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The fact is, by increasing the roll centre height in one axle, you are increasing lateral load transfer from the direct lateral force component, while at the same time you are decreasing lateral load transfer from roll angle component.
Bear in mind that the roll moment arm is the perpendicular distance between the CG of the sprung mass and the roll axis. The overall effect will depend upon roll centre heights and roll stiffnesses, and a definitive conclusion will require a deeper analysis.
One thing we can tell without any deep analysis is that increasing the roll centre height in one axle decreases the lateral weight transfer on the opposite axle, everything else kept constant.
This happens because raising the roll centre in any axle will approximate the roll axis to the sprung weight CG. This will decrease roll angle component, but since the roll centre height of the opposite axle will not be raised, the direct lateral force component will not increase and the overall effect will be a reduction in weight transfer on that axle.
Figure 9 shows a contour plot of lateral weight transfer sensitivity lateral weight transfer divided by lateral acceleration on both axles of an open wheel single-seater.
To obtain these, I created a MATLAB routine to calculate the total lateral weight transfer from our previous discussion, keeping the front and rear roll stiffnesses equal and constant while varying front and rear roll centre heights.
The input data were based on the manuals from the manufacturer of an important formula category. By analysing Figure 9 you can see that lateral load transfer is very sensitive to changes in roll centre height.
For example, if you investigate what would happen to the weight transfer in both axles if you held rear roll centre height constant at 30 mm while increasing the front roll centre height, you would see opposite effects happening on front and rear tracks weight transfer would decrease in the rear axle while increasing in the front.
Now do the same, but picking a front roll centre height and imagining a vertical line instead. As you see, when we increase front roll centre height, the lateral weight transfer decreases on the rear axle while increasing on the front.
Conversely, if you increase rear roll centre height, lateral load transfer increases on the rear axle and decreases on the front axle. Can you see the trend?
When you increase roll centre height in one axle you increase the overall lateral load transfer on that axle, while decreasing it on the opposite axle.
This leads as to believe that the roll centre height gain is higher than the decrease in the roll moment arm. The change in this arm with roll centre heights will depend on the wheelbase and weight distribution.
For this case, roll moment arm decrease with roll centre heights was smaller than the increase in roll centre heights themselves.
In my time in Baja, I have done calculations of the type for vehicles that had roughly the same weight distribution and wheelbases of approximately mm.
The results were the same. This component is the easier to control. By looking at the equation, you can see that the weight transfer component from roll angle can be altered by changes in front or rear roll stiffnesses, roll moment arm or weight distribution.
As we discussed, we should input consistent units into the equation to obtain meaningful results. The sprung mass used was kg, which gives a weight of With a CG height of mm and the minimum roll centres specified in 3 mm, which is very low, the moment arm will be mm.
With those values, the gravity term will be Do you see how small it is compared to the roll stiffness of the car? You might not be convinced of the insignificance of this term by arguing that those values were obtained for a very light car with a very low CG.
Figure 10 shows the plot of the roll angle component versus gravity term. Varying the gravity term from Nm to Nm resulted in a difference of only 0.
Bear in mind that these values were obtained for a fairly heavy race car with an unreasonably high CG, and this is only one of three weight transfer components.
At this moment, you should be convinced of the irrelevance of the gravity term on roll angle weight transfer component.
This basically rules out weight distribution as a way of controlling roll angle component. We now have roll moment arm and roll stiffnesses to play with.
From our previous discussion on direct force weight transfer component, you know that to change roll moment arm you need to play with roll centre heights, which will ultimately affect that weight transfer component in the opposite way you want.
Another reason to rule out changes in roll moment arm is that, because it directly multiplies the proportion of roll stiffnesses, it will have the same effect on both axles whether is to increase or decrease lateral load transfer.
For setup, we look into changing the lateral load transfer in one axle relative to the other, to affect balance. This makes changes in roll moment arm to control roll angle component useless.
Roll stiffness can be altered by either changing ride stiffness of the suspension vertical stiffness or by changing the stiffness of the antiroll bars.
Ride stiffness can be altered by either changing springs or tyre pressures tyre pressure affects tyre stiffness, which contributes to the overall ride stiffness.
This is generally not the first option to take because of the effect that it has on other aspects of the car. For the sake of example, ride stiffness controls ride height, which has strong effects on aerodynamics of ground effect cars almost every race car with relevant aerodynamics design.
Another example would be the effect of ride stiffness on wheel hop frequency. Hence, springs and tyre pressures should only be changed when other aspects need modification, but not only roll stiffness itself unless the vehicle has no antiroll bar.
The most reasonable option would be changes on antiroll bar stiffness. This can be done in multiple ways.
The hardest one would be to change the bar itself, though there are some antiroll bars that have adjustable stiffnesses, eliminating the need to replace bars.
These adjustable bars generally have blade lever arms, as the one shown in figure By rotating the lever arms, its area moment of inertia in bending is changed, hence altering its stiffness.
Figure 12 shows a finite element stress analysis, with colours closer to yellow and green indicating higher stresses. Some race cars have push-pull cables connected to the bars that allow the driver to change roll stiffnesses from inside the car.
Figure 13 shows the contour plots of lateral weight transfer sensitivity as a function of front and rear roll stiffnesses.
These data were obtained for the same open wheel car analysed in figure 9, but this time front and rear roll centres heights were held constant and equal, while roll stiffnesses varied.
If you compare figures 13 and 8, you will see that, while lateral weight transfer changes with roll centre heights along contours defined by lines that have the same inclination, the effect is different with respect to roll stiffnesses, as the lines that limit the contours have different inclinations.
If you represent the rear roll stiffness as proportion of front roll stiffness in a line plot, the result will be a straight line, with an inclination equal to the proportion between the roll stiffnesses.
If you represent multiple proportions, you will have multiple lines with different inclinations. Do you see where this heading?
Lateral load transfer in one axle will change with the proportion of the roll stiffnesses on that axle, not the roll stiffnesses themselves. This can be confirmed by adopting the conclusions from the analysis of figure 10, where we agreed that the gravity term is negligible for roll angle lateral weight transfer component.
If we use , the remaining roll angle component will be: If we keep the roll moment arm constant, then roll angle lateral load transfer component in one track will obviously be a function of the ratio between the roll stiffness on that track and the total roll stiffness of the car.
The term between brackets in the equation above is the roll rate distribution or roll stiffness distribution for a given axle, and it will ultimately control the elastic lateral load transfer component.
So far, we have discussed the influence of each component in lateral load transfer in isolation. For this analysis, only the rear axle was considered.
The front and rear roll centres heights were kept equal, but varied from 3 mm to the CG height mm. Roll stiffnesses were input in the form of roll rate distribution, varying from 0 to 1.
Figure 14 shows the contour plot. Figure 14 can lead us to very interesting conclusions. First notice that there are two particular regions in the plot, where any changes to one of the components will produce no sensitive effect on weight transfer.
This is characterised by the green region in the graph. If we define , the rear roll rate distribution and , the sprung weight distribution on the rear axle, then the lateral load transfer equation for that axle can be rewritten to give: Substituting the values on the terms inside the brackets, we have: But if we assume that front and rear roll centers have the same height, then the moment arm will be given by: This shows that when weight distribution and roll rate distribution are equal, for a horizontal roll axis, the sprung weight load transfer component will be independent of roll centres heights.
If , and will have the term inside brackets resulting in. This will tell us that lateral load transfer on a track will become less dependent on the roll rate distribution on that track as the roll axis gets close to the CG of the sprung mass.
This conclusion is somehow trivial, as we know that roll moment arm decreases as roll axis gets closer to the sprung mass CG and roll rate distribution only affects the roll angle lateral load transfer component.
If your driver complies about oversteer in the slowest corners, it means that the front axle is generating higher lateral force than the rear. By the methods presented here, the simplest solution would be shifting roll rate distribution to the front, by either stiffening the front antiroll bar or softening the rear.
In order words, the goal would be to reduce lateral load transfer in the rear axle in comparison to the front axle. If that was the case, you should work on the roll centres heights instead, and then adjust suspension parameters accordingly.
Bear in mind that lateral load transfer affects the balance through tyre load sensitivity the tendency of the tyres to generate higher lateral forces at a decreasing rate with higher vertical loads.
Notice that this is just one possibility and other parameters might be investigated as well. Bear in mind that all the analysis done here was for steady-state lateral load transfer, which is why dampers were not mentioned at all.
I hope this article was useful to you, and that you have enjoyed reading it. Please, leave a comment below , to let me know what you liked most in this article or what else you would like to know about the subject, or even some criticism or any knowledge you might want to share.
Also, if you liked this post, please share it on Twitter or Facebook , and among your friends. The information you are discussing on these articles is invaluable for students of vehicle dynamics.
Thank you for the article! Thank you very much for that, Lucas! Coments like yours are a great fuel to keep going with the articles! I hope to see you around!
A well written article Rodrigo with plenty of food for thought. One thing we can tell without any deep analysis is that increasing the roll centre height in one axle decreases the lateral weight transfer on that axle, everything else kept constant.
This happens because raising the roll centre will approximate the roll axis to the sprung weight CG. This will decrease roll angle component, but since the roll centre height of the opposite axle will not be raised, the direct lateral force component will not increase and the overall effect will be a reduction in weight transfer.
Thanks for the compliments and for the feedback! You are absolutely right. There was a typing error on the text, I apologise for that.
What I mean to say was:. I already fixed it on the text. Once again, thank you for you help, all the support is much appreciated. Rodrigo, this is SUCH a great article.
I truly believe if that understanding vehicle dynamics will make someone a better driver. Are you familiar with any of his work?
This has been the best racing year of my life. Thank you very much for such a nice comment, man! Always good to know that my work is being helpful somehow.
Also I have a glance at your website, and you have great content over there! I must say you got a new follower! One point I want to explore further is the relationship between roll stiffness and ride stiffness as your statement further below seems at odds to me.
This is the case for most of the ride motions related to low-frequency inputs e. ARBs will have an effect on quarter car ride motions and hence, ride stiffness , for example, hitting a kerb.
Regarding the effect of ARBs to upset the chassis, yes, they can affect the balance of the whole car through their important effect on ellastic load transfer component of a track.
Keep in mind that ARBs change the roll stiffness distribution of the whole car, and hence can alter the load distribution on the tyres, affecting the ability of the car to produce a yaw moment either to control or stabilize the car.
So obviously weight transfer is important for us. I hope you cover soon the transient load transfer, i have the doubt about if that topic includes laplace transforms.
Thanks for the support! Whether it includes Laplace transforms or not, it depends how you solve the time dependent equations. If you do this numerically, Laplace is not necessary.
If you want to solve the equations analytically, then Laplace might be a useful tool. The actual cause is the lateral acceleration and the inertial force it produces on the car.
In my understanding, slowing down the weight transfer gives the car a greater capacity to generate lateral acceleration and yaw rates since the inside tires will be loaded for a greater amount of time.
This would translate in a better response of the car, isnt that right? You can do both, a delay in body roll can be achieved by increasing damping, and a delay in weight transfer can be achieved by increasing roll inertia which would make the elastic transient weight transfer component slower, and therefore, the whole weight transfer as well.
Hi Rodrigo, at first I would like to thank you for your effort! Great articles, lot of knowledge, very well written! One question, what is going on when RC is under the ground?
So Kinematic Load Transfer Component should be subtracted from the total lateral load equation? Thank you very much, Swider!
Always good to know that RCD is being useful to people somehow! Regarding your question, yes, if you have your RC below the ground, you get a minus sign on the geometric weight transfer component, because it starts to act opposite to body roll.
That is, the assumption that unsprung weight transfer is independent of sprung weight transfer is no longer valid. Well it does matter.
If anti-roll bars are too stiff, than the suspension will gradually become dependent, as the bar will tend to rotate and lift the inner wheel in the corner instead of twisting.
Thanks for the reply. It will be interesting to see if that helps. Hi Rodrigo, thanks for your blog. Most books that approach the subject of lateral load transfer point out that the equations are derived for a vehicle with rigid axles.
But then I read in a lecture note from one of my professors that the contribution of the lateral load transfer of the unsprung mass for an independent suspension ist a function not only of the center of gravity of the unsprung mass, but also of the height of the pole.
I would like to know if you there really is this difference mentioned above for independent suspensions vs. The only thing I can see different is that with solid axles the geometric component is zero, because the roll centers are at the ground.
The bit about the lecture note from your professor got me thinking, though. What do you mean by height of the pole?
Can you send me these notes? The roll center of a solid axle depends on how the axle is located and is usually well above ground.
With a Watts Link, for example, the roll center is on the propeller axis. With a Panhard Rod, the roll center height varies depending on the angle of the Panhard Rod to the ground.
You can see pit crews making changes to the height of one end of the Panhard Rod during pit stops to tune the handling. With leaf spring location, the roll center is determined by the location of the spring mounting points on the car body and the center line of the axle.
Actually, not every author makes this assumption. And then because of these notes from my professor I got confused. Sure, I can send you them, but they are in portuguese.
As far as I can tell, your name does sound brazilian, am I right? Hi … Very good explaination But I am having a trouble in the load transfer by kinetic and elastic part please clarify this Basically the load from tire goes to the chassis via linkages kinematic elements and springs elastic elements.
But here I see u have taken load from tire to go completely in springs or in linkages depending on the case. Am I wrong somewhere???? Well, lateral load transfer begins at every part of the car that has mass hence, it begins in the whole car.
The inertial forces act on both sprung and unsprung masses. However, the inertial force acting on the sprung mass is transmitted to the tires through both, suspension linkages geometric load transfer component and springs and ARBs elastic load transfer component.
Can you also post another article on the factors affecting Longitudinal Load transfer? Que beleza de artigo cara!!
A really good read, Rodrigo! It is not very often that you find articles that explain the math behind fully. I pretty much understood what you have had to say.
But, I got stuck at the equation when you explicitly added MCG sorry for not using the subscript to Mphi and later divided it by the track width to find the load transferred over a particular axle.
If it does why do we have to explicitly add MCG later on. And thanks again for the article. However, this conclusion makes 0 sense based on the facts you presented.
You determined in your article that roll angle influenced mainly by roll stiffness is the main contributing, and easiest to manipulate, factor in load distribution.
So, if the car is experiencing oversteer, your goal should be more load distribution rearward. While I agree that an increase in front roll stiffness would be a potential cure for this situation, I disagree in your summation that the goal here would be to reduce lateral load transfer in the rear.
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