mass properties and automotive lateral acceleration 2011 weigand

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


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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

Author’s 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 value significantly greater than the maximum lateral acceleration level. 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” or “tripped” 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 “drift angle” relationship of the vehicle tires front-to-rear, and the lateral force balance front-to-rear due to those drift angles. The lateral force/drift angle relationship is dependent upon normal load, so the most significant mass property with regard to directional stability is the vehicle static weight distribution.

However, the static normal loads are dynamically modified in response to directional “disturbance” forces. Such disturbances generate initial inertial lateral 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 suspension roll resistance. Such changes readjust the lateral force/drift angle relationship front to rear, and thereby generate a secondary directional reaction. If the secondary reaction is such as to augment 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 parameters are the vehicle mass properties of weight, and weight distribution (longitudinal, lateral, and vertical c.g.).


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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: 𝒎𝒂 = 𝑭𝒇 + 𝑭𝒓 (EQ. 1.1)

ROTATIONAL: 𝑰𝜶 = 𝑭𝒇𝒍𝒇 − 𝑭𝒓𝒍𝒓 (EQ. 1.2)

Equation 1.2 shows that the turning situation involves mass properties other than just weight and center of gravity; the rotational moment of yaw inertia (“I”) will have an important effect on the turn-producing forces (“Ff” and “Fr”) whenever there exists some appreciable angular acceleration (“α”). 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 with 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 Equation 1.1, and how the lateral forces therein are influenced by the weight and center of gravity. To do this, one must begin by considering just how these lateral acceleration / turn-producing forces are generated. The generation of these forces is a bit more complex than might first be supposed; the relationship is not simply a case of applying Coulomb’s friction law.


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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 Coulomb’s Friction Law: “F = µN”. 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 coefficient “µ” is itself a function of the normal load as per Equation 2.11

𝝁 = 𝒃 −𝒎𝑵 (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

1 Reference [11], page 127. The 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 Coulomb’s law by substitution for “µ” results in the normal load / potential lateral force relationship of Equation 2.2:

𝑭 = (𝒃 − 𝒎𝑵)𝑵 (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 transfer in a turn. This matter of weight transfer will bring to the fore the role of the c.g. in determining lateral acceleration.


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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 3.12

:

𝑭𝒂𝒙𝒍𝒆 = (𝒃 −𝒎𝑵𝒊)𝑵𝒊 + (𝒃 −𝒎𝑵𝒐)𝑵𝒐 (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

2 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:

𝑵𝒊 = 𝑾𝟐−𝑾𝒂𝒚 �

𝒉𝒄𝒈𝒕� (EQ. 3.2)

𝑵𝒐 = 𝑾𝟐

+ 𝑾𝒂𝒚 �𝒉𝒄𝒈𝒕� (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:

𝑭𝒂𝒙𝒍𝒆 = �𝒃 −𝒎�𝑾𝟐−𝑾𝒂𝒚 �

𝒉𝒄𝒈𝒕��

𝑾𝟐−𝑾𝒂𝒚 �


+ �𝒃 −𝒎�𝑾𝟐

+ 𝑾𝒂𝒚 �𝒉𝒄𝒈𝒕��

𝑾𝟐


(EQ. 3.4)

From this equation, further equations for the maximum axle lateral acceleration limits of slide and overturn3 can be determined, Equations 3.5 and 3.6, respectively4

𝒂𝒚𝒔𝒍𝒊𝒅𝒆 =−𝟏 +�𝟏−𝟒�

𝟐𝒎𝑾𝒉𝒄𝒈𝟐

𝒕𝟐��𝒎𝑾𝟐 �−𝒃�

(𝟒𝒎𝑾𝒉𝒄𝒈𝟐 𝒕𝟐⁄ )

(EQ. 3.5)

:

𝒂𝒚𝒐𝒗𝒆𝒓𝒕𝒖𝒓𝒏 = 𝒕 𝟐𝒉𝒄𝒈⁄ (EQ. 3.6)

3 Reference [15], Section V.A. Reference [7], page 311. 4 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 g’s).

Note that slide would not have set in until a significantly higher acceleration level (“ay” = 0.8 g’s, 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 g’s, 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 / 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 location of the 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:

𝑾𝒇 = 𝑾 (𝒍𝒓 𝒍𝒘𝒃⁄ ) (EQ. 4.1)

𝑾𝒓 = 𝑾 �𝒍𝒇 𝒍𝒘𝒃⁄ � (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 location of the 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 g’s, 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 won’t 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 g’s 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 g’s! 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 g’s, respectively5

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 Fédération Internationale de l’Automobile) vehicles tended to have longitudinal weight distributions of about 45/55 to 40/60

.

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5 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.

. 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

6 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 gravity7

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

. 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):

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 kg) at 7 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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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)8

. 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)9. 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)10

Figure 5.2 - SPRUNG MASS IN ROLL AT 0.5 g’s LATERAL

, 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:

8 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. 9 See Appendix E for roll stiffness determination methodology. 10 Reference [6], pages 191-192.


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Given all this information, the roll angle “θs” of the sprung mass under a 0.5 g lateral acceleration “ay” can be determined from the fact that the roll resistance (roll angle “θs” times the roll stiffness “kroll”) has to equal the roll moment (lateral force “Ws ay” times the roll height “hr”) at equilibrium11

Roll Angle x Roll Stiffness = Lateral Force x Roll Height

:

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

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

θs = 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 “θs” 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 “θs” more than the 3.9 degrees12

Figure 5.3 – ROLL GEOMETRY

:

11 Reference [3], pages 133 and 136. 12 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 reality13 than that initially considered may be expressed14

:

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

Where:

θs = 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, g’s.

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 radius15

Wusr = The rear axle unsprung mass weight, lb.

), ft.

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 RC’s in roll) are all still

13 Even with all this complication, 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. 14 Reference [3], page 136. 15 Ibid, page 134.


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unaccounted for. Still, the roll angle value as determined by Equation 5.1 may have all the 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 “θs” 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, let’s 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:

𝑵𝒊𝒇 = 𝑾𝒇

𝟐− 𝜽𝒔𝒌𝒇𝒓𝒐𝒍𝒍/ �

𝒕𝒇𝟐� (EQ. 5.2)

𝑵𝒐𝒇 = 𝑾𝒇

𝟐+ 𝜽𝒔𝒌𝒇𝒓𝒐𝒍𝒍/ �

𝒕𝒇𝟐� (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, let’s 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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𝑵𝒊𝒓 = 𝑾𝒓

𝟐− 𝜽𝒔𝒌𝒓𝒓𝒐𝒍𝒍/ �

𝒕𝒓𝟐� (EQ. 5.4)

𝑵𝒐𝒓 = 𝑾𝒓

𝟐+ 𝜽𝒔𝒌𝒓𝒓𝒐𝒍𝒍/ �

𝒕𝒓𝟐� (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 Fiesta’s 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 Fiesta’s 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 Fiesta’s 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 chapter’s 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 performance16

16 Race car design may even include a fair amount of directional instability for low speed sharp 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.

.


21

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 g’s, as illustrated by Figure 5.417

Figure 5.6 – ROLL GAIN, DEGREES ROLL PER G’s LATERAL ACCELERATION

; note that the little Fiesta S has more than double the roll gain of any of the high performance sports cars shown:

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)18


.



22

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 “α” 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:

𝑰𝜶 = 𝑭𝒇𝒍𝒇 − 𝑭𝒓𝒍𝒓

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 “𝜔” 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


23

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 oscillation19, 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: “…oscillation…is critical to the feel of the car. …a high decay rate might...be described as “twitchiness”…a vehicle that would wiggle around as the driver turned into a corner”20

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 beat

. It is best that this transient phase be as short in duration (“response time”) as possible.

21

19 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..

; how well they did with respect to roll and yaw transients is illustrated in Figures 5.4, and 6.1/2, respectively.

20 Reference [8], pages 38-39. 21 Ibid, page 37.


24

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 “g’s”) 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 doesn’t 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 vehicle’s 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 & Track22

Figure 6.3 – “STANDARD” SLALOM COURSE

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.

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:



25

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 6.423

Figure 6.4 – ROLL STEER EFFECTS

. 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:

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

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


26

function is proportional to, but has some small lag time to, the steering angle function24. “Slip angle” is a misnomer like term “shock absorber” is when used to refer to a damper. The tires don’t 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 angle”25

Figure 6.5 – DRIFT ANGLE AND LATERAL FORCE

, which is a practice which this paper will now adopt:

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 “ψ” 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

24 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. 25 Reference [1], page 73.


27

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 change26; 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 vehicle27

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 gyration

.

28

26 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.

. In SAE J670e standard symbolism this quantity is depicted as “k2/ab”. The values of the yaw radius of gyration and the pitch radius

27 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. 28 Reference [19], page 55.


28

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 possibilities29

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



29

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 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.


30

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 skidpad30, and General Motors used a 216 ft (66 m) diameter skidpad to obtain the lateral acceleration results of Figure 7.231

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 g’s on the large pad, versus 1.23 g’s 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.

. 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.

32

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:

30 Reference [16], page 16. 31 Reference [8], page 40. 32 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?


31

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”.


32

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 vehicle’s 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 obtained33

:

Figure 7.2 – MAX. LATERAL ACCELERATION RESULTS 33 Reference [8], page 40.


33

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 Z5134

“…bad pavement sent its’ wheels bounding…minor bumps or irregularities threw the car off on a momentary tangent…ride was annoyingly harsh…straight-line stability was practically nonexistent.”

(which was corroborated by a number of other sources) :

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 didn’t help that GM was undoubtedly designing to lower 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 g’s for the Z51, and 0.85 g’s for the base Corvette35! This contrasts sharply with the 0.95 g’s GM supposedly attained with the Z51 on a 108 ft (33 m) radius skidpad36. 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 g’s (skidpad radius not stated, but M/T currently uses a 100 ft/30.5 m radius pad)37. Road & Track, on a 100 ft (30.5 m) radius pad, put the 1984 base Corvette at 0.842 g’s38

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

. All of this underscores the variability of max lateral acceleration testing possible due to skidpad and methodology variations.

34 Reference [17], page 64. This paper’s author recognizes such criticism as being reminiscent of his own experience with his 1999 Firebird. 35 Ibid, pages 65 and 68. 36 Reference [8], page 40. 37 Reference [8], page 36. 38 Reference [20], pages 57 and 58.


34

spring was fiberglass39. 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 bolts40. 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 hood41

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 reduction

.

42

.

39 Reference [18], page 30. 40 Ibid, page 31. 41 Ibid, page 29. 42 Ibid, page 30.


35

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 seat-belted in, yet who fail to survive anyway, which still represents a disconcertingly large proportion of all fatalities at almost 12%43

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)

.

44

. The SSF is exactly the same as “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛” as calculated by the rigid model Equation 3.6 in Chapter 3:

𝑺𝑺𝑭 = 𝒕𝟐𝒉𝒄𝒈

(EQ. 8.1)

Where:

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

“g’s”) 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 can’t 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 “θs” which will reduce the “hcg” by some amount “dz” and cause the sprung weight to shift laterally by some 43 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. 44 Reference [2], page 1.


36

amount “dy”, all of which is among a number of suspension dependent changes which will render the real “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛” somewhat different (undoubtedly less!45

Still, while the difference between the actual “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛” 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 analysis

) than that predicted by the SSF.

46

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 “g’s”:

. As part of the NHTSA’s “New Car Assessment Program” (NCAP), the NHTSA began providing SSF based rollover resistance ratings for new cars in 2001.

Figure 8.1- TILT TABLE RATIO TEST

45 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 “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛” or the SSF becomes “( 𝑡

2− dy)/(hcg − dz)”.

46 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.


37

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 g’s); 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 ground47. Reportedly, real vehicle TTR measurements are about 10% to 15% less than SSF calculations48

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 g’s. 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)

.

49

Figure 8.2 – SIDE PULL RATIO TEST

. However, the SPR is regarded as being somewhat more realistic than the TTR as the vehicle weight vector stays perpendicular to the ground plane.

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

47 Reference [7], page 317. 48 Reference [15], Section V.A., page 2. 49 Ibid.


38

the “J-turn” are both open loop maneuvers that don’t 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 Honda50

Figure 8.3 – FISHHOOK TEST GROUND PATH

.

NHTSA’s 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 “termination”51


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 vehicle’s “inside” tires (fail), or

51 Hopefully not the driver’s.


39

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 repeatable52

.



40

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 with have an interacting effect expressed in a quantity called Poisson’s Ratio.

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 Poisson’s Ratio53. This is to be expected, as most materials are isotropic54

Figure 9.1 – TIRE TRACTION “CIRCLE” (ELIPSE)

and tires are anisotropic. The relationship with respect to tires can be illustrated by the following diagram:

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

53 Siméon Denis Poisson, 1781-1840. 54This 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).


41

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 actual ellipse 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 and rear axles55

(Eq. 9.1a)

:

𝑭𝒚𝒇 = ��𝒃 −𝒎𝑵𝒊𝒇�𝑵𝒊𝒇�𝟐 − 𝑭𝒙𝒇𝒊𝟐 + ��𝒃 −𝒎𝑵𝒐𝒇�𝑵𝒐𝒇�

𝟐 − 𝑭𝒙𝒇𝒐𝟐

Or

𝑭𝒚𝒇 = �𝑭𝒚𝒇𝒊𝟐 − 𝑭𝒙𝒇𝒊𝟐 + �𝑭𝒚𝒇𝒐𝟐 − 𝑭𝒙𝒇𝒐𝟐 (Eq. 9.1b)

𝑭𝒚𝒓 = �[(𝒃 −𝒎𝑵𝒊𝒓)𝑵𝒊𝒓]𝟐 − 𝑭𝒙𝒓𝒊𝟐 + �[(𝒃 −𝒎𝑵𝒐𝒓)𝑵𝒐𝒓]𝟐 − 𝑭𝒙𝒓𝒐𝟐

(Eq. 9.2a)

Or

𝑭𝒚𝒓 = �𝑭𝒚𝒓𝒊𝟐 − 𝑭𝒙𝒓𝒊𝟐 + �𝑭𝒚𝒓𝒐𝟐 − 𝑭𝒙𝒓𝒐𝟐 (Eq. 9.2b)



42

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 deformation56, 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 maximum the 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 and 12 degrees. However, most driving on public roads rarely exceeds 5 degrees or 0.3 g’s lateral acceleration57

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 physics

. 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 g’s, 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).

58

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.

) 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.

56 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. 57 Reference [1], page 78. 58 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, and instead is mainly based on a regression analysis of reams of empirical tire data.


43

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 “ΔFy/Δψ” 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” region59

Figure 9.2 - CORNERING STIFFNESS: LINEARIZED

.

LATERAL FORCE / DRIFT ANGLE FUNCTION 59 Reference [14], page 126.


44

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 speed60

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, ψ” functions which serve to represent an infinite variation; any change in normal load alters the “Fy, ψ” relation, but these five example curves may suffice as the intermediate possibilities can be approximated by interpolation:

). While all these factors are significant, only normal load is fundamental and germane to the topic of this paper.

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



45

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)61

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



46

kg) and weight distribution of 54/46. This means that at 0.5 g’s 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 g’s 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).

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 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 simplicity’s 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 ““ψfi = 4.0 deg, ψfo = 5.8 deg, ψri = 4.0 deg, ψro = 4.5 deg”. Note that the inner and outer drift angles are different, which they would 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 increasing lateral acceleration the outer wheels become ever more dominant), which the Fiesta is known to be.


47

The drift angles at front and rear determine the turn center of the vehicle and the attitude or “yaw angle” (“β”) the vehicle adopts with respect to the tangent line to the turn circle. If the vehicle is oversteering, as is the case with the Fiesta, then “ψf > ψr” and the vehicle adopts a “nose out” attitude with respect to the tangent line as depicted in Figure

9.562

Figure 9.5 – UNDERSTEERING TURNING GEOMETRY

.

As noted in Chapter 7, an understeering vehicle on the skidpad will not hold to the constant radius of the pad as velocity is slowly (minimizing longitudinal force effect) increased; it will “nose out” and take a path of an ever increasing spiral outward. Since the Fiesta is an understeering vehicle it should display such behavior via the models presented in this paper. For the purpose of demonstrating the fidelity of those models, let’s say that the Fiesta’s speed is slowly increased to a lateral acceleration of 0.7 g’s.

For 0.7 g’s lateral acceleration in steady-state equilibrium the lateral traction forces required at the axles are 868.49 lb (393.9 kg) front and 741.09 lb (336.2 kg) rear. From Equation 5.2 the lateral acceleration of 0.7 g’s corresponds to a roll angle of 6.4 degrees.

62 For “Darwin” (Ackermann) turning geometry information see Appendix C.


48

From Equations 5.2 through 5.5 for a roll angle of 6.4 degrees the normal loads become “Nfi = 0.00 lb, Nfo = 1260.85 lb, Nri = 70.41 lb, Nro = 988.29 lb” (0.0, 571.9, 31.9, 448.3 kg).

Using Equation 2.2 these dynamic normal loads correspond to potential lateral traction forces of “Fyfi = 0.00 lb, Fyfo = 877.12 lb, Fyri = 82.51 lb, Fyro = 795.26 lb” (0.0, 397.9, 37.4, 360.7 kg), or on a potential force per axle basis “Fyf = 877.12 lb, Fyr = 877.77 lb” (397.9, 398.1 kg). These per axle potential traction forces exceed the required for dynamic equilibrium @ 0.7 g’s lateral forces of 868.49 lb (393.9 kg) front and 741.09 lb (336.1 kg) rear.

Again 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 = 0.00 lb, Fyfo = 868.49 lb, Fyri = 69.67 lb, Fyro = 671.42 lb” (0.0, 393.9, 31.6, 304.6 kg). Also again assuming that the tires of the Fiesta were those of Figure 9.4, by interpolation of that figure the corresponding drift angles are determined to be “ψfi = 0.0 deg, ψfo = 14.5 deg63

Note that at 0.7 g’s that the front inner wheel has lifted off the ground and, judging from the traction forces and drift angles, the vehicle is just about ready to slide nose first off the skidpad. This is severe understeer, and the Fiesta’s designers seem to have done everything in their power to make it so. The most basic thing was the weight load bias to the front, but they also had a MacPherson independent suspension at the front with a non-independent beam axle suspension at the rear (an independent suspension system generally allows roll camber changes that have an adverse effect on lateral traction). Of course, a lot of the oversteering factors are the natural result of a FWD/front engine configuration, but to complete the picture the track at the front is wider than at the rear (more understeer), the roll axis is nose down to the front (yet more understeer), the front roll stiffness is greater at the front than the rear (even more understeer)

, ψri = 3.7 deg, ψro = 9.0 deg”. Note that the average drift angle at the front axle (7.25 deg) is greater than the average drift angle at the rear axle (6.4 deg), but comparing just the dominant outer wheels results in an even more disparate 14.5 deg front vs. 9.0 deg rear, all of which is understeering indicative.

64

The vehicle concept explains this; the Fiesta was just a little front wheel drive pulling a relatively big box behind. In anticipated usage that box could be filled to the brim with people and cargo, with possibly more cargo strapped to the roof and towed along behind.

….

63 Note that this is “off the chart” of Figure 9.4 and in treacherous territory. The total traction coefficient “μ” at this point is only about 0.69, but 15 degrees drift is extreme. However, this is probably the result of not having a real set of tire characteristics; the “b” and “m” coefficients came from Reference [11] for an unspecified tire, while Figure 9.4 came from Reference [5] and is for a 6.00x16 tire at 28 psi. The Ford Fiesta S ran on 155SR12 tires, possibly with a large pressure differential front to rear (another over/under steer factor). 64 Curiously, even with all that bias toward understeer, this author calculates an initial cornering stiffness differential of only about 2% front to rear (front bias) growing to about 27% at 0.7 g’s (slide).


49

With such loadings, possibly right up to the 5622 lb (2550 kg) GVWR (and maybe beyond!), the c.g. could migrate quite a distance aftward from its’ normal location. Apparently the design intent was to ensure consistent handling characteristics no matter how the little grocery getter was loaded, and the consequent reduction of the maximum lateral acceleration level was not of much significance.

Now that is seems that the Ford Fiesta reaches its’ lateral acceleration limit near 0.7 g’s, sliding off the skidpad nose first, let’s see how well Equation 3.5 predicted this occurrence. First, the equation for “ayslide” from Chapter 3:

𝒂𝒚𝒔𝒍𝒊𝒅𝒆 =

−𝟏 + �𝟏 − 𝟒�𝟐𝒎𝑾𝒉𝒄𝒈

𝟐

𝒕𝟐 ��𝒎𝑾𝟐 � − 𝒃�


Into this equation the front end values “b” (1.2), “m” (0.0004), “Wf” (1240.7 lb), “hcg” (22.74 in), and “tf” (52.52 in) are input:


−𝟏 + �𝟏 − 𝟒�𝟐(𝟎.𝟎𝟎𝟎𝟒)(𝟏𝟐𝟒𝟎.𝟕)𝟐𝟐.𝟕𝟒𝟐𝟓𝟐.𝟓𝟐𝟐 ��(𝟎.𝟎𝟎𝟎𝟒)𝟏𝟐𝟒𝟎.𝟕

𝟐 � − 𝟏.𝟐�

(𝟒(𝟎.𝟎𝟎𝟎𝟒)(𝟏𝟐𝟒𝟎.𝟕)𝟐𝟐.𝟕𝟒𝟐 𝟓𝟐.𝟓𝟐𝟐⁄ )

= 0.8252 g’s

So, if the Fiesta actually did slide off line at around 0.7 g’s then the rigid model Equation 3.5 would be definitely optimistic, as expected. However, at 0.7 g’s the lateral traction force potential at the front axle (877.12 lb) still exceeded the required traction force (868.49 lb) to maintain heading, although by the slightest of margins; the slide point would actually be just a little more than 0.7 g’s, say 0.71 g’s.


50

10 – DIRECTIONAL STABILITY

Vehicle directional stability is a subject that was initially developed for, and by, the aircraft industry starting from that industry’s earliest days65. Although the advent of the automobile preceded the airplane, it would be more than 35 years before the science of stability would even start to be applied to ground vehicles66

What held back the development of directional stability theory as applied to automobiles was the lack of information regarding tire behavior. The behavior of an aircraft in flight is determined by aerodynamic forces, and the effort to understand such forces was present at the very dawn of the aircraft; Orville and Wilber Wright built a wind tunnel to study aero effects long before their first aircraft ever left the ground

. Stability as it pertains to a vehicle can have many aspects, mainly yaw (directional) stability, roll stability, and pitch stability. The emphasis of this paper on lateral acceleration narrows the focus to just directional stability and roll stability, and to a limited extent the latter has been dealt with previously in Chapters 6 (transient effects) and 8 (rollover).

67. The behavior of automobiles is predominantly determined by tire behavior, and the first tire testing machine was possibly the rotating steel drum tire tester of Becker, Fromm, and Maruhn circa 193068. These German researchers generated tire data as a prerequisite for their investigation of the great automotive problem of the time: steering “shimmy”. In this they were following up on the French researcher George Broulheit who had identified the tire characteristic of “slip angle” in his investigation relating to shimmy69. A notable follower in the footsteps of these Europeans was R.D. Evans of the Goodyear Tire and Rubber Company who continued the investigation of the physical properties of tires via another drum tire tester70

The war years of 1939-1945 brought most research into tire and automotive behavior to a halt. However, in Germany at Junkers Aircraft the researchers Von Schlippe and Dietrich developed a simple structural model of the pneumatic tire which after the war would tie in with the research of Dr. A.W. Bull and (later) S. A. Lippmann at US Rubber. Essentially this had to do with the lag time of drift angle formation after application of a side force being a

.

65 Orville and Wilber Wright thought a certain amount of instability was unavoidable; they regarded part of the pilot’s role was to serve as a stability augmentation system. This design approach became less and less viable as aircraft performance increased. F.W. Lanchester was one of the first to investigate airplane stability analytically; he published his book on the subject, Aerodonetics, in 1908. 66 Reference [14], page 123. 67 Ibid, page 101. 68 Becker, G.; H. Fromm, and H. Maruhn, Schwingungen in Automobillenkungen, Krayn, Berlin, 1931. 69 Broulheit, G.; “La Suspension de la Direction de la Voiture Automobile: Shimmy et Danadinement”, Société des Ingénieurs Civils de France, Bulletin 78, 1925. 70 Evans, R.D.; “Properties of Tires Affecting Riding, Steering, and Handling”, Journal of the Society of Automotive Engineers, 36(2):41, 1935.


51

function of the distance rolled, and hence of velocity and time. This meant that, for highway speeds and small drift angles, the transient phase of the tires adjusting to the lateral forces could be neglected; modeling of vehicle behavior for just the steady state condition would still have broad validity. This opened up the avenue to all the steady state modeling investigations into vehicle directional stability which would subsequently occur.

One of the first to attempt a mathematical analysis of automotive stability was Prof. Yves Rocard of the Sorbonne in 1954. Prof. Rocard’s model was 4-wheeled but rigid; no weight transfer or roll effects were included. In 1955 M.A. Julien and G.J. Arnet presented a paper based on an analysis of a less simplified model in that it separated the vehicle into a sprung and an unsprung mass. Maurice Olley at General Motors had begun research into the “shimmy” problem in the early 1930’s, and by the 1950’s he and his GM compatriots had progressed to major revelations regarding automotive stability, benefiting greatly from the tire work of Evans, Bull, Gough71, and others. Also around this 50’s period William F. Milliken, David W. Whitcomb, and Leonard Segel of the Cornell Aeronautical Laboratory72 would do a great deal to advance the understanding of tire behavior and automotive directional stability, including the construction of the AF-CAL on-road tire tester73

Since the heady period of the 1950’s many new approaches to understanding automotive directional stability have been taken. Mathematical vehicle modeling and empirical tire research, utilizing newer and ever more realistic test machines, would continue, but computer simulations would come to play an ever more prominent role. At first the computer simulations were largely through the use of analog devices intended to serve in validation of theory, and vice versa. However, analog computers were limited in capability, and were to be replaced by more versatile digital machines. Since early digital machines were typically slower than analogs, for a while in the early 1970’s hybrid computers found favor. As the computational speed and other characteristics of digital computers rapidly improved

. The stability model utilized by the CAL investigators appears to have been limited to the “bicycle” model, so the effects of lateral weight transfer and roll were again not inherently accounted for, but they can be integrated into the results after the fact (somewhat).

71 V.E. Gough was a major researcher into tire behavior at the Dunlop Tire Research Center in England during the 1950’s/early 1970’s; he is best known for the “Gough Diagram” which brings all the major characteristics of a tire together into a single “convenient” diagram (see W. Steeds Mechanics of Road Vehicles, page 223, for further info). 72 Originally founded in 1943 as the Curtiss-Wright Airplane Division Research Laboratory, the lab was donated to Cornell University in 1946. In 1972 the laboratory became the Calspan Corporation, a publically traded company. 73 This device was CAL constructed under contract to the US Air Force to study the behavior of aircraft nose wheel tires; later it was put to more general use. Testing tires on the road eliminated the artificialness of the tire-drum contact, but was limited to the study of free rolling tires at speeds up to 40-50 mph (64-80 kph). It was later superseded by the Calspan TIRF (TIre Research Facility) in 1973, an advanced flat-bed tire tester which is in use up to the present day.


52

they came to dominate the simulation world, initially as just mainframes but ultimately also as “personal computers”. The change in platforms also coincided with corresponding changes in simulation software; specific purpose built software was to a large extent supplanted by general multipurpose dynamic simulation and finite element analysis codes.

The result has been a modern plethora of highly detailed and complex modeling attempts. A full understanding would involve a long exposition involving a good deal of higher math and complex algorithms which are beyond the scope of this paper (and beyond this author’s capability as well). Therefore, this paper will address the subject through a simple “analysis” that will cover most automotive cases, especially passenger cars, and arrive at a limited result with the limitations as fully identified as possible.

The greatest single complicating factor with regard to automotive directional stability is that an automobile must be “dynamically” stable, not just “statically” stable. The defining difference between those two conditions is that a “statically” stable system may show the accepted stable state characteristic of returning to equilibrium position after the application of a disturbance input, but in returning may “overshoot” the equilibrium position and then execute a reverse correction back to the equilibrium point once again, which results in an even greater overshoot, and so on. Although the system initially exhibits what would be regarded as stable behavior, that is not the full story; that initial behavior degenerates into an ever increasing series of oscillations ultimately leading to a loss of control. Such a system is statically stable, but dynamically unstable.

The cause of this curious dichotomy in behavior is the presence of an energy source. A spring displaced from its’ equilibrium position will return to that position when released, although it may take a large number of oscillations to do so if not critically damped. However, if the same spring is subject to energy input from a vibrating ground plane, then it will never settle back to its’ equilibrium position; it will vibrate about that position indefinitely as long as the energy input continues (or, if the energy input is at the spring system’s resonance frequency, the oscillations may grow in amplitude until the spring breaks).


53

Figure 10.1 – STATICALLY STABLE, DYNAMICALLY UNSTABLE

So, whether a system is fully, i.e. dynamically, stable or not has to do with the presence of an energy source. The traditional high school physics example of a stable system, such as the one consisting of a marble in a bowl, is limited as that example does not involve a source of energy input; the stability exhibited by the displaced marble in ultimately coming back to rest in the center of the bowl is only a “static” stability. Vehicles are quite different; as a system they have a source of energy input in the form of an engine or motor. The energy input from that engine or motor is in large part reflected in the kinetic energy of the moving vehicle, which is essentially equal to “1

2𝑚𝑒𝑉2”74

That is, a vehicle may be dynamically stable in a certain speed range, and unstable in another, usually higher, range. For this paper the involvement of speed in the stability question will only be lightly dealt with in the course of the following simple exposition

, so it should not be surprising that a full automotive directional stability analysis should hinge on the parameter of “speed” (velocity).

75

Consider the case of a vehicle traveling in a straight line on a level (slightly crowned) road just before encountering a “disturbance” such as a sudden change in the road crown

. The matter of automotive directional stability will be approached initially from the standpoint of an old and incomplete “analysis”, a “static” stability analysis, which has to do with the lateral force balance front-to-rear and the consequent drift angle relationship of the vehicle tires front-to-rear. The lateral force/drift angle relationship is dependent upon normal loads and the lateral inertial reaction forces through the c.g., so the most significant vehicle mass property with regard to directional stability is the weight distribution.

74 To be truly equal to all the kinetic energy inherent in the vehicle, not just the translational energy but also the energy of all the rotational components, the mass symbol is the effective mass “me”. 75 For a more complete treatment see Reference [14], pages 123-229. Speed not only indicates the presence of kinetic energy for oscillations to feed on, but also affects traction directly. However, one would think that the slight dependency of the traction coefficient on speed would have little effect on directional stability which depends on a front to rear balance, and such traction dependency would affect front and rear equally.


54

(highly exaggerated) resulting in a lateral (with respect to the vehicle) component of the weight vector acting through the c.g.76

Figure 10.2 – LATERAL DISTURBANCE FORCE “Fd”

:

Most disturbance forces act through the c.g., or generate a reaction force acting through the c.g. Aerodynamic disturbance forces tend to act through the center of pressure (CP), which for most modern automobiles is located just forward of the c.g, so the effect is much the same as if the aero force had been at the c.g. At any rate, since the disturbance force is taken as acting laterally though the c.g., the location of the c.g. is very significant to the nature of the vehicle reaction. To illustrate the possibilities of that reaction, consider the case of three vehicles of varying longitudinal weight distribution, with suspension and tire characteristics equal all around, encountering the same lateral disturbance force “Fd”77

The forward c.g. bias case (“lf < lr”) develops larger drift angles at the front tires then at the rear in response to the disturbance force “Fd”; this action results in the forward c.g. vehicle steering away from “Fd” generating an inertial reaction “ma” that tends to counteract the effect of the disturbance. Therefore this “understeering” action is considered to be “stable”. The mid c.g. case (“lf = lr”) develops equal drift angles all around in response to “Fd”; this results in no turning action, just a sideways shift, no inertial reaction is generated. The rear c.g. bias case (“lf > lr”) develops larger drift angles at the rear tires than at the front in response to “Fd”; this results in a steering into the disturbance force “Fd” generating an inertial reaction “ma” that tends to augment the disturbance. This in turn causes an even sharper turning, generating a larger inertial reaction. This sort of “oversteering” action is therefore considered to be “unstable” as it tends to “feed upon itself”.

.

76 Reference [14], page 134 & 143. 77 Ibid, page 169. Reference [6], page 56.


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Figure 10.3 – THE THREE CASES: STABLE, NEUTRAL, AND UNSTABLE

When forces such as “Fd” and “ma” act at the c.g. some weight transfer and roll is created, especially in the oversteering case. The static normal loads are dynamically modified in response to the weight transfer and roll, which generally tends to amplify the vehicle response, as does camber changes and other effects. However, for directional stability, it can be seen from the above illustration that the primary parameters are the vehicle mass properties of weight, weight distribution (longitudinal and lateral c.g.), and the yaw inertia. To this list may also be added the vertical c.g. and roll inertia, as it has been already shown how those mass properties affect the normal loads at the tires, and thereby the drift angles.

The essence of automotive directional stability may be gleaned from the illustrative example just given of the three vehicles of varying weight distribution. However, such an “explanation” falls far short of an authentic analysis. Such an analysis begins with an assessment of the geometry of a vehicle undergoing a slight turning action, either as the result of a lateral disturbance force or an input from the steering wheel.


56

From the geometry of the situation the differential equations of motion are written as if the vehicle were a free body moving in space, but generally in only 2 DOF (1 translational, 1 rotational); these equations consist of the summation of forces in the lateral direction, and the summation of moments about the vehicle c.g., in accord with the concept of “Dynamic Equilibrium”78

Figure 10.4 – “BICYCLE MODEL” TURNING GEOMETRY

. The analysis leading to the following discussion was essentially (other modeling results may have been blended in) based upon the “bicycle model”; the geometry of which is presented in Figure 10.4:

The resulting differential equations are solved in accord with the appropriate classical math methodology. Depending on the complexity of the model, the resulting expressions may include:

78 Although Newton’s (1643-1727) Second Law is the basis for the solution of the problem, the concept of “Dynamic Equilibrium” results from the work of Jean le Rond d’Alembert (1717-1783), and is sometimes referred to as “D’Alembert’s Principle”.


57

1) “Understeer Coefficient”, symbolized as “Kus” and expressed in radians or degrees. This presents a more general way (than comparing specific drift angle values “ψf” and “ψr”) of classifying a vehicle condition as stable (understeering): “Kus > 0”, neutral: “Kus = 0”, or unstable (oversteering): “Kus < 0”, and can take various forms79

, one of which is:

𝑲𝒖𝒔 = �𝑾𝒇

𝑪𝒔𝒇− 𝑾𝒓

𝑪𝒔𝒓� (Eq. 10.1)

2) “Characteristic Speed”, provides a means of comparing designs

with respect to the degree of understeer present80. Mathematically, it is the speed at which the steering angle “δ” to make a turn of radius “r”81

is equal to “2lwb/r” (that is, the steering angle is twice that determined by the low speed Darwin steering geometry “lwb/r”, see Figure C.2):

𝑽𝒄𝒉𝒂𝒓 = �𝒈𝒍𝒘𝒃𝑲𝒖𝒔

(Eq. 10.2)

3) “Critical Speed”, a speed at which a vehicle can switch from initially stable (understeer) to unstable (oversteer) behavior:

𝑽𝒄𝒓𝒊𝒕 = �𝒈𝒍𝒘𝒃−𝑲𝒖𝒔

(Eq. 10.3)

4) “Steering Angle”, as required at the front wheels to make a steady state turn of radius “r” (see Figure 10.4):

𝜹 = 𝒍𝒘𝒃𝒓

+ 𝑲𝒖𝒔𝒂𝒚 (Eq. 10.4)

5) “Static Margin”, a determinant like “Kus” for classifying a vehicle condition as stable (understeering): “SM > 0”, neutral: “SM = 0”, or unstable (oversteering): “SM < 0”; but as a dimensionless fraction

79 Reference [14], pages 161-164. The formulation used is different from that of Eq. 10.1, but the essence is the same. 80 Ergo, both vehicles must be essentially understeering. 81 This turn radius is measured from the origin or center of the turn to the vehicle c.g.


58

with the advantage of relating the longitudinal c.g. placement to the “Neutral Steer Point” for that condition (C.G. – N.S.P. = SM ˟ lwb)82

:

𝑺𝑴 = 𝒍𝒓𝑪𝒔𝒓−𝒍𝒇𝑪𝒔𝒇𝒍𝒘𝒃�𝑪𝒔𝒇+𝑪𝒔𝒓�

(Eq. 10.5)

There are a number of other significant relationships, as well as variations on the above, which result from the “bicycle model”83

analysis or are obtained from similar analyses that differ in the complexity of the model and/or the details of the approach. The advantages of obtaining such relationships from an analysis of the equations of dynamic equilibrium are:

1) Insights are obtained, such as the existence of a critical speed (Eq. 10.3) or the exact role played by the longitudinal c.g. location (Eq. 10.5) in obtaining stability for a particular design possessed of unequal tire characteristics (cornering stiffnesses) front to rear, which would be much more difficult to obtain otherwise. Obviously, the greater the magnitude of a positive Kus or SM the more stability is present. However, if the goal is to obtain the maximum amount of lateral acceleration, then the closer these values are to zero then the closer the design is to its’ maximum lateral acceleration capability given a particular set of tire characteristics (which is a lot more sophisticated judgment than the “50/50” longitudinal weight distribution conclusion of Chapter 4).

2) Whole areas within the realm of linear behavior can be determined as stable or unstable without detail calculations of the sort carried out in Chapter 9 where considerable work went into the investigation of only two points in the Ford Fiesta’s operational envelope: the “4-up” condition at 0.5 g’s and at 0.7 g’s. Given enough computational resources, it would be possible to investigate a vast number of such points so as to gain an understanding of the lateral acceleration / directional stability performance for a specific vehicle throughout its’

82 This is all very similar to aircraft stability theory wherein the relationship between the center of gravity and the center of pressure is expressed in terms of percent of the “Mean Aerodynamic Chord” (MAC). Typical passenger car SM values may range between +0.03 to +0.07. 83 The utilization of the “bicycle” model means that lateral “weight transfer” and roll effects are neglected. Also, the analysis is based on linearized tire functions, so the results are limited in applicability to the lower levels of the performance spectrum under 0.4 g’s (i.e., the “linear” portion of the “Fy, N, ψ” curves).


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operational envelope, regardless of whether the region is linear or not. However, that is appropriate for detail design; in the conceptual design stage the sort of theoretical relationships obtained herein are the most useful.

In conclusion, it is good idea to review the limitations of the theoretical relationships obtained. Remember that the model did not inherently include the effects of “weight transfer” or roll84

, that the model was based on low speed (less than 0.4 g’s) large radius (greater than 24.0 ft or 7.3 m) turns in the steady state condition which means small angles (less than 10 deg or 0.175 rad); all of which means that whatever value the relations have is restricted to the linear tire behavior zone. Given all those restrictions and the various simplifications inherent in the theory, it’s amazing that the results are as useful (at least for passenger car design) as they are.

84 Therefore there is also no accounting for the effects of “roll steer” or “roll camber”, in addition to the effects covered in Chapter 5. These and other effects (aligning torque variation with lateral load, etc.) constitute a large area of design not covered in this paper.


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11 – SAFETY

Just what “safety” is, and what is to be considered “safe”, is ultimately a matter of philosophy. However, once a safety standard has been established, then the matter comes into sharper focus; either the standard is complied with or it is not, and a question regarding compliance can be settled by empirical means. However, there still remains the question as to whether the standard is adequate.

That automotive lateral acceleration, rollover, transient behavior, and directional stability all have to do with safety is understood, but no standards of performance in these areas have established, at least not as a federal regulation with all the force of law. That leaves possible only some general statements regarding those automotive characteristics and safety.

So, in general, the higher the maximum lateral acceleration the safer a vehicle should be, although there will always be some possible occurrence of reckless driving and/or adverse road situations resulting in an accident no matter how many g’s the vehicle may record on the skidpad. It should also be obvious with regard to safety that transient behavior in maneuver should be as smooth and as short-lived a phase as possible, and that with regard to directional stability road going vehicles sold to the general public should maintain a healthy dose of understeer throughout the operational envelope.

Regarding rollover, however, there is something more to be said with regard to safety than just some vague statement urging a high level of performance. Just as with lateral acceleration, raising the level at which out-of-control behavior ensues is well and good, but again no matter how high the lateral acceleration threshold there will always be cases of exceedence. When a vehicle slides off course into guard rails, other vehicles, etc., the passengers have the protection afforded them by carefully engineered crush zones and restraints such as air bags and seat belts; when a vehicle rolls over the passengers essentially have only that protection as may be afforded by divine providence.

As noted in Chapter 8, roll-over accidents in the US for 2009 accounted for a disproportionately large percentage of vehicle accident fatalities even when properly seat-belted in, about 12% or 2821 deaths85


. If the unbelted 5475 that died in roll-over accidents in 2009 had been belted in, then a fair percentage of them would have died anyway, adding substantially to that 12% figure. This is a situation that has been notable for a long period of time, and the obvious expedient of strengthening the roof structure, thus completing the


61

supposedly inviolate nature of the passenger “crash cage”, has been consciously and deliberately avoided by the US automotive manufacturers86

In 1973 Federal Motor Vehicle Safety Standard (FMVSS) 216, Roof Crush Resistance, went into effect. This was a very limited test of roof strength as it involved a quasi-static

for four decades.

87 pushing at constant (slow) speed of a large area steel plate against the most robust portion of the roof in a test with a total force equal to 1.5 times vehicle empty weight88

Of course, a great many people have died (not to mention those suffering quadriplegia, paraplegia, and various other spine and brain injuries) during the struggle to obtain a roof crush standard, and it is certain many more will do so despite the revised standard as such limited quasi-static loadings in no way reflect the magnitude of the dynamic loadings actually experienced in real rollover accidents. This has been a matter of common knowledge for as long as the matter has been in discussion; this is evidenced by the fact every motor sport governing body in existence that sanctions, or did at one time sanction, racing of production motor vehicles required additional structure to resist roof crush.

. That standard was so obviously inadequate that finally in 2009 (36 years later) an improved standard was forced through, which involved applying a force of 3 times the vehicle weight. This new standard will begin to be phased in by 2013 with 100% compliance required of each manufacturer by 2017. Even then, vehicles of GVWR exceeding 10,000 lb will be exempt.

NHRA, NASCAR, SCCA, and many other such organizations have a long history of requiring that “roll bars”, often integrated into complete “roll cages”, of a very definite quality be installed on vehicles used in competition. The effectiveness of such reinforcement has been observed innumerable times in crashes which involved high speed dynamic impacts to the roof structure but resulted in nothing more than a shaken driver. That private organizations, and very often even private individuals, have long achieved that which seems impossible for the federal government and major corporations to do gives cause for concern.

86 Volvo has been traditionally an outstanding exception with regard to roof crush standards, not only in comparison to US manufacturers, but in comparison with automotive manufacturers in general. Three years after buying Volvo in 1999, Ford told Volvo executives that their view on one of the most contentious areas of automotive safety was out of step with Ford's and had to change. Ford executives, certain e-mail messages suggest, were concerned that Volvo's view on roof strength would be used against Ford in rollover cases, a potentially expensive liability concern. 87 This ignored the dynamic nature of most rollover accidents. SAE J996 is a dynamic drop test, but the automotive industry fought against the application of such a test as a federal standard. 88 The wording was “Empty weight, or 5,000 lb, whichever is less”, not GVWR. Vehicles in excess of 6,000 lb GVWR were excluded altogether from need to comply with the standard.


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12 – CONCLUSIONS

This paper has concentrated on the influence of mass properties on certain aspects of automotive performance associated with acceleration in the lateral direction, and in so doing it was inevitable that various non-mass properties parameters would be encountered, if only to note their influence and then exclude them from further discussion. Therefore, if the reader wishes to obtain a more complete and thorough understanding of automotive performance in this regard, then he is referred to the many fine texts listed in the “References” section of this paper (References [1], [7], and [14] are highly recommended).

The mass properties conclusions (plus a few non-mass properties conclusions) to be drawn from this paper are, in chapter by chapter order:

Chapter 1 – Consideration of a turning vehicle free body in plan view (2D) highlights the significance of translational (weight) and rotational (yaw) inertias in determining the magnitudes of the lateral traction forces at the tire/road contact patches necessary to accomplish the maneuver.

Chapter 2 – Vehicle weight and its’ distribution determines the magnitude of the static normal loads at tire/road contact patches; the traction potential that can be generated at those patches varies in accord with those normal loads. The coefficient of traction is itself a function of normal load, thereby resulting in a highly nonlinear relationship between normal load and traction.

Chapter 3 - A simple two dimensional study of the weight transfer along an axle due to lateral acceleration produces two equations: one for the approximate lateral acceleration at which slide will occur, and one for the approximate lateral acceleration at which rollover will occur. These accelerations, in g’s, are numerically equal to the coefficient of traction “μ” necessary to reach those accelerations. The most significant mass property parameters for these lateral acceleration limits are vehicle weight and c.g. height.

𝒂𝒚𝒔𝒍𝒊𝒅𝒆 =−𝟏 +�𝟏−𝟒�

𝟐𝒎𝑾𝒉𝒄𝒈𝟐

𝒕𝟐��𝒎𝑾𝟐 �−𝒃�


(EQ. 3.5)

𝒂𝒚𝒐𝒗𝒆𝒓𝒕𝒖𝒓𝒏 = 𝒕 𝟐𝒉𝒄𝒈⁄ (EQ. 3.6)


63

Chapter 4 - Studying maximum lateral acceleration from just the simple models utilized up to this point places limits on the conclusions drawn. These limits were encountered, some perhaps implicitly, throughout the development of the theory. To keep things straight in mind it is best to explicitly enumerate the conclusions and the limits on those conclusions now. The limitations are:

1) Steady-state turning: constant angular velocity, no braking or drive torque at wheels, no change in turn radius.

2) Conventional automotive configuration: four wheels, two axles in tandem.

3) No roll; suspension characteristics not taken into account (front / rear roll stiffness, compliance, camber change).

4) The front and rear track dimensions are equal. 5) The tire characteristics (“b”, “m”) are equal all around.

These limitations circumscribe the utility of the conclusions. However, the conclusions are still of value as the weight and c.g. are the most basic (but not sole) determining factors of maximum lateral acceleration levels. High lateral acceleration levels can be obtained, even in apparent violation of the following conclusions, by tailoring of suspension characteristics, tire characteristics, and track relationship front to rear. Violation of the conclusions for maximizing lateral acceleration is often done because there are other important design criteria (directional stability, longitudinal acceleration, braking, ride) the pursuit of which requires compromise. However, design cannot stray too far from the following conclusions if lateral acceleration is of any priority:

1) Minimize weight, the lighter the load on the tires the better the relative traction. 2) Locate the vehicle c.g. so that there is an even static weight/area load on all tires (except in case of special purpose vehicles like some race cars that make left or right hand turns exclusively). 3) Minimize c.g. height; the lower the c.g. the less “weight transfer” for a given lateral acceleration, and so the more even the dynamic load on the tires, laterally and longitudinally, which results in greater traction. 4) Maximize the track, which reduces the effect of “weight transfer”, again tending to even the dynamic loads and thereby increase traction. 5) Maximize “b” (increase tire contact area and tread compound “stickyness”), and minimize “m” (brace the tread, keep it flat against the ground).


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The mass properties conclusions may be more succinctly stated: for maximum lateral acceleration a minimum weight and a 50/50 weight distribution is the ideal. However, there are other performance criteria which militate against this “perfect balance”; thus necessitating the manipulation of other parameters to attain an acceptably high lateral acceleration level without “perfect balance”. In such a case, it should be remembered that, no matter how high the lateral acceleration level, it will never be as good as what could have been achieved had the weight balance been “perfect”.

Chapter 5 – In Chapter 3 a “rigid” (no roll) model was used to develop the equations for the limiting lateral accelerations"𝑎𝑦𝑠𝑙𝑖𝑑𝑒”and “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛”; in Chapter 5 the inclusion of roll effects in the vehicle model revealed that roll adversely affects (reduces) these lateral acceleration limits due to a lateral shift in sprung mass c.g. toward the “outer” wheels. Therefore, to the five factors previously identified as necessary for maximizing lateral acceleration levels can now be added a sixth:

6) Minimize roll, which causes an adverse lateral shift in the sprung weight (and often a similar shift in the unsprung weight if the suspension is of an independent type).

The measure of sprung mass roll under lateral load is called “roll gain”. The roll gain for the 1980 Fiesta, which seems limited in lateral acceleration to about 0.71 g’s, calculated out to about 9.2 deg/g, while the 1984 Corvette Z51 had an empirically determined lateral acceleration of 0.95 g’s and a roll gain of 2.0 deg/g89

; this tends to illustrate the association of maximum lateral acceleration with minimum roll.

Chapter 6 – The transient condition represents a period of transition (“response time”) from one steady state condition to another steady state condition, and the design objective is to minimize the duration and smooth the character of this transition as much as possible. To this end the mass properties objectives are the minimization of the roll (“Ix”) and yaw (“Iz”) inertias, plus the minimization of the “Oscillation Center” location factor “X” which involves the yaw radius of gyration “Kz” and the longitudinal c.g. locating lengths “lf” and “lr”:

X = Kz2/( lf x lr)

In its’ simplest form, this is yet another argument for minimizing the yaw radius of gyration “Kz” (the numerator) while getting the weight distribution as close to “50/50” as possible (maximizing the denominator “lf x lr ”).

89 Reference [8], pages 39-40. The reported figures were GM test results.


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Chapter 7 – The terms “oversteer”, “neutral steer”, and “understeer” are introduced and defined as characteristics of a basic vehicle handling tendency which is observable in a skidpad test. Also measurable in a skidpad test is the maximum lateral acceleration capability of a vehicle. Because of the possibility of lateral asymmetry in vehicle construction, the most important of which would be an offset lateral (not on the vehicle centerline) c.g., the skidpad test must be run at least twice, once CW and once CCW, with the results averaged. The skidpad size (radius, diameter) is very significant, as the larger the skidpad the higher the maximum lateral acceleration obtained due to decreased angularity and possibly greater aerodynamic download (race cars). A 100 ft (30.5 m) radius is the most common, and probably constitute the minimum acceptable size, for passenger car testing; much larger sizes are appropriate for vehicles designed to operate while generating significant down force (race cars).

It should go without saying that skidpad, and all other performance testing, should only be carried out for a vehicle in a standard, reproducible weight condition (curb weight plus driver, etc.). However, the obvious is often the most frequently overlooked, which would tend to account for the wide discrepancies in some reported skidpad test results for the same vehicle90

Chapter 8 – There are numerous means to obtain an empirical rating for a vehicle’s rollover resistance. However, the most significant rating used is calculated, the “SSF” (Static Stability Factor), which is numerically equal to the “𝑎𝑦𝑜𝑣𝑒𝑟𝑡𝑢𝑟𝑛” calculation in Chapter 5:

.

𝑺𝑺𝑭 = 𝒕𝟐𝒉𝒄𝒈

(EQ. 8.1)

The SSF is used by the NHTSA as part of the NCAP for rollover resistance ratings because there is a strong correlation between the SSF and accidents involving vehicle rollover91

90 Reference [8], pg. 40: ‘84 Corvette Z51 rated at 0.95 g’s (GM test). Reference [17], pg. 65; ’84 Corvette Z51 rated at 0.84 g’s (C/D test). Test weights are not reported, but even “standard” curb weights can vary by 100 lbs or more.

. However, that correlation is such that it can be improved. For instance, it has been noted that the correlation improves when certain other parameters are added to the regression analysis, such as vehicle wheelbase. However, because the link between such other parameters and rollover can’t be incontrovertibly demonstrated, as it can for c.g. height and track width, those other parameters are not used in the determination of rollover rating of “one



66

to five stars”92. So, use of the SSF results in certain anomalies, such as the Ford Pinto and the Chevrolet Vega obtaining the same SSF, yet one was significantly more inclined to be involved in rollover accidents than the other93

It is largely because of this indication of “something lacking” in rollover rating methodology, and manufacturer’s pressure supporting an empirical “dynamic” method, that the NHTSA has come to include the results of the “Fishhook” test along with the SSF (heavily biased toward the SSF) in establishing NCAP rollover ratings. Yet this is hardly an ideal situation; the Fishhook test is time consuming, arduous, and expensive. Consequently the NHTSA has taken the expedient of not actually subjecting every vehicle to the test; some are rated according to “similarity” with other vehicle’s which have undergone the test, based on vehicle configuration, suspension type, etc. This is hardly an ideal state of affairs, and it is hard to see where a significant improvement in methodology is obtained by including a dynamic test when implemented in such a fashion.

.

In the interest of improving the NHTSA NCAP rollover resistance rating system, while reducing effort and expense, the following modest proposal is made. The SSF is only of value as a “first order estimate” of a vehicle’s resistance to rollover, and is very conservative in nature (actual roll over occurs much more easily)94

. It doesn’t take into account the basic suspension characteristics that so strongly influence roll as demonstrated in Chapter 5. That situation would be largely corrected if the formula for SSF as used for rollover resistance rating were revised to:

SSF = 𝒕𝟐−𝒅𝒚𝒉𝒄𝒈−𝒅𝒛

(EQ. 12.1)

The symbols “dy” and “dz” represent the dimensional changes due to roll as was illustrated in Figure 5.3 (“dy = hr cos θs”, “dz = hr sin θs”):

92 Such “precise” calibration is typical of all NCAP ratings. 93 The Ford Pinto and the Chevrolet Vega are US automotive products of the 1970’s era; the NHTSA rated vehicles for rollover resistance solely on the basis of the SSF during the years 2001-2003. However, the Pinto and Vega were included in studies leading to the adoption of the SSF as a rating tool. NHTSA began to use the combination of the SSF and dynamic maneuver for establishing rollover resistance ratings in 2004. 94 Reference [7], page 312.


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Figure 5.3 – ROLL GEOMETRY

As those dimension changes are dependent upon the roll angle “θs” that angle must be determined using Equation 5.1. That leaves only the question of what lateral acceleration level “ay” to use in the Equation 5.1 computation of “θs”; this author suggests use of that value of “ay” obtained by the original first order approximation. However, caution must be taken to ensure that the “θs” value obtained is not beyond the physical capability of the automobile in question to achieve. The limiting value of “θs” at the point the vehicle sprung mass rolls into contact with the suspension snubbers can be determined by an easy study of the suspension system geometry, or by simple empirical means.

The result of this modest proposal should be a significant increase in correlation95

Chapter 9 – An automobile’s lateral behavior is almost totally dependent upon its’ tires drift angles, and a tire’s drift angle is totally dependent upon the magnitude of the normal, lateral, and longitudinal forces incumbent upon the tire; all these forces are weight and mass properties (c.g., roll / pitch / yaw inertia) driven to varying degree. The most important distinction to keep clear in one’s mind when dealing with tire forces is the difference between

and a considerable improvement in productivity and expenditure if the Fishhook test is consequently abandoned for rating purposes. To counter manufacturer objections that to rely on even the modified SSF alone for establishing rollover resistance ratings is still not fully taking all pertinent factors into account, the presence of supposed mitigating factors such as the installation of Electronic Stability Control (ESC) could be given the same “weight” as the Fishhook test’s “pass” or “fail” had in previously influencing the overall combined rating.

95 The correlation can possibly be further improved in that the “dy, dz” dimensional changes apply to the sprung mass c.g. and are here applied to the vehicle c.g. as a good approximation. This was done to keep the matter simple, as simplicity is one the advantages of SSF usage. However, at the expense of somewhat more calculation, the corresponding total vehicle dimensional changes can be easily determined.


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a potential traction force and a required traction force. Up to the point of slide the required traction forces will be less than the potential, and the required force distribution among the tires will reflect the potential force distribution.

The drift angle distribution will reflect the required force distribution, or vice versa, depending upon the perceived order of cause and effect. In accord with the drift angles at the wheels the resultant geometry will determine the vehicle path (maneuver radius “r”) and the vehicle attitude (yaw angle “β”). There are many factors which influence the tire traction force / drift angle relationship, but the most fundamental is normal load which relates back to vehicle weight and weight distribution (c.g. location).

Chapter 10 - The terms “oversteer”, “neutral steer”, and “understeer”, previously introduced in Chapter 7 as relating to observable handling behavior, acquire additional significance in Chapter 10 as the terms are now related to vehicle directional stability. The subject of vehicle stability is immense and complicated beyond the scope of this paper; only an overview of ground vehicle stability, to the extent that it furthers basic understanding of the roll of mass properties, is attempted.

The essence of vehicle directional stability is a matter of obtaining sufficiently “understeering” behavior by loading the front wheels, both statically and dynamically, so as to ensure that the front wheels will always be operating at greater drift angles than the rear wheels. Unfortunately, this also ensures that the front axle will have a lower "𝑎𝑦𝑠𝑙𝑖𝑑𝑒” than would be the case otherwise. So, to the six factors necessary for maximizing lateral acceleration levels can now be added a seventh:

7) Minimize stability to the bare minimum necessary to maintain control on the skidpad. The means used to enhance stability (“nose down” roll axis inclination, etc.) do so by ensuring a “weight bias” to the front (especially the front outer wheel in the dynamic case) thus violating the 11th Commandment “Thou shall keep thy contact patch pressures as even as possible all around if thou wish for thy vehicle to hath maximum lateral acceleration”.

One of the key relationships resulting from a directional stability analysis and highlighted in Chapter 10 is the “Understeer Coefficient”. This coefficient is symbolized as “Kus” and expressed in radians or degrees; the formulation for this coefficient is:

𝑲𝒖𝒔 = �𝑾𝒇

𝑪𝒔𝒇− 𝑾𝒓

𝑪𝒔𝒓� (Eq. 10.1)


69

Since the coefficient is based on a front (subscript “f”) to rear (subscript “s”) comparison of the lateral force (assumed to be in proportion to the front “Wf” and rear “Wr” vehicle weight distribution) to cornering stiffness ratios, stability (understeering) can readily be determined (“Kus > 0”) for whole areas within the linear tire region.

Another key relationship is the “Static Margin”, which is symbolized as “SM” and

expressed as a portion of the vehicle wheelbase:

𝑺𝑴 = 𝒍𝒓𝑪𝒔𝒓−𝒍𝒇𝑪𝒔𝒇𝒍𝒘𝒃�𝑪𝒔𝒇+𝑪𝒔𝒓�

(Eq. 10.5)

The greater the magnitude of a positive SM the more stability is present. However, if the

goal is to obtain the maximum amount of lateral acceleration, then the closer SM is to zero the closer the design is to its’ maximum given a particular set of tire characteristics. This a more sophisticated judgment than the Chapter 4 conclusion that a “50/50” longitudinal weight distribution is the means to maximum lateral acceleration; the SM takes weight distribution and tire characteristics into account.

Chapter 11 – This chapter, termed “Safety”, was not originally envisioned as a part of this paper; although safety issues are inherent in all aspects of vehicle performance. This paper’s topic is the role mass properties play in regard to lateral acceleration performance; any associated safety issues were to be given just passing mention. However, it was impossible to study the matter of vehicle rollover without becoming aware of the issue of roof crush resistance, and the horrific role the automobile industry has played for decades in denying the obvious96

The essence of the problem is that once you dance with the Devil it’s hard to get free. The auto industry drifted into a situation of inadequate roof strength during a period of a general unawareness of safety issues, and to forthrightly admit now the truth about roof strength would open the door to thousands, if not millions, of lawsuits by the injured and/or the relatives of the deceased. So the industry continues to stonewall against the inevitable while thousands more suffer the adverse effects.

. Hence the creation of this chapter as a sounding board for the author’s opinion, even though the issue really has little to do with mass properties, but much to do with human cupidity.

The solution to all this is a clean slate; legislation establishing a mandate for the NHTSA to rigorously enforce new realistic dynamically based roof crush standards, while limiting the 96 Denying the obvious has long been a tradition in the automotive industry. Maurice Olley relates in his memoirs how Harold Wilson, head of the Automobile Manufacturers Association, in an address to the Surgeon General’s Conference in November of 1959, denied that automobile exhaust was in any way responsible for air pollution (Milliken, Chassis Design, Principles and Analysis, page 577.)


70

liability incurred by the industry for all previously incurred claims, is the obvious and rational solution. However, since it is obvious and rational, it probably has little chance of occurring anytime soon.


71

REFERENCES

[1] Bastow, Donald; Car Suspension and Handling, Plymouth, Devon; Pentech Press Ltd., 1980.

[2] Boyd, Patrick L.; “NHTSA’S NCAP Rollover Resistance Rating System”, Washington, DC; NHTSA Paper 05-0450, June 2005.

[3] Costin, Michael; and David Phipps, Racing and Sports Car Chassis Design, Cambridge, Mass.; Robert Bentley, 1967.

[4] Campbell, Colin; Design of Racing Sports Cars, Cambridge, Mass.; Robert Bentley, 1976.

[5] Campbell, Colin; The Sports Car, Cambridge, Mass.; Robert Bentley, 1969.

[6] Campbell, Colin; New Directions in Suspension Design, Cambridge, Mass.; Robert Bentley, 1981.

[7] Gillespie, Thomas D.; Fundamentals of Vehicle Dynamics, Warrendale, PA; SAE R-114, 1992.

[8] Grable, Ron; “Formula One Chevy: A Technical Overview of the World’s Best Handling Production Car”, Motor Trend, pages 37-40, March 1983.

[9] Highway Loss Data Institute, Insurance Institute for Highway Safety, “Rollover and Roof Strength”, www.iihs.org, March 2011.

[10] King-Hele, Desmond; “Erasmus Darwin’s Improved Design for Steering Carriages – and Cars”, Notes Record Royal Society 56, London, pages 41-62, 2002.

[11] Lamar, Paul; “More About Cornering Power”, Road & Track, pp. 127-131, October 1969.

[12] Martinez, Rosendo; “Vehicle Cornering Stability and C.G. Limits”, Los Angeles, CA; SAWE #1930, 1990.

http://www.iihs.org/�


72

[13] McKibben, Jon; “Group 7 Competition, Two Seat Racing with No Holds Barred”, Car Life, pages 20-23, January 1968.

[14] Milliken, William F.; and Douglas L. Milliken, Race Car Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995.

[15] NHTSA, “Rollover Prevention Docket No. NHTSA-2000-6859 RIN 2127-AC64”, 2000.

[16] Simanaitis, Dennis; “Road Testing at R&T”, R&T Road Test Annual, pp. 12-16, 1982.

[17] Sherman, Don; “Chevrolet Corvette, Four C/D Apostles Bring Home Seven Revelations”, Car and Driver, pages 63-68, October 1983.

[18] Smith, Kevin; “The New Corvette”, Road & Track, pages 25-39, March 1983.

[19] Wiegand, B.P.; “Mass Properties and Automotive Vertical Acceleration”, Los Angeles, CA; SAWE #3521, 2011.

[20] No Byline, “Corvette vs. Ferrari 308GTBi Quattrovalvole, vs. Porsche 928S, vs. Porsche 944”, Road & Track, pages 53-59, August 1983.


73

AUTHOR’S BIOGRAPHICAL SKETCH

Brian Paul Wiegand, now retired, was a Senior Weight Engineer and Mass Properties Handling Specialist for the Mass Properties Analysis and Control Group of Northrop Grumman Corporation, Bethpage, NY. He is a 1972 graduate of Pratt Institute, Brooklyn, NY, and a licensed Professional Engineer registered in the State of New York (# 58470). He continues to be an active member of the Society of Allied Weight Engineers and of the Society of Automotive Engineers. He has presented three SAWE papers: “Mass Properties and Automotive Longitudinal Acceleration” (SAWE #1634, 1984), “The Basic Algorithms of Mass Properties Analysis & Control” (SAWE #2067, 1992), and “Automotive Mass Properties Estimation” (SAWE #3490, 2010). He has also published two articles: “The Weight and C.G. Implications of Obtaining Maximum Automotive Lateral Acceleration Levels” (SAWE Journal Weight Engineering, Winter 1982-’83), and “The Mystery of Automotive POI Values” (SAWE Journal Weight Engineering, Spring 2011). Recently he reaffirmed his long-standing interest in automotive engineering by attending the SAE Seminar “Vehicle Dynamics for Passenger Cars and Light Trucks” (Troy, MI; August 11-13, 2009)


74

APPENDICES


75

APPENDIX A – SYMBOLISM

a Translational acceleration, usually in gravity units, also can mean the same as “lf” per SAE J670e (in, cm).

ay Lateral translational acceleration, usually in gravity units.

ayslide Lateral translational acceleration, usually in gravity units, sufficient to cause loss of traction and sliding.

ayoverturn Lateral translational acceleration, usually in gravity units, sufficient to cause vehicle overturn (rollover).

α The Greek lower case letter “alpha” meaning rotational acceleration, usually in radians/second2.

b The basic tire coefficient of traction (dimensionless), also can mean the same as “lr” per SAE J670e (in, cm).

β The Greek lower case letter “beta” used to represent the “yaw angle” (angle between the vehicle centerline and the tangent line to the turning circle) of a maneuvering vehicle.

Cs Cornering stiffness, or “ΔFy/Δψ”, in lb/deg, kg/rad, or kN/rad. See Figure 9.2.

Csf The cornering stiffness of the tires on the front axle.

Csr The cornering stiffness of the tires on the rear axle.

C/D Car and Driver magazine is a major US reviewer and tester of new production automobiles; it was founded as “Sports Car Illustrated” in 1955.

CW The clockwise direction of rotation.

CCW The counter-clockwise direction of rotation.

CG Center of gravity.

CP Aerodynamic center of pressure.

C4 The fourth generation of the General Motors, Chevrolet Division, Corvette sports car, which is now in its’ sixth generation (C6).

DI The Dynamic Index, equal to the sprung mass pitch radius of gyration squared divided by “lf ˟ lr” (dimensionless).


76

δ The Greek lower case letter “delta” used to symbolize the steer angle (degrees, radians)

dy The lateral shift of the c.g. of the sprung mass in roll (in, cm).

dz The vertical decrease of the height of the sprung mass c.g. in roll (in, cm).

F A force, as in “F = ma” (lb, kg).

Fd A lateral disturbance force (lb, kg).

Ff A lateral force in the ground plane acting at the front axle (lb, kg).

Fr A lateral force in the ground plane acting at the rear axle (lb, kg).

Fx A tire traction force acting in the longitudinal direction (lb, kg).

Fy A tire traction force acting in the lateral direction (lb, kg).

Faxle A force in the ground plane acting at an axle (lb, kg).

Fxf The longitudinal force in the ground plane acting at a front axle (lb, kg).

Fxfi The longitudinal force in the ground plane acting at a front “inner” wheel (lb, kg).

Fxfo The longitudinal force in the ground plane acting at a front “outer” wheel (lb, kg).

Fxr The longitudinal force in the ground plane acting at a rear axle (lb, kg).

Fxri The longitudinal force in the ground plane acting at a rear “inner” wheel (lb, kg).

Fxro The longitudinal force in the ground plane acting at a rear “outer” wheel (lb, kg).

Fyf The lateral force in the ground plane acting at a front axle (lb, kg).

Fyfi The lateral force in the ground plane acting at a front “inner” wheel (lb, kg).

Fyfo The lateral force in the ground plane acting at a front “outer” wheel (lb, kg).

Fyr The lateral force in the ground plane acting at a rear axle (lb, kg).

Fyri The lateral force in the ground plane acting at a rear “inner” wheel (lb, kg).


77

Fyro The lateral force in the ground plane acting at a rear “outer” wheel (lb, kg).

GM General Motors Corporation, once the largest automotive manufacturer in the world, and one of the world’s largest manufacturing enterprises, was founded in 1908. On 10 July 2009 a new “General Motors Corporation” was created from the bankrupt remains of the original company.

GTP Grand Touring Prototype was an automotive racing category created under the jurisdiction of the IMSA.

HRM Hot Rod Magazine is the oldest existing magazine (US) dedicated to the modification and customization of production vehicles; it was founded in 1948.

hcg The height of the center of gravity above the ground plane (in, ft, cm, m).

hr The distance between the RC and the CG (in, ft, cm, m).

hrc The height of a roll center above the ground plane (in, ft, cm, m).

hrrc The height of the rear roll center above the ground plane (in, ft, cm, m).

hfrc The height of the front roll center above the ground plane (in, ft, cm, m).

I A vehicle rotational inertia, units lb/in2, lb/ft2, kg/cm2, kg/m2.

Ix The vehicle roll rotational inertia, units lb/in2, lb/ft2, kg/cm2, kg/m2.

Iz The vehicle yaw rotational inertia, units lb/in2, lb/ft2, kg/cm2, kg/m2.

IMSA The International Motor Sports Association is a major sanctioning body for automotive competition, and was founded by John Bishop (formerly of the SCCA) and his wife Peggy in 1969.

K A radius of gyration, usually pitch per SAE J670e (in, ft, cm, m).

Kus The understeer coefficient “�𝑊𝑓

𝐶𝑠𝑓− 𝑊𝑟

𝐶𝑠𝑟�”, allows for a determination of stability

in the linear region of the vehicle operational envelope. Units: degrees or radians. See Equation 10.1.

Ky The sprung mass pitch radius of gyration (in, ft, cm, m).

Kz The vehicle yaw radius of gyration (in, ft, cm, m).

kroll The total vehicle roll stiffness (lb-ft/deg, Nm/deg).


78

kfroll The roll stiffness at the front axle (lb-ft/deg, Nm/deg).

krroll The roll stiffness at the rear axle (lb-ft/deg, Nm/deg).

lf The distance from the front axle to the c.g. (in, cm).

lr The distance from the front axle to the c.g. (in, cm).

lwb The “wheelbase” distance from the front axle to the rear axle (in, cm).

MIRA The Motor Industry Research Association (Ltd) was founded 1949 in the UK and is now a major automotive test facility with aerospace, rail, and other branches.

M/T Motor Trend magazine is a major automotive publication (US) founded in 1949.

m Mass, as in “F = ma”, or the rate of tire traction coefficient decrease with normal load.

N The normal load at a tire/road contact patch (lb, kg).

NASCAR The National Association for Stock Car Auto Racing was founded by Bill France Sr. in 1947. It sanctions and governs various auto racing events, and is the largest sanctioning body for “stock car” racing in the United States.

NCAP The NHTSA’s New Car Assessment Program was created in 1979 in response to Title II of the Motor Vehicle Information and Cost Savings Act of 1972.

NHRA The National Hot Rod Association was founded by Wally Parks in 1951. It is one of the world’s largest auto sports governing organizations, specializing in drag racing.

NHTSA The National Highway and Traffic Safety Agency is a subdivision of the US Department of Transportation (DOT). It’s mission statement is “Save Lives, Prevent Injuries, Reduce Vehicle Related Crashes”.

Ni The normal load at an “inside” wheel tire/road contact patch (lb, kg).

No The normal load at an “outer” tire/road contact patch (lb, kg).

Nfi The normal load at an “inside” front wheel tire/road contact patch (lb, kg).

Nfo The normal load at an “outer” front tire/road contact patch (lb, kg).


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Nri The normal load at an “inside” rear wheel tire/road contact patch (lb, kg).

Nro The normal load at an “outer” rear tire/road contact patch (lb, kg).

OC The “Oscillation Center” is a point in vehicle plan view about which the vehicle initially tends to rotate during the transient phase of a maneuver.

θs The Greek lower case letter “theta” with subscript “s” meaning roll angle of the sprung mass (weight), units in degrees or radians.

RC Roll center, which is a point about which the sprung mass will rotate under lateral acceleration due to suspension kinematics.

R&T Road & Track magazine is an automobile enthusiast’s magazine founded at Hempstead, NY, USA in 1947; it was perhaps the most technically orientated of all major automotive publications.

r The radius of a turn (ft, m).

rrf The rolling radius at the front axle (in, cm).

rrr The rolling radius at the rear axle (in, cm).

SAE The Society of Automotive Engineers (now “SAE International”).

SAWE The Society of Allied Weight Engineers, which is the premier professional society for engineers involved in mass properties analysis and control.

SCCA The Sports Car Club of America is a descendent of the Automobile Racing Club of America founded in 1933 by brothers Miles and Sam Collier, which was dissolved in 1941 at the entry of the US into WW II. In 1944 it was reconstituted as the SCCA, and began sanctioning road racing events in 1948.

SPR Side Pull Ratio, a test to determine the lateral acceleration rollover point.

SSF This represents the “Static Stability Factor”, which is calculated as “t/2hcg” (dimensionless). The SSF is numerically equal to “ayslide”, and also numerically equal to the coefficient of traction necessary for overturn to occur without the aid of “tripping”, etc.

SM The “Static Margin” is not only a measure of stability, but allows for determination of the location of the longitudinal c.g. in order to achieve that degree of stability. See Equation 10.5.


80

TC The “Turn Center” which is the origin of a vehicle maneuver path radius.

TTR The “Tilt Table Ratio”, a test to determine the lateral acceleration rollover point.

t Vehicle track width, from tire tread center to tire tread center (in, cm).

tf Vehicle front track width, from tire tread center to tire tread center (in, cm).

tr Vehicle rear track width, from tire tread center to tire tread center (in, cm).

μ The Greek lower case letter “mu” meaning the tire coefficient of traction “μ = b – mN” (dimensionless).

V Vehicle velocity, sometimes called “speed” (mph, ft/sec, kph, m/sec).

Vchar The “characteristic velocity” provides a means of comparing the degree of understeer present in one design to the degree of understeer present in another design; see Equation 10.1. Units: ft/sec, m/sec, mph, or kph.

Vcrit The “critical velocity” is a speed at which a vehicle can switch from initial understeer (stable) to oversteer (unstable) behavior; see Equation 10.3. Units: ft/sec, m/sec, mph, or kph.

W A vehicle weight, usually “curb weight plus driver” (lb, kg).

Wf The portion of vehicle weight borne by the front axle (lb, kg).

Wr The portion of vehicle weight borne by the rear axle (lb, kg).

Ws The weight of the vehicle sprung mass (lb, kg).

Wsf The weight of the front axle vehicle sprung mass (lb, kg).

Wsr The weight of the rear axle vehicle sprung mass (lb, kg).

Wus The weight of the vehicle unsprung mass (lb, kg).

Wusf The weight of the vehicle unsprung mass at the front axle (lb, kg).

Wusr The weight of the vehicle unsprung mass at the rear axle (lb, kg).

X A quantity similar to DI, but calculated for the total vehicle from “Kz2/( lf ˟

lr)”, and is dimensionless.

X,Y,Z Center of gravity coordinates (in, ft, cm, m).


81

ψ The Greek lower case letter “psi” used to symbolize the drift (“slip”) angle (degrees, radians).

ψfi The Greek lower case letter “psi” used to symbolize the drift (“slip”) angle at the front inner (“fi” subscript) tire (degrees, radians).

ψfo The Greek lower case letter “psi” used to symbolize the drift (“slip”) angle at the front outer (“fo” subscript) tire (degrees, radians).

ψri The Greek lower case letter “psi” used to symbolize the drift (“slip”) angle at the rear inner (“ri” subscript) tire (degrees, radians).

ψro The Greek lower case letter “psi” used to symbolize the drift (“slip”) angle at the front outer (“fo” subscript) tire (degrees, radians).


82

APPENDIX B – LATERAL ACCELERATION PROGRAM

The following computer program was used to generate the data for all the graphs in the first few chapters of this paper. It was written for the Commodore 2001 “PET” minicomputer97

and, with slight modifications, may be used with any other computer utilizing BASIC. If the single axle focus of this program is changed to reflect two axles separated longitudinally by some wheelbase length “lwb”, and the normal loads “Ni” and “No” are determined for each axle on the basis of Eq. 5.2/5.3 (front) and Eq. 5.4/5.5 (rear), and if provision is made to calculate the necessary lateral forces at each axle to generate lateral acceleration “ay” and rotational acceleration “α” along with the associate slip angles, then the basis for a lateral acceleration (maneuvering) simulation program will be obtained. However, the “devil is in the details”!

“MAXG/LAT”

1 PRINT “PRG FOR MAX LAT ACCEL: TIRE FORCES @ AY”

2 PRINT “WIEGHT & CG EFFECTS PRIMARY”

3 PRINT “UNITS: LBS, INCHES”

4 PRINT “INPUT AXLE WT LOAD, TIRE COEFFS, CG HEIGHT, TRACK”

5 INPUT W, B, M, HCG, T

6 AA=2*M*W

7 AB=HCG^2

8 AC=T^2

9 AD=(AA*AB)/AC

10 AE=(AA/4)-B

11 AF=SQR(1-(4*AD*AE))

12 AS=(AF-1)/(AD*2)

13 AOT=T/(2*HCG)

14 PRINT “MAX LAT ACCEL (SLIDE), MAX LAT ACCEL (OVERTURN), IN G’S”

15 PRINT AS,AOT

16 PRINT “DECISION: INPUT 1 FOR MAX ACCEL ITERATION, 0 FOR FORCE @AY”

17 INPUT DEC

97 This represented the state of the art in personal computers circa 1979.


83

18 IF DEC>0 THEN GOTO 3

19 PRINT “INPUT AY IN G’S”

20 INPUT AY

21 WTF=(W*HCG*AY)/T

22 NS=W/2

23 NI=NS-WTF

24 N0=NS+WTF

25 FI=(B-M*NI)*NI

26 FO=(B-M*NO)*NO

27 FAXLE=FI+FO

28 PRINT “AY, INNER FORCES (NORM & LAT), OUTER FORCES (NORM & LAT) , FAXEL”

29 PRINT AY,NI,FI,N0,F0,FAXEL

30 PRINT “DECISION: INPUT 1 FOR FORCE ITERATION, 0 FOR END”

31 INPUT DEC

32 IF DEC>0 THEN GOTO 19

33 END


84

APPENDIX C - STEERING

In 1758 Erasmus Darwin (1731-1802), future grandfather of Charles Darwin (1809-1882), invented the so-called “Jeantaud-Ackermann” steering system (Figure C.1d) in order to rectify some of the shortcomings of the “axle” steering system (Figure C.1b) then in use for horse drawn carriages, coaches, and wagons98. The main shortcoming of the axle steering system was that the steered wheels had to be significantly smaller in radius than the non-steered wheels in order to clear the body/chassis at full lock; this incurred significantly greater road shock transmission as such shock varies inversely with the size of the wheel99

Figure C.1 – STEERING GEOMETRY SYSTEMS

. Another shortcoming was that axle steering reduced the vehicle resistance to overturn when cornering. A third shortcoming, which would not be readily apparent until the birth of the automobile, was that the encountering of a bump in the road by one of the steered wheels would generate a large impact moment about the axle pivot point. The drivers of some of the earliest automobiles would experience these moments as severe “kick-back” through the tiller.

98 Reference [10], page 41. 99 Reference [19], pp. 20-21 & 118-130.


85

At around 1816, Darwin’s steering innovation was apparently independently reinvented by a German carriage manufacturer named George Lankensperger. A certain Rudolf Ackermann became acquainted with Lankensperger and his invention and, acting obstensibly as Lankensperger’s agent, obtained an English patent (number 4212) in his own name in 1818. However, the geometry of the steering arrangement in Ackermann’s patent illustration was both extreme (control arm inclination with respect to the longitudinal axis was only about 6.5 degrees, while Darwin’s arrangement involved angles of 23 to 30 degrees) and difficult to discern. Consequently, “Ackermann Steering” became associated with the vastly inferior geometrical layout of Figure C.1c100

In 1878 French carriage maker Charles Jeantaud corrected the “Ackermann Steering” back to the original (Darwinian) concept of Figure C.1d. However, to this day the steering geometry in question is still known as “Ackermann Steering”, although Rudolf Ackermann, despite all his scheming perfidy, never profited from his patent. Life is not only unfair, but often ironic.

(zero degrees inclination of the steering arms with the longitudinal axis).

The following diagram is of “Darwin Steering” geometry in a low speed (no drift angles) turn, which is the only kind of turn for which such geometry is truly valid:

Figure C.2 – DARWIN STEERING GEOMETRY (LOW SPEED TURNS)

100 An even earlier (1714) claimant for this flawed but significant alternative to axle steering may be a certain “M. Du Quet” who possibly indicated such steering on his wind-driven carriage; see Reference [10], page 61.


86

APPENDIX D – DERIVATION OF EQUATION 3.5

By inspection of Figure 3.2, it can be seen that the lateral acceleration level for an axle at which slide will be incipient is the point “A” where the lateral inertial load “W ay” is just equal to the lateral potential traction force (with weight transfer) of Equation 3.4:

𝑾𝒂𝒚 = �𝒃 −𝒎�𝑾𝟐−𝑾𝒂𝒚 �


𝑾𝟐−𝑾𝒂𝒚 �


+ �𝒃 −𝒎�𝑾𝟐


𝑾𝟐


Implicit in this is the assumption that the tires will have the same lateral traction parameter values on both the “inside” and the “outside” ends of an axle in a turn (“b = bi = bo”, “m = mi = mo”), which is a reasonable assumption except for some circle track race vehicles. With that in mind, the above simplifies into:

W𝒂𝒚 = 𝑾𝒃− 𝒎𝑾𝟐

𝟐− 𝟐𝒎�𝑾𝒉𝒄𝒈𝒂𝒚

𝒕�𝟐

This can be divided through by “W” and rearranged to become:

�𝟐𝒎𝑾𝒉𝒄𝒈

𝟐

𝒕𝟐�𝒂𝒚𝟐 + 𝒂𝒚 − �𝒃 −

𝒎𝑾𝟐

� = 𝟎

This is a quadratic equation of the form “aX2 + bX + c = 0” and can be solved by using the standard equation for the quadratic roots:

𝑿 =−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

So, by substituting “ay” for “X”, etc., and ignoring the negative root which has no physical meaning here, the result is obtained:


−𝟏 + �𝟏 − 𝟒�𝟐𝒎𝑾𝒉𝒄𝒈

𝟐

𝒕𝟐 � ��𝒎𝑾𝟐 � − 𝒃�


Q.E.D.


87

APPENDIX E – ROLL STIFFNESS DETERMINATION

For an independent suspension, the roll stiffness at that particular axle line is101

𝒌𝒓𝒐𝒍𝒍 = 𝒌𝒄𝒔𝒕𝟐

𝟏𝟑𝟕𝟓 (EQ. E.1)

:

Where:

kroll = The roll stiffness at an axle line, lb-ft/degree.

kcs = The effective suspension spring & tire stiffness at a wheel, lb/in.

t = The track width at the axle line, in.

Exempli gratia, the 1980 Ford Fiesta S had a MacPherson strut front suspension with effective suspension spring and tire stiffness at a wheel of 123.9 lb/in, and a front track of 52.5 in, so the front roll stiffness is102

kroll = 123.9 (52.5)2 / 1375 = 248.4 lb-ft/degree

:

103

For a non-independent suspension, the roll stiffness at that axle line is

104

𝒌𝒓𝒐𝒍𝒍 = �𝒌𝒔𝒕𝒔𝒃𝟐��𝒌𝒕𝒕𝟐�𝟏𝟑𝟕𝟓�𝒌𝒔𝒕𝒔𝒃𝟐+𝒌𝒕𝒕𝟐�

(EQ. E.2)

:

Where:

kroll = The roll stiffness at an axle line, lb-ft/degree.

ks = The effective suspension spring stiffness at a wheel, lb/in.

tsb = The “spring base” distance between springs along an axle, in.

kt = The spring stiffness of a tire on the axle, lb/in.

t = The track width at the axle, in.

Continuing with the example, the Fiesta had a dead beam / coil spring rear suspension with effective suspension spring and tire stiffness “kcs” of 142.7 lb/in (assuming a “kt” of

101 Reference [14], page 603. 102 Reference [6], page 191. 103 Ibid, page 192 quotes 218.3 lb-ft/deg (296 Nm/deg), which is used throughout Chapters 5 and 9 of this paper. 104 Reference [14], page 604.


88

1800 lb/in, by reverse calculation the “ks” is 155.0 lb/in). The rear track was 52.01 in105

kroll = (155.0 (36.0)2)(1800(52.01)2) / (1375(155.0(36.0)2 + 1800(52.01)2)) = 140.3 lb-ft/deg

, and the rear spring base is 36.0 inches (as estimated by this author), so the rear roll stiffness is:

However, the Fiesta also had a roll bar (a.k.a. “stabilizer bar”, “anti-roll bar”) at the rear, and its’ resistance to roll of 37.0 lb-ft/deg must be added in to the quantity just calculated:

kroll = 140.3 lb-ft/deg + 37.0 lb-ft/deg = 177.3 lb-ft/deg106

It should be noted that both the roll resistances calculated herein for the 1980 Ford Fiesta S (1.1 liter, European version) are both about 14% greater than figures quoted by Reference [6]; the reason for this is unknown, but for the sake of comparability of results the quoted figures were used for calculations within the body of this paper.

105 Reference [6], pp. 191-192. 106 Ibid, page 192 quotes 154.9 lb-ft/deg (210 Nm/deg), which is used throughout Chapters 5 and 9 of this paper.


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