When it comes to how springs work, I think most of you are probably familiar with the relationship between load, spring rate, and displacement, and may have even calculated it yourselves at some point. But figuring out how fast a spring moves is probably not something most people are familiar with.
Because of that, you hear all sorts of imaginative claims out there, such as, “Adding preload makes a spring move faster,” or “If you install a spring with a higher natural frequency on a car, the spring will move faster.”
But is that really true?
Today, let’s take a look at what factors actually determine how fast a spring moves!
Let’s say you’re driving straight, and you hit a large obstacle that sends a sudden, significant impact into only the front-left tire.
Strictly speaking, the body of the car would also move upward slightly in this situation. But to keep the calculation simple, let’s ignore the vertical movement of the body and assume that all the energy input into the front-left tire is used to compress the front-left suspension.
Now, let’s calculate how fast the spring moves as it rebounds after being compressed by the impact.
The conditions for the car are as follows:
Suspension motion ratio: 1
Front-left unsprung mass: 30 kg
Spring compression caused by the impact: 30 mm
Wheel-rate spring rate: 490 N/mm
First, the formula for the spring’s velocity is:
v = x√(k/m) where: v: velocity (m/s) x: spring compression (m) k: spring rate (N/m) m: mass (kg)
Since we need to use consistent units, we first convert the spring compression of 30 mm to 0.03 m, and the spring rate of 490 N/mm to 49,000 N/m.
Then:
v = 0.03 × √(49000/30) = 0.03 × 40.410394702 = approximately 1.2
So, we can see that the spring moves at approximately 1.2 m/s, or 1.2 meters per second.
In reality, various non-conservative forces come into play, such as damper force, friction, hysteresis losses in the damper bracket’s rubber mount and suspension bushings, and so on. So, if you actually measured it, you would get a slightly lower value.
But if we look at the pure mass-spring system and ignore those factors, this is how the speed is determined.
So why does this happen???
The reason is that, as a property of springs, the elastic potential energy stored in a compressed spring is converted into kinetic energy when the spring extends again.
The formula for elastic potential energy is:
E = 1/2kx²
The formula for kinetic energy is:
E = 1/2mv²
Since these two energies are equal,
1/2kx² = 1/2mv²
Solving this equation for v gives us the formula we started with:
v = x√(k/m)
From this equation, we can see that the greater the amount of compression caused by the impact, the higher the spring rate, and the lower the unsprung mass, the faster the spring will move as it extends again!
By the way, what happens if we include the fact that the body of the car also moves upward?
The calculation gets a little more complicated, but let’s work it out while we’re at it.
If only one wheel is hit, we also have to calculate the change in body angle, which makes things complicated. So, let’s assume that all four wheels experience the same impact at the same time.
The sprung mass is assumed to be 1,000 kg.
・Total elastic potential energy for all four springs
E = 1/2×49000×0.03²×4 = 88.28 J
・Velocity ratio between the sprung and unsprung masses
Let’s use uppercase V for the velocity of the sprung mass and lowercase v for the velocity of the unsprung mass.
Likewise, let uppercase M represent the sprung mass and lowercase m represent the unsprung mass.
V = (m/M)v = {(30×4)/1000}v = 0.12v
・Kinetic energy
The total kinetic energy is the sum of the kinetic energy of the sprung mass and the unsprung mass:
E = 1/2MV²＋1/2mv²
Solving this equation for v gives us:
v = √{2E/(m＋m²/M)}
Substituting the numbers:
v = √{2×88.28÷（120＋120²÷1000)}
= 1.14455231423
Therefore:
Velocity of the sprung mass, V = approximately 0.14 m/s
Velocity of the unsprung mass, v = approximately 1.14 m/s
The speed at which the spring extends is the difference between the velocity of the sprung mass and the velocity of the unsprung mass, so:
1.14 - 0.14 = 1 m/s
In other words, the spring moves at 1 meter per second.
This is slightly slower than the 1.2 m/s we calculated at the beginning.
That’s because some of the input energy gets used up lifting the body.
Now, in both cases, the only things we use to calculate the velocity are displacement, mass, and spring rate.
Strictly speaking, this calculation gives us the velocity at the instant the spring starts extending from its maximum compression. The spring’s velocity varies depending on its position in the stroke, since the motion can be treated as the projection of uniform circular motion.
But even when we take that into account, the factors that determine the speed are ultimately still displacement, mass, and spring rate.
If the additional displacement caused by the impact remains the same at 30 mm, adding preload does not increase the elastic potential energy associated with that additional displacement.
Therefore, preload does not directly affect how fast the spring moves. (Except when the spring reaches full extension.)
The natural frequency of a spring by itself is determined by F = 1/2π√(k/m).
For example, a spring with a smaller mean coil diameter and fewer coils—that is, a spring made from a shorter length of wire—has a lower effective mass m, which increases its natural frequency. In other words, if you look at the spring itself, a lighter spring moves faster.
However, in an actual car, the spring is moving a sprung mass and an unsprung mass that are far heavier than the effective mass of the spring itself. In this case, the speed at which the spring moves is determined by the amount of energy involved in moving those masses.
So, ultimately, if the spring rate and the mass being moved remain the same, the spring will move at the same speed whether you use a spring with a higher natural frequency or one with a lower natural frequency.
Well, strictly speaking, using a lighter spring does make the car itself slightly lighter, so in that sense, the spring might end up moving a little faster.
But… the difference in spring weight between different brands is usually only around 500 grams per spring at most.
You might hear someone say, “This spring is lighter, so it moves faster and is therefore better!”
But if you put a 500-gram weight on the lower control arm and conduct several blind tests, and the person can’t consistently tell whether the weight is there or not, then I’d say the practical difference is negligible.
That’s about how much we’re talking about here.
So, that’s the story of how fast a spring moves.
Displacement. Mass. Spring rate.
Come on, everyone, say it with me: displacement, mass, spring rate!
Those are the factors that determine how fast a spring moves.
That said, this isn’t something you’ll have much opportunity to use when setting up a car.
Inputs from lateral and longitudinal G forces may feel fast, but they aren’t nearly as instantaneous as the kind of impact we assumed above. So, the stroke velocities caused by pitching and rolling are literally orders of magnitude slower than under the conditions we just looked at.
So, this is probably more of a piece of knowledge for suspension nerds like me. But hey, it might at least help answer the question, “Is this really true?” Haha.