Beam Axles - Front, Rear or both.

[Z]

BEAM-AXLE (3) . REAR - VARIATION ON B-A(2).
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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

[Z]

BEAM-AXLE (4) . REAR - WITH SIMPLER BEAM.
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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

[Z]

BEAM-AXLE (5) . FRONT - WITH SIMPLE BEAM.
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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

[rwstevens59]

Z,

I've been quiet lately on the forum but reading intently. Thanks for all of the work to produce the above. Lots to study!

Ralph

[Jay Lawrence]

Thanks Z, lots of very cool information that I sure didn't learn at uni

I realise you didn't intend to delve into details with these systems but I do have a couple of queries:
In order to get the desired n-line slopes (for the rear) you have depicted some quite far forward cylindroid locations. Do you feel that this may jeopardise ideal driver placement to the extent that it may invalidate the concept? To get the 40/60 distribution I imagine you would have to run some very short linkages, or else run them alongside and underneath the driver (which may not be legal, and has other negative effects).

What are your thoughts on mounting the engine on the rear axle/arm with regard to driver controls? I imagine an engine moving about would have negative effects on throttle/clutch/gear control. Also there's the rule (assuming it's still a rule) that the intake must be hard mounted to the chassis. I guess one could argue that the suspension is part of the chassis but I could see it being an issue.

I don't think the front beam plus flat floor underneath would be allowed/safe. I'd imagine you'd have to have another floor over the top of the beam. Was this your intention? If so I envisage a humongous front end (they're already quite large for the most part, given the template restrictions).

Anyway, apologies if this comes across as obtuse, just trying to absorb.

[Z]

Jay,

"[In FSAE] To get the 40/60 distribution I imagine you would have to run some very short linkages, or else run them alongside and underneath the driver..."

All of these recent Beam-Axle drawings are meant to be quite general, and apply to a whole range of cars, not just FSAE.

For example, along with Ralph's examples from a few pages back (ie. racing on very rough tracks!), I have also got PM's from Baja students, and other more general enquiries. So, I drew the B-A(2) and B-A(3) sketches with quite long links so that they would suit long travel suspensions, much longer than is needed in FSAE.

In short, all the sketches would work fine in FSAE even if the physical links were MUCH shorter than sketched. Even with, say, 10" (250 mm) long links, the full bump or droop travel of 1" (25 mm) only changes the slope of the links by 1:10 (at most!), so much shorter links are NO PROBLEM.

And the Cylindroid itself is a "virtual" thing, so it can be placed anywhere, even off at "infinity".
~~~o0o~~~

"What are your thoughts on mounting the engine on the rear axle/arm with regard to driver controls? I imagine an engine moving about would have negative effects on throttle/clutch/gear control. Also there's the rule (assuming it's still a rule) that the intake must be hard mounted to the chassis. I guess one could argue that the suspension is part of the chassis but I could see it being an issue."

Yes ... the main problem is the Rules...

Again, considering the smoothness of FSAE tracks, I see NO PROBLEM with having an engine rigidily mounted to one of the "triangular" Axles, either the "Twin Beam-Wing" sketch, or B-A(2) or (3). With the engine mounted at the front of the triangle, nearer to the Body, it moves much less than the wheels when they go over bumps. And since no bumps in FSAE... (Or, looking at it the other way, many cars have won FSAE with effectively NO suspension at all...) And given that throttle/clutch/gear control can all be done via cables (and often are), I see no problems there either.

In fact, I reckon the ideal candidate for such Beam-Axle-mounted-drive would be the electric cars! Mount the heavy but compact motor(s) near the front of the Axle "triangle", just under the driver's back (ie. where the gearbox is in TBW sketch), and then take a chain or toothed-belt reduction back to the no-CV-driveshafts.

Importantly, the above amounts to "Outboard-Drive", so longitudinal n-lines have to be appropriately chosen. These are shown in B-A(1) as n2, but for good "antis-" they should have a LOWER slope, closer to that of n1. So either use a linkage similar to the TBW sketch, or if using linkages B-A(2), (3), or (4), then P should be lower, about the same level as R.
~~~o0o~~~

"I don't think the front beam plus flat floor underneath would be allowed/safe. I'd imagine you'd have to have another floor over the top of the beam. Was this your intention?"

Yes. The bottom-floor would be part of the aero-undertray. Any "cockpit-floor" above the Beam would be for the Rules.

"If so I envisage a humongous front end..."

Well, NO DIFFERENT to most Double-Wishbone FSAE cars.

Most every FSAE DW car out there today has a "stepped floor". This rises up roughly 100 mm from the seat-base to the footbox, just to provide suitable attachment "nodes" for the lower wishbones. Then there is a R&P that necessarily sits above this floor so that the tie-rods can get out to appropriate pick-ups on the upright, and keep away from the wheel-rim. And then another "cockpit-floor" above the R&P, to comply with Rules.

I figure many DW cars have this cockpit-floor at about 150 mm above ground (ie. at R&P location). Anyone care to share their numbers?

Anyway, a B-A(5) type Beam-Axle, albeit bent into a very wide "V" for FSAE, should be able to fit under a similar height cockpit-floor. Note the Pitman-Arm and tie-rods can be in front of, or behind, the beam, and the beam cross-section can be a widish, lowish, RHS, as sketched. Add the 350 mm high foot-box template, and the underside of the steering-shaft and Bevel-Gear-Box can be less than 500 mm above ground (the template has big semi-circular cut-outs at top and bottom for the steering).

It all looks very similar to most DW cars I have seen. From a quick check of photos, these have the top of their FRH at 600-700 mm above ground. Certainly, the tops of most noses at the front-axle-line are well above the tops of 13" front wheels (= 500+ mm diameter).

Z

[Kevin Hayward]

The raised cockpit floors are a really interesting case. If we make a rough assumption that there is around 50kg of stuff in the nose of the cockpit (drivers legs, front chassis, pedals, shocks etc.) a rise of 100mm will cause a COG height rise of nearly 20mm for a competitive car.

In perspective this is equivalent to dropping the COG of a CBR by 83mm, dropping a 5kg rear wing by 1m, dropping the drivers COG by 67mm.

I wouldn't say that it is desirable to have this stepped floor. If we accept that it needs to be there then a beam may be as good as a double wishbone setup, but maybe there are ways to avoid the step altogether.

Maybe that will come at a weight penalty, which makes for an interesting question. How much weight could you add before lowering the COG by 20mm is not worth doing?

Also please note that the usual reason for lowering COG for racecars is solely for increased tyre friction while cornering (i.e. load depenadant tyres). With FSAE there are many places on a track where the vehicle is limited by wheel lift, something that is greatly affected by COG height and very often not modelled properly by students (i.e. proper inclusion of jacking affects and transients).

Kev

[mech5496]

Once more, thank you Erik for the (really onteresting) discussion, I believe that you give a rather useful insight, even to people not remotely interested in beam axles. Kev, the issue you raised (pun intented) is way overlooked by teams, albeit 50kg seems a bit overexagerrated IMO. Stepped floors are a bit harder to manufacture, but (for most double wishbone cars at least) offer "better" mounting points for the lower A-Arms. Moreover, I believe that one could lose a bit of the CoG height by simply putting the pedalbox/driver feet lower again, either by creating a nose-mounted "bucket" or a 2-step chassis. Another -or maybe even more- concerning issue though, and overlooked by almost everyone, is drivers feet plus pedalbox, impact attenuator, front wing etc. mounted more than half a meter in front of the front axle, and the effect of all that weight to your moment of inertia...

[Kevin Hayward]

Harry,

50kg was a bit much, because the legs pivot at the hip. Lowering the feet by 100mm will only lower the leg COG by around 50mm. A persons COG is roughly at the top of the hips. The legs account for a lot of weight of a person. Large bones, significant muscle mass. The torso has a lower density due to the lungs. I made an assumption of 30kg for legs, 10kg for chassis, 10kg for stuff. We might end up with something closer to 15kg effective lowering of the legs, 15kg of chassis and associated. Lowering 30kg by 100mm still ends up with a pretty large COG effect. The effect of a stepped nose is more detrimental the lighter the car gets.

It is useful to do the calcs as to what a 10mm drop in COG is worth in time/points when compared to 1kg of weight. Can't do it using Optimum Lap, but just about any decent simple lap time sim will give a coarse value to go by. Tyre temps will change this, but generally works both ways for COG and weight. Because the tyres are run under the ideal temp adding weight/COG will cause less of a drop off in performance than indicated by purely the tyre load sensitivity. Poor modelling of the transients and jacking will cause an overestimate of maximum cornering which will underestimate the effect of COG height in parts of the track where the car lifts both insides. It is also very dependent on tyre load sensitivity. Engine power also comes into play. Lower powered cars are more sensitive to total mass relative to COG. By extension drag and downforce also affect the results.

From the sims it is very interesting how many kilos you can add for each 10mm of COG drop (you can add even more if you use pounds). When you compare this to the weight of the design changes required to lower the nose you get the design guidance you are after.

I would also add that any laptime sim that doesn’t include the effect of COG height is not worth running for anything but the most basic decisions (i.e. final drive).

Getting back to the orginial question, beam axles front/rear or both. From the results of calculations I have seen (and performed) I would not be using a beam at the front of a car, and not be having any step to the front chassis. If you accept that the step is required (i.e. for aero) then you might as well run a beam, with mounting points back to the front roll hoop.

Instead I think the best option is probably a well designed five link (or 3 links and a wishbone) to improve contact patch migration (relative to the car) during during steering. Harder to design than double a-arms, very easy to build. Direct acting shocks of course.

Kev

[mech5496]

Kevin, thanks for your thoughts. Indeed CoG height vs. mass is an interesting debate and something we worked quite a lot on last year to quantify (as well as other parameters). I still believe though that a "double step" chassis (or a way to get feet/pedalbox low) is the best compromise, as your A-Arm mounting points would be placed to offer more straight load paths.

A bit offtopic, but an interesting resource, is the following link, especially towards the end:

http://msis.jsc.nasa.gov/sections/section03.htm

I know that there are enough teams out there trying to model all car parts/weights but often "forget" the driver as part of the game.