MECH 4420 Homework #5 Due Friday 3/23/2018 in class (Note: checko due 3/9/2018) Steady State Handling The steady-state handling results that we developed in class give a lot o insight into what happens in the linear region o the tire curve. In racing, however, the vehicle attempts to operate at the limits o handling. Thereore, it is important to include the saturation o tire orce and the dependence o this saturation on load when examining the handling characteristics over the ull range rom parking lot maneuvers to the riction limits. To do this, there are a number o available tire models that handle load dependence, saturation and a wealth o other things. One model that produces surprisingly good correlation with experiments is the Magic Formula tire model, so named since the ormulation has no underlying physical rationale (this is oten called the Pacjeka tire model ater one o its developers, Hans Pacejka). Pacejka parameters are a common means o communicating tire data between tire suppliers and vehicle manuacturers. The Magic Formula or side orce as a unction o slip angle is: 1 = Dsin( C tan ( BF)) F y 1 F = (1 E) a + ( E / B) tan ( Ba) each o the coeicients in the model is described as a unction o vertical load (and can be modiied to also become a unction o camber angle): C = 1.30 3 2 D = a F + a F + a F E = a 0 BCD = a sin( a 6 F 3 2 + a 1 7 4 F tan + a 2 1 8 ( a F The coeicient D represents the peak side orce while the product o coeicients BCD represents the cornering stiness. One set o real data (which requires vertical orce in kn and slip angle in degrees and gives side orce in N) rom SAE Paper 870421 is: a0 = 0 a3 = 1078 a6 = 0 a1 = 22.1 a4 = 1.82 a7 = 0.354 a2 = 1011 a5 = 0.208 a8 = 0.707 1) Once we put empirical data into a ormula like this it is easy to lose sight o what is going on. To prevent this, plot the ollowing to get an idea o the tire behavior predicted by this model: a) Plot side orce as a unction o slip angle or normal loads o 2, 4, 6 and 8kN. b) Plot the cornering stiness (BCD) as a unction o normal load rom 0 to 8kN. Explain using this graph how lateral load transer could be used to change the tire cornering properties on one o the axles. c) Plot the peak side orce (D) as a unction o normal load rom 0 to 8kN. Explain how the maximum traction should dier between cases o minimal and large lateral load transer. 5 ))
2) Using MATLAB, develop a program that calculates the steer angle required or any level o lateral acceleration. The low o the program should parallel the development in Gillespie and in class but substitute this tire model. The structure should look like: Calculate lateral acceleration (as a unction o Velocity) Calculate lateral tire orces (ront and rear) Calculate roll angle Calculate normal load on all 4 tires Find peak lateral orce at each tire (see check-o) Get slip angles at ront and rear axles (see check-o) Calculate steering angle The MATLAB script should solve or the slip angle given inner and outer normal loads and the combined lateral orce. 3) Inspired by a particularly thrilling lecture in vehicle dynamics, you decide to devote your weekends to racing and want to adapt your car to the purpose. You decide to get a handle on the characteristics o your car by running it through the simulator developed above. Plot the steer angle as a unction o lateral acceleration up to the limits (watch out or tire lit-o as you change parameters a tire that lits o the ground should not be producing orce though the existing script does not check or this). Does the car understeer or oversteer in the linear region? What is the understeer gradient (approximated rom the graph)? Does it display limit understeer or limit oversteer? What is the peak lateral acceleration? What is the limiting actor? Assume that your car has the ollowing parameter set and it initially has the ront roll bar installed. Assume also that your test track has a radius o 50m. m = 1475 kg L = 2.58 m a = 1.16 m b = 1.42 m h = 0.6 m h t cg = h = t r = 0.1m = 1.44 m = 232 Nm/deg (springs only) = r r 480 Nm/deg (springs and anti - roll bar) = 250 Nm/deg (springs only)
4) You re bumming. This doesn t seem like it is going to win you many autocrosses. Figuring that the ront anti-roll bar might be causing you problems, you decide to remove it. Plot the new steering angle required as a unction o the lateral acceleration. Does the car understeer or oversteer in the linear region? What is the understeer gradient (approximated rom the graph)? Does it display limit understeer or limit oversteer? What is the peak lateral acceleration? What is the limiting actor? 5) Thinking this is going well, you decide to put the anti-roll bar you removed rom the ront on the rear axle. Does the car understeer or oversteer in the linear region? What is the understeer gradient (approximated rom the graph)? Does it display limit understeer or limit oversteer? What is the peak lateral acceleration? What is the limiting actor? Does this seem like a good modiication? 6) You decide to buy an additional anti-roll bar o equal stiness so that you have bars on both the ront and the rear. Does the car understeer or oversteer in the linear region? What is the understeer gradient (approximated rom the graph)? Does it display limit understeer or limit oversteer? What is the peak lateral acceleration? What is the limiting actor? 7) From this coniguration, assume that you decide to modiy the suspension kinematics. Can you make a roll center height change that gets you around the corner aster? I so, what do you do? 8) Finally, assume you decide to make some major modiications to the car and can move the c.g. location (though not the height) anywhere you want and add total roll stiness up to 600 Nm/deg divided as you want between ront and rear. What should you do to get around the corner as ast as you can? What is your rationale or this? Plot the steering angle versus lateral acceleration. What is the peak lateral acceleration? Just as a point o interest, i you were to get seriously into racing, you would want to be able to consider several other modiications as well that might come in handy on race day. One o these is changing the tire pressure, which alters the tire characteristics (cornering stiness and peak). Another is to add wedge, cranking up the pre-load on a rear spring to create a diagonal imbalance in the tire normal orce distribution. In many racing circuits, there are also suspension modiications that can be made. In NASCAR, the rear roll center height can be adjusted by changing the track rod o the suspension. The key to understanding the challenge o racing is to see that the tire properties change as the tires heat up and wear. Since your ront tires are steering while your rear tires carry all o the tractive orces in acceleration, they wear at dierent and rather unpredictable rates. Thus, the tricky balancing act you have done on this assignment only holds or one point in time. As a crew chie, you need to make the call on how much to change each parameter so that the car will run as well as it can between pit stops. You also have rather imprecise data rom which to work i you have any data at all. I you make the wrong call well, that s racing, and you ll get em next week.
HW #5 Check O These unctions will be needed to complete Problem #2. a) Write a matlab unction to calculate Fymax, given Fi, Fo (using the Pacejka model). This is airly straightorward to write using basic logic. However you want to do it in a more sophisticated manner, thereore use the matlab unction minsearch. This requires that you write a matlab unction (which minsearch will call) that takes in alpha, Fi, Fo, and returns Fy such as: unction Fy=calc_max_y(alpha,Fi,Fo) insert code here Fy=-(Fy_inner+Fy_outer) Note: the negative sign because minsearch is looking or a minimum thereore, you must lip the Pacejka curve. Thereore minsearch will actually return Fy_max (use abs ater minsearch to convert to max Fy). b) Write a matlab unction to calculate alpha, given Fi, Fo, and Fy For this use the Matlab unction solve or ero This requires that you write a matlab unction (which solve will call) that takes in alpha, Fi, Fo, Fy_desired and returns Fy such as: unction Fy=calc_Fy(alpha,Fi,Fo,Fy_desired) insert code here Fy=Fy_inner+Fy_outer-Fy_desired c) Check the unctions or parts (a) & (b) by writing a script that plots Fytot (or a given Fi and Fo) vs. alpha and veriying Fymax and the alpha or Fy_desired! The script will use calc_fy (with Fy_desired=0), minsearch calling calc_max_fy and solve calling calc_fy (with the given Fy_desired). Show this to me in class or your check-o: plot(alpha,fy_tot) hold on plot(alpha_or_fymax,fymax, *b ) plot(alpha_or_fydes,fydes, *r ); hold o
Things to be careul about: The minimum value or F is 0 (maximum F is F) I F is ero then Fy at that tire is 0 Have to take ront and rear axles separately I you are going to have several m-iles and unctions that use a large set o variables, it is oten beneicial to place these variables in a separate m-ile called load_pacejka_parameters.m or example. Then in any m-ile or unction that requires these parameters you simple include the line: load_pacejka_parameters Be careul that the variables in load_pacejka_parameters have unique names so as to not accidentally rename a variable in another m-ile. Capital variable names usually help. This is not too necessary or this assignment, but a useul tool to keep in mind when running simulations with lots o variable deinitions that are used in multiple m-iles.