Q45tech from my archives. The links are still valid.
True a bar's torsional stiffness is a 4th power function of diameter [assuming the MODULUS is the same] [ NOT THE SQUARE AS YOU MENTIONED} but in the Q/J case the bar attachments at the midpoint so it just happens to be a 1/4 reduction from bar stiffness to wheel stiffness. The sway bar if calculated in isolation is 4 times stiffer [the front bar is 664 lb/in but for every 1" of wheel movement the bar only twist 1/4"] so we say the bar's effective wheel rate is 166 lb/in. People skip over this lots of times and novices hear 600 pound springs/bar and wonder why their car has only 150 pound springs/bar. A 10-20% change can be dramatic depending on how close the designers got in the first place see below:
http://www.rqriley.com/suspensn.html http://www.eibach.com/ERS_Worksheet.htm
Another thing missed is the angle from vertical must be corrected by the Cosine x spring rate.
One of the things that owners miss is that the way the sway bars are coupled to the lower arm [in the middle] results in a 1/4 coupling ratio. Slack in the bushings is created by limiting the ability to tighten the assembly nuts so the bars essentially are not there for the first 1-1.5" of wheel travel while the 1/4-3/8" of total slack is taken up.Placing a metal washer [or two] between the bottom nut [on the front] and bottom and top nut [on the rear] and the bushing retainer allows the bushing to be preloaded.This makes the bar act immediately vs the OEM design......effectively doubling [or any amount of adjustment in between] the bar stiffness at the full 4" of wheel travel.Obviously getting the bar too closely coupled will affect harshness so you must make FINE FINE adjustments in 1/4 turn of the nut increments [you will feel the resistance to tightening]! Always set the bar preload with you or your weight in the drivers seat. It will only be correct at one weight load but we are speaking of very fine PRECISION adjustments.When everything is correct [all new bushings and parts] the system can be set so that you can feel a major difference in real roll couple between a full and 1/4 tank of gasoline [this is a 90 pound change]. The Q like magic gets transformed.The stock front 29 mm bar is roughly equal to the 166 pound in springs.The A [20 mm] rear bar adds roughly 40 pounds of additional stiffness to the rear 120 pound springs. IRS systems have a bit of extra rear roll due to the higher roll center and you should not attempt to remove all of it as the toe compensation design [under deceleration/braking] will be compromised.
When you have a 53/47 weight balance [on a 4350 lb car] the correct term is the larger rear bar would make the transient handling slightly less understeering.Oversteering only occurs when the front wheel C of F exceeds the rear.Only when the rear stiffness exceeds the front or you exceed the C of F of the rear tires by accelerating in water [while trying to turn at the same time] can oversteer occur.The roll stiffness of the front is roughly 250 in/lb vs rear Q45t 135 in/lb or 165 in/lb with a 20 mm bar quite a ways from 235 lb/in which would be the neutral balance point.Even the solid 24 mm Stillen Bar [no longer available] just [exceeds]/equals this neutral point [by 15 lbs]. It was assumed that the bar would be used as part of a redesign which included 20 pound stiffer rear springs so the neutral point wouldn't be exceeded!The front of a Q has the ability to reach 0.78G while the rear can get to 0.92 G [due to less weight on the rear tires [slope of C of F vs slip angle vs weight on tire].....the 120 pounds lighter load on each rear static plus the almost 100 pound lower corning load ~~~~ 220 lbs.
The safe [dry] rule of thumb is never install a rear sway bar equal to or > the rear spring stiffness: and a 20 mm bar is only 34% of spring if perfectly coupled with solid metal bushings...the more typical rubber OEM bushings slack will reduce the 1st inch [of wheel/body travel] to almost no added stiffness.....then the bar will kick in progressively. So under a fully side to side swing [-3 to zero to +3] the 20 mm might get close to 85/366 or 23% total stiffness increase. {ON A GOOD DAY}
Story of interest:
http://www.autospeed.com/A_0477/page1.html One important point to note, one which is often misunderstood, is that regardless of the amount of roll allowed by the suspension design, the actual degree of weight transfer remains unchanged. This is only affected by the track, C of G height and cornering acceleration. So, as with most design features in anything mechanical, the selection of roll axis position is a compromise: too low and we get excessive roll, too high and other undesirable handling traits surface. In practice, the com- promise varies with different types of car but always such that some roll occurs. Lowering the C of G is another technically possible way of reducing the roll couple, but this can only be done to a certain extent, due to the boring necessity of leaving comfortable space for the occupants.The roll couple that such compromise leaves must be resisted by the car's springs, which leads us to roll stiffness. This term is defined such that the degree of roll is equal to the roll couple divided by the roll stiffness. Stiff springing obviously reduces roll and hence increases the roll stiffness, but if this is the criterion for selecting spring rates we will usually end up with an uncomfortable ride over normal road irregularities, so the anti-roll bar was developed to ease the situation. The anti-roll bar is a torsion bar (torsion spring) connecting the suspension systems on each side of the vehicle in such a way as to allow both wheels to respond unhindered to two-wheel bumps, such as a ridge across the road. But if the wheels try to move independently, as with a single-wheel bump, or in opposite directions when the car rolls then the anti-roll bar resists this tendency. Roll is reduced as intended but comfort suffers as the effective spring rate of each wheel is increased in the individual single-wheel bumps, although the combined spring rate of the two wheels is unchanged over joint disturbances. Again a compromise must be reached between the requirements of minimum roll and good response to road shocks. Roll bars, unlike the springs, are undamped (theoretically damping could be incorporated, although the manufacturers have generally concluded that it is not worthwhile). This is another reason for limiting the influence of the anti-roll bar, as oscillations might occur if the undamped bar is too stiff. It is well known that the under/over steering characteristics of a car can be substantially modified by tuning the springing and anti-roll bar stiffnesses, altering the roll stiffnesses of each end. While the vehicle as a whole has a certain roll stiffness, this is made up of the separate roll stiffnesses at the front and back, which may be quite different. For example, let's consider the case of a beam axle pivoted on the chassis at its mid point, and devoid of any form of springing. As unlikely as this layout seems, you may see it fitted to the front of some tractors, because it has good terrain-following properties. Now, because the chassis is completely free to rotate about the pivot point of this beam axle, then no roll stiffness is provided at this end of the machine and so all the stiffness needed must be available from the other end. This lack of any roll stiffness means that body roll cannot cause any weight transfer to the outside wheel, and hence as the total weight transfer must be the same anyway, the other end must obviously be subjected to proportionally more. Tyres have the interesting property that although they are capable of supporting higher cornering forces when subject to higher vertical loading, this does not go up in proportion. In other words, the co-efficient of friction is reduced as more weight is placed on them. In practice this means that weight transfer reduces the combined cornering force capable of being developed by the pair of tyres at one end of the car. Now as we have seen, the weight transfer at either end of the vehicle can be controlled to some extent by altering the roll stiffnesses of one or other, or both ends. Therefore, the tyre slip angles needed to produce the required cornering forces can be adjusted by modifications to the wheel springing, thus giving us the means to alter the under/over steering properties. Dampers, too, have their part to play in the extremely complex interrelations .....""
http://www.ctv.es/USERS/softte...g.htm
Since the midpoint of lower arm where the bars attachometer only moves [1/4] the total vertical is a max of about 1.75" on a lift with the wheels drooping about 1".