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!
A Simple One-Wheel Model
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 the maximum speed the spring reaches 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: 49 N/mm
First, the formula for the spring’s maximum velocity is:
v = x√(k/m)where: v: maximum 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 49 N/mm to 49,000 N/m.
Then:
v = 0.03 × √(49000/30) = 0.03 × 40.410394702 = approximately 1.2 m/s
So, in this idealized model, the spring reaches a maximum extension velocity of approximately 1.2 m/s.
At the instant of maximum compression, of course, the spring’s velocity is zero. It then accelerates as the stored elastic energy is converted into kinetic energy, reaching its maximum velocity as it passes through the equilibrium position.
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 somewhat different value.
But if we look at the pure mass-spring system and ignore those factors, this is how the maximum 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
At the point where the stored elastic potential energy has been converted into kinetic energy,
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 higher the maximum speed the spring reaches as it extends again!
What If the Body Moves Too?
By the way, what happens if we include the fact that the body of the car also moves?
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 = 88.2 J
・Velocity ratio between the sprung and unsprung masses
Let’s use uppercase V for the speed of the sprung mass and lowercase v for the speed of the unsprung mass.
Likewise, let uppercase M represent the sprung mass and lowercase m represent the total unsprung mass.
As the springs extend, the sprung and unsprung masses move in opposite directions. From conservation of momentum, the magnitudes of their velocities are related by:
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 = √{2×88.2÷（120＋120²÷1000)}= approximately 1.14 m/sTherefore:
Speed of the sprung mass, V = approximately 0.14 m/s
Speed of the unsprung mass, v = approximately 1.14 m/s
Here’s the important part: as the spring extends, the sprung mass and unsprung mass are moving in opposite directions. The body moves upward while the unsprung mass moves downward.
So the speed at which the spring itself extends is the relative speed between its two ends. Since those ends are moving in opposite directions, their speeds are added:
1.14 + 0.14 = approximately 1.28 m/s
In other words, in this model the spring reaches a maximum extension velocity of approximately 1.28 m/s.
Interestingly, that’s slightly faster than the approximately 1.2 m/s we calculated when we assumed that the body didn’t move at all.
That might seem strange at first, but the reason is simple. We’re talking about the speed at which the spring changes length—that is, the relative speed between its two ends. In the first calculation, only the unsprung mass was allowed to move. In the second calculation, the sprung mass and unsprung mass move away from each other in opposite directions, so both motions contribute to the spring’s extension velocity.
Now, in both cases, the quantities involved in determining the velocity are displacement, mass, and spring rate.
Strictly speaking, the velocities calculated above are the maximum velocities reached as the spring passes through its equilibrium position. At maximum compression, the velocity is zero, and 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.
What About Preload and Natural Frequency?
If the additional displacement caused by the impact remains the same at 30 mm, adding preload does not change the relationship between that additional displacement, the spring rate, and the masses being accelerated.
Therefore, for the idealized linear spring system we’re considering here, preload does not directly determine how fast the spring moves. (Except when the spring reaches full extension or other boundary conditions come into play.)
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 can respond at a higher natural frequency.
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 primarily by the displacement, spring rate, and the masses involved in the motion.
So, ultimately, if the spring rate and the masses being moved remain the same, simply choosing a spring with a higher natural frequency by itself does not make the suspension move dramatically faster.
Well, strictly speaking, using a lighter spring does make the car itself slightly lighter, so in that sense there can be a very small difference.
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 main factors that determine how fast the spring moves in the idealized systems we’ve looked at here.
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 much slower than under the impact 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.