spring tech

Spring rates are a little tricky.  The spring rate is the the amount of weight that will cause the spring to compress one inch.

Front suspension:  Cutting a straight coil spring will increase the spring rate in direct proportion to the amount of the spring cut, i.e., cutting one coil off of a 5-coil spring will increase the spring rate 20%.

To determine the actual spring rate at the wheels ("wheel rate"), the leverage of the suspension arm has to be taken into account.

For instance, if a lower control arm ("LCA") has the coil spring located exactly halfway between the wheel and the chassis pivot, then when the wheel moves up one inch, the spring will only compress one half inch.  So the wheel rate is now twice the spring rate.

But the LCA also acts as lever, multiplying the torque on the spring.  The formula, for a completely upright spring is wheel rate = spring rate X (motion ratio)².

[The complete formula is Wheel Rate = ( (MotionRatio^2) * SpringRate ) * sin (Spring Angle).  But since our springs are pretty close to vertical, I didn't worry about the spring angle.]

According to my measurements, the total arm length (L2) from the centerline of the chassis pivots to the centerline of the lower ball joint is 18.5".  The distance from the center of the spring pocket to the chassis pivots (L1) is 11".  This results in a 1.68:1 leverage factor, or a motion ratio of .59.

So the effective spring rate, or wheel rate, is actually only 35.3% of the nominal spring rate.  On a truck with motion ratio of .59, a 1,000 lb front spring will have the same rate as a 353 lb leaf.

Rear suspension:  With leaf springs on a solid axle, the wheel rate is calculated a little differently.

d3 = Distance between spring centerlines
d4 = Distance between tire centerlines
The motion ratio is d3/d4.

For our trucks, the motion ratio is 48/66 = .73  (squared = .5329)

So with the stock front springs rated at 650 lbs, the wheel rate is 229.4.  The stock rear leafs are 180.

My RUSlow (Hypercoil) front springs have a 1,100 lb rating.  My (old style) Hotchkis leafs have a 278 lb rating.  So my truck's wheel rates are 388 front, 278 rear.

Yet my truck feels oversprung in the rear.  Using the math below, I have calculated the unsprung weight on each front wheel to 1,249 and on each rear wheel to be 825.

Comparing some possible setups:

 F rate F whl rate R rate R whl rate F-R Ratio rlbs/rate F lbs/rate R Stock 650 229.4 180 96 1.27 5.4 4.6 Hotchkis (old) 960 338.9 278 148 1.21 3.7 3.0 Hotchkis (new) 825 291.2 243 129 1.20 4.3 3.4 My current setup 1,100 388.3 278 148 1.39 3.2 3.0 1,100 388.3 243 129 1.60 3.2 3.4 1,100 388.3 180 96 2.16 3.2 4.6
These calculations just confused me more.  It looks like the front and rear rates are matched pretty well.

Not knowing what to do with this data, I did some more research.

calculating optimal spring rates:

The optimal spring rate equation:

SR = WR / (MR*ACF), where

SR = desired spring rate

WR = wheel rate

MR = motion ratio

ACF = angle correction factor

The following will solve these in sequence.

spring frequency:

Rancho offers the following guidelines when assuming a suspension frequency ("Hz" = Hertz = cycles per second = "cps"):

"For rock crawling up to 15 mph, start with .75 Hz on the front & .93 Hz on the rear.
For 4wd accessible trails up to 30 mph, start with 1.1 Hz on the front & 1.375 Hz on the rear.
For general on & off-road driving, start with 1.35 Hz on the front & 1.688 Hz on the rear."

Another site states "Typical frequencies range from 1 cps for large soft sedan to 2.5 for a racecar without aerodynamic down force."  On corner-carvers.com, one post stated "For a combination street/auto-x/track car I'd start at ~1.8-2.0Hz, then after you've driven it decide where to go."  And another stated "As a general rule, the resonant frequency of your rear suspension needs to be 10-15% higher than the front suspension to have level ride at 60-80mph."  And another Web site explained:

". . . there is a good reason not to have equal frequencies. The inputs are experienced at the front wheel and then some time later at the rear.  If you have equal frequencies then there will be mismatch in the phase of the damped curves, driven by the velocity of the car and the wheelbase length. This will force a pitching motion.  To avoid this, the rear frequency is generally higher by some multiple of the front.  Guideline values from Olney, Milliken, and SAE references support a range of 1.2 to 1.3 times the front frequency to achieve a flat ride."

Another site:

"The ride quality normally associated with the vehicle's response to bumps is a factor of the relatively low frequency bounce and rebound movements of the suspension system. Following a bump, the undamped suspension (without shocks) of a vehicle will experience a series of oscillations that will cycle according to the natural frequency of the system. Ride is perceived as most comfortable when the natural frequency is in the range of 60 to 90 cycles per minute (CPM), or about 1 Hz to 1.5 Hz. When the frequency approaches 120 CPM (2 Hz), occupants perceive the ride as harsh. Consequently, the suspension of the average family sedan will have a natural frequency of about 60 to 90 CPM. A high-performance sports car will have a stiffer suspension with a natural frequency of about 120 to 150 CPM (2 to 2.5 Hz)."

Taking the above recommendations into account, I will use 1.8 Hz front, 2.0 Hz rear.

If you want to measure actual frequency,

To calculate the suspension or spring frequency, check out this low-tech method from Smithees Race Car Technologies (great site):

"The bounce test measures the spring frequency directly, without calculation. Sometimes the spring frequency cannot be calculated accurately, e.g., large diameter coil spring working on a poor motion ratio, or leaf spring live rear axle. With leaf springs, calculation of the spring rate and spring frequency is impractical.

Smithees "Bounce Test" measures spring frequency quite accurately, in a matter of minutes, and the numbers can be used in the Smithees Weight Transfer Worksheet. The procedure is a no brainer, but exceptionally valuable in gaining a feel for what's going on.

The "bounce test" results are a key part of the easy to measure numbers we need to begin a suspension handling project. (If the car has coil overs or Mcpherson strut suspension, the "bounce test" cannot be done. But this is OK, because for these suspensions, the spring frequencies will calculate quite accurately from the spring motion ratios.)

Procedure

1. Remove the shocks from the car, and disconnect the anti-roll bar(s) if fitted.

2. Put the car on a level hard surface (eg concrete floor)

3. Seat a driver in the car (we need car at road going weight)

4. Start bouncing the car at one end - push down sharply, let the car bounce, and then push it down again to keep the bouncing motion going continuously. Look up at a sweep second hand clock and start counting the number of bounces in one minute. I usually count for 30 seconds and then double it. This number is the "cycles per minute".

5. Re-check a couple of times, to see that you can come up with a repeatable number. There can be problems. For instance with independent front suspension, it is just as valid to push down on one guard only, or push at the front, both sides together. If suspension is binding on one side, you will notice this, and it must be repaired.

6. If the end you are not bouncing is moving around a lot, you will not get a good figure - there is "some cross" over between front and rear. You may have to put the shocks back in that end you're not bouncing."

[Note:  it seems to make more sense to do one end at a time, leaving the anti-sway and shocks attached on the other end, and only disconnecting the bottom shock mounts]ant to try to measure f

wheel rate:

Wheel rate = (SF/3.128)2 * W, where

SF = suspension frequency and

W = sprung weight

So the only thing that needs to be determined now is the sprung weight for each wheel.  That's not easy.

Unsprung weight is that part of a vehicle's total weight that is not on the springs.  Sprung weight is the total weight minus the sprung weight.  The sprung weight is the weight that each shock is supporting (i.e. the corner weight less unsprung weight).  Unsprung weight is the tire and wheel, hub, upright, brakes, and about half the weight of the shock and control arms.  For the rear suspension of a truck, the unsprung weight is the tires and wheels, the entire rearend, and half the weight of the shock and traction bars.  The anti-sway bars are pretty much supported by the frame mounts.

I calculated unsprung weight by assuming 5,000 lbs, with a 56/44 weight distribution front-to-rear.  The stock weight distribution is 57/43, but I have removed the class II hitch, the spare, and the rear bumper (-208 lbs), as well as relocated the 75 lb battery to the rear.

Since both the driver and the fuel tank are on the left side of the truck, it has to be unbalanced side-to-side.  Because I am not even thinking about trying to run different spring rates on each side, I am just going to assume the same weight on each side.

As a point of reference, JJ's SVT weighed a stock Lightning with a 145 lb driver/jockey and 1/2 tank of gas (a full 25 gal. tank weighs 156 lbs @ 6.25 lbs/gal).  His findings:

Total = 4,894

LF 1,477 / RF 1,301 = 2,778 (56.8%)
LR 1,080 / RR 1,036 =  2,116 (43.2%)

The main things to account for in calculating unsprung weight are the wheels, the control arms, the rotors, the spindles, and the rear axle.

5,000 lbs * .56 =  2,800 front axle weight, /2 = 1,400 lbs per side

- 72 lbs wheel/tire

- 11 lbs 1/2 lower control arm

- 5 lbs 1/2 upper control arm

- 23 lbs spindle (BellTech; stock is ___ lbs)

- 27 lbs rotor/hub (Brembo; stock is 24 lbs)

- 12 lbs caliper (Brembo; stock is 17 lbs)

- 1 lb 1/2 shock

= 1,249 lbs front axle unsprung weight

5,000 lbs * .44 = 2,200 rear axle weight, /2 = 1,100 lbs per side

- 72 lbs wheel/tire

- 170 lbs 1/2 axle

- 17 lbs rotor

- 5 lbs caliper

- 10 lbs traction bars (estimated)

- 1 lb 1/2 shock

= 825 rear axle unsprung weight

Plugging this back into the formula (SF/3.128)2 W, and using the target 1.8F/2.0R frequencies from above, the optimum wheel rates are:

Front: (1.8/3.128)2 * 1,249 = 413 lbs

Rear: (2.0/3.128)2 * 825 = 337 lb

motion ratio:

angle correction factor:

The angle correction factor is used where the spring is not vertical.  The angle correction factor is the cosine of the angle.  For instance,

 Angle (degrees) 0 3 6 9 12 ACF 1 0.9986 0.9945 0.9877 0.9782
Fortunately, the rear ACF is 1 and the front spring angle is close enough to vertical (less than 5 degrees) that it is not worth calculating -- at least on a farm implement.
So, going back to the formula

optimal spring rate = wheel rate/motion ratio

and plugging in the numbers, we get

Front optimal spring rate = (413/.59) =  700 lbs/inch

Rear optimal spring rate = (337/1) = 337 lbs/inch

This makes absolutely no sense to me.  According to this information, I need to stiffen the rear up substantially, yet move back to the stock front springs.

If the data is recalculated at Rancho's recommended 1.35F/1.688R, the numbers are as follows:

Front: (1.35/3.128)2 * 1,249 = 232 lbs  (232/.59) =  393 lbs/inch

Rear:  (1.688/3.128)2 * 825 = 240 lb (240/1) = 240 lbs/inch

That's even softer than stock up front and about new Hotchkis rear stiffness in the back.

The data still makes no sense to me.

Further research is required.

A leaf spring calculator is here.  Another is here.  The Hotchkis leafs are 56.5" eye-to-eye, 2.5" wide, and .5" thick.  There are two leaves.  The first calculator shows 275 lbs/in.  The second shows 288 lbs.  If the checkbox for "diamond/tapered ends" on the first is not checked, the results between the two are nearly identical.

The stock (99-02) leaves are the same length and width, but there are 5, each .30" thick.  That works out to 142 lbs/in on the first calculator and 156 lbs/in on the second.

The Hotchkis springs with one leaf removed calculate at 130 lbs, very near what the stock 99-02 leaves calculate to.

The stock springs with one leaf removed work out to 110 lbs, and 80 lbs with two leaves removed.