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# Determine spring rates based on natural frequency2

## Determine spring rates based on natural frequency

(OP)
Is the correct?
Below is the calculation I used to determine my spring rates for auto cross racing. My target natural frequency is 2.2 in the front and 2.5 in the rear. The calculation is three steps, finding the wheel rate, find the wheel rate in series with the spring rate of the tire and using this value to calculate the natural frequency.

The equations:
The wheel rate Kw
Kw=Ks*(MR)^2

Where:
Ks = the coilover Spring rate
MR = the Motion Ratio

The wheel rate with the contribution of the tire spring rate

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

Where:
Kw' = combined spring rate
Kt = the spring rate of the tire
Kw = the spring of the wheel (the contribution of the coilover spring acting through the suspension)

The Natural Frequency

NF = 1/2π(Kw'/Msm)^1/2

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

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

Front Corner NF Calcs

Selecting a front spring rate of 140,101 Newtons per Meter (N/M) (800 lbs/in)
Kw=Ks*(MR)^2
Kw = [(140,101 N/M)*((.7334)^2)]
Kw = 75,357 N/M
Kw' = (Kw*Kt)/(Kw+Kt)
Kw' = (75,357 N/M * 318,030 N/M)/(75,357 N/M + 318,030 N/M)
Kw' = 60,922 N/M
NF = 1/2π(Kw'/Msm)^1/2
NF = (.1592)*[(60,922 N/M)/312 Kg]^1/2
NF = 2.22 Hz

Rear Corner NF Calcs

Selecting a rear spring rate of 210,152 Newtons per Meter (N/M) (1200 lbs/in)
Kw=Ks*(MR)^2
Kw = [(210,152 N/M)*((.6822)^2)]
Kw = 97,804 N/M
Kw' = (Kw*Kt)/(Kw+Kt)
Kw' = (97,804 N/M * 318,030 N/M)/(97,804 N/M + 318,030 N/M)
Kw' = 74,801 N/M
NF = 1/2π(Kw'/Msm)^1/2
NF = (.1592)*[74,801 N/M)/294 Kg]^1/2
NF = 2.54 Hz

Artificer of control system engineering

Replies continue below

### RE: Determine spring rates based on natural frequency

So, are your unsprung masses supposed to be the total M_us ? They seem a bit high to me, but you oughta know better than me.

Also, your rear to front frequency ratio seems a bit low, based on cars I know about with 'good' ride.

What kind of travel will you accept for, lets say, a 4 g vertical input with these rates? Or will yo trailer your vehicle there...

### RE: Determine spring rates based on natural frequency

(OP)
cibachrome:

#### Quote (cibachrome)

So, are your unsprung masses supposed to be the total M_us ?

Sorry, I messed that up. I meant the sprung mass.

#### Quote (cibachrome)

Also, your rear to front frequency ratio seems a bit low, based on cars I know about with 'good' ride.

From what I've gathered it's good practice to set the rear 10% faster than the front. The idea is to keep the upsets in phase with the front. I believe this is for a better ride. That being said my object is to improve the traction. I'm less concerned about ride quality, but would still like to be able to drive it on street to get to and fro the autocross events. Should the front be faster than the rear?

#### Quote (cibachrome)

What kind of travel will you accept for, lets say, a 4 g vertical input with these rates? Or will yo trailer your vehicle there...

Now your over my head. I don't know what you mean by "a 4 g vertical input". Please explain or you suggest a link to an explanation? I'd still like to be able to drive to the autocross events. I'll tolerate a harsh ride, but I don't want to get myself in trouble on the streets or highways.

Bill

Artificer of control system engineering

### RE: Determine spring rates based on natural frequency

The 10% 'rule' dooes not apply to competition vehicles, and is often ignored in performance variants of production cars with RWD.

The reason is that a high rear wheelrate makes traction worse, and nobody cares about maintaining a flat ride in a competition. Depending on the excellence of your suspension design and the type of event you may gain improvements in traction by stiffening the front up, this stabilises the body a bit better and so upsets the rear axle less.

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

### RE: Determine spring rates based on natural frequency

(OP)
Thanks Greg.

If I set the front ride frequency at ~3 Hz should the rear be ~10% less?

My design goal is for improved autocross track performance. The speeds on course are less the 60 MPH and the surface is generally pretty flat with the minor disturbances you find in parking lots.

The suspension on the Saturn Sky has long short arm double wishbones on all four corners and it's rear wheel drive. The power output at the crank is ~190 HP. It weighs 3000 lb with me in it. I'm classed for street tires.

I've searched the bulletin boards looking for how others have setup their cars, but I've not found much published except stuff about modifying the stock setup and a few who run stiffer springs, but its with the front ride frequency lower than the rear.

Any help is greatly appreciated.

Bill

Artificer of control system engineering

### RE: Determine spring rates based on natural frequency

I think the reason I've read for lower front frequency is to prevent, or at least reduce, "pitch" when proceeding over a bump.
At least one of the old civilian texts ( Colin Campbell, Puhn ?? ) went into a little detail showing what frequencies would be "right" to create a stately pitch-free ride over a single bump at various speeds.

### RE: Determine spring rates based on natural frequency

For a non aero car with a sensible suspension geometry you will maximise the grip from a tire if you can minimise the changes in vertical load it sees. Consequently the simplest way to set up spring rates for a given circuit is to make them as soft as you can without hitting the jounce and rebound springs or stops.

That may not be a practical approach for a weekend racer but at least it is a start. so figure out how to measure your suspension travel, drive a good lap, see if you are using most of your linear range spring travel up on each wheel.

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

### RE: Determine spring rates based on natural frequency

(OP)

#### Quote (Greg)

figure out how to measure your suspension travel

I may do this next year, but for now I'm modifying my coilovers so I can easily change the springs. Part of this process is selecting a starting spring rate.

I'm steering towards stiff springs to start with due to the natural of autocross courses. Autocross courses are short, in the half mile range with 15 to 20 transitions (turns). One of the typical elements is a slalom. I'd like the chassis to be fully recovered from each upset before entering the next. With soft springs you may use more of the available travel, but it takes longer for the chassis to return to its neutral position. I'm thinking that a with stiff suspension would recover faster.

How stiff? Since most of the time on course is at slower speeds the aero affects are negligible. So somewhere below 3 Hz would be a good starting point with the front set 10% faster than the rear.

Now this gets back to my poorly worded original post. Do the tire spring rates need to be used when calculating the wheel rates?

Artificer of control system engineering

### RE: Determine spring rates based on natural frequency

I never thought about the frequency when selecting springs (motorcycle experience only) - the main thought is to keep the suspension off the bottoming stops and keep the steering geometry in the correct range. As with Greg, I want to use the softest springs possible that are consistent with this.

The spring rates that you are discussing sound excessively high. For your rear wheel rate, 1200 lb/in corrected for the motion ratio gives 558 lb/in at the wheel. In the absence of weight distribution information, assume your 3000 lb car is equally spread around all four corners. The static compression would be only 1.34 inches (from completely unloaded, to static load on the wheel, only compresses the wheel 1.34 inches). There is no way that is anywhere close to being optimum. The spring rate is probably at least double what it needs to be. Use the springs and the geometry to keep fore/aft pitching in check, use antiroll bars to keep body roll in check and balance the roll stiffness front to rear, use the low-speed damping circuits to keep oscillations in check. The suspension has to act as suspension by moving and soaking up irregularities - not a substitute for bolting the axle to the frame.

A friend of mine is a pretty good autocrosser, and he routinely beats up cars with modified suspension using a stock Mazda 3.

If you are going to spend money on suspension, spend it on dampers - and beware that the majority of aftermarket suspension components out of the box are crap.

When you are doing that slalom, when the suspension is fully compressed on one side and unloaded on the other side and you change directions, it is the job of the low-speed circuits in the dampers to limit the rate at which the body roll transfers from one side to the other. If you insist on using the frequency approach, keep in mind that antiroll bars add to the frequency in roll and the effect can be substantial. The frequency in roll need not (and will not) equal the frequency straight up and down.

### RE: Determine spring rates based on natural frequency

Properly calculated ride frequencies need a 4Dof model, allowing pitch and bounce of the body on the wheelrates, connected to the unsprung mass riding on the tire rates.

But nobody does it that way, most R&H engineers use the simple 1dof equation, some, including me use the 2dof of sprung corner mass, wheelrate, sprung mass and tire rate.

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

### RE: Determine spring rates based on natural frequency

(OP)

#### Quote (BrianPetersen)

The spring rates that you are discussing sound excessively high.

I think its all relative. My car presently has spring rates of 211 lb/in in the front and 285 lb/in in the rear. This equates to ride frequencies of 1.12 and 1.34 Hz. The car is RWD, has understeer, but does a reasonable good job of getting around the autocross course.

I've ridden in my competitor's MAZDA Miata (RWD). His car is a few hundred pounds lighter than mine and has 700 lb/in springs in the front. You can really feel the tires grip as it goes from turn to turn. It feels like the tires are attached to the track with thick rubber bands. This car has a ton of grip and his times reflect that. For tires he uses Hankook R-S3. I’ve already bought a set for my car.

Getting back to the ride frequency, what will be the results of going from 1.1 Hz to 2.2 Hz? The only way to know is to try it and measure the results. I know the springs are just one component of the suspension and I'm not looking for a silver bullet, but just trying to optimize my starting point.

I know a Solo II champion that uses 2Dof to calculate ride frequencies and suggest using ride frequencies in the 2.2 in the front and 2.5 in the rear for autocross racing. His car was AWD.

Bill

Artificer of control system engineering

### RE: Determine spring rates based on natural frequency

The current state of autocrossing at the top level (where wins and trophies are sometimes decided at the 0.001 second level) has you well into the realm of diminishing returns in many areas. That's a margin on the order of one inch, spread out over maybe a 3/4 mile long course.

By way of illustration, the really serious people in classes where the cars can still be street driven will make sure that the windshield washer bottle is empty if complete removal of the bottle itself is prohibited. Others will spend more on each individual shock than I spent on a full set of Koni single-adjustable Sports for my daily-driver/occasional autocross and once-in-a-while track day Mustang.

Norm

### RE: Determine spring rates based on natural frequency

Personally I'd leave the springs alone and fit adjustable shocks, remove the rear sta bar, add a couple of mm to the front sta bar, if traction is the problem.

If braking is the problem, do the opposite with the sta bars.

Really without information this sort of exercise is just a harmless hobby that enriches the part suppliers.

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

### RE: Determine spring rates based on natural frequency

(OP)
Jeff:

Thanks for the spreadsheet calcs! Yes that helps. The masses I listed are the sprung not unsprung. The unsprung for the front is 108 lbs and the rear is 107 lbs.

Interesting! GM made three different sets of springs for the Kappa platform (SKY/SOLSTICE).
Model codes:
FE2 - front 128 lbs/in, rear 177 lbs/in
FE3 - front 174 lbs/in, rear 228 lbs/in
ZOK - front 211 lbs/in, rear 285 lbs/in
Plugging these numbers into your spreadsheet I find the difference between the GM front to rear natural frequencies (NF) and the time to travel NF at 60 MPH is:
FE2 - .02 Hz delta
FE3 - .02 Hz delta
ZOK - .02 Hz delta

It appears the GM engineers are using something very close to this flat ride theory to calculate the front spring rates.

Thanks again,
Bill

Artificer of control system engineering

### RE: Determine spring rates based on natural frequency

Maybe try goal-seeking the speed to suit 0.00 Hz delta . . . in any case I'd expect better ride quality if you aren't cranking up the damper stiffnesses just to address pitch behavior.

Norm

### RE: Determine spring rates based on natural frequency

One thing that the corner frequency approach ignores is the mode shapes of the 2 pitch/bounce modes. Usually this is described by the location of the pitch centre (which would be at infinity for a pure bounce mode, and at the cg for pure pitch).

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

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