Matlab/Octave script to show the variation of Lateral Force and Aligning Moment with Slip Angle (0 to 15 deg)

1a, These questions test conceptual knowledge. Answer in a sentence or two in your own words. A textbook definition is not necessary

• Calculate the tire stiffness. Tire size – 275/40R18, Free radius is 13.33 inches and loaded radius is 12.36 inches
• Explain in your own words what slip ratio and slip angle are (explain, don’t give the formula)
• What is the risk of having too much mechanical trail? (think in terms of driving at the limit)
• You know lateral force is proportional to slip angle. Can a tire produce lateral force at zero slip angle?
• In your own words, explain tire load sensitivity and how increasing the rear roll bar size can induce oversteer (again, use your own words)

2a, This challenge tests your understanding of the brush tire model for lateral force and aligning moment
This exercise should be done in octave. Turn in your code

Write an Octave script to show the variation of lateral force and aligning moment with slip angle (0 to 15 deg)
Use the brush tire model we developed in class. Develop your script based on the algorithm we looked at in class. These are your deliverables

• Plot (use subplot) showing lateral force vs slip angle
• Plot showing aligning moment vs slip angle
• Error free Octave code

1a.

• 275/40R18, Free Radius = 13.33 in & Loaded Radius = 12.36 in || Deflection equals to Free radius-loaded radius=13.33-12.36 = 0.97 in || So, Vertical Stiffness equal to Vertical Load/Deflection. If we know the total weight on particular tyre then we could easily calculate the tyres vertical stiffness.
• Slip Ratio tells you about how much your wheel angular velocity is different from forward velocity of tyre. Suppose, Slip ratio is 1, it means angular velocity of wheel reduced to zero during braking results full lock up condition. Slip Ratio is 0, means that wheel angular velocity equals free rolling angular velocity of tyre. At Slip 0.15-0.20, tyre can provide maximum braking/tractive force, beyond that the braking/tractive force is approx constant. Slip Angle =Slip angle defines as the angle between a wheel's heading direction and the wheel travel direction. This slip angle generated due to the moment caused by Lateral force and pneumatic trail which multiplies togethor. Generally slip angle caused during turning, as as we steer the wheel the friction force hold the tyre tread generating lateral force and deflecting the tread in the direction of turn.
• By giving too much mechanical trail, the driver could not feel that when tyre looses grip suppose when we are driving at slip angles where we are getting peak lateral forces then the self-aligning torque is acting at contact patch due to pneumatic and mechanical trail. Due to high slip angle there is a chance of loosing grip and it should be sensed by the driver through reduction of steering force or effort as reduction happens because of pneumatic torque is reduced at high slip angles. If the reduction is higher then driver will know that he is about to loose grip and if the reduction is low then he is having the grip. But, what happens that with too much mechanical trail, will give adequate amount of aligning torque even when there is no pneumatic trail!! So, by this driver's will assume that he is getting aligning torque which means tyre has a grip but actually tyre looses grip.
• Yes due to camber thrust caused by leaning the wheel into or out of the turn. At zero slip angle, if camber is given to the wheel it will generate cornering/lateral force due to camber thrust(force purely due to camber)
• Tyre Sensitivity - The corenering characteristics of a tire vary with vertical load. The tire however does not follow the conventional friction theory where the friction force is proportional to vertical force/load. Lateral force increases with vertical load upto a particular slip angle beyond that the lateral forces started to saturates. If you increase further vertical load then the reduction in lateral force will results in understeer or oversteer behaviour. Suppose you are turning left, then due to inertial force vehicle tends to roll on right side, and weight on the right wheel increased but if the vertical weight increases beyond particular slip angle then the wheel will not get proportional lateral force due to tire sensitivity, in replace the tire will experience reduction in estimated lateral force from vertical force. If this reduction happens in front side it will be understeer and if on rear side it will be oversteer.
• ARB Anti Roll Bar, If the bar will be more stiff then it will transfer or we can say adds more vertical weight on tires resulting reduction in lateral load as per vertical load due to tire sensitivity. If ARB is stiffer in front than rear then Understeer and if ARB is stiffer in rear then Oversteer will occur.

2a. Octave File for Lateral Force and Aligning Moment v/s slip angle (.m)-

https://drive.google.com/open?id=1OyL5Df_rAr4aWyTYKfDn9Tpry7UAv9Hv

Lateral Force v/s slip angle curve pic -

Aligning Moment v/s slip angle curve pic -