, like how the front impact effects the rear springs. Take a look at what happens when we set the frequencies equal. We see that because the car hits a bump, the front and rear are misaligned. In fact, at about 0.6 seconds, the front is moving back down, and the rear is still up! This is kind of the pinnacle of an "unsettled car".

One thing we can do is increase the rear (or decrease the front) suspension frequency in hopes that we get the front and rear movements aligned, and thus the car feeling "settled".

Here we see that by the time the car is starting to settle (~0.5-1 second), it is settling as a whole. Both front and rear are at nearly the same position. This is good! The graphs I'm presenting here are done with a simple calculator, so we kind of take a "guess-and-check" approach. For me, I expected the average speed to be around 70 mph on my local track (2.2 miles, ~ 1:50 lap times = 72 mph average speed). Setting rates of 1.65Hz front and 1.8Hz rear looked to be a pretty good balance

[Optional]: Keep in mind there's more to this than simply "aligning" the front and rear axle. Stiffening up the rear = less rear grip. Less rear grip means more rotation. This might be good for a FWD or AWD prone to oversteer. Alternatively, you may have a high-power, short wheelbase RWD car that may be prone to oversteer. For this scenario, we may soften the rear rates to increase rear grip.

## Translating Frequency to Spring Rates

We started with frequency as it is the most normalized scale to gauge suspension stiffness. For example, 500 lb/in springs might seem really stiff, but on a 10,000lb vehicle that might be soft. Frequency allowed us to separate factors like weight of the vehicle and suspension design from stiffness. It's normalized. 1Hz in one vehicle will behave roughly the same as 1Hz in another vehicle, even if there spring rates are totally different.

However, ultimately we need to translate frequency to a real number -- spring rates. There's a few pieces of information you'll need in order to accurately calculate the appropriate spring rates. The overall vehicle weight, the front and rear motion ratios, and an approximation for unsprung weight. I'm not going to include all of that here, but a good example on how to get this is available here: eibach.com/america/en/motorsport/products/suspension-worksheet

The hardest part here is getting accurate unsprung weight and motion ratios. A small error can make a big difference. If you're following along with the calculator (again, download link here: www.dropbox.com/s/re6mgeznu1pn7z6/Suspension%20Frequency%20Analysis.xlsx?dl=0), it's time to move on to the "Front Suspension Calculator Tab". The numbers in the spreadsheet are some estimates I used for my GTI. Take all these numbers with a grain of salt--I highly advise going through your own car and the process, OR, finding someone who has accurately calculated it! Corner unsprung weight varies a lot based on tire, brake, and suspension setup. For me, I estimate this to be around ~110 lbs. (40 lbs wheel and tire, 40lbs between rotor and caliper, 30lbs misc (strut, knuckle, halfshaft, etc).

Inputting this into the calculator- take a look at row 12- we want to find the cell closest to our target frequency -- 1.65Hz. In cell J12, you can see a frequency of 1.68Hz is calculated, which corresponds to a spring rate of 8kg/mm. Since springs are generally only to the nearest 1kg/mm, we have some leeway in our math. Additionally, it's always a good idea to do a sanity check with some background info of what coilover rates come in other cars. I actually compiled a list of coilover spring rates a while back. It's available here: www.golfmk7.com/forums/showthread.php?t=17884

You'll see that this 8kg/mm is reasonable for a front rate. Following the same process for the rear, you'll see about 9kg/mm corresponds to a 1.82Hz (close to our 1.8Hz target). This rate is a bit further off from what most coilovers use, but I swear I checked my math up and down on this one! See, most coilovers simply make the rear softer because there is less weight out back. But the truth is, we need stiff springs for 2 reasons: rotation and motion ratio. The first we already explained. The second, motion ratio, essentially means that a small movement in the wheel translates to a large force on the spring. To compensate, that rear spring needs to be stiff!

## Does the math actually work?

Well, I actually purchased a coilover kit from BC Racing with a front spring rate of 8K and a rear spring rate of 9K. I also added the "Swift Spring" option to ensure I would have high quality linear springs. (It's a little known fact that most springs are actually progressive, BC even hints at this in their installation manual when discussing preload). I will neglect speaking of the overall coilover quality, but instead focus on the effects of the spring rates. I chose BC for this reason: custom spring rates and valving at minimal cost. Let's start with one of my favorite visuals:

Notice anything interesting about this mid-corner picture? Look at the front and rear! My car sits perfectly level when parked at about 3/8" wheel gap. Here we see the rear has just a bit of compression in it, and the front has about 3/4" of compression. What does this mean? The springs are effectively translating the force to the front of the vehicle, maximizing front grip, exactly what a FWD needs for maximum power-out grip!

But really, how is it? Well, I'll be honest. On street, it's stiff. With street tires, it's also a little prone to oversteer, and you have to really watch trail braking. However, when you put the car on some R-compounds, like the current Maxxis RC-1 that I run, the car feels PERFECT. I mean wow, the balance is nearly RWD-level. The car has monster turn in, and feels extremely neutral mid corner. Understeer is relatively gentle as you get on throttle, and the lift-off oversteer is equally impressive. While on street tires that effect could be a bit too much at times, on track tires it's PERFECT. I cannot accurately describe my joy in how this setup has turned out.

I think my dad really said it best when he drove my car: "Don't change a thing on the suspension. The car is dialed. It's perfectly neutral, so easy to drive-- so easy to drive fast." Keep in mind, he's a motorsport instructor. Still not convinced? My unbiased track instructor was a Spec Miata driver and told me, "Wow! This car really handles well!" How many GTI drivers can say that?? The point is this: While I may have started with a base model GTI, I think it's safe to say the car is well on its way to Time Attack success!