Most of these calculations are usually used for the design of single seater circuit cars and so they do not acount for large chassis flex, loose bushes ect ect so you can use them mearly to point you in the right direction. and understand the influences of certain geometries.
Secondly lost of these constants used require analysis that needs to be found from manufacturers such as roll centres and vertical centre of gravity heights although with some sensible thought you can estimate. roll centres are critical and if you can find the heights from BMW then you will have a resonable start.
Okay this is quite hefty but heres the process for the weight transfer, by knowing the weight transfers you can alter the loading on each tyre aiding in greater grip through the corners. using the loads calculated you can adjust oversteer to underster by adjustment of the roll bar. by principle it should be noted that these calculations allow the static camber required to be known which is of greater improtance to cornering grip than how STIFF the roll bar is.
okay time for the maths:
Terms used:
w = total mass of the car and driver
WF = total mass front
WR = total weight rear
UWF = total unsprung weight front
UWR = total unsprung weight rear
UGF = unsprung CoG height front
UGR = unsprung Cog height rear
TF = Track front
TR = track rear
CF = height of front roll centre
CR = height of rear roll centre
SGF = Sprung CoG height front
SGR = sprung CoG height rear
WmF = front wheel movement ( max travel )
WmR = rear wheel movement ( max travel )
SmR = relative front spring movement
SmR = relative rear spring movement
SWF = total sprung weight front
SWR = total sprung weight rear
Sw = total sprung weight
UtF = unsprung weight transfer front
UtR = unsprung weight transfer rear
CtF = weight transfer via roll center front
CtR = weight transfer via roll centre rear
WDR = proportional unsprung weight on rear axle
TM = Mean track of sprung weight
CM = mean roll centre of sprung weight
GM = mean CoG of sprung weight
LM = mean roll moment of sprung weight
St = weight transferred due to sprung mass
Wt = total weight transfer
ArF = front roll resistance due to springs
ArR = rear roll resistance due to springs
BF = front roll bar rate
BR = rear roll bar rate
BrF = front roll resistance due to bars
BrR = rear roll resistance due to bars
Fr = total front roll resistance
Rr = total rear roll resistance
Drf = front roll stiffness, proportional to total roll stiffness
DrR = rear roll stiffness, proportional to total roll stiffness
WtF = total front weight transfer
WtR = total rear weight transfer
^2 = squared
okay, now the maths is actually quite simple. by changing values around such as the roll bar stiffness you will be able to mathmaticaly calculate its effect on how your car will respond to these changes, you could also use it to experiment with different spring stiffnesses etc etc, easiest way is build an excel sheet. i would attach mine but that would be too easy.
REMEMBER TO KEEP UNITS THE SAME USE METERS AND CONVERT KILOGRAMS TO NEWTONS
TOTAL SPRUNG WEIGHT FRONT & REAR
SWF = WF - UWF
SWR = WR - UWR
SW = SWF + SWR
we can now calculate the unsprung weight tranfer ( unsprung means wheels, tyres, hubs, anything thats mass isnt suspended on springs)
UtF = UWF x UGF / TF
UtR = UWR x UGR / TR
our first insight into geometry, by increasing the track we can reduce weight transfer, although most people won so lets continue...
WEIGHT TRANSFER VIA ROLL CENTERS ( either contact BMW or calculate, the calculations are too complex for me to bother writhingin with primitive keyboard functions)
This is the element of sprung mass that is reacted onto the outer tyres directly through the roll centres.
CtF = SWF x CF / TF
CtR = SWR x CR / TR
WEIGHT TRANSFER VIA SPRUNG MASS
this is the mass of the car that rolls around the axis between front and rear roll centres. It should never be thought of a s front and rear or individual height as it has no knowledge of the system it operates in. there for we must calculate roll centers and Cog for it indipendently, these will be refered to as mean.
PROPORTION OF SPRUNG MASS ON REAR AXLE
WDR = SWR / SW
this tells us the distance of the sprung mass Cog between front and rear axles as a percentage. we now now where along the wheelbase the centre of sprung mass is so we can calculate its mean track. mean roll centre and mean CoG using the proportion (percentage)
MEAN TRACK
TM = ((TR - TF) x WDR) + TF
MEAN ROLL CENTRE
CM = ((CR - CF) x WDR) + CF
finding the mean CoG is extremely dificult and for exact pin point requires solid modeling however I have a rough calculatn that is precise enough for us.
GM = ((SGR - SGF) x WDR) + SGF
Now we can calculate the moment between mean roll center about which our sprung mass will rotate and the mean CoG
LM = GM - CM
Congrats if youve gotten this far, we know have all the data to calculate the sprung weight transfer
TOTAL SPRUNG WEIGHT TRANSFER
St = SW x LM / TM
TOTAL WEIGHT TRANSFER FOR THE VEHICLE
Wt = UtF + UtR + CtF + CtR + St
What it is important to learn from this is that St is not calculated for front and rear as it is indipendent. its value is inversley proportionat to the mass transfered via the roll centers
GOLDEN RULE
The ratio of front to rear sprung weight transfer is directly proportionat to the ratio of front to rear roll resistance.
In other words the end of the car that is stiffest will receive the major prt of sprung weight transfer and its exact share is determined by its relative stiffness to the stiffness of the other end of the car.
This is important for creating desirable handling characteristics, too stiff at the front and the rear loading will drop too much causeing it to go light and spin, on the contary too soft and the rear is over loaded and it will spin.
ROLL CONTROL
we have to know the instalation ratio of the spring, in other words for ever centimeter of wheel travel how much does the spring compress, you can find this calculation every where so I have excluded it as MacPherson struts are individualy angled, and the spring rate are an influencing factor.
ROLL RESISTANCE OF SPRINGS
ArF = (SF / (Wmf / SmF)^2 x TF^2 / 2 x pi / 180
ArR = (SR / (WmR / SmR)^2 x TR ^2 / 2 x pi / 180
The pi / 180 converts from radians to degrees.
ROLL BARS
here you can simply get the information from supplier or if you fancy designing your own heres what you need..
AGULAR RATE OF ROLL BAR
Angular rate = 19700 x (OD^4 - ID^4) / Bar length
must do for each bar
(19700 is a conderived from average shear modulus of steel, for solid bars omit ID, Note tubular bars are stiffer dependent upon design)
NOTE: angular rate is inversley proportional to length ie double the length = half the stiffness
LINEAR RATE OF ROLL BARS
this is to find the effect of the lever arm of the ARB, this distance is to the pick up point for the drop link.
Roll bar rate BF = Angulr rate / (lever length^2 x pi / 180)
must do for each bar
EFFECT OF ARB ON ROLL RESISTANCE
BrF = BF x (Wmf x BmF)^2 x TF^2 x pi x 180 = roll resistance of front bar
BrR = BR x (WmR x BmR)^2 x TR^2 x pi x 180 = roll resistance of rear bar
DISTRIBUTION
TOTAL FRONT ROLL RESISTANCE
Arf + BrF = Fr
TOTAL REAR ROLL RESISTANCE
ArR + BrR = Rr
we can now start to calculate the loadings on each axle
Distribution of sprung weight transfer is proportional to ratio of front to rear roll resistance. therefore
DrF = FR / (Fr + Rr) = front proportion of roll resistance
DrR = 1-DrF + rear proportion of roll resistance
DYNAMIC WEIGHT TRANSFER APPLICABLE TO EACH END
WtF = (St x DrF) + CtF + UtF = total weight transfered to outer front wheel
WtR= (St x DrR) + CtR + UtR = total weight transfered to outer rear wheel
these are important values as by manipulation roll and spring stiffness we can change the proportional split, also investigate how close to the max tyre load we are getting, by adjusting these values you can alter your car set up from oversteer to balnced to understeer!
WEIGHT TO OR FROM FRONT
Wt to Front = WtF-(Wt x (WF / W))
Wt to Rear = WrR - (Wt x (WR / W))
NOTE; these calculations refer to one G corners
we can now look at the loadingof each wheel on each axle.
Total Outer Front Tyre LOAD = (WF / 2) + WtF
Total inner front tyre load = (WF / 2 ) - WtF
Total outer rear tyre loads = (WR / 2) + WtR
total inner rear tyre load = (WR / 2 ) - WtR
Almost done
We can now find out our roll created by the set up we have.
some note should be made that having a vehicle that is stiff by no means makes it fast. infact most calculations I have done for people have shown that the £1000 coilover kit they bought made them between 5 and 10 miles an hour SLOWER through the same corner compaired to the standard roll.
ROLL ANGLE
Roll angle = (SW x LM) / (Fr + Rr)
this is for a 1 g corner
Congratulations you made it through.........
Optimal front tyre camber for the optimal lateral loads capable of the tyre is -1.8 degrees. So if we can calculate the roll we will get in a 1g corner we can calculate the static camber required to get to the majic -1.8 degrees in a 1g corner. Word of warning, rear wheels transmit power and to get best power traction requires the largest contact patch, so running huge static cambers at the rear will mean less ability to put power down so as with all suspension design, compromise, its upto you how you run the system dut in terms of high power single seater circuit racers - 0.5 is usualy suficient.
anyway hope it brings light to some but possibly confusion to others. I can answer peoples questions but unfortunatly I cnt do the maths for you. some people may use it but for others try to understand that this is the mimum level of calculation required so dont think that because some company made it and its really expensive that they give a toss wether in reality it makes yourcar slower.
