[python] Vython: What is the smallest turning angle that is acceptable?

All,

Looks like Dirk was on to something with the turning radius problem!

It seems as though the best geometry might be narrow side-to-side, and long front-to-back (i.e. long along the long axis of the bike).

For example:

G = ground link, Y = distance from G to C, S = side links, C = coupler link

G = 20 cm, Y = 25 cm, S = 25.4 cm, C = 11.4 cm

Oh, new formulas by the way -- I started writing an email earlier today, but it's too late to finish it, so I'll paste it at the end of this one.

So, anyway, as you can see Y > G, and I think the steering might get even better as G gets smaller (Y can't get much bigger than 25 cm), but we need to determine the smallest acceptable turning angle. It will be limited by one side link's pivot with the coupler link colliding with the middle of the other side link (making the side links bow-legged would help) as the coupler link shrinks along with G (it needs to be smaller than G to create a toe-in which places the virtual pivot near the front wheel hub). Sorry this must be confusing without pictures, I've been thinking about it a lot so it all makes sense to me.

The geometry given above (G=20,Y=25,etc.) has a roughly 50 degree turning angle, which is probably plenty for anything but trying to walk the bike or making an extremely tight turn with your feet on the ground (not sure if that's possible on a low-racer anyway, but I do that on my BikeE sometimes).

But I think we should try to define a minimum turning angle, and a maximum for Y (explained in what I wrote earlier, which is below), as well as the diameter of the pivots and the width of the side arms (10 mm? 30 mm?). Armed with that I can figure out the minimum for G and do some more calculations and modeling to see how the steering looks.

Actually I'm confused and tired as usual by the time of night that I have some free time to write about this stuff... Bedtime!

Michael

---

Here's what I wrote earlier:

"Dimensional constraints of four bar linkage" is a fancy way of saying "How big can it be?"

Hello everyone!

I've been incredibly busy these past few weeks! So not much progress on the project, although it is always bouncing around in my head!

On Sunday, I got together with a math buddy, and he helped me to understand the math on this page: http://iel.ucdavis.edu/chhtml/toolkit/mechanism/fourbar/fourbarpos.html (the same link I posted before). It turns out that the end result is not as simple as it appears -- it needs to be solved by a computer to get numbers! So although he was able to make a pair of equations that give the side link angles when given the coupler link angle, it needs to be solved by a computer program. Then, it is easy to figure out the position of the virtual pivot.

So if anyone would like these, I can type them up, however right now they are of no use to me, as it is much easier to just build a cardboard model! I use cardstock (flat cardboard, like a notebook cover, as opposed to corrugated) and some small nails for pivots. It is easy to change which holes the nails connect in order to alter the geometry.

So I think the next step is to define some constraints -- maximum dimensions for the Y axis (long axis of the bike), and for the x axis, using a plane which is tilted to be perpendicular to the steering axis (66 degrees P4, 62-65 degrees Tom Traylor).

I'd love to get some input, here's a crude stab at it:

Maximum Y: 25 cm or 10"

Must be small enough to fit under the seat while allowing adequate ground clearance -- actually, it would be quite simple to have the side links parallel to the ground (or at any convenient angle), but have the actually pivots angled -- if this isn't obvious, draw a picture and imagine the path that the side links will make -- it is the angle of the pivot points which determines the angle of the virtual pivot, not the angle of the side links. So let's say 25 cm or 10" -- any longer would be rather cumbersome and probably too heavy -- plus we are getting too close to the rear seat mounting point (measuring on P4, since I am a little gnome with a height of 178 cm / 70" and I must use 24" wheels!
< : ) | = <-- supposed to look like a happy little gnome!). It would be better if it was shorter, perhaps 15 cm/6" -- 20 cm/8" at most. Remember this is the total Y when the wheel is centered, which is shorter than the side link lengths (because they are at an angle).

Maximum X: 40 cm or 16"

This is about the seat width, maybe a little more, but not much. Better to not go wider than that, right?

----

So after I get some feedback, I will model a few geometries, and report back!

I think I'll start with the ground link (X dimension, maximum 40 cm), and for several lengths (40 cm / 30 cm / 20 cm) try several Y dimensions (like 25 cm / 20 cm / 15 cm), for each combination, calculate the side link angle and coupler link length that would result in a virtual pivot to be located near the front wheel hub when the wheel is centered (very simple math), then model with cardboard and make some rough notes about how the virtual pivot moves when steering and how the wheel moves.

Well it is either start now or do the dishes, so here goes:

Let:

G = ground link

Y = distance from ground link to coupler

R = wheel radius + small gap

S = side links

C = coupler link

Then:

GIF image



(Typical -- figuring out the formulas took me about 10 minutes at most, but it ended up taking twice as long to make them look "just so" in Photoshop! And I still don't like the way that they look... There's just no pleasing some people!)

So I can just plug in some values and try several geometries with my cardboard model!

Here goes a few:

For all of the following,

R = 33 cm

Vy is the Y axis offset of the virtual pivot with respect to the wheel axis
Vx is the X axis offset of the virtual pivot with respect to the wheel axis

Which should be about right for a 24" wheel, plus a gap and little extra for the thickness of the coupler link.

G = 40, Y = 25, S = 26.4, C = 22.8:
        10 degree turn: Y = 24, Vy = 6.3, Vx = 12.7
        20 degree turn: Y = 22.7, Vy = 13.3, Vx = 19.3

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