Spring Rates & Suspension Frequencies - Plus Calculator!

When we talk about suspension settings and setups, you often hear spring rates and damping levels thrown around without context or proper validation. Much like spouting peak horsepower without talking about peak torque or better yet the torque curve, spring rates and damping are rather meaningless without taking your suspension frequency into account.

Sprung Corner Mass (kg):


Frequency:


Motion Ratio:

Spring Rate: kg/m

Spring Rate: lbs/in

Natural Frequency

Before we can examine suspension frequency, we need to introduce the concept of natural frequencies. Every [elastic] object, material, etc has a certain speed of oscillation that will occur naturally when there are zero outside forces or damping applied. This natural vibration occurs only at a certain frequency, known as the natural frequency.

Suspension Frequency

Derived straight from Natural Frequency, is the suspension frequency found in cars. This is how fast the suspension travels up and then back down to the same point when you drive over a bump. If cars did not have shocks/dampers, the springs would continue to bounce up and down at this rate for quite some time.
By examining the suspension frequency, we are able to fairly accurately predict the handling characteristics of the suspension and how it will 'react' to driver inputs, road surface feedback, brake/acceleration loading, and downforce loading.
To calculate suspension frequency for an individual corner, you need Mass and Spring rate:

f = 1/(2π)√(K/M)

f = Natural frequency (Hz)
K = Spring rate (N/m)
M = Mass (kg) 

When using these formulas, it is important to take Mass as the total sprung mass for the corner being calculated. That is, the axle weight divided by two, minus an estimated or measured unsprung mass for that corner (things like wheels, tires, brakes, control arms, suspension components etc. all contribute to unsprung mass. Anything that is not supported by the springs).

So now we have a calculable frequency, which has next to no science behind what frequency to use on your car beyond proven empirical findings from decades of racing.
Choosing a specific frequency for your application is still a little beyond my current knowledge base, however the following is a good guideline for choosing your starting point for calculations:

0.5-1.0Hz Passenger cars, typical OEM
1.0-1.5Hz Typical lowering springs
1.5-2.0Hz Rally Cars
1.5-2.5Hz Non-Aero racecars, moderate downforce Formula cars
2.5-3.5Hz Moderate downforce racecars with up to 50% total weight in max downforce capability
3.5-5.0+Hz High downforce racecars with more than 50% of their weight in max downforce

 One of TopGear's many tests showcases some of the differences between various race cars that are likely to have completely different suspension frequencies. The Touring car in the middle likely has a 3.0-3.5Hz suspension frequency with it's simple aero package and race tires, while the F1 car will have a frequency in excess of 5.0Hz, as its downforce produced and ultimate grip level is far greater. The bike will run off the same principles as a car's suspension.

One of TopGear's many tests showcases some of the differences between various race cars that are likely to have completely different suspension frequencies. The Touring car in the middle likely has a 3.0-3.5Hz suspension frequency with it's simple aero package and race tires, while the F1 car will have a frequency in excess of 5.0Hz, as its downforce produced and ultimate grip level is far greater. The bike will run off the same principles as a car's suspension.

While much of this may seem rather arbitrary, there is some reason behind selecting frequencies. The higher grip levels that are available to you, the higher frequencies you will be able to and be forced to use. Things like bushing stiffness, tire compound, tire width, downforce, etc. all contribute to grip, and increasing any one of those will allow/require you to run higher suspension frequencies. For OEM passenger cars and rally cars, the available grip levels are extremely low, and therefore the suspension frequency has to remain low to provide as much mechanical grip as possible. On race cars with wide, race compound tires, solid bushings, and high levels of downforce, the frequency is significantly increased to take advantage and work with the available traction.
Like adjusting any suspension component: increasing the stiffness on one corner, axle, or side of the car, will reduce the available mechanical grip for that corner, axle, or side. It is for that reason that you will want to run as low a suspension frequency as possible for the current vehicle setup. Any higher and you are sacrificing mechanical grip, but too low and the car will respond too slowly for the grip available in the tires.

I will use our Saab 9-5 Time Attack car as a working example to provide some better understanding:
~1270kg empty curb weight, ~1360kg with driver and fuel, and ~680kg of downforce at 250km/h. The front unsprung corner weight is ~385kg, and the rear is ~205kg.
Front downforce will consist of a large front splitter and diffuser, ducted hood venting, and vented wheel wells.
Rear downforce will be the new BMSPEC rear wing, and a moderately sized diffuser with a full flat floor and side skirt extensions.

The first thing to notice is that the mass is vastly different from the front and rear of the car. This is why frequency is much more valuable than spring rate: it provides a relatively similar 'feel' for different corner masses. If we want the front and rear to behave the same, we will run similar frequencies in the front and rear, and for this reason we will run the same frequency from one side to the other on the same axle.

mission raceway august-11.jpg

Without any downforce, we run near identical frequencies in the front and rear of the car to keep the car balanced. With stiff bushings, race compound tires, and a fairly stiff chassis from the factory, a 2.0Hz frequency will promote good mechanical grip while resisting and controlling the load levels produced by the race compound tires.

However, with an aero package that all begins to change. Adding downforce means increasing the suspension frequency to compensate for the increased loads at higher speeds.

We will be running higher downforce in the rear than the front to try and shift our load balance toward the rear of the car, closer to an optimal 50/50, and to overcome the inherent lift produced by the sedan's body style.
I will touch more on balancing aero, and how this differs for AWD and RWD platforms in a separate aerodynamics article.

On the front we will be increasing from 2.0Hz (no aero), to 2.5Hz, and in the rear we will bump up the frequency to 3.0Hz to accommodate the higher amount of rear downforce. This imbalance of 2.5/3.0 will become extremely balanced at higher speeds when downforce starts becoming a bigger factor, but will still promote a healthy amount of rotation in low speed corners.

Spring Rates

Once the desired frequency has been established for a given corner/axle, you can now start calculating proper spring rates. This is the entire purpose of establishing our suspension frequency, and the end goal is to determine the best spring rates for your application, vehicle, and specific corner weights.

The only added piece of information required, in addition to the previous formula, is the motion ratio of the suspension type. This is the ratio between how much the spring travels relative to the wheel. Different suspension types will have a different range of possible values, and ultimately you will have to take measurements to determine your exact motion ratio throughout your suspension's entire range of travel.
Now with Motion Ratio, Mass, and Frequency in hand, we can insert that all into our formula below to provide us with the corresponding spring rate:

K= (4π²F²M)/mr²

f = Natural frequency (Hz)
K = Spring rate (N/m)
M = Mass (kg) 
mr = Motion Ratio (Spring:Wheel)

 

While the above formula itself is very easy to manage, it can be tedious to calculate out multiple corner weights and multiple frequencies to see how the desired spring rates will change. To simplify our lives and yours, we've created this spring rate calculator below:

Daniel Routley

DRTuned Media, Langley, BC