The quantification of automotive directional stability may be expressed through various stability metrics, but perhaps the most basic of these automotive stability metrics is the “Understeer Gradient” (Kus). The Understeer Gradient (in degrees or radians per unit gravity) appears extremely uncomplicated when viewed in its most common formulation. This metric appears to depend only on the front and rear axle weight loads (Wf, Wr), and on the front and rear axle cornering stiffnesses (Csf, Csr). However, those last quantities vary with lateral acceleration, and the nature of that variation is dependent upon many other parameters of which some of the most basic are: Total Weight, Sprung Weight, Unsprung Weight, Forward Unsprung Weight, Rear Unsprung Weight, Total Weight LCG, Sprung Weight LCG, Total Weight VCG, Sprung Weight VCG, Track, Front Track, Rear Track, Roll Stiffness, Front Roll Stiffness, Rear Roll Stiffness, Roll Axis Height, Front Roll Center Height, and Rear Roll Center Height. Note that exactly half of these automotive directional stability parameters as listed herein are mass properties. The purpose of this paper is to explore, through a skidpad simulation, the relative sensitivity of automotive directional stability (as quantified through the Understeer Gradient) to variation in each of the noted vehicle parameters, with special emphasis on the mass property parameters. The simulation is constructed in a spreadsheet format from the relevant basic automotive dynamics equations; the normal and lateral loads on the tires are determined as the lateral acceleration is increased incrementally by a small amount (thereby maintaining a “quasi-static” or “steady-state” condition). The normal loads are used for the calculation of the lateral traction force potentials at each tire, with the required (centripetal) lateral traction forces apportioned accordingly. From those required (actual) lateral tire forces the corresponding tire cornering stiffnesses are determined; this determination is based upon a tire model developed through a regression analysis of tire test data. This construction of a fairly comprehensive lateral acceleration simulation from basic automotive dynamic relationships, instead of depending upon commercial automotive software such as “CarSim” (vehicle model) and Pacjeka “Magic Formula” (tire model), constitutes a unique aspect of this paper; this return to basics hopefully provides a clearer view and understanding of the results than would be the case otherwise. Even more unique is this paper’s emphasis on, and exploration of, the role specific mass property parameters play in determining automotive directional stability.

1. Society of Allied Weight Engineers, Inc.
Aerospace • Marine • Offshore • Land • Allied Industries
SAWE
Mass Properties and Automotive
Directional Stability
SAWE Paper 3765
2021 SAWE Tech Fair
Brian Paul Wiegand, PE
Northrop Grumman Corporation, Retired

2. SAWE
Background
Have written papers concerning automotive performance and the Mass Properties
significance to that performance:
• Acceleration (0-60 MPH, Quarter-Mile ET, Top Speed).
• Braking (60-0 MPH Distance).
• Lateral acceleration (Max Lat G’s, Roll Stiffness, Roll Gain).
• Road Shock and Vibration (Transmissibility/Gain, Gyroscopic Reaction, Road Contact).
• Ride (Pitch and Bounce, Frequency of Motions, “Flat Ride”).
• Crash Survival (Peak Deceleration, Head Injury Criterion, Neck Injury Criterion).
Now a paper on another aspect of automotive performance and the Mass
Properties significance to that performance:
• Automotive Directional Stability (Understeer Gradient)
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3. SAWE
Automotive Directional Stability Theory
• Although the development of the automobile preceded that of the
aircraft, the development of automotive directional stability theory
lagged that of aircraft by about 35 years.
• Primary forces that determine the dynamic behavior of aircraft are
aerodynamically generated by pressure differentials acting over the
aerosurface areas. In contrast, the primary forces that determine the
dynamic behavior of automobiles are traction forces acting over the
tire-to-ground contact areas:
• First wind tunnel: National Physical Laboratory, UK, 1903.
• First tire test machine: researchers Becker, Fromm, and Maruhn, GR, 1931.
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4. SAWE
Automotive Directional Stability Theory
• 1925 French researcher George Broulheit, develops “slip angle” concept.
• 1931 German researchers Becker, Fromm, and Maruhn develop first tire
testing machine.
• 1934 Maurice Olley of General Motors, develops “over/under/neutral
steer” behavior concepts, formulates “skidpad” testing for behavior.
• 1956 Leonard Segel of the Cornell Aeronautical Laboratory advances
modern automotive directional stability theory:
• “Theoretical Prediction and Experimental Substantiation of the Response of the
Automobile to Steering Control”, presented at the Institution of Mechanical
Engineers, London, UK.
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Key Historical Theory Development Points

5. SAWE
Significance:
Side Load results in
Lateral Traction Force
“Fy” causing Slip Angle
“ψ”
Automotive Directional Stability Theory
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• Automotive Directional Stability begins with the Tire.
• Most fundamental Tire concept is “Slip Angle” (ψ):

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Automotive Directional Stability Theory
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
 “Lateral Traction Force vs. Slip Angle” behavior @ a
Constant Normal Load:
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Significance:
•Slip Angle = f (Lateral
Traction Force “Fy”)
•Cornering Stiffness =
“∂Fy/∂ψ”

7. SAWE
Automotive Directional Stability Theory
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“Lateral Traction Force vs. Slip Angle” behavior @
Varying Normal Load:
Ref.: Campbell, Colin; The Sports Car,
Robert Bentley Inc., Cambridge,
Massachusetts, 1969, pg. 165.
Significance:
Happens to be tire of the
subject vehicle utilized by
this paper for simulated
stability testing.

8. SAWE
Automotive Directional Stability Theory
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 Stability Behavior reveals itself under Side Load “Fd”:
Ref.: Taborek, Jaroslav J.;
Mechanics of Vehicles,
Penton Publishing Co.;
Cleveland, OH, 1957, pg. 23.
Significance:
Understeer,
Neutral Steer,
Oversteer.

9. SAWE
Automotive Directional Stability Theory
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 That was Stability Theory as understood by Maurice
Olley et al circa 1934. It has certain deficiencies:
•Does not show the variation in stability
with speed.
•Does not include the effects of roll.
•Is only qualitative, not quantitative.
•Does not allow for variance front to rear in
suspension and/or tires.

10. SAWE
Automotive Directional Stability Theory
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 Circa Leonard Segel’s 1956 paper a number of
quantitative measures, or metrics, of Automotive
Directional Stability were developed. The most
significant is the “Understeer Gradient” (Kus) in deg/g:
Ref.: SAE J670_JAN2008, para. 11.3.3.1, “Understeer/Oversteer Gradient”, symbol “U”.

11. SAWE
Automotive Directional Stability Theory
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“Don’t let anyone persuade you that the skidpad
tests don’t mean anything. They mean pretty
much everything if we will just take the trouble to
interpret them.”
Maurice Olley
Ref.: Milliken and Milliken; Chassis Design Principles and Analysis, SAE R-206, pg 49. This quote was
found in a letter from Olley to a colleague that was written sometime in the early 1930’s.

12. SAWE
Automotive Directional Stability Theory
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Skidpad Testing:
Significance:
Centrifugal Force supplies
Side Load -
Fy = m a
a = V2/R
Side Load reveals Stability
behavior.
R
V

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Automotive Directional Stability Theory
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Construct Skidpad Simulation of a subject
vehicle undergoing Stability Testing.
Conduct Sensitivity Analysis for each of
18 parameters:
Total Weight, Sprung Weight, Unsprung Weight, Forward Unsprung
Weight, Rear Unsprung Weight, Total Weight LCG, Sprung Weight LCG,
Total Weight VCG, Sprung Weight VCG, Track, Front Track, Rear Track,
Roll Stiffness, Front Roll Stiffness, Rear Roll Stiffness, Roll Axis Height,
Front Roll Center Height, and Rear Roll Center Height.

14. SAWE
Automotive Directional Stability Theory
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Skidpad Simulation:
Significance:
Simulation is of Steady-State
condition, driven by “ay”
incrementation, terminates
when “ay = apotential” at front
or rear axle.

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Automotive Directional Stability Theory
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Subject Vehicle; Automobile Model:
1958 Jaguar XK150S
Nominal Parameter Values:
Wt =3639.80 lb
Wf = 1748.03 lb Wr = 1891.76 lb
Ws = 3117.93 lb
Wusf = 214.03 lb Wusr = 307.85 lb
LCG = 53.01 in VCG = 22.91 in
LCGs = 51.82 in VCGs = 24.62 in
husf = 12.62 in husr = 12.68 in
RG = 16.10 deg/g
Krollf = 276.59 lb-ft/deg
Krollr = 167.78 lb-ft/deg
Lwb = 102.00 in
tf = 51.75 in tr = 51.25 in
hrcf = -7.8 in hrcr = 14.4 in
hr = 20.66 in hra = 3.48 in
Nominal Mass Properties:

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Automotive Directional Stability Theory
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Subject Vehicle; Tire Model:
1958 Dunlop RS4 6.00×16 bias ply
Significance:
Data “pick-off”
of “Fy, ψ” points
at every 200 lb
in Normal Load
“N”.

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Automotive Directional Stability Theory
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Subject Vehicle; Tire Model:
1958 Dunlop RS4 6.00×16 bias ply
Significance:
Construction of a
Tire math Model
as appropriate for
Stability Analysis.

18. SAWE
FOR N = 0: ψ = 0
Cs = 0
FOR N = 200: ψ = 0.0000007298 Fy
3 - 0.0001649884 Fy
2 + 0.0373856159 Fy - 0.0569243091
Cs = (0.0000021894 Fy
2 - 0.0003299768 Fy + 0.0373856159)-1
FOR N = 400: ψ = 0.0000002013 Fy
3 - 0.0000831348 Fy
2 + 0.0238845288 Fy- 0.1077319068
Cs = (0.0000006039 Fy
2 - 0.0001662696 Fy + 0.0238845288)-1
FOR N = 600: ψ = 0.0000000900 Fy
3 - 0.0000470133 Fy
2 + 0.0176483212 Fy - 0.1059394999
Cs = (0.00000027 Fy
2 - 0.0000940266 Fy + 0.0176483212)-1
FOR N = 800: ψ = 0.0000000520 Fy
3 - 0.0000321034 Fy
2 + 0.0152816867 Fy - 0.1398755826
Cs = (0.000000156 Fy
2 - 0.0000642068 Fy + 0.0152816867)-1
FOR N = 1000: ψ = 0.0000000314 Fy
3 - 0.0000214695 Fy
2 + 0.0139331283 Fy - 0.1532355494
Cs = (0.0000000942 Fy2 - 0.000042939 Fy + 0.0139331283)-1
FOR N = 1200: ψ = 0.0000000181 Fy
3 - 0.0000128501 Fy
2 + 0.0130700607 Fy - 0.1445554791
Cs = (0.0000000543 Fy
2 - 0.0000257002 Fy + 0.0130700607)-1
FOR N = 1400: ψ = 0.0000000134 Fy
3 - 0.0000104467 Fy
2 + 0.0138562141 Fy - 0.1328149449
Cs = (0.0000000402 Fy2 - 0.0000208934 Fy + 0.0138562141)-1
FOR N = 1600: ψ =0.0000000233 Fy
3 - 0.0000198687 Fy
2 + 0.0175605669 Fy - 0.0919030834
Cs = (0.0000000699 Fy
2 - 0.0000397374 Fy + 0.0175605669)-1
FOR N = 1700: ψ = 0.0000000458 Fy
3 - 0.0000376466 Fy
2 + 0.0220521855 Fy - 0.0917899852
Cs = (0.0000001374 Fy
2 - 0.0000752932 Fy + 0.0220521855)-1
FOR N = 1900:
ψ = 0.000000326337885 Fy
3 - 0.000191622271760 Fy
2 + 0.044829711256039 Fy -
0.030653568375158
Cs = (0.000000979013655 Fy
2 - 0.00038324454352 Fy + 0.044829711256039)-1
Automotive Directional Stability Theory
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Subject Vehicle; Tire Model:
1958 Dunlop RS4 6.00×16 bias ply
Significance:
This is the resultant
Tire math Model as
appropriate for
Stability Analysis >>

19. SAWE
Automotive Directional Stability Theory
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Subject Vehicle; Tire Model, Traction Coef.:
Significance:
The Tire Lateral Traction
Coefficient Model is utilized
to find the tire traction force
potentials >>>>>>>>>>>>
μ = (b – m N)
Ref.: Lamar, Paul; “Aerodynamics & the Group Seven Racing Car”, The Aerodynamics of Sports & Competition Automobiles,
Proceedings of AIAA Symposium; Los Angeles, CA, 20 April 1968.
Pif = 40 psi Pir = 45 psi
bf = 1.0779 lb/lb br = 1.0570 lb/lb
mf = 0.000297 lb-1 mr = 0.000269 lb-1
Kzf = 1329 lb/in Kzr = 1471 lb/in

20. SAWE
Automotive Directional Stability Theory
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Sensitivity Analysis:
•“One-At-a-Time”(OAT) Method:
Modify input value, get new output value.
•“Input-Output Regression” Method:
Fit a linear line to input/output; R2 measure of linearity.
•Parameter Coupling/Uncoupling:
Some coupling must be maintained if results are to be correct
(Ex.: Sprung Wt, Sprung Wt VCG, Sprung Wt LCG, Unsprung Wt
VCG, etc. affect Roll Gain).
Some coupling must be broken if results are to be correct (Ex.:
Goland & Jindra coupled LCG to fwd/aft spring stiff balance).

21. SAWE
Automotive Directional Stability Theory
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Sensitivity Analysis Results:
•Top 4 Directional Stability parameters are
Mass Properties.
• Most influential parameter is Total Vehicle
Weight LCG; next most influential is
Sprung Weight LCG.
•Out of the 18 parameters, Mass Properties
rank 1,2,3,4,8,10,11,13, and 16 in Stability
Significance.

22. SAWE
Automotive Directional Stability Theory
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Sensitivity Analysis Results :

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Automotive Directional Stability Theory
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Q&A (5 min)

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Automotive Directional Stability Theory
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Back-up Slides

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Automotive Directional Stability Theory
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The study of aircraft directional stability long preceded
the existence of any aircraft; Sir George Cayley (1773–
1857) published On Aerial Navigation in three parts
over 1809-1810. Frederick William Lanchester (1868-
1946) published Aerial Flight in two parts over 1906-
08, and George Hartley Bryan (1864–1928) published
Stability in Aviation in 1911. All of these studied the
principles of flight long before powered aircraft became
a reality. The Wright’s were well aware of the works of
Cayley, and possibly of Bryan, but initially they still flew
a highly unstable “Wright Flyer” (Kitty Hawk, 1903).

26. SAWE
Automotive Directional Stability Theory
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• “Understeer Gradient” (Kus) is a quantitative stability metric expressed in
radians or degrees per “g”.
• “Characteristic Speed” (Vchar) is a metric, calculated at zero lateral
acceleration, for comparing understeering designs with respect to the exact
degree of understeer present.
• “Critical Speed” (Vcrit) is a speed, calculated at zero lateral acceleration, where
an oversteering vehicle can switch from initially controllable (slight oversteer) to
very unstable (high oversteer) behavior.
• “Steady-State Steering Angle” (δss) steering angle required to make steady-
state turn of radius “R” at “ay”; is an indicator of under/over steer and of steering
responsiveness.
• “Static Margin” (SM), is a quantitative stability metric like “Kus”, but is expressed
as a dimensionless fraction. It constitutes a direct borrowing from aircraft stability
theory, with traction forces substituted for aerodynamic forces.
Five Most Common Stability Metrics

27. SAWE
Automotive Directional Stability Theory
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1958 Jaguar
XK150S
Suspension
Geometry

28. SAWE
Automotive Directional Stability Theory
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1958 Jaguar
XK150S Mass
Properties as
Needed for
Skidpad
Simulation

29. SAWE
Automotive Directional Stability Theory
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Typical Variation
in Understeer
Gradient (Stability)
with Lateral
Acceleration (“ay”
in g’s):
Ref.: Wong, Jo Yung; Theory of Ground Vehicles, John Wiley & Sons Inc.; Hoboken, NJ, 2008, pg. 375.

30. SAWE
Automotive Directional Stability Theory
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Road undulations result
in lateral force vectors
through the C.G.
Wind gusts result in
lateral force vectors
near the C.G.

31. SAWE
Automotive Directional Stability Theory
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