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Mark Ortiz Automotive is a chassis consulting service primarily serving oval track and road racers. This newsletter is a free service intended to benefit racers and enthusiasts by offering useful insights into chassis engineering and answers to questions. Readers may mail questions to: 155 Wankel Dr., Kannapolis, NC 28083-8200; submit questions by phone at 704-933-8876; or submit questions by e-mail to: firstname.lastname@example.org. Readers are invited to subscribe to this newsletter by e-mail. Just e-mail me and request to be added to the list.
THE ANTI CONTROVERSY
A number of people have asked me to write a response to an article in the March 2009 Racecar Engineering by Danny Nowlan, in which the author makes some assertions that challenge conventional thinking (if there is such a thing) about anti-squat and other geometric pitch and roll resistance phenomena, and contradict some of what I say in my "Minding Your Anti" video. I will try to do that this month.
Let me begin by emphasizing that I intend no disparagement to Mr. Nowlan's integrity, intelligence, or intentions, nor to those of any of my colleagues in the field of vehicle dynamics, even if they have reached different conclusions than I have about how things work. One might suppose that vehicle dynamics, being merely Newtonian physics, would not be controversial or hard to understand for educated people in the twenty-first century. But any one who has delved into the field will have noticed that there is a lot of disagreement, and sorting through all the contradictions is anything but easy. In this situation, surely anybody who makes an honest effort to understand the laws of nature, and to deliver honest goods and services to the public, deserves the benefit of the doubt regarding their intentions.
I should also mention that Mr. Nowlan explicitly invited responses in his article, from any who might disagree with what he was saying.
Some readers will not have seen Mr. Nowlan's article. I will summarize his assertions, as faithfully as I can, in my own words:
1. We must not be slaves to orthodoxy. We must be willing to think things through from first principles, and entertain any conclusions that may result, even if they contradict what accepted authorities say.
2. The car has only one center of mass. It is not a string of separate masses, and should not be analyzed as such.
3. All dynamic effects should be analyzed as forces and moments about that single center of mass.
4. The laws of physics act similarly in all directions. Therefore, we should be able to use fundamentally similar methods to analyze front-view and side-view suspension geometry.
5. In front-view geometry, the point where the force line intercepts the c.g. plane is the roll center.
6. Similarly, in side view, the point where the side-view force line intercepts the c.g. plane may be taken as a pitch center.
7. All forces exerted by the tire upon the road, and by the road upon the tire, act at the ground plane, not at the wheel center.
8. Any time a tire is exerting a longitudinal force, it is necessarily transmitting a torque.
9. Therefore, there is no dynamic difference between systems with inboard and outboard brakes, or between systems with sprung and unsprung differentials.
10. Accepted theory asserts that with inboard brakes or sprung diffs, forces are applied to the car at wheel center height.
11. Accepted theory thus predicts ample longitudinal anti for such systems, but data traces from actual cars contradict this. Suspension displacements are very nearly what we would expect from pitch moments about the ground, reacted entirely by the car's springs, implying little or no anti, or a low pitch center.
Taking these more or less in order, I agree with #1. Willingness to re-examine and question accepted theory is the difference between science and dogma, and is an important foundational principle of human liberty in general.
#2 is correct, almost. That is, the car is a collection of masses that are rigidly attached to each other, and can therefore be considered a single body – except that when the car has a suspension system, some of the attachments within it are not rigid, and some components can move with respect to others, albeit in a (hopefully) limited and carefully-controlled manner. For some purposes, rigorous analysis requires us to treat these masses separately. Moreover, the masses are separate to different degrees in different modes. The nuances of this will make a good future article or newsletter.
However, we can examine the principles of anti effects using an assumption that all of the car's mass is sprung, and the masses and inertias of unsprung components are zero. Real cars aren't like that of course, but imagining so greatly simplifies the discussion and the associated equations.
Since we will be presenting equations, it is time for some definitions.
First, the vehicle axis system:
Per SAE convention, x is longitudinal, y is transverse, and z is normal or vertical. Unless otherwise stated, sign conventions are: x positive forward; y positive rightward; z positive downward. Unless otherwise stated, vehicle origin is per SAE aerodynamic axis system: on the ground plane, midway along the wheelbase, and centered with respect to the front and rear tracks. However, for particular purposes, we may use local origins and sign conventions may deviate, so it is necessary to pay attention to context. Even then, x will always be longitudinal, y transverse, and z vertical, in some sense that is appropriate to the context.
SAE vehicle axis convention takes as origin a point on the roll axis, directly below the sprung mass center of mass or c.g. I don't care for that, because the points in question move and are subject to uncertainties. Actually, there is no origin that is completely immune to some uncertainties and relative movements, but for our purposes here, we will use the SAE aerodynamic convention.
x, y, and z may denote coordinates in this axis system, or linear displacements in this axis system.
Next, angular quantities about the axes:
φ (phi) is roll, or angular movement about an x axis, conventionally positive rightward, or clockwise as seen looking forward.
Mx or Mφ is a moment in roll, or about an x axis.
θ (theta) is pitch, or angular movement about a y axis, conventionally positive rearward, or clockwise when looking from left to right.
My or Mθ is a moment in pitch, or about a y axis.
ψ (psi) is yaw, or angular movement about a z axis, conventionally positive rightward, or clockwise when looking down.
Mz or Mψ is a moment in yaw, or about a z axis.
L is center spacing between two wheels of a pair under consideration.
Lx is wheelbase.
Ly is track.
rt is tire loaded radius. For simplicity, we will ignore distinctions between loaded and effective tire radius.
F is force. F with an x, y, or z subscript is force in an x, y, or z direction.
m is mass.
a is acceleration.
F = ma. When m is expressed in pounds, a should be in g's.
H is height.
Hcg is height of the center of mass or center of gravity (c.g.).
Hrc is roll center height.
Hpc is pitch center height.
dx/dz is the instantaneous rate of change of a point's x coordinate with respect to change in its z coordinate: the first derivative of x displacement with respect to z displacement.
dy/dz, similarly, is the first derivative of y displacement with respect to z displacement.
dx/dz and dy/dz are also the instantaneous slopes or inclinations, from vertical, of the contact patch center's path of motion in side and front view respectively.
A force line is a notional line of action for the vector sum of an x or y ground-plane force at a tire contact patch and the induced z-direction support force within the suspension system that results from angularities in the suspension linkage. In front-view geometry, the force line is the line from the contact patch center to the front-view instant center. The front-view force line is an instantaneous perpendicular to the contact patch center's path of motion in front view, and has a slope or dz/dy, relative to ground plane, equal to the contact patch center's dy/dz. The side view force line is an analogous construction in side view, whose slope is the contact patch center's dx/dz. There are some subtleties to assigning that dx/dz, which we will address.
The front-view and side-view force lines define a plane, which we may call a force plane.
Returning to Mr. Nowlan's assertions, #3 isn't necessarily correct. In particular, it is incorrect to suppose that forces at the contact patches act along the force lines or force planes of the suspension geometry, and therefore exert moments about the center of mass according to their vertical or perpendicular distance from the center of mass. This misconception stems from a misunderstanding of what the force lines and planes represent.
The car is not a body floating in space, acted upon by angular forces in the force planes. Nor is it an airplane with four wings or control surfaces each exerting a lift and a drag force. It is a body with a center of mass above the ground plane, supported through compliant suspension at four points in the ground plane, accelerated in the x and y directions by forces in the ground plane. These forces do not act on the car as a whole along the force planes; they act on the car as a whole along the ground plane. The ground-plane forces induce support forces within the car, in the suspension systems, according to the slopes of the force planes. These induced forces create moments within the vehicle
that may oppose or exaggerate the suspension displacements in roll and pitch, and can alter the distribution of tire normal forces.
This may seem merely a rhetorical distinction, but it's not. In particular, the anti-roll and anti-pitch moments do not depend on the proximity of the force plane to the center of mass. Rather, they depend only on the spacing of the wheels, the magnitude of the ground-plane forces, and the slopes of the force planes. The anti-roll and anti-pitch moments do not depend in any direct way on where the center of mass is.
#4 is correct. The laws of physics do act similarly in all directions, and we should be able to apply fundamentally similar methodology for both side-view (pitch-related) and front-view (roll-related) phenomena. However, #5 is incorrect, and consequently so is #6. The roll center is not where a single wheel's force line intercepts the c.g. plane. Correspondingly, applying similar thinking to the side view is likewise incorrect.
Indeed, we cannot meaningfully speak of a roll center or pitch center for a single wheel, even in simple two-dimensional modeling. Even in 2D, we need two points of support to resist an overturning moment, and it is the combined effects at these two support systems that create any moment. Likewise, we cannot meaningfully speak of right-side or left-side roll, or front-end or rear-end pitch.
What is the roll center, then? It is a notional coupling point for transmission of y forces between the notional front or rear axle and the sprung mass. Its height is the height at which the centrifugal inertia force for that end of the car would generate a roll moment exactly equal to the anti-roll moment induced by the linkage geometry, so that no roll would result.
Importantly, the roll center is not any sort of coupling point for z forces. It is analogous to a horizontal link, or a roller in a vertical slot, not a pin in a hole. When the car is in motion, a sustained z force does not induce a geometric roll moment. Engineers were confused about this for years, partly because when a car is tested sitting still on a shop floor, with no slip plates under the tires, asymmetries in its anti-roll actually will produce roll moments in response to z forces. But when the wheels are free to track in and out as they roll along, that effect goes away. This is why we roll a car before we scale it.
The equation for the roll moment Mx or Mφ produced by a total wheel pair y force Fy acting at a height H is:
Mx = FyH (1)
The equation for the anti-roll moment induced in the suspension by right and left wheel ground-plane y forces FyR and FyL is:
Mx = (FyR(dy/dz)R – FyL(dy/dz)L)Ly/2 (2)
Or, in English, the geometric anti-roll moment is equal to the difference of the two induced support forces, times half the track. (The sum of the two support forces is the net jacking force for the wheel pair.) Stated a bit differently, the moment in roll for the wheel pair induced by the linkage geometry has to equal the difference in the support forces induced, reacted on the half-track. If our modeling gives us some other value for this moment, there has to be an error in our modeling.
Note also that this equation includes no term representing c.g. location.
Combining equations (1) and (2), for a condition where H is the height of the roll center, Hrc, and the roll moment and the geometric anti-roll moment are equal in magnitude:
FyHrc = (FyR(dy/dz)R – FL(dy/dz)L)Ly/2 (3a)
Solving for Hrc:
Hrc = ((FyR/Fy)(dy/dz)R – (FyL/Fy)(dy/dz)L)Ly/2 (3b)
This can also be written:
Hrc = (Ly(FyR/Fy)(dy/dz)R – Ly(FyL/Fy)(dy/dz)L)/2 (3c)
Or, in English, the portion of the total y force at the right wheel, times the track, times the right wheel's force line slope, minus the portion of total y force at the left wheel, times the track, times the left wheel's force line slope, all divided by two, is the roll center height.
We should note that when both wheels have anti-roll, i.e. when both force lines slope upward toward the center of the car, those two force line slopes are mathematically of opposite sign, so the difference of the two products in the large parentheses ends up with a larger absolute value than either product by itself: the two products either add, because we're subtracting a negative, or add negatively, because we're subtracting a positive from a negative.
Doing the same thing graphically, we construct what I call a resolution line. This is a vertical line in our front-view drawing, whose y position relates to the distribution of y force between the right and left tires. If, for example, the right tire is estimated to make 70% of the cornering force for the pair, the resolution line is 70% of the track from that wheel (30% of the track from the left wheel). We
then find the two points where the front-view force lines intercept this resolution line, average those heights (add them and divide by two), and that's the roll center height.
My video includes illustrations.
How does this compare with Bill Mitchell's force application point method? He draws a vertical line at the c.g., finds the force line intercepts of that, and does a weighted average of those, with the weighting based on the FyR/FyL distribution.
Turns out that the FAP method gives the same answer as the resolution line method, when the c.g. is midway between the wheels. It also gives the same answer when the force lines have equal and opposite slope, i.e. when they cross in the middle of the car, even if the c.g. is offset. When the c.g. is laterally offset, and the suspension is also markedly asymmetrical, it creates an error. If the c.g.
offset is small, the error is too small to measure experimentally on a kinematics and compliance rig. It would become a concern in a car that has a large lateral c.g. offset, and markedly asymmetrical
independent suspension – for example, perhaps a Supermodified with independent suspension, in a rolled condition. (Such cars once existed, but Supermodifieds are now required to have beam axles at both ends.)
The oldest method of assigning a roll center is to take the front-view force line intersection as the roll center. The force line intersection method can also give a correct answer, but the only time it necessarily does is when the force line slopes are equal and opposite. The more asymmetry we introduce in the geometry, the more wildly anomalous this model tends to become. The most obvious anomalies occur when one wheel has slight anti-roll and the other has slight pro-roll, so that the force line intersection lies a long way outside the vehicle. The intersection can then be far above or below the ground, despite the fact that the suspension clearly generates little net pro-roll or anti-roll moment. And if we further suppose that z forces generate moments about this point, the modeling anomalies can get really outrageous. Finally, in some cases the force lines are parallel, and they have no intersection.
Since the roll center does not transmit z forces, it doesn't matter what its lateral location or y coordinate is. I call it undefined. If you like to think of it as being under the c.g., that's fine. If you like to think of it as midway between the wheels, that's fine. Just don't think of it as something that can transmit force vertically.
Regarding #7 through #11, it is true that the car and the road can only apply force to each other where they touch, i.e. at the contact patches. However, this does not mean that conventional thinking on anti-squat is incorrect, although it may be that some accepted presentations could be clearer.
It is also true that if there is an x force at the contact patch, there is a torque at the wheel, with the exception of the case where the x force is induced by the tire running at a slip angle when cornering.
However, the torque may or may not act through the suspension linkage, and accordingly may or may not contribute to the Mθ in the suspension.
Let's imagine a free-body diagram for the wheel and stub axle in an independent rear suspension, in side view, when the wheel is propelling the car. The tire exerts a rearward force upon the pavement, and the pavement exerts an equal and opposite forward force Fx upon the tire, which propels the car. The wheel and axle assembly exerts an equal forward force on the rest of the car (remember we are imagining that the wheel assembly doesn't have any mass of its own). The remainder of the car, acting through the linkage and upright and bearings, exerts an equal and opposite rearward inertia force upon the axle. Thus the sum of the wheel and axle assembly's forces is zero, and there is a couple, equal to the wheel's torque My, which must be reacted somehow, for the sum of the moments to also be zero.
My = Fxrt (4)
In most independent suspensions, the torque is applied through the drive shaft. The drive shaft is jointed so that it can only apply rotational force. It applies this force directly to the axle. It does not apply rotational force to the upright. The shaft sees a rearward torque at its outboard end and a forward torque at its inboard end. The stub axle at the diff, driving the axle, sees a rearward torque from the axle. This in turn reacts through the diff, the pinion bearings, and the transaxle mounts.
This torque is thus applied to the car as a whole, but the load path does not include the suspension linkage.
Imagining the free-body diagram for the upright, then, we have a forward force (thrust force) exerted at the wheel bearings, but no torque applied at the wheel axis. Both longitudinal links, or side-view
projected control arms (SVCA's), exert rearward force upon the upright and forward force upon the sprung structure. These forces are both smaller than the total Fx at the wheel center, and both in the same direction. Their magnitudes are inversely proportional to the distances from the hub or wheel center, and there are induced z forces within the suspension depending on the magnitude of the x forces and the SVCA angularities. The overall z force Fz induced is Fx times the dx/dz value at the wheel center:
Fz = Fx(dx/dz) (5)
Why not the dx/dz at the contact patch? Actually, the dx/dz at the contact patch is the same! If we constrain the wheel rotationally (lock it) at the inboard end of the drive shaft, and then move the suspension, the wheel doesn't rotate with the upright, and the point at the static contact patch center stays directly below the wheel center through the entire travel. Thus, its dx/dz is the same as that of the wheel center. This differs from a situation where torque is applied to the upright and does act through the linkage, as with an outboard brake. In that case, the dx/dz that matters is that of the contact patch center, assuming the wheel rotates with the upright. The force line then runs from the contact patch center to the instant center, and has a slope equal to the contact patch dx/dz for a wheel
that rotates with the upright. Also, the forces at the SVCA's are opposite in direction, with the force on the lower opposite to and greater than the Fx at the contact patch, and the one on the upper in the same direction as the contact patch Fx and smaller than the force on the lower: clearly a different picture, and one that will usually produce a different induced z force.
Kinematics and compliance testing consistently confirms that it matters mightily whether torque reacts through the linkage or not. The only case where it does not matter is where the upright does not rotate in side view as the suspension moves (SVCA's parallel to each other).
With independent suspension or inboard brakes, the force line has the same slope as a line from the wheel center to the side-view instant center (SVIC), not the slope of a line from the contact patch center to the SVIC. This is what some authors mean when they say that the force acts at the wheel center. They don't mean that the rearward pitch moment Mθ is equal to Fx(Hcg – rt). It's still FxHcg. So conventional theory does not predict huge anti-squat for independent suspension. On the contrary, anti-squat by conventional theory is generally very modest for independent suspension.
So, in #11, Mr. Nowlan has misunderstood the theory he is attempting to refute, which is in fact confirmed by the data trace he presents.
Taking Terry Satchell's chapter in Milliken and Milliken's Race Car Vehicle Dynamics as an example of a conventional presentation, when Fx is considered as acting at wheel center height, a
different rule for determining percent anti is used, and the result is a correct calculation when the full method is taken together.
If we insist on considering the force line as passing through the instant center, that method is correct. I prefer to say that the force line passes through the contact patch center but has the same slope as a line from wheel center to instant center. I then use the same rule for percent anti as for a system where the torque does act through the linkage. I find this a slightly clearer way of thinking about the matter, but both methods give the same answer.
At this point, we should probably review what is meant by percent anti-squat, since Mr. Nowlan seems a bit confused about that. Percent anti-squat is the percentage by which the anti effect reduces rear suspension compression, compared to what would occur with zero anti. It is not the same as percent anti-pitch.
Here is the equation for percent anti-squat Pas, per my method:
Pas = (Lx(dx/dz)/Hcg)*100% (6a)
Or, by Mr. Satchell's method:
Pas = (((rt + Lx(dx/dz)) – rt)/Hcg)*100% (6b)
It will be apparent that (6a) and (6b) are algebraically equivalent.
Graphically, what Mr. Satchell and others do is shift the force line up to the wheel center, but then determine anti-squat based on where the force line intercepts the front axle plane, not relative to the ground and the c.g. height, but relative to a point a tire radius above the ground and a point a tire radius above the c.g. In other words, they draw the same picture I do, except everything is moved up a tire radius, and the final answer is the same.
Mr. Nowlan poses an example where Lx(dx/dz) = .20Hcg (force line intercepts front axle plane at 20% of c.g. height), and the c.g. is in the middle of the wheelbase, so that the height of the force line at the c.g. plane is 10% of c.g. height. He says that the c.g. plane/force line intersection should be considered the pitch center, and also the anti-squat is 10%.
Actually, the anti-squat is 20%. But this doesn't mean the car pitches 20% less than it would with no anti. It means the rear compresses 20% less, and the front extends the same as it would with no anti-squat. Remember, there is no such thing as rear pitch. Pitch is the relative displacement of the front and rear, or alternatively, the angular displacement resulting from that. For the simplest case to analyze, where the front and rear have equal wheel rates in pitch as well as equal static weight, if the front and rear displacement with no anti is identical in magnitude, and we call that absolute displacement z, then with no anti, the relative displacement front to rear is z + z, or 2z. With the anti, the relative displacement is z + .80z, or 1.8z. That's 10% less than 2z. So, for this simplified case, a 20% squat reduction produces a 10% pitch reduction.
Note, however, that we could move the c.g. forward or back, and if we don't change the springs, we still get the same deflections in response to a given z force, and the same anti-squat. If we make just the front springs softer, the pitch increases, because the front end rises more, but the squat and anti-squat stay the same.
Therefore, we cannot say that the c.g. plane/force line intercept can be called a pitch center, nor that it necessarily tells us anything definite about what the car will do.
Mr. Nowlan does not elaborate on what he recommends when both front and rear wheels are making x forces, but definitely neither will the average of the c.g. plane/force line intercepts serve as a pitch center, nor will a weighted average (except when the c.g. is midway along the wheelbase), nor will the side-view intersection of the force lines.
Well then – is the idea of a pitch center analogous to a roll center useful or relevant at all? I think it is. And I think we can assign it using a method closely resembling our front-view approach. Of course, this will only work if the front-view approach is correct, and if we adapt it correctly.
One issue we need to address is that the term "pitch center" is already in use in another area of vehicle dynamics, with a meaning not analogous to a roll center. Ride engineers speak of pitch centers and bounce centers. The pitch center is a center of rotation about which the car oscillates in
response to sequential perturbations at the front and rear, as when going over a short speed bump. It is determined by the relationship of the front and rear static deflections or natural frequencies, and does not relate to pitch moments created by x accelerations, or opposing or exaggerating pitch moments induced within the suspension as a result of x ground-plane forces.
What would a pitch center analogous to a roll center be? It would be a notional coupling point between sprung mass and wheel pair suspension, for the total x force of a right or left wheel pair, with front/rear x force distribution close to actual for the situation being modeled. It would have a height Hpc such that the total Fx for that side of the car, applied at a height Hpc, would produce no net pitch, because the geometry would produce an equal and opposite anti-pitch moment. The equations for this would be of the same form as equations (3a), (3b), and (3c):
FxHpc = (Fxf(dx/dz)f – Fxr(dx/dz)r)Lx/2 (7a)
Hpc = ((Fxf/Fx)(dx/dz)f – (Fxr/Fy)(dx/dz)r)Lx/2 (7b)
Hpc = (Lx(Fxf/Fx)(dx/dz)f – Lx(Fxr/Fx)(dx/dz)r)/2 (7c)
Or, in English, the portion of the total x force at the front wheel, times the wheelbase, times the front wheel's force line slope, minus the portion of total x force at the rear wheel, times the wheelbase, times the rear wheel's force line slope, all divided by two, is the pitch center height.
For anti-squat in a rear-drive car, Fxr/Fx = 1, and Fxf/Fx = 0. With all-wheel drive, Fxf/Fxr depends on the front/rear thrust ratio. In braking, Fxf/Fxr depends on the overall front/rear retardation force ratio.
With sprung diffs or inboard brakes, dx/dz is the value for a wheel that does not rotate with the upright. For outboard brakes, live axles, or odd cases such as Humvees and VW Transporters where the upright contains drop gears, dx/dz is the value for a wheel that does rotate with the upright or axle.
Doing the work graphically, we construct a vertical resolution line, positioned according to the Fxf/Fxr relationship. For propulsion with rear drive, the resolution line is at the front axle. For braking, it is aft of the front axle by a distance equal to the front overall retardation percentage times the wheelbase.
Force lines are as described above: from the contact patch to the SVIC where torque does react through the linkage, and where torque does not react through the linkage, through the contact patch center and parallel to a line through the wheel center and SVIC.
We then find the intercepts of the front and rear force lines with the resolution line, average these, and that's our pitch center height.
The left and right pitch centers define a pitch axis, analogous to a roll axis. The left and right Fx values, times the respective Hpc values, divided by the respective wheelbases, are the left and right geometric load transfers. The total Fx for both sides, times the moment arm of the c.g. about the pitch axis, divided by the total angular elastic pitch resistance rate, is the pitch angle. The right and left angular elastic anti-pitch moments, divided by the respective wheelbases, are the right and left elastic load transfers.