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TIRE MODELS FOR THE DETERMINATION
OF VEHICLE STRUCTURAL LOADS

BY

Robert Thomas Jane

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

I982

ABSTRACT

TIRE MODELS FOR THE DETERMINATION
OF VEHICLE STRUCTURAL LOADS

By

Robert Thomas Jane

This thesis develops a tire model which estimates the magnitude
and frequency of forces applied to the wheel spindle during low-speed
tire/rough road interactions. A simple tire test to obtain input
parameters for the model, and tests which validate the model are
presented. The model is then used with a quarter car simulation

to illustrate the utility of the model. Finally the limitations

of the model for higher speed applications are examined.

ACKNOWLEDGEMENTS

I would like to thank my major professor and good friend,
Dr. James E. Bernard, for his assistance and encouragement through-
out my graduate studies.

I would also like to thank Jan Swift for her organizational
assistance and typing of this dissertation.

I would also like to thank my parents, Clarence and Caroline

Jane, for the support and guidance they have given me.

1'1

TABLE OF CONTENTS

LIST OF FIGURES ........................................... iv
CHAPTER l - Introduction .................................. l
CHAPTER 2 - Literature Survey ............................. 2
CHAPTER 3 - The Model ..................................... 8
CHAPTER 4 - Model Validation .............................. 17
CHAPTER 5 - Vehicle Simulation ............................ 2l
CHAPTER 6 - Spin Constraint ............................... 28
CHAPTER 7 - Conclusions ................................... 34
LIST OF REFERENCES ........................................ 35

10.

ll.
l2.
l3.

l4.
l5.

l6.
l7.

LIST OF FIGURES

Point contact model ................................. 3
Fixed footprint model ............................... 3
Radial spring model ................................. 5
Radial spring model on 2" x 4” bump ................. 7
Force-deflection test data for a l4" Goodyear
Pl85/75-Rl4 tire at 28 psi .......................... l0
Table of vertical force as a function of

vertical spring deflection .......................... ll
Radial spring model without shear force effects ..... l3
Closeup of tire profile using shear force

algorithm during a bump/tire impact ................. l6
Measured and computed vertical tire forces for

a T4" Goodyear Pl85/75-Rl4 tire at 28 psi ........... l9
Measured and computed longitudinal tire forces

for a l4" Goodyear Pl85/75-Rl4 tire at 28 psi ....... 20
Quarter car model ................................... 22
Vertical tire forces, impact speed 3 ft/sec ......... 24

Wheel spindle displacement, impact speed

3 feet/sec .......................................... 25
Car body displacement, impact speed 3 feet/sec ...... 26
Longitudinal force biasing. Test data from a

l4“ Goodyear P185/75-Rl4 at 28 psi .................. 29
Longitudinal tire force due to tire slip ............ 31

Effect of vehicle speed on longitudinal tire
force ............................................... 32

iv

CHAPTER I

Introduction

Directional response simulations of passenger cars have been
in use since the l950's. Since the trajectory of a vehicle is
almost entirely dependent on the forces and moments applied to
the vehicle from the road, the representation of the tire-road
interface is of great importance. 'By and large, these directional
response simulations have been limited to smooth road studies,
and a variety of tire models have been developed for this purpose
[4]. '

More recently, structural analysts have developed models which
can reproduce modal properties of vehicles quite faithfully.

These models are valuable in that the free vibrational modes of
the automobile are indicators of structural performance. But

an additional use of these models is the calculation of vibrational
response, and or the stresses resulting from loading on the
vehicle through the tires. This capability depends on additional
information, the forces applied to the structure by the tires.

This thesis was motivated by this need for tire forces generated
during rough road vehicle simulations to be used as input to these

complex vehicle models.

CHAPTER 2

Literature Survey

Two types of mathematical models of tires have been developed and
presented in the literature. One type is the finite element model.
This model is an excellent representation of a tire, but because of
its complexity and high computing cost it is not used for tire-
vehicle-system simulations. The remaining tire models are semi-
empirical, they depend on the tire test data for their utility.
Three of these models will be discussed here, the so-called point
contact model, the fixed footprint model, and the adaptive foot-
print model.

Figure l presents the point contact model. This model has been
extensively used in vehicle simulations to predict vertical forces
for tires traversing smooth road surfaces. Davis, in Reference 7,
claims that the point contact model is useful only if it is operated

on a ground surface that exhibits the following characteristics:

(l) the road profile cannot have any step changes in ele—
vation or slope.

(2) the elevation and slope of the road profile within the
tire contact patch can be defined by a plane tangent to

the ground at the ”ground contact point”.

Fvertical = Kp'

   

F

longitudinal

Figure l. Point contact model.

n
E K
=l

b'5i

TFvertical = i

—- Flongitudinal

  

Figure 2. Fixed footprint model.

To overcome some of the shortcomings of the point contact
model, the fixed footprint, or ”brush" model was developed. In
Reference 4 the fixed footprint model is discussed along with its
use in vehicle simulations. As shown in Figure 2, the model con-
sists of a number of parallel springs evenly spaced across the
tire contact patch. The model has been used to successfully compute
the vertical and longitudinal forces created by a tire slowly moving
over an irregular road surface. But since the computation of longi-
tudinal force depends on the local slope of the terrain, the fixed
footprint model has its limitations. For example, the fixed footprint
model predicts no longitudinal force upon encountering a step elevation
change in terrain such as a curb.

A third formulation is the so-called adaptive footprint, or
radial spring model. A diagram of the model is shown in Figure 3.

The model assumes that:

(l) the tire is a thin disk that deforms only in the radial
direction.

(2) the terrain is undeformable.

(3) the radial force-deflection relationship for the tire

being modelled is known for a rigid planar surface.

Two procedures have been used to calculate forces for this model.
A brief discussion of each follows:
The first method redefines the terrain contacting the tire model

through the use of an ”equivalent ground plane" that reflects the

   
      
  

 

A Fvertical

    

longitudinal

Figure 3. Radial spring model.

 

original elevation and slope of the terrain. Models of this type
were presented by McHenry [6], and latter by Davis [7]. In
McHenry's model, the tire is represented as a disk composed of
nonlinear radial springs. The springs are spaced 4 degrees apart
in the wheel disk, and have identical load-deflection charac-
teristics that match the flat terrain properties of a nonlinear
point contact model.

If a tire encounters irregular terrain, as in Figure 4, the
tire model computes the forces as follows: At each point in time
the individual radial spring deflections are computed. A vector
summation of these deflections, is used in conjunction with the
load-deflection properties of the radial springs to compute the
resultant radial force vector, ”equivalent ground contact point”,
and the ”equivalent ground plane” of the tire. The resultant
radial force vector is assumed to act through the ”ground contact
point“, on a line through the wheel center.

It is important to note that the equivalent ground plane
formulation assumes the net force vector acts through the wheel
center. Since this assumption was not in agreement with our
measurements, we were led to consider a second approach presented
in [4], which does not require the resultant tire force to act
through the wheel center. This model and some details of the
supporting test program are presented in following sections of

this thesis.

 

 

 

F iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

 

 

CHAPTER 3
The Model

Figure 3 is a schematic diagram of an adaptive footprint model.

Algorithms used to compute forces at the tire/road interface for
such a model have been presented in the literature by McHenry [6],
Davis [7], and Captain [4]. The method employed here uses the
radial spring deflections for force calculations, and neglects
damping, as our goal is the simulation of low speed impacts.
Higher speed impacts, where damping would be expected to be more
important, will be discussed in Chapter 6.

The total force on the wheel hub can be expressed as the sum

of all the forces created by the radial springs:

_,_I
in

II M 3
0-l
—h

(3.l)

To calculate the individual force contributions from each spring,

one might be led, as in [4], to use a linear relationship:

df. = K * E. (3.2)

Where Kr represents a linear spring constant and 5} is the radial

deflection of the ith spring. Since the tire model must support

 

the normal load, MS, of the tire at equilibrium, the following

relation results:
n
NS = 2 K * 6- - k (3.3)

where k is a unit vector in the vertical direction. This equation
does not allow the value of Kr to be determined because the
individual spring deflections, E}, are not known at the trim
condition. Additional information is required to solve for Kr'
Either the dynamic spring rate of the tire at equilibrium, or the
contact patch length at equilibrium could be used. But if the
dynamic spring rate is specified, the model's contact patch will
not necessarily have the correct length, and if the contact patch
length is specified, the resulting dynamic spring rate may be in
error.

To deal with this situation, a nonlinear force-deformation
relation was developed. This was done as follows: Consider a
simple test in which the vertical tire force of a free rolling
tire is measured as a function of wheel center height, as in
Figure 5. The experimental data can be related to the sum of the
vertical radial spring deflections, as shown in Figure 6, by
simulating the same test using the radial spring model, thus
yielding the summation of vertical spring deflections corres-
ponding to the measured forces. Now vertical forces can be
computed for any terrain by summing the vertical component of

the radial spring deflections and finding the force from Figure 6.

 

to
UI
O
0

hi

6‘

.3

Q
lllllllllll

1593
Fz : _
(Lbs.) -

18510“

500—

 

 

Deflection (in.)

Figure 5. Force-deflection test data for a l4" Goodyear
Pl85/75-Rl4 tire at 28 psi.

ll

 

 

o TTTTIFTIIITTrTlTTTIITIITI

0 50 100 150 800 850

Summation of vertical spring deflections (in.)

Figure 6. Table of vertical force as a function of vertical spring
deflection.

 

l2

Defining the vertical force versus total spring deflection
in this way assures the correct dynamic spring rate as well as
the proper contact patch length at the trim condition.

It is reasonable to assume that longitudinal forces can be com-
puted using a similar algorithm. However, from the free rolling
tire test results, it was found that a linear spring constant

could be used to compute the longitudinal tire force:
n .
F = E K * 5, . 1 (3.4)

where i is a unit vector in the longitudinal direction. It was
found that setting the magnitude of KX to ninety percent of Kr
yielded excellent agreement between the test data and computed
longitudinal forces.

An obvious question to be asked at this stage concerns the
number of radial springs required to meet the objectives of the
model. Its clear that the answer is not ”one” - the point spring
model has been shown to be inadequate for rough road simulations.
On the other hand very large numbers of springs lead to increased
computational expenses.

Related questions arise concerning enveloping. For example,
if the deflection of each radial spring is computed by finding
the intersection of the undeformed tire disk with the road profile,
the awkward situation shown in Figure 7 may result. Computing
forces in this manner assumes only radial deformation of each
spring and ignores the forces and moments that can result if

there is a stretching of the tire perimeter.

 

 
   

 

 

14

Since the stiffness of the tire perimeter precludes slope
discontinuities such as those illustrated in Figure 7, the
following algorithm was developed. Initially the deflection of
each spring is computed based on the intersection of the un-
deformed tire disk with the road profile. Then a comparison is
made between the vertical position of the endpoints of neigh-

boring springs:
52. = ZTIREi - ZTIRE1._1 i=2, n (3.5)

where ZTIREi represents the vertical position of spring i's endpoint
with respect to the road. If the magnitude of 621 is greater than

a given value, DELTA, then the endpoint position of springs close

to spring i will be modified. The value of DELTA is determined

by limiting the stretching of the tire circumference:
DELTA = R * de (3.6)

where R is the undeformed tire radius, and do is the angle between
adjacent springs. This allows a maximum arc length of /2'*

DELTA for each arc segment before the following algorithm is used
to modify the tire profile. If a tire impacts a sharp bump, and
the magnitude of 521 exceeds DELTA, then the M springs adjacent

to spring i are to be modified, with M being computed by:

 

M = (3.7)

 

IS

The spring endpoints being modified are always lower than spring
i. For example, if a tire is going up a bump, then the springs

before spring i are modified using the following equation:

6Z-

_ T
ZTIRE’Q "' ZTIREQ + W (IL-I
IQ, .

)2 2:1, M (3.8)
where t is a counter that moves to the left or right of spring i.

Figure 8 illustrates the resulting smoothing created when a
tire encounters a 2" by 6" bump. Note the greater movement of
the springs close to the object with a decreasing effect on the
springs located away from the obstruction. In this example,
6Z1 becomes greater than DELTA when i=2l, or between the twenty
and twenty-first springs. By computing M as previously dis-
cussed, it is found that the seventeen springs before spring 2l
must be modified.

Using this enveloping algorithm, with 200 radial springs
evenly spaced across the 2 radian arc of the tire model led to
realistic tire profiles. Computing the tire profile with this
intuitively based approach to account for tread shear force led
to very good correlation with test data. This will be discussed

in the next section.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ooooooooooo

 

 

 

 

l\\\\\\\\\\\\\\\\\\\\) l

 

 

 

 

 

r/

___i=21
_. t=l

CHAPTER 4

Model Validation

To validate the modified radial spring model, rolling tire
tests were performed at the Highway Safety Research Institute in
Ann Arbor, Michigan, using a flat bed tire testing machine [6].
The flat bed test machine is instrumented to measure all of the
forces and moments created by a tire rolling along the test bed.
To simulate rough road conditions, various sized wooden blocks
were placed on the test bed and allowed to move under the tire
while the wheel spindle height was held constant. The tests were
made using a bed speed of three ft/sec, the standard speed of the
test machine.

To gather parameters for the modified radial spring model,
the vertical force vs. deflection was measured for a Goodyear
Pl85/75-l4 radial tire, inflated to 28 psi. The input data was
gathered by varying the wheel spindle height, and measuring the
resulting vertical force as the tire rolled along the flat bed
of the test machine. Figure 5 presents this data.

To validate the modified radial spring model output for
rough road simulations, the flat bed test machine was used to

gather test data. Road disturbances were provided by wooden blocks

l7

 

l8

placed on the bed of the test machine. The dimensions of

the blocks used were two inches high by six inches long, and

one inch high by six inches long. In both cases the wooden blocks
were of sufficient width to eliminate any lateral force generation.
Both vertical and longitudinal tire forces were measured using

a strip chart recorder to record the forces.

Figure 9 presents the measured test data and the computed
model output for vertical force versus longitudinal position as
the tire encounters each of the wooden blocks. The vertical force
output was expected to be symmetric since the bump was symmetric
and because the wheel spindle was held at a fixed height. The
Figure shows a static normal load of 800 pounds was present in
the tire, and the peak vertical force was generated as the wheel
spindle passed over the center of the bump, occuring at x=l5.5
inches. The symmetric plots of the model output and the tire
test results are in close agreement.

The results of the tire testing program also suggested that
a linear spring constant Kx would be sufficient to compute the
longitudinal forces for the radial spring model. A comparison of
longitudinal tire force calculated using this linear spring are
presented in Figure TO. The figure incidates that the free
rolling test data and simulation produce symmetric, very similar
results. As the leading edge of the tire first impacts the bump,
a negative or rearward force is produced. Then as the wheel
center passes directly over the bump, the longitudinal force
passes through zero and becomes positive, or forward, as the tire

begins to move down off the bump.

 

19

N
U!
Go
«9

  
 
  
 
  

.1 . x x x x measured test data
1 for l" x 6" bump
« ' ' ' ' measured test data
~ for 2" x 6" bump
- ''''' computed radial sprin
LGBB-— model output l" x 6"
- - --- computed radial sprin
FZ d model output 2" x 6"
(Lbs ) 1
1586—
lDDB-

 

 

SEE-I [llUTITU—TITTI'IFTHTIIUII'IUII'TFrIl

D 5 ID 15 28 25 30 35
Spindle X position (in.)

Figure 9. Measured and computed vertical tire forces for a l4" Goodyear
Pl85/75-Rl4 tire at 28 psi.

gump

9
bump

 

 

20

   

 

 

400- . 0
arm-
F
X ‘
(Lbs.)
0—. I
-299._ computed 2"x6" bump
--------- computed l"x6" bump
- “u" measured l"x6" bump
. 0- 00 measured 2"x6" bump
-436 TTITIIUFT'iffrU'TIrI'IUFUITT—Virliilj
B 5 10 IS 29 25 30 35

Spindle X position (in.)

Figure l0. Measured and computed longitudinal tire forces for a l4"
Goodyear Pl85/75-Rl4 tire at 28 psi.

 

 

CHAPTER 5

Vehicle Simulation

To illustrate the utility of the modified radial spring
model, a so-called quarter car model simulation was implemented.
Figure ll presents a schematic diagram of the model. The vehicle
model has two degrees of freedom, the vertical position of the
wheel, the unsprung mass, and the vertical position of the car
body, the sprung mass. These two quantities are represented by
Zw and Zb respectively.

A variety of tire models could be used to compute the forces
at the tire/road interface. For purposes of comparison here, two
tire models were used, a simple point contact model, and the
modified radial spring model. The parameters input to both of
these tire models are representative of l4“ radial car tires,
and are presented in Table 5.l. The vehicle parameters are
presented in Table 5.2, indicating a simple approximation for a
3200 pound car.

0

TABLE 5.l Tire Parameters

Radial spring model .................. Figure 5

Point contact model .................. Kp=l000 Lb/in.

2l

22

 

 

 

 

 

 

 

—— MB
V/
Zb K c
M11
N F (t)

2

Figure ll. Quarter car model.

 

 

23

where Kp is a linear spring constant used to compute vertical

force for the point model.

TABLE 5.2 Vehicle Parameters

K (suspension spring stiffness) ............... l00 lb/in

C (suspension damping coefficient) ............ 2.2 lb-sec/in
MB X 9 (weight sprung mass) ................... 700 lbs

MN X 9 (weight unsprung mass) ................. lOO lbs

To study the effect of the tire model on transient response,
a low speed impact with a 2” high by 6” long bump was simulated.
Figures 12 through l4 present the computed output for the quarter
car model. Figure l2 presents the resultant vertical force created
at the tire/road interface. The Figure indicates that the vertical
force spike of the point model begins when the wheel center is
directly over the leading edge of the road obstruction, and ends
as the wheel center passes the end of the obstruction. The force
which occurs immediately after the center line of the tire reaches
the bump, is the product of the point spring rate and the height
of the bump. The force output of the radial spring model has a
more realistic time history with the vertical force building up
before the wheel center reaches the bump, and decaying after the
wheel center has passed over the end of the obstruction.

The resulting rigid body motions are shown in Figures l3

and 14. The Figures indiciate that the higher frequency content

 

24

modified radial spring model

 

point contact model

“.
DDDDDDDDDDDD
OOOOOOOOOOOOOO
IIIIIIIIIIIIIII
UUUUUUUUUUUUUUUUUUUUUUUUUUUU

T

 

 

1000—

 

Time (sec.)

Vertical tire forces, impact speed 3 (feet/sec.).

Figure l2.

 

 

 

.b

H m U
lLlllllllllllllll1111111]

I
g...

Figure 13.

25

 

modified radial spring model

0
v”
--..aa---'-

................ point contact model

 

 

 

O
H
to

Time (sec.)

Wheel spindle displacement, impact speed 3 (feet/sec.).

 

 

26

............... point contact model

 

“\‘ modified radial spring model

 

 

Figure l4.

Time (sec.)

Car body displacement, impact speed 3 feet/sec.

”I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

27

of the point model leads to enhanced excitation of the wheel hop
mode of vibration, and less excitation of the car body. The more
realistic force-time history created by the radial spring model
leads to smaller wheel displacement; and larger car body motion.
All of the work so far presented has been of low speed
testing and simulations. The next section is concerned with

higher speed impacts.

 

CHAPTER 6

Spin Constraint

In the free rolling tire tests presented in Figures 9 and 10,
the tire was unconstrained by any moments applied about the spin
axis by the tire testing apparatus. The angular spin rate, m, was
able to vary as the tire traveled over the test block. In parti-
cular, the longitudinal velocity of the wheel center relative to
the ground was held constant, and the spin rate, w, increased
as spindle-to-ground distance is decreased, thus avoiding longi-
tudinal stretching of the tire contact patch. This section will
consider similar tests in which the wheel spin rate was held
constant.

Figure l5 presents the longitudinal force test data for a l4"
radial passenger car tire impacting the 2" x 6" block with spin
rate held constant, and the free rolling test data of the same
impact. The symmetric plot of the free rolling tire contrasts the
biased forces measured when the tire was not permitted to spin up
as it encountered the wooden block. As the rolling radius of the
tire is shortened by the tire/block interaction, the constant
angular velocity of the tire, w, maintained by the flat bed test
machine servo drive becomes smaller than the spin rate required
for no longitudinal slip. As a result, the force bias caused by

this longitudinal slip is in the negative or rearward sense. The

28

 

29

O
X
(Lbs.)

750 —————————————— constant wheel
‘ spin rate

     

 

 

X spindle location

Figure 15. Longitudinal force biasing. Test data from a l4" Goodyear
Pl85/75-Rl4 tire at 28 psi.

30

vertical tire force, on the other hand, did not show substantial
changes when the spin rate of the tire was constrained to remain
constant during tire testing.

A simple simulation was used to aid in understanding these
results. In this simulation longitudinal brake force was assumed

to be a function of the longitudinal tire slip, S, where:

S = l - Rw/U (6.l)

R is the average length of the radial springs in the contact
patch; w is the wheel spin rate, and U is the longitudinal velocity
of the wheel spindle.

Using a simple u-slip curve, shown in Figure l6, to compute
the brake forces caused by longitudinal tire slip, a simulation of
the tire impacting the 2" x 6" bump at various speeds was implemented.
The results of this simulation are presented in Figure l7.

The simulation of the low speed impact indicated a sym-
metric longitudinal force as the tire crossed the bump, in good
correlation with the free rolling tests. But as the speed, U, is
increased, the simulation predicts a force bias similar to the
one created during the constrained wheel spin testing. This non-
symmetric force output is expected because as the longitudinal velocity
is increased, the tire/wheel assembly does not have sufficient time
to spin up while the tire is in contact with the test block. This
inability to spin up creates a deformation of the tire contact patch,

and therefore large brake forces due to longitudinal slip of the

 

 

31

750:

J

l

 

U1

0

O
“I

11

m
01
O

l

 

I
(U
UT
0
llllllllLIllllIllll

 

 

I
p...

Slip

Figure l6. Longitudinal tire force due to tire slip.

 

 

 

32

 

 

 

 

19°01 ------- 1000 ft/sec.
: -------------- 50 ft/sec.
‘ 3 ft/sec.
'1000 TIITIITTTITIIIllIlTllTll'
o 10 20 30 4° 5°
X spindle location (in.)
Figure l7. Effect of vehicle speed on longitudinal tire force.

 

33

tire. The highest velocity, which effectively holds the tire‘
spin rate constant during the tire/bump impact, leads to a result
very similar to the results measured while testing with the wheel
spin rate constrained.

The simulation predicts little longitudinal force biasing
for tire/bump impacts with longitudinal velocity less than ten

feet per second.

 

CHAPTER 7

Conclusions

The modified radial spring model yields accurate vertical and
longitudinal tire forces created during tire/rough road inter-
action at low speeds. The model requires a minimum of input data,

which can be gathered by measuring the vertical force-deformation

 

relationship of the tire.

The longitudinal forces generated by the model at high speeds
are expected to be inaccurate due to the significant force biasing
created by longitudinal tire slip. Further work should include
high speed testing to confirm the effects of wheel spin rate on

tire forces.

34

 

LIST OF REFERENCES

 

 

LIST OF REFERENCES

Segel, L., "Theoretical Prediction and Experimental Sub-
stantiation 0f the Response of the Automobile to Steering
Control,“ Proc. Auto. Div., The Instn. Mech. Engrs., N0. 7,
p. 310 (l956-l957).

Ellis, J., "The Dynamics of Vehicles During Braking,”
Symp. Cont. of Vehicles, Instn. Mech. Engrs., Proc., pp.
20-29 (l963). .

Bernard, J.E., Segel, L., and Wild, R.E., "Tire Shear Force
Generation During Combined Steering and Braking Maneuvers”,
SAE paper_N0. 770852.

Captain, K. M, Boghani, A. B. , and Wormley, D. M, "Analytical
Tire Models for Dynamic Vehicle Simulation", Vehicle System
Dynamics 8 (l979) pp. l- 32.

 

Rasmussen, R.E., and Cortese, A.D., "The Dynamic Spring Rate
of Rolling Tires", SAE paper No. 680408.

McHenry, R. et al., "Vehicle Dynamics in Single Vehicle
Accidents", Cornel Aeronautical Laboratories, Report No.
VJ-225l-V-3, December, l968.

Davis, D.C., ”A Radial Spring Terrain-Enveloping Tire Model”,
Vehicle System Dynamics 3_(l974), pp. 55-69.

 

35

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