Weight Jacking

[P^squared]

Hello everyone !

My name is Prathamesh Patalay and I'm studying at the Maharashtra Institute of Technology, Pune.
Because of packaging constraints and poor local rim choices, we're needing to compromise on some aspects of the steering/suspension wheel assembly.

I was looking into the effect of weight jacking, but couldn't find a mathematical relationship to determine the height through which the tyres move w.r.t the car.

We also run a Quaife helical gear type LSD in the rear. So we cannot afford to lift the inner rear in turns.

Here's what I tried :

Terminology :

ster = steer angle
cast = castor angle
r = tyre radius
s = scrub radius

Weight jacking occurs only when we have caster. If there's no caster any amount of scrub won't cause any weight jacking.

Therefore assuming zero scrub, and 90 deg castor, some steering angle, ( rear view of the car.There's no steering action, only dH )
we get change in height dH = r * ( 1-cos(ster))

Adding castor into the equation, we notice that the change in height is proportional to the perpendicular distance between the tyre contact point( ideally ) and the steering axis.
Therefore dH = ( 1-cos(ster)) *(r*sin(cast))

Further adding scrub into the problem, it increases the perp. distance between contact point and steering axis to sqrt[ (r*sin(cast))^2 + (s)^2 ] .

then dH becomes --> (1-cos(ster)) * sqrt[ (r*sin(cast))^2 + (s)^2 ]

Entering values, the answers come close to 2~3 mm.(inner goes down, outer lifts) Assuming the car to have ~25mm droop travel, the change in vertical load on the tyres isn't that big to be problematic.

But I have some reservations about my maths. First thing is, the equation doesn't make sense for the case of zero caster angle with some trail ( caster wheels ). Also the inner wheel first goes down, and later starts rising( although I think this is correct ). How do I take the tyre's tread width into account ??

Also Go-karts and cars with spools use this phenomenon to their advantage to unload the inner rear. How far do they need to take the parameters to get some significant benefit ?? Or can we not do it without a solid rear ARB ??

Criticisms towards my approach are definitely welcome !! :D

Sorry for the long post, but thanks for reading.

[Adam Farabaugh]

Kingpin inclination is as significant to this effect as caster.

[P^squared]

Can anyone please suggest any corrections ?

[BillCobb]

I suggest you parse out spindle length, wheel offset, wheel camber angle and tire overturning moment from the scrub radius parameter. This will show you the contributions of each factor when you get a working simulation on down the road. To get a more reasonable, believeable, and accurate predictor of all these effects, you will need to add the rear suspension roll stiffness and all 4 tire spring rates. Then compute the moment balance using statically indeterminate numerical methods with roll angle as the final outcome. If your frame is not torsionally stiff, there's even more complexity. If your input is a steer angle (or better yet a steer moment), the reaction gets much more real and thorough and of course, more valuable.