@martinbo
@Giannis, quite simply: love your work man. Very interesting explanation. Please tell me if I'm getting this right:
- The clamps are the points of inflection.
- Depending on the bushing in the clamps (flexible or rigid) you will either get rotation at the point of inflection or not.
And now the question
which one do you think is the case with the new XC90 suspension as shown - fig 1 or fig 2 in your drawings?
Now having understood your explanation, I am expecting that fig 1 is the case. I also expect that there has to be bushing in the spring clamps in order to damp out spring vibration from the axle subframe.
Simply put, figure 1 describes how the transverse leaf spring works. I drew the deflected shape for the case when the left wheel is loaded (moves upwards). In that case, the right wheel will also move, thus the leaf spring works as an anti-roll bar as well.
What figure 2 shows, is how the leaf spring would deform, if the two connections with the chassis were to work like full clamps, that is to allow absolutely no displacement and no rotation of the leaf spring. If that were the case, then the leaf spring between the two clamps would have no reason to exist. That is what you described in post #101.
How this works: If you were to clamp the outer leaf spring, as per figure 2, then let's say that the bending moment (
M) that the clamp would need to resist, would be 10 Nm. Given the thickness of the leaf spring and that maximum stress (
σ) equals (
M*
t)/(2*
I), where
I is the moment of inertia of the leaf spring and
t is its thickness. Maximum stress would be quite significant!
Now, if the clamp would allow rotation, thus be a pin, then at that point the bending moment of the leaf spring would be zero (fundamental principle of statics). The sign of the bending moment of the leaf spring would be different to the left of the pin (connection to the chassis) compared to that to the right, therefore the curvature left of the connection to the chassis would be of opposite sign to that to the right. Then, at the exact point of the connection with the chassis, where the curvature changes from [+] to [-], it's exactly zero at that point. The point of zero curvature is called point of inflection. (So, yes, the clamps are the points of inflection)
Since at the point where the leaf spring is connected to the chassis, no bending moment exists, the only thing that is exerted on the leaf spring is a (shear) force, equal to half the weight of (on)the axle. Right?
Since at the critical point, where the leaf spring is connected to the chassis, only a shear force is exerted to the spring itself, it's a much better situation that having bending moment as well.
As for this:
Depending on the bushing in the clamps (flexible or rigid) you will either get rotation at the point of inflection or not.
The point of having a bushing is to allow for some flexibility, so that the rubber material absorbs unwanted vibrations. Since there is deformable rubber material between the (whatever it is made of) leaf spring and the metal part of the clamps, rotation can be observed. De facto. Because rubber is deformable (*). If you didn't want any rotation, the leaf spring (or any other sub-frame/ element) would have to be directly bolted to the chassis - a la BMW F10 M5 and F80 M3 (**)
(*) Such rubber parts/ bushings are present everywhere a suspension components meets the chassis. At the front, the control arms have bushings all around and so has the shock absorber/ spring assembly, at its top part where it meets the strut tower. This part is called the top mount and looks like this:
(**) I've been hearing that exactly because the rear subframe of the recent M cars is directly bolted on the chassis as to "allow for better feeling of the road", every other bushing (including the rear control arms and those of the differential) are now much much thicker and softer, or else the chassis would crack and/ or the ride would be unbearable. Talk about marketing at its best
