Suspension design based on understeer gradient

[Deadly_panda]

Is understeer gradient the ONLY major factor influencing suspension design? and if so how much of an understeer gradient is required/optimum and how is it determined?

[Deadly_panda]

Is understeer gradient the ONLY major factor influencing suspension design? and if so how much of an understeer gradient is required/optimum and how is it determined?

[BillCobb]

Nope.

I'd say about 3rd down the list (maybe 4th).

Next question....

[Vittorio]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by BillCobb:
Nope.

I'd say about 3rd down the list (maybe 4th).

Next question.... </div></BLOCKQUOTE>

Next question: What are the first, second and maybe third factors?

[BillCobb]

In order of priority (sort of), my recommendation would be
1) Its gotta live (durability).
2) Its gotta be within the budget (cost and manufacturability and timing).
3) Its gotta fit (package and envelope)
4) Meets the rules (materials, brakes, steering, springs, joints, bushings, dampers, fasteners, etc)

I would never design based on an understeer target because these cars are 'special'. Instead, design for control sensitivity (g/100 deg SWA), low front AND rear axle sideslip gradients (deg/g), adequate kingpin torque gradient and sign [front suspension], satisfactory roadwheel angle range (turn circle), camber stiffness, and stable brake reactions (anti's and toe change). Drivers can adapt to a low or slightly oversteering car, it will be fast in the corners, comfortable in the slalom and aggressive under power.

You will need to provide your frame dude (or dudess)designer with acceptable attachment stiffnesses, too. A great suspension dangled from a soggy frsme is a soggy doggy. That include the steering system.

Besides, its hard to get an 'understeering' car response out of a rearward weight distribution without either: a significant tire or wheel size difference or, some ugly geometry ratios, compliances, or tire lateral load transfer distribution configs.

[Z]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by BillCobb:
In order of priority (sort of), my recommendation would be
1) Its gotta live (durability).
2) Its gotta be within the budget (cost and manufacturability and timing).
3) Its gotta fit (package and envelope)
4) Meets the rules (materials, brakes, steering, springs, joints, bushings, dampers, fasteners, etc) </div></BLOCKQUOTE>
Agreed on above (and rest of Bill's post).

I'd probably put 2,3,4 as equal first, because without them you're not going to have ANY fun at all.

Durability then sets how long your fun lasts at the comp.

Also, "5. It's gotta be quick and easy to build", because that gives you more fun before comp. http://fsae.com/groupee_common/emoticons/icon_smile.gif

Z

[mech5496]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Z:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by BillCobb:
In order of priority (sort of), my recommendation would be
1) Its gotta live (durability).
2) Its gotta be within the budget (cost and manufacturability and timing).
3) Its gotta fit (package and envelope)
4) Meets the rules (materials, brakes, steering, springs, joints, bushings, dampers, fasteners, etc) </div></BLOCKQUOTE>
Agreed on above (and rest of Bill's post).

I'd probably put 2,3,4 as equal first, because without them you're not going to have ANY fun at all.

Durability then sets how long your fun lasts at the comp.

Also, "5. It's gotta be quick and easy to build", because that gives you more fun before comp. http://fsae.com/groupee_common/emoticons/icon_smile.gif

Z </div></BLOCKQUOTE>

And after that in case of not-so-rare mishaps...http://fsae.com/groupee_common/emoticons/icon_wink.gif

[Deadly_panda]

fine ... agreed on the above....
but the thing that got me going was that every aspect from roll resistances, camber, roll steer to aligning torque have a direct influence on the understeer gradient ... so will understeer gradient (which needs to be positive for understeer) be the sole factor determining the value of these parameters or is there something else to it?

As in if we have a rearward weight distribution (as BillCobb said) which causes oversteering (negative understeer coefficient)....can we compensate it with by altering the other parameters (camber, roll resistance etc) so that the gradient is increased....and if that is what decides the setting of these values ... what value of understeer gradient must we aim for?

[Z]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Deadly_panda:
... what value of understeer gradient must we aim for? </div></BLOCKQUOTE>
Deadly,

For FSAE aim for zero, or slightly negative (ie. small O/S, as Bill said).

More importantly, U/S gradient is an end result, or effect, of which the other factors you mentioned are the causes. You can have many different combinations of those causes giving the same final effect.

But even more importantly, understeer gradient is a concept beloved by US auto makers (mainly thanks to Ralph Nader and the Chevy Corvair!). Have you ever heard of a modern US production car with sporty handling? http://fsae.com/groupee_common/emoticons/icon_smile.gif

You don't want an understeering bland-mobile in FSAE.

Z

PS. If your car was driving on very bumpy roads you would want very soft suspension with no camber change, etc. But, fortunately, no bumps in FSAE so your job is much easier. Your job is mostly about the other things mentioned above (points 1-5), with U/S gradient as the final tuning AFTER the car is built (via small changes to tyre pressure/toe/camber, springs, etc.).

[Deadly_panda]

ok so we want slight oversteer.....

then what will be the factor for determining the front and rear roll rates and thereby the spring to be used (since we can vary the position of wheels wrt cg in side view, wheelbase and front and rear trackwidths)?

and in kinematic analysis we aim at finding camber change and minimizing it. IS that done solely to maintain the best configuration during cornering?

[Zac]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">But even more importantly, understeer gradient is a concept beloved by US auto makers (mainly thanks to Ralph Nader and the Chevy Corvair!). Have you ever heard of a modern US production car with sporty handling? </div></BLOCKQUOTE>

The Chevy Volt actually.

[DougMilliken]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Zac:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Have you ever heard of a modern US production car with sporty handling? </div></BLOCKQUOTE>
The Chevy Volt actually. </div></BLOCKQUOTE>
I won't comment on "sporty handling" which is very subjective, but in terms of measurable quantities, here's a story I've told often. I own a couple of US-spec station wagons (one from a US brand, other Japanese brand) that are near neutral steer. One of them is so close that an increase of 2 psi (30-&gt;32psi) on the rear tires changes the linear behavior from directionally unstable (wandering on the freeway) to very slightly stable (not wandering). Makes a big difference in driver workload, much easier to keep lanes on boring long straight roads.
The same cars stay very near neutral all the way to the limit--no need to adjust the steering as speed is increased on a constant radius circle.

Design judge questions: Why would front wheel drive station wagons be near neutral steer, when the sedan variants of the same platform are understeer and push/plow at the limit? And what must be happening when I add 2 psi to the rear tires?

[BillCobb]

As a matter of fact, based on thousands of measurements over the years, there ar NO production cars having linear range oversteer and very few (only exotic, limited one of a kind deals) that have enough oversteer to be unstable in open loop control. The lowest numbers are generally above +1.5 deg/g You can understand why this is. Different tire sizes, wheel rim width, and pressure are used with varying degrees of success to correct the architectural induced oversteer trait. Some brands (which you probably would guess to be neutral steer) actually have rather large measured understeer values. Some have substantial amounts of roll understeer to 'fix' these problems, too, for which they then exhibit traits in limit handling maneuvers that paying customers should never have to deal with.

Sorry, but even recognized race cars in the big series have comfortable understeer levels because they need to be fun to drive in a lerge speed range without the threat of closed loop control mistakes. Adding large amounts of engine power also demands them to be understeering.

So, the narrow focus for FSAE should prescribe a car control strategy that avoids geometric steer factors because tires don't 'hear' steer/slip angle inputs at high cornering levels. Camber, yes, weight changes, yes, longitudinal slip changes, yes. Also, aligning moment reactions (steering compliance) is a bad deal because these stabilizing moments return to low or negative values at the limit, too. If roll rate frequency and yaw rate natural frequency are very different, then some bizare sign changes can occur as yaw rate frequency changes with speed (squared) and roll rate frequency does not. I.E. your roll understeer will exhibit roll oversteering effects.

The dilemma of how to produce a comfortable driving racing car with a high max lateral capability is further swayed by the speed range it operates in. A plot of steering gain (g/100 deg SWA) vs. speed for several understeer values ought to give you an idea why some understeer is beneficial. Not too much, just the right amount...

One other thing. The traditional liturgy of cornering compliances and understeer is defined only for steady state handling. BUT, a theoretical update of the handling equations to include the dynamics of a tire (all of them: Fy, Mz and Mx) and body roll response would be beneficial to a FSAE car design. Getting the data for these components is the challenge. Perhaps the TTC could run a few plays for the tire elements if there was enough interest.

[BillCobb]

Doug: Is that a question or is that a challenge to answer? The answer is actually quite simple and elegant and based on control systems theory. And it can be tested subjectively and objectively (which we have obviously done) and can be corrected.

Anybody need the answer envelope or should I hold off?

[Price]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by DougMilliken:
Design judge questions: Why would front wheel drive station wagons be near neutral steer, when the sedan variants of the same platform are understeer and push/plow at the limit? And what must be happening when I add 2 psi to the rear tires? </div></BLOCKQUOTE>

I'll take a stab at it

Front wheel drive stationwagons are more neutral than their sedan counterparts because of added weight at the extreme rear of the car. This shifts the weight balance to the back on a usually front heavy front wheel drive sedan. Most cars, excluding higher performance cars, come stock with the same size wheels an tires, so presumably the same cornering stiffness, so closer to a 50/50 balance should give more neutral steer.

The increase in tire pressure in the rear increases the cornering stiffness which provides more of a stabilizing force for small slip angles during your 'wandering' maneuvers. I would assume it would also respond to slip angles faster because the carcass would be a little stiffer. This makes the system respond faster, and seem more pointed.

Was I close?

[BillCobb]

Nope. First paragraph is just plain incorrect. FWD wagons are very understeering (as measured, btw), because they have to still be understeering when you load up the family, the dog, the groceries, the tents and the beer cooler. Most suspensions have load dependent roll steer, especially for wagons, so that when load is added, the suspension changes the ride height. Side view trailing arm angles changes and roll steer is added. A delta of 10 to 15% is about average for these types of vehicles (from some nominal starting value). The result is a wagon that has the same understeer when loaded as when unloaded.

The situation Doug is talking about is a wander at a non-zero steer frequency) often roadway induced and in closed loop driver control.

The 2nd paragraph is a bit closer. But the 'problem' is caused by a phenomenon originating in the front of the vehicle.

A solution to this using Bode analysis to determine the cause makes for a great thesis or project report. And its not oversteer, its a localized instability in the yaw rate transfer function under certain circumstances.

Now keep going. What are the circumstances? What are the down sides to a LOT of understeer?

[wil oberlies]

In response to Dougs question...I'm afraid I have to go for the low-hanging fruit and say CG location must be slightly more aft on the wagon then the sedan, and the increase in tire pressure must increase the lateral force available per unit slip angle, resulting in a net increase in aligning moment about the CG. How's that for a run-on sentence? Edit: I posted at the same time as Price...but one more thought; the CG must be higher on the wagon, resulting in more roll steer?

[js10coastr]

Doug, is that a "Design Judge" question... or a question for the design judges?

I've never raced a station wagon (or driven one for that matter), so I do have to ask if the station wagons run a different rear spring rate compared to the sedans.

[Drew Price]

I know from driving people around for our school's SafeRide service ('07 Prius) that when people get in the back, the steering effort goes up slightly compared to unloaded.

More weight in the back --&gt; more pitch towards the rear --&gt; increased castor angle in the front suspension --&gt; more steering effort (more centering effect).

[Z]

Too many issues to cover fully here, so just briefly...

Doug's question regarding the wagon is one of those "How long is a piece of string?" things. Too many unspecified variables. Anyway, here are two factors that come to mind.

1. The wagon probably has better aerodynamic stability than the sedan, because of a bigger "aero-rudder". It possibly (?) also has slightly more rear aero downforce, or less lift, possibly as a side-effect of increased drag. This, combined with the higher pressure in the rear tyres, stiffens the "tyre-rudder".

2. The wagon will have a mass-distribution longitudinal principle axis that slopes more upwards-to-rear, than the sedan. This will influence transient response, and possibly couple with the driver/steering to give a "wandering" behaviour.
~~~~~0o0~~~~~

Bill, you say in your earlier post "Drivers can adapt to a low or slightly oversteering car, it will be fast in the corners, comfortable in the slalom and aggressive under power.". I agree with this, and think FSAE cars benefit by being slightly "loose" (ie. O/S-ish).

Later on you suggest "some understeer is beneficial"? Did I misinterpret something?

Z

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Price:
The increase in tire pressure in the rear increases the cornering stiffness </div></BLOCKQUOTE>

http://s3.amazonaws.com/kym-assets/entries/icons/original/000/000/554/facepalm.jpg?1282626490

Dangerous to get into the mindset that more inflation/stiffness = more response.

[BillCobb]

Z:

I believe the Endurance Course task would be more 'enjoyable' if the car was a bit understeerng. For the Autocross Course, a car whose oversteer is greater than its negative Ackernman Gradient (oh boy, gonna have to demo that one) could e faster. But, now the driver would have to be much more conditioned to operate it underal conditions. Without a lot of practice feeling and sensing the effects, it would be a handfull. A lot of drivers in pretty suits and shoes, but probably not a lot of hours. I wonder what a decent simulator would be able to teach them...

[BillCobb]

What I was trying to say was that if the car had a negative oversteer number that is less than its Ackerman gradient at a decent speed then you would like it. Else, no joy.

X_F: Yeah the presumption of higher stiffness with higher pressure at thoses pressures is questionable. The tire is probably over the top, especially those tires.

What I'm trying to eleicite from the group is a System Engineering solution to the 'wander' situation. You know, r(s)/delta = (r1*s + r0)/ (d2*s^2 + d1*s +d0) for a simple bicycle model having a wheelbase, cornering compliances and running a speed. Examine the yaw by steer phase characteristics vs frequency for a few different speeds in a range of say 50 to 140 kph for a few different cornering compliance recipes. When we say that front compliance is bad for you, we don't mean your liver. You'll see why. And the rear has got to be good too.

This is New School Vehicle Dynamics. Its Math based. The cars are good right off the Hauler as they say. Old School stands for guessing, changing parts and parameters for unknown reasons and having to stop when the green flag drops because you're out of time.

There is a closed form solution to this. Extra points for WHY. Show all work. Hint, use an algebraic solver like Maple or Derive etc. Its much handier and can spew the results right into Matlab. A GUI with sliders and numeric boxes gets a free table dance.

[Warpspeed]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by DougMilliken:

Design judge questions: Why would front wheel drive station wagons be near neutral steer, when the sedan variants of the same platform are understeer and push/plow at the limit? And what must be happening when I add 2 psi to the rear tires? </div></BLOCKQUOTE>

If the sedan and station wagon platforms are otherwise of identical design and geometry, perhaps the only difference being that the station wagon could be fitted with a higher rear spring rate to cope with a greater anticipated load range.
That would result in greater rear roll stiffness, and hence reduce the understeering tendency at all speeds compared to the sedan variant.

I am thinking your station wagon might be suffering from a lack of rear lateral stiffness, most probably due to some significant tire sidewall flex.
If the vehicle receives a lateral disturbance, that could very slightly yaw the vehicle, and cause the front wheels to yaw along with everything else.
The resulting slight change in vehicle direction would reinforce the original lateral disturbing force creating some oversteering high speed directional instability.
A bit more air at the back should stiffen up the tire sidewalls and increase rear lateral stiffness. Although I would be rather surprised if adding only 2psi would be enough to really solve the problem.

[Zac]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Price:
The increase in tire pressure in the rear increases the cornering stiffness </div></BLOCKQUOTE>

http://s3.amazonaws.com/kym-assets/entries/icons/original/000/000/554/facepalm.jpg?1282626490

Dangerous to get into the mindset that more inflation/stiffness = more response. </div></BLOCKQUOTE>

I see this every day. With respect to IP changes on cornering stiffness it is highly load dependent. With an increase in IP, you'll see a decrease in low load CS (rounding out your footprint) and increase your higher load CS (keeping your FP center line from lifting). What's really interesting is that if you plot this for several loads you'll see an inflection point where multiple curves intersect.

[DougMilliken]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by js10coastr:
Doug, is that a "Design Judge" question... or a question for the design judges? </div></BLOCKQUOTE>
Justin, it seems that it was more of a question than I first imagined! To me, design judge questions are ones that can be answered/discussed at different levels...

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">FWD wagons are very understeering (as measured, btw), because they have to still be understeering when you load up the family, the dog, the groceries, the tents and the beer cooler. </div></BLOCKQUOTE>
Bill, maybe you never tested a '92 Corolla Wagon DL (5-speed)? When it was new, mine really was neutral steer all the way through a constant radius test--almost no steering wheel angle change from low speed to the limit.

I ran informal tests on different sized circles including a 220 foot traffic circle on Grand Island. Driver only and no cargo. With steering ratio of about 18:1(??) your suggested minimum of +1.5 deg/g would be about 20 degrees of steering wheel angle at the low Ay limit for this car, and I certainly would have noticed that much change. Of course it could be horsed around a little with power off & on, but not much (not much power!) Suspension is struts all around with no anti-roll bars, rear has parallel lateral links.

This has been a little workhorse of a car, the handling and steering system are not that great--it was a surprise to find that it was neutral steer in the constant radius test.

I haven't seen detailed suspension (K&C) data so can't be sure of other contributing factors. It seems likely that (compared to the sedan) the extra rear weight and stiffer rear springs (for load capacity) contribute to this result.

Wander -- because of the surprising neutral steer result from the constant radius test, my simple explanation for the highway speed wander was lack of directional stability (weather vane). The wander is random in direction, with equal tire pressures the car won't "point". I tried a few different combinations of pressures and 30/32 F/R in the little tires (175/70-R13) was all it took to subjectively reduce the wander. With no tire data, it seemed pretty likely that this raised the rear cornering stiffness a small amount, enough to add some static stability.

Of course there could be more subtle things going on and I defer to Bill who has done much more work in this area (on-center handling) than we have.

[js10coastr]

Ahh ok http://fsae.com/groupee_common/emoticons/icon_smile.gif

I had the springs and the weight part figured out... I was stuck in "rocket mode" though with the aero, where more surface area in the back produces a more "stable" vehicle; but then there are "over-stable" rockets that weather-cock into the wind too http://fsae.com/groupee_common/emoticons/icon_smile.gif.

Maybe I should trade in the sedan for a wagon, would probably make the commute a bit more exciting... although, I don't think any car could make 10 mph traffic exciting http://fsae.com/groupee_common/emoticons/icon_biggrin.gif

[Warpspeed]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by DougMilliken:
Wander -- because of the surprising neutral steer result from the constant radius test, my simple explanation for the highway speed wander was lack of directional stability (weather vane). The wander is random in direction, with equal tire pressures the car won't "point". I tried a few different combinations of pressures and 30/32 F/R in the little tires (175/70-R13) was all it took to subjectively reduce the wander. With no tire data, it seemed pretty likely that this raised the rear cornering stiffness a small amount, enough to add some static stability.

</div></BLOCKQUOTE>

Doug, I remember a friend that had a small Japanese sedan that had some truly horrid high aspect cross ply rear tires, and it felt strangely unstable at any speed.
Stationary, you could push the rear of the car sideways by hand, and it would wobble like a jelly on those tires, although they did not appear visually to be under inflated.
At freeway speeds it would wander and felt really dangerous, even to a passenger.
Much later he fitted some radials at the back, which made it much more sure footed.
Your anecdote triggered a very powerful recollection of that car.

[DougMilliken]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Much later he fitted some radials at the back, which made it much more sure footed.
Your anecdote triggered a very powerful recollection of that car. </div></BLOCKQUOTE>
Did your friend ever rotate those relatively high cornering stiffness radials to the front (with the low cornering stiffness bias tires on the rear)? From memory, the radials could have 2x the cornering stiffness of bias tires and, with them swapped, the car might have been even more "memorable"! Back when radials were replacing bias ply tires, there were probably many mis-matched sets like this, and some very scary cars. Tire stores posted rules (if only two radials, must be on the rear), but very few people understood why this was important.

Apologies to the originator of this thread, we seem to have hijacked it!

[BillCobb]

Whenever I have to inter-mix radials with biased tires (like on an ex-wife's Mercedes, Jag, or BMW), I put the radials on the right side. (you know, righty tighty, lefty loosy). For girl friends it's the other way around (loose in, tight off). Ain't that true, Blue?

Happy New Year !

[Warpspeed]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by DougMilliken:
Did your friend ever rotate those relatively high cornering stiffness radials to the front (with the low cornering stiffness bias tires on the rear)?

The car might have been even more "memorable"! </div></BLOCKQUOTE>
Hahaha... I am sure you are right Doug!!

At the time it was the rear that felt diabolically unstable, so that was the end we were concentrating on.
I cannot recall what the front tires were, but I am fairly certain they were also cross plies from a different manufacturer.

That car felt awful, almost as though something was broken or adrift in the rear suspension, but both leaf springs and shackles appeared to be intact. The very obvious lateral wobble was all in the sidewalls.
Ever since, I have become totally paranoid about rear lateral stiffness.

But getting back onto the topic of understeer gradient......

My simple understanding is, that it is possible (within reason) to set almost anything up on a skid pad for a gentle understeering gradient by tuning only the relative front/rear roll stiffness.

But the real test is a figure of eight course, or a slalom, where there is a very sudden directional change from cornering at the limit one way, to cornering at the limit the other way.

Rather like tacking a yacht up wind, everything undergoes a very dramatic and sudden change of direction. That change can be smooth, or chaotic, if load transfer at front and rear are not suitably in phase and magnitude with each other. And to get that happy cooperation between front and rear, roll axis inclination is the key.

I am desperately trying to get my head around all this, and any assistance would be most gratefully received.

And a Merry Christmas to all.

[PatClarke]

Cobb,

You are an evil man http://fsae.com/groupee_common/emoticons/icon_wink.gif http://fsae.com/groupee_common/emoticons/icon_biggrin.gif http://fsae.com/groupee_common/emoticons/icon_smile.gif

Best seasons wishes

Pat