Beam Axles - Front, Rear or both.

[rwstevens59]

To continue...

Body Rotation followed by body translation along a horizontal line to realign the body panhard link body attachment point with its fixed arc of motion.

BODY ROTATION.jpgBODY_TRANSLATION.jpg

The body is now rolled one degree to the right and has also translated.

I would appreciate your thoughts, experience in doing a 2D layout and constructive criticisms.

Several caveats:

Yes, I use SolidWorks daily in my real world job but will not have a seat of Solidworks anytime soon to carry on what is essentially a hobby.

For the same reason I do not, for solid axle suspensions, plan on investing in any of the 3D kinematics programs that actually work.

Performing the same task analytically in three dimensions leaves me with no pictures and I unfortunately need pictures to get my head around what is happening.

Thanks,

Ralph

[rwstevens59]

Or...

Would it be more correct or conventional to roll the solid axle WRT the body??

AXLE ROLL WRT BODY.jpg

[rwstevens59]

Better picture
AXLE ROLL WRT BODY_2.jpg

[rwstevens59]

A comparison showing the discrepancies of the two methods used so far.

AXLE ROLL WRT BODY_3.jpgBODY_TRANSLATION.jpg

So what I am clearly struggling with here is a clear definition of beam axle suspension roll. Dixon's definition of body roll is suspension roll plus axle roll on tires equals total body roll. Where he appears to measure the respective angles at the kinematic roll center.

I am clearly and purposefully neglecting axle/tire roll here because my only interest is in the motion path(s) of the axle WRT the body. I am just trying to form a clearer picture of axle motions, scrub and relative magnitudes of axle steers and link positions when moved from measured static positions.

[Z]

Ralph,

Well, it was only a little over a month ago that I said I would post a sketch and some more words on this "in the next week or so...". And I actually have most of the words written, and the sketch is half finished on the drawing board (honest!).

Unfortunately, there was that whole thing with the changed Rules that came up. And a few other questions via PMs. And Geoff has started an interesting thread that I have already started a looong reply to.. And I've got some boring but urgent business that must be finished by end of next week (deadlines!). And an old-boys' reunion tonight. Ahhh..., this "retirement" stuff is hard work!

Anyway, I'll do my best to get that sketch finished by end of next week. I think that once you get the gist of Ball's "Cylindroid" (aka Plucker's "Conoid"), then the 3-D Kinematics of 2 DoF joints (ie. Beam-Axle to Car-Body) becomes a lot easier to understand.

Z

[rwstevens59]

One last iteration. I have also attempted stepping around the conventional kinematic roll center while making the appropriate rotation point change between each step as the roll center has moved.

As you can see by the panhard bar link length change this method is very close but certainly not perfect.

Rear Roll Center.jpg

[rwstevens59]

Ralph,

Well, it was only a little over a month ago that I said I would post a sketch and some more words on this "in the next week or so...". And I actually have most of the words written, and the sketch is half finished on the drawing board (honest!).

Unfortunately, there was that whole thing with the changed Rules that came up. And a few other questions via PMs. And Geoff has started an interesting thread that I have already started a looong reply to.. And I've got some boring but urgent business that must be finished by end of next week (deadlines!). And an old-boys' reunion tonight. Ahhh..., this "retirement" stuff is hard work!

Anyway, I'll do my best to get that sketch finished by end of next week. I think that once you get the gist of Ball's "Cylindroid" (aka Plucker's "Conoid"), then the 3-D Kinematics of 2 DoF joints (ie. Beam-Axle to Car-Body) becomes a lot easier to understand.

Z

Thank you Z. At your convenience as always.

Ralph

[Z]

FOUR N-LINE CONSTRAINTS = TWO REVOLUTE DoFs = ONE CYLINDROID.
================================================== ============

I have been wanting to give some more information about the 3-D kinematics of beam-axles for some time. Unfortunately, to do this properly requires quite a bit of preliminary theory, because most students will have never learnt any of it (ie. because of "failure of the education system", the 2-D-only approach used by the automotive-cottage-industry, etc., etc.). So a lot of work required before we can get to any practical applications, and many other things to do...

However, prompted by Ralph's specific questions above, I will just dive into the 3-D explanation of that particular example here, to give anyone interested in this subject a "taste" of it. I might come back to some of the more rigorous foundations later.
~o0o~

THE FOUR N-LINE CONSTRAINTS - The mechanical linkage of Ralph's rear suspension is shown at the top-left of the sketch below. I have distorted the geometry a bit to make it more general. The two main "links" are the Car-Body and the Axle-Housing. These are interconnected by four smaller links, namely the 2 x Trailing-Links, 1 x Slider, and 1 x Panhard-Bar. These four minor links determine the four n-line constraints between the Body and Axle, shown as n-TL1, n-TL2, n-SL, and n-PB. As explained many times before, an "n-line" is simply a straight-line along which there can be NO relative movement between the two bodies.

As noted in an earlier post these four n-lines can ALWAYS be ALL intersected by two other straight-lines. (Note that there can be "degenerate" cases where the intersections are "at infinity", and sometimes there can be infinitely many straight lines intersecting all four n-lines.) Here the two straight lines doing the intersecting are labelled ISA-'R' and ISA-'P'. The four n-line intersection points on each of these ISAs are shown as little white "balls". Note how the two Trailing-Link n-lines are deliberately drawn so that they do NOT intersect each other behind the car (ie. they are mutually "skew"). Nevertheless, it is still quite easy to find the line ISA-'R'.

The two ISAs are, of course, "Instantaneous Screw Axes" (see much Z-ranting elsewhere). In this particular case their screws have a "thread-pitch" = zero (ie. shown as "p = 0"), so they are "Revolute" joints (ie. like simple hinges). ISA-'R' might be called the "Roll Revolute", but it is only APPROXIMATELY so, because it is not necessarily exactly parallel with the centre-line of the car (hence the quote marks around the 'R'). Similarly, ISA-'P' might be called the "Pitch Revolute", but even more roughly so, because it is not very lateral to the car. More on this below...
~o0o~

THE TWO REVOLUTE DEGREES of FREEDOM - Any body that is completely free to move in 3-D space has 6 DoFs. Therefore, putting 4 x constraints on the body leaves it with (6 - 4) = 2 DoFs. Importantly, there are "two infinities" of different ways of specifying these 2 DoFs. At the right of the sketch is shown JUST ONE WAY of doing this, albeit a neat and easy to find way.

Here the Axle can move wrt the Body, and AT THE INSTANT, either as a pure rotation about the Roll-Revolute ISA-'R', or as a pure rotation about the Pitch-Revolute ISA-'P', or as two small pure rotations about BOTH these revolutes AT THE SAME TIME. In this last case, the Axle's motion wrt the Body is a SCREWING motion (ie. both rotating and translating) about another ISA that lies on the unique Cylindroid that is determined by the two revolutes. More details below...

Note that this "conceptual mechanical linkage" of two revolutes could equally have the central T-shaped-link, which interconnects Body and Axle, reconfigured so that ISA-'R' is between the central-link and Body, and ISA-'P' is between the central-link and Axle. For motions of Axle wrt Body, these two different mechanical configurations (ie. Body-P-R-Axle, or Body-R-P-Axle) are identical AT THE INSTANT. However, they will behave differently after significant finite displacements. This is worth keeping in mind when designing actual linkages like this.

Also note that, in general, any 2-DoF joint can be physically implemented as FOUR x 1-Degree-of-Constraint n-lines IN PARALLEL (eg. 4 x ball-ended-links, as at top-left of sketch), OR as TWO x 1-Degree-of-Freedom joints IN SERIES (eg. 2 x revolutes, as at right of sketch). Similarly, a 3-DoF joint can be done as 3 x (1-DoC) n-lines in parallel, or 3 x (1-DoF) revolutes in series. Or a 1-DoF joint, such as an "independent suspension" can be done as a "Five-(1-DoC n-line)-Links-in-Parallel", or as a "Single-(1-DoF revolute)-Swing-Arm".

Interestingly, less ingenious engineers prefer the "multi-link in parallel" approach (see most suspensions), whereas Nature prefers the serial approach (see most skeletons). Roboticists have tried both, but wisely follow Nature when persuing maximum versatility.
~o0o~



THE ONE CYLINDROID - A glimpse of this creature of 3-D Kinematics is shown in the middle of the linkage at the right of the sketch, with its "spine" lying on the common perpendicular of the above two revolute axes. A more detailed, close-up view of the Cylindroid is shown at the bottom-left of the sketch.

Actually, only a "core" taken from the Cylindroid is sketched, and you should imagine it extending outwards to infinity in all directions perpendicular to its spine. It is a "ruled-surface" formed by a straight-line generator that is always perpendicular to the spine, and moves sinusoidally up-and-down the spine as it rotates about it. This straight-line generator is shown more explicitly at each end of the spine, with these two positions being mutually perpendicular when viewed along the spine. So, after rotating 180 degrees, the generator is back to where it started.

Thus, the entire ruled-surface of the Cylindroid exists between two parallel planes that are at each end of the spine, and that are perpendicaular to it. So, when viewed from a large distance, the Cylindroid looks like an almost flat surface, but with a little wrinkle in the middle that is its spine.

In 3-D Kinematics the Cylindroid is probably second only in importance to the ISA/Motion-Screw. Hopefully by now most of you students will be aware that ANY 3-D MOTION of any one body, with respect to any other body, is best described by the ISA that exists between them. This ISA is also an extremely simple concept to understand, being nothing more than "a nut moving on a bolt". All of the straight-lines lying on the Cylindroid, and intersecting its spine, are ISAs. So the Cylindroid is formed from "a single infinity of ISAs".

The "thread-pitches" of these ISAs vary sinusoidally according to the ISA's rotational position around the spine. It is possible for all ISAs on a particular Cylindroid to be Left-Handed, or all to be Right-Handed, or, as in this example, some to be LH and others to be RH. The dividing lines between LH and RH ISAs are, quite obviously, ISAs of zero-pitch, namely revolutes. The magnitudes of the "thread-pitches", measured as distance-translated per radian-of-rotation, of all the ISAs on the Cylindroid are easily and uniquely determined, but I will leave that for another time...

But for now, a quick quiz:
Q1. Which of the ISAs on the above Cylindroid are RH, and which are LH?
Q2. Over what range do their pitch magnitudes vary?

Answers are quite easy...
~o0o~

While the Cylindroid is almost as ubiquitous a creature as the ISA, it is, like the teenage werewolves and vampires of modern movies, rather shy. So only keen-eyed, or well trained, geometric hunters can spot it. One of its first recorded sightings, in the early 1800s, was by William Rowan Hamilton, the Irish mathematician who also discovered Quaternions. Later, in the middle 1800s, the German mathematician Julius Plucker (pronounced Ploo-ker) tracked some down, and called them his Conoids. Shortly after, still in the middle 1800s, another Irish mathematician, Robert Ball, snared a few of them, and called them Cylindroids.

Because it is such a common creature it is always being rediscovered. So in the 1960s the Australians Jack Phillips and Ken Hunt stumbled across some of them in a small field of Kinematics. (They then did their homework and found that these creatures had been sighted many times before, as above.) In this case they found that whenever ANY three bodies are in relative motion, then the three ISAs that exist for the relative motion of each pair of those bodies ALWAYS lie on the same Cylindroid (eg. picture three asteroids floating about in deep space, and their mutual ISAs, 1-2, 2-3, and 3-1).

But the Cylindroid is also found in 3-D Statics, an entirely different field to Kinematics. Here, any three WRENCHES (ie. Force-Screws, and I hope all you students also understand this concept by now) that are in mutual equilibrium also ALWAYS lie on the same Cylindroid.

And it is also found in many, many other places...
~o0o~

More coming next post (10K char limit!!!).

Z

[Z]

FOUR N-LINE CONSTRAINTS = TWO REVOLUTE DoFs = ONE CYLINDROID. (Last bit...)
================================================== =======

Getting back to Ralph's suspension, since this is a 2-DoF joint between the Axle and Body, then there is ALWAYS a unique Cylindroid associated with that joint. Thus ANY instantaneous relative motion of Axle wrt Body must ALWAYS be about ONE of the ISAs lying on the Cylindroid. So as noted earlier, the motion can be about ISA-'R', or about ISA-'P', or about any ONE of the other ISAs that lie on the Cylindroid.

Repeating this last point for emphasis, a very small rotation about ISA-'R', TOGETHER with a very small rotation about ISA-'P', is EQUAL to a small SCREWING motion (ie. = rotation + translation) about ONE of the other ISAs that lie on the Cylindroid. The particular ISA that results from the two small rotations about ISA-'R' and ISA-'P' depends on the relative sizes of said rotations. (In a more rigorous explanation we would have to speak of relative "rotational velocities", namely dTheta/dts, because "rotational position vectors" DO NOT COMMUTE!)

Back to this specific example, a pure "Roll" rotation of Axle wrt Body, about the particular ISA that is purely LONGITUDINAL to the car in plan-view, will have a small amount of "screwing" (ie. as the Axle "rotates" about the ISA, it also "translates" along it, like a nut on a bolt). Similarly, a pure "Pitch" rotation about the ISA that is purely LATERAL in plan-view, will also have some "screwing".

More importantly, because the spine of this particular Cylindroid is not vertical (ie. the "plane" of the Cylindroid is tilted away from horizontal), ANY rotation about any of the ISAs that are not exactly horizontal (ie. all but one of them) will have some vertical component of rotation. Therefore, just from a glance at the Cylindroid, we see that almost all motions of the Axle wrt Body have some degree of axle-steer.

In general, for this sort of beam-axle 2-DoF joint, it is a good idea to try to keep the plane of the Cylindroid close to horizontal. That way any motion of Axle wrt Body has no, or negligible, axle-steer. However, as long as the Cylindroid's spine is not tilted TOO far away from vertical, or if it is tilted in the "right direction", then the amount of axle-steer might be acceptable. In practice it is a matter of knowing the numbers.
~o0o~

Bottom line, the "usefulness" of the Cylindroid is that just by taking a quick glance at it, we can get a good feel for different suspension behaviours.

EG1. "Whoa..., that rear-beam-axle's Cylindroid is leaning backwards far too much! The car is going to have terrible roll-oversteer."

EG2. The two axles on skateboards have Cylindroids designed to give very large roll-understeer. In fact, the skateboard rider rolls the Body (ie. tilts the "deck" in to the turn) to make the skateboard steer.

EG3. A double-wishbone (or "5-link") suspension WITHOUT the toe-link is a 2-DoF joint (ie. with Body fixed, the Upright can rotate about the ~vertical "steer-axis", OR it can rotate about the ~horizontal "instant-axis", OR it can move about both together). So there is a Cylindroid in there! Adding the toe-link determines which of the single-infinity of ISAs on the Cylindroid becomes the unique ISA for motion of Upright wrt Body.
~o0o~

Enough for now...

Questions welcome, and maybe some sketches of different beam-axle-linkages and their Cylindroids later...

Z

[rwstevens59]

Z,

Whew, thank you for putting in the effort to keep me on the path to learning more and more about 3D kinematics. There was no way I was pulling the above out of Jack Phillips book on my own.

Many more questions and comments to follow.

For now I just wanted to express my gratitude for the work you have done to produce the above.

Ralph