Understeer budget spreadsheet 2

[GregLocock]

Placeholder for a discussion with cibachrome. I'm looking for a non-proprietary spreadsheet to put the linear range understeer budget together, or failing that just the equations. I found this

https://projects.skill-lync.com/projects/Final-Project-83167

, which is pretty complex, and does a lot of things right. Unfortunately it is wrong in several places, and the polarity of the various K&C measurements is not defined.

The relevant SAE papers are 670078 and 760713. Milliken kind of wanders past the subject, but doesn't set the equations down in one neat form.


Cheers

Greg Locock


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[cibachrome]

% By your command: (But not a spreadsheet). This is Fortran to Matlab for a vehicle you might recognize). I % % use spreadsheets only for cakes, pizza and cookies.

%A few caveats:

% Watch the sign conventions. Chassis engineers are confused by negative tire stiffnesses so we coddle them by % being positive thinkers.

% K % C coefficient conventions are (+) for all understeer effects and (-) for oversteer effects regardless of % front or rear designation.

% Force & Moment based K & C coefficients are 'per 1000 N' or 'per 100 Nm' to avoid the ridiculously small
% values likely to be obfuscated by those having exceptional brilliance.

% Unit conventions are N, Nm, deg, kg.
%----------------------------------------------
% Cornering compliance budget after simulation has completed.

CAF = 1489.373 ; CAR = 1401.010 % Tire cornering stiffnesses per tire (N/deg)
CCF = 0.313 ; CCR = 0.323 % Tire camber stiffnesses per tire (N/deg)
NAF = 50.082 ; NAR = 42.897 % Tire aligning moment stiffnesses per tire (Nm/deg)
CGF = 47.657 ; CGR = 43.392 % Tire camber stiffnesses per tire (N/deg)
NGF = 4.766 ; NGR = 4.339 % Tire alinging moment camber stiffnesses per tire (Nm/deg)

L = 2915.000 % WHEELBASE (mm)

FYFperg = 9778.8 % front axle sideforce per g from simulation (N/g)
FYRperg = 8430.9 % Rear
NFperg = -403.16 % Front axle aligning moment per g from simulation (Nm/g)
NRperg = -317.93 % Rear

WF = 972.000; WR = 885.000 % AXLE MASS (kg)
WUF = 99.000; WUR = 99.000 %UNSPRUNG MASS (kg)

EF = 0.001; ER = 0.055 %ROLL STEER (deg steer/deg roll)
GF = 0.789; GR = -0.720 %ROLL CAMBER (deg camber/deg roll)
KF = 1085.000; KR = 573.000 %ROLL STIFFNESS (front roll stiffness (Nm/deg)

EYF = 0.191; EYR = 0.045 %L.F. STEER (deg/1000 N)
GYF = 0.291; GYR = -0.328 %L.F. CAMBER (deg/1000 N)
ENF = 0.774; ENR = -0.113 %A.T. STEER (deg/100 Nm)
GNF = 0.022; GNR = 0.023 %A.T. CAMBER (deg/100 Nm)

cc_WGT_f = 9.806*WF/2/CAF; % Cornering Compliances from weight contributions
cc_WGT_r = -9.806*WR/2/CAR;

cc_RBAT_f = -(NFperg + NRperg)/2/L/CAF; % Cornering Compliances from rigid body Mz (F)
cc_RBAT_r = -(NFperg + NRperg)/2/L/CAR; % Rear

ROLLpg = 4.47; % Roll gradient (deg/g)

cc_RS_f = ROLLpg * EF ; % Front roll steer contribution (deg/g)
cc_RS_r = ROLLpg * ER ; % Rear

cc_RC_f = ROLLpg * GF * CGF/CAF; % Roll Camber Front (deg/g)
cc_RC_r = ROLLpg * GR * CGR/CAR; % Rear

cc_LFDS_f = EYF *(FYFperg-9.806* WUF )/2/1000; % From Front L.F. deflection steer
cc_LFDS_r = EYR *(FYRperg-9.806* WUR )/2/1000; % Rear

cc_LFDC_f = GYF *CGF*(FYFperg-9.806* WUF )/2/1000/CAF ; % Front L.F. Deflection Camber
cc_LFDC_r = GYR *CGR*(FYRperg-9.806* WUR )/2/1000/CAR ; % Rear

cc_ATDS_f = - ENF *NFperg/2/100; % Front Aligning Moment Deflection Steer
cc_ATDS_r = - ENR *NRperg/2/100; % Rear

cc_ATDC_f = - GNF *CGF*NFperg/2/CAF/100; % Front Aligning Moment Deflection Camber
cc_ATDC_r = - GNR *CGR*NRperg/2/CAR/100; % Rear


DF = cc_WGT_f + cc_RBAT_f + cc_RS_f + cc_RC_f + cc_LFDS_f + cc_LFDC_f + cc_ATDS_f + cc_ATDC_f; % Front Cornering Compliance Sum
DR = cc_WGT_r + cc_RBAT_r + cc_RS_r + cc_RC_r + cc_LFDS_r + cc_LFDC_r + cc_ATDS_r + cc_ATDC_r; % Rear Cornering Compliance Sum

K = DF + DR; % DR is a negative number (always oversteering). Total Vehicle Understeer (deg/g)

cc(1,= [cc_WGT_f cc_RBAT_f cc_RS_f cc_RC_f cc_LFDS_f cc_LFDC_f cc_ATDS_f cc_ATDC_f];
cc(2,= [cc_WGT_r cc_RBAT_r cc_RS_r cc_RC_r cc_LFDS_r cc_LFDC_r cc_ATDS_r cc_ATDC_r];

figure('numberTitle','Off','MenuBar','None','Name','Cornering Compliance Element Summary',...
'Position',[200 430 800 700])

colormap('hsv')
h=bar(cc);
L=legend(h,'Wgt/Tires','Rigid Body Mz','Roll Steer','Roll Camber','Fy Steer','Fy Camber','Mz Steer','Mz Camber');
set(L,'FontSize',7,'location','North')
legend('boxoff')

vref(1.5)
ylabel('Cornering Compliance Values (deg/g)')
set(gca,'XTick',[])

text(1,-1.25 ,'Front')
text(2,-1.25 ,'Rear')
text(.9,-1.4 ,'Computed From 1^o Linear Tire Stiffnesses')

href(get(gca,'YTick'))
xlabel(['DF = ' num2str(round2(DF)) ' DR = ' ...
num2str(round2(DR)) ' K = ' ...
num2str(round2(K)) ' deg/g ' ])

 

[GregLocock]

Ahem. Thank you very much. It almost runs in Octave straight out the box!

Cheers

Greg Locock


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[cibachrome]

Here is my assessment of the data from your spreadsheet race car. Not a big surprize probably. Giant rear tires, rear roll oversteer, questionable rear aligning moment camber oversteer coefficient.

BTW: I didn't have enough roll rate info so I assigned steady state roll gain (2.5 deg/g, 2.0 Hz. and .6 damping ratio) just to drive the roll steer and roll camber effects.

 

[GregLocock]

Oh, it was a really odd setup, probably as a result of all the errors in the spreadsheet (factor of 2 on weight/tires for a start)

Rollgain was 3.8 deg/g if you care. thanks.

Cheers

Greg Locock


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[TMAPV]

Thanks for sharing! Quick question, how do you measure some of the K&C? For example, your code has:

cc_ATDS_r = -ENR *NRperg/2/100 ;

Where ENF is Mz Steer, NRperg is the front axle aligning moment per g, so I understand why we have a division by 100 in the equation, but not sure where the 2 comes fro, as to me it implies that you're taking moment deflection steer only based on one tire?

Also, what kind of simulation do you run to get Nperg and FYperg?

 

[cibachrome]

ENR is per tire. CAF, NAF, NGF etc are per tire. FYFperg and NFperg come from a total vehicle dynamics simulation model that reports the sum of FYF and FYF from Both tires. BTW, it's not uncommon to have 2 different ENF's measured from left and right wheels, especially if you study the effect resulting from large steer angles and aymmetric steering systems. Also note the non-linearity in the overall curve shapes. Either you install a function representing this effect (as I do) or you need to try to figure out what the correct single ENF value to choose as a result of the iterative process used to get the tire slip angles at any 'moment' in time (! Good one Bill). This curve shape is a good judge of a vehicle's character and 'quality' BTW. It is the convolution of mechanical compliance in the lower steering parts (bushings, tierods, gear mounting, R&P gear clash and pushaway forces, caster setting), the steering assist system (pressure, valve shape, plumbing parts, electronic feedback controller gains) and the upper steering system (U-joints, U-joint angles, and I-shaft isolator type and durometer).

All the K&C data comes from different types and brands of machinery (MTS, Anthony Best, etc) owned by vehicle manufacturers, suppliers (Morse Measurements, etc). A modern wheel aligner with electronic readouts can also be used to get many of these coefficients. You would think that determining the forces per g would be easy as pi (you know: weights, g's and a calculator app. But that's where all the propeller heads stop. There are other actors in the play besides Fy: Mz, Mx, Fx just to name the usual ones. So, you need to actually perform some sort of total vehicle simulation to figure out the true state of the system. The Matlab script shown above is usually just the first cut to investigate and validate proposed handling targets. Obviously tires are not completely linear, and compliances (especially ENF) are not user friendly. I use just a simple model with 4 nonliner tires, a non-linear steering system and all these compliance terms. And, we validate with all the ISO road tests: step, frequency response, max-lat, sine sweep, constant radius, etc.

 

[GregLocock]

I tended to use these models as a first cut when setting a vehicle up, before we had sims (or cars). So for NFperg I'd use

%%GL try this instead
normalcy_bodge=1.02; % a non-tunable parameter
FYFperg = g*WF*normalcy_bodge; % front axle sideforce per g (N/g) with normalcy bodge
FYRperg = g*WR/normalcy_bodge; % Rear
SAFperg=FYFperg/CAF/2;% front slip angle per g
SARperg=FYRperg/CAR/2;% rear slip angle per g
NFperg = -NAF*SAFperg*2; % Front axle aligning moment per g (Nm/g)
NRperg = -NAR*SARperg*2; % Rear

the 1.02 is an architecture dependent number, when in doubt use 1.

Note that the 2s cancel out but SAF is wrong if you leave it in which could cause trouble if re-used.

Officially back then we wouldn't build an ADAMS model until we had a prototype, that always seemed a bit daft to me. Now we officially run ADAMS right through so these simple models are used less than they were. But for a quick look they are invaluable. Coupled with a bicycle model you can quickly generate enough data to keep the development engineers on their toes, without getting bogged down in the details.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[TMAPV]

Thanks cibachrome. I think in a more complex simulation even having left and right side with their own slip angle calculation would give a slightly different value of the ENF contribution towards understeer, due to load transfer sensitivity, but I doubt the magnitude is even worth looking over.

A question for you clever people: It's often mentioned that a neutral car would "theoretically" achieve higher lat acc in steady state, both slip angles peak at the same time. Is this even true for a non-simplified 'bicycle' model with, because as soon as we incorporate for example roll understeer (positive or negative) into the model, we can change the total understeer without changing the slip angles through the lat acc level. (As slong as this new understeer contribution doesn't actually change cornering stiffness at any end)

 

[cibachrome]

Your question will take me 2 replies because my dogs are looking at me funny.

First: The more 'complex' simulation with L & R suspension contributions to ENF doesn't give you a lot of extra contribution because:

1) The ENF comes mostly from NET TIEROD load not separate loads. Hence the rack stiffness, rack mount, rack tooth engagement [rack "push-away" from pinion load], power steering assist compliance (hydraulic valve profile / torque sensor), and I-Shaft joints and joint angles and even tilt wheel mechanisms are the source of ENF in the Total Vehicle understeer budget.

2) Any local compliance at the corners may have a slight influence on the vehicle because of the rigid body net yaw moment because of the tire Mz outputs (which are already modified by load transfer and nonlinear tires).

3) Even with the best of the best column I-shaft bits, the further away you are from a straight shot to the gear's pinion from the steering wheel, the higher the compliance will be. Even though double Cardan's can be set up for CV displacement characteristics, the MOMENT transmission characteristics resulting from the shaft's finite joint angles crank up the compliance. And it gets much more 'soft' as joint angles increase beyond 20 degrees or so.

4) Ackermann geometry can influence the ENF at high sideforce levels as it affects net tierod load gradient, too.

5) Caster obviously is mixed up in the ENF deal, too. Caster angle installed to align the steering wheel torque gradient with the peak front sideslip gradient increases the tierod loads, so more compliance angle (higher ENF) and more understeer via high front cornering compliance.

6) As Greg mentioned, ENF and it's usual non-linear traits are a bad choice for tuning a vehicle to get a satisfactory understeer level. Spongy rack bushings, low hydraulic pressure, wide valve profiles, and jello I-shaft isolators do get you to a legal level, but the effect wears off when the tierod loads go south [underseer drops] and the road feel levels become distaseful. But, that's how MANY high rent badges get the job done.

 

[cibachrome]

OK, the coyotes have been shooed away.

As far as the 'neutral' car is concerned, At some stage of it's life and operation, you can find a vehicle which is 'Neutral Steer'. I'm thinking of a sedan with 5 passengers and a compact spare tire tire in the rear and pulling about 0.30 g's Ay. The literature is filled with vehicle models which only have tire Fy elements (even if they are nonlinear front and rear tires). But, realistically, the vehicle has more than just Fy influences acting on it. In the case you mention, front and rear roll steer present in the understeer budget, doesn't change the tire slip angles, but it does modify the AXLE'S sideslip gradient, hence it changes the 'understeer' [Front Cornering Compliance - abs(Rear Cornering Compliance).

Meanwhile, your 'neutral steer' vehicle probably suffers from sluggish transient response characteristics because it has a low natural frequency. The damping factor for the yaw velocity and sideslip responses is very high. This results in long response time (phase lag for us Controls folks). If they are too long (longer than a drivers response perception), the vehicle becomes ugly to drive. So, Understeer is a benefit to vehicle performance because it raises the damped natural frequencies by reducing the critical damping parameter.

Should I mention that 'roll steer' as a Cornering Compliance element is the convolution of g's by steer and roll by g ? So if at max lat you are out of g's by steer, then Zero times EF and ER are still Zero. Hense roll steer is patch only for linear range, down the road, driving pleasability IMHO.

Meanwhile, roll freguency is almost always independent of speed, while yawrate and sideslip have speed^2 dependency. Now with given sprung c.g. and roll axis heights, your net sprung roll inertia (sprung inertia + Mr^2 terms) and your roll stiffness values, you can produce a sinful coupling of roll frequency with yawrate or sideslip frequency. It would be called 'Q' if you were in a Controls arena. [No its not the Head of MI6). At some speed they may cross over and scramble your roll understeer into roll oversteer effect. Hence, you see sleds which can go VERY fast equipped with ROLL OVERSTEER in the back end. Of course this increases the rear cornering compliance which has to be lowered by specifying tires with much higher cornering stiffness, adding cost, poorer ride quality, replacement headaches, and low mu troubles. Plus the bigger tires reduce any advantage your weight distribution may have provided.

Answer to your question though, is: You changed the understeer by means of adding some roll steer because you changed the axle sideslip angle gradient(s), not the tire slip angles. It's dependent on lateral acceleration via the convolution with roll, so it's officially a member of the Cornering Compliance recipe.

BTW: another common blooper I've seen in my days, is a change in the steering gear ratio intended to raise the gain of a low steering sensitivity vehicle (g's/100 degSWA at a designated speed). They lower the ratio (say 16:1 to 14:1) because the car is not 'quick' enough. But, guess what ! The lower ratio develops higher ENF because of the increase in internal gear loading, and the steering sensitivity still goes down! Like apples ? How do you like THEM apples ?

 

[GregLocock]

Roll steer has its own problems. In Australia we often drive at high speeds over very bumpy roads. If you put a lot of roll steer in then the car becomes very darty on these irregularities.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[TMAPV]

Greg: Makes sense. I guess those Germans have it easy with their smooth roads and can afford to run a tonne of roll steer to make up for putting their engine in the wrong place eh?

cibachrome: Thanks for clearing that up, makes you think why people seek neutral steer when it doesn't actually deliver the max lat acc or transient response... :S

I guess the roll steer convolution makes sense, if our acceleration by steer lost its interest at the limit, the this doesn't feed into our roll by acceleration (output of one into input of the other) and therefore no roll or its roll steer effects from kinematics?

So with the frequency coupling, the goal would be to raise the roll frequency to get away from yaw? Since MR^2 is a component of total inertia, the moment arm between CG height and RC height @CG comes into play. My understanding is then that raising the RCH @CG could minimise Ixx and hence raise its natural frequency, assuming we've already met roll stiffness goals. In the physical world, what does the frequency crossover 'look' like? Because it's obviously not that the rear wheels actually turn out of the corner instead of into the corner (as per design), right?

 

[cibachrome]

I think you have it for most of my spewing. Just to be clear: At a peak tire force level the derivative of force by slip is just about zero, so adding any more road wheel steer angle derived from sprung mass rolling is not going to elicit any more force. Fz and Camber can usually add a bit more.

The manifestation of the sideslip by g, and or yaw velocity by steer frequency proximity can make a vehicle tail-happy and produce a head-toss phenomenon. Wear a neck brace and you'll be fine.

I once started showing this a few dozen yeas ago (1982 Camaro) and the result was a new interest in Frequency Response road testing. Since then, an interesting Matlab exercise to to illustrate via simulation how all this interacts has kept me busy on snowy nights.