1997 A Vehicle Dynamics Tire Model For Both Pavement and Off-Road Conditions PDF
1997 A Vehicle Dynamics Tire Model For Both Pavement and Off-Road Conditions PDF
1997 A Vehicle Dynamics Tire Model For Both Pavement and Off-Road Conditions PDF
SAE TECHNICAL
PAP 970559
Jeffrey P. Chrstos
JPC Engineering
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both tire slip angle and longitudinal slip ratio. This composite
slip is then used in a force saturation function, controlled by five
shaping parameters. From the force saturation function, tire
lateral and longitudinal forces are computed.
The STIREMOD equations are based on a composite slip
formulation, which is basically a quadratic function of lateral
and longitudinal slip. Lateral slip is expressed as the ratio of the
side slip velocity of the tire patch relative to me longitudinal
speed of the tire patch, which is the equivalent of die tangent of
the tire patch slip angle, a Longitudinal slip is defined as the
ratio S of the differential tire patch to ground longitudinal
velocity divided by the longitudinal velocity of the wheel hub
coefficient of friction out in me region where, on a firm surface, relative to the ground.
we would expect reduced coefficient of friction described by me
limiting slide value. Composite Slip
where the sensitivity coefficient; K3 is used as a fitting parameter properties: 1) as slip increases from zero there is a positive
in STTREMOD to accommodate the asymmetrical lateral force slope; 2) the function ratio reaches an asymptote at high slip
response under traction versus braking conditions. The initial conditions. The roots of the numerator and denominator
patch length apois dependent on normal load (Fz), rated polynomials can be located to give a wide variation in the shape
of the saturation function to accommodate me full range of
design load (FZT), tire width (Tw) and tire pressure (Tp): pavement and off-road surface conditions. This function is a
ratio of polynomials that define a load normalized composite
force:
The first function gives an initial linear slope of aligning Lateral/Longitudinal Stiffness Transition
moment as a function of lateral slip ratio which then falls off at
higher slips due to the denominator quadratic in composite slip.
The shaping coefficient G, allows fitting the peak and fall off of
aligning torque at high slips. The second function and shaping Camber force stiffness is also reduced as a function of the
coefficient G2 allow accounting for combined cornering and force saturation function:
braking effects on aligning moment due to tire patch lateral Camber Force Stiffness Transition
offset.
where µx,µy Uy are the equivalent of transition coefficients of The Metz model is based on an exponential function of slip
friction, µpx,µpy are the peak coefficients of friction, a is angle, with the parameter relating to cornering stiffness being a
the tire slip angle and S is me longitudinal slip ratio. In terms function of vertical load. The Metz exponential model is given
by the following equations:
of friction ellipse interpretation, µpx,µpy define the limit
force ellipse conditions, and the above equations provide the
transition throughout the saturation region. Under paved
surface conditions the limit slip coefficients of friction are
referred to as slide coefficients of friction and are typically 10 Where: A is the equivalent of maximum lateral force (sliding
30% below the low slip peaks depending on speed as defined friction); B is the equivalent of cornering stiffness (1/deg); C
by the Kux and Kuy parameters. By setting Kux and Kuy to & D are empirical coefficients for variation of B with Fz
negative values, tire forces can also be caused to increase (1/deg); m is an empirical exponent; a is tire slip angle
(deg);
beyond me peak transition region, which can be used to FZT rated tire load (lbs).
produce forces due to surface deformation under high slip
conditions as will be discussed subsequetly. To make STIREMOD emulate an off-road tire model,
According to Calspan convention [16] the peak force saturation function shaping parameters can be determined
coefficients of friction are a function of normal load: to match the exponential shape of Metz's model. The following
derives equivalent shaping parameters C1 through C5 for the
case of pure cornering. These shaping parameters can then be
used in STIREMOD to predict tire forces during off-road
operation.
To compute the equivalent C1S for a particular set of
Metz coefficients, equation 16 must be set equal to equation 17.
where SNT is the measurement skid number (i.e., 100 x Metz's equation for Fy is a function of slip angle (a) in
coefficient of friction) while SN0 is the skid number of the degrees, while STTREMOD's equation for F is a function of
simulated surface.
composite slip which are nearly proportional as illustrated in
In the slip to slide transition region two other saturation Figure 2 since a = tan or for small slip angles. The slope of s
functions are also defined. The longitudinal stiffness versus a at a given normal load is designated 'DsDa'. Metz's
coefficient Kc merges to the lateral stiffness coefficient Ks equation for Fy can now be put in the domain of composite slip
for symmetry in the limit locked wheel condition: by:
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Therefore:
Shear displacement is the length over which the soil has been
compacted. The normal pressure can be expressed as normal
load per unit area:
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Now, we note that shear stress p is nominally proportional to order of one foot. The characteristic distance for tire/soil
normal pressure as illustrated in Figure 3 [9-11]. The ratio of interaction under furrowing conditions is suspected to be on me
mese two quantities gives the ratio of composite horizontal force order of several feet to tens of feet based on typical marks left
to normal load: by vehicles in off road encounters, e.g., [12].
Given the development of tire force due to large slip
accompanied by soil displacement or plowing, we must now
This last relationship gives normalized tire/soil horizontal force determine the relationship between force and slip. This will be
which is the basic output of STIREMOD. defined by recent measurements discussed further on.
THE FORCE AND MOMENT MEASUREMENT
For dry soils, me shear strengm (p) is proportional to the
normal stress on the sheared surface (?) and the angle of internal PAVED SURFACE CHARACTERISTICS - The tires
shearing resistance ofthe material (F): used to demonstrate parameter identification on paved surfaces
are P205/65R15 steel belted radials (these tires were part of a
simulation evaluation program conducted by the National
Highway Traffic Safety Administration [15,16]). Twenty-six
Here, we will assume that there is an equivalent coefficient tires were purchased from a single manufacturing batch in an
friction corresponding to the ratio of shear strength to normal attempt to minimize the tire-to-tire variability. Six of these tires
stress: were sent to Smithers Scientific Services, Inc., and were tested
on their MTS Flat-Trac II [17] flat-belt tire force and moment
measurement machine. A complete description of this testing
For sand, Wong [9] reports from measurements a ratio on the and its data analysis can be found in [18], and the complete test
order of 0.7. matrix is documented in Appendix A of this reference.
Now consider the relationship of shear stress (p) to shear Five types of tests were used to compute the tire model
parameters. Each was run at three normal loads: 560, 930, and
displacement (As). Wong [9] suggests a general exponential 1300 lbs (note: It is preferable to test a tire at four or five
relationship for plastic soils (e.g., sand, saturated clay, dry normal loads. Many of the load varying parameters are
snow): described by second order polynomials. Much more reliable
curve fits are achieved when there are more than three, the
minimum, normal loads used). Measurements during each test
included: slip angle, slip ratio, inclination angle, belt speed,
spindle height, lateral, longitudinal, and vertical force, and
In the above expression, Xc is a characteristic
compaction overturning and aligning moment. The five test types were:
distance. In loose soils such as sand, when the medium is
compacted under pressure without significant disturbance, the 1.Quasi-static steering - At each normal load, with zero
compaction distance is on the order of 0.1 feet. For our purpose camber angle and a belt speed of 30 mph, slip angle is
here, where we are considering plowing or furrowing of the soil swept at 1 deg/sec between ± 15°.
surface, me compaction distance will be much longer, perhaps
on the order of feet or tens of feet. 2.Quasi-static braking/driving - At each normal load, witii
zero camber angle and a belt speed of 30 mph, longitudinal
The exponential soil shear strength response as a function slip ratio is swept at 33 percent per second between ± 50
of distance is analogous to tire side force lag which is really a percent.
distance function related to the tire rolling a characteristic length 3.Quasi-static discrete cambering - At each normal load, with
as the tire patch assumes its new force/slip operating condition. zero slip angle and a belt speed of 30 mph, camber angles
Thus, for each update interval or frame time, Ts, in the vehicle of -6, -4, -2, 0, 2, 4, 6 degrees.
dynamics, me tire moves an incremental distance Ax based on
its velocity, U: 4.Discrete sinusoidal steering - At 930 pounds normal load,
zero camber angle and a belt speed of 30 mph, with
amplitude of a = 0.8° discrete sinusoidal slip angle
frequencies of 0.13, 0.63, 1.25, 1.88, 2.50, 3.13, 3.38, 3.75,
4.38, and 5.00 Hz.
The shear force development time constant, Tc for the tire
5.Discrete sinusoidal loading - At 930 pounds normal load,
under soil furrowing conditions will then be given by: zero camber angle and a belt speed of 10 mph, with
amplitude of ± 2.4 mm axle height varied sinusoidally at 1
Hz.
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described above. The computation of the parameters has been B1y, B3y , B4y Calspan Peak Lateral Force versus Normal Load Parameters
automated using a program written in the MATLAB© [23] - These parameters relate the peak lateral normalized
language. The following describes the procedure used to force (FY/FZ) to tire normal load in equation 13. Quasi-static
compute each parameter in the order that they are computed. steering test; are used (at Fz = 560, 930, and 1300), and
for each test the average of the peak positive and negative normalized
Tw Tire Contact Patch Width (in) - The tire contact patch lateral force is computed. A second order polynomial
width is measured using a tope measure with the vehicle at its is fit through the three peak normalized forces, and B1y,
curb load. Tw is men used in the initial tire patch area (ao) B3y, B4y are determined. determined.
computation.
A0, A1 , A2 Calspan Cornering Stiffness versus Normal Load
Tp Tire Inflation Pressure (psi) - This is the tire inflation Parameters - These parameters relate on-center cornering
pressure as tested. Tp is then used in the ao computation. stiffness to tire normal load. Quasi-static steering tests (at F2 =
560, 930, and 1300) are used, and the cornering stiffness of each
Fn- Tire Design Load - This is the tire design load as marked is computed for a slip angle range of 2°. A second order
on the tire side wall.
polynomial is fit through the three cornering stiffness, and A0 ,
RR Tire Rolling Radius at Test Loading (ft) - The tire rolling A1, A2 are determined.
radius is the effective tire radius under free rolling conditions at
normal driving load. Using data from the 0.13 Hz discrete PLYSTEER Slip Angle Offset for Zero Lateral Tire Force
sinusoidal steering test (at Fz = 930 lbs), rolling radius is (rad) - STIREMOD offsets the slip angle to account for tire
computed from: plysteer. PLYSTEER is the slip angle where the lateral force is
zero for the quasi-static steering tests. Using the regression
results from the determination of the cornering stiffness
parameters A0 , A1 and A3, the average intersection with the
slip angle axis is computed.
Uax , Uoy, Coefficient ofLateral and Longitudinal Friction of Kµx Coefficient of the Decay in Longitudinal Friction with
Experimental Test Surface -This is the Directional skid number Increasing Slip Ratio - Kux is the slope of the normalized
ofthe pavement on which the vehicle tests were run. A nominal longitudinal force (FX/FZ) versus slip ratio at high slip ratios.
number of 0.85 is chosen for dry pavement. Quasi-static braking tests (at Fz = 560, 930, and 1300) are used,
and a straight line is fit to normalized lateral force versus slip
CS/FZ Calspan Coefficient for Longitudinal Tire Force ratio data above slip ratio of 0.2 (chosen to be past the peak
Stiffness - CS/FZ is computed from the quasi-static braking runs longitudinal force). These three slopes are averaged to compute
K10.
(at FZ = 560, 930, and 1300). For each test, a straight line is fit
to the normalized longitudinal force versus longitudinal slip Km Coefficient ofthe Decay in Lateral Friction with Increasing
between zero slip and the slip ratio corresponding to 75% of the
peak longitudinal force (note: the 75% of the peak longitudinal Slip Angle - Kpy, is the slope of the normalized lateral force
force limit for the curve fit is chosen as an approximate average (FY/FZ) versus slip angle at high slip angles. Quasi-static
steering tests (at Fz = 560, 930, and 1300) are used, and a
region where the Fx / Fz verses slip ratio curve is linear, and straight line is fit to normalized lateral force versus slip angle
could be changed for a particular tire). The slopes of the three data above slip angles of 10° (chosen to be past the peak lateral
tests are averaged to compute CS/FZ. force). These three slopes are averaged to compute Kuy.
B1x , B3x , B4x Calspan Peak Longitudinal Force versus Normal A3 , A4 Calspan Camber Stiffness versus Normal Load
Load Parameters-These parameters relate the peak longitudinal Parameters - These parameters relate on-center camber
normalized force (FX/FZ) to tire normal load in equation 13. stiffness to tire normal load using equation 8. Quasi-static
Quasi-static braking (at Fz = 560, 930, and 1300) are used, and discrete cambering tests (at Fz = 560, 930, and 1300) are used,
for each test the average of the peak normalized lateral force is and the camber stiffness of each is computed about zero camber
computed. A second order polynomial is fit through the three angle. A second order polynomial is fit through the three
peak normalized forces, and B1x, B3x, B4x are determined. camber stiffness, and A3-, A4 are determined.
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C1, C2, C3, C4, C5 Shaping Coefficientsfor Force Saturation U is the vehicle speed in ft/sec. Solve for KTL and substitute
Function -The shaping parameters C1, C2, C3, C4, C5, can not in U = 44 ft/sec (the tire test speed). TSPRINGR
be determined independently, nor can they be determined from
a single tire test. A non-linear iterative estimation procedure is Tire Spring Rate (lb/ft) - From the Discrete Sinusoidal
used to determine the best (in a least squares sense) values for Loading Tests, data from a 10 mph, and a 1 Hz loading
C1, C2, C3, C4, C5. The estimator uses data from quasi-static frequency, tire normal load is regres ed against axle height.
steering, braking, and driving tire tests at three normal loads, The
and uses STIREMOD to compute the tire forces. This estimator
is run after all of the load varying parameters have been results of the above identification procedures are summarized
determined. in Figure 4 for several parameters. The overall force
and moment fits are summarized in Figure 5. OF -ROAD
Ka Coefficient of Elongation of Tire Contact Patch Due to
Longitudinal Force - This rameter is used to allow
asymmetry in the tire's braking/driving while cornering
predictions. It is computed during the non-linear curve fitting
used to determine the shaping parameters C1, C2, C3, C4, C5
(see above for description ofprocedure).
G1 , G2 Aligning Moment Shaping Parameters - Parameter G1
determines thshape of the aligning moment curve versus slip
angle for zero longitudinal slip. A non-linear iterative estimation
procedure is used to determine the best (in a least squares sense)
values for G1. The estimator uses data from quasi-static steering
tire tests at three normal loads, and uses STIREMOD to
compute the tire forces. This estimator is run after all of the
load varying parameters have been determined. G2 adjusts the
aligning moment for longitudinal tire forces. A non-linear
iterative estimation procedure is used to determine the best (in a
least squares sense) values for G2. The estimator uses data from
the combined steering/braking/driving tire tests at the 930 pound
normal load, and STIREMOD to compute the tire forces. This
estimator is run after all of the load varying parameters have
been determined.
us into the right ball park. Typical friction data often reported
for off road surfaces gives coefficients less than for paved
surfaces. For example, tire/surface friction values for gravel and
sand have been reported in the region of 0.4-0.6 [9,24]. These
values are for braking, and low slip conditions where the soil is
not being significantly disturbed. More relevant are
measurements reported by Delays and Brinkman [12] made by
towing a compact passenger car at different slip angles over sod
under free rolling tire conditions. At a slip angle of 39° they
obtained an effective friction coefficient value as high as 1.13
with moist sod, while at 90° slip they achieved a maximum
value of 0.89. Typical values were about 0.5 on the average
with the vehicle being towed at 10-15 mph. They report a slight
trend of increased motion resistance with decreased firmness of
soil, and variable forces which can be attributed to local
irregularities of the ground surface. In comparing their results
with sinkage models they also suggest that the high coefficients
are probably due to bulldozing (i.e., plowing or furrowing)
effects. Finally, Delays and Brinkman comment that towing
force increase with slip angle is consistent with the assumption
that "motion-resistance force is proportional to the projection of
the vertical tire/soil interface area in the direction of motion."
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ACKNOWLEDGMENTS
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REFERENCES
17.Chrstos, J.P. and Grygier, P., A Instrumentation and Field G1 , G2 - shaping coefficients for tire aligning torque
Testing of 1994 Ford Taurus GL for NADSdyna
Evaluation, NHTSA Draft Report, 1996. K1 - coefficient for aligning moment dependence on vertical
load
18.Chrstos, J. P., A Parameter Measurement and Computation
Procedures for 1994 Ford Taurus GL NHTSA Draft Ka - coefficient for tire patch length dependence on
Report, 1996. longitudinal force
19.Chrstos, J. P., Evaluation of the VDANL and VDM RoAD
Vehicle Dynamics Simulations, NHTSA Draft Report,
KC,KS - longitudinal and lateral stiffness coefficients
1996. Km - aligning moment stiffness
20.Pottinger, M. G., Flat-Trac II Machine Helps in Tire Force, Kx - coefficient for cornering stiffness dependence on
Moment Measuring, Rubber and Plastic News, March 19, longitudinal force
1990.
K - camber stiffness
21.Lee, S., A Development of New Dynamic Tire Model for
Improved Vehicle Dynamics Simulation, Ph.D.
Dissertation, The Ohio State University, 1994. KUX , Kuy - coefficients for limit slip change in coefficient of
friction
22.Bernard, J. E. and Clover, C. L., Tire Modeling for Low
Speed and High-Speed Calculations, SAE Paper No. Mz - tire aligning moment
950311, 1995. Pz - average tire patch normal pressure
23.MATLAB© - Reference Guide, The Mathworks, Inc., July S - longitudinal slip
1993.
SNo,SNT- - skid numbers (%) for simulated surface and tire
24.Warner, C.Y., et al., Friction Application in Accident test respectively
Reconstruction, SAE Paper 830612, Society of Automotive
Engineers, Warrendale, PA. Tc - tire/soil shear force development time constant
T - tire pressure
NOMENCLATURE
Ts - simulation frame time or sampling interval
ap - tire patch length under traction/braking conditions U - wheel hub forward speed
apo - static tire patch length Tw - tire patch width
A - Metz model off-road coefficient of freedom Xc - characteristic compaction distance
A0, A1, A2 - quadratic coefficients for lateral stiffness Yy - camber stiffness
coefficient
a - lateral slip angle
A3, A4 - quadratic coefficients for camber stiffness
coefficient ?x - tire longitudinal movement during simulation frame
time Ts
ATP - tire patch area
As - soil shear displacement
B - Metz model off-road cornering stiffness
f - soil angle of internal shearing resistance
B1x,B3x,B4x,B1y,3y,B4y
- quadratic coefi nts for longitudinal (x) and y - tire camber angle with surface
lateral (y) coefficients of friction C - Metz
µ - tire/surface coefficient of friction
of -road tire model shaping coefficient C1 ,...,C5 -
polynomial coefficients for STIREMOD saturation function D µpx,µpy - peak longitudinal (x) and lateral (y) tire/surface
- Metz coefficients of friction
off-road tire model shaping coefficient DsDa - slope µx,µy ~ Peak to slide transition coefficients of friction
of compsite slip (ó) versus slip angle (a) Fx,Fy,FZ - longitudinal, î - soil shear strength
lateral and vertical tire forces FXesl - estimated p - tire/soil shear stress
longitudinal force used to increase cornering stif nes under ó - composite slip
hard braking conditions FZT - rated t - tire relaxation distance time constant