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Mark Ortiz Automotive is a chassis consulting service primarily serving oval track and road racers. This newsletter is a free service intended to benefit racers and enthusiasts by offering useful insights into chassis engineering and answers to questions. Readers may mail questions to: 155 Wankel Dr., Kannapolis, NC 28083-8200; submit questions by phone at 704-933-8876; or submit questions by e-mail to: mortiz49@earthlink.net. Readers are invited to subscribe to this newsletter by e-mail. Just e-mail me and request to be added to the list.

DAMPING RATIO

I would like to ask you an advice about the way to choose the right damping coefficients for a given race car.

I normally work using damping ratio as a index of how much damping I am using. Literature references about mechanical vibrations state that a common sense value for damping ratio should be 0.7, but I have seen a lot of very fast racecars running with damping ratios of more than 1 at least at low speed.

Very often, in these aero cars, there is then a limited difference in slope between low and high speeds of the shaft. Talking with guys who work on post rigs I heard that they choose a damping ratio of about 0.6 but they calculate it in a quite wide range of speeds, so this number is probably an average of what the dampers do in both low and high speed. But how to calculate the relationship between the two areas?

What is your thought about setting the correct damping on a damper and about building its curve from scratch? What is the influence of aerodynamic load on the car?

The really short answer is yes, 0.6 to 0.7 is optimal.

A slightly longer answer is that the matter is controversial, but 0.6 to 0.7 is about midway in the range that various people like.

A better answer than either of those would be that the concept of damping ratio is meaningful, but the behavior of actual cars is so complex that describing it in terms of a damping ratio is only a very crude approximation, albeit a useful one.

For readers who haven't heard of damping ratio, a bit of explanation will be helpful. The damping ratio is a measure of how stiff the damping is, for the setup and the mass of the car. The stiffness of

a damper can be expressed as its damping coefficient (C). This is the amount of force the damper produces per unit of shaft velocity. It thus has units of force/velocity – in English units, pounds force per inch per second, or lbf-sec/in; in metric units, newtons per meter per second, or N-sec/m. It corresponds to the slope of the force versus velocity line when we test a damper on a shock dyno.

The damping ratio (ζ, zeta) is the ratio between the damping coefficient we have (C) and the damping coefficient that would give us what is called critical damping (C_crit). That is,

ζ = C/C_crit(1)

Critical damping is the least damping that will make the system non-oscillatory – that is, will make it return to its neutral or undeflected position after being disturbed, in the least time possible without any overshoot whatsoever, i.e. with no oscillation whatsoever.

For a given spring constant (k) and mass (m),

C_crit = 2√(km) (2)

A professor I work with at UNC Charlotte told me he did some research on damped oscillatory systems and found that the system returns to a reasonable semblance of rest quickest when the damping ratio is such as to allow moderate overshoot: between 0.4 and 0.8. 0.6 would be right in the middle of that range. That is, the system may oscillate a little, but it returns to a condition of near-zero deflection, and near-zero velocity, sooner than it would if critically damped. If the system is more lightly damped than this, it oscillates perceptibly for a significant number of cycles, and stays perceptibly in motion longer.

All this theory has its limits, however, when we try to apply it to actual cars. Engineers like to think in terms of damping coefficients and ratios, because these are used in equations that model oscillatory behavior. If you don't have an input for damping coefficient, you can't use the equations.

Trouble is, real shocks don't have constant damping coefficients. Some come pretty close, but many are deliberately made non-linear. Not only are they made non-linear, often even relatively linear ones are deliberately made stiffer in extension than in compression, or sometimes vice versa. Some are deliberately made position-sensitive. That is, they have different force/velocity characteristics depending on where the piston is in the tube.

Is this complicated enough yet? We're just getting started. Not only is the force/velocity relationship often non-linear, but force is not exclusively a function of velocity at all – even a highly non-linear function. When we valve shocks stiffly, they start to act like springs some of the time: they sometimes make forces in the same direction as velocity rather than the opposite direction. At such times, the damping coefficient goes negative, and so does the damping ratio. Far as I can ascertain, this happens because oil is compressible, and the metal parts in shocks are not perfectly rigid. When the shock is making serious force and the piston reverses direction, the compressibility

of the fluid and the expandability of the tube make the fluid on the high-pressure side of the piston spring back. Only when the fluid and tube are done springing back does fluid start to flow through the piston in the direction the designer intended.

Short of this extreme, non-rigidity of fluid and metal may not reverse direction of flow through the piston, but may reduce or increase the rate of flow. Reduction of flow occurs when the piston is gaining speed after a reversal of direction; increase of flow occurs when the piston is slowing down.

In general, it is undesirable to have the shock acting like a spring. Ways to reduce such effects include using a bit more bleed, using lower operating pressures, and making the parts stiffer. Unfortunately, more bleed may conflict with our need for low-speed damping. Lowering working pressure means either making the damper larger in diameter, or giving it a greater motion ratio and a longer stroke. For a given material, making anything stiffer usually means making it heavier. So there are inescapable tradeoffs between making the damper work well and making it small and light.

As if this weren't enough, the spring constant k, as a wheel rate, isn't necessarily constant either. It is often deliberately made to vary with suspension displacement. In some cases the damping coefficient C varies as k varies; in some cases not.

For example, we may use a bump rubber to get a rising k as the suspension enters some range of displacement. In that case, k increases and C does not, so ζ decreases. Or, we might use rocker geometry to make the motion ratio of a coilover increase as the suspension compresses. In this case, the damping and the springing get stiffer together.

It might be supposed that in the latter case, the damping ratio at least stays the same as the suspension stiffens. Nope.

Consider a case where the rocker geometry is such that the coilover-to-wheel motion ratio is 0.5 at static position, and increases by a factor of √2 at some amount of compression, making it about 0.7. What happens to k? It doubles. If the shock is linear, what happens to C? It also doubles. And what happens to C_crit? It goes up with √k, meaning it only goes up by a factor of √2. So what happens to the damping ratio, C/C_crit? It increases by a factor of √2. If it was 0.7 at static, it increases to 1.0.

Using some combination of third spring or ride-only snubber, shock shaft bump rubbers, and rising-ratio rocker geometry, it is possible to arrive at a combination that would give a nearly constant ζ through the full range of travel, if the shock is linear. However, most cars won't be this way, and there is no guarantee that a constant-ζ setup would give us a faster car.

As to the influence of downforce, again there's a short, simple answer: it doesn't do anything. However, as with the previous short, simple answers, the real story isn't nearly so simple.

If we imagine that we have a hypothetical car, with linear springing and damping, and we just add an aerodynamic downforce, what happens? The suspension compresses, but k, C, and m all stay the same, so C_crit and ζ don't change either. In particular, adding downforce is not the same as adding mass.

In the real world, however, when we decide to build a downforce car, everything changes. The days are long gone when people would take an existing car and just bolt on some wings. Downforce cars today are heavily dependent on ground effects, which in turn are heavily dependent on height and

attitude of the car's underside with respect to the road surface. In order to keep the underside from moving much with respect to the road, we have to spring the car very stiffly. If we want to have at least a bit of compliance in the lower speed ranges, we will design in some amount of rising-rate effect.

So although downforce in itself does not affect damping ratio, the design implications of pursuing downforce, especially under current rules, affect damping ratio a lot.