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.

This is certainly an interesting topic for which there is a lot of misinformation around - you've picked out one of very few resources that actually described anti-geometry accurately, although I'm not sure I agree with some of Z's other views I would love to hear Scawen's input on it.

From my understanding:

You are correct in assuming the suspension upright travel can be described as a rotation about an instantaneous axis.

If you only had a single wishbone, the instant axis would be a line going through the two connecting points on the chassis. This is hopefully obvious - if you try to move the upright, it is constrained to the chassis in those two points, and so rotated about them. Having two wishbones however means your instant axis moves with suspension travel (hence 'instantaneous'). You can consider the upper wishbone and lower wishbone as two separate planes in 3d space, and the line defined by the intersection of those two planes is the instant axis for that corner at the specific wheel position.

Forces at the contact patch then act on the chassis in the direction of a vector from contact patch to the instant axis. i.e. if the instant axis is above ground at the point where it is in line with the wheel axle, you have an anti roll jacking effect (i.e. if the front view instant centre is above ground). Similarly if the instant axis is above ground at the point where it is directing in front of the contact patch, you have anti dive or anti squat (jacking), depending on if the wheel is braked/driven. (the point is also called the side view instant centre).

As such it's not actually that difficult to model - you could create look up tables for the front and side view instant centre positions with wheel travel, and use cosines to apply the appropriate jacking force as a proportion of the contact patch forces. I would imagine this is already implemented considering how high the instant centres look on BF1 particularly, but that is a question for Scawen.

Out of interest, are you a part of the FSAE competition?

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.

Yeah he certainly knows his stuff. I must admit CM isn't really taught in the same way anymore as far as I can tell - I didn't really understand the term when I first read his posts. You are right though, the automotive industry seems to have gone towards instant centres and roll centres, which is fine as long as you realise what they actually signify - sadly very few people seem to. A perfect example of this is the design spec sheet you have to fill in as part of the competition, which includes 'roll centre migration'. Really you are only interested in the angle from contact patch to roll centre (effectively the n-line angles for left and right tyres), so why it is specified in terms of lateral and vertical displacement, as if minimising migration is the target, is a nonsense.

You are right though, I suppose the ISA location is constant relative to the wheel centre for a given wheel travel - so a lookup table could still work with tyre contact patch deflection taken into account afterwards. It wouldn't work for suspension damage though! I'd suggest there are two ways to calculate it in real time - geometric and force based. Creating planes defined by the wishbones and calculating the line through which they intersect would surely take minimal processing power. Alternatively you could construct a suspension link matrix for the 6 links that constrain the upright (direction vectors and outboard positions), and invert to calculate the loads in each link for upright equilibrium given a contact patch force/moment input. Those loads could then be separated into their vector components and summed to find the overall jacking force applied to the body. You would have to continuously compute the link matrix and its inverse though which I would imagine would be significantly more computationally expensive.

I understand the force based method is actually more accurate, as the toe link (and possibly push/pull rod?) can also provide some jacking forces. I haven't really got my head around that though.

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?

Sorry for the delay getting back to you. Yes I am a part of FSAE - just had a big deadline last week compiling a cost report/eBOM for the car, and also preparing for our car launch on Weds, so been a busy time!

From my limited experience I think you are right r.e. modern vs old VD teachings though. I didn't take the vehicle dynamics module but have a copy of the notes and it is very much a long list of equations that don't truly explain their limitations or assumptions. Seems like a good module to get an understanding of the basics and the terminology, but less good if you actually have to design anything.

Good point on the jacking effect due to toe - I remember now, it is essentially a function of your caster and kingpin inclination angles. I was probably thinking of the effects of bump steer through suspension travel on jacking.

Interesting thoughts on analysing full independent suspension. I haven't really looked into it in any detail as the added cost/mass/complexity/compliance don't really justify the move from double wishbone when you're only dealing with max +-30mm suspension travel. I'd have to have a bit more of a think on how that works and how you could calculate them if this is true.

There are a number of cars that in some way create a 'virtual' suspension. As far as I know, it's generally done on FWD cars to give zero scrub radius (I presume what you mean when you say the effect of kingpin inclination) and so minimal torque steer. I'm not sure the Citroen DS link you gave is an example of that though - that appears to just be a standard suspension with the two outboard joints positioned to give zero kingpin inclination and zero scrub radius?

On your last paragraph, you are absolutely right about tie rod orientation needing to point towards ISA for zero bump steer. The difference in length of the tie rod vs the swing arm will also have an effect on the linearity of the bump steer. You are right about the push/pull rod too, I think I was just getting confusing - I haven't really looked at this stuff since last year when I was working on the geometry!

By interconnected suspension are you talking about his longitudinal z bars? They are an interesting concept, although again I'm not convinced of it's benefits in FSAE due to the limited suspension travel and generally smooth tracks. His designs also don't take into account the need to package the bars around underfloor aero and carbon fibre monocoques - although I'm sure he still believed both are unnecessary

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.