Beam Axles - Front, Rear or both.

[Jay Lawrence]

Z,

To repeat Ralph's sentiment: thanks for that! This is all (unfortunately) quite brain frying in a way, and I don't claim to properly understand it yet, so this question may sound quite silly:

I'm confused by the way you have modelled Ralph's ISA-'R' wrt the Panhard bar. On the ISA-'R' portion of your 2 revolutes (RHS of sketch) there doesn't appear to be any effect from the location of the Panhard bar. I would have thought that the Panhard bar link would allow roll but at the expense of providing some level of lateral axle movement (unless we assume that the Panhard bar is located on some frictionless joint at the centre of the 'diff'). i.e. I would think that the ISA-'R' line/revolute would be angled in plan view

[Loz]

Jay

A few things to note.

... I have distorted the geometry a bit to make it more general.

I would guess that to enable general discussion, the sketch is an adapted version of Ralph's system which includes ISA's drawn at more convenient locations (i.e. easier to interpret).

The effects of the panhard bar link are taken into account in the series swing-arm/revolute joint mechanism.
Which is what the following statement pertains to:

Here the Axle can move wrt the Body, and AT THE INSTANT, ...or as two small pure rotations about BOTH these revolutes AT THE SAME TIME.

But importantly, we are effectively talking about infinitesimally small time steps and analysis at a singular instant (implicit in the term "Instant screw axis"). At another instant the ISA's have shifted to take new directions. Which is where the cylindroid comes into play. At each instant the two revolute ISA's can shift up or down the cylindroid spine with the sinusoidal variation defining the cylindroid.

This is a main reason that it is necessary to consider the suspension mechanism in 3D and not 2D because things like the ISAs aren't necessarily fixed in space, especially in the case of these multi-link mechanisms with more than 1 DOF.

Loz

[Kevin Hayward]

Z,

Thanks again for your drawings and descriptions. I find it much easier to work through numerical methods through simulation, a side effect of studies in computer science. I never liked looking at 3d problems using 2d sketches. Although with improved 3D CAD packages the geometric methods are very easy to setup. Creating the cylindroid in a 3d sketch in Solidworks was a trivial exercise and it reveals quite a lot very quickly. Not prepared to give up on the numerical methods in design purely for the fact that iteration and analysis of results can be very easily automated and reviewed in bulk. Please note I am not talking about plugging away at OptimumK (or similar kinematics programs). They are fantastic to use, and I am pretty happy with what we achieved at OptimumG. However, not being able to program sweeps and solution search routines limits the effectiveness of these tools for fast design. It was the reason we included a excel spreadsheet input. This allowed the use of a spreadsheet to aid in the creation of suspension points in the first place. Much better to make your own geometry code and explore the solution space.

...

It should be noticed that for a simple four bar / symmetrical rear end the two ISA lines have been stated as the pitch and roll axes of the suspension system. In which case the tilting of the cylindroid can be controlled by altering the slope of the roll axis (which for this case will lie on the centre-plane of the car). I am reluctant to mention this simplification. I only do so to show how you can see existing (incomplete) knowledge of kinematics as drawn in a few texts applies in special cases. It does become a useful simplification in a few design cases.

I find design of these mechanisms to be purely a case of working through the unknowns. Keep adding your constraints until only one variable is changing the desired output. In the case of a beam axle with 4 links (8 points, 24 variables) you can make choose the following contstraints to simplify the problem (excluding spring/dampers):

- Assume symmetry (24 to 12 variables)
- Try and get as much lateral direction on the links as possible to provide inherrent lateral stiffness (12 to 10 variables)
- Given packaging use the lower mounts on the beam to be inboard, the upper to be outboard.
- Move lower mounts on the beam inboard as much as possible
- Move upper mounts on the beam outboard as far as possible
- Keep your beam mounts as close to the same plane as possible to improve inherrent beam stiffness (10 to 6 variables)
- Design to your anti-squat requirments / desires (6 variables to 5 variables)

Once this is done you are only playing with your 4 chassis mounts. Add a few more constraints based on the chassis:
- Width from template and structural design (5 to 3 variables)
- Decide height of "roll axis" (3 to 2 variables)
- Fix either lower or upper chassis mount based on packaging, ability to put a decent hardpoint, alternatively make one link as long as possible befor erunning into strength concerns (2 to 1 variable)

The only thing you are left to alter is the ratio of length of the top arms vs. the lower. This one variable will have direct control over the rollsteer of your beam. Sweep a value using geometry software, choose the result you want. Alternatively (or addittionally) use a geometric approach to view the cylindroid.

The problem with stating a solution method you use is that others will read it as the only way to solve a problem. However change the constraints just a little and you need to apply a new method. Hence the disconnect with how-to design texts and dynamics literature. In order to design you will need to introduce constraints, the trick is making sure you don't over-constrain the problem, and force yourself into a sub-optimal solution (i.e. we must have double a-arms with push-rods).

Make sure you have all your KPI's decided before you even start:
- Target component stiffness (really important for a beam, toe stiffness is likely more important than camber stiffness and could swamp the effects of mild rollsteer quite easily)
- Target component strength (make all the links close to planar with the car's centre planes and watch the axial loads climb through the roof. Good luck keeping it stiff and together.)
- Target pitch jacking (anti-squat)
- Target roll jacking (roll axis height)
- Target roll steer

I'm not sure if this adds anything to this discussion, but design of a good beam system is a lot easier than a double a-arm. You start out with a lot less variables in the first place. Assuming symmetry and non steering we have 18 for the double a-arm vs 12 for the beam. Of those variables for the beam it is quite easy to eliminate (or severly restrict) a number of them simply through structural concerns. One thing that is not often mentioned is that on a beam you can add some extra mounting points to decouple your heave and roll rates. i.e. Springs closer in will have the same heave rate, but softer roll. So if we compare the following symmetrical suspensions (double for asymmetric):

- Double a-arm with push/pull rods, coilovers, and an anti-roll bar (~43 variables)
- Beam with direct acting coilovers and an added variable for altering distance between spring mounting for tuning roll/heave characteristics (19 variables)

Structurally a beam is no harder to design than an upright, and the links can just be big tie-rods. Welding can be easier than an upright or a-arms as you have a bit of space to get to everything, and you wont want super thin sections. Weight differences of the components are negligable. The beam requires a lot less support structure (i.e. 6 chassis mounting points vs. 14 for the double a-arm), so a beam system will be lighter overall (quite significantly lighter). Add in the fact that the chassis mounting points will be closer to the COG of the car and your chassis structure can be a lot shorter as well.

Every team should consider the option and do some preliminary designs. It takes very little time to do the early studies. If you then decide to go double a-arm at least you ahve a good idea of the trade-offs you have made in cost, weight, ease of manufacturing, component count in order to get whatever you wanted from the double a-arm.

Kev

[Mitchell]

Structurally a beam is no harder to design than an upright, and the links can just be big tie-rods. Welding can be easier than an upright or a-arms as you have a bit of space to get to everything, and you wont want super thin sections. Weight differences of the components are negligable. The beam requires a lot less support structure (i.e. 6 chassis mounting points vs. 14 for the double a-arm), so a beam system will be lighter overall (quite significantly lighter). Add in the fact that the chassis mounting points will be closer to the COG of the car and your chassis structure can be a lot shorter as well.

Every team should consider the option and do some preliminary designs. It takes very little time to do the early studies. If you then decide to go double a-arm at least you ahve a good idea of the trade-offs you have made in cost, weight, ease of manufacturing, component count in order to get whatever you wanted from the double a-arm.

Kev

UQ Racing has shifted to a beam axle this year for the reasons in bold. The rear of the chassis is now just the regulated roll hoop support bars.

We are using a peg and slot (hidden behind the bottom of the shock) and 4 link.

rear beam.jpg

The biggest saving is in chassis structure, with the change in unsprung being (at least for us) negligible.

It has also improved access to the engine and general working on the rear of the car, which can be removed with only 5 bolts.

Mitchell

[rwstevens59]

Z,

To repeat Ralph's sentiment: thanks for that! This is all (unfortunately) quite brain frying in a way, and I don't claim to properly understand it yet, so this question may sound quite silly:

I'm confused by the way you have modelled Ralph's ISA-'R' wrt the Panhard bar. On the ISA-'R' portion of your 2 revolutes (RHS of sketch) there doesn't appear to be any effect from the location of the Panhard bar. I would have thought that the Panhard bar link would allow roll but at the expense of providing some level of lateral axle movement (unless we assume that the Panhard bar is located on some frictionless joint at the centre of the 'diff'). i.e. I would think that the ISA-'R' line/revolute would be angled in plan view

Jay Lawrence,

Maybe it would help if I explain my thought process while studying my own 2D drawings and why the need for 3D thinking becomes obvious albeit not necessarily intuitive for one not trained to 'spot cylindroids'.

Some conditions first as per my 2D drawings:

The trailing links are parallel longitudinally in plan view at static setup.
The trailing links slope down from the body mounts to axle mounts at 5 deg. Both links are parallel in side view i.e. not skewed.
The panhard bar is located in front of the axle and about 3.5 in. below axle centerline and is level (i.e. in the center of its arc of travel) at static setup.
The live axle pinion nose angle is at zero degrees i.e. level with the ground as set by the torque arm slider at static setup.

Now lets attempt to roll the axle with respect to the body about some point, axle centerline or axle roll axis and roll center as described in Milliken, Olley and a host of other chassis reference texts.

You are correct as the axle rolls WRT body the axle will move laterally due to the panhard bar arc radius.

Simultaneously the side view trailing links are moving in opposite directions along their side view arc paths causing the axle to steer in plan view.

But wait...if the axle is moving laterally in rear elevation view are not the trailing links also moving in arcs in plan view WRT their respective body mounts.

And if that is true than the side view arc paths of the trailing links are being skewed by the action of the panhard lateral movement so we can not say that the trailing links follow their side view arc paths the moment any roll takes place.

To make matters worse if all of what has been said up to this point is true than the steer of the axle has moved the axle out of parallel with the panhard bar causing an additional albeit small further lateral movement of the axle due to the now angled panhard bar in plan view.

OK. Now that your head hurts as much as mine you can see why two wheels on a pipe connected to a body with four links is anything but simple as it would appear.

We now have two choices (maybe more, but the only ones I can think of) set up our coordinate system(s) and use analytic geometry coupled with an iteration routine to plot our points through some arbitrary magnitude of axle roll with respect to body or start to learn more about 3D kinematics as shown by Z to give us a picture of what is happening at any instant in time.

Being a 'picture guy' first before heading for the computer I much prefer the latter to develop my 'gut feel' for the motions first.

Lastly, I am not trying to design this suspension. It already exists in hundreds of cars that race here in the U.S. and I am simply doing an exercise in reverse engineering to better understand the layout as well as dispel some of the setup voodoo I hear all the time.

So that, to date, is my thinking on the interactions and complications of thinking about a beam axle in 2D vs. 3D.

Corrections, observations and criticisms welcome as always.

Ralph

[Jay Lawrence]

Thanks Ralph. In reality, for me it is easy to crack open Solidworks, model it up and see how it all works, but I guess like you I'm trying to use some crayons (which gets tricky for all the inter-related constraints you've mentioned).

Mitchell, thanks for the image. I'm interested to see how your car works out. Is there a better picture that shows things a bit clearer? Bit hard to see how it all works with all those black links. Love the majestic magenta highlights by the way

[Z]

Jay,

"... all ... quite brain frying in a way, and I don't claim to properly understand it yet,..."

Like most other things, the first step to understanding these things is to learn how to "talk the talk".

So now you know that there is a thing called "the cylindroid"...
and it is intimately related to 2 DoF Kinematic joints...
and you can also find it in double-wishbone suspensions...
but it also is common in 3-D Statics, with Wrenches and stuff...
and it sort of looks like a long threaded rod (of variable thread-pitch, apparently?) that spins around a short axle called the "spine"...
and ...,

Well, that is a pretty good start.

Hopefully by now the concept of Motion and Force Screws is also starting to become more of an everyday thing. I reckon all this would be a lot easier if all the "Whizo Suspension Programs (now in all-new, full-colour, 3-D!!!)" would show these Motion and Force Screws as standard. The "understanding" would then sink in without you even realising it. The cylindroid, of course, is simply made up of a lot of these "screws" spun around the spine. So any genuinely 3-D program that claims to model beam-axle linkages should also show where the cylindroid is (perhaps just in an abreviated way as in my sketch).

But ..... I am not aware of any suspension program that shows the ISAs yet... Come on guys...
~o0o~

"I'm confused by the way you have modelled Ralph's ISA-'R' wrt the Panhard bar...
... I would think that the ISA-'R' line/revolute would be angled in plan view."

I hope Loz answered that. The Panhard-Bar n-line ("n-PB" at top-left of sketch) intersects ISA-'R' just to the right of the differential. So, yes, ISA-'R' is at a slight angle, being on the car centreline near the Slider, but angling outwards-going-backwards, to pass to the right of the diff.

(I should note that I started the sketch about a month ago, then got interupted with a bunch of other stuff. When I got back to finishing the sketch I decided to change a few things, kind of forgot a few bits (at top-left, the "Body" should have an "arm" reaching back to the PB...), drew a few bits in slightly the wrong places, etc., etc...)

To help with the general understanding I will try to do some more sketches of more conventional (ie. symmetric) beam-axle linkages, suitable for FSAE cars, ... err... soon.

In these cases the quite conventional description, which shows a side-view of the various link-axes (ie. n-lines) intersecting at a "Pitch-centre", and maybe also shows the "Roll-axis" of the beam, is pretty much all you need. In such cases the cylindroid's spine is the short line, perpendicular to the Roll-axis, that joins the Roll-axis (ie. the revolute = ISA-R) to the Pitch-centre (ie. the revolute = ISA-P, but seen only as a point in side-view).
~~~~~o0o~~~~~

Kev,

"It should be noticed that for a simple four bar / symmetrical rear end the two ISA lines [of zero thread-pitch] have been stated as the pitch and roll axes of the suspension system. In which case the tilting of the cylindroid can be controlled by altering the slope of the roll axis (which for this case will lie on the centre-plane of the car). I am reluctant to mention this simplification. I only do so to show how you can see existing (incomplete) knowledge of kinematics as drawn in a few texts applies in special cases. It does become a useful simplification in a few design cases."

Yes, as I wrote in the last paragraph above. I do agree that this simplified approach (ie. side-view only) is almost all you need to get a good design. The extra 3-D information is only necessary to check what happens towards the ends of the wheel travel, when everything goes skew, and only for cars with long wheel travel (ie. off-road), or maybe when using very short links.

I want to do a few sketches of variations of your (ECU's) four-link rear end. I reckon it is a good layout for FSAE, for a whole lot of reasons. But there are also lots of little variations that can be made that may make it easier to do in a different car (ie. with different engines, whatever). The thought processes involved in making the variations, while constantly going back to the 3-D Kinematics, then becomes useful for solving a whole range of other problems.

Err..., soon ... hopefully.
~~~~~o0o~~~~~

Mitchell,

Now you have to start driving it like you stole it!

And then keep ironing out those little bugs...
~~~~~o0o~~~~~

Ralph,

"We now have two choices ... use analytic geometry ... or start to learn more about 3D kinematics."

I remember a class with Jack Phillips (late-1970s?) where we were discussing something very 3-D-ish (I think about the ISAs spread around a "regulus" = hyperboloid of revolution?). The next class was the following morning. He walked in with a model he had made up overnight, consisting of a ~half-dozen, metre long dowels of wood (maybe 10 mm diameter), some bits of fret-sawed plywood to hold the dowels together, and, no doubt, quite a few lengths of different coloured string.

The "answers" became very clear, very quickly.

For a four-bar beam-axle linkage, I might start with a box of toothpicks (the ones that are sharp at both ends) and two matchbox-sized blocks of polystyrene foam to represent Body and Axle. All at about 1/50 or 1/32 scale (metric or imperial). Plasticine could be used instead of the PS-foam. Then I might upgrade to shish-kebab skewers about 20 cm long x 3 mm diameter. Pretty soon I would be wanting to use Jack's 1 metre long dowels, and a scale of about 1/4, or even 1/2...

Two advantages of this approach for FSAEers.
1. The 3-D problems become very easy to see, and hence easier to understand.
2. Good hands-on practice for building the real car!
~o0o~

Another thing I have been wanting to mention for some time is your particularly short Panhard-Bar. Since your cars spend most of their time cornering to the left, this PB will mostly be in tension, being pulled by the Body's centrifugal force horizontally to the right. So, like a pendulum turned sideways, the PB, with the Body on the end of it, will always be trying to "hang" horizontally (ie. when cornering).

The shortness of the PB means that any relatively small up-or-down movements of the Body's PB-attachment-point (wrt Axle), changes the angle of the PB quite a lot, and so results in quite a large restoring force from the PB that tries to get the Body-PB-attachment-point back to its "comfortable" position where the PB is horizontal. This restoring force that tries to pull the Body down-or-up, naturally has an equal and opposite reaction-force pulling the Axle, near the right-wheel, up-or-down.

I would have to watch the cars' racing before conjecturing what sort of effects the above has on handling. But my guess now is that if the PB was on the other side of the car, where it would be in compression, then the cars would be UNDRIVEABLE! The unstable "inverted pendulum" action of the PB would, during high lateral-G cornering, either pull the left-side of the Body hard down on its bump-stops, or it would flip the left-side of the car upwards, with the PB possibly rotating a full 180 degrees (unless stopped by the dampers or whatever).

So the PB location and size makes sense. Unless the cars start cornering to the right!

Z

[rwstevens59]

Z

And here is a quick example of a class of car run here in the states with a short panhard bar mounted on the left of axle centerline, totally decoupled four trailing links and a highly compliant torque arm mount.

While it flies in the face of convention, some seem to make it work!

A lot more differences than the suspension I am interested in, but too long to go into now.

tonystewartLatemodel.jpg

[Z]

Ralph,

Interesting! It looks like the left-side PB was trying to flip the left-side of Body upwards (as I suggested above), but was stopped by the left-side suspension droop-stop.

Also, the action of the PB pushing the left-side of car upwards seems to have had the reaction of planting the LR wheel (pushing it down onto the road) as seen by the wrinkling of the LR tyre wall, which, presumably, it is still giving forward thrust.

(Edit: So effect of the PB+droop-stop is LR hard on ground, but LFront lifted OFF ground?)

I guess if the driver doesn't mind the Body moving around like that, then the car IS driveable.

Lots of ways of skinning a cat...

Z

[Jay Lawrence]

Thanks Z.
I understand the concept of the cylindroid and ISA's, but the practical implementation and relation to a given system is still taking its time getting into my head, but I will get there.

Can't remember if I've asked this before, but why do you talk about centrifugal force?