how to design a bell crank for push rod suspension

Scott,

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How undesirable do you find ARB's?
...Can you expand on your position on the subject?

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If less than one-tenth of my posts here are on this general subject (in truth, probably many more), then I still have well into triple figures of posts ranting about it! :)

So, very briefly, of the many ways to interconnect wheels with springs, the "ARB" (or "lateral-U-bar", which interconnects "end-pairs" of wheels) is one of the worst. Yes, it does stiffen the Roll-mode, which can be beneficial for some types of car (sometimes, but not always), but it also EQUALLY stiffens the Twist-(aka-Warp)-mode, which is generally VERY BAD for all cars.

A much bettter choice of wheel-pair interconnection is the "longitudinal-Z-bar", which interconnects "side-pairs" of wheels. (Note: "Z" is the shape of the bar in plan-view, and not any relation to me. :) "U-bars" resist opposite movements at their ends (eg. up+down), and "Z-bars" resist same movements (eg. up+up, or down+down).)

"Side-pair-Z-bars" are good because they stiffen Heave and Roll, while NOT affecting Pitch and Twist. They also make it easy to adjust Elastic-Roll-Moment-Distribution (~ LLTD) to allow easy adjustment of US/OS. And packaging complexity is similar to ARBs. In principle, a complete (and very good!) suspension for a car can be made from two side-pair-Z-bars, and a third end-pair(=lateral)-Z-bar, typically at the heavier end of the car. No more is needed!

The only real downside of side-pair-Z-bars is that almost no one in the entire auto-world knows about them! Oh, ... and if, say, someone in F1 stumbles onto them, and they start winning races because of them, then the organisers start jumping up and down, shouting "It's FRIC-ing cheating!!!", and they immediately ban them as being against the spirit of motorsport. ("FRIC" was a hydraulic Front-Rear-Inter-Connect ... Z-bar.)

Aaaarghhhh, ... progress!!!
~o0o~

Getting back to FSAE, it is desirable for the car to have the lowest possible CG-Height, to allow the narrowest possible Track-Width for better slalom speed. Bumps are almost non-existent on the tracks, but some damped suspension movement is beneficial to suppress "bouncing on the tyres', which can start by "stick-slip" of the tyres during hard cornering.

So, what sort of "springing" to use?

Since the major masses are distributed mainly along the centreline of the car (eg. IA and pedals at front-centre, driver's bum in middle-centre, engine at rear-centre), it follows that to have a low CG, this car-centreline should run at a minimum ride-height, and it should NOT be allowed to move up-down much. So it should be STIFFLY SPRUNG.

So end-pair(=lateral)-Z-bars are well suited. These, of course, are also known as "third-springs" (actually seventh and eighth), and work well at controlling changing aero loads. More below in reply to Claude, but rising-rate is GOOD here!

That only leaves the Roll-mode to control, since Twist-mode can be left soft. So side-pair-Z-bars can be added. Or, for easier packaging, simple, softish, corner-springs can be used (ie. they only connect to one wheel, so simplest type of spring).

Note that with appropriate suspension kinematics there is NO disadvantage (on the smooth FSAE tracks) of having a soft Roll-mode. Here "good kinematics" are those that give ~100% "camber recovery". So, beam-axles, or independent suspensions with Front-View-Virtual-Swing-Arm = ~half-Track-Width. With these the car can have low CG and also considerable Roll-Angles, but nothing touches the ground (since main masses along car-centreline and any undertray can be "unsprung"), and tyres maintain high grip because always close to zero camber.

Note that for very bumpy roads the preferred kinematic behaviour is ZERO camber recovery, mainly because of the gyroscopic forces involved. But that ain't FSAE!
~o0o~

More coming next post.

I haven't had time to read your links, but may add more later after checking them.

BTW, you can "Search" this Forum for my ranting on "Z-bars", starting back in 2005 (there was a sketch in one of the posts back then).

Z

Claude,

Posts are arriving like buses. Nothing for ages, then all at once! :)
~o0o~

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1. ... main reason why variable motion ratio are used is getting the most of aeromaps.

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As I touched on in above post, I see rising-rates as being GOOD for the so-called "third-springs". These, in effect, control the ride-height of the centreline of the car at its front and rear. They can thus be used beneficially to change the "attitude" of the car (ie. rake, or F&R heights) at different speeds. For example, large rake at low speed for better downforce, but less rake at high speed for less drag and less axle-busting DF, and so on. Similary, such rising-rates are good on motorbikes, where these "centreline" springs have no influence on cornering.

But the problem with rising-rates, especially the agressively rising-rates found on motorbikes, is when they are used on the corner-springs. Here, whenever the car is cornering, there is a "Lateral Load Transfer" OFF the inside-wheels, and ONTO the outside wheels. The changing spring-rates then make the inside of the car-body necessarily LIFT MUCH MORE than the outside falls. So the CG necessarily LIFTS, and it lifts quite a lot if the rising-rates are similar to those on motorbikes.

Note that a high CG is NO problem when travelling in a straight line. Even during longitudinal accel/braking it is not too bad, because wheelbases are typically much longer than track-widths. Really, the only important time to have a low CG is when cornering. So rising-rate corner-springs = BAD.

Note also that a common solution to the problem of the inside of the car lifting too much during cornering is to "droop limit" the corner-springs. Look at the force-deflection curve of such springing, and see that it gives an agressively FALLING-RATE! The springs are very stiff in extension, but softer in compression. So the car "jacks down" in corners, which is good because it lowers CG height.
~o0o~

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2. Simplified problem:
... driver complains about oversteer.
... is the car too lazy or too nervous?

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Well, several problems there, so many possible solutions.

For the "too lazy or too nervous" problem, I would start (and possibly end) with toe-adjustments. Especially at the rear. Toe-in for stability, toe-out for agility. This is something that can, and should, be done on any and all cars. Even cars with NO suspension can have their toe-angles adjusted. For high-speed racing the next step is to look at aero-stability, mainly by having aero-DF behind CG. This is not so important in FSAE, because it is very low-speed racing!

For the "oversteer" problem, there are a whole host of solutions, starting with changes to tyre-sizes, pressures, +++. The very last thing that might need adjusting is LLTD. As noted by Matt earlier (U of Cincinnati), this can be done very easily with some coil-spacers stuffed into the corner-springs. And as noted by JT A. directly above, in FSAE conditions ... who cares about "optimum" pitch and heave frequencies!?

FSAE teams should focus on building a car that:
1. Can RELIABLY drive 30 kms, at an average speed of ~50 kph.
2. Then, chase less mass, and more aero-downforce.
3. Then, chase less cost (roughly equal to quicker build), and less fuel-consumption.

Fine details such as optimum spring-rates for optimum handling/aero-platform-control are a long way down the list.
~o0o~

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3 ... a motion ratio of 10?

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Just using round numbers. (I also had D = 0.5 m of upward wheel travel, which was only half-way up the parabolic bump!)

Also, MR = 10 follows the line of thinking that if a bit more is a bit better, than much more must be much better! :)
~o0o~

Final BTW for now.

Students wanting to learn more about "energetic" approaches to solving Mechanics problems can google "Lagrangian" (1700s), "Hamiltonian" (1800s), and Principles of "Virtual Work", or "Least Action".

However, IMO these are NOT very useful for solving FSAE type problems because they do not like friction, which dissipates energy, nor the irregular input of energy, such as a driver getting on and off the throttle. Better is to use Newton's Laws directly. Very easy these days, because computers do all the boring work for you.

(Hint: Simply time-step NII, "F causes P-dot". Or, spelling it out in detail,
DO
dP = F * dT
LOOP.)

Z

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Originally Posted by Z

Scott,

If less than one-tenth of my posts here are on this general subject (in truth, probably many more), then I still have well into triple figures of posts ranting about it! :)

BTW, you can "Search" this Forum for my ranting on "Z-bars", starting back in 2005 (there was a sketch in one of the posts back then).

Z

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Hi Z,

Well aware, versed, and very appreciative, of your writings on the board as well as RCE. But if in all of that you ever restricted yourself to just the spring/bar tradeoff ala conventional production cars, I must have missed it because I'd surely have remembered.

Please do, if you have the time, read thru those two threads I linked - there is mathematical treatment of the subject as well as my comic book antics.

Thanks,

Scott

Scott,

Thanks for the entertaining posts on ARB-rates. Also good to see a forum intended, I guess, for amateur rev-heads, that has more in-depth calculations than most of this forum intended for (wannabe?) engineering students.

I agree there is not much good advice out there on how to calculate such spring-rates. This boils down to modern society's almost total abandonment of DEFINITIONS. Poor Euclid tried so hard to show how it should be done, but now he is totally ignored. Anyway, once you give a clear definition of what you want to know, say, an ARB's contribution to Roll-rate, then the right "equation" is there in front of you.

Also amusing are those units. Ahh..., I keep thinking of the Mars Climate Orbiter. A billion dollar rocketship that was going quite well, made it all the way to Mars, and then ... came in a little low and got burned to a crisp! The main contractor, Lockheed Martin, used your good-ol'-fashioned imperial units, while NASA did its figuring in those new-fangled metric thingies. Poor confused little rocketship never had a chance. :)

Getting back to things like Roll/Pitch/Twist-modes, even with metric units there is the problem of angles. Degrees? Radians? Other? The French did manage to squeeze 100 "gradians" into a right-angle. My solution is to abandon angles, and measure the suspension modes with LINEAR dimensions only.

As briefly as I can:

Single-Wheel-Mode - Defined as displacement, Dn, vertically in Car-coordinates, with +ve = "up", of the given wheelprint "n", in units such as metres, cms, mms, or inches/feet/fathoms/cubits...

Two-Wheel Axle-Bounce-Mode - Defined as HALF THE SUM of vertical displacements, Dl and Dr, of the two wheelprints "l" and "r", again in linear units. So "Axle-Bounce = 1/2 x (Dl + Dr)". This is the displacement that a "third-spring" would feel, and would have to react against. This gives the AVERAGE vertical displacement of the two wheels.

Two-Wheel Axle-Roll-Mode - Defined as HALF THE DIFFERENCE of vertical displacements, Dl and Dr, of the two wheelprints "l" and "r", AGAIN IN LINEAR UNITS. So "Axle-Roll = 1/2 x (Dl - Dr)". Easy! This is the distance each wheel moves AWAY from the average given above, with one wheel moving up, the other down. Only tricky bit here is deciding which wheel gets the negative sign. But same problem in Single-Wheel-Mode, in deciding whether up or down is positive. You can bet that if half the world chooses one way, then the other half will go the other way!

So "Roll" is conveniently defined as a distance, just like a single-wheel displacement. In fact, it is quite straightforward to convert back and forth between Axle-Bounce&Roll displacements, and Single-Wheel-Left&Right displacements, using simple arithmetic. To convert to "roll-angle", simply throw in the half-track dimension, with Roll-Angle = ~Axle-Roll/(T/2), in radians (using "small angle approximation").

Continuing in a similar manner:
Four-Wheel Twist-Mode - Defined as "Twist = 1/4 x (+Dfl - Dfr - Drl + Drr)". In words, it is the "averaged amount" by which the FL&RR diagonal-pair go up (+ signs), and the FR&RL diagonal-pair go down (- signs). (IIRC, this gives twice as big a value as you put in one of your posts, where you only had one diagonal-pair moving.) Again, this approach gives a simple LINEAR measure of Twist (aka Warp) motion. This is convenient, because it is not obvious how such a Twist motion can be described with an angular measure. For example, would it be the difference of the angles of the two axles in end-view, or difference in angles of the two side-pairs in side-view?

Similarly, the Heave, Pitch, and Roll displacements have essentially the same equations as above, just with different + and - signs. Heave has all pluses, thus giving the "average" vertical displacement of all four wheels. Pitch and Roll have two pluses and two minuses, in the obvious way.

Not so obvious is that each Dn can have a weighting coefficient in front of it. My preference is to have all four coefficients summing to 1, with these replacing the 1/4 in front of the brackets. So the Heave-Mode for a 60%R-weight car might have coefficients of 0.3 for the rear-wheels, 0.2 for the front-wheels, thus always making Heave a direct measure of CG vertical movement wrt ground (err, to first order approximation).

The next step is to find the "spring-stiffnesses" of each of these modes (2 or 4 wheel). This really depends on what you want from such numbers, but the bottom line is that if you clearly define the quantity, then the appropriate equation should fall straight out of the definition.
~o0o~

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...the spring/bar tradeoff ala conventional production cars...

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Well, I cannot be totally opposed to ARBs, because one of my favourite cars of all time has them. But it is hardly a "conventional" car!

The front-drive Citroen DS (launched ~1955) has front and rear ARBs, of conventional U-bar shape. After that things get a little UNconventional. The DS has NO CORNER SPRINGS. It does have an "oleo-pneumatic" unit at each corner, with these being best described as very rugged, gas-pressurized dampers. The (nitrogen) gas is typically at around 50 bar (~750 psi) when the unit is at full "extension", but rises towards 200 bar (~3,000 psi) under full load (= heavily loaded car hitting big bump).

But, importantly, the L and R units at each end of the car are hydraulically connected such that each "end-pair" of these dampers acts as a lateral-Z-bar. That is, each end-pair only resists "Axle-Bounce" at their end of the car. They provide no "spring" resistance to Axle-Roll (or 4-wheel Twist) motions, although they do provide damping resistance. Taken together, the two end-pairs only control Heave and Pitch of the car. Furthermore, each end-pair senses the centreline height of the car at their end, via a mechanical link connected to the centre of their end's ARB, and a simple hydraulic valve then maintains a set ride height (...with a ~3 second delay) regardless of changing loads.

So, taken in terms of earlier discussions, the car has only four "springs". Each end of the car (F & R) has a reasonably conventional lateral-U-bar (= ARB), together with an oleo-pneumatic lateral-Z-bar, which has a smoothly rising-rate (via the increasing gas pressure).

But it is perhaps even simpler than that. The rear ARB is really quite thin and offers little Roll resistance (so 4-wheel Twist is also very soft). It seems to be there mostly to provide the height-sensing function. And with the front Track being considerably wider than rear (1.5 m vs 1.3 m), and the front ARB being a lot thicker than the rear, its seems that the front ARB provides almost all the anti-Roll for the car. (One day I should measure these numbers.) So the DS is not far from a 2F-1R three-wheeler, with only 3 springs!

But wait, there's more! In terms of conventionally accepted VD-wisdom, it gets even weirder. The front suspension is equal-length, parallel wishbones. So the nominal "RC" starts at ground level, then goes below ground during cornering (whilst shooting off to infinity!). More accurately, the outer-wheel's lateral-n-line slopes down-to-centre during cornering, and it slopes down quite a lot because of the highish roll-angles. So the front kinematics are "pro-roll" (in Mark Ortiz's terminology), and they jack the front of the car down.

Conversely, the rear suspension is pure trailing-arm, so its outer-wheel lateral-n-line slopes up-to-centre during cornering, giving progressively more "anti-roll", which also jacks the car up. Since the suspension is so soft, this gives a noticeable change in pitch-angle during hard cornering (nose down, so lowering the CG). It also moves the Total-LLTD rearwards as cornering G's increase, so presumably moving the car's handling more towards oversteer at the limit. But, in practice, the DS "corners on rails". The long wheelbase (3.125 m) and very soft suspension, especially at the rear (both in Bounce and Roll), means it is almost impossible to unstick the rear.

Waaaaay too complicated to work? Well, as noted, it only has two springs (the ARBs), plus the four, extra-heavy-duty, gas-pressurized, dampers. The suspension control-arms are also simple and strong. As for its dynamic capabilities, the early cars did win the 1959 Monte Carlo Rally, and countless other rallies like it. In later years Citroen focussed more on proving the car in the harsher, long-distance, off-road rallies. To quote the head of the competitions department at that time, Marlene Cotton,"Whenever the roads were really bad, that was when the DS was happiest...".

So, yes, I can accept ARBs, when they are done right. :)

Z

Thank You Z.

I was looking for a different heuristic, but I appreciate your response.

About our old gang on that board - Very Serious "rev-heads" laboring to understand how to get the most out of our stupid front wheel drive cars. When I first started there was a setup orthodoxy, not uncontested, but no why beyond this is what works. A group of us worked out the why very publicly and had great fun doing it. Like every community, it has run thru a life cycle and currently exhibits the barest signs of life. Kind of like this board. Students graduate and move on with life, Honda track rats move up in the world of hardware, expend the sum of their motivation, succumb to the normal mandates of life, etc. We're lucky if we get to ride the peak of a wave, but we feel it's absence after keenly.

Maybe thinking and working really are in the main driven by necessity, and relatively few of us carry a necessity within us. A love of learning joined to the opportunity to apply that learning in pursuit of pleasure and fulfillment is a magic combination. My most basic motivation is not sucking with respect to the competition and the bliss of being in the zone behind the wheel of a sorted car.

I can understand how one day it all seems trivial. Wrestle a big vehicle dynamics problem to the ground and after the satisfaction fades the next one isn't as all consuming. Win, and forget, enough races and one day you find yourself asking what it means to you now and maybe even what it ever meant. When you look around and most or even all of your old friends are gone it gets lonely and eery. And you feel your accumulated philosophy as a barrier between yourself and the new people.

Beyond numbers this is all really about souls.

:)

Scott

JTA,

I judge much more what the students have between their ears that what the car has between its tires.

A team with the lack of understanding with a car with nonadjustable dampers, direct mounted from A-arm to chassis, and no ARB's will do poorly with me.

A team with the same lack of understanding, and a car with expensive 4 way adjustable dampers money can buy, front and rear titanium blade electronically adjustable ARBs, pushrod or pullrod bellcrank actuation, and their design report says that their ride and roll frequencies are somehow "optimized" will do miserably because the ratio exploitation of resources divided by resources will be even worse.

If you come to Formula Student and you put wings you are expected to understand aerodynamics. Putting wings because it is faster (and because it is cool- yes, I heard that) won’t do the team any good.

Coming to FS without an aeromap is ridiculous. If you do aero, do aero. I expect the team to show me a 5 D aeromap of downforce, drag, aerobalance, yaw moment, pitch moment and roll moment Vs front and rear ride height, yaw angle, roll angle and steering angle and, yes, speed.
I expect that CFD simulations are made with rolling floor and rotating wheels.
I expect the team to exlain why and by how much the CFD is too optimistic compared to reality.
I expect the team to show me how they use their aeromaps to simulate and develop the car on track.
Last year at FSG a team show me their 3rd front and rear springs. I ask them why they had such device. They told me “because it helps to maintain the downforce distribution aerodynamic platform (a concept they could not define). I asked them if they had a aeromap at least downforce and downforce distribution (aerobalance) Vs front and rear ride height. They didn’t. How do you think they did with me in design?

I agree that like tires and aerodynamic, dampers are complex models that are difficult to understand, measure model, exploit. That said, there are basic simulations that can be run in the time and frequency domain to see the transmissibility and the phase shift between road input, suspended and non-suspended mass accelerations. There are basic “mechanical grip” 101 simulations that should be run before you decide your suspension stiffness and damping. They won’t be perfect but they will be useful. I expect teams to do such simulations before they even think about going on a 4 or 7 post rigs if they have that luck (more and more teams do)

If the only think a team know is: "softer ARB in rear = better rear grip, stiffer ARB in rear = worse rear grip" but can't explain why” they should stay home.

As far as the debate between the direct actuation and pull push rod it is more about the criteria that was used to make that choice than the choice itself. The existing resources are part of this criteria.

If you team is only 2 or 3 members forget it: it is not reasonable. Not at the competitive level that Formula Student is today. If you want to go in the playground with the big kids you need to stop wearing shorts.

Some whiners will tell me they do not know how to acquire such knowledge. Quick story to illustrate my reply. A few years ago during a meeting I ask my team “How do you define a good programmer?” One of them (A guy who has been at OptimumG for now 10 years) answered “How good are you with Google? Because most of the answers are there". This forum, the many, many books available on the market (books list in the sticky section of this forum), the OptimumG seminars are some of the many ways to acquire such knowledge. It is there.

Again, this is not racing; this is an engineering and project management competition. If there was a design event in Formula One it is not sure that the winner would be Mercedes or Ferrari.

Dear Z,
Thanks for the reply.
Just for clarification purposes. I did understand the Newtons Second Law approach you adopted earlier. In fact I feel your Newtons approach is definitely more explanatory and
correct in comparison to the KE method you adopted. ( We conclude that the upward force F that acted on the mass M over the whole of the distance D = 0.5 m, is given by,
F = W/D = 5/0.5 = 10 Newtons. This does not have to be true. Just the integral of Force times displacement in the interval b/w 0 and 0.5 should be 5. Telling you nothing about
the instantaneous force. Although I do realise you have done it to make life easier.)

(<- Think about why "parabolic" implies "constant acceleration".) I guess this is because y = kx^2( where y is along the bump height and x along the car's forward velocity)is the eqn of a parabola and upon differentiating it twice w.r.t time we
would end up with y(double dot) = 2k*( x*x(double dot) + x(single dot )^2). Assuming you don't accelerate or decelerate while on the bump, i.e maintain the same 'x' component of velocity, you are left with
y(double dot) = 2k*(x(single dot)^2) , where x(single dot) is a constant.

Yikes!!! That is a lot of force to accelerate a small mass. And it will certainly increase wheel loads on the uphill side of bumps, and keep the wheel hanging in the air over the downhill sides!
This I do agree. But if you were to consider two cars with different springs but same damping ratio ( i.e b1/sqrt(k1*m1) = b2/sqrt(k2*m2) , where b is the damping coefficient) won't the car with the stiffer spring come back to it's steady state load distribution
quicker than the on with softer springs. Isn't that desirable? How do you come to a reasoning on where you should draw that line b/w better response and road holding ?

Always welcome to criticisms.
Adi

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But if you were to consider two cars with different springs but same damping ratio

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Must be time for my annual(?) plug for our Olley book, "Chassis Design" SAE R-206, (C) 2002. As well as this thread, an email just came in from a retired Swiss engineer interested in some Olley history, and friend wants a copy of the book for his new hire to read...

For a simple linear damper, Olley presents an analysis that ends up with a plot of ratios and a line for equal damping of sprung and unsprung. Figure 6.25 page 353. Perhaps someone here can reproduce the plot, with their FSAE car spotted in.

Z,

".....Here, whenever the car is cornering, there is a "Lateral Load Transfer" OFF the inside-wheels, and ONTO the outside wheels. The changing spring-rates then make the inside of the car-body necessarily LIFT MUCH MORE than the outside falls. So the CG necessarily LIFTS, and it lifts quite a lot if the rising-rates are similar to those on motorbikes."

Did you calculate how much the CG would raise with rising rates?

Claude

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Originally Posted by Claude Rouelle

Did you calculate how much the CG would raise with rising rates?

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Not the answer to your question, but a related story:

I measured CG rise on a C3 Corvette cornering on a circle. This version (1968 - 1982) had high roll centers F and R (rear fixed-length half-shafts were also the "upper control arms" for the independent rear suspension). Roll was around an axis that connected (approximately) the tire contact patches on the outside pair of wheels. There was droop on the inside wheels, but negligible bump travel on the outside. CG height increase was approx 1/2 the amount of droop (rebound) on the inside suspensions.