Cut Springs and Spring Rate

This has been bugging me for quite a while:

"What happens to the spring rate when you cut it shorter?"

Since i'm a metric guy, spring rate is measured in kg/mm. A 5 kg/mm spring will be one mm shorter if a 5 kg weight is put on top of it while it's still on planet Earth. Stuff tends to behave differently if you measure it extra-terrestrially. Put 10 kg over it and it'll be two mm shorter from its free-weight length. Also we're talking about linear-rate springs here. Progressive-rate springs complicate matters.

My puny logic first thought that spring rate should remain the same as you don't change the spring material when you cut it. Turns out i was wrong. The spring rate increases when you cut it shorter. Googled about this problem last night and found this link. It's an online calculator which tells you the spring rate with some input variables like the number of coils, spring diameter, wire diameter, etc. The formula is below:

K=(G*d)/(8*C³*N)
Where,
K = Spring Rate
d = Wire Diameter
N = Number Of Active Coils
G = Modulus of Rigidity
C =Spring Index, a ratio of Spring Diameter (D) divided by the Wire Diameter (d)

We can see from the formula above that Spring Rate is inversely proportional to the Number of Active Coils (K ~ 1/N). Since cutting a spring basically reduces N, Spring Rate would increase with the same proportion! Example, my Ground Control coilover spring is 6 kg/mm and has 200mm free length. If i cut it 25% shorter to 150mm, i'd also increase the spring rate by 25% to 8 kg/mm.

An analogy i use is to compare two steel rods of the same diameter. One is half the length of the other. The shorter one would be more rigid, right? It bends less than the longer one when given a same load transversely. Wound those rod to the make a spring of the same diameter, you'll end up with one spring half the length of the other.

If we play with the variables, we'll end up with below summary:

1. Higher Modulus of Rigidity yields higher Spring Rate

This is a matter of aluminum spring vs steel spring of the same dimension give different spring rate

2. Increasing Wire Diameter yields higher Spring Rate

You should be able to imagine this intuitively. In case you can't, from the formula above, K is directly proportional to (d/C³) where C³ is (D/d)³ and we end up with K ~ d⁴. This reads to Spring Rate increases to the power of 4 for every increment of Wire Diameter.

3. Increasing Spring Diameter yields lower Spring Rate

By logic, to increase spring diameter, you need to get longer wire. Longer wire is less rigid. Less rigid means lower Spring Rate. Enough? If you prefer the difficult way, K is inversely proportional to C³ , again, where C³ is (D/d)³ and gives us K ~ 1/D³. This reads to Spring Rate reduces by root of 3 for every increment of Spring Diameter.

and finally, back to our main discussion..

4. Reducing Number of Coils yields higher Spring Rate

See above.

This should be equally important to AE86 and Charmant owners:

- Stock OEM Spring Front: 380mm free-weight length and 1.8 kg/mm

- Stock OEM Spring Rear: 360mm free-weight length and 2.2 kg/mm

I didn't measure them myself.. I got this from an internet forum so take it with a grain of salt.