Race car daping A nuber of issues ago I wrote an article on exploring approaches to specify a race car daper. This article is the second in that series and we shall be following on fro where we left off. In particular in this article we will be exploring the use of daping ratios and what they tells us and ore iportantly how we can use it. efore we get into this let s recap where we are at with the first article. In the first article I presented the exaple of controlling a quarter car ass being subjected to a sinusoidal road input. In that article we explored what daping would have to be used to control this and what we could do via an inerter. The conclusion of this was that daping alone wouldn t get the job done and an inerter would only work at a specified frequency. However we did leave the article on the note that we could approxiate what we wanted to do by approxiating the job with daping ratios. This is where we will take up the discussion. As a precursor to what we will discuss let e go through the two schools of daping I have encountered over y career. The first school dictates all the daper does is control the oscillation of the wheel and that s it. The second school regards the daper as an essential eleent of the setup of the race car and in addition to controlling the oscillation it is used as a vital tool to control chassis attitude and control tyre loads. During y career I have varied between both schools of thought and I trust what we will discuss here ight provide soe answers to this conundru. Let s talk about who we can characterise dapers through daping ratios. As before we will be using the ¼ car odel approxiation to draw our line in the sand. I realise that a real car functions with a chassis and four contact patches. However bare with e this is a very useful approxiation. To refresh our eories Let s look at the quarter car odel, Fig-1 quarter car odel.
Looking at this we have a syste with 4 degrees of freedo. ody oveent and velocity and wheel oveent and velocity. To solve this properly we need to start solving eigenvalues and eigenvectors and this doesn t lend itself to nice hand calculations. At this point we need to ake soe approxiations. The approxiation that we are going to ake is, << << T (1) t Please note all the rates we are about to quote are going to be in wheel rates. The astute reader will instantly recognise that this is perfectly valid for a road car, but we could run into trouble with an open wheeler applications where the spring rate of the chassis are of the sae agnitude as the tyre, or in a V8 Supercar or NASCAR where the unsprung asses are quite high. However bare with e. What we are constructing here is a rough rule of thub. You ll see its power as we carry on. In this case the governing equations of the sprung ass reduce to, x = x C x () What this eans in layan s ters is that the acceleration of the sprung ass is the su of the spring rate of the daper ties the daper oveent and the daping rate ties the daper velocity. The daping rate is the slope of the daping curve at that particular daping velocity. Let e illustrate this in Figure, Fig- Evaluating a daper slope fro a typical force vs daping curve
The beautiful thing about equation () is that we can now do a laplace transfor on this. The end results are shown below, C = s + s + (3) If we copare this to the ideal second order syste with natural frequency ω and daping ratio ζ it is seen that, ω = s + ζ ω s + (4) When equations (3) to (4) are copared we know have the tools to specify our desired daping rate in ters of daping ratios and natural frequency, = ω (5) C = ω C ζ = ω ζ (6) This is the first point of the analysis. Equation (5) specifies the natural frequency in rad/s of the syste (To convert to Hz divide by approxiately 6.83185), and equation (6) eans a daper rate based on the daping ratio that is desired can be specified. It s at this point when calculating you need to be really precise with your units. Everything here needs to be in strict SI units. That is asses in kg, Spring rate in N/ and daping rate in N//s. Also all rates are in Wheel rates not at the daper. I ake absolutely no apologies for this. esides as far as I a concerned any easureent that has the slug as it s easure of density is intellectually flawed. efore we get into this in further detail why don t we do a worked exaple of this. For this particular case let s outline the following variables,
Table-1 Values for ¼ car exaple Ite Value Motion Ratio (Daper/Wheel) 1 Spring Rate 175 N/ (1lbf/in) Daping Rate 1333 N//s ¼ car ass 157 kg The daping rate was actually calculated fro Fig- and I would invite the interested reader to repeat the calculation. Let s now work through the calculation of the natural frequency and the daping ratio, ω = MR 175 157 MR C 1 1333 ζ = = ω 157 33.386 b = 1 = 33.386rad / s =.3 (7) As far as I a concerned the real power of this is that it gives us a powerful way of non diensionalising what is going on with a daper. If you characterise a daper this way the oent you do a spring change you can specify a daper, which in theory has identical characteristics. I should also add this applies both in bup and rebound. Obviously when back calculating for rebound when re working your nubers reeber to ultiply the slope by 1, if using a standard force vs daper curve, that shows bup positive and rebound negative. Now that we have gone to all this trouble to calculate daping ratio what does all this actually ean. To get to the heart of the atter we need to return to what our quarter car is going to do when we apply a step input to it. The daping ratios will actually tell you what the car is going to do. This is illustrated in Fig-3,
Fig-3 Second order syste response to a step input. Reviewing this graph is very interesting. At low daping ratios the ¼ car odel is decidedly underdaped and it takes a long tie to return to an equilibriu state. However think about what this is ideal for. If we are hitting a bupy bit of the circuit we actually want the car to do this. When the daping ratios hit about.5.7 the car s oscillation is rearkable decreased. Not so great with dealing with a bups but ideal for controlling the body when we are pitching and rolling. Daping ratios greater than this are ideally placed to force the car to alost act like a spring as we discussed in our previous articles. However be warned there are soe consequences for over doing it. On the back of this I cae up with a rough daper guide that I actually outlined in y first article. Now that we have discussed this in further detail let e outline it in further detail. Table Rough outline to daping ratios Daping Ratio Range What this applies to.3.4 Ideal for filtering out bups.5 1. This deals with body control. 1. + This deals with extree body control/driving teperature into the tyres. Ared with this knowledge let s look at the first daper curve we saw in Fig-. Assuing the sae nubers fro Table 1, I ll work out the daping ratios for both bup and rebound. This is presented in Table 3
Table-3 Daping ratios for Daper presented in Fig- Velocity (/s) Daping ratio in bup Daping ratio in rebound 1.4.95 13.3.6 5.616.77 38.175.31 5.167.86 63.174.31 Table 3 presents soe enlightening insights into what this daper is trying to do. First things first the daping ratios fro tell e iediately this is a high downforce car. The high daping ratios are tell tale signs this is a high downforce car. The high daping ratios iediately suggest that body control is paraount. Looking at the bup at 13/s the daping ratio jups to.3. This indicates the daper engineer is trying to give soe feel to the car as well as load the tyres. eyond this range the dapers blow off to a low ratio to allow the car to ride the bups. In rebound fro 13 5 /s the daping ratio is.7. This tells e body control is still paraount. eyond the daping ratios blow off to.3. This tells e this is designed for the bups. I think the reader is fast starting to get the idea that daping ratio is not just a useful paraeter but it can tell you an awful lot about what you want both the chassis and the tyre to do. As a rough rule of thub the higher the daping ratio the better it is for both controlling the chassis and putting teperature into the tyre. The lower it is the better it is for riding bups. However the question has to be asked how do we deterine what we want and need. One is experience and a bit of infored intuition. Let e illustrate with an exaple say you have a low downforce car going over a circuit with a lot of undulations. In general ters for a car without a lot of downforce you would actually like lowish daping nubers in rebound Say daping ratios in the order of.3.5. However let s just say your rounding out a turn with an undulation. It could be desirable to actually have a higher daping ratio in rebound to keep the car connected with the ground? I not saying this is what you need to do I just asking the question so you can think about it. The second ethod is to use siulation software to aid us in deterining this. Indeed it was this very question of evaluating the appropriate dapers that lead to the creation of ChassisSi. Indeed when I was evaluating what lap tie algorith to use it was this reason that I disissed the pseudo static lap tie approach. To illustrate what I a taking about let e evaluate too totally different types of dapers for the front of an F3 car. The results of the siulation is shown below in Fig 4,
Fig-4 Siulation results for two totally different dapers This is a bit of an obvious exaple because this was a very bupy circuit and I set the second daper very stiff in both bup and rebound, but this none the less shows how you can use siulation to look at different configurations and see the ipact they can have on vehicle perforance. For exaple in this trace we can see very quickly the ipact this has had on loads, corner speed and teperatures (The red is the baseline and the black is the daper change). I would wager the loss in teperature is due to the fact the apex corner speed has dropped by k/hr. In closing then while we haven t achieved the grand unified theory of race car daping we nonetheless have delved into how useful the daping ratio can be a very effective tool. Not only does it non diensionalise the way we can look at daping but we can use it too force the behaviour of the race car to get it too what we wanted too. Is this a agic bullet? No it isn t however if properly utilised it is a very powerful tool to investigate what is going on with the car. It will also give us soe good for thought for the next instalent in this series on race car daping.