<all> I UNDERstated the case for swaybars... sorry everyone :>

["Aaron Bohnen" <bohnen@xxxxxxxxxxxxxxxx>]

Hi Gary,

Thanks for the mail:
 
> Nice summary, but isn't it to the 4th power of diameter, as that's what I
> remember off the tech books?

You're right, of course. The parameter of interest is the polar moment of 
inertia of the cross-sectional area. That parameter does have the radius in 
it raised to the 4th power. My fault for the wrong information - I was 
being much too theoretical about it. A typical fault of an academic. 

I'm a bit ashamed. I ought to have been deriving the stiffness relationship 
from the classic deformation of a cylindrical bar equation:

angle of rotation = (applied torque * length)/(section polar moment of 
inertia * material shear modulus)

or 

Phi = (TL)/(JG)

and coming to the conclusion that the ratio of the stiffnesses was of 
course equal to the ratio of the sections' polar moments of inertia - that 
parameter being dependent on the 4th power of the diameters. This is the 
correct conclusion.

Instead I forgot that little gem and went back to basic principles for a 
crack at the portion of the total torque carried by the outside 1.5 mm of 
my 15.5 mm bar (the 'additional' diameter added when I upgraded the bar 
from the original 14 mm unit.) Well, the result of that is that the outside 
1.5 mm of bar carries 33.4% of the total bar torsion. My mistake was 
allowing that fraction's similarity to the ratio of the third powers of the 
bar diameters (namely 1.36 to 1) to seduce me with its' similarity to the 
fractional torque that I had calculated. This wrongly convinced me of 
something I should have known much better about.

I guess I should learn not to drink and derive.

Well, anyway, to make a long story short, I am very sorry to have 
mislead anyone. In retrospect, at least I understated the case for larger 
bars!

To summarize, the torsional stiffness of a uniform, solid cylindrical bar 
is proportional to the 4th power of the diameter, the shear modulus of the 
material, and inversely proportional to the length of the bar.

So what does all of this mean to would-be swaybar upgraders?

Well, it means that my 15.5 mm bar is actually 50.3% stiffer than the 
original 14 mm bar. And that the 18 mm M5 rear bar is 173% stiffer than the 
stock 14 mm that came on my car. etc. etc.

IMPORTANT PART COMING UP NEXT:

What's the kernel of truth about swaybar diameters that we can learn from 
all of this? 

Well, there are two. The first is:

1. Every little bit more diameter makes a big difference to the torsional 
stiffness.

The second is:

2. The amount of difference each little bit more makes decreases 
dramatically the bigger the bar is to start. In other words, in order to 
increase the torsional stiffness of a small bar by a given fraction you 
need to increase the diameter much less than you would need to increase the 
diameter of a larger bar to increase its' stiffness by the same fraction. 

To see this second kernel of truth in action, let's consider a hypothetical 
situation...

- ----------example------------
Bob has a 10 mm swaybar in his car (maybe not a BMW, but this is just an 
example) and wants a swaybar with twice the stiffness. The stiffness 
increase he sees is proportional to the ratio of the fourth powers of the 
diameters of the two bars. He needs to solve the following equation:

(original diameter^4)/(new diameter needed^4) = 2

With his original bar of 10 mm diameter, it turns out Bob needs an 11.89 mm 
diameter bar if he desires a swaybar of twice the original units' 
stiffness. Bob's new swaybar is 1.89 mm larger in diameter than his old 
one.

Bill, on the other hand, has a BMW M5 with an 18 mm diameter swaybar. He 
also wants twice the stiffness - for his swaybar that is. He needs to 
solve the same equation as Bob. With Bill's original bar diameter of 18 mm, 
however, it turns out that in order for him to get a bar of twice the 
stiffness he must find a 21.41 mm unit. Bill must increase his swaybar 
diameter by 3.41 mm.
- ---------end example----------

So what this means is that if you have small swaybars they are easy to 
upgrade. Generally you can use the same mounts as you've got if the 
increase in diameter is small, as it is likely to be between 'i' and 'e' 
models, etc. etc. If you have large swaybars and you want to upgrade them 
you will generally have to upgrade/strengthen mounts as well, etc.

Sorry big boys. This time the little boys win! 

One last point - the axial stiffness of the cylindrical bar is 
proportional only to the second power of the diameter. In other words, 
increasing the diameter of a cylindrical bar increases it's end-to-end 
compressive stiffness by the second power, but increases the bending and 
torsional stiffnesses by the 4th power. Fatter is stiffer, but more so in 
twisting than in bending or compression. Draw what conclusions you may 
about that little gem of wisdom! I figured it might be of more than 
passing interest, and some considerable chagrin to the Firebird, Camaro and 
Mustang owners - and maybe to the Bubbas here in BMW-land as well. Take my 
drift guys? :>

best regards to all,

Aaron
___________________________________________________________
Aaron Bohnen                     email: bohnen@domain.elided
- -Ph.D. Student, Civil Engineering Department, U.B.C.
- -Technicraft Engineering Services

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