Mass Properties and Automotive Lateral Acceleration Rev B

Rev. B SAWE Paper No. 3528 Category Number 31.0

MASS PROPERTIES AND AUTOMOTIVE LATERAL ACCELERATION

By

Brian Paul Wiegand, P.E.

For Presentation at the 70th Annual International Conference

of the Society of Allied Weight Engineers, Inc.,

Houston, TX, 14-19 May 2011

Permission to publish this paper, in full or in part, with credit to the Author and to the Society, may be obtained by request to:

Society of Allied Weight Engineers, Inc. P.O. Box 60024, Terminal Annex

Los Angeles, CA 90060

The Society is not responsible for statements or opinions in papers or discussions at its meetings. This paper meets all regulations for public

information disclosure under ITAR and EAR


TABLE OF CONTENTS

Chapter: Page:

Table of Contents ... i

Abstract ......... ii

1 Introduction..... 1

2 Tire Behavior: Lateral Force Generation ..... 3

3 Weight Transfer Along an Axle: A Two-Dimensional Model . .. 5

4 Weight Transfer Between Axles: A Three-Dimensional Model .................... 11

5 Weight Transfer Between Axles: A Sprung Model .. ..14

6 The Transient Condition ... ...... ..... 22

7 The Steady State Condition 30

8 Rollover . 35

9 Tire Behavior: Slip Angles . .40

10 Directional Stability .......... 50

11 Safety ........ 60

12 Conclusions ................ 62

References................ 71

Authors Biographical Sketch.......... 73

Appendices.. ....... 74

A Symbolism.. ......75

B Lateral Acceleration Program ... .....82

C Steering .... .. 84

D Derivation of Equation 3.5 .. 86

E Roll Stiffness Determination ..87

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ABSTRACT

There are a number of automotive performance aspects which are associated with accelerations in the lateral direction: maneuver (transient and steady state), roll-over, and directional stability. For each of these automotive performance aspects certain mass property parameters play significant roles; it is the intent of this paper to make explicit exactly how those mass property parameters affect each of those automotive performance aspects.

With regard to maneuver, the maximum lateral acceleration which can be attained in steady-state turning is an important index of performance and safety. The obtaining of high maximum lateral acceleration levels has inherent vehicle weight and center of gravity (longitudinal, lateral, and vertical) implications. However, before attaining a steady-state condition, a turning maneuver must first go through a transient phase. When the transient phase is included in the full maneuver picture, the previous list of significant vehicle mass properties parameters acquires two more members: the mass moments of inertia about the roll and yaw axes.

For modern passenger vehicles, the lateral acceleration point at which roll-over can occur is generally at a level significantly greater than the maximum lateral acceleration. That is, a modern car will tend to slide out of control long before there is a possibility of overturn. Accidents involving rollover generally occur because the vehicle was flipped by obstacles in the roadway, not because the vehicle traction was great enough to reach the critical lateral acceleration level. However, the level at which rollover could occur is still an important index of safety, and the most significant mass property for the determination of that level is the vertical center of gravity.

Lastly, there is the matter of directional stability, which has to do with the lateral tire traction force balance front-to-rear, and the front-to-rear drift angle relationship of the vehicle tires due to those forces. The lateral force/drift angle relationship is dependent upon normal load, so the most significant mass properties with regard to directional stability are the vehicle weight and static longitudinal and lateral weight distribution.

However, the static normal loads are dynamically modified in response to lateral directional disturbance forces. Such disturbances generate lateral initial inertial reactions at the vehicle c.g.; the consequent roll moment not only causes lateral changes in the normal load distribution, but also longitudinal changes due to the front-to-rear suspension roll resistance balance. Such changes readjust the initial lateral force/drift angle relationship front-to-rear, and thereby affect the lateral inertial reaction. If this reaction augments the effect of the original disturbance, then the vehicle is termed unstable or oversteering; if the reaction is such as to diminish the effect of the original disturbance, then the vehicle is termed stable or understeering. Therefore, for directional stability, the primary mass property parameters are the vehicle weight, and total weight distribution (longitudinal, lateral, and vertical).

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1 - INTRODUCTION

Maneuver, the case of an automobile undergoing directional change, is a situation of general plane motion: translation plus rotation. Initially, as a vehicle begins a directional change, the situation is as depicted in Figure 1.1:

Figure 1.1 GENERAL PLANE MOTION: AUTOMOBILE IN TURNING

Note that the radius of the turn is to be considered large enough so that simplification, by ignoring the angularity which would require resolution of the forces into X and Y components, is plausible; therefore the forces producing acceleration are to be considered essentially purely lateral in orientation.

For this case, the principle of dynamic equilibrium requires the following relationships between forces, moments, and the accelerations produced thereby:

TRANSLATIONAL: = + (EQ. 1.1) ROTATIONAL: = (EQ. 1.2) Equation 1.2 shows that the turning situation involves mass properties other than just weight and center of gravity; the yaw rotational moment of inertia (I) will have an important effect on the turn-producing forces (Ff and Fr) whenever there exists some appreciable angular acceleration (). Such angular acceleration is associated with the

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transient phase of initiation or termination of a turn, or with application of the accelerator or brakes in a turn, or with a turn of varying radius. The transient condition will be dealt with at length in Chapter 6.

Consider for the moment only the steady-state condition of constant angular velocity; it is in this steady-state condition (or as close to it as can be reasonably approximated on a skidpad) that the maximum lateral acceleration level is to be obtained. Therefore, in this limited case, the matter reduces to just a consideration of how the lateral forces are influenced by the weight and center of gravity (no I term)1:

= = + Substitute Wt/g for m, and V2/R for a:

= + The dynamic equilibrium also requires a moment balance about the CG:

= = =

Substitute this expression for Fr in the force equation and do a little manipulation:

= + = + = + = Now solve for Ff, and Fr, then substitute Wf for Wt lr/lwb, Wr for Wt

lf/lwb, and ay for V2/gR:

= (EQ. 1.3) = (EQ. 1.4)

So, for steady-state turning the lateral tire traction force(s) at each axle need only equal the weight load at each axle times the lateral acceleration in gs. However, the generation of those tire forces is a bit more complex than might first be supposed; the matter is not simply a case of applying Coulombs friction law

1 Reference [1], pp. 199-201.

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2 TIRE BEHAVIOR: LATERAL FORCE GENERATION

Automobiles produce all primary direction controlling forces at the tire/road interface. As noted, the force generation is not necessarily in accord with Coulombs Friction Law: F = N. Because of the nature of rubber pneumatic tires, the tractive force (F) and normal load (N) relation is nonlinear. Empirical studies show that for a tire the lateral traction coefficient is itself a function of the normal load as per Equation 2.12:

= (EQ. 2.1) The coefficients b and m are particular to the type of tire concerned. The b coefficient is the basic coefficient of traction and is dependent upon the type of tire material and road surface and is directly proportional to the magnitude of the contact area. The m coefficient is a measure of the decrease of contact area due to tire distortion under lateral load (and, therefore, is an inverse measure of tire structural stiffness). Such contact area decrease, due to distortion, leads to decreased lateral force generation potential from what otherwise may be expected. The nature of this distortion is depicted in Figure 2.1:

Figure 2.1 CONTACT AREA DECREASE UNDER LATERAL LOAD

2 Reference [11], page 127. Longitudinal traction force potential also decreases under increasing normal load, but in accord with a somewhat different mechanism; for more info see Reference [7], page 57.

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Combining Equation 2.1 and Coulombs law by substitution for results in the normal load/potential lateral force relationship of Equation 2.2:

= ( ) (EQ. 2.2) Graphically, this function may be depicted for a typical set of coefficient values as

shown in Figure 2.2:

Figure 2.2 NORMAL LOAD / LATERAL TRACTION RELATIONSHIP

Knowledge of this function is basic but cannot, by itself, be used to determine, even roughly, the maximum lateral acceleration potential of a vehicle because there is at least one very significant modifying factor: weight transfer3 in a turn. This matter of weight transfer will bring to the fore the role of the c.g. in determining lateral acceleration.

3 Weight transfer is a bit of a misnomer; some say a more appropriate term would be load transfer.

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3 WEIGHT TRANSFER ALONG AN AXLE: A TWO-DIMENSIONAL MODEL

The lateral force generation potential for an axle is essentially only a matter of adding the lateral force generation potentials of the tires. This should be readily discernable from the axle force Equation 3.14:

= ( ) + ( ) (EQ. 3.1)

In the static case the normal loads would be equal, Ni = No. However, it is not the static case, but that of dynamic equilibrium in a steady-state turning situation, in which we are interested. In such a case, a weight transfer moment occurs which alters the lateral force generation potential by decreasing the normal load on the tire closest to the turn center (inner tire) and increasing, by an equivalent amount, the normal load on the tire furthest from the turn center (outer tire). This situation is as depicted in Figure 3.1:

Figure 3.1 LATERAL WEIGHT TRANSFER IN TURNING

4 Reference [11], page 127.

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From the information given in this figure, expressions for the normal loads, including the effect of weight transfer for the inner and outer tires, may be determined: Equations 3.2 and 3.3, respectively:

= (EQ. 3.2) = + (EQ. 3.3)

The quantity W ay (hcg/t) is the weight transferred. Substituting Ni and No from Equations 3.2 and 3.3 into Equation 3.1 for Ni and No, respectively, produces the two-dimensional model Equation 3.4 which gives us the axle lateral traction force potential taking weight transfer into account:

= +

+ +

(EQ. 3.4)

From this equation, further equations for the maximum axle lateral acceleration limits of slide and overturn5 can be determined, Equations 3.5 and 3.6, respectively6:

= + ( ) (EQ. 3.5)

= (EQ. 3.6) 5 Reference [15], Section V.A. Reference [7], page 311. 6 For derivation of Eq. 3.5 see Appendix D. The value for the overturn acceleration is also approximately (given the simplifications inherent in the analysis) equal numerically to the coefficient of traction at overturn. Reference [12] therefore suggests that the greatest coefficient of traction anticipated to be encountered in vehicle use (the coefficient depending on tire and road surface) be used as a design criterion to establish t and/or hcg so that it would be impossible for the vehicle to rollover from traction forces (comment: this tends to occur regardless for passenger vehicles). Note, the result of Eq. 3.6 is called the rollover threshold by Reference [7].

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Examination of a plot of Equation 3.4 for typical variable values, such as Figure 3.2, allows for greater understanding of the mechanism of the lateral traction force phenomena:

Figure 3.2 LATERAL TRACTION POTENTIAL W/ WEIGHT TRANSFER, SINGLE AXLE

Note that as lateral acceleration levels increase, the lateral traction force potential decreases. This is due to the effect of weight transfer. Simultaneously, the inertial loading, or the force required to achieve ay, increases. At some lateral acceleration level, the potential and inertial forces will be equal, i.e., the function plot lines will intersect (point A). After this point, dynamic equilibrium can no longer be maintained and slide sets in (ay = 0.7 gs).

Note that slide would not have set in until a significantly higher acceleration level (ay = 0.8 gs, point B) had the phenomenon been one of lateral traction force potential without weight transfer, i.e., if the tire loadings stayed even. Also note that for these typical variable values, the lateral acceleration limit of overturn is much larger (ay = 1.4 gs, point C) than that of slide. In such a case, the slide acceleration is truly the maximum lateral acceleration possible.

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The point of intersection of the traction potential and inertial loading functions, i.e., the slide acceleration, is given us directly by means of Equation 3.5. Observe what happens if this equation is plotted for increasing total (axle) weight as in Figure 3.3:

Figure 3.3 SLIDE AND OVERTURN LATERAL ACCELERATION vs. AXLE LOAD

It would seem that not only is it beneficial to keep the tire loadings even (i.e., minimize weight transfer), but also to keep the loadings as light as possible in order to achieve the highest maximum lateral acceleration (slide) levels.

The reason for this last observation is that while increasing weight does produce increasing lateral traction potential, in accord with the non-linear Equation 3.4, the inertial loading also increases in accord with the Newtonian law F=ma. In other words, with increasing weight, inertial forces increase more rapidly than lateral force generation capacity; this is due to the decreasing traction factor b mN.

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Plotting Equations 3.4 and 3.5 while varying the center of gravity height produces Figure 3.4:

Figure 3.4 SLIDE & OVERTURN ACCELERATION vs. C.G. HEIGHT

Note that on this figure, not only does the slide lateral acceleration level decrease with increasing hcg but the overturn acceleration drops also. This is because hcg is the weight transfer and overturn moment arm.

For the variable values used, the overturn acceleration level drops so precipitously that at about an hcg value of 60 inches, the overturn acceleration curve intersects with the slide acceleration curve. For yet higher c.g. values, the overturn acceleration would be the maximum lateral acceleration limit as it would be lower in value than the slide acceleration level; this would be a most unsafe condition.

For most vehicles, and certainly any modern passenger car design, such an extreme situation need not be of concern; the slide acceleration is normally the maximum lateral acceleration limit. The purpose of Figure 3.4 is to point out the fact that, in order to obtain maximum lateral acceleration levels, the c.g. height should be kept at a minimum.

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From this consideration of the physics of a single axle, the next step towards the reality of a conventional four-wheeled vehicle configuration would be to consider the case of two axles in tandem.

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4 WEIGHT TRANSFER BETWEEN AXLES: A THREE-DIMENSIONAL MODEL

A conventional vehicle has two axles in tandem, and its maximum lateral acceleration level is the acceleration level of whichever axle reaches its slide point first. To determine the vehicle maximum lateral acceleration level, therefore, an equation for lateral force generation must be written for each axle. The portion of vehicle weight that can be assigned to each axle equation is determined from the total vehicle weight and the longitudinal center of gravity. Using the notation of Figure 1.1, the weights (axle loadings) to be apportioned to the front and rear axle are to be in accord with Equations 4.1 and 4.2:

= ( ) (EQ. 4.1) = (EQ. 4.2)

Using the same values for weight (4000 lbs. total vehicle weight, 2000 lbs. per axle initially), c.g. height, track, and tire coefficients as were used to generate all previous figures, the effect of varying the longitudinal c.g. on the maximum lateral acceleration (slide) may be seen from Figure 4.1:

Figure 4.1 SLIDE ACCELERATION vs. LONGITUDINAL WEIGHT DISTRIBUTION

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Note in Figure 4.1 that as the c.g. moves rearward from the 50/50 position, the vehicle maximum lateral acceleration decreases, as the rear axle goes into slide at lower and lower acceleration levels. For instance, at 35/65 the maximum lateral acceleration level that could be attained by the vehicle would only be 0.637 gs, the rear axle governing.

If this were the case with an actual vehicle, then the logical thing to do would be to change the rear axle parameters so as to acquire a higher rear axle slide point. Due to the limitations inherent in this paper there are various suspension changes that wont be discussed, but changing the rear axle tire behavior so that m is 0.00035 instead of 0.0004 is well within the scope of this paper, and could possibly be accomplished by a wider wheel, higher tire inflation pressure, a different tire size and/or type, etc. Any such changes that would make m equal 0.0035 would also make the 35/65 vehicle competitive with the previous 50/50 vehicle, with a maximum lateral acceleration of 0.69 gs for each.

However, if the same modifications leading to an m of 0.00035 were carried out at both axles for the 50/50 weight distribution then that vehicle configuration would corner at a maximum lateral level of 0.745 gs! The conclusion drawn from this is that, if all other things are equal, then the 50/50 weight distribution vehicle will always attain a higher lateral acceleration than a vehicle of uneven weight distribution. This is the rationale behind the commonly encountered statement that a 50/50 weight distribution is the ideal for vehicle design. Note that this is analogous to the findings for a single axle: the maximum lateral acceleration level is highest when the tire loadings are as even as possible (i.e., without weight transfer, point B on Figure 3.2).

In the light of the above, some of the results of a comparative testing reported in an August 1983 issue of Road & Track should not be surprising. The 1984 Corvette and the 1984 Porsche 944, when at driver onboard test condition weights of 3390 lb (1537.7 kg) and 2940 lb (1333.6 kg) respectively, were found to have exactly a 50/50 longitudinal weight distribution; in this condition on the skidpad these vehicles attained maximum lateral accelerations of 0.842 and 0.821 gs, respectively7.

Given such apparent adherence to a 50/50 weight distribution among sports cars, one would expect all race cars to be of a 50/50 weight distribution, but that is not the case. Group Seven (a former racing classification sanctioned by the Fdration Internationale de lAutomobile) vehicles tended to have longitudinal weight distributions of about 45/55 to 40/608. This was the result of performance considerations other than lateral acceleration being factored into the design such as braking and longitudinal acceleration. Also, the mathematical model from which the 50/50 longitudinal weight distribution ideal was obtained was a very

7 Reference [20], page 58. Presumably the skidpad was of 100 foot (30.5 m) radius. It is also assumed that the Porsche 944 was a 1984 model as it was not so stated. 8 Reference [13], pages 20-23.

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simplified model that did not include the modifying effects of steering, transients, aerodynamics, and directional stability. When all of that is factored in, then a design can stray somewhat from the ideal 50/50 weight distribution, but still not too radically if obtaining high maximum lateral acceleration levels is of any concern, at least for conventional designs.

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5 WEIGHT TRANSFER BETWEEN AXLES: A SPRUNG MODEL

The two dimensional model for weight transfer along an axle as used in Chapter 3 was highly simplified; there was no consideration of the effects resulting from the presence of a suspension. By design, a suspension allows for deflection under vertical loads, and consequently there is some deflection under longitudinal loads (dive, squat) and some deflection under lateral loads (roll). The deflection under lateral loads modifies the weight transfer results from that obtained using just the previous simple moment balance equations Eq. 3.2 and Eq. 3.3.

The exact nature of the effects resulting from the roll deflection under lateral load depends upon the type of suspension, in particular the suspension roll center (RC) height (hrc) and roll stiffness (kroll), and upon the mass property parameters of weight and center of gravity9. It is the suspension at the front of the vehicle, and the suspension at the rear, which acting in concert determines the roll response to a lateral load. To illustrate the matter, consider Figure 5.1 which depicts the roll axis and c.g. situation of a 1980 Ford Fiesta S (1.1 liter, European version):

Figure 5.1 1980 FORD FIESTA S (EURO VERSION) ROLL AXIS

The vehicle is in the 4 up condition, which is the curb weight condition of 1609.4 lb (730.0 kg), plus 4 occupants at 165.3 lb (75 kg) each, and a small amount of baggage at 28.7 lb (13.0 kg), resulting in a total vehicle weight of 2299.4 lb (1043.0 kg, GVWR is 1160

9 The effects of damping upon roll are neglected in this analysis. For information on this aspect of roll see Reference [1], page 83.

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kg) at c.g. coordinates of 41.4, 0.0, 22.74 inches (105.2, 0.0, 57.8 cm); the wheelbase is 90 in (228.6 cm), and in this condition the front/rear longitudinal weight distribution is 54/46. The total unsprung weight is determined to be 161.0 lb (73.0 kg) at c.g. coordinates of 45.0, 0.0, 9.6 inches (114.3, 0.0, 24.4 cm)10. From all this some simple weight accounting establishes the sprung weight and its c.g. coordinates:

The front suspension roll stiffness kfroll is 218.3 lb-ft/deg (296.0 Nm/deg), and the rear suspension roll stiffness krroll is 154.9 lb-ft/deg (210.0 Nm/deg), which makes for a total stiffness about the roll axis of 373.2 lb-ft/deg (506 Nm/deg)11. The front suspension roll center height hfrc is 7.28 inches (18.5 cm), and the rear suspension roll center height hrrc is 7.52 inches (19.1 cm)12, which by proportioning makes the roll axis height under the sprung mass c.g. 7.39 inches (18.8 cm). Looking at the vehicle cross-section through the sprung c.g., the situation is as depicted:

Figure 5.2 - SPRUNG MASS IN ROLL AT 0.5 gs LATERAL

10 The unsprung mass c.g. is assumed to be at mid-wheelbase longitudinally, at centerline laterally, and at the rolling radius height of the 155SR12 tires. 11 See Appendix E for roll stiffness determination methodology. 12 Reference [6], pages 191-192.

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Given all this information, the roll angle s of the sprung mass under a 0.5 g lateral acceleration ay can be determined from the fact that the roll resistance (roll angle s times the roll stiffness kroll) has to equal the roll moment (lateral force Ws ay times the roll moment arm hr) at equilibrium13:

Roll Angle x Roll Stiffness = Lateral Force x Roll Moment Arm

Into this simple equation we can plug all the necessary parameters as drawn from the previous discussion and solve for s:

s x 373.2 lb-ft/deg = 2138.4 lb x 0.5 g (23.73 in 7.39 in) / (12 in/ft)

s = 3.9 deg

However, it can be seen from Figure 5.3 that there are at least two complications that makes the roll angle determination not quite so simple. One complication is that the roll height hr decreases by an amount dz during the rolling action, which would tend to make the roll angle s less than the 3.9 degrees calculated. Another complication is that the rolling motion moves the sprung mass c.g. off centerline laterally by an amount dy, which would tend to make the roll angle s more than the 3.9 degrees14:

Figure 5.3 ROLL GEOMETRY

13 Reference [3], pages 133 and 136. 14 Ibid, page 134. The first complication is implicitly neglected, and the second complication is explicitly neglected.

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Finally, there is the further complication that the unsprung masses at the front and rear axles also make their contributions to the sprung roll moment as transmitted through their linkages (if the suspension is of an independent type, otherwise the unsprung mass moments are absorbed internally). From this fact, and the geometry of Figure 5.3, a more complex reality15 than that initially considered may be expressed16:

s kroll = Ws ay hr Cos s + Ws hr Sin s + Wusf ay rrf + Wusr ay rrr (EQ. 5.1)

Where:

s = The sprung mass roll angle, degrees.

kroll = The total vehicle roll resistance, lb-ft/deg.

Ws = The weight of the sprung mass, lb.

ay = The lateral acceleration, gs.

hr = The sprung mass roll moment arm, ft.

Wusf = The front axle unsprung mass weight, lb.

rrf = The front axle unsprung mass vertical c.g. (approx. the

rolling radius17), ft.

Wusr = The rear axle unsprung mass weight, lb.

rrr = The rear axle unsprung mass vertical c.g. (approx. the

rolling radius), ft.

Even with all this complication, the resulting value for the roll angle will still just be approximate as matters such as free surface effect of liquids, lateral shift of the unsprung mass c.g.(s), and various secondary deflections (including shift of the RCs in roll) are all still unaccounted for. Still, the roll angle value as determined by Equation 5.1 may have all the

15 The resultant roll angle determination is still approximate, if for no other reason than the fact that the effect of free surface movement of liquids has not been considered. However, in high performance vehicles sometimes fuel tank baffles, etc., have been provided to ameliorate free surface effects. 16 Reference [3], page 136. The best (perhaps only) way to solve this equation is iterative. 17 Ibid, page 134.

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accuracy that is needed for early design studies. Since this equation does not allow for an easy analytical solution a numerical solution is recommended. A simple iterative spreadsheet approach yields 4.6 degrees, which seems reasonable, but probably is a bit low considering that free surface effects, etc., were not taken into account.

Now that the roll angle s has been reasonably estimated, the weight transfer effect of roll, the normal loads Ni and No, can be determined at the front and rear axles. First, lets consider the front IFS suspension which is of the MacPherson strut type. Based on the sprung weight c.g. location, the front sprung weight Wsf is determined to be 1160.2 lb (526.3 kg). That, plus fact that the front suspension roll center height hfrc is 7.28 inches (18.5 cm), the front track is 52.52 inches (133.4 cm), the front unsprung weight Wusf is 80.5 lb (36.5 kg), and the front unsprung weight c.g. height rrf is 9.6 inches (24.4 cm), can all be depicted in the following diagram:

Figure 5.4 1980 FORD FIESTA S FRONT STRUT SUSPENSION IN ROLL

The appropriate equations for Nif and Nof, taking the sprung mass roll and the relative roll stiffnesses into account, are now:

= / (EQ. 5.2) = + / (EQ. 5.3)

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Plugging the appropriate values into Equations 5.2 and 5.3 produces the following results:

Nif = (1240.7/2) 4.6 (218.3)/(52.52/24) = 161.47 lb, or 73.24 kg

Nof = (1240.7/2) + 4.6 (218.3)/(52.52/24) = 1079.23 lb, or 489.48 kg

If Equations 3.2 and 3.5 were used, i.e. if roll stiffness was not considered, the results for Nif and Nof would have been:

Nif = (1240.7/2) 1240.7 (0.5) (23.73/52.52) = 340.06 lb, or 154.25 kg

Nof = (1240.7/2) + 1240.7 (0.5) (23.73/52.52) = 900.64 lb, or 408.52 kg

Now that the front axle has been considered, lets turn our attention to the rear axle, which is non-independent of the dead beam type. The rear sprung weight Wusr is determined to be 978.2 lb (443.7 kg), the rear suspension roll center height hrrc is 7.52 inches (19.1 cm), the rear track is 52.01 inches (132.1 cm), the rear unsprung weight Wusr is 80.5 lb (36.5 kg), and the rear unsprung weight c.g. height rrr is 9.6 inches (24.4 cm), as depicted in the following diagram:

Figure 5.5 1980 FORD FIESTA S REAR BEAM SUSPENSION IN ROLL

The appropriate equations for Nir and Nor, taking the sprung mass roll and the relative roll stiffnesses into account, are now:

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= / (EQ. 5.4) = + / (EQ. 5.5)

Plugging the appropriate values into Equations 5.4 and 5.5 produces the following results:

Nir = (1058.7/2) 4.6 (154.9)/(52.01/24) = 200.55 lb, or 90.97 kg

Nor = (1058.7/2) + 4.6 (154.9)/(52.01/24) = 858.15 lb, or 389.25 kg

If Equations 3.2 and 3.5 were used, i.e. if roll was not considered, the results for Nif and Nof would have been:

Nir = (1058.7/2) 1058.7 (0.5) (23.73/52.01) = 287.83 lb, or 130.56 kg

Nor = (1058.7/2) + 1058.7 (0.5) (23.73/52.01) = 770.87 lb, or 349.66 kg

Note that without incurring roll, the little Fiestas normal loads in a 0.5g maneuver would have been much more even front-to-rear on the heavily loaded outside wheels. The effect of the Fiestas relative roll stiffnesses, which is the result of the spring rates and moment arms, is to have much more of the roll moment resisted by the front suspension than by the rear. Consequently, when roll is taken into account the Fiestas normal loads are much more skewed toward the front outer wheel. This is by design; the suspension roll centers (roll axis slanting down towards the front) and spring rates (greater roll stiffness at the front) were carefully chosen to create this effect.

This may seem peculiar in light of the previous chapters admonition that for maximum lateral acceleration the normal loads should be kept as even all around as possible. However, the Fiesta was not a high performance vehicle; what the designers were intent upon was the obtaining of a high degree of stability for a rather prosaic little grocery getter to be driven by the semi-conscious general public. The absorption of most of the weight transfer under lateral load by the front suspension is conducive to maintaining directional stability for reasons to be dealt with in Chapter 10. Race car drivers, as opposed to the general public, are assumed to be highly skilled and alert, at least when in competition, and so the compromise point in design between stability and performance is shifted for racing vehicles toward performance18.

18 Race car design may even include a fair amount of directional instability for sharp low speed maneuvers, and a changeover to increased directional stability for high speed sweeping maneuvers; this has to do with the fastest way to get around a road race course.

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Speaking of performance in the current context of roll is a natural lead-in to the subject of roll gain. Roll gain is the steady state equilibrium amount of roll, usually in degrees, per lateral acceleration, usually in gs, as illustrated by Figure 5.619; note that the little Fiesta S has more than double the roll gain (by this authors calculation) of any of the high performance sports cars shown:

Figure 5.6 ROLL GAIN: DEGREES ROLL PER Gs LATERAL ACCELERATION

The lower the roll gain the less weight transfer due to lateral c.g. shift and the harder a car can corner without incurring adverse suspension camber angles; the result is flatter and faster cornering. This can be achieved by increasing roll stiffness, preferably through the use of anti-roll bars, but general suspension stiffening will also suffice at the expense of ride quality. However, a better means to this end may be by the reduction of the roll moment arm by roll center manipulation (good) and/or c.g. height reduction (best)20.

19 Reference [8], pages 37-38. 20 Reference [1], page 49.

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6 THE TRANSIENT CONDITION

The subject of roll gain is closely inter-related with the topic of transient response in maneuver; there are a number of aspects to the subject of transient response of a vehicle in maneuver, and roll gain is just one such aspect. As noted in Chapter 1, there is an angular acceleration associated with the transient condition at the initiation or termination of a turn, or with the application of the accelerator or brakes in a turn, or with a turn of varying radius. The yaw inertia of the vehicle I times this angular acceleration represents an inertial moment which must be equaled by moments generated by forces at the tires in order for the transition from straight ahead to turning to occur:

= The above, where Fflf > Frlr, is the case when the vehicle is initiating a maneuver as was depicted in Figure 1.1. When the vehicle comes out of the turning maneuver there will be a less intense shorter lived reversal of this situation, i.e., Fflf < Frlr. Then, as is often the case in slaloms, chicanes, or lane changing maneuvers, a turning action in the opposite direction may commence causing a situation of Fflf > Frlr once again, only now the forces will be pointing in the direction opposite from before!

With all this fluxing of forces and moments inflicted upon a damped spring-mass system it should not be surprising that there would be some oscillatory behavior observable. Figure 6.1, which is a plot of vehicle yaw velocity versus time for the transient phase at the commencement of a turn leading up to a steady state condition, illustrates such yaw oscillation behavior and its two phases, rise and decay, which sum to the total vehicle response time:

Figure 6.1 TRANSIENT YAW RESPONSE TO TURN INITIATION

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For the same specific vehicles as noted in Figure 6.1, the over plot of transient responses is illustrated by Figure 6.2:

Figure 6.2 COMPARATIVE TRANSIENT YAW RESPONSE FOR FOUR VEHICLES

Of course, along with this yaw oscillation there would be some corresponding roll oscillation21, and maybe even a little pitch oscillation. All these oscillations result in fluctuating demands upon the tire contact patches, shifts in suspension linkage, and confusing sensory inputs to the driver: oscillationis critical to the feel of the car. a high decay rate might...be described as twitchinessa vehicle that would wiggle around as the driver turned into a corner22. It is best that this transient phase be as short in duration (response time) as possible.

Transient behavior is so important to the feel and capability of a vehicle that General Motors conducted a special effort during the development of the fourth generation Corvette (C4) to acquire superior transient behavior with respect to various foreign sports cars renowned for their handling prowess. Exhaustive testing of a Ferrari 308, a Porsche 928, and a Datsun 280ZX established the performance targets that GM was determined to beat23; how well they did with respect to roll and yaw transients is illustrated in Figures 5.6, and 6.1/2, respectively.

21 Reference [7], page 319. Scale estimation of the example roll oscillation plot indicates a roll response time of 0.324 seconds for an unspecified vehicle.. 22 Reference [8], pages 38-39. 23 Ibid, page 37.

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The data for all the sports car figures was obtained by instrumented runs on a skidpad. For the roll gain measurement of the roll angle and corresponding lateral acceleration (in gs) was taken after all transient effects had died out and the vehicles were circling the pad in a steady state condition. For all transient data appropriate steering angle was quickly dialed in and held constant along with velocity until all fluctuation died out and steady state attained.

Such skidpad testing allowed for the obtaining of data useful in making engineering determinations. Less rigorous in producing useful data, but more common for vehicle comparisons with regard to transient behavior, is the slalom. Slalom consists of a course of evenly spaced traffic cones, or other such items, laid out in a straight line. Such a course doesnt allow for a vehicle settling in to a steady state condition; running the course keeps a vehicle in a series of alternating transient conditions. A vehicle is taken on a timed run though the obstacle course as quickly as possible, often for multiple times in alternating directions in order to average out any directional (wind, gradient) or driver anomalies. The resultant average time and/or speed for completion of the course is taken as indicative of how beneficent or malevolent the vehicles transient behavior is.

As with all test results care must be taken to assure apples-to-apples comparison. General Motors, Motor Trend, and Hot Rod Magazine all favor a 600 foot (182.9 m) slalom course, while MIRA and Road & Track24 favor a 700 foot (213.4 m) course. And, of course, even when the length is the same there are many other possible variances, like cone spacing. There just is no universal standard slalom course, although the course depicted in Figure 6.3 comes closer to that designation than any other.

Figure 6.3 STANDARD SLALOM COURSE

To get the transient characteristics of response time (rise + decay) and roll gain (degrees/g) for the four sports cars presented earlier GM utilized a skidpad for two reasons:


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1. Skidpad results tend to be less driver dependent. 2. It was necessary to reach a steady-state termination.

However, with proper instrumentation hard data can also be obtained from slalom course testing. Colin Campbell relates how roll steer and steering lag characteristics can be obtained from empirical tracking of steering angle and rear wheel slip angle; a plot of such tracking would look as follows in Figure 6.425. Because the front slip angles closely mimic the front steering angles, and the angles are similar on left and right sides, only the front steering angle and the consequent rear slip angle are shown for clarity:

Figure 6.4 ROLL STEER EFFECTS

As directional input is made at the steering wheel, the front wheels create steering angles with respect to the velocity vector; the exact nature of those angles being dependent on the steering geometry of the vehicle (see Appendix C). The tires, being elastic, distort as the wheels turn and run at some slip angles to the new direction vector. The slip angle

25 Reference [4], page 58. Reference [7], page 323 shows a similar plot for left side front and rear wheels (tires).

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function is proportional to, but has some small lag time to, the steering angle function26. Slip angle is a misnomer, like the term shock absorber when used to refer to a damper. The tires dont actually slip but, through a process of continual deformation at the road contact patch with rotation, give the impression of slippage; the only real slippage occurs in a small portion of the total tire/road contact area (as shown). It is for this reason Donald Bastow suggests the term drift angle in lieu of slip angle27, which is a practice which this paper will now adopt:

Figure 6.5 DRIFT ANGLE AND LATERAL FORCE

As the driver turns the steering wheel the front wheels begin to turn, forming a steering angle with the vehicle velocity and a drift angle builds up as the tire at the ground contact area distorts. Associated with this distortion is a lateral force vector Fy which begins to push the front of the vehicle away from the original velocity direction toward the new direction of the front wheels.

This incipient turning action generates a translational lateral inertial reaction through the sprung mass c.g. and a rotational reaction about the vertical axis through the c.g. It is these inertial reactions which prevent the vehicle response to steering input from being

26 A reasonable estimate for this time lag might be the time it takes for the wheels to go through a half rotation or so depending on the type of tire, inflation pressure, and other factors. 27 Reference [1], page 73.

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instantaneous; there are a number of small lag time contributions involved: play in the steering, deflection of the tires, compliance of suspension bushings, etc.; but the vehicle weight, yaw inertia, and sprung mass inertia are the biggest factors.

Note that it is Case (A) of Figure 6.4 that gives the most precise response to steering input. In Case (A) the time lag between steering input and rear drift angle is at a minimum, and the drift angle closely follows the steering angle with only slightly less magnitude (which is the essence of understeering behavior). The worst response to steering input is Case (D) which is very imprecise with high lag and a general overreaction; in this case even after the driver has reversed course and is turning right the rear end is still trying to go left! This is a case guaranteed to take out quite a few cones on the slalom!

The mass properties palliatives for adverse steering behavior such as typified by Case (D) is to reduce roll and yaw inertias (and perhaps move the vehicle c.g. forward). The sprung mass roll inertia reduction would reduce roll gain, keeping suspension linkages closer to optimum positions and improving ground contact. Reducing vehicle yaw inertia would decrease response time, both rise and decay portions, and decrease lateral force demand/fluctuation at the tires. Such reductions and proper balance would do much to ensure a minimum of drama in the transient phase from steady state straight-ahead to the steady state turning condition.

It is important at this point to take special note of the aligning moment, or the lateral force Fy time the pneumatic trail d. The effect of this torque is create an automatic reversal of any driver induced course change28; when the driver relinquishes the wheel after a turn it is the aligning moment which causes the wheel to self-center and return course to straight ahead (more or less). Fluctuating forces at the tires during the transient stage of a maneuver will not only affect the motion of the vehicle directly, but will cause fluctuations in aligning moment and consequent behavior of the steering wheel, resulting in greater difficulty for a driver to control the vehicle29.

The relationship of the rotational inertia, represented by the radius of gyration K (I = K2M), to the location of the vehicle longitudinal center of gravity as given by lf and lr, is neatly encapsulated by the quantity K2/(lf x lr) which is known as the Dynamic Index (DI) when K is the pitch radius of gyration30. In SAE J670e standard symbolism this quantity is depicted as k2/ab. The values of the yaw radius of gyration and the pitch

28 Some additional aligning moment arm may be added by that suspension characteristic known as castor; which is the angle of the kingpin to the vertical. 29 At extreme maneuver, i.e. racing conditions, the aligning moment sometimes reverses itself, causing an effect something like the reversal of controls in WW II era aircraft when inadvertently entering the transonic flight realm. This is another reason why race car drivers, like fighter pilots, are a special breed. 30 Reference [19], page 55.

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radius of gyration are highly interrelated as both derive from the common basis of the vehicle mass distribution, but Kz (yaw radius of gyration) is always greater than Ky (pitch radius of gyration) for two reasons:

1. The vehicle plan view about the yaw axis is always greater than the vehicle elevation view about the pitch axis.

2. The yaw inertia is generally the inertia of the entire vehicle mass, but the pitch inertia most commonly referred to is the inertia of just the vehicle sprung mass.

Just as the DI or Ky2/(lf x lr) is an important factor in the pitch motion of the sprung mass, the quantity Kz2/( lf x lr) (or X) is an important factor in the yaw motion of the entire vehicle as it determines an oscillation center (OC) about which the vehicle will initially tend to pivot. For the value of this important yaw motion factor there are three possibilities31:

1. Kz2/( lf x lr) = X < 1 2. Kz2/( lf x lr) = X = 1 3. Kz2/( lf x lr) = X > 1

Each of these possibilities corresponds to a certain physical reality with a particular location for the oscillation center or pivot point:

Figure 6.6 TRANSIENT EFFECT OF YAW INERTIA & LONGITUDINAL CG

31 Reference [1], pages 32-33.

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For Case (1) the yaw inertia is relatively small, equivalent to Wf (lf X0.5)2 + Wr (lr X0.5)2, and the oscillation center is at distance lr X aft of the c.g. As soon as the front wheels begin to steer the tendency to pivot about the OC will generate lateral reaction forces at the rear tires in the same direction as the forces will be when the steady state condition is reached; there will be no reversal of forces from transient to steady state. This case would represent a short and very smooth transient period.

For Case (2) the yaw inertia is as if the front and rear axle loads represented equivalent masses actually concentrated at the respective axle lines. The tendency to pivot about the rear axle means that steering angle input at the front wheels will not immediately generate lateral reaction forces at the rear tires; the will be some small lag time.

For Case (3) the yaw inertia is relatively large, equivalent to Wf (lf X0.5)2 + Wr (lr X0.5)2, and the oscillation center is at distance lr X aft of the c.g. As the front wheels begin to steer the tendency to pivot about the OC will generate lateral reaction forces at the rear tires in the opposite direction as the forces will be when the steady state condition is reached; there will be a reversal of forces from transient to steady state. This case would represent a long and fluctuating transient period; this has been described as creating an uneasy sensation of floating at the rear of the vehicle.

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7 THE STEADY STATE CONDITION

The transient condition has terminated and the steady state has begun when the sprung mass has attained some constant roll angle, the vehicle has reached some constant yaw velocity, and the associate fluctuations of traction forces at the tires have ceased. This is the condition generally sought after while circling the circumference of that test course known as the skidpad.

Like the slalom, there is no official standard course. Road & Track uses a 200 ft (61 m) diameter skidpad32, and General Motors used a 216 ft (66 m) diameter skidpad to obtain the lateral acceleration results of Figure 7.233. The size of the skidpad is critical, because the larger the radius the less angularity is involved in the resolution of the forces at the tires, and thus the higher the maximum acceleration possible. Furthermore, if the vehicle in question is a race vehicle designed to generate aerodynamic down force, then the higher velocities required on the larger radius skidpads (a = V2/r) brings those aerodynamic qualities into greater account, again making a higher maximum acceleration possible.

The aero effect is such that Car and Driver magazine recommended testing race cars such as the 1981 Lola T600, an IMSA GTP race car, on a skidpad of 372 ft (113.4 m) radius, as opposed to a more prosaic 141 ft (43 m) radius pad. The Lola, which came replete with an aerodynamic quality called ground effects, pulled 1.42 gs on the large pad, versus 1.23 gs on the small one. How well the Lola would have done on the large pad without ground effects is open to question, but it is very likely it still would have bested its figure from the 141 ft (43 m) pad. The point here is that, whenever confronted by empirically obtained lateral acceleration figures, it is wise to ascertain how they were obtained, and the size of the pad is the most basic consideration.34

Although it has been stated there is no officially recognized standard skidpad, the 100 ft (30.5 m) radius pad as depicted in Figure 7.1 comes close to being a standard by virtue of its popularity:

32 Reference [16], page 16. 33 Reference [8], page 40. 34 Information regarding all relevant considerations is seldom given, but still very important. For instance, what was the nature of the skidpad surface (asphalt, concrete, etc.) and whether that surface was clean or dusty, damp or dry. Also, exactly how was the skidpad test conducted; are the results an average of multiple readings taken in runs going in both clockwise (CW) and counter-clockwise (CCW) directions? And, of course, what was the condition of the vehicle at the time of test; was the vehicle in some standard and reproducible configuration?

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Figure 7.1 STANDARD SKIDPAD

Figure 7.1 reveals the nature of another interesting result that can be obtained from skidpad testing other that the maximum lateral acceleration that a vehicle is capable of. That other result is the basic nature of the directional stability of the vehicle tested; that basic nature is characterized as understeering, neutral, and oversteering behavior. These are terms having to do with directional stability, and a full discussion of directional stability and what is involved therein is reserved for Chapter 10, but a quick discussion of the behavior patterns depicted in Figure 7.1 is in order at this point.

When a vehicle is circling around the skidpad along the 100 ft (30.5 m) radius in a steady state condition, after the maximum lateral acceleration has been recorded, then another type of test may be carried out. Holding the steering angle input constant, the vehicle velocity may be slowly increased and the vehicle response noted. If the vehicle attitude changes, taking a nose in/tail out position with respect to a tangent to the circle, and changes course so as to spiral in toward the center of the circle (- - - path), then the vehicle is said to oversteer.

If, however, the vehicle adopts a nose out/tail in attitude with respect to a tangent to the circle, and changes course so as to spiral out away from the center of the circle (oooo path), then the vehicle is said to understeer. If the vehicle tends to neither oversteer nor understeer but instead continues to circle the pad at constant 100 ft (30.5 m) radius ( path), then the vehicle is said to be neutral.

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As should now be apparent, these terms oversteer, understeer, and neutral; are characterizations of a basic vehicle handling tendency. This handling character is determined by a number of factors, but the most basic is the vehicle longitudinal mass distribution. The oversteering character is associated with an aft weight bias. The understeering character is associated with a forward weight bias. And, of course, the neutral handling character is associated with a vehicle c.g. near the midpoint of the wheelbase.

The trouble with this last condition is that neutral handling tends to not be constant; as a vehicle is operated fuel is consumed, loading varies, and onboard objects can shift positions. All of this variation in condition results in a shifting longitudinal c.g. location during operation, which in turn can cause the vehicle handling character to shift, perhaps from neutral to understeer then back to neutral on the way to oversteer again Such change in a vehicles basic handling character can be disconcerting to the driver and is therefore inherently dangerous. For reasons better dealt with in Chapter 10, it is generally best if a vehicle maintains a degree of understeering character throughout its operational envelope.

It was on a skidpad somewhat larger than the standard skidpad of Figure 7.1 that GM obtained its data from the target vehicles for its design of the 1984 Corvette. That skidpad was of 108 ft (33 m) radius and, along with the roll gain info of Figure 5.6 and the transient response data of Figures 6.1/2, the lateral acceleration info of Figure 7.2 was obtained35:

Figure 7.2 MAXIMUM LATERAL ACCELERATION RESULTS


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Note that once again the Corvette Z51 severely trounced its target opposition, as it did in almost all objectively measurable categories of performance. However, the numerical test advantage the Corvette Z51 demonstrated perhaps may have been obtained by too narrow a focus on certain performance aspects to the neglect of other less quantifiable aspects. In contrast to the highly enthusiastic review by Motor Trend, Car and Driver had this to say about the Corvette Z5136 (which was corroborated by a number of other sources) :

bad pavement sent its wheels boundingminor bumps or irregularities threw the car off on a momentary tangentride was annoyingly harshstraight-line stability was practically nonexistent.

The problem seems to have been that GM had narrowly focused on beating the targeted competition with regard to a number of performance criteria, at which it succeeded, but thereby neglected the refinement for which some of the competition is famous. It probably also didnt help that GM was undoubtedly designing to much lower cost targets than most of the competition.

Car and Driver obtained a maximum lateral acceleration figure on a 141 ft (43 m) radius skidpad of 0.84 gs for the Z51, and 0.85 gs for the base Corvette37! This contrasts sharply with the 0.95 gs GM supposedly attained with the Z51 on a 108 ft (33 m) radius skidpad38. In the very same issue that Motor Trend reported the GM figure, Motor Trend reported the result of their test of the Z51 as 0.92 gs (skidpad radius not stated, but M/T currently uses a 100 ft/30.5 m radius pad)39. Road & Track, on a 100 ft (30.5 m) radius pad, put the 1984 base Corvette at 0.842 gs40. All of this underscores the variability of max lateral acceleration testing possible due to skidpad and methodology variations.

All of the above indicates that if the 84 Corvette was not as overwhelmingly superior in maximum lateral acceleration as GM and M/T indicated, it was still better than most of its competition. The Corvette performance had been achieved by GM in large part by the mass properties engineering tactics of weight and c.g. height reduction. Basic Corvette construction consisted of the traditional (for Corvette) fiberglass body and steel frame, but much weight and c.g. reduction was obtained through material substitution: suspension control arms and uprights were forged in 6061 T-6 aluminum (saving 36 lb/16.3 kg over the previous configuration), the brake calipers were also aluminum, and the rear transverse leaf

36 Reference [17], page 64. This papers author recognizes such criticism as being reminiscent of his own experience with his 1999 Firebird. 37 Ibid, pages 65 and 68. 38 Reference [8], page 40. 39 Reference [8], page 36. 40 Reference [20], pages 57 and 58.

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spring was fiberglass41. Cross-members were reduced on the steel frame by tying the transmission and the differential together via an aluminum C-section beam; this saved weight and allowed for the entire driveline to be installed with only four bolts42. Completing the materials picture was the use of magnesium for the air cleaner housing and valve covers; urethane sheet molding compound was used for the immense hood43.

Yet, despite all this intense focus on weight reduction, GM missed the 3000 lb (1360.8 kg) Corvette target curb weight by 150 lb (68 kg)! The 84 Corvette weight was somewhat lighter than most of the vehicles used to formulate its design specification goals, but still disappointing in view of all the aluminum, magnesium, plastics, and high strength steels that had gone into its construction; there would seem to have been ample scope left for further weight reduction44.

41 Reference [18], page 30. 42 Ibid, page 31. 43 Ibid, page 29. 44 Ibid, page 30.

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8 ROLLOVER

As noted in Chapter 3, rollover (overturn) seldom occurs for modern passenger cars as the result of lateral acceleration resulting from tire traction forces because the slide acceleration point is generally reached first (page 8, top). However, this does not mean that vehicle rollover is not a concern. Although rollover was involved in less than 3% of passenger vehicle accidents in the US for 2009, rollover was involved in about 35% of all fatalities (23,437 total fatalities, 8,296 roll-over fatalities). Of the 8,296 fatalities about 66% failed to wear seatbelts, with many resultant ejections from the vehicle; it can be assumed that some would have survived had seatbelts been utilized. However, that leaves 34% who were properly belted in, yet who fail to survive anyway, which still represents a disconcertingly large proportion of all fatalities at almost 12%45.

Therefore, even though rollover is unlikely, it still merits serious concern. The NHTSA used to (NCAP 2001-2003) rate vehicles for rollover resistance based solely on a mathematically derived figure of merit called the Static Stability Factor (SSF)46. The SSF is exactly the same as as calculated by the rigid model Equation 3.6 in Chapter 3:

=

(EQ. 8.1)

Where:

SSF = A figure numerically equal to the lateral acceleration for overturn (in

gs) as calculated by Eq. 3.6.

t = The average vehicle track width, front plus rear divided by two.

hcg = The vehicle center of gravity height above the ground plane.

Of course, the accuracy of using the SSF as a means of comparison between vehicles depends on the accuracy of the track measurements and c.g. measurement, and for the latter it is crucial that all vehicles be in the same weight condition when measured. Even with all due care in measurement, the SSF cant be a totally accurate means of comparison between vehicles for reasons touched on in Chapter 5 and illustrated in Figure 5.3, under lateral inertial load a rolling movement of the sprung mass will occur through some angle s which will reduce the hcg by some amount dz and cause the sprung weight to shift laterally by some

45 Reference [9], page 1. Per the NHTSA, roll-over accidents have on average accounted for 10,000 deaths per year over the decade from 2000 through 2010. 46 Reference [2], page 1.

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amount dy, all of which is among a number of suspension dependent changes which will render the real somewhat different (undoubtedly less!

47) than that predicted by the SSF.

Still, while the difference between the actual and the SSF is significant, the SSF is simple to determine and exhibits a strong statistical correlation with the incidence of roll-over accidents, as determined by NHTSA statistical analysis48. As part of the NHTSAs New Car Assessment Program (NCAP), the NHTSA began providing SSF based rollover resistance ratings for new cars in 2001.

Other measures of rollover resistance include passive tests such as the Tilt Table Ratio (TTR), which is depicted in Figure 8.1. Clinometers are utilized to measure the degree of tilt of the table and of the vehicle sprung mass; the degree of table tilt corresponds to a certain amount of lateral gs:

Figure 8.1- TILT TABLE RATIO TEST

47 As discovered in Chapter 3 the drop in c.g. height dz due to roll is probably negligible, but the lateral shift in c.g. dy significantly diminishes the track dimension t/2 significantly; essentially or the SSF becomes (

)/( ).

48 Reference [15], page 37 (Section V.A.). Reference [2], page 1, states that the SSF was chosen as the basis for vehicle rollover ratings because it highly correlated with actual crash statistics; it can be measured accurately and inexpensively and explained to consumers, and changes to vehicle design to improve SSF are unlikely to degrade other safety attributes. Note: the degrade other safety attributes is a reference to possible c.g. height reduction by manufacturers through reduction of roof and roof support structure, thereby reducing crush resistance. However, increased rigor and wider applicability of the FMVSS 216 roof crush resistance standard has since made such reduction unlikely.

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TTR is the ratio of the lateral force (parallel to the table top) at onset of high side wheel lift off to vehicle weight (and therefore equivalent to gs); for a rigid vehicle model the TTR should be the same as the SSF calculation. For a real vehicle the TTR results should be more realistic (i.e., less) than the SSF as most of the complications of roll, deflections, and liquid free surfaces will be present. However, TTR is regarded as likely to be a little different from a real rollover because the whole vehicle is tilted; the physical situation is not the same as if the lateral g level were attained by forces at the tires while the vehicle was on level ground49. Reportedly, real vehicle TTR measurements are about 10% to 15% less than SSF calculations50.

There is also another empirical determination, the Side Pull Ratio (SPR), which involves winching in of a cable acting through the vehicle c.g. in order to measure the force required to cause two wheel lift on a side. That force is divided by the vehicle weight to produce the SPR metric in gs. Again, the result should be near identical to the SSF for a solid body vehicle, and very close to the TTR for any real vehicle (and therefore also about 10 to 15% less than SSF)51. However, the SPR is regarded as being somewhat more realistic than the TTR as the vehicle weight vector stays perpendicular to the ground plane.

Figure 8.2 SIDE PULL RATIO TEST

All the roll-over resistance metrics mentioned so far are essentially static in nature. However, there are at least two dynamic methods for rollover resistance determination, the J-turn and the Fishhook. Since 2004 the NHTSA has been using the results of the SSF calculation plus the Fishhook test to arrive at rollover resistance ratings for new vehicles. The Fishhook and

49 Reference [7], page 317. 50 Reference [15], Section V.A., page 2. 51 Ibid.

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the J-turn are both open loop maneuvers that dont really provide a direct measure of roll resistance, but do give an indication of vehicle dynamic response under high lateral loads. The NHTSA grades vehicles on a pass or fail basis for the Fishhook maneuver, and combines that result in a complex formulation that favors the SSF results to produce a rating of one to five stars.

The Fishhook name comes from the shape of the path taken by the vehicle during the test. The Fishhook invokes the rollover tendency of a vehicle by approaching as close as possible to actual rollover through a rather harsh maneuver. The Fishhook uses steering inputs that approximate the steering a panicked driver might use to regain lane position after dropping two wheels off the roadway onto the shoulder, but is performed on a level pavement with a rapid initial steering input followed by an over correction. The original version of this test was developed by Toyota, and variations of it were adopted by Nissan and Honda52.

Figure 8.3 FISHHOOK TEST GROUND PATH

NHTSAs test version includes roll rate measurement in order to time the counter-steer to coincide with the maximum roll angle each vehicle takes in response to the initial steering input. The test utilizes an automated steering system programmed with inputs intended to compensate for differences in vehicle steering gear ratio, wheelbase, and stability properties. To begin, the vehicle is driver controlled in a straight line. The driver releases the throttle, coasts to the target speed (which starts around 35 mph/56 kph and increased in 5 mph/8 kph increments for each run until termination53 is achieved), and then activates the auto-pilot which commences the maneuver. The test runs conclude when a termination condition is achieved involving two inch or greater lift of the vehicles inside tires (fail), or

52 Reference [2], page 4. 53 Hopefully not the drivers.

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if the vehicle completes the final run at maximum speed of 50 mph/80 kph without lift (pass). If needed, further testing is undertaken to confirm the exact speed at which lift occurs, and that the lift point is repeatable54.


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9 TIRE BEHAVIOR: SLIP ANGLES

As noted and diagramed in Chapter 6, the lateral traction force generated at the tire/road contact area is associated with a certain slip (drift) angle. Of course, the contact patch is also often called upon to generate longitudinal forces (acceleration and braking) as well, and all these forces interact just as stresses acting in different orthogonal directions in a material sample have an interacting effect expressed in a quantity called Poissons Ratio55.

The relationship between lateral and longitudinal forces (analogous to stresses) and drift angle (analogous to strain) is not as clean-cut as the situation expressed in Poissons Ratio This is to be expected, as most materials are isotropic56 and tires are anisotropic. The relationship with respect to tires can be illustrated by the following diagram:

Figure 9.1 TIRE TRACTION CIRCLE (ELIPSE)

Any combination of lateral and longitudinal traction forces is possible as long as the resultant total traction force does not exceed the maximum traction possible represented by the bounds of the outer circle. In the situation depicted in the first quadrant of the traction circle, the acceleration force of 539.8 lb (244.8 kg) allows for a lateral left hand traction force of up to 762.6 lb (345.9 kg). If the centrifugal force needed at this time to complete some maneuver actually goes up to that level, then the resultant force will be 934.3 lb (423.8 kg), and no further increases in acceleration forward or to the left will be possible; the tire is at

55 Simon Denis Poisson, 1781-1840. 56This would be true regarding traditional engineering materials, excluding the new burgeoning field of composites which have a directional structure, as do tires (though on different scales).

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force saturation. If the driver hits the gas to go faster, the tire will spin helplessly making huge clouds of white smoke (burnout!) and the vehicle will slide off the road to the right.

If a lateral traction force of 762.6 lb (345.9 kg) is actually required of the tire in this situation of also supplying an acceleration force of 539.8 lb (244.8 kg), then the tire will be running at a drift angle of 8 degrees. If the driver lifts off the gas, then the drift angle will snap back to about 5.5 degrees. Conversely, if the driver were taking a turn which caused a lateral force load at his rear wheel(s) of 762.6 lb (345.9 kg), and he stepped on the gas so as to acquire an acceleration force of 539.8 lb (244.8 kg) at the rear wheel(s), the rear drift angle would go to 8 degrees; the rear end would step out adopting a new attitude and smaller turning radius. This effect is called throttle steering, and in some automotive sporting events (gymkhana, autocross, drifting) a driver may make as much use of the throttle to steer as the steering wheel.

Simplifying the traction field as a circle instead of the more realistic ellipse57 allows for an easier determination of how much traction potential is available for lateral acceleration. The Pythagorean relationship between lateral and longitudinal traction forces can be used to revise the equations for the potential lateral traction force at the axles. Essentially Equation 3.1 becomes for the front axle:

= + (Eq. 9.1a)

Or

= + (Eq. 9.1b) And for the rear axle:

= [( )] + [( )] (Eq. 9.2a)

Or

= + (Eq. 9.2b) 57 Reference [11], page 129. Even the ellipse is a simplified model of a more complex reality.

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This can be made even more representative of reality by explicitly taking weight transfer into account in determining the lateral force potential; Equations 5.2/3 (front) and 5.4/5 (rear) for the determination of Ni and No front and rear can be substituted into Equations 9.1 and 9.2. However, the result is cumbersome and there is little point in doing so other than to create a computer simulation of automotive maneuvering. For such a simulation it would also be important to also know how the drift angle varies as a function of lateral and normal loads, which is a topic only touched on so far.

As is apparent from Figure 9.1, as lateral force increases the tire deformation58, i.e. the drift angle, increases until the deformation reaches a limit and any further demands upon the tire will tend to result in 100% slippage; the vehicle will tend to go into an uncontrolled skid. Thus the limit of tire deformation determines the maximum lateral resistance possible and restricts the range of the drift angles; at present for passenger car tires the maximum b and drift angle seems to be around 1.0 g and 12 degrees. However, most driving on public roads rarely exceeds 5 degrees or 0.3 gs lateral acceleration59. For racing vehicle tires (like those used in Formula 1) the present limits (without aerodynamic loading) seem to be about 8 degrees and 2.0 gs, but the softer compounds that allow for such greater traction do not give the sort of mileage expected of road tires (race tires are designed to last only a few hundred miles or less).

Since the drift angle/lateral force relationship is dependent upon quite a few parameters, it is common to look at functions which constitute only a partial differential of the total relationship (for which no one has yet established a complete definitive formulation based on physics60) in order to achieve a degree of understanding. If the tire drift angle/lateral force partial differential function is plotted the result looks like Figure 9.2.

The lateral force increases with drift angle (or vice versa) on a shallow curve, reminiscent of stress-strain curves, up to a deformation limit. At that point the situation is very unstable and will either quickly return to the safety of the useful traction region or result in 100% slip (a slide) at the tire/road contact patch.

58 Conversely, in the literature one often reads of increasing deformation (drift angle) causing an increase in the lateral force. It is a matter of perspective; if a lateral force is applied to a vehicle running in a straight line then the tires will develop drift angles and the vehicle will deviate from the straight line, but if a driver twists the steering wheel of a vehicle running in a straight line then the resultant drift angles will result in a lateral force and again the vehicle will deviate from the straight line. The line between cause and effect is often difficult to discern. 59 Reference [1], page 78. 60 Hans Bastiaan Pacjeka, Professor Emeritus at Delft University of Technology in the Netherlands, has developed a tire model called the Magic Formula because it is based on relatively little underlying physics; instead it is mainly based on a regression analysis of reams of empirical tire data.

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If the appropriate portion of the useful region is linearized by the fitting of a straight line to the original shallow curve, then a simplified lateral force/drift angle relationship (the slope of the fitted line) can be obtained. This quantity is commonly called the Cornering Power of the tire, which is yet another misnomer like shock absorber (damper) or slip angle (drift angle). It would seem more appropriate to call this quantity the Cornering Stiffness. Such linearized tire relations are often used for simulations/studies of automotive dynamic behavior. Note that a little earlier it was demonstrated by use of the traction circle how the addition of longitudinal traction force for acceleration could decrease the cornering stiffness Fy/ at a tire (762.6 lb/5.5 deg vs. 762.6 lb/8.0 deg); braking force additions likewise tend to decrease the cornering stiffness (Cs) of the affected tire(s).

The problem with simulations/studies based on linear models is that tire behavior beyond the proportional limit is not taken into account. Although road going passenger vehicles seldom venture into the nonlinear region between proportional limit and deformation limit, racing vehicles frequently do. A simulation/study of race car behavior would seem to require a non-linear curve fit to the entire (both linear and transition portions) useful traction region61.

Figure 9.2 - CORNERING STIFFNESS: LINEARIZED LATERAL FORCE / DRIFT ANGLE FUNCTION


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If matters were just as simple as Figure 9.2 then understanding of automotive dynamic behavior, and how mass properties influences that behavior, would be very easy. The potential or maximum lateral force that a tire can supply, and the drift angle associated with that force, is dependent on many parameters. That the lateral traction force potential is influenced by longitudinal forces is a subject already touched on, but the lateral force potential is also influence by normal load, camber angle (which can change with roll), roll steer (which can be the result of normal load and camber change with roll, but toe in/out can also change with roll), tire type (size, carcass type and material, rubber type, tread design, aspect ratio), inflation pressure, wheel rim width, road material and surface (smooth, rough, dusty, etc.), weather (rain, snow, ice), temperature (both ambient and of the tire itself), and the speed of the vehicle (all basic tire coefficients of traction are slightly speed dependent; the same lateral force will produce a slightly smaller drift angle at high speed than at low speed62). While all these factors are significant, only normal load is fundamental and germane to the topic of this paper.

To illustrate the effect of normal load on lateral resistance and drift angle a plot such as Figure 9.3 is often used. Note that it is essentially like Figure 9.2 except that there are now a large set of Fy, functions which serve to represent an infinite variation; any change in normal load alters the Fy, relation, but these five example curves may suffice as the intermediate possibilities can be approximated by interpolation:

Figure 9.3 LATERAL TRACTION vs. DRIFT ANGLE FUNCTIONS AT NORMAL LOAD


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Only the useful traction region is shown; 13 degrees is about the maximum drift angle/deformation limit. Note that as normal load increases the lateral traction force necessary for a certain drift angle increases as well, but at a decreasing rate just as was the case for Equation 2.2/Figure 2.2. Actually, Figure 9.3 is a poor way to illustrate this behavior, which is better shown by the following real data plot of lateral force vs. normal load for an actual 6.00x16 tire inflated to 28 psi (193 kPa)63:

Figure 9.4 LATERAL TRACTION vs. NORMAL LOAD

AT DRIFT ANGLE

Figure 9.4 clearly shows how increasing normal load will increase the lateral force necessary to cause the same amount of deformation (drift angle), but at a decreasing rate and only up to a point. That point is the deformation limit, and traction demands beyond that point are likely to result in an out-of-control slide of the vehicle.

This figure provides the ability to demonstrate the directional effect of weight transfer/roll under lateral acceleration. Recall that the 1980 Ford Fiesta S in the 4 up condition as modeled in Chapter 5 had static axle loads of 1240.7 lb (562.8 kg) front and 1058.7 lb (480.2 kg) rear, as determined from the total vehicle weight of 2299.4 lb (1043.0


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kg) and weight distribution of 54/46. This means that at 0.5 gs lateral acceleration in steady-state equilibrium the lateral traction forces at the axles must be 620.35 lb (281.4 kg) front and 529.35 lb (240.1 kg) rear (assuming a turn radius large enough that the effects of angularity requiring the resolution of forces into X & Y components can be neglected). From Equation 5.2 it was determined that for the Fiesta the lateral acceleration of 0.5 gs translated into a roll angle of 4.6 degrees. From Equations 5.2 through 5.5 it was determined that for a roll angle of 4.6 degrees the normal loads (now dynamic) become Nfi = 161.47 lb, Nfo = 1079.23 lb, Nri = 200.55 lb, Nro = 858.15 lb (73.2, 489.5, 91.0, 389.3 kg respectively).

Assuming tire characteristics of b = 1.2 and m = 0.0004 and using Equation 2.2 these dynamic normal loads correspond to potential lateral traction forces of Fyfi = 183.33 lb, Fyfo = 829.18 lb, Fyri = 229.57 lb, Fyro = 735.21 lb (83.2, 376.1, 104.1, 333.5 kg), or on a potential force per axle basis Fyf = 1012.5 lb (459.3 kg), Fyr = 964.78 lb (437.6 kg). These per axle potential traction forces easily exceed the required lateral forces of 620.35 lb (281.4 kg) front and 529.35 lb (240.1 kg) rear for dynamic equilibrium, and there is no need to resort to Equations 9.1a through 9.2b to make a determination whether the combined lateral and longitudinal forces would exceed available traction because, for simplicitys sake, there are no longitudinal forces in this example (although in reality there would always be some longitudinal forces due to rolling resistance and the need to counter that resistance with some driving forces to keep the vehicle going around the skidpad at constant velocity).

Assuming the actual forces generated at each tire will be in proportion to the potential forces possible at that tire, the actual lateral forces work out to be Fyfi = 112.33 lb, Fyfo = 508.02 lb, Fyri = 123.86, Fyro = 405.49 (51.0, 230.5, 56.2, 183.9 kg). Assuming for this example that the tires of the Fiesta were those of Figure 9.4, by interpolation of that figure (and not one like Figure 9.1 because again there is no X and Y forces requiring combination into resultant forces via use of Equations 9.1a through 9.2b) the corresponding drift angles are fi = 4.0 deg, fo = 5.8 deg, ri = 4.0 deg, ro = 4.5 deg. Note that the inner and outer drift angles are different, which they would tend not be if there were no weight transfer. Also note that the average drift angle at the front axle (4.9 deg) is greater than the average drift angle at the rear axle (4.24 deg), which is indicative of an understeering vehicle (and the front to rear drift angle comparison is even more so indicative if limited to just the outer wheels; with inc

Mass Properties and Automotive Lateral Acceleration Rev B

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