## Spring Brake

Understanding damping ratios and the notion of critical damping

**BY CLAUDE ROUELLE**

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### Racecar suspension is a complex science, and requires much more than just an understanding of the dampers, but getting to grips with them is a good place to start

The damper is the brake of the spring.’ In this article, we will review that theory, as well as the pros and cons of some oversimplified quarter-car simulations.

If you would put a mass on a spring with no damper and push the mass down by 60mm (as seen on the green trace in **Figure 1**), then release your force, the mass will oscillate indefinitely. We should say nearly indefinitely because ultimately the movement will stop, mainly due to the spring material’s internal resistance, but also because of air friction. Note that in this example you allow the spring to work in tension and move not only down but also up by 60mm. As we will see later that is not the case in most car suspensions.

The mass position, as a function of time, can be described by a sinusoid called harmonic motion.

The equations are as follows:*z=Z_0 e^(-ζω_n t) sin (ω_D t+φ)*

With the undamped period being*ω_N=√(K/m) in rad/sec* and the undamped frequency *f =1/2π √(K/m)* in Hertz (Hz)

The critical damping coefficient is*C_crit=2√Km in N/(m/sec)*

The damping ratio is *ζ=C/C_crit =C/**(2√Km)*, which is dimensionless.

The damped natural period is *ω_D=ω_N**√(1-ζ^2 ) in rad/sec*, and the damped frequency is *f =1/2π √(K/m) √(1-ζ^2 ) *in Hertz (Hz) with Z0 being the initial amplitude (in meters), t the time in seconds, φ the phase difference (in radian), K the spring rate (in N/m), m the mass (in kg), and C the damping in N/(m/sec).

Be careful with the units here. Many make the mistake of forgetting to convert their spring stiffness from lb/ft into N/mm rather than Newton per meter. Likewise for the damper, that is in Newton per meter per second.

## Critical Damping

That’s it for the mathematical formulae, but what does it mean in practice? Let us start with the notion of critical damping and a simple example. You go into your garage and push down hard on your car bonnet/hood. The front part of the car’s suspended mass goes down. Let it go, releasing the downward force and the car will return to its initial position with a certain number of oscillations of different amplitudes. If the front suspension is critically damped, it will return to its initial state as quickly as possible without any overshoot. That is what 2√Km means, and it is represented by the blue trace in **Figure 1**. The damping ratio is one.

If your suspension is less damped, let us say at 10 or 20 percent of the critical damping (damping ratio of 0.1 or 0.2) such as shown by the orange trace in **Figure 1**, it will take a while for the suspension to get back to its initial state.

Here’s a simple illustration, and a trick for all racecar engineers. There are circuits where there is an annoying bump in one of the straights, usually above a tunnel passing under the racetrack. That bump did not exist in the first few years of the circuit’s existence, but then with temperature variation and seasonal fluctuations the tarmac moves and creates the bump. As it is in a straight, approached at a relatively constant speed, neither in a braking or acceleration zone, it has no influence on car performance and is just an annoyance for the driver.

Have a look at the damper’s potentiometer signals: if it takes more than one, or even worse, more than two oscillations after that unique bump for the dampers to stabilise, you know your suspension is underdamped. You do not know (yet) if the lack of damping is in compression or rebound, low or high speed, but at least you know you have insufficient damping.

**Figure 1: Amplitude vs time for a simplified one degree of freedom (DOF) mass on a spring ****with different damping ratios**

The other specific case is the one of a critical damping of 0.7 (as shown by the red trace in **Figure 1**). That is the damping that will bring the mass back to its initial stage as quickly as possible, albeit with the ‘cost’ of an overshoot. If response to a steering input, or returning back as quickly as possible to a given ride height after a bump is the main target, usually a damping ratio of 0.7 is ideal.

We will not go into too much mathematics here, but you will notice that 0.7 is half of the square root of two. That is not a coincidence. It is worth noting that the mass goes back to its initial state in less than half of the time it takes with the critical damping.

The black trace in **Figure 1** with a damping ratio of 1.8 shows an over-damped suspension. As we will see later, over-damping does not make sense in terms of tire mechanical grip consistency, but it could make sense in terms of aerodynamics, especially with cars requiring a constantly low ride height.

## Minimum Drag

Many years ago, that kind of damping was used in rear suspension rebound on Superspeedways in NASCAR. Aeromaps show that often a relatively high front ride height and an exceptionally low rear ride height, (of about 3mm) give minimum drag. On tracks like Talladega and Daytona, minimum drag is a main target, more so than handling and downforce. However, the cars had to go through technical inspection with quite high static ride heights.

The trick was to use soft rear springs with just enough pre-load to maintain the rear part of the suspended mass at its legal static ride height for technical inspection. The springs used had low bumps and exceptionally high rebound damper settings.

As soon as the car left the pit lane, the rear ride height went down under the effect of downforce, banking, and every single little track bump. It went down until the rear springs were coil-bound, reaching an exceptionally low rear-ride height. So low in fact, and with such stiff

suspension, that the suspended mass frequencies were so high they made the drivers’ vision blurred. The cars were faster but difficult to drive and unsafe. When NASCAR cottoned on to the trick, it imposed specific sealed dampers for these circuits.

As for spring stiffness, the choice of damping is both a matter of art and science. Usually, passenger cars have a damping ratio of 0.3 (when ride and comfort are the main targets) to 0.5 (when response is the main target), while racecars are around 0.6-1.0.

Some high-end road sports cars have active, controlled damping using signals from sensors such as steering, lateral acceleration, and yaw rate (gyro) and a damping ratio in the 0.3 region in a straight line for comfort, as much as 1.5 for good turn-in response and back to 0.5

at the corner apex. Defining such active dampers and, even more, their controller, requires serious R&D competencies, budget, and time. Consequently, they are outlawed in most racing categories.

When I am asked what damping ratio to choose, I often suggest trying 0.7 to start. That is because it will give you the quickest ride height recovery. You will probably not beat the lap record the first time you put your new car on the track with that, but you will not be ridiculous. That is the simplified answer. Oversimplified in fact, because it does not tell you if you need the same damping ratio in bump and in rebound, at low and high speed, or even how to define low and high damper speed.

To obtain a better answer you need a two degree of freedom simulation, as presented with the quarter-car model in **Figure 2**.

**Figure 2: Two degree of freedom quarter-car model with suspended and non-suspended masses, tire stiffness and damping, and suspension stiffness and damping**

Mu is the non-suspended mass, Ms is the suspended mass (both in kg). Kt is the tyre stiffness, and Ks the suspension stiffness, both in N/m. Ct is tyre damping and Cs suspension damping, both in N/(m/sec).

The road profile, Zt (in meters) is an input, while the vertical moment of the non-suspended Zu and the suspended mass Zs (also in meters) will be outputs.

Most academic quarter-car models are presented this way, but they have their flaws. In my next article I will explain what a good quarter-car model should be, but for now will just explain why such models are unsuited to real suspension design and development.

The following remarks are numbered with the same reference on **Figure 2**.

### When I am asked what damping ratio to choose, I often suggest trying 0.7 to start. That is because it will give you the quickest ride height recovery

## Tire Modelling

We all have seen slow-motion video of a race tire hitting a succession of kerbs. The tyre takes off, lands a few kerbs later, and does not stick with the kerb’s surface. And yet on most academic quarter-car models, the tyre spring can work in extension!

That’s the first major issue. Secondly, only professional team engineers with access to detailed tire models, or able to undertake complex tire tests (when allowed), will have access to such essential information and a good knowledge of a tire’s vertical stiffness. It is vertical load, speed, pressure, camber, temperature, slip angle, slip ratio, and also excitation frequency sensitive.

I was once told by a tire manufacturer that a race tire has no damping. Clearly, everything is relative. It’s true that, compared to a suspension damper, the tyre has little damping, but I know that if I let a wheel and tire fall on the ground from a given height, there will be a logarithmic decrement of the successive amplitudes. So, there clearly is some damping in a tyre, but again only specific tests allow us to acquire such tyre damping characteristics.

It is worth noting that on most seven post rigs the tires are not rotating, and therefore have no speed, no slip angle, and no slip ratio. As such, the tire stiffness and

damping are not representative of reality. Most seven post rigs are still useful, but we need to remember that their test results have to be analyzed in relative comparison.

And no more than the tire does the suspension spring work in extension stiffness. We never have seen a spring welded to a spring platform, at least not on a racecar! There is one exception to this though: suspensions with torsion bars where the wheel movement can be controlled in both up and down directions.

A good quarter-car model should (but most will not) consider that the suspension movement could be limited by the minimum spring length in full compression.

Moreover, these simplified models do not consider several other important aspects of suspension stiffness: the spring’s possible non-linear stiffness (in fact, there isn’t any linear spring stiffness); the possible assembly of two or more springs of different stiffnesses; the spring gap (in full drop it is possible that the spring is not in contact with the spring platform); the spring preload; the use of a bump stop (also called a bump rubber) and its gap or pre-load, and finally possible bump rubber hysteresis (that is both stroke and stroke speed sensitive).

Most simplified quarter-car models just use a constant damping, C. The race engineer will quickly find that approach irrelevant because a graph of damper dyno test data will show different damping in compression and rebound, as well as different damping in low and high-speed conditions. That is without even considering damper force vs speed hysteresis.

Then there are the basic mechanical constraints: minimum and maximum eye-to eye damper length should be considered.

Road inputs, with or without a rotating wheel (in that case, at a given speed), or vertical acceleration vs time as the one induced by an actuator at the tire contact patch on a seven-post rig will be input options.

The following road profile also needs to be considered: sinusoidal, rectangular, and ramp bumps and, most importantly, random, or stochastic bumps.

Most simplified models do not consider inputs on the suspended and non-suspended masses such as lateral or longitudinal weight transfer, vertical acceleration from the track banking or slope, or aerodynamic downforce.

## Challenges Ahead

These considerations show just some of the limitations of most quarter-car simulations. There are other challenges too, such as the choice of numerical integrator, especially when we have non-linear and discontinuous inputs. Examples of discontinuous inputs are a bump rubber suddenly becoming active, a damper being fully extended, a tire off the ground, or a fully compressed spring. And all that without considering the possible input of braking or acceleration torque, or slip angle and tire transient model with relaxation length.

In the next article, we will examine the challenges and benefits of a good quartercar model, and answer the main question that you might use one for: between ride, aerodynamic downforce consistency, tire mechanical grip and response, what are we looking for, and how do we decide how best to make compromises?