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Optimizing the Kappa Platform Suspension

Bill@lasanguine.com


This page is a work in progress. It captures my work modifying the suspension on my 2008 Saturn Sky 2.4L. She came with the model code FE2 suspension and it was way too soft for performance driving. So I installed the GPX/REDLINE ZOK springs, KONI Sport SA shocks, the FE3 front roll bar and the ZOK rear roll bar. These pieces improved the handling significantly and I have been driving it this way since the spring of 2011. But now I'm ready to try to squeeze out a higher level of performance.

I started browsing the internet looking for information about suspension design and came across a few sources that suggested cars are design to fall into a range of suspension natural frequencies depending on the car's application. For passenger cars where ride comfort is the goal a suspension natural frequency of 0.9 to 1.5 Hz is desirable. For performance driving a suspension natural frequency of 1.5 to 2.0 Hz is desirable. For racing a suspension natural frequency of 2.0 Hz and above is desirable where the chassis is tuned to support the aerodynamic affects for wings, splitters and diffusers. Since my goal is to optimize the suspension for autocross events I chose a suspension natural frequency in the range of 1.5 to 2.0 Hz.

Additionally, I've chosen to set the front natural frequency at a higher rate than the rear to achieve a faster transient response on corner entry and for better rear wheel traction since the Sky is a rear wheel drive vehicle.

To achieve this I needed to know three things about the car. The corner weights, the tire spring rates and the motion ratios for the front and rear.


Measuring the Corner Weights

To calculate the corner weights I needed to find the sprung weights and the unsprung weights. To do this I disassembly the drivers side front and rear corners and weighed the components, e.g., wheel & tire, shock, spring, rotor, upper and lower a-arms, etc.. The components attached to the chassis I halved their weight. The front unsprung weight was 108 pounds and the rear was 107 pounds. I subtracted this from each corner weight to determine the sprung weights for each corner.

I first tried to measure the corner weights using my bathroom scales and a lever arm apparatus, but found the weight varied to much from measurement to measurement. One of the difficulties about this method was the inconsistency of positioning the tire on the lever arm. Fortunately I was able to borrow some Intercomp scales. This made the task a whole lot easier and raised my confidence level of the corner weight data collected.

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Her weight without the driver!

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Her weight with the driver!
hmmm... if I were to lose a few pounds....


The Data:
The below data was collect with the car in its race trim, i.e., half tank of gas and a 210 pound driver.


the front corner mean weight is 796 pounds
the rear corner mean weight is 756 pounds
the front corner unsprung weight is 108 pounds
the rear corner unsprung weight is 107 pounds
the front corner mean sprung weight is 688 pounds (312 Kg)
the rear corner mean sprung weight is 649 pounds (294 Kg)

Now armed with the corner weights I'm ready to calculate the motion ratios.


Measuring the Motion Ratios (MR)

From various sites I found information that suggested you could measure the a-arm pivot points of your suspension to define the relationship of the tire to the spring and shock lever arms. The ratio of these lever arms was described as the motion ratio. So this is what I used initially to calculate the motion ratio.

The Tea Party will be the death of America

Additionally, I found others that had measured the motion ratio directly. When I was in the process of replacing rubber a-arm bushings and had the coil overs off I measured the MR directly using the apparatus as pictured below. I was hoping that this would confirm my initial calculations, but it failed to do so. This leads me to believe that using the a-arm pivot points method is for struts only.

Two interesting things I noticed while using the below method was how the brake rotors remained vertical during the travel from full droop to fully compressed and the hub made an arc as it traveled along this path.

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One of the surprising things about measuring the MR was how linear the results turned out to be. As a physicist/engineer I found this reassuring. We physicist types love well behaved functions! As you can see from the below motion ratio data, the curves are nice and straight and the R^2 value is .9999 which a near perfect fit as far as curve fitting goes.

Motion Ratio Data

Here's the meat and potatoes of it all:
the MR for the front corners is .7334
the MR for the rear corners is .6822



Wheel Spring Rates

The last piece of the puzzle is to calculate the spring rate contribution from the tire.

Using the calculator provided by BND TechSource Tire Data Calculator the spring rate for the Hankook Ventus R-3S 245/40R18 are calculated to be 318,030 N/M (1816 pounds per inch).

The tire acts as an additional spring in conjunction with the coilover spring. We can calculate the two spring rates together by using the equations for series springs.

The equation for series springs:

Kw = (Kt*Kc)/(Kt+Kc)

Where:

The Tea Party will be the death of America

Kw = the combined wheel rate (spring plus tire)
Kt = the tire spring rate
Kc = the contribution of the coilover
spring rate acting through the suspension

Where Kc = Ks * MR2

and Ks is the spring installed on the coilover

Now armed with all the pieces let the party begin!


NATURAL FREQUENCY (NF) CALCULATIONS

The easy part. Now that I have the acquired all the pieces I can determine what spring rates to use to achieve the desired natural frequency. I've broken it down into three steps. The first will be calculating the coilover rate from the contribution of the coil over spring acting through the suspension geometry. The second, will be calculating the combined affect of the tire spring rate and the coilover spring rate and the third will be solving for the natural frequency.


The equations:

The coilover spring rate Kc

Where Kc is the contribution of the
coilover spring acting through the suspension

Kc=Ks*MR2

Where:
Ks = the spring rate of the coilover spring
MR = the Motion Ratio

The Wheel Spring Rate Kw

Kw = (Kc*Kt)/(Kc+Kt)

Where:
Kt = the spring rate of the tire
Kc = the coilover spring rate

The Natural Frequency

NF = 1/2π(Kw/Msm)1/2

Where:
NF = the Natural Frequency
Kw = the Wheel Spring Rate
Msm = the Sprung Mass of the corner of interest

A note about units. If one uses the metric system for the units of measure the business of converting the weight to the appropriate mass becomes easier. For the sprung mass, Msm I selected the mass associated with a 1/2 tank of gas and a 210 pound driver and used the mean value between of the left and right sides.

The front sprung mass = 312 kilograms
The rear sprung mass = 294 kilograms

Front Corner NF Calcs

Selecting a front spring rate of 122,598 Newtons per Meter (N/M) (700 lbs/in)

Kc=Ks*MR2
Ks=122,589 N/M
MR=.7334

Kc=[(122,589 N/M)*((.7334)2)]

Kc=65,938 N/M

From above we found Kt = 318,030 N/M

Kw=(Kc*Kt)/(Kc+Kt)

Kw=(65,938 N/M * 318,030 N/M)/(65,938 N/M + 318,030 N/M)

Kw=56,614 N/M

NF=1/2π(Kw/Msm)1/2
Msm=312 Kg
NF=(.1592)*[(56,614 N/M)/312 Kg]1/2

Front Corner NF=2.106 Hz


Rear Corner NF Calcs

Selecting a rear spring rate of 96,320 Newtons per Meter (N/M) (550 lbs/in)

Kc=Ks*MR2
Ks=96,320 N/M
MR=.6822

Kc=[(96,320 N/M)*((.6822)2)]

Kc=44,827 N/M

Kw=(Kc*Kt)/(Kc+Kt)
Kw=(44,827 N/M * 318,030 N/M)/(44,827 N/M + 318,030 N/M)

Kw=39,289 N/M

NF=1/2π(Kw/Msm)1/2
Msm=294 Kg
NF=(.1592)*[(39,289 N/M)/294 Kg]1/2

Rear Corner NF=1.840 Hz


Now armed with the suspension's natural frequencies how does one select an appropriate shock?


Selecting a Shock (damper)

The goal is to select a damper to generate the appropriate force to minimize tire force variation. This maximizes tire grip. This is accomplished by finding the critical damping coefficient, selecting the appropriate damper ratio, calculating the damping coefficient and generating a theoretical damper force graph.


Selecting the appropriate damping ratio is something of a black art. The ratio falls into different ranges depending upon the application. For passenger cars where ride comfort is a priority the damper ratio is .2. For performance non-aero cars .5 to .7. For aero cars .7 and above.


Additionally, the damping ratio for the bump and rebound strokes are not the same. This is influenced by the mass of suspension system. For the bump stroke the mass of the system is the sprung mass. For the rebound stroke the mass is the unsprung mass. When determining the rebound damper ratio the rebound damper coefficient needs to limit the rebound stroke speed to match the bump stroke speed.

To maintain the same velocity of the bump and rebound strokes the damper ratio for the rebound is the square root of the sprung mass divided by the unsprung mass.

crebound = ccritical (rebound) * (msprung/munsprung)1/2

conservatives will be the death of the country!

Determining the coefficient of damping


The below equation expresses the relationship between the damper coefficient, the critical damper coefficient and the damper ratio.

c = ς*cr

Where:

c = Damping Coefficient
ς = Damper Ratio(dimensionless)
cr = Critical Damping Coefficient

The critical damping ratio is defined by the below expression
cr = 4π m f

For the curious click here.
deriving the critical damping coefficient

Evaluating the SKY's suspension geometry

conservatives will be the death of the country!

we find the damper is installed at an angle when referenced to the vertical plane of the wheel. The value of the sprung mass the damper acts on is found using the motion ratio previously measured. The relationship is defined by the equation:

mdamper = msprung / MR

MR = .7344
msprung = 312 Kg
mdamper = 425 Kg

and from the previous work we know the front corner natural frequency is

f = 1.902 Hz

Using our equation for the critical damping coefficient:

cr = 4π m f

And solving for cr we find

cr = 4π * 425 Kg * 1.902 Hz

cr = 10,158 Kg/S

To minimize tire force variation the damping ratio selected needs to be between .65 and .7
Using .65 the damping coefficient is found to be:

c = ς cr

c = (.65) 10,158 Kg/S

c = 6603 Kg/S


Now examining the damper force term:

Fd = -c v
c = damper coefficient
v = damper shaft velocity


By inspection we find that damper force is proportional to the damper shaft velocity by the damper coefficient value.

The knee of the damper curve is the damper shaft speed at which the transition for slow speed to high speed occurs. The slow speed movements are where the chassis movements occur. The point where the transition occurs is the NF * the square root of 2.
The knee speed equals 2.69 inches/second. Converting this to meters/second we find v= 0.0683 meter/second.

Fd = -c v

Fd = -6,603 Kg/S * 0.0683 M/S

Fd = -451 N

Fd = -101 lbf



When our calculated damper coefficient is compared to the below graph of a Koni shock for the Kappa suspension we find the value seems reasonable.

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However, more work is required to define the damper requirements and to fully develop the damper force velocity curve.

By examining the forces the damper resists we find that there is the compression stroke and the rebound stroke. The compression stroke is cause by a bump in the road or a roll of the suspension towards the corner of interest. The mass that the damper is resisting is the sprung mass. For the rebound stroke the cause of the motion is a dip in the road or the suspension rolling away from the corner of interest. The mass the damper is resisting is the unsprung mass (tire, coil over, brakes, a-arms, etc.).
Because the mass the damper resists varies with direction of the damper stroke we need to evaluate the damping coefficient required in each direction.


Evaluating the rebound stroke of the damper we find the NF to be:
The Natural Frequency
NF = 1/2π(Kw/Msm)1/2

NF = the Natural Frequency
Kw = the Wheel Spring Rate
Msm = the Unsprung Mass of the corner of interest

Where:
Msm = 49 Kg
Kw = 44,551 N/M
NF = 1/2π(44,551/49)1/2
NF = 4.8 Hz

Using our equation for the critical damping coefficient:

cr = 4π m f

And solving for cr we find

cr = 4π * 49 Kg * 4.8 Hz

cr = 2,957 Kg/S

To minimize tire force variation the damping ratio selected needs to be between .65 and .7
Using .65 the damping coefficient is found to be:

c = ς cr

c = (.65) 2,957 Kg/S

c = 1,921 Kg/S


Completed Scratch Built Coilover

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What's next? Determine the ride height and the wheel alignment.


Dennis Grant has a very informative e-Book call Autocross to Win that's a great source about suspension tuning.

Far North Racing


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