Deadlines in school were always so much "fun"

Toe-angles can cause jacking even if you have zero caster, KPI etc. When travelling straight ahead, toe-angles try to steer the vehicle in the direction the tyre is pointing, but because the wheels are pointed in opposite directions, the tyres try to steer the vehicle in opposite directions, so they cancel each other (if they have equal amount of grip). But because tyres with toe angle try to steer the car, they constantly create lateral and longitudal forces at the contact patches. And if the suspension n-lines are not horizontal, the lateral and longitudal forces created by toe-angles cause jacking.

The ISA's can also be used to model the movements of solid axles. Those only need 4 links to locate the axle and give it 2DoF, so it's possible to find the two ISA's. And if I have understood it correctly, you would only need to know the orientations of those two ISA's to be able to tell how much bump steer the solid axle has. I'm currently trying to find a copy of book Freedom in Machinery by Jack Phillips.

With the effect of KPI, what I mean is how it affects the tyres camber angle during steering inputs. If we imagine a suspension with zero camber, caster, trail, scrub etc, but with some amount of kingpin inclination. During steering input KPI angle causes the top of the inside tyre to tilt into the corner, so it can help keep the inside tyre more vertical during body lean (same effect as with caster angle). But, the top of the outside tyre is actually tilted away from the corner, so rather negative effect(so opposite to caster effect). KPI angle was developed to minimize the kickback caused by the offset between steering axis ground intersection point and tyre contact patch center.

That Citroen DS suspension is indeed a suspension with zero kpi-angle and zero scrub radius, but instead of using virtual geometries it archieved it with two simple L-shaped control arms. That's the main reason I think it's rather elegant solution. Those L-shaped arms also make it possible to have rather large steering angles.

Good article about steering geometries by Erik Zapletal
http://www.driftforum.pl/viewtopic.php?f=125&t=22850

And this one is about Toe-angles
http://www.driftforum.pl/viewtopic.php?f=125&t=22852

With interconnected suspension I'm talking about any suspension where any two wheels are interconnected, even lateral anti-roll bar is interconnected spring (U-spring, resists movement in different directions). Z-bar is just any type of spring that resists movement when the wheels try to move in same direction. There are almost unlimited ways to build suspension interconnections, for example using torsion bars (coiled or straight), leaf springs (could be steel, carbon fibre, fiberglass, wood etc), air, hydraulics (with or without a need for pumps), etc. etc or combinations of any of those. The tyres can also be connected in many different ways. It's quite simple to build a compact system where you can have separate springs for roll, pitch and heave. Suspension interconnections are also in no way a new inventions. Citroen 2CV was the first commercially manufactured vehicle with longitudal interconnection and it was designed in 1930's (it used very soft springs, and that is why it has such a big body movements). And even thought 2CV suspension is very simple (it could be simplified even further), some VD books have completely misunderstood it's working principles. There seems to be very little good literature about interconnected suspensions. US Patent 6702265 Balanced suspension system is good read if you're interested about them.

Even Z says that interconnected suspension is not necessarily needed in fsae but there has been some teams that have tried it, if I remember correctly one team was UWA in 2012(?). There has been winning FSAE cars with so stiff springs, that they in practice had no suspension at all. But it's worth thinking about interconnected suspensions, because in practice they are no more complex than normal suspensions and they can even be simpler than conventinal suspensions. Modern racecars have also started to use those so called third springs in aero cars. What is the purpose of those thirds springs? They resist wheel movement in same direction, so they are actually just lateral z-springs, but they are just added alongside the already existing four corner springs and two antiroll bars. So better name might be seventh and eight springs.

I have built few simplified models of interconnected suspensions to better understand how they work (technic lego pieces are good tools ). Simplest interconnected suspension only needs 3 z-springs. It would need:
-1 longitudal spring between left side front and rear wheel (Simplest might be torsion bar. In FSAE, linked air bags could also be interesting (2 compact bags or cylinders+hose+schrader valve per side), use air compressor at pits to fill, for fine tuning effective spring stiffness and LLTD move bags inboard or outboard at suspension arm)
-1 longitudal spring between right side front and rear wheels
-1 lateral spring between front or rear wheels. (Centrally pivoted leaf spring might be easiest to build, or could be torsion bar or some other type. Can be used to adjust rake angle (would be very useful in normal cars)

Those longitudal springs control lateral roll, and 4 wheel heave motions, but have no effect on longitudal pitching. The lateral spring controls only longitudal pithcing and 4 wheel heave, but has no effect on lateral movements. One interesting feature of that style of suspension is, that even when using very, very stiff springs (think lego model), the wheels can still freely follow the contours of the ground, so more grip (twist mode is soft). Think how the tyre contact patch loading changes with interconnected springs when hitting bumps and then compare it to a car with normal suspension. LLTD is adjusted simply by changing the relative leverage ratios that the front and rear wheels have on the longitudal spring (can be any ratio between 100%F-0%R to 0%F-100%R, and that LLTD ratio stays constant no matter what shape the ground is (not counting the additional effects of longitudal weight transfer caused by acceleration or braking). So 3 interconnected springs to have better suspension control than with 6 or 8 conventional springs.

With stiffly sprung racecars with 4 corner springs and 2 anti-rollbars, the shape of the road can have very large effect on the tyre loading and lateral load transfer. Because the LLTD can change several times in single corner depending on the shape of the road, it can make a vehicle that is difficult to drive.

Z likes underfloor aero very much. One of his fsae consepts is vehicle with front and rear beam axles with aero undertray mounted directly to the axles. http://www.fsae.com/forums/showthread.php?1324-Beam-Axles-Front-Rear-or-both . Z's take on carbon fibre seems to be that it's just one possible building material along many others for building parts. It's more about how much time and money is needed to build a part that is good enought for the job it's required to do and at what point the money and time invested start to give only diminishing returns. His opinion seems to be that any vehicle that can be built quickly to maximise testing time and driver training time is good starting point.

Last edited by Juge_, Mon 29 May 2017, 11:22.

Older way of CM teaching seemed to be much more about geometric thinking and visualizating the problem and then doing calculations. Whereas, at least from my experiences, todays way seems to be much more centered around algebraic equations. Older way of designing also seems little bit more logical, where you start from defining your main goals, then thinking what sort of ways you have to archieve those goals and then step by step you get closer to the detail work, constantly thinking how the steps you take affect the big picture. Modern way seems to be more about jumping straight to the detail work. Older CM books seems to also be much more rigorous in cleardy defining the terminology they use so that all the readers understood what the writer had meant. In modern VD literature, just how many different definitions there are just for the roll centre alone.

I remember reading in some older VD book, where the author tried to explain why he thought that the roll center should stay as stationary as possible. I don't remember the exact wording, but it went along something like: He built a car and liked how predictable the handling was. He wanted to know why it felt good, so he drew the suspension in 2-D view at different positions of wheel travel, and noticed that the roll center stayed almost stationary. And because of that he concluded that roll center migration was bad. I personally think that when other authors later wrote their books about VD, they just referenced that original book and just said that roll center migration is bad. And when enought books said that same thing, it sort of became industry standard.

One way to experience how suspension toe-angles cause jacking is by using roller skates. When you are rolling along, keep your knees straight, spread legs apart and point your feet directly forward. Now your legs are the lateral n-lines and their angle is rather steep. When you turn your feet inwards (toe-in), your body is jacked up. When you turn your feet outwards (toe-out), your body is jacked down (might cause some pain). So the toe-angles constantly cause lateral and longitudal forces.

I think it might be a good idea to model the independent suspension as a 5 link system. 4 links that define the suspension arms and the fifth would be the toe-link. If there are four randomly oriented straight lines of infinite length (for example n-lines) in 3-D space, it's always possible to draw at least two straight lines that intersect all those four lines even when those lines don't touch each other. Those two intersecting lines are the ISA's for those 4 n-lines. Easy to visualize by using 4 pieces of string randomly oriented in 3-D space (between two chairs fox example). Now I just need to learn how to actually calculate those intersections

Main reason why I think it would be a good idea to model any suspension as a 3-D 5-link system, is that for example with actual 5-link suspension, it is very unlikely that you can find a 3-D plane that contain the four joints of any two of those links, so the way usually presented in VD books of two intersecting planes can't be used to find the ISA. But by using the orientation of the 4 n-lines of the suspension arms in 3-D space, it is possible to calculate the location of the two ISA's for the upright, and so calculate the lateral and longitudal contact patch n-lines. So if we model the suspension as a 5 individual links, and then have damage, locations of those pickup points and distance between them might change because bent suspension arms, but the n-line would still be straight line between outboard and inboard joints and so we could still calculate the ISA location.

I think that atleast audi uses those 3-D n-lines to create the virtual steering axis for their A8 front suspensions to eliminate the effect of kingpin inclination. It's just a 5 link suspension (4 links for suspension arms and the fifth for steering). The original Citroen DS had a rather elegant solution to that same problem. http://www.citroen-ds-id.com/index.html?ds/DS_Pivot_Hub.html

I think that the toe-link on itself doesn't cause jacking, but the toe-link can cause the upright to rotate around the steering axis either by bump-steer or by not being stiff enough and that rotation then causes changes in toe-angles, and so also causes changes in jacking. After some thinking about how the 3-D orientation of the tie-rod affects bump steer, I've started to think that as long as the n-line of the tie-rod points directly at the suspension up-down ISA, there will be no bump-steer and so no changes in toe-angles. For example swing arm suspension, where the tie-rod inner ball joint is directly at the swingarm ISA. If the tie-rod is shorter than the swing-arm (like in double wishbones), but points to the ISA when the suspension is at rest, there would be no bump-steer at the middle, but it would slowly increase when suspension goes up or down. With double wishbones, it should be possible to build it in a way, that when the steering rack is centered, tie-rods n-lines point constantly to the suspension ISA during suspension movement, so no bump-steer when driving straight ahead. But because during steering, the inner ball joint moves in relation to the rest of the suspension, so the tie-rods n-line could no longer point to the ISA, and you would get bump-steer during suspension travel. And so to get realistic bump-steer curves, the suspension has to be modeled in 3-D because the 2-D approach does not take into account the orientation of the ISA. I'm pretty sure that push/pull rods don't have effect on the jacking, seeing as they basically just act as a damper rod extension so that the damper can be moved to some other location and orientation (and push/pull rods add unsprung weight and compliance)

Another very interesting subject Z likes to write about is the interconnected suspensions.

Are you part of the FSAE?

Yeah, the way Z writes is rather, um, interesting, but the more I read his writings the more I have come to the conclusion that he knows his way around classical mechanics (CM) very well. I've had basic education in CM, but I'm very rusty in those calculations and I'm trying to relearn all those things. I agree with Z, in that if you want to learn about vehicle dynamics, study classical mechanics. CM has been good enough to take humans into moon and back, among other things, so I think that it's good enought for designing cars.

Yes, I can visualize the movement of the ISA in double wishbone suspension. Swing arm suspension is just good way to visualize the instantaneous screw axis, because it's location is constant and you can physically touch the ISA (it's the hinge). You can use your arms when trying to visualize the movement of the upright. Double wishbones just form virtual swing arms. Main reason for why the manufacturers started using double wishbone suspensions because it allows for long virtual swing arms in small package.

Lets imagine 3 different swing arm suspension types:
Pure leading or trailing arm (like in Citroen 2CV) that has its hinge line purely lateral wrt. car body.
Pure lateral swing arm (vw beetles rear) with hinge line purely longitudal
Semi trailing (many cars in rear) or semi leading arm (ford twin i-beam front). Hinge line is at some angle between lateral and longitudal wrt car body.

-Pure leading or trailing arm suspension has longitudal jacking but no lateral. (amount of longitudal jacking changes depending on if the car has inboard or outboard brakes or drive)
-Pure swing arm has lateral jacking but usually no longitudal.
-Semi-trailing or semi-leading usually has jacking in both directions. For example double wishbones usually fall into this category, the virtual lateral and longitudal swing arms are just a lot longer.

The orientation and location of the instantaneous screw axis wrt car body tells us directly how much jacking the suspension has in lateral and longitudal direction because it tells at what angle the longitudal and lateral n-lines need to rise to intersect the ISA. It also tells how the upright moves in relation to the car body. (It screws around the instantaneous screw axis. In the uprights case, the thread pitch of the ISA is zero, so no longitudal wrt ISA movement, and the virtual swing arm length can change so the radius of the rotation is not constant). For some reason, the vehicle dynamics industry seems to refuse to use the term instantaneous screw axis. Think about how a nut moves along a bolt.

I'm wondering how much prosessing power it would take to calculate the angles of the n-lines in real time because when the tyres deforms during driving, the contact patch center also changes its location wrt upright, so the angle of the n-line also changes. Or would it be possible to take it even a step further, and calculate the screw axis location wrt car body in real time (maybe using those SAE paper equations for 5-link suspension, wishbones can be modeled as 5-link) and use that to calculate the n-lines. So if you bend your suspension, it would still act realistically.

I'm not part of the FSAE competition, just an engineer with general interest in automotive side.

I have been doing some studying about suspension geometries, and some time ago I came along a interesting thread at fsae.com forums. In that thread, a user named "Z" explains rather simple way of calculating suspensions jacking forces (aka. anti-geometries) using classical mechanics. And if I'm not mistaken, LFS currently does not model suspension jacking forces?

http://www.fsae.com/forums/showthread.php?4063-Jacking-force and read the posts by username Z.

I'll try to give brief description of those calculations. He uses the suspension linkages neutral-lines (n-lines), and because of that, you only need the angle of the lateral and longitudal lines, to be able to calculate the lateral jacking (anti-roll), and the longitudal jacking (anti-dive and anti-squat).

What are the n-lines and how do you find them? N-lines are straight lines in space, that have no motion along them. If you have force directed directly along a n-line, there will be no movement in the linkage. If there is angle between the force and the n-line, the force will cause the linkage to move.

For example, in front view of double-wishbone suspension, the line you draw from the tyres contact patch to the linkages instantaneous screw axis, ISA (more commonly known in vehicle dynamics as a instant centre (IC), but that term is wrong), is an N-line (also the wishbones themselves are n-lines, no movement along them). If you know the angle of the n-line wrt to the cars body, and the horizontal force of the tyre, you can use vector algebra parallelogram to calculate the vertical component. It is that vertical component (wrt car body) that causes the jacking. The jacking tells us how the weight transfer is distributed between the suspension linkages and the springs. If there is no jacking, all of the weight transfer is taken by the springs. If there is upwards jacking, some of the weight transfer is carried by the suspension linkages and some by the springs. And because the springs don't have to carry so much load, they are less compressed, so less body roll.

For example with outside tyre, if the N-line rises towards the center of the car, it causes upward jacking, and if the n-line goes downwards, the jacking pushes the car down when the tyres experience lateral force.
And for the inside tyre, if the n-line rises towards the car centerline, it causes downwards jacking. In suspension front view, with upwards sloping (towards car center) n-lines (the so-called roll center is above ground. Roll-center is not very good term) outside suspension linkage pushes the body upwards, and inside linkage pulls the body down, so the body does not roll so much (anti-roll). If the n-lines are horizontal wrt car body, tyres contact patch force won't cause jacking and all of the weight transfer is taken by the springs. In sideview the same principle works, if the front suspension n-line rises towards the rear, when the front tyres experience rearward longitudal loading, (from braking, or from toe-in or toe-out angles) the suspension linkage tries to lift the front end up (aka. anti-dive). For the rear axle, if the line rises towards the front, during braking it tries to pull the rear down, but during acceleration it tries to lift the rear up.

So in short:
-You only need the angle of the lateral and longitudal n-lines for the wheel to be able to calculate its jacking. And you can calculate it separately for each wheel, so you get realistic jacking behaviour.
-During suspension movements the angle of the n-line wrt car body can change (double wishbone suspension, McPhersons etc.), so the jacking force can also be different at different points of suspension travel. It's rather easy to calculate how the n-line angle changes during suspension movement.
-If you know the lateral and longitudal n-line angles, you can easily see how the toe-in or toe-out constantly cause jacking because toe-angles cause lateral and longitudal forces on tyres.

In the thread http://www.fsae.com/forums/showthread.php?5786-Side-View-Suspension-geometry-interpretation/page3 post 22, Z also points to a SAE paper "Suspension Analysis with Instant Screw Axis Theory", by C. H. Suh, SAE Paper 910017. Suh uses Screw theory to describe independent suspensions uprights movement in 3d space.

Book about the Screw theory can be found from the openlibrary.org: A treatise on the theory of screws by Robert Ball. It's a rather old book, but those theories are used in modern robotics design. https://ia601409.us.archive.org/24/items/atreatiseontheo00ballgoog/atreatiseontheo00ballgoog.pdf

That screw theory is rather interesting, because if I have understood it correctly, the up-down motion of the suspension upright in 3-D space can be descibed as a rotation around a instantaneous screw axis.

Quote from MicroSpecV :

Basically like 2 syringes connected together with a tube, push one end and the other end rises/falls. Ain't it?

Yeah, basically like 2 syringes. And if you add third syringe with spring behind it to the tube, you get some springing if both original syringes push in at the same time.

Racon:

I'm not saying that we should copy the exact suspension of 2cv. I'm trying to say, that the basic idea behind the 2cv suspension is good (interconnected springs), and that it's shame that modern manufacturers have forgotten about it, or think that it could only be made with overly complicated hydraulic or electronic systems. Even those hydraulic systems could be made in very cheap and simple way, with no pumps and water+glycol mix. British leyland used this very simple hydraulic version.

And here are some videos of 2cv racing, rather nice cornering with the 125mm wide tyres
https://www.youtube.com/watch?v=BA0nChmkeRI
https://www.youtube.com/watch?v=hAocmae9vyE

Actually interconnected suspensions are not a new thing at all. First mass produced vehicle to use interconnected suspension (mechanical) was developed in the 1930's, but was released after ww2. That vehicle was Citroen 2CV. I think I should point out, that the reason the 2CV has rather large body movements, is that Citroen used very soft springs because of the very bad roads in france.

I think it's a shame that modern automobile manufacturers have lost the knowledge of interconnected suspensions. Modern cars use 4 corner spring and 1 or 2 antirollbars, so 5 or 6 springs in total. And racing vehicles have started to use those so-called third springs in front and rear, so 7 or 8 springs in total. What's slightly funny, is that with simple mechanical interconnected suspension, you would only need 3 springs in total to control all suspension modes.

Even the 2CV suspension could be made simpler. Simplest interconnected suspension could be made with 3 z-springs. A z-spring is spring that resists wheel movement in same direction, but wheels are free to move in opposite directions. For example centrally pivoted leaf spring, or torsion bar in z-shape (antirollbars are u-springs, springs that resist movement in opposite direction)

You would need:
-1 longitudal z-spring between left side front and rear wheels (for example simple torsion bar)
-1 longitudal z-spring between right side front and rear wheels.
-1 lateral z-spring between right and left wheel (can be mounted to front or rear axle, or you can have lateral z-springs at both axles. And by making this spring adjustable, you could have simple self-leveling suspension.)

With that suspension:
-roll is controlled by the longitudal z-springs
-pitching is controlled by the lateral z-spring
-heave is controller by the longitudal and lateral z-springs
-twist is completely soft, and this is very good thing, because with twist soft suspension, your lateral load transfer distribution between front and rear wheel is determined with the geometry of the z-bar (=how much leverage front and rear wheel have to the spring) and it can be anything between 100%/0% to 0%/100%, so you can freely adjust the handling balance. With modern stiffly sprung cars, the handling balance is pretty much dictated by the twist in the road.
-travelling comfort would be much better with interconnected suspension than with conventionally sprung.

It's not very complicated to even have a mechanical suspension with fully separated suspension modes with separate springs for pitch, roll and heave.

Pitch: Wheels at front and rear move in opposite directions (braking, acceleration)
Roll: Wheel at left side move in opposite direction than right side wheels
Heave: All wheels move in same direction
Twist: Wheels in opposite corners move in same direction.

Good way to understand the basic principles behind interconnected suspensions, is to build a model using legos.

Interesting feature of interconnected suspensions is that even if you installed extremely stiff springs, so that you would have no roll/pitch/heave, your tyres would still be able to follow the undulations of the terrain. One vehicle type with need for maximum traction, has extremely stiff suspension (=no suspension) and has soft twist mode, is farm tractor. Rear axle solidly mounted to the frame, and front axle with central hinge (front axle works as a very stiff lateral z-spring).