Deriving Car Slip Angles - Tire Slip - WARNING: Maths

Understanding oversteer, understeer, and load transfer all relates back to slip angles, so it's important to understand where the equation for slip angles comes from. This video derives the equation used for slip angles, as shown in the book "Automotive Engineering Fundamentals" by Richard Stone and Jeffrey K. Ball. The derivation of slip angles requires an understanding of basic dynamics, the relationship between slip angle and lateral load, and static free body diagrams of a car.
Related Videos:
Slip Angles - • Slip Angles - Tire Tra...
Understeer vs Oversteer - • Understeer vs Overstee...
Traction Circle - • Traction Circle - Expl...
Tire Load Sensitivity - • Why Do Wider Tires Hav...
Proof of F=m*V^2/r:
www.nuffieldfoundation.org/pra...
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[@7150285]

"5% of you are still with me" LOL!

[@logmegadeth72]

These types of videos are my favorite out of all the ones you upload.

[@MichaelGedies]

Well done! How short and sweet that was! Being in uni physics right now I followed you to the end. Very clear presentation, nice job.

[@Brando4774]

As someone nearing completion of a mechanical engineering degree and an avid car enthusiast, I've always been really interested in the maths of vehicle dynamics but never had the time to open a textbook (outside of class!) and teach myself. Just wanna say thanks for such a great video, you've given me plenty to think about in terms of how I drive and modify my car! As someone also studying a pure mathematics degree alongside my engineering, I can really appreciate you going to the effort to prove these things as you go, you strike me as someone who should have studied pure math as well! You can be assured that someone is watching and learning something so please keep it up :)

[@cobra454tim]

I understood about 70% good lesson on what happens with rear slips angles and how it affects the front, over steer and under steer Thank you!

[@BeyondtheTrial]

i enjoy videos like this where you go into detail of equations

[@ryanwearsjeans9169]

Awesome video! Keep them coming. I've learned so much from all these whiteboard videos. Can't thank you enough for the fantastic and free education on this stuff.

[@aladdindorado3608]

I love the math. Please make a video explaining the maths of the 4 stroke engine. The calculus and differential equations. Thank you so much.

[@billwilmington8284]

All these equations and formulas are making my head spin!

[@Travelwith_me]

The derivation was very well explained. 🎯

[@ricksays7133]

Some mechanical engineering professor teaching vehicle dynamics out there will use your video to explain this topic. I haven't touched physics and engineering subjects for a while and I managed to understand everything in the video.

[@Turteng]

Physics = the coolest subject! Awesome video! Math is not actually hard and boring once you understand it, people don't usually like it because they don't understand and give up on trying.

[@fercm97]

I saw a lot of videos of this... and I needed a ecuations for my project and u got it

[@Morwola]

Maybe I'm not wrong, if I say that 97% of people hate Maths and Physics. They say they are boring, unuseful and too complicated. Whenever I take a sit down in a Physics lesson, I feel that I'm the only one from the class who is actually looking forwards the upcoming 45 mins. Physics is definitely the most interesting subject of all times. Combined with practicular Maths, your opportunities are endless and they are open open for you to explore. Dont's listen to those who say Maths is not for KZbin. This is what Engineereing Explained should mean (although, I know knew that you'd probably loose that 97% of your subscribers :) ). To sum up, me and probably some of your viewers would be happy if you could come up with videos like this later. And don't tell us, that substituting equations is the most advanced Maths in engineeering, so don't be shy, show us some deep shit. :D

[@KingdomOfMog1]

Morwola id say alot more the 3% of people enjoy physics and math

[@CDP135Z]

Morwola I absolute love physics! Might be why I'm working to a degree in EE.

[@Morwola]

Kingdom Of Mog Well, I go to the 10th grade, a Maths-specified class. There are 34 students in my class. There are maybe 3 people (included myself) in the class who like Physics. That's 9% only in a MATHS class, and I wouldn't like to calculate the percentage in the other class, who learn foreign languages primarily. Maybe you are right, but the only thing I hear from people is "Why do we need to learn this bullshit?".

[@stevehash158]

Morwola BHAHA obviously if you get a Q like that its called Ignorance everything in this world is build on numbers or by numbers.. When i hear Q like "why do we need this shit" lol i am just like smiling to myself and thinking "they simply dont know any better" a.k.a ignorance =)

[@stevehash158]

Morwola well said my friend math/physics will get you the jobs that pays alot go for it man its brilliant stuff..

[@claudiucarbunaru2149]

thank you! good math, good physics, good structure of presentation! as A. E. I enjoyed!

[@MostlyLoveOfMusic]

very good explanation my man

[@BboyDaquack]

taking dynamics now. love this kind of video xD. If you can I'd love to see you derive more crazy math stuff that can be applied to real world car stuff.

[@Robby94LS]

Now THIS is one I'm going to have to watch a few times..... LoL. I 100% understand the use of all this, it would be REAL nice for suspension tuning, but it's WAY over my head. But hey, the way I've learned everything has been hanging with people over my head and playing catch up! Great vid as usual!!

[@ericseewald3683]

Im not a Automobile Engineer but i could follow your explanation. Very Interesting, do they use this type of calculation in Motorsports?

[@mohammedroushdy1321]

I wanted to press the like button many times👍🏼👍🏼👍🏼👍🏼👍🏼 The video really helped me ❤

[@najiaahmadi6867]

You have been incredibly helpful

[@pgr92]

Great video really helps us inderstand the physics and math behind the things we like, just one quick question R refering to radius will this be established according to the point of reference or has to do with an specific measure depending on the vehicle ? Thanks!

[@sabaripandian116]

easy understandable derivation. i also want to know about slip of tire in road and their effects in wear of tire. thnk bro...

[@footballshowItaly]

Thanks! Nice explanation

[@deeplato5647]

This is awesome in 2021. Come back to 2015 Fenske Style!!

[@stampede92]

Engineering Explained Hey! Awsm Vid, do you know were can I find ATV tyre data?

[@Overkill14]

This is deffinately over the scope of my High school physics class, in which I'm constantly relating everything to cars and it all makes perfect sense once I do that

[@mightygarhem7]

4 years later, engineering student now? :)

[@Overkill14]

No lol. I’m now a welder/industrial mechanic.

[@whothou]

@@Overkill14 1 year later, a working for NASA as an engineer now? :)

[@krttikagoel7818]

rlly good vid thanks

[@richardcrilley11]

I think you do a great job and I think you have a great future. I do think you could slow down a bit on some of your videos, especially the ones with equations. Give people a chance to write them down and work through them. Also maybe write a little bigger, perhaps use two whiteboards. After you make one of these You Tubes, just look at it and make sure people are able to see it. Once again, you do a much better job than I could so keep them coming.

[@ashishverma3796]

Superb

[@scrawford9]

After watching this video, I realize I'm dumb as shit.

[@stevenolson95]

Awesome!

[@Backyardsmash]

This video reminded me how much I need to touch up on my math

[@Deadpan_Stan]

well as a second year engineering student I was able to follow this

[@SquiggleSquared]

I am an incoming physics and Mechanical Engineering student interested in automotive engineering. will it be helpful to look over and review the book you site in the beginning? Thanks!

[@SCOTTS11B]

Can you make a video on engine braking and how it works to slow down a vehicle?

[@hansdewyse1361]

Hi, First of all, great video as always Second, I was wondering if in one of your next videos you could explain the importance of the roll center heigth, and why it's important to keep it as low as possible Thanks!

[@sonicstep]

I do not know of it being stated anywhere that roll-centre should be kept as low as possible. It is a chassis geometry tuning option in RC racing. Whether it is set low middle or high front or rear is a user choice based track topography and its conditions.

[@ajayv304]

It shud be kept as low as possible and as much nearer to CG of Vehicle. Roll centre height plays important role in the development of JACKING force when a Vehicle is cornering. And this Jacking force is important in keeping the vehicle's Suspension in not compressing too much.

[@vatsalkansara4003]

Hey! I think the entire mass of the vehicle should be used and not just the mass on the rear tire in the formula for lateral force on the rear tire, since in the expression for centripetal force we use 'm' as the entire mass of the car. Please correct me if I am wrong.

[@rabbitbabbit7311]

Can you do a video on 3 link independent suspension

[@Cotygarcia20]

This is great, I'm an entry level automotive engineering student and I have yet to get anywhere near these sorts of equations as I'm still taking calculus and other basic courses. I've been itching to look at these sorts of equations just because the math finally means something purposeful as well as interests me. My question from this video is how does all of this math translate to the actual components of the car and adjusting the components to get what you want out of the car? (Dampening, spring load, camber, toe, caster etc.) also how would wheel width make a difference?

[@logmegadeth72]

Coty Garcia If you're interested in more of the gory details, there are books like the one he mentioned that you can download.

[@agustinlavia]

I have a question. I'm using Motec telemetry analysis software and I want to calculate the slip angles for the tyres. I have the velocity vectors for the car, toe angles and steering angle. Can I calculate the slip angles using that? Would it be correct anyway? Thanks!

[@alinush27]

Can anybody explain to me on short what does the "cornering stiffness" means ? I always run into it when studying vehicle dynamics and I honestly couldn't find a good explanation on the subject Thank you !.

[@faisalarefin1263]

2 Questions,First,In the 8th step,You wrote 3 lines/Equations,And You pointed out that with the 2nd Equation we could figure out a lot of stuff such is my oversteering ot understeering,Well I'd like to know,What kind of Result is telling me my car is oversteering and what kind of result is telling me my car is understeering.Second questionThere's a third line,which i don't think you really explained in the 8th step,but still A big thank you for teaching something real as often as you do

[@mohamedarif578]

Thanks man

[@James58094]

You are great!

[@killerontheroad280]

Wow, this video gets uploaded as I'm home just after I dropped out of Calculus

[@ahardy1017]

kevinhrz93 Hahaha this isnt even calculus...

[@markgriz]

ahardy 10 9th grade algebra at best

[@CDP135Z]

markgriz Nope, definitely physics. Algebra based physics, but physics none the less. I'm certainly not an engineer, but I'm on my way to it and I had to pay attention to follow, so don't feel bad kevinhrz93.

[@killerontheroad280]

The point of my comment was to point out the irony of me dropping my calculus class as there is a warning sign about math in the title of the video, haha.

[@markgriz]

***** Just pointing out that the "math" part of this is nothing more than rearranging variables, so you shouldn't be scared by the "maths" warning. It hardly rises to the level of Calculus. Of course, you need to have a decent understanding of physics to put the equations together in the first place. But simply rearranging variables is nothing more than division and multiplication

[@laurg17]

How can you derive the optimal slip or drift angle for any given coefficent of friction? Thank you

[@iamnotjobless]

I am planning to buy the new Subaru WRX STi(2014 or 2015). But since the import laws in my country are a little too rigid, I opted to buy a WRX and work on it. However the problem I am facing is that the new WRX STi JDM(the one i wanted to import) runs on the EJ20 and the WRX on ther other hand is paired with the FA20 like in the BRZ. Which one should I get?

[@diegobula4597]

Question: If i have the cornering force vs Slip Angle graphic is the same lateral force vs slips angle one ?

[@ajaykudtarkar7801]

please a video on yaw rate sensor

[@jp2654361]

hey jason , i want to know how different is heavy vehicle(TRUCK) engineering from automobile(CAR) engg and if possible make some vids on trucks or buses, thank you

[@apekshasamarth2835]

Could u please tell me which factors should be considered while deciding the wheelbase and trackwidth for a car?? -(SUPRA car)

[@rezasaatchi1401]

How do you obtain "R" (instant rotation radius) while it depends on slip angles and at the same time the slip angles are dependent on "R" ?

[@30second_riddles]

can you please explain how can we find the cornering stiffness without using tyre data.

[@harshasaraf524]

Hello sir can you please tell me value of cornering stiffness for slick tires

[@iurieysi9103]

can someone help me please? if the slip angle is the difference between the direction the wheel is pointing and the direction the wheel is moving, which are both known vectors, why do we need to calculate it? or am I missing something?

[@abhishekdixit2318]

On Rear Wheels which side of wheel get more Slip angle at Cornering (Inner or Outer) on Rear Side ??

[@ashwanthvh6812]

can you help me in deriving camber angle please....??? not measuring camber angle but deriving it..

[@hruthiktejaereddy1999]

Is self-aligning torque produced by this lateral force and slip angle.

[@louisthefool]

Could you please explain how you get the value of the cornering stiffness ?

[@kushagrapandey1171]

is there any way to know cornering stiffness or just i hav to dig some tire data.

[@sankaranajay8059]

Is there any way to determine the cornering stiffness of the tyre

[@mrweisu]

Why adding a thicker rear sway bar can reduce FWD understeer? Which parameter changed? weight on rear tire, cornering stiffness? I don't get it.

[@rizwan7860]

How is the tire stiffness determined?

[@chase9692]

Super thank you.

[@Wisenyctophile]

Please suggest me best books for automobile engineering

[@DarkMan2555]

Hey Jason, could you do a video on what determines a car's redline RPM? I think I know the answer, but I thought it would be interesting if you explained it.

[@aleksandar.tepsic]

DarkMan2555 The redline RPM of an engine is a factor of safety below the engine's failure RPM, with fuel cut being the next measure of safety. The valvetrain (valve float) and strength of the connecting rods limit the maximum RPM of an engine.

[@DarkMan2555]

Aleksandar Tepsic I read somewhere that ignition could be cut as a different safety measure.

[@aleksandar.tepsic]

DarkMan2555 That would work as well, yes, any measure to cease further combustion and increase in RPM. I do believe cutting fuel supply is widely used though over other methods.

[@ziavrasc]

so.. whats the exact difference between a(angle) and s (slip ratio) especially on the diagram?

[@AbdulBasit-hm4mq]

what it is cornering stiffness? any other formula not in terms of slip angle?

[@pkonneker]

Math has been kicking my butt lately, haha. I like this video, though.

[@christiaanroos4479]

Would be nice if you could explain why and how you are able to use the sum of forces and moment equations so more non engineers can understand this, i hope :P

[@JoshTheEngineer]

Christiaan Roos You can think about/visualize the sum of forces and sum of moments equations in a pretty simple way. I'll explain both of them below. First we'll start with the sum of the lateral forces, as seen in Equation 3 on the board. Let's just start with something moving around in a circle. The force due to centripetal acceleration from to this circular motion is Fcg. This is acting at the center of gravity, or CG. If this were the only force on this object, nothing would be keeping it from staying at the same radius. This is where the tires come in handy. The sum of both of their lateral forces counteracts this, keeping the car at a constant radius R, in this case. In order for the car to not increase or decrease the radius of the turn, the forces from the front and rear tires needs to be equal and opposite the force Fcg. It's easiest to think about the car in a sort of 'snapshot' when figuring out the forces acting on it. This is essentially what Figure 2 on the board is doing. It's a snapshot of the top view of the car. We know there is an outward force because it's driving in a circle. We also know that the tires produce a lateral force. The sum of the forces equations becomes the simple task of saying which ones are acting to the left and which ones are acting to the right. There's one little issue with the force equation though. The lateral force on a tire acts perpendicular to the wheel plane, which is why in a turn, the lateral forces don't actually act on the same axis as Fcg. The actual force equation would have a cosine term acting on the lateral force for each tire. However, we can use the small angle assumption, where the cosine of a small angle is approximately equal to one. This is where Equation 3 comes from. Moving onto the moment equation, or Equation 6. First of all, when you think moment, think of torque. Same thing, different word. A moment is caused by a force acting at a certain distance away from an axis of rotation. Make sure you're looking at Figure 3 on the board. This is a side view, not a top view. When we say the sum of the moments is zero, it essentially means that nothing is rotating, or spinning around its center of gravity. Go outside and take a look at your car sitting on the street. It's probably not spinning out of control. It's most likely just sitting there, not moving. Since it's not moving, we know that the sum of all the moments acting on the car is zero. All the moments trying to rotate it one way are balanced by all the moments trying to rotate it the other way. From the side view drawing, you can see the force Wf trying to spin the car clockwise about the CG. However, there is another force, Wr, that is trying to spin the car counter-clockwise about the CG. Note, however, that in Jason's calculations, he takes the moment about the front tire. Using this logic, the force Wr (acting at a distance L) is trying to spin the car counter-clockwise about the front tire, while the force of the car due to gravity (acting at a distance a) is trying to spin the car clockwise about the front tire. Because the car isn't rotating about this particular axis, the sum of these moments is zero. If anything here made things less understandable, let me know and I'll fix it for you!

[@karthikrao7553]

bro how do you determine cornering stiffness?

[@sainathdevagoni652]

How can i get the cornering stiffness of the tyre??

[@tongzhao8380]

Underlying assumptions: Eq. 1 and Eq. 3 assumes small slip angle (nothing going crazy); Eq.2 and Eq. 4 assumes constant spinning rate of vehicle (no angular acceleration); Eq.6 assumes no pitching is happening (no front dipping or lifting).

[@AndikaRamaMaulana]

Where i can find that book? That automotive fundamental. I'm studying mech engineering, and this one really interest me. Again, thanks for the explanation;)

[@AKDaBear]

Andika Rama Maulana you can find it on SAE international for 110 dollars

[@logmegadeth72]

Andika Rama Maulana not to be that guy, but you can find a bunch of good ones if you search automotive engineering on TPB :)

[@AndikaRamaMaulana]

Thankyou very much!

[@RahulDutta1992]

try Genta vehicle dynamics ;)

[@steves8014]

OMG! Now I know to trust the engineers who developed the T1 bars for my C5 Z06.

[@jabez_kun6243]

"this is a basic equation" umm no I love the very informative vid and would like to use what I've learned

[@TaleDreamer]

The proof for F = mV^2/R is no longer available. Can you please cite a new source? Thank you.

[@torque2802]

If this is meant to be applied to a car how would you find the radius of the corner or is it based on steering angle and how much that turns the wheels? Also how do you find out the cornering stiffness for a tyre?

[@adityabhargava8685]

i have same dbout ... could youbplease help me with that if you have figured it out

[@martinwink2063]

In what unit is corner stiffness expressed?

[@ananthkaimal5941]

this makes sense but ,,Is this true that cornering stiffness is completely a property of Tyre

[@rabbitbabbit7311]

How to calculate cornering stiffness?

[@spidey5324]

Is slip angle the same as drifting angle?

[@MrNickelbacker]

Hey Engineering Explained I think I understood what you did to this point but at 5:56 I can't quite follow. Why does mg*a have to be equal to Wr*b? Assuming the center of gravity is exactly in the middle (50:50 weight distribution and therefore a = b), Wf and Wr would be the same and both half of the total weight which is mg. So wouldn't that mean, that in this case mg*a = *2* * Wr*b? Or did you want to say Wr*L = mg*a or how you wrote it: 0 = Wr(L) - mg*a? Could you or anyone else explain, where I make a mistake, please?

[@rahulbball9395]

+SimBaa he's taking the moment about the front wheel to the only forces causing moment are mg and wr.

[@siciliandragon340]

is there any way to find cornering stiffness using basic tire information like aspect ratio?

[@siciliandragon340]

Me after experience : No you cannot find it unless you have some sort of tire data. Thankyou.

[@vishnukumar4531]

Why did u say that the total moment on the car is 0 ie: why did u equate the torque in the front and rear wheels @ 4:03

[@AKDaBear]

Hi man, first of all nice video. Recently I test drove a BMW r1200gs (motorcycle), which has a cardan shaft. I was wondering if you could do an explanation video on this. I know you mainly focus on cars, but I think it would be extremely helpful if you could do an in-depth video on cardan shafts (on for example BMWs).

[@logmegadeth72]

AKDaBear FWIW, a cardan shaft is pretty much a simplified version of a CV shaft.

[@AKDaBear]

David F. Thanks, but I'm wondering why BMW uses a cardan shaft on their bikes instead of a chain which other companies use, like the pros/cons of using a cardan vs chain etc

[@logmegadeth72]

AKDaBear I believe it's mainly due to the losses in power transfer. A cardan shaft will experience less loss as opposed to a chain, but a chain is cheaper to produce and manufacture. That's why you tend to see cardan shafts on more expensive bikes like a BMW, whereas you usually see chains on other more affordable bikes.

[@AKDaBear]

David F. Thanks! Do you know something about the maintenance/durability of the cardan vs chain? For example, I believe the Dakar riders have chains apposed to cardan, would that be because of durability or stress on the cardan or something else?

[@logmegadeth72]

AKDaBear Probably easier to maintain/repair like you said. I feel like it would be a pain in the butt to tear down and repair or even replace a cardan shaft in the middle of a race, that's assuming you have the part(s) with you. If a chain breaks its much easier to slap on a new one. You just wrap the chain around the two gears/cogs connect the master link and tension it, and you're ready to go.

[@racervee9529]

Suscrito!

[@SuzhouChen]

I m a PhD student of Mech Eng....why the hell am I watching this after a long day?!?!?!?!

[@chandimosomi1958]

You legend

[@ashwinkumarsabat6282]

Can you help me in calculating transmissibility ratio of shock absorber

[@BrandonKent136]

take a vibes course. good luck

[@SHiFTDI3S3L]

I failed math in HS, not for lack of trying, but I still failed it. I'm going to have to watch this a few more times.

[@chamarajayasekera6570]

Where can i find the Cornering stiffness value

[@abhishekkrishnan7570]

calculate karke apne dimag me milega

[@gyw648]

Cornering stiffness is usually provided by your co-workers (if you work for an auto maker), or obtained by experiments.

[@nikazoran2558]

that information may be determined by testing tire . The most known tire testing lab are Smithers , ARdL and STL

[@neel9456]

2:20 why for small angles????

[@oliviercloutier1975]

God I love these.

[@rajdungrani3073]

how to find cornering stiffness

[@jlbmanetti]

For what kind of problem is usefull use this equation? Because when you linearizes the Fy x Slip Angle graph using the Cornering Stiffness you're loosing alot of data.

[@JoshTheEngineer]

João Luiz Barros Manetti There are some simplifying assumptions used in this analysis, but the most important one is the assumption of small slip angles. This assumption gets used in Equation 1 (as you mentioned, and I'll get back to that), and also in the sum of the forces and sum of the moments equations. To reiterate what Jason mentioned in the video, the analysis is using something called 'small angle approximations'. This assumption says that for small enough angles, the cosine of the angle is approximately equal to one. Recall from trigonometry that the cosine of zero is one, so the cosine of any angle close to zero will also give a value very close to one. This makes the sum of the forces and sum of the moments equations simpler, because now instead of having the tire lateral force multiplied by a cosine term, you simply have the tire lateral force (times one). Now on to the cornering stiffness part of your question. By definition, the cornering stiffness is the slope of the lateral force versus slip angle curve, at zero slip angle. Again, as Jason mentioned, in a region close to the origin (at small slip angles), the lateral force versus slip angle curve is actually quite linear. The first part of Equation 1 on the board is just saying that if you give me the cornering stiffness of the tire and the slip angle, I can tell you what the tire lateral force is. Similarly, the second part of Equation 1 is saying if you give me the tire lateral force and the cornering stiffness, I can tell you the slip angle. Think about the implications of this equation though, looking at the first part of Equation 1. The cornering stiffness is defined as the slope of the linear portion of the lateral force versus slip angle curve. What if I ask you to solve for the tire lateral force after giving you a cornering stiffness and a slip angle of, say, 45 degrees (this is exaggerated of course). At 45 degrees, we are definitely not in the linear portion of the curve anymore. For all tires, this means that you will over-predict the lateral force at a given slip angle, and the equation won't be of much use anymore. To sum this all up, you can see that small slip angles is one of the most important limiting assumptions in this analysis. There are others that I didn't go over, but this should at least help a little bit. While this may seem restrictive, most passenger cars operate at low slip angles. You'll obviously need to be more careful when looking at race cars, etc.

[@paulbeever9379]

If you could do this with only the data available in a real car, that would be interesting (cornering stiffness is not available, especially to ASIL D). The maths would be to a higher level though, having read papers on the topic.

[@alex59248]

you said if i have a question write it in the comments: how do i math?

[@rajeshrahul52]

how to calculate cornering stuffiness dude