# Thread: Beam Axles - Front, Rear or both.

1. Originally Posted by Z
To me, there is undoubtedly great confusion above. Inertial forces are described as both "real" and "fictitious"! In fact, they are so "fictitious" that apparently they cannot hurt you!
Siding this... If you are hit with a baseball bat, what causes the pain?

2. Originally Posted by Z

Does anyone else think that my use of reductio ad absurdum (a few posts ago) was incorrect?
Z
Sorry for me to bugger in, but I wanted to 'hear' that, the incorrectness; i've seen some other posts with several lines with incorrect logic/ values etc.. Maybe you just want to test us. Thats what a teacher should be, if he meets the right students.
I've understood a lot of what I know now from you ambitious posts; I have read and still read you posts.
If there's use of offtopic, I,ve said to write stuff

Thanks

TUIasi Racing

3. Originally Posted by Markus
If you are hit with a baseball bat, what causes the pain?
Markus,

The indoctrination is strong in you.

As all small boys know, if the the bat is moving very slowly, but unstoppably (ie. like a hydraulic press), then it slowly and gently pushes you out of the way. So, NO PAIN.

However, if the bat is moving very quickly, then Inertial forces hold your body in place, and the bat's (EM) contact forces squash the impact zone against the opposing Inertial forces. So, MUCH PAIN!
~o0o~

A good example of the absurdity of the Modern Education system is seen in the teaching of Special and General Relativity (SR and GR).

About 1915, Einstein based GR on the Equivalence Principle. Briefly, the EP says that Gravitational and Inertial forces are indistinguishable. The usual thought experiment used to describe this is as follows:

You are standing in a small room with no windows. You feel yourself being pulled downwards against the floor. The EP says there is no way that you can tell if there is a very large mass, say, the Earth, just below the floor pulling you down with Gravitational forces, or if the room is out in deep space and there is only a small rocket engine under the floor that is accelerating the room upwards at 1 G, and you are feeling the resultant Inertial forces acting downward.

So, by this very Principle, if Gravitational forces are "real", then the INDISTINGUISHABLE Inertial forces should be just as "real"! If it walks like a duck...

Furthermore, if the very large mass just below the floor happens to be a Neutron star, then your body feels exactly the same "forces" on it as if the upward acceleration of the room is extremely large, perhaps caused by a mega-Mythbusters'-style explosion. Namely, your body is squashed extremely flat! And the last thing that goes through your mind, is ...
~o0o~

Here is an example like the Hi-G one above. ("G" = Gravity, or acceleration, or same-same!?).

The Mythbusters' Rocket-Propelled-Truck-Hits-Stationary-Car. (Ignore that the boot of the car is filled with explosives, because they don't go off.) In particular, watch the slow-motion at about 45 seconds, and even better at times 1:20 and 2:20.

Even with no narration, you know when the film is slo-mo just from the behaviour of the truck and car. IMPORTANTLY, if those slo-mo bits were "real-time" speed, with the truck travelling at only a few mph, then the truck would simply push the car forwards and there would be very little damage. Maybe only a cracked tail-light.

BUT (!!!) it is very obvious that those sections are happening very quickly, with very high accelerations. And hence also very high "Inertial forces". It is also very obvious that something, let's call it "The Force of Inertia", is holding the front of the car almost stationary, while the truck progressively crumples the rear of the car.

If there are NO Inertial forces pushing leftward on the front-half of the car, in opposition to the truck pushing rightward, then NO squashing of the rear-half of the car.
~o0o~

The absurdity of the Modern Education system is that for most of the time between 1915 and now, the ideological dislike of an Inertial reference frame like an "Absolute Space" was put before good reason (see my previous posts). So nowadays GR is taught with little reference to the above Equivalence Principle (ie. the EP is very quickly skipped over, or even ignored) and Inertial forces are called "fictitious".

The real problem, IMO, is that the expert Teachers cannot say "We DO NOT KNOW how all this stuff really works. Nevertheless, here are some different, and inconsistent, hypotheses...". Being "experts", they want you to believe that they know everything, and have all the answers.

The big question is that now that the Higgs Field/Mechanism/Ether provides a slightly more justifiable way of explaining Inertial forces as being "real", will the Teachers start teaching so? Here I should note that these Higgs ideas only started to be developed in the 1960s, and although they fit in reasonably well with the "Standard Model" (ie. the one with Quarks, Gluons, etc.), they were considered speculative until those few blips on the computer screens in 2012.

My guess is that you will continue to be taught the absurdities until the current generation of Teachers get old and retire, or die. Perhaps when the students who are now growing up with the Higgs ideas become Teachers, then maybe things might change. But given the scarcity of well-reasoned thinking nowadays, I doubt it.

Idiocracy, here we come...

Z

4. BEAM-AXLE KINEMATIC DESIGN.
=======================

A few months/pages ago I posted on "The Cylindroid" and its connection with Beam-Axle kinematics. That post started with a given particular suspension linkage (ie. Ralph's Beam-Axle example), and that SPECIFIC LINKAGE was then ANALYSED to get some idea of how it behaves.

That is, we first identified the four "idealised" n-lines of the given physical linkage. Those four n-lines then determined a unique Cylindroid. That Cylindroid told us ALL the possible ways the Axle could move with respect to the Body, namely all the ISAs, and thus ALL the possible n-lines for the axle. Finally, this knowledge of the Cylindroid allowed us to predict how that particular suspension would behave when travelling over bumpy roads, namely its bump-steer, etc., or how it would respond to horizontal forces at the wheelprints.

(Note that the Axle's behaviour when subject to horizontal wheelprint forces (ie. due to acceleration, braking, or cornering) was NOT explicitly covered in that earlier post. Very briefly, the "anti-" effects of these forces are roughly as described in the "Jacking Forces" thread. However, note that Beam-Axles "jack" DIFFERENTLY to Independent-Suspensions, albeit according to exactly the same principles of Mechanics! I have covered this in a recent PM and might include it in a further post...)

BUT (!!! ) the above process does not work when designing a NEW suspension, because there is NO GIVEN "specific linkage" to start from.

So, how do we go about SYNTHESIZING a new Beam-Axle suspension to suit a given problem, such as, say, FS/FSAE?

Well, I would suggest simply running the above "analysis" process backwards.
~~~o0o~~~

Namely,

STEP 1. - First, decide what sort of behaviour is wanted. This includes the amount of bump-steer wanted (if any!), the amounts of "anti-" squat, dive, or roll wanted, and consideration of any other desirable or undesirable Kinematic behaviour (...more details below).

STEP 2. - Next, determine a range of Cylindroids that characterise the above desirable motions of Axle wrt Body.

STEP 3. - Finally, from an appropriately chosen Cylindroid, choose four suitable n-lines that can be approximated by real, physical, links between Axle and Body. Choose these n-lines and their links so that they fit in well with the rest of the car's "big-picture" design, and also so that they work well structurally.
~~~o0o~~~

This is all easier understood by working through some examples. So below are five posts and sketches that go through this SYNTHESIS process. That is, from a high-level list of "wants" -> to detailed practical implementation.

Please note that all this was a bit rushed (ie. Silly-Season fast approaching, and exploding laptop power supply not helping...). So the "practical examples" should be taken only as a rough indication of what might work, but are not necessarily completely accurate in detail.

Also I haven't explicitly justified every decision made in the words written below. This is due to lack of space and time, but not due to lack of reasons. Any questions regarding the "reasoning" will be happily answered...

Z

5. BEAM-AXLE (1) . REAR - CONCEPTUAL.
==================================
We wish to SYNTHESIZE a Rear Beam-Axle design. Following the above advice, we start by considering the Kinematic behaviour we want.

STEP 1. DESIRABLE KINEMATIC BEHAVIOUR.
======================================
1. AXLE ROTATIONS.
=================
1.1 STEER (ie. Axle rotation about vertical Z-axis) - This is a very important behaviour to consider. Imagine trying to drive fast around a bumpy corner while a "gremlin" is randomly steering your rear wheels. NOT desirable!

So, as a first approximation, we want NO steering of the Axle under any of its possible Kinematic motions (ie. NO "Yaw" rotation of Axle, wrt Body).

From a simple consideration of rotation vectors (strictly speaking, rotational "velocity" vectors) we see that the Axle can only steer wrt Body if it rotates about an ISA that has some vertical component. This means that we want ALL possible ISAs for the motion of the Axle to be horizontal wrt Body. It follows that the spine of the Cylindroid should be vertical, or at least close to vertical.

Importantly, note that because of manufacturing tolerances, etc., no Kinematics can work EXACTLY as expected. Nevertheless, the GOAL here is for horizontal ISAs. Any deviations from this goal can be considered later. Also note that we are NOT considering "compliance steer" here. In practice, this might be the biggest problem, but also to be considered later...

1.2 CAMBER (ie. about longitudinal X-axis) - The nature of a Beam-Axle fixes this behaviour. The Camber-Change of the whole Axle, and also of each individual wheel, is simply, and always, the change in the two wheelprint heights divided by Track-Width. Fortunately, this is the best possible Camber-Change that can be expected when driving on a smoothish road (ie. one that is essentially a straight-line between the two wheelprints).

(As noted elsewhere, on very bumpy roads (ie. with short wavelength bumps), the Camber-Angle between road-surface and wheel, at any instant, is set by the L-R position of the bump under the wheel. There is nothing a passive suspension can do to control this angle. Also, any quick changes in Camber can give very large gyroscopic forces. So, on very bumpy roads, best is end-view wheelprint movement that is vertical, and NO Camber-Change at all.)

1.3 CASTOR (ie. about lateral Y-axis) - This has some importance for a front-axle, because of its influence on the Steer-Axis, but less importance for a rear-axle. It is connected to longitudinal "antis-", as covered below. It can have a BIG influence on "wheel-hop" during braking. This is a big subject, I think already discussed elsewhere, but NOT hugely important in FS/FSAE.

Without going into too much detail, we can say that we do not want very large Castor-Changes, but "moderate" amounts of it are acceptable. It is MUCH LESS important to control than Steer.
~o0o~

2. WHEELPRINT TRANSLATIONS.
============================
We skip the consideration of translations of the whole Axle here. These have some influence on behaviour of the whole car, but not enough to warrant a detailed discussion. Instead, we consider only the movements at the two wheelprints, because that is where the biggest forces are acting.

2.1 VERTICAL (ie. bump/droop of wheelprints in Z-direction) - Controlled by Spring-Dampers, so consider elsewhere...

2.2 LONGITUDINAL (X-direction) - This has a BIG influence on the "Anti-Pitch" behaviour, either during acceleration (ie. "anti-squat"), or during braking (ie. "anti-lift" for rear-axle). For FS/FSAE an "anti-squat" value between 0 and 100% is suitable for AutoX and Enduro. For the Acceleration event anti-squat of 100+% can be beneficial.

Note that there is a BIG, BIG, difference between having Inboard or Outboard Drive/Braking (more details below). Also, longitudinal jacking behaviour of Beam-Axles is essentially the same as with Independent Suspensions, and as covered in the "Jacking" thread (... unless front and rear Beams are somehow connected, eg. UWA!, in which case LESS of a problem).

2.3 LATERAL (Y-direction) - This determines "anti-Roll". Put simply, lateral n-lines that are horizontal give 0% anti-Roll, and loosely translate as "Roll-Centre on ground". Lateral n-lines that slope steeply up-to-car-centre give positive anti-Roll. If, in end-view, the n-lines intersect the Body's CG, then 100% anti-Roll, and NO ROLL through corners. This is acceptable for Beam-Axles, because they DO NOT JACK in the same way as Independent Suspensions (maybe more explanation later...).

BUT (!!!) steeply sloping lateral n-lines (ie. = high RC) implies a lot of lateral wheel scrub when driving over bumpy roads. This lateral wheelprint movement then either shakes the Body sideways, or shakes the wheelprints sideways, possibly pushing them over their "Fy limit", or a bit of both. As a sweeping generalisation, it is best to have anti-Roll = 0 - 100% (ie. "RC" between ground and CG height), but closer to 0% if expecting bumpy ground, especially in corners.
~~~~~o0o~~~~~

STEP 2. FIND THE CYLINDROID.
============================
From the above desirable Kinematics we determine a suitable Cylindroid.

For minimal adverse steer effects, the Cylindroid should have a vertical spine, wrt Body.

Because of the general L-R symmetry of FS/FSAE Dynamic events, the Cylindroid's spine should be on the L-R symmetry plane of the car. Also for symmetry reasons, the two ISAs at the ends of the Cylindroid's spine, which are always mutually perpendicular, should be oriented laterally and longitudinally to the car, and they should be of zero thread-pitch (ie. "revolutes").

We call the lateral ISA the "Pitch-ISA" for motion of the Axle wrt Body, and designate it "P" in all these sketches. Similarly, we call the longitudinal ISA the "Roll-ISA" and designate it "R". So, in plan-view we have R running down the centreline of the car, P is perpendicular to the centreline, and the spine appears as a dot. But we do not yet know P's longitudinal position, nor do we know the heights of R or P above ground.

To determine R's height above ground we consider an end-view, as at the bottom-left of the sketch below. As per the "Desirable Kinematics" above, we choose lateral wheelprint n-line slopes such that R is slightly above ground, say about 100 mm. For typical FS/FSAE CG-heights (~300 mm) this gives an anti-Roll effect that reduces the cornering Roll couple carried by the springs by about 1/3 from the case with horizontal n-lines (= ground level "RC"). So softer springs can be used. But these n-line slopes do not introduce too much lateral scrub, so are acceptable on moderately bumpy tracks.

To determine P's longitudinal position and height above ground we consider a side-view, as at bottom-right of sketch. For appropriate longitudinal "anti-" behaviour we want the longitudinal wheelprint n-lines to slope slightly up-to-front of the car.

IMPORTANTLY, note that for Outboard-Drive/Braking, this n-line is n2, but for Inboard-Drive/Braking it is n1. This is because in the Inboard case the DRIVESHAFT AND CV-JOINTS ARE PART OF THE "SUSPENSION" LINKAGE that transmits forces from the wheelprints to the Body, so they MUST be considered when determining the correct n-lines. For the Inboard case the correct wheelprint n-line (ie. n1) is always parallel to the n-line through the wheel's axle, namely n0, because the driveshafts and CVs always keep the "wheel-leg" (ie. from axle to wheelprint) in a vertical position.

In short, the side-view n-lines shown in the sketch give quite large anti-squat/lift (~150+%) for Outboard-D/B (= n2), but only moderate anti-squat/lift (~50%) for Inboard-D/B. The sketched Kinematics are more suited to Inboard-D/B, namely a "De-Dion Axle" with Inboard brakes. For many reasons, but no space to cover here, it would also work acceptably with Outboard brakes.

So, finally, based on the above reasoning, we have placed P, and thus also the Cylindroid's spine, at about the half-wheelbase position. Note, however, that the Cylindroid could also be placed a LONG way in front of the car, albeit with P much higher above ground. Or the Cylindroid could be BEHIND the car, with P possibly underground. The position shown here was mainly chosen to make the sketches easier, although it is also well suited to some of the practical linkages below. But it is worth thinking about the alternative longitudinal positions.

In summary, a Cylindroid has been chosen that has its top-most ISA P lateral to the car, and its bottom-most ISA R longitudinal. In most suspension design literature, these P&R axes ALONE are used for Beam-Axle design discussions. For symmetric designs, such as here, this P&R-only approach is good enough. The additional knowledge of the Cylindroid given here "merely" gives a deeper understanding of the Kinematics, and helps explain what happens when everything goes cockeyed, as in Ralph's earlier examples.
~~~~~o0o~~~~~

(More coming, 10k limit!!!)

Z

6. BEAM-AXLE (1). REAR - CONCEPTUAL. (Last bit...)
=====================================

STEP 3. FIND A SUITABLE PHYSICAL LINKAGE.
=====================================
As noted earlier, because this is a 2 DoF joint, we have to find four n-lines to act as the 4 Degrees-of-Constraint of the Axle wrt Body.

The Cylindroid offers us many, many, potential n-lines to choose from. Our choice is guided as follows.
1. The n-lines, and their subsequent real links, should connect to convenient positions on the Axle and Body.
2. The n-lines should NOT pass through regions that are occupied by other important stuff, such as the engine or driver, nor should they be in inaccessible regions, such as a long way from the car, or underground.
3. The four n-lines should be configured in space to provide good rigidity to the Axle's "constraint". This is a similar problem to designing a stiff spaceframe structure. It can be summed up by saying that the n-lines should be widely spaced and as orthogonal to each other as possible. Having two n-lines that are close together and almost parallel makes one of those n-lines almost redundant. (Similarly, long narrow triangles in spaceframes are BAD design!)

A methodical process for selecting the n-lines is as follows.

From symmetry arguments again, it makes sense to have n-lines/real-links that are mirrored on left and right sides of the car. This reduces the problem to finding only two n-lines on one side of the car. Each of these n-lines MUST pass through both of the P and R axes. So, pick any one point somewhere on P on one side of the car, and another point somewhere on R, and then draw a line through these two points for one n-line. Do similar for the other n-line.

Thus, only FOUR INFINITIES of choices. And MOST of them are easily eliminated! Sift through these choices to see which best meet the criteria listed above. Done!
~o0o~

The next three sketches all have identical Cylindroids to the one in this concept sketch. They differ only in the choice of n-lines and their real, physical implementation. Obviously, many other variations are possible. Just follow the above process...

Z

(Edit: See also Kevin's post from a few pages back for a different, but also quite similar, approach to reasoning your way from a big-picture list of "wants", down to nitty-gritty mechanical hardware.)

7. BEAM-AXLE (2) . REAR - PRACTICAL.
================================
This layout starts with a triangular Axle structure similar to the first "Twin Beam-Wing" sketch I did on this thread (page 3 current Forum).

As explained in that post, that rear linkage only gives ~0% anti-squat, and it only manages that because of the slope of the final-drive chain run. Using that sort of linkage for an Inboard-Drive, De-Dion Axle would give quite large PRO-SQUAT during acceleration. However, the linkage in the sketch below could be quite easily retro-fitted to the "Twin Beam-Wing" car with minimal changes to its Axle and Body structures. This linkage would give positive anti-squat and the other desirable behaviours described above when run as a De-Dion (ie. with Inboard-Drive).

The approach here to finding suitable n-lines for the links is as follows. Two points are chosen on P, which are symmetrically disposed left and right of the car centreline. Two points are also chosen on R, one in front of, and the other behind, the rear-axle-line. These four points are then connected by four n-lines, such that each point on P and R is intersected by two n-lines.

The result is effectively a single "tetrahedral link" that connects Axle to Body, as seen at bottom-left of sketch. The wide base of this tetrahedral link on both the Axle and Body gives a stiff and strong connection.

In the more practical example shown at the top-left of sketch there are two "ball-ended-links" following the two rearward n-lines, and a single, smaller, "3 BJ wishbone" that acts as the two forward n-lines. As a general rule, the real links only have to follow their n-lines for a short distance. But the longer the real link, then the less the n-line changes its orientation as the suspension moves through its full range.

An advantage of this layout is that all connections to the Body are at a single bulkhead near the car's CG, which in FS/FSAE might be the Main Roll Hoop. Hence, a structurally simple and efficient chassis is possible.

The apparent disadvantage of the large triangular Axle structure is not so bad, because it stiffens the Axle from bending "in steer", and it carries some of the ground-to-Body forces closer to the MRH. Also, a low mounted engine can be positioned "inside" the triangle, the forward-most part of the Axle doesn't move around much (ie. much less than the wheelprints) so doesn't interfere much with other parts located there, and an "underwing" surface can be conveniently fixed directly to the bottom of the "triangle".

Z

8. BEAM-AXLE (3) . REAR - VARIATION ON B-A(2).
========================================
This layout is very close to B-A(2) above. This Axle has the same three attachment points as before, but its structural shape is changed from a triangle to a "Y". Either Axle shape would work equally well in either of these two sketches. The choice of Axle shape depends on other packaging requirements.

The two major differences with this layout relate to the way the same four n-lines have been realised with different links.

In B-A(2) the two rearmost n-lines, which intersect R behind the car, have links that go from their Axle connections FORWARD to the Body. Here the links for these two n-lines go from the Axle REARWARD to a single connection on the centreline of the Body, and behind the axle-line.

This, of course, is only desirable if there is some suitable Body structure in this aft position. I would suggest this layout for a car that has a "backbone" chassis, and is perhaps a front-engine-rear-drive, with a diff or transaxle mounted between the rear-wheels. This way the rearmost suspension point on the Body can share the strong chassis structure already needed for the final drive.

(Edit: This De-Dion B-A would well suit a Lotus-7/Clubman type car, especially if built with a backbone chassis (as per the 1960s Lotus Elan).)

The second difference is that the two front n-lines are now realised by a "Ball-in-Tube" joint, rather than the earlier wishbone. See detail at bottom-left of sketch. The 4-DoF, or 2-DoC, "B-T" joint is Kinematically very similar to the wishbone. Both produce a "planar-pencil" of n-lines that amounts to all the straight-lines that pass through a single point, and lie in a flat plane, and look somewhat like the spokes of a wagon-wheel.

In both cases P always lies somewhere in this plane. So P always lies in the plane of the wishbone, or it lies in the plane that passes through the centre of the B-T Ball and is perpendicular to the centre-axis of the Tube. The main Kinematic difference is that the n-lines of the B-T joint maintain the same angle (wrt Body here) throughout its range of movement, whereas a wishbone's n-lines change their angle as the wishbone pivots on the Body.

Practically, the B-T joint is very compact, but not suited to large travel. In the layout shown the B-T joint would only require a short travel, much less than the vertical wheel travel. In the production car world such a joint would be made as a one-piece rubber bush, which combines the rotation of the Ball plus the short axial plunge of the Tube.

Also worth noting here, is that this layout works as if the Body is connected via the R-revolute to a middle-link, and the middle-link is then connected via the P-revolute to the Axle. This is the OPPOSITE way around to all the previous sketches, which are closer to Body-P-R-Axle. The advantage of this layout is that R is thus very stable wrt Body, throughout the suspension's range of motion. This means less chance of adverse Axle-steer effects in the middle of bumpy corners.

Also note that the big separation between the two Body attachment points (ie. at front and rear of R, and along the Body's strong "backbone") implies good potential stiffness of the linkage against Axle-steer effects. But, as always, the "stiffness" of any chain is governed by the stiffness of its weakest link.

Z

9. BEAM-AXLE (4) . REAR - WITH SIMPLER BEAM.
========================================
This layout keeps the the same two rearward n-lines/links as B-A(2) (ie. those that intersect R behind the rear-axle-line), but it uses a simpler Axle structure. Now the Axle is just a straightish tube connecting the two wheels via "uprights" at each end. (And these uprights can be conveniently used to directly mount a rear wing.)

With the forward end of the triangular Axle structure gone, there must now be two new n-lines to get full control of the Axle. These new n-lines/links attach to the Axle at the tops of the uprights at each wheel, and then go forward to attach to the Body, probably near the MRH for FS/FSAE cars. The n-lines then continue forward to eventually intersect R a long way in front of the car. I believe, from photos, that ECU's current car uses something like this.

The plan-view, at top-right of sketch, shows how the four n-lines/links form a "W" shape to control both longitudinal and lateral forces between Axle and Body. The wider the connection points on both Axle and Body (ie. in the lateral direction), then the stiffer will be the Axle's "steer" control, wrt Body.

A big advantage of this layout is that, like B-A(2), the only connections to the Body are near the MRH. Thus no chassis structure is necessary behind the MRH, other than mandated by the Rules. Also like B-A(2), having multiple attachment points for the links on the Body allows the height and position of P to be adjusted quite easily. This allows easy variation of anti-squat between different events.

If a P-revolute and Cylindroid closer to the rear-axle is considered acceptable, then only two BJs on the Body are needed. These BJs will be on P as it passes through the Body (see bottom-left of sketch). Note that in side-view this gives much steeper n-lines for Outboard-Brakes (ie. n2 in B-A(1)), and a quite short "Side-View-Virtual-Swing-Arm". The steeper n-lines will give high levels of anti-lift during braking, which, in itself, can be advantageous.

BUT (!!!) the short SVVSA can possibly result in severe "wheel-hop" during braking. This depends on many factors, too many to cover here, but CAUTION is advised. At worst, the problem is solved by fitting Inboard-Brakes.

Z

10. BEAM-AXLE (5) . FRONT - WITH SIMPLE BEAM.
=========================================
Lastly, for now, here is similar thinking to above applied to a front Beam-Axle.

This layout shares the same Kinematics as the "Twin Beam-Wing's" Model-T-Ford layout way back on page 3 (see link in B-A(2) post above). Namely, the Cylindroid is squashed flat like a disc, and sits a short distance behind the front-axle-line. Whenever a Cylindroid is squashed flat like this, all its ISAs form a planar-pencil of revolutes. So any motion of the Axle, wrt Body, is a pure rotation about a horizontal axis that passes through the centre of the Cylindroid (ie. intersection of P and R).

(BTW, in the original TBW sketch, the main BJ connecting Axle and Body has 3-DoF, or 3-DoC, so can be represented by any three orthogonal n-lines passing through the centre of the ball. The Peg-in-Slot joint provides only one lateral contraint (ie. it is 5-DoF), so is represented by a lateral n-line through its contact point. Any straight-line intersecting all four of these n-lines is a revolute joint for the relative motion of Axle and Body (ie. = ISA of zero thread-pitch). Thus all such revolutes pass through the BJ-centre, and lie in the plane defined by the BJ-centre and the P-S n-line.)

In practice, and especially if the chassis's Front-Bulkhead is behind the front-axle-line, I prefer the Model-T style layout of the TBW sketch. It is simply more simple, and rugged. However, if the car has a FB forward of the front-axle-line, then the layout in this sketch can work well. This layout also allows the Cylindroid and P to be moved much further rearward for less anti-dive, or for P to be moved up or down for further anti-dive tweaking.

But note that the Cylindroid of the TBW is very stable throughout the range of suspension movement. The Cylindroid below will move around a bit as the suspension moves.

Perhaps the main advantage of this layout is the very simple Axle structure. The Axle is simply a straight-tube connecting the two wheels' King-Pins. Having the Axle pass through the Body allows the Body to have a flat-floor for easy build. For low ride height FS/FSAE cars the Axle might have a slight bend in the middle, to form a very wide "V". This lowers its centre section and gives more room for the foot-box template to pass over it.

The plan-view of this layout illustrates how well chosen n-lines give strong and stiff Kinematic constraint to the Axle, especially in steer as seen here. The four n-lines, and their real links, are spaced as widely and orthogonally as possible, while still attaching to convenient positions on the Axle and Body. The side-view shows that under hard braking all the links are in tension, which avoids the buckling problems that can occur when links are in compression.

Also in side-view, the links form a "Watt's linkage" to give good control of Axle Castor. The instantaneous Castor-change is a rotation about P, which is acceptable as shown. Moving P rearwards gives less Castor-change. In general, moderate levels of Castor-change are acceptable, but DO NOT LET TRAIL GO NEGATIVE! Very large motions of this linkage towards both full bump and droop will INCREASE Castor, and so also increase Trail, so it is quite safe (ie. it just gives more "self-centring" at full travel).
~~~o0o~~~

STEERING - Also shown at bottom-left of the sketch is a suitable steering-linkage for this front Beam-Axle.

I strongly suggest considering the inverted "Tractor King-Pins" for steering. Or the Citroen-2CV style king-pins that have been used recently by UWA. The fact that a Beam-Axle can use "revolute" steer-axes gives it several advantages over the more common double-wishbone layouts, namely the possibility of using stiffer and lower friction roller bearings. See this SLT-Swing-Arms post and sketch for more details. Bottom line here is that just because everyone else uses tall and bulky "uprights", with teeny-weeny little BJs at their ends, does NOT mean they are the best solution.

A key feature here is that the vertical shaft of the Bevel-Gear-Box should be allowed to slide axially up-down. This shaft is then spring-preloaded downwards to always keep the larger crown-gear in tight mesh with the horizontal-shaft pinion-gear. This eliminates all backlash between the teeth, which is the bane of all FSAE Rack-and-Pinions I have ever seen. Bizarrely, the prior-art of every production car R&P ever made shows just how easy it is to solve this problem (ie. spring-preload)!!!

Below the BGB is a "flex-disc" type UJ that accomodates any horizontal motions of the Axle, wrt Body/BGB. These motion are small, so large angle UJs are NOT needed, and the flex-disc type has less potential backlash. Below this is a splined-shaft to allow for the significant vertical motions of the Axle. Again, backlash should be kept to a minimum here, perhaps by using off-the-shelf "ball-splines". The "UJ+spline" can be replaced by a single "plunging-CV-joint", but again, aim to minimise stiction and slop!

Finally, the Pitman-Arm and linkage out to the wheels gives potentially very good Ackermann control. I suggest using quite long Steer-Arms and Pitman-Arms (ie. at least 100+ mm) for stiff and precise steer-angle control. However, making the PAs a bit shorter than the SAs will make it easier to get the right Ackermann. Also, with the same BGB layout, the PAs can point rearward, and the SAs can point forward, as in the TBW sketch. This has further potential advantages for Ackermann...

Enough for now...

Z