Hi all,
Looking for some help from mechanical engineers or other suspension gurus in understanding non-ideal swaybar shapes and/or the role of bushing attachments.

First, here's the idealized case. The bar ends are perpendicular to the torsion rod, which is held in place by bushings placed just inside the bar ends, allowing the rod to deform torsionally but keeping it straight:
For a bar with a circular cross-section, the "spring rate" (deflection of rod ends for given amount of force) is:

S (lbs/in) = (pi*G*d^4)/(32*L*R^2)
where:
R = Length of end perpendicular to L (torque arm - inches)
L = Total length of bar (inches)
d = Diameter of bar (inches)
G = modulus of elasticity of bar material (psi)

Now suppose you have a bar that looks like this:
R = Length of torque arm outside bushings (inches)
L = Total length of bar (inches)
c = Length of offset center section (inches)
r = Length of torque arm inside bushings (inches)

So there's an "offset" section of the bar, but it's placed inboard of the bushings. This is the general shape of the S2000's rear bar.

We wish to use same formula as above, using:
Leff = Effective length of torsion bar
Reff = Effective length of torque arm

What are Leff and Reff? It seems clear (correct me if you disagree) that Leff = L, but do we use Reff = R, Reff = R + r, or something in between? For the S2000 rear bar, the calculated difference is quite substantial: we have R=8.5in and r=1.25in, so the predicted stiffnesses differ by a factor of (9.75/8.5)^2 = 1.32, i.e. a 32% difference.

Intuitively, I'd lean toward using Reff = R, since the bushings help prevent the "offset arms" (of length r) from deflecting. On the other hand, I could see where the center section isn't necessarily restricted to deforming strictly in torsion -- doesn't the twisting of the little section of bar between R and r exert a force on that little lever?

One data point: Swift Springs, in their thread (link) just used Reff = R, Leff = L. (They don't say this explicitly, but I've recreated their numbers using my measurements and this assumption.)

Any thoughts?

 

I'm going to say that it is a fairly good approximation to use R and L from your diagram as you would use them in the formula for the ideal bar. However, the complex bar will be a little stiffer than the formula predicts, because the center section has to bend, or the "short arms" have to twist, as one end of the center section moves up and the other end moves down. Because it has to twist AND bend, instead of just twist, more energy is stored for a given angle, therefore the spring rate is higher.

 

Thanks for the reply! Not sure I'd agree that the effective spring rate would be *higher* -- this would imply that not only is Reff < R+r, but Reff < R.

My intuition:
1) To whatever extent the short arms deflect for a given applied force, they are acting as extensions of the longer arms (Reff > R), which lowers the overall spring rate.
2) To whatever extent the short arms do NOT deflect, but twist instead, then they are acting as extensions of the torsion bar (Leff > L), which again lowers the overall spring rate (but not as dramatically).

Our rear bar actually looks more like this:
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^Actually maybe that's not a bad idea. I'd been assuming Leff = L, but perhaps it's closer to say Leff = L+r (or even L+2r), and keep Reff = R.

 

Yes, I agree with your last post. With the configuration shown, there is no bending moment in the section between the bushings, thus Reff = R. There is torsion, so I would use Leff = L + 2r.

 

Thanks Suspension! Yeah, the more I look at it the more I think that's the best approximation. I'll re-run my numbers and see how they come out.

I'd still love to hear anyone else's thoughts... Agree? Disagree? Any engineers ever seen or done any similar analyses?