U.S. patent application number 09/286671 was filed with the patent office on 2002-02-07 for model-based method for determining the road handling performance of a tyre of a wheel for a vehicle.
Invention is credited to MANCOSU, FEDERICO, SANGALLI, ROBERTO.
Application Number | 20020014114 09/286671 |
Document ID | / |
Family ID | 8236609 |
Filed Date | 2002-02-07 |
United States Patent
Application |
20020014114 |
Kind Code |
A1 |
MANCOSU, FEDERICO ; et
al. |
February 7, 2002 |
MODEL-BASED METHOD FOR DETERMINING THE ROAD HANDLING PERFORMANCE OF
A TYRE OF A WHEEL FOR A VEHICLE
Abstract
A method for determining the road handling of a tire, comprising
descriptions of the tire by means of a first,
concentrated-parameter, physical model and by means of a second,
finite-element model, a simulation on the second, finite-element
model of a selected series of dynamic tests and an application to
the first physical model of equations of motion suitable for
representing the dynamic tests in order to obtain first and second
frequency responses of selected quantities; a comparison between
the first and second frequency responses of the selected quantities
for determining the concentrated parameters of the first physical
model and physical quantities indicative of the drift behavior of
the tire.
Inventors: |
MANCOSU, FEDERICO; (MILANO,
IT) ; SANGALLI, ROBERTO; (BRUGHERIO, IT) |
Correspondence
Address: |
FINNEGAN HENDERSON FARABOW
GARRETT & DUNNER LLP
1300 I STREET N W
WASHINGTON
DC
200053315
|
Family ID: |
8236609 |
Appl. No.: |
09/286671 |
Filed: |
April 6, 1999 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60092594 |
Jul 10, 1998 |
|
|
|
Current U.S.
Class: |
73/146 |
Current CPC
Class: |
B60C 99/006 20130101;
B60C 99/003 20130101; G01M 17/02 20130101; G06F 30/15 20200101 |
Class at
Publication: |
73/146 |
International
Class: |
G01M 017/02 |
Foreign Application Data
Date |
Code |
Application Number |
Apr 7, 1998 |
EP |
98830209.7 |
Claims
1. Method for determining the road handling of a tire of a wheel
for a vehicle, said tire being comprised by selected mixes of
rubber and reinforcing materials, said method comprising: a) a
first description of said tire by means of a first,
concentrated-parameter, physical model, said first physical model
comprising a rigid ring which represents the tread band provided
with inserts, a belting structure and corresponding carcass portion
of said tire, a disk which represents a hub of said wheel and
beading of said tire, principal springs and dampers connecting said
rigid ring to said hub and representing sidewalls of said tire and
air under pressure inside said tire, supplementary springs and
dampers representing deformation phenomena of said belting
structure through the effect of a specified vertical load and a
brush model simulating physical phenomena in an area of contact
between said tire and a road, said area of contact having a dynamic
length 2a, b) a definition of selected degrees of freedom of said
first physical model, and c) an identification of equations of
motion suitable for describing the motion of said first physical
model under selected dynamic conditions, characterized in that it
comprises d) the definition of said concentrated parameters, said
concentrated parameters consisting of the mass M.sub.c and a
diametral moment of inertia J.sub.c of said rigid ring, the mass
M.sub.m and a diametral moment of inertia J.sub.m of said disk,
structural stiffnesses K.sub.c and structural dampings R.sub.c
respectively of said principal springs and dampers, and residual
stiffnesses K.sub.r and residual dampings R.sub.r respectively of
said supplementary springs and dampers, wherein said structural
stiffnesses K.sub.c consist of lateral stiffness K.sub.cy between
said hub and said belt, camber torsional stiffness K.sub.c.theta.x
between said hub and said belt and yawing torsional stiffness
K.sub.c.theta.z between said hub and said belt, said structural
dampings R.sub.c consist of lateral damping R.sub.cy between said
hub and said belt, camber torsional damping R.sub.c.theta.x between
said hub and said belt and yawing torsional damping R.sub.c.theta.z
between said hub and said belt, said residual stiffnesses K.sub.r
consist of residual lateral stiffness K.sub.ry, residual camber
torsional stiffness K.sub.r.theta.x and residual yawing torsional
stiffness K.sub.r.theta.z, and said residual dampings R.sub.r
consist of residual lateral damping R.sub.ry, residual camber
torsional damping R.sub.r.theta.x and residual yawing torsional
damping R.sub.r.theta.z, e) a description of said tire by means of
a second, finite-element model comprising first elements with a
selected number of nodes, suitable for describing said mixes, and
second elements suitable for describing said reinforcing materials,
each first finite element being associated with a first stiffness
matrix which is determined by means of a selected characterization
of said mixes and each second element being associated with a
second supplementary stiffness matrix which is determined by means
of a selected characterization of said reinforcing materials, f) a
simulation on said second, finite-element model of a selected
series of virtual dynamic tests for exciting said second model
according to selected procedures and obtaining transfer functions
and first frequency responses of selected quantities, measured at
selected points of said second model, g) a description of the
behaviour of said first physical model by means of equations of
motion suitable for representing the above dynamic tests for
obtaining second frequency responses of said selected quantities,
measured at selected points of said first physical model, h) a
comparison between said first and said second frequency responses
of said selected quantities to determine errors that are a function
of said concentrated parameters of said first physical model, and
i) the identification of values for said concentrated parameters
that minimize said errors so that said concentrated parameters
describe the dynamic behaviour of said tire, j) the determination
of selected physical quantities suitable for indicating the drift
behaviour of said tire, and k) the evaluation of the drift
behaviour of said tire by means of said physical quantities.
2. Method according to claim 1, characterized in that said selected
physical quantities are the total drift stiffness K.sub.d of said
tire, in turn comprising the structural stiffness K.sub.c and the
tread stiffness K.sub.b, and the total camber stiffness
K.sub..gamma. of said tire.
3. Method according to claim 1, characterized in that it also
comprises c) a definition of said brush model, said brush model
having a stiffness per unit of length c.sub.py and comprising at
least one rigid plate, at least one deformable beam having a length
equal to the length 2a of said area of contact and at least one
microinsert associated with said beam, said microinsert consisting
of at least one set of springs distributed over the entire length
of said beam, said springs reproducing the uniformly distributed,
lateral and torsional stiffness of said area of contact.
4. Method according to claims 1 and 3, characterized in that said
degrees of freedom referred to at previous point b) are composed
of: absolute lateral displacement y.sub.m of said hub, absolute yaw
rotation .sigma..sub.m of said hub and absolute rolling rotation
.rho..sub.m of said hub, relative lateral displacement y.sub.c of
said belt with respect to said hub, relative yaw rotation
.sigma..sub.c of said belt with respect to said hub and relative
rolling rotation .rho..sub.c of said belt with respect to said hub,
absolute lateral displacement y.sub.b of said plate, absolute yaw
rotation .sigma..sub.b of said plate and absolute rolling rotation
.rho..sub.b of said plate, and absolute lateral displacement
y.sub.s of the bottom ends of said at least one microinsert.
5. Method according to the claim 1, characterized in that said
selected series of virtual dynamic tests referred to at previous
point f) comprises a first and a second test with said tire blown
up and not pressed to the ground, said first test consisting in
imposing a translation in the transverse direction y on the hub and
in measuring the lateral displacement y.sub.c of at least one
selected cardinal point of said belt and the force created between
said hub and said belt in order to identify said mass M.sub.c, said
lateral stiffness K.sub.cy, and said lateral damping R.sub.cy, said
second test consisting in imposing a camber rotation .theta..sub.x
on said hub and in measuring the lateral displacement of at least
one selected cardinal point of said belt y.sub.c and the torque
transmitted between said hub and said belt in order to identify
said diametral moment of inertia J.sub.c, said camber torsional
stiffness K.sub.c.theta.x, said camber torsional damping
R.sub.c.theta.x, said yawing torsional stiffness K.sub.c.theta.z
and said yawing torsional damping R.sub.c.theta.z.
6. Method according to claims 1 and 5, characterized in that said
selected series of virtual dynamic tests referred to at previous
point f) also comprises a third and a fourth test with said tire
blown up, pressed to the ground and bereft of said tread at least
in said area of contact, said third test consisting in applying to
said hub a sideward force in the transverse direction F.sub.y and
in measuring the lateral displacement y.sub.c of said hub and of at
least two selected cardinal points of said belt in order to
identify said residual lateral stiffness K.sub.ry, said residual
lateral damping R.sub.ry, said camber residual stiffness
K.sub.r.theta.x, and said camber residual damping R.sub.r.theta.x,
said fourth test consisting in applying to said hub a yawing torque
C.sub..theta.z and in measuring the yaw rotation of said hub and
the lateral displacement y.sub.c of at least one selected cardinal
point of said belt in order to identify said residual yawing
stiffness K.sub.r.theta.z and said residual yawing damping
R.sub.r.theta.z.
7. Method according to claims 1 and 3, characterized in that it
also comprises m) an application to said first physical model of a
drift angle .alpha., starting from a condition in which said at
least one beam is in a non-deformed configuration and said brush
model has a null snaking .sigma..sub.b, n) the determination of the
sideward force and the self-aligning torque that act on said hub
through the effect of said drift and which depend on the difference
.alpha.-.sigma..sub.b and on the deformation of said at least one
beam, o) the determination of the deformation curve of said at
least one beam, p) an application of said sideward force and said
self-aligning torque to said second, finite-element model in order
to obtain a pressure distribution on said area of contact and q)
the determination of the sideward force and the self-aligning
torque that act on said hub through the effect of said drift
.alpha.0 on said first physical model, that depend on the pressure
distribution calculated in the previous step p), r) a check, by
means of said pressure distribution obtained in the previous step
p), that said sideward force and said self-aligning torque are
substantially similar to those calculated in previous step q), s) a
determination of the sideward force and of the self-aligning torque
for said angle of drift, and t) repetition of the procedure from
step m) to step s) for different values of the drift angle .alpha.
to obtain drift, force and self-alignment torque curves, suitable
for indicating the drift behaviour under steady state conditions of
said tire, and u) the evaluation of the steady state drift
behaviour of said tire.
8. Method according to claim 1, characterized in that it also
comprises i) a simulation of the behaviour of said first physical
model in the drift transient state by means of equations of motion
reproducing selected experimental drift tests, and ii) the
determination, with a selected input of a steering angle imposed on
said hub, of the pattern with time of the selected free degrees of
freedom of said first physical model, of the sideward force and of
the self-aligning torque in said area of contact in order to
determine the length of relaxation of said tire.
9. Method according to claim 1, characterized in that said first
elements of said second, finite-element model have linear form
functions and their stiffness matrix is determined by means of
selected static and dynamic tests conducted on specimens of said
mixes, whereas the stiffness matrix of said second elements is
determined by means of selected static tests on specimens of said
reinforcing materials.
10. Tire for a wheel of a vehicle, said tire being made from
selected mixes of rubber and reinforcing materials and comprising a
carcass, a belting structure, a tread band provided with inserts,
shoulders, sidewalls, beads provided with bead wires and bead
fillings, said tire being representable by means of a first,
concentrated-parameter, physical model and a brush model with a
road, characterized in that said concentrated parameters comprise
structural stiffnesses K.sub.c consisting of lateral stiffness
K.sub.cy, camber torsional stiffness K.sub.c.theta.x and yawing
torsional stiffness K.sub.c.theta.z, structural dampings R.sub.c
consisting of lateral damping R.sub.cy, camber torsional damping
R.sub.c.theta.x and yawing torsional damping R.sub.c.theta.z,
residual stiffnesses K.sub.r consisting of residual lateral
stiffness K.sub.ry, residual camber torsional stiffness
K.sub.r.theta.x and residual yawing torsional stiffness
K.sub.r.theta.z, and residual dampings R.sub.r consisting of
residual lateral damping R.sub.ry, residual camber torsional
damping R.sub.r.theta.x and residual yawing torsional damping
R.sub.r.theta.z, said tire also being representable by means of a
second, finite-element model comprising first elements with a
selected number of nodes, suitable for describing said mixes, and
second elements suitable for describing said reinforcing materials,
said concentrated parameters being identified by means of a
selected series of dynamic tests on said second, finite-element
model and represented by equations of motion applied to said first
physical model, said tire having construction characteristics
substantially equivalent to said concentrated parameters which
describe the dynamic behaviour of said tire and enabling the
determination of selected physical quantities suitable for
indicating the drift behaviour of said tire for evaluation of said
tire in relation to its road handling.
11. Tire according to claim 10, characterized in that said selected
physical quantities are the total drift stiffness K.sub.d of said
tire, in turn comprising the structural stiffness K.sub.c and the
tread stiffness K.sub.b, and the total camber stiffness
K.sub..gamma. of said tire.
12. Tire according to claim 11, characterized in that the total
drift stiffness K.sub.d and the total camber stiffness
K.sub..gamma. are within the following value ranges:
K.sub.d=500-2,000 [N/g]K.sub..gamma.=40-3,500 [N/g]where
g=degree.
13. Tire according to claim 11, characterized in that the
structural stiffness K.sub.c and the tread stiffness K.sub.b are
within the following value ranges: K.sub.c=8,000-30,000
[N/g]K.sub.b=150-400 [N/g]where g=degree.
Description
[0001] This application is based on European Patent Application No.
98830209.7 filed on Apr. 7, 1998 and U.S. Provisional Application
No. 60/092,594 filed on Jul. 10, 1998, the content of which is
incorporated hereinto by reference.
[0002] This invention relates to a method for determining the road
handling of a tire of a wheel for a vehicle.
[0003] At the present time, to determine the road handling
performance of a tire, the manufacturers of pneumatic tires are
obliged to produce numerous physical prototypes in order to
experimentally evaluate the effects of the various design
parameters on the drift behaviour of the tire, under steady state
and transient state conditions. The experimental tests are
conducted according to iterative procedures, that are largely
empirical and based on experience and are also extremely demanding
in terms of time and cost.
[0004] Furthermore, automobile manufacturing companies are
insisting more and more frequently that the makers of pneumatic
tires come up with tires with extremely precise technical
characteristics as early as the initial stages of vehicle study and
dynamic behaviour forecasting.
[0005] In such a position, the tire manufacturers are finding it
very difficult to respond satisfactorily and with the necessary
flexibility to the various market demands.
[0006] The object of this invention is to provide a scientific
methodology with which to identify the performance characteristics
of a tire in relation to road handling, on the basis of previously
defined design specifications.
[0007] The above object is achieved according to this invention by
a method for determining the road handling of a tire of a wheel for
a vehicle, said tire being made from selected mixes of rubber and
reinforcing materials, said method comprising:
[0008] a) a first description of said tire by means of a first,
concentrated-parameter, physical model, said first physical model
comprising a rigid ring which represents the tread band provided
with inserts, a belting structure and corresponding carcass portion
of said tire, a disk which represents a hub of said wheel and
beading of said tire, principal springs and dampers connecting said
rigid ring to said hub and representing sidewalls of said tire and
air under pressure inside said tire, supplementary springs and
dampers representing deformation phenomena of said belting
structure through the effect of a specified vertical load, and a
brush model simulating physical phenomena in an area of contact
between said tire and a road, said area of contact having a dynamic
length 2a,
[0009] b) a definition of selected degrees of freedom of said first
physical model, and
[0010] c) an identification of equations of motion suitable for
describing the motion of said first physical model under selected
dynamic conditions, characterized in that it comprises
[0011] d) the definition of said concentrated parameters, said
concentrated parameters consisting of the mass M.sub.c and a
diametral moment of inertia J.sub.c of said rigid ring, the mass
M.sub.m and a diametral moment of inertia J.sub.m of said disk,
structural stiffnesses K.sub.c and structural dampings R.sub.c
respectively of said principal springs and dampers, and residual
stiffnesses K.sub.r and residual dampings R.sub.r respectively of
said supplementary springs and dampers, wherein
[0012] said structural stiffnesses K.sub.c consist of lateral
stiffness K.sub.cy between said hub and said belt, camber torsional
stiffness K.sub.c.theta.x between said hub and said belt and yawing
torsional stiffness K.sub.c.theta.z between said hub and said
belt,
[0013] said structural dampings R.sub.c consist of lateral damping
R.sub.cy between said hub and said belt, camber torsional damping
R.sub.c.theta.x between said hub and said belt and yawing torsional
damping R.sub.c.theta.z between said hub and said belt,
[0014] said residual stiffnesses K.sub.r consist of residual
lateral stiffness K.sub.ry, residual camber torsional stiffness
K.sub.r.theta.x and residual yawing torsional stiffness
K.sub.r.theta.z, and
[0015] said residual dampings R.sub.r consist of residual lateral
damping R.sub.ry, residual camber torsional damping R.sub.r.theta.x
and residual yawing torsional damping R.sub.r.theta.z,
[0016] e) a description of said tire by means of a second,
finite-element model comprising first elements with a selected
number of nodes, suitable for describing said mixes, and second
elements suitable for describing said reinforcing materials, each
first finite element being associated with a first stiffness matrix
which is determined by means of a selected characterization of said
mixes and each second element being associated with a second
supplementary stiffness matrix which is determined by means of a
selected characterization of said reinforcing materials,
[0017] f) a simulation on said second, finite-element model of a
selected series of virtual dynamic tests for exciting said second
model according to selected procedures and obtaining transfer
functions and first frequency responses of selected quantities,
measured at selected points of said second model,
[0018] g) a description of the behaviour of said first physical
model by means of equations of motion suitable for representing the
above dynamic tests for obtaining second frequency responses of
said selected quantities, measured at selected points of said first
physical model,
[0019] h) a comparison between said first and said second frequency
responses of said selected quantities to determine errors that are
a function of said concentrated parameters of said first physical
model, and
[0020] i) the identification of values for said concentrated
parameters that minimize said errors so that said concentrated
parameters describe the dynamic behaviour of said tire,
[0021] j) the determination of selected physical quantities
suitable for indicating the drift behaviour of said tire, and
[0022] k) the evaluation of the drift behaviour of said tire by
means of said physical quantities.
[0023] To advantage, said selected physical quantities are the
total drift stiffness K.sub.d of said tire, in turn comprising the
structural stiffness K.sub.c and the tread stiffness K.sub.b, and
the total camber stiffness K.sub..gamma. of said tire.
[0024] According to a preferred embodiment, said method also
comprises
[0025] l) a definition of said brush model, said brush model having
a stiffness per unit of length c.sub.py and comprising at least one
rigid plate, at least one deformable beam having a length equal to
the length 2a of said area of contact and at least one microinsert
associated with said beam, said microinsert consisting of at least
one set of springs distributed over the entire length of said beam,
said springs reproducing the uniformly distributed, lateral and
torsional stiffness of said area of contact.
[0026] Preferably, said degrees of freedom referred to at previous
point b) are composed of:
[0027] absolute lateral displacement y.sub.m of said hub, absolute
yaw rotation .sigma..sub.m of said hub and absolute rolling
rotation .rho..sub.m of said hub,
[0028] relative lateral displacement y.sub.c of said belt with
respect to said hub, relative yaw rotation .sigma..sub.c of said
belt with respect to said hub and relative rolling rotation
.rho..sub.c of said belt with respect to said hub,
[0029] absolute lateral displacement y.sub.b of said plate,
absolute yaw rotation .sigma..sub.b of said plate and absolute
rolling rotation .rho..sub.b of said plate, and
[0030] absolute lateral displacement y.sub.s of the bottom ends of
said at least one microinsert.
[0031] According to another embodiment, said selected series of
virtual dynamic tests referred to at previous point f) comprises a
first and a second test with said tire blown up and not pressed to
the ground, said first test consisting in imposing a translation in
the transverse direction y on the hub and in measuring the lateral
displacement y.sub.c of at least one selected cardinal point of
said belt and the force created between said hub and said belt in
order to identify said mass M.sub.c, said lateral stiffness
K.sub.cy, and said lateral damping R.sub.cy, said second test
consisting in imposing a camber rotation .theta..sub.x on said hub
and in measuring the lateral displacement of at least one selected
cardinal point of said belt y.sub.c and the torque transmitted
between said hub and said belt in order to identify said diametral
moment of inertia J.sub.c, said camber torsional stiffness
K.sub.c.theta.x, said camber torsional damping R.sub.c.theta.x,
said yawing torsional stiffness K.sub.c.theta.z and said yawing
torsional damping R.sub.c.theta.z.
[0032] Preferably said selected series of virtual dynamic tests
referred to at previous point f) also comprises a third and a
fourth test with said tire blown up, pressed to the ground and
bereft of said tread at least in said area of contact, said third
test consisting in applying to said hub a sideward force in the
transverse direction F.sub.y and in measuring the lateral
displacement y.sub.c of said hub and of at least two selected
cardinal points of said belt in order to identify said residual
lateral stiffness K.sub.ry, said residual lateral damping R.sub.ry,
said camber residual stiffness K.sub.r.theta.x, and said camber
residual damping R.sub.r.theta.x, said fourth test consisting in
applying to said hub a yawing torque C.sub..theta.z and in
measuring the yaw rotation of said hub .sigma..sub.m and the
lateral displacement y.sub.c of at least one selected cardinal
point of said belt in order to identify said residual yawing
stiffness K.sub.r.theta.z and said residual yawing damping
R.sub.r.theta.z.
[0033] According to another embodiment, said method also
comprises
[0034] m) an application to said first physical model of a drift
angle .alpha., starting from a condition in which said at least one
beam is in a non-deformed configuration and said brush model has a
null snaking .sigma..sub.b,
[0035] n) the determination of the sideward force and the
self-aligning torque that act on said hub through the effect of
said drift .alpha. and which depend on the difference
.alpha.-.sigma..sub.b and on the deformation of said at least one
beam,
[0036] o) the determination of the deformation curve of said at
least one beam,
[0037] p) an application of said sideward force and said
self-aligning torque to said second, finite-element model in order
to obtain a pressure distribution on said area of contact and
[0038] q) the determination of the sideward force and the
self-aligning torque that act on said hub through the effect of
said drift on said first physical model, that depend on the
pressure distribution calculated in the previous step p),
[0039] r) a check, by means of said pressure distribution obtained
in the previous step p), that said sideward force and said
self-aligning torque are substantially similar to those calculated
in previous step q),
[0040] s) a determination of the sideward force and of the
self-aligning torque for said angle of drift, and
[0041] t) repetition of the procedure from step m) to step s) for
different values of the drift angle .alpha. to obtain drift, force
and self-alignment torque curves, suitable for indicating the drift
behaviour under steady state conditions of said tire, and
[0042] u) the evaluation of the steady state drift behaviour of
said tire.
[0043] According to another preferred embodiment, said method also
comprises:
[0044] i) a simulation of the behaviour of said first physical
model in the drift transient state by means of equations of motion
reproducing selected experimental drift tests, and
[0045] ii) the determination, with a selected input of a steering
angle imposed on said hub, of the pattern with time of the selected
free degrees of freedom of said first physical model, of the
sideward force and of the self-aligning torque in said area of
contact in order to determine the length of relaxation of said
tire.
[0046] To advantage, said first elements of said second,
finite-element model have linear form functions and their stiffness
matrix is determined by means of selected static and dynamic tests
conducted on specimens of said mixes, whereas the stiffness matrix
of said second elements is determined by means of selected static
tests on specimens of said reinforcing materials.
[0047] With the method according to this invention, three main
results are obtained:
[0048] 1. determination of the links between physical parameters of
the tire and its structural properties;
[0049] 2. determination of the steady state drift behaviour of the
tire, without the need to build prototypes at this stage;
[0050] 3. determination of the transient state behaviour of the
tire, when a generic law of motion is imposed on the hub, without
the need to build prototypes at this stage.
[0051] These results have been achieved by the production of a very
simple, first physical model, with only nine degrees of freedom,
that manages to make allowance for the majority of the structural
characteristics of the actual tire.
[0052] The structural characteristics of the tire are reproduced in
the first physical model by means of an appropriate condensation of
concentrated equivalent masses, stiffnesses and dampings.
[0053] In practice, the concentrated-parameter model is equivalent
to a kind of dynamic concentration of the complex finite-element
model, summarizing all its dynamic characteristics in a low number
of mass, damping and stiffness parameters.
[0054] More specifically, it has been proven that this
correspondence may be held valid in the range of frequencies
between 0 and 80 Hz.
[0055] The method enables identification of the structural
parameters needed for the complete description of the first
physical model using simulated virtual numerical tests with a
second, extremely detailed model built from finite element models
(F.E.M.) reproducing the behaviour of the non-rolling tire (not
drifting).
[0056] One of the major advantages of the method according to the
invention is that it partly dispenses with the need to construct
physical prototypes and the resultant experimental tests in the
iterative process of tire determination, replacing this approach
with virtual prototyping.
[0057] The tire's design parameters (characteristics of the mixes,
inclination of the threads, shape of the sidewalls, width of the
belt, etc.) are directly fed into the second, finite-element model
which is extremely detailed.
[0058] The concentrated parameters of the first physical model are
identified by minimizing the difference between the vibrational
dynamic behaviour of the second, finite-element model of the
non-rolling tire and the corresponding response given by the first
physical model.
[0059] The identification procedure defined comprises various
operations that are executed in a precise, pre-established order.
Starting from the transfer functions obtained by means of a series
of virtual dynamic tests conducted on the non-rolling
finite-element model, the masses, stiffnesses and dampings of the
concentrated-parameter model are determined, providing a better
description of the dynamic behaviour of the tire.
[0060] Thus the identification procedure enables a link to be
established between the design parameters (fed into the second
finite-element model) and the condensed structural properties
(contained in the first model with nine degrees of freedom), which
is extremely useful in the construction of a tire.
[0061] The method consists in linking the design parameters of the
tire, characteristics such as the mixture and the belt, to the
structural parameters, for instance the structural stiffness and
camber stiffness of the tire, because the quantities appearing in
the model used have a physical significance. This means that these
quantities are directly linked to the design parameters, in other
words the model used is a physical model. By so doing, any change
to the design parameters of the tire leads to a change of the
parameters of the predictive physical model of the tire and this
change, in turn, produces a variation of the tire's structural
parameters.
[0062] The model permits identification of the structural
parameters starting from dynamic analysis made on the second,
finite-element model of the non-rolling tire. One requirement of
the concentrated-parameter model is, in fact, that it be predictive
of actual behaviour of the tire.
[0063] One of the main advantages of the method according to the
invention is that the concentrated parameters are not identified by
means of experimental tests on prototypes, but by means of virtual
dynamic tests on the finite-element model of the non-rolling
tire.
[0064] The method according to the invention uses a model of
contact between tire and road that enables forecasting of the drift
curves at steady state. Also fed into the brush model, in addition
to the longitudinal and transversal stiffness of the inserts of the
tread, was their torsional stiffness, these stiffnesses being
identified by means of numerical simulations on the second
finite-element model, without any need for experimental tests.
[0065] The method according to the invention enables drift curves
to be determined
[0066] by applying a drift angle .alpha. to the first physical
model;
[0067] created in the area of contact on account of the drift are a
sideward force and a self-aligning torque that act on the first
physical model as forces acting on the hub and cause a lateral
displacement and a snaking motion of the plate of the brush
model;
[0068] because of the snaking motion .sigma..sub.b and the lateral
displacement of the plate, the forces set up in the contact area
are modified; this results in a variation of the unconstrained
degrees of freedom, among which those of the plate, and therefore
of the forces acting of the first physical model.
[0069] With this procedure, after a certain number of iterations, a
point is reached at which the degrees of freedom of the first
physical model settle about a steady state value. In this
situation, the sideward force and the self-aligning torque created
in the area of contact and from which the drift curves may be
obtained are determined.
[0070] The method according to the invention also enables transient
state drift behaviour of the tire to be evaluated, making allowance
in the brush model for the dynamic deformations undergone by the
inserts of the tread in this stage.
[0071] In this way, the length of relaxation of the tire while
drifting is determined upon variation of the running conditions
(speed, vertical load, drift angle, etc.). This procedure is also
implemented without any need for experimental testing.
[0072] Characteristics and advantages of the invention will now be
described with reference to an embodiment of the invention,
illustrated indicatively and by no means exclusively in the
accompanying drawings, where:
[0073] FIG. 1 shows a concentrated-parameter physical model of a
tire used in a method for determining the road handling of a tire
of a wheel for a vehicle, constructed according to the
invention;
[0074] FIG. 2 shows a finite-element model of a tire used in the
method according to the invention;
[0075] FIGS. 3-6 are schematic representations of test procedures
with a non-rolling tire to which the concentrated-parameter
physical model of FIG. 1 is subjected;
[0076] FIGS. 7-15 are graphs showing the results of the tests
illustrated in the FIGS. 3-6, obtained from the finite-element
model of FIG. 2 which describes a selected real tire;
[0077] FIGS. 16-21 depict modes of vibration of the finite-element
model describing the selected real tire;
[0078] FIG. 22 is a flow diagram of a procedure for determining the
steady state drift curves of the concentrated-parameter physical
model of FIG. 1;
[0079] FIG. 23 is a schematic representation of a brush model
associated with the concentrated-parameter physical model of FIG.
1;
[0080] FIG. 24 depicts a contact pressure distribution determined
using the finite-element model describing the selected real
tire;
[0081] FIG. 25 shows details of the brush model of FIG. 1;
[0082] FIGS. 26 and 27 are graphs illustrating the results obtained
with the brush model of FIG. 1;
[0083] FIG. 28 shows schematic representations of the brush model
of FIG. 1;
[0084] FIG. 29 depicts another contact pressure distribution
determined using the finite-element model describing the selected
real tire;
[0085] FIGS. 30 and 31 are drift curves obtained from the
concentrated-parameter physical model describing the selected real
tire;
[0086] FIGS. 32 and 33 are schematic representations of drift
transient state test procedures to which the concentrated-parameter
physical model of FIG. 1 is subjected;
[0087] FIGS. 34, 35 and 36 are further schematic representations of
the brush model of FIG. 1;
[0088] FIGS. 37-52 illustrate the drift transient state test
results obtained from the concentrated-parameter physical model
describing the selected real tire; and
[0089] FIG. 53 depicts a distribution of forces acting on a beam of
the tire brush model.
[0090] Illustrated in FIG. 1 is a concentrated-parameter physical
model, indicated generically with the numeral 1, reproducing the
drift behaviour of a tire of a wheel, made from selected mixes of
rubber and reinforcing materials.
[0091] The physical model 1 comprises a rigid ring 2 which
represents a tread band provided with inserts, a belting structure
and corresponding carcass portion of the tire, and a rigid disk 3
representing a hub of the wheel and beading of the tire. The model
1 also comprises principal springs 4, 5 and 6 and principal dampers
7, 8 and 9 which connect the rigid ring 2 to the hub 3 and
represent sidewalls of the tire and air under pressure inside the
tire. The model also comprises supplementary springs 10, 11 and 12
and supplementary dampers 13, 14 and 15 which represent phenomena
of deformation of the belt through the effect of a specified
vertical load.
[0092] Associated with the physical model 1 is a brush model 20
which simulates physical phenomena present in an area of contact
between tire and road. The brush model 20 comprises a rigid plate
21 under which a system representing the tread is applied. The
system is preferably bidimensional and comprises numerous parallel,
deformable beams 22 orientated longitudinally, the ends of which
are hinged to the plate, and numerous microinserts, or rows of
springs, 23 arranged in parallel. In this particular case, there
are three deformable beams 22 whilst there are 5 rows of
microinserts 23 associated with each beam. The bottom ends of the
microinserts 23 of the brush model interact with a road or the
ground 24. The model reproduces the local deformations occurring
inside the area of contact and represents the uniformly distributed
lateral and torsional stiffnesses of the portion of tread in the
area of contact.
[0093] The rigid ring 2 has an equivalent roll radius r [m], mass
M.sub.c [kg] and diametral moment of inertia J.sub.c [kg*m.sup.2].
The rigid disk 3 has mass M.sub.m [kg] and diametral moment of
inertia J.sub.m [kg*m.sup.2].
[0094] The principal springs 4, 5 and 6 have structural stiffnesses
K.sub.c, respectively comprising lateral stiffness K.sub.cy [N/m]
between hub and belt, camber torsional stiffness K.sub.c.theta.x
[Nm/rad] between hub and belt and yawing torsional stiffness
K.sub.c.theta.z [Nm/rad] between hub and belt.
[0095] The principal dampers 7, 8 and 9 have structural dampings
R.sub.c, respectively comprising lateral damping R.sub.cy [Ns/m]
between hub and belt, camber torsional damping R.sub.c.theta.x
[Nms/rad] between hub and belt and yawing torsional damping
R.sub.c.theta.z [Nms/rad] between hub and belt.
[0096] The supplementary springs 10, 11 and 12 have residual
stiffnesses K.sub.r, respectively comprising residual lateral
stiffness K.sub.ry [N/m], residual camber torsional stiffness
K.sub.r.theta.x [Nm/rad] and residual yawing torsional stiffness
K.sub.r.theta.z [Nm/rad].
[0097] The supplementary dampers 13, 14 and 15 have residual
dampings R.sub.r, respectively comprising residual lateral damping
R.sub.ry [Ns/m], residual camber torsional damping R.sub.r.theta.x
[Nms/rad] and residual yawing torsional damping R.sub.r.theta.z
[Nms/rad].
[0098] The residual stiffnesses and dampings permit allowance to be
made for the variation of local stiffness due to deflection of the
tire. The lateral and residual yawing stiffnesses K.sub.ry and
K.sub.r.theta.x connect the bottom end of the rigid ring to the
plate, as also does the residual camber stiffness K.sub.r.theta.z.
In some cases, the camber deformation of the plate .rho..sub.b
(absolute rolling rotation) is not taken into consideration, so
that connecting the second end of the spring 11, which represents
the residual camber stiffness, directly to the plate is tantamount
to connecting it to the ground. In these cases, the stiffness
K.sub.r.theta.x already incorporates the effect due to camber
deformability of the brush model.
[0099] The equivalent system has stiffness per unit length c.sub.py
and the contact area has a dynamic length 2a and dynamic width
2b.
[0100] With the equivalent system, allowance may be made both for
the deformability of the inserts in the tread and for the different
speeds between a point of the insert in contact with the road
(assuming there is adhesion, this point has a lateral velocity
y'.sub.s=0) and the corresponding point on the belt. Three factors
play a decisive role: the coefficient of friction at the interface
between wheel and road, the normal pressure distribution and
stiffness of the inserts in the tread.
[0101] Shown in FIG. 1 is an absolute trio of reference axes
O-X-Y-Z having versors i, j, k, where the origin O coincides with
the centre of the hub with the tire non-deformed, the X axis lies
in the plane of the hub and is of longitudinal direction, the Y
axis is perpendicular to the X axis and the Z axis is vertical.
[0102] The degrees of freedom of the physical model 1 are:
[0103] absolute lateral displacement y.sub.m of the hub, absolute
yaw rotation .sigma..sub.m of the hub and absolute rolling rotation
.rho..sub.m of the hub,
[0104] relative lateral displacement y.sub.c of the belt with
respect to the hub, relative yaw rotation .sigma..sub.c of the belt
with respect to the hub and relative rolling rotation .rho..sub.c
of the belt with respect to the hub,
[0105] absolute lateral displacement y.sub.b of the plate, absolute
yaw rotation .sigma..sub.b of the plate and absolute rolling
rotation .rho..sub.b of the plate.
[0106] A further degree of freedom is:
[0107] absolute lateral displacement y.sub.s of the bottom ends of
the microinserts.
[0108] This degree of freedom has the objective of reproducing the
sideward forces created under the contact and which are linked to
the relative displacements between the top and bottom ends of the
microinserts. In the case of perfect adhesion of the microinserts,
with the tire not drifting, y.sub.s=0.
[0109] Motion of the physical model is described by assuming small
displacements and small rotations of the hub.
[0110] The tire is described using a Finite-Element Model (F.E.M.)
30 depicted in FIG. 2. The finite-element model 30 comprises first
elements (bricks or shells or multilayer composites) with a
selected number of nodes, having appropriately selected form
functions, preferably of the first or second order and, even more
preferably, linear, and second elements suitable for describing the
reinforcing materials. Each first element has a first stiffness
matrix which is determined using a selected characterization of the
mixes and a second supplementary stiffness matrix which is
determined using a selected characterization of the reinforcing
materials.
[0111] More than ten different types of mixes are usually to be
found inside a tire. Their elastic properties are fed into the
finite-element model after selected static and dynamic tests are
conducted on specimens of the mixes.
[0112] The static tests consist of tensile, compression and shear
tests in which the test conditions, forces and elongations applied
are established in relation to the properties of the mixes measured
previously (hardness, etc.).
[0113] The Mooney-Rivlin law of hyperelasticity was taken as the
constitutive law. This law describes the specific deformation
energy in relation to the derivatives of the displacements
(deformations), separating the form variation energy from the
volume variation energy (deviatoric and hydrostatic part of the
stress tensor).
[0114] The coefficients of the constitutive law are calculated in
such a way as to minimize the difference between the experimental
and the calculated deformation energy.
[0115] The dynamic tests are conducted by applying first a static
predeformation to the specimens and then an oscillating load with
frequency in the range from 0.1 to 100 Hz. The dynamic modulus of
the mixture is thus detected as the complex ratio between stress
and deformation. As the frequency is changed, the modulus and
relative phase between stress and deformation are measured.
[0116] The reinforcing materials used are fabrics and of metallic
type. In the tires for automobiles, metallic cords are used only
for the belts and fabric cords for the carcass and the outermost
belt (zero degrees) located just under the tread band. The metallic
cord is subjected to pulling until it breaks and to compression in
order to obtain by experimental means the characteristic of the
cord from critical load to breaking. This characteristic is
implemented in the finite-element model. The fabric cord is also
subjected to pulling until it breaks.
[0117] The second elements that describe the reinforcing materials
are defined within the first elements (bricks, for instance) of the
finite-element model 30, by assigning the geometrical disposition
of the fabrics, the orientation of the cord, the spacing between
the single cords (thickness) and the experimentally obtained
traction and compression characteristic of the cord. These
characteristics, in relation to the dimensions of the brick element
taken, are resumed in a supplementary stiffness matrix which is
overlaid on the mixture stiffness matrix, enabling extraction of
the cord tensions.
[0118] A frequency domain analysis is performed, wherein a
linearized response is determined to a harmonic excitation based on
the single degrees of freedom of the physical model. The response
is obtained by resolving a matrix system, complete with mass,
damping and stiffness matrices. Linearization of the matrices is
performed at the end of the preliminary static determination stages
so that non-linear behaviour of the actual tire is implicitly taken
into account.
[0119] By defining the isotropic linear viscoelasticity, the
damping and stiffness matrices in relation to frequency may be
built. It follows that the relation between stresses and
deformations is considerably influenced by the elastic behaviour (a
higher modulus of elasticity corresponds in particular to higher
frequencies) and the damping of the mix.
[0120] In the case of FIG. 2, the finite-element model
comprises:
[0121] 17,500 elements (16,000 defined by the user+1,500 generated
autonomously, needed for definition of the constraints of the
contact between the tire and the rim it is fitted on and the
road);
[0122] approx. 36,000 nodes (19,000 defined by the user+approx.
17,000 generated autonomously, needed for resolution of the
hydrostatic part of the stress tensor);
[0123] approx. 74,000 degrees of freedom (19,000 nodes.times.3
translational degrees of freedom for each node+17,000 degrees of
freedom associated with the elements for resolution of the
hydrostatic part of the stress tensor).
[0124] The equations of motion of the concentrated-parameter
physical model 1 of the non-rolling tire are obtained using the
Lagrange method.
[0125] The unknown parameters of these equations are identified by
comparing the vibrational dynamic response determined using the
finite-element model with that obtained from the
concentrated-parameter physical model.
[0126] The above-described degrees of freedom (independent
variables or generalized coordinates) of the physical model are
contained in a vector x organized as follows:
x={y.sub.m .rho..sub.m .sigma..sub.m y.sub.c .rho..sub.c
.sigma..sub.c y.sub.b .rho..sub.b .sigma..sub.b}.sup.T (1.1)
[0127] The kinetic energy, expressed through the independent
variables of the model, is as follows: 1 E c = 1 2 M m y . m 2 + 1
2 M c ( y . m + y . c ) 2 + 1 2 J m . m 2 + 1 2 J m . m 2 + 1 2 J c
( . m + . c ) 2 + 1 2 J c ( . m + . c ) 2 ( 1.2 )
[0128] The potential energy, expressed through functions of the
independent variables of the model, is as follows: 2 V = 1 2 K cy y
c 2 + 1 2 K c x c 2 + 1 2 K c z c 2 + 1 2 K ry ( y p - y b ) 2 + 1
2 K r z ( s p - b ) 2 + 1 2 K r x ( r p - b ) 2 ( 1.3 )
[0129] where
[0130] y.sub.p is the absolute lateral displacement of the bottom
point of the belt;
[0131] s.sub.p is the absolute yaw rotation of the belt;
[0132] r.sub.p is the absolute camber rotation of the belt.
[0133] The relations linking the physical variables to the
independent variables are as follows:
y.sub.p=y.sub.m+y.sub.c+r*.rho..sub.m+r*.rho..sub.c
s.sub.p=.sigma..sub.m+.sigma..sub.c
r.sub.p=.rho..sub.m+.rho..sub.c
[0134] When these relations are inserted in the potential energy
equation, the potential energy in relation to the generalized
coordinates is obtained. The dissipation energy D is similar in
form to the potential energy V.
[0135] If Lagrange's theorem is applied to the kinetic energy
expression (1.2), the inertia terms [M]* {umlaut over (x)} are
found, where the general mass matrix [M] is: 3 [ M ] = [ M m + M c
0 0 M c 0 0 0 0 0 0 J m + J c 0 0 J c 0 0 0 0 0 0 J m + J c 0 0 J c
0 0 0 M c 0 0 M c 0 0 0 0 0 0 J c 0 0 J c 0 0 0 0 0 0 J c 0 0 J c 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ( 1.4
)
[0136] The matrix includes the concentrated parameters of the model
M.sub.m, M.sub.c, J.sub.m and J.sub.c.
[0137] On derivation of the expression of the potential energy V
with respect to the vector of the independent variables, a general
stiffness matrix [K] is obtained (1.5): 4 [ K ] = [ K ry K ry r 0 K
ry K ry r 0 - K ry 0 0 K ry r K ry r 2 + K r x 0 K ry r K ry r 2 +
K r x 0 - K ry r - K r x 0 0 0 K r z 0 0 K r z 0 0 - K r z K ry K
ry r 0 K cy + K ry K ry r 0 - K ry 0 0 K ry r K ry r 2 + K r x 0 K
ry r K ry r 2 + K r x + K c x 0 - K ry r - K r x 0 0 0 K r z 0 0 K
c z + K c z 0 0 - K r z - K ry - K ry r 0 - K ry - K ry r 0 K ry 0
0 0 - K r x 0 0 - K r x 0 0 K r x 0 0 0 - K r z 0 0 - K r z 0 0 K r
z ]
[0138] The structural and residual stiffnesses of the model are
included in this matrix.
[0139] Finally, on deriving the expression of the dissipation
energy D with respect to the derivative vector before the
independent variables, a general damping matrix [R] is obtained
(1.6): 5 [ R ] = [ R ry R ry r 0 R ry R ry r 0 - R ry 0 0 R ry r R
ry r 2 + R r x 0 R ry r R ry r 2 + R r x 0 - R ry r - R r x 0 0 0 R
r z 0 0 R r z 0 0 - R r z R ry R ry r 0 R cy + R ry R ry r 0 - R ry
0 0 R ry r R ry r 2 + R r x 0 R ry r R ry r 2 + R r x + R c x 0 - R
ry r - R r x 0 0 0 R r z 0 0 R c z + R c z 0 0 - R r z - R ry - R
ry r 0 - R ry - R ry r 0 R ry 0 0 0 - R r x 0 0 - R r x 0 0 R r x 0
0 0 - R r z 0 0 - R r z 0 0 R r z ]
[0140] The structural and residual dampings of the model are
included in this matrix.
[0141] The vector of the forces F contains the forces and torques
acting on the physical model in the different situations the model
is found in, as will be illustrated later.
[0142] In short, the equations of motion are obtained in matrix
form:
[M].{umlaut over (x)}+[R].{dot over (x)}+[K].x=F (1.7)
[0143] In this equation, the contributions linked respectively to
the degrees of freedom of the hub, the belt and the tread may be
discerned: 6 [ M m m M m c M mq M c m M cc M cq M qm M qc M qq ] x
_ + [ R m m R m c R mq R c m R cc R cq R qm R qc R qq ] x . _ + [ K
m m K m c K mq K c m K cc K cq K qm K qc K qq ] x _ = { F m F c F q
} ( 1.8 )
[0144] To identify the concentrated parameters of the physical
model, a series of virtual dynamic tests, simulated on the
finite-element model 30, is defined, in which the tire is variously
excited and under different conditions so as to highlight the
contribution of each of the terms that has to be identified.
[0145] For each test, the finite-element model is used to determine
the frequency response of particular quantities measured at precise
points of the tire.
[0146] To obtain the frequency responses of the
concentrated-parameter physical model, the equations of motion for
the selected tests are defined.
[0147] For each test and for each frequency response of the
quantities taken into consideration (displacements of the points
considered, constraining reactions, etc.), the difference is
computed between the results obtained from the finite-element model
and those obtained using the concentrated-parameter physical model.
This difference is considered an error and is defined as
err={v-v.sub.FEM} (1.9)
[0148] An objective function is then also defined, consisting of
the weighted sum of the differences found on each channel (each
channel corresponds to a quantity considered) between the responses
of the concentrated-parameter physical model and those of the
finite-element model. The objective function may be expressed as
follows: 7 f = i = 1 N channels ; v i - v i FEM r; + P i (1.10)
[0149] Finally each stiffness and damping parameter of the physical
model is attributed that value by means of which the objective
function is minimized, where the vector of the weightings P.sup.-
enables the various channels to be given different weightings.
[0150] To identify the values of the three stiffnesses and the
three dampings present between the rigid ring and the hub, two
virtual tests are conducted on the tire blown up and not pressed to
the ground. Also obtained at this stage of the identification are
the values for the mass and diametral moment of inertia of the belt
that most closely adapt behaviour of the model to that of the
actual tire. In particular, the value of the mass of the belt
M.sub.c is determined in the first test and the value of the
diametral moment of inertia of the belt J.sub.c in the second
test.
[0151] The mass of the hub M.sub.m, on the other hand, is
determined by imposing the conservation of the total mass of the
tire: M.sub.tot=M.sub.c+M.sub.m. A first sharing of the masses is
then made with a view to defining the first-attempt values of the
mass of the belt and its diametral moment of inertia. These values
will then be modified in the identification stage in order to
optimize dynamic behaviour of the model.
[0152] A first test (test A) on the finite-element model consists
in exciting the blown-up tire not pressed to the ground by imposing
a translation in the Y direction on the hub. The lateral
displacement at particular points of the belt of the tire and the
force set up between hub and belt are measured. A schematic
representation of the first test is shown in FIG. 3 depicting belt
32, hub 33 in transverse section and a connection spring 34, which
represents the lateral stiffness and damping between belt and hub.
Also shown are an arrow 31 representing the displacement imposed on
the hub in the Y direction, and a side view of the belt 32, in
which the cardinal points N, E, S and W are shown. As the belt is
symmetrical, the lateral displacement of one cardinal point only of
the belt 32 is considered in this test.
[0153] A first comparison test (A) is carried out on the
concentrated-parameter tire model, as will be illustrated later.
When the frequency responses of the quantities measured in these
first two tests (lateral displacement of a cardinal point of the
belt and force between belt and hub) are compared and the errors
minimized, the following concentrated parameters of the model 1 may
then be determined:
[0154] the lateral stiffness between hub and belt K.sub.cy,
[0155] the lateral damping between hub and belt R.sub.cy,
[0156] the mass of the belt M.sub.c.
[0157] A second test (test B) on the finite-element model consists
in exciting the blown-up tire not pressed to the ground by imposing
a camber rotation on the hub. The lateral displacement at
particular points of the belt (camber rotation) and the torque
transmitted between hub and belt are measured. A schematic
representation of the second test is shown in FIG. 4 depicting the
belt 32, the hub 33 in transverse section and a connection spring
35, which schematically represents the camber torsional stiffness
and damping between belt and hub, respectively K.sub.c.theta.x and
R.sub.c.theta.x. Also shown are an arrow 41 representing the
rotation imposed on the hub about the X axis, and a side view of
the belt 32, in which the cardinal points N, E, S and W are
indicated. As the belt is symmetrical, the lateral displacement of
one cardinal point only of the belt 32, selected indifferently
between points N and S, is considered in this test.
[0158] A second comparison test (B) is carried out on the
concentrated-parameter tire model, as will be illustrated later.
Again in this case, when the frequency responses of the quantities
measured in these second two tests (lateral displacement of a
cardinal point of the belt and torque between belt and hub) are
compared and the errors minimized, the following concentrated
parameters of the model 1 may then be determined:
[0159] the diametral moment of inertia of the belt J.sub.c,
[0160] the camber torsional stiffness between hub and belt
K.sub.c.theta.x,
[0161] the camber torsional damping between hub and belt
R.sub.c.theta.x,
[0162] the yawing torsional stiffness between hub and belt
K.sub.c.theta.z (by symmetry with K.sub.c.theta.x)
[0163] the yawing torsional damping between hub and belt
R.sub.c.theta.z (by symmetry with R.sub.c.theta.x).
[0164] The connection between hub and belt is made in like manner
for both the camber rotations and the yaw rotations and it is
possible to consider that the behaviour of the belt not pressed to
the ground is axial-symmetrical.
[0165] A third test (test C) on the finite-element model consists
in exciting the tire blown up, pressed to the ground and bereft of
its tread (rasped) at least in the area of contact by the
application to the hub of a force in the Y direction. The lateral
displacement of the hub and particular points of the belt are
measured. A schematic representation of the third test is shown in
FIG. 5 depicting the belt 32, the hub 33 in transverse section, a
connection spring 36 (in addition to the one described above,
indicated with numeral 35) which schematically represents the
residual, lateral and camber stiffnesses and dampings between belt
and ground 24. Also shown are an arrow 51 representing the force
imparted, an arrow 52 representing the lateral displacement of the
hub and a front view of the belt 32, in which the cardinal points
N, E, S and W are indicated. In this test, the lateral displacement
of the N and E cardinal points of the belt 32 is considered.
[0166] A third comparison test (C) is carried out on the
concentrated-parameter tire model, as will be illustrated later.
When the frequency responses of the quantities measured in these
third two tests (lateral displacement of the hub and of the N and E
cardinal points of the belt) are compared and the errors minimized,
the following concentrated parameters of the model 1 may then be
determined:
[0167] the residual lateral stiffness K.sub.ry,
[0168] the residual lateral damping R.sub.ry,
[0169] the residual camber stiffness K.sub.r.theta.x,
[0170] the residual camber damping R.sub.r.theta.x.
[0171] A fourth test (test D) on the finite-element model consists
in exciting the tire blown up, pressed to the ground and bereft of
its tread (rasped) at least in the area of contact by the
application to the hub of a yawing torque about the Z axis. The yaw
rotation of the hub and the displacement at particular points of
the belt are measured. A schematic representation of the fourth
test is shown in FIG. 6 depicting the belt 32, the hub 33 in
transverse section, a connection spring 37 (in addition to the one
described above, indicated with numeral 35) which schematically
represents the residual yawing stiffnesses and dampings between
belt and ground 24. Also shown are an arrow 61 representing the
yawing torque imparted, an arrow 62 representing the rotation of
the hub and a front view of the belt 32, in which the cardinal
points N, E, S and W are indicated. In this test, the lateral
displacement of the E and W cardinal points of the belt 32 is
considered.
[0172] A fourth comparison test (D) is carried out on the
concentrated-parameter tire model, as will be illustrated later.
When the frequency responses of the quantities measured in these
fourth two tests (yaw rotation of the hub and displacement of the E
and W cardinal points of the belt) are compared and the errors
minimized, the following concentrated parameters of the model 1 may
then be determined:
[0173] the residual yawing stiffness K.sub.r.theta.z,
[0174] the residual yawing damping R.sub.r.theta.z,
[0175] During the third and fourth tests, in addition to the
identification of the residual quantities, the stiffness and
damping values already identified in the first and second tests are
also modified in order to better describe the transfer functions
given by the finite-element model. The stiffness and damping values
between hub and belt are therefore modified slightly from those
identified earlier.
[0176] To obtain the frequency responses of the physical model 1
needed for the identification of its concentrated parameters, the
equations of motion relative to the four tests described above are
obtained.
[0177] The tests A and B are carried out with the tire blown up and
not pressed to the ground, imposing displacements on the hub.
[0178] The free degrees of freedom (generalized coordinates) of the
physical model are:
x.sub.i.sup.A-B={y.sub.c .rho..sub.c .sigma..sub.c y.sub.b
.rho..sub.b .sigma..sub.b}.sup.T (1.11)
[0179] whereas the constrained degrees of freedom (constrained
coordinates) are:
x.sub.u.sup.A-B={y.sub.m .rho..sub.m .sigma..sub.m}.sup.T
(1.12)
[0180] As stated above, only a translation in the Y direction is
imposed on the hub in test A and therefore
.rho..sub.m=.sigma..sub.m=0, whilst only a camber rotation is
imposed on the hub in test B and therefore y.sub.m=.sigma..sub.m=0.
Furthermore the tire is raised off the ground and the residual
stiffnesses and dampings relative to the tread cancel each other
out.
[0181] Under these conditions, from the general matrices of mass,
stiffness and damping reported above for the concentrated-parameter
physical model, a mass matrix [M].sup.A-B, a stiffness matrix
[K].sup.A-B and a damping matrix [R].sup.A-B are obtained.
[0182] When a partition is made of the matrices in relation to the
free and constrained coordinates, equations of motion in scalar
form are obtained from (1.8): 8 { ( M m + M c ) y m + M c y c = F
ym ( J m + J c ) m + J c c = C m ( J m + J c ) m + J c c = C m (
1.13 )
[0183] where F.sub.ym, C.sub..rho.m and C.sub..sigma.m represent
the forces and torques imposed on the hub.
[0184] These equations govern the motion of the system in the
simulated tests A and B.
[0185] For the displacements imparted to the hub, the equations
are: 9 { M c y c + R cy y . c + K cy y c = - M c y m J c c + R c x
. c + K c x c = - J c m ( J m + J c ) m + J c c = C m ( 1.14 )
[0186] The frequency responses must now be determined of the
quantities measured in the tests A and B to be able to compare the
frequency responses given by the finite-element model with the
frequency responses of the concentrated-parameter model.
[0187] In the test A, with the physical model, the frequency
response of the lateral displacement of the belt is reconstructed,
taking into consideration the first equation of the system (1.14).
The displacement imparted on the hub is:
y.sub.m.sup.A=A.e.sup.i.OMEGA.t (1.15)
[0188] the complete differential equation to be resolved is as
follows:
M.sub.c.sub.c.sup.A+R.sub.cy{dot over
(y)}.sub.c.sup.A+K.sub.cyy.sub.c.sup-
.A=M.sub.c..OMEGA..sup.2.A.e.sup.i.OMEGA.t (1.16)
[0189] The solution of this equation, having a sinusoidal force
agent equivalent to y.sub.m, is of the type:
y.sub.c.sup.A=B.e.sup.l.OMEGA.t (1.17)
[0190] By means of substitutions and simplifications, the frequency
response is obtained for the degree of freedom y.sub.c upon
variation of .OMEGA., that is to say the frequency response of the
lateral displacement of the belt with respect to the hub. 10 y c A
= M c 2 A ( - M c O 2 + i R cy + K cy ) t ( 1.18 )
[0191] As the finite-element model provides the absolute
displacement of the points of the belt, the frequency response of
the absolute lateral displacement of the belt may be determined
from the absolute lateral displacement: 11 y c _ ass A = y c A + y
m A = [ M c 2 ( - M c 2 + i R cy + K cy ) + 1 ] A t ( 1.19 )
[0192] Upon computing the difference between the lateral
displacement of the physical model and that provided by the
finite-element model, an error is obtained that is a function of
M.sub.c, K.sub.cy and R.sub.cy. In the identification stage, this
error is minimized, that is to say, values are chosen for M.sub.c,
K.sub.cy and R.sub.cy that make the error minimum.
[0193] In the test A, the frequency response of the force
transmitted between hub and belt provided by the finite-element
model is also compared with that provided by the
concentrated-parameter physical model. This force is equal to:
F.sub.hub-belt.sup.A=(K.sub.cy+i..OMEGA..R.sub.cy).B.e.sup.i.OMEGA.t
(1.20)
[0194] Again with this quantity, when the difference is computed
between the frequency response of the force determined with the
finite-element model and that obtained using the
concentrated-parameter physical model, a second error is obtained
that is also function of M.sub.c, K.sub.cy and R.sub.cy. The
above-stated quantities are obtained upon minimization of this
error.
[0195] In the test B, with the physical model, the frequency
response of the rotation of the belt is reconstructed and the same
method of procedure as the test A is then followed. The second
equation of the system (1.14) is considered, starting from a camber
rotation imparted to the hub of the type:
.rho..sub.m.sup.B=C.e.sup.i.OMEGA.t (1.21)
[0196] the frequency response is obtained of the absolute camber
rotation of the belt: 12 c _ ass B = c B + m B = [ J c 2 ( - J c 2
+ i R c x + K c x ) + 1 ] C i t ( 1.22 )
[0197] and, from this, the frequency response of the rotation of
the lateral displacement of the S point of the belt is
determined:
y.sub.c.sub..sub.--.sub.ass-south
.sup.B=.rho..sub.c.sub..sub.--.sub.ass .sup.B.r (1.23)
[0198] Also determined is the frequency response of the torque
transmitted between hub and belt:
C.sub.hub-belt
.sup.B=(K.sub.c.theta.x+i..OMEGA..R.sub.c.theta.x).D.e.sup.-
l.OMEGA.t (1.24)
[0199] Upon computing the difference between the displacement
y.sub.c.sub..sub.--.sub.ass.sub..sub.--.sub.south .sup.B and the
torque C.sub.hub-belt of the physical model and those given by the
finite-element model, an error is obtained that is a function of
the quantities J.sub.c, K.sub.c.theta.x and R.sub.c.theta.x. In the
identification stage, this error is minimized and values are
identified for the above-mentioned quantities.
[0200] In the tests C and D, with the tire blown up, pressed to the
ground and bereft of tread, the free degrees of freedom and the
constrained degrees of freedom of the physical model are determined
in the two test situations and an axial force is imposed on the hub
in the Y direction, in the test C, and a yawing torque about the Z
axis, in the test D. When these conditions are put into the
equations of motion (1.7) and (1.8), the equations of motion for
the free degrees of freedom of the physical model are obtained and
the complex vector of the free degrees of freedom is determined.
This vector contains both the modulus and the phase of the free
degrees of freedom in question.
[0201] In the test C, from this vector, the frequency responses of
the lateral displacement of the hub, of the N point and of the E
point of the belt are determined. In the identification stage, the
difference is computed between the frequency response obtained from
the finite-element model and that obtained from the
concentrated-parameter physical model and a second error is
obtained which is a function of K.sub.ry, R.sub.ry, K.sub.r.theta.x
and R.sub.r.theta.x. On minimizing this error, the above-mentioned
quantities are determined.
[0202] In the test D, from the above-mentioned vector, the
frequency responses are determined of the yaw rotation of the hub
and of the displacement of the E point of the belt in relation to
the quantities K.sub.r.theta..sub.z and R.sub.r.theta.z. These
quantities are determined by minimizing the error resulting as the
difference between the frequency response provided by the
finite-element model and that provided by the
concentrated-parameter physical model.
[0203] Once the concentrated parameters of the physical model have
been identified, it is possible to determine some quantities that
allow an evaluation, or an initial approximation at least, to be
made of the behaviour of the tire. These quantities are the total
drift stiffness of the tire K.sub.d, comprised in turn by the
structural stiffness K.sub.c, the tread stiffness K.sub.b and the
total camber stiffness K.sub..gamma. of the said tire.
[0204] The stiffness K.sub.d is an important parameter for the
purposes of definition of the tire in that it provides an
indication of the effect that the design parameters have on the
drift behaviour.
[0205] To obtain K.sub.d, the free and constrained degrees of
freedom are partitioned:
x.sub.l={y.sub.c .rho..sub.c .sigma..sub.c y.sub.b .sigma..sub.b}
(1.25)
[0206] A further partition is made of the vector x.sub.l into
internal (of the belt) and external (of the tread) degrees of
freedom:
x.sub.l={x.sub.e x.sub.i}.sup.T={{y.sub.b .sigma..sub.b} {y.sub.c
.rho..sub.c .sigma..sub.c}}.sup.T (1.26)
[0207] The stiffness matrix (1.5) is then modified following these
partitions and the equations of motion can be written as follows
for the free degrees of liberty: 13 { [ K ee ] x _ e + [ K ei ] x _
i = F _ e [ K ie ] x _ e + [ K ii ] x i = F _ i = 0 _ ( 1.27 )
[0208] Remembering that no external forces act on the degrees of
internal freedom, the second equation of the system (1.27) may be
resolved in relation to the external degrees of freedom, the result
inserted in the first equation and the latter made explicit in
relation to the external coordinates x.sub.e alone. At this point a
stiffness matrix is defined for the tire structure, limited to the
external degrees of freedom: 14 { [ K ee ] x _ e - [ K ei ] [ K ii
] - 1 [ K ie ] x _ e = F _ e x _ i = - [ K ii ] - 1 [ K ie ] x e (
1.28 )
[0209] The resolving equation is:
[{circumflex over (K)}.sub.ee]x.sub.e=F.sub.e (1.29)
[0210] wherein:
[{circumflex over
(K)}.sub.ee]=[K.sub.ee]-[K.sub.ei][K.sub.ii].sup.-1.[K.s- ub.ie]
(1.30)
[0211] is a matrix (2,2). The first element of the first row of
this matrix represents the total stiffness of the tire K.sub.d:
K.sub.d={circumflex over (K)}.sub.ee(1,1) (1.31)
[0212] With the stiffnesses of the tire with concentrated
parameters known from the identification procedure, it is possible
to calculate K.sub.d.
[0213] The total camber stiffness K.sub..gamma. of the tire
provides an indication of the tire's ability to exit from a
longitudinal track, namely parallel to the direction of movement of
the tire, made on the road when the tire is moving with rectilinear
motion. The method according to the invention enables calculation
of K.sub..gamma. without the need to make and test a tire
prototype, as is generally done.
[0214] To determine K.sub..gamma., a virtual test, practically
equal to the experimental one, is carried out consisting in
imposing a selected camber angle on the hub, with a null drift
angle, and in measuring the sideward force created on the hub.
[0215] The free and constrained degrees of freedom are
identified:
x.sub.l.sup..gamma.={y.sub.c .rho..sub.c .sigma..sub.c y.sub.b
.sigma..sub.b} (1.32)
x.sub.v.sup..gamma.={y.sub.m .rho..sub.m .sigma..sub.m .rho..sub.b}
(1.33)
[0216] By imposing a camber angle and leaving all the other
constrained degrees of freedom unaltered, the sideward force is
determined by resolving the matrix equation:
[K]x=F (1.34)
[0217] where the matrix [K] is the (1.5) and F a vector containing
the forces acting on the tire. Figuring among these are the force
and torque due to the deformations undergone by the inserts of the
tread in the assumption that there is perfect adhesion. These two
contributions are: 15 F yb = 2 c p a y b M zb = 2 3 c p a 3 b (
1.35 )
[0218] As the previous two terms are functions of two of the nine
degrees of freedom of the concentrated-parameter model (y.sub.b and
.sigma..sub.b), they must be expressed in function of these degrees
of freedom and thus be brought to the left-hand side of the equals
sign in the matrix equation reported above (1.34), so that they too
contribute to determining the stiffness matrix of the system.
[0219] The partitioned matrices K.sub.ll.sup..gamma.,
K.sub.lv.sup..gamma., K.sub.vl.sup..gamma. and K.sub.vv.sup..gamma.
are known: 16 [ K vv ] y = [ K ry K ry r 0 0 K ry r K ry r 2 + K r
x 0 - K r x 0 0 K r z 0 0 - K r x 0 K r x ] ( 1.36 ) [ K vi ] y = [
K ry K ry r 0 K ry 0 K ry r K ry r 2 + K r x 0 - K ry r 0 0 0 K r z
0 - K r z 0 - K r x 0 0 0 ] ( 1.37 ) [ K lv ] y = [ K ry K ry r 0 0
K ry r K ry r 2 + K r x 0 - K r x 0 0 K r z 0 - K r y - K ry r 0 0
0 0 - K r z 0 ] ( 1.38 ) [ K ll ] y = [ K cy + K ry K ry r 0 - K ry
0 K ry r K ry r 2 + K r x + K c x 0 - K ry r 0 0 0 K c z + K r z 0
- K r z - K r y - K ry r 0 K ry - 2 c p a 0 0 0 - K r z 0 K r z - 2
3 c p a 3 ] ( 1.39 )
[0220] From these, the vector of the forces acting on the
constrained degrees of freedom may be calculated using the
equation:
F.sub.v.sup..gamma.={-[K.sub.vl].sup..gamma..[K.sub.ll].sup..gamma.-1.[K.s-
ub.lv].sup..lambda.+[K.sub.vv].sup..gamma.}x.sub.v.sup..gamma.
(1.40)
[0221] Remembering that:
[{circumflex over
(K)}].sup..gamma.=-[K.sub.vl].sup..gamma..[K.sub.ll].sup-
..gamma.-1.[K.sub.iv].sup..lambda.+[K.sub.vv].sup..gamma.
(1.41)
[0222] is a matrix (4,4), the first element of the second row,
which is the total camber stiffness of the tire, is determined:
K.sub..gamma.={circumflex over (K)}.sup..gamma.(1, 2) (1.42)
[0223] To demonstrate the validity of the method according to the
invention, the results obtained are brought into the procedure for
the identification of the concentrated parameters of the physical
model of a tire of the 55 range (H/C section ratio of 0.55)
manufactured by the Applicant. In the graphs of FIGS. 7-15, the
frequency response obtained with the concentrated-parameter
physical model is represented with a continuous line, while the
frequency response obtained with the finite-element model during
the tests A, B, C and D is represented with asterisks.
[0224] FIG. 7 is a graph of the frequency response of the
displacement of a point of the belt obtained during the test A from
the physical model and from the finite-element model, whereas FIG.
8 is a graph of the frequency response of the force created between
the hub and the belt, again in the test A.
[0225] The values of the concentrated parameters identified with
the test A are reported in the following Table I.
1TABLE I Parameter Symbol Value identified Linear stiffness between
hub and belt.sup.9 K.sub.cy 5 e.sup.5 [N/m] Linear damping between
hub and belt R.sub.cy 116.906 [Ns/m] Belt mass M.sub.c 7.792
[kg]
[0226] It may be observed from the graphs of FIGS. 7 and 8 that the
frequency response has a single resonance peak. Corresponding to
this peak is a mode of vibration of the tire illustrated in FIG.
16. This mode of vibration maintains the tread band substantially
rigid and may therefore be described by the concentrated-parameter
model, that simulates the sidewall and the belt with a rigid ring.
The parameters identified are therefore valid in a set of
frequencies ranging from 0 to 100 Hz, since modes of vibration that
considerably deform both sidewall and belt appear with frequencies
that are higher than this.
[0227] FIG. 9 is a graph of the frequency response of the
displacement of a point of the belt obtained during the test B from
the physical model and from the finite-element model, whereas FIG.
10 is a graph of the frequency response of the torque created
between the hub and the belt, again in the test B.
[0228] The values of the concentrated parameters identified with
the test B are reported in the following Table II.
2TABLE II Parameter Symbol Value identified Torsional stiffness
between hub and belt K.sub.cex/K.sub.cez 3.3 e.sup.4 [N/m]
Torsional damping between hub and belt R.sub.cex/R.sub.cez 8.217
[Ns/m] Diametral moment of inertia of belt J.sub.c 0.373
[kg*m.sup.2]
[0229] As in the previous test, again in this one there is a single
resonance peak corresponding to the modal deformation of the tire
shown in FIG. 17.
[0230] In the test C, consideration also needs to be given to the
normal load bearing on the tire. Three loads corresponding to three
standard working conditions were applied to the tire: a reduced
load of between 2,500 and 3,000 N, an intermediate load of between
3,500 and 4,800 N and a high load of between 5,100 and 6,500 N.
[0231] FIG. 11 is a graph of the frequency response of the lateral
displacement of the hub obtained from the physical model and from
the finite-element model during the test C, with a normal load of
2,914 N bearing on the tire, whereas FIGS. 12 and 13 are graphs of
the frequency response of the lateral displacements of the E and N
points of the belt, again in the test C.
[0232] The values of the concentrated parameters identified in the
test C are reported in the following Table III.
3TABLE III Parameter Symbol Value identified Residual linear
stiffness K.sub.ry 665964 [N/m] Residual camber stiffness R.sub.r0x
11461 [Nm/rad] Residual linear damping R.sub.ry 2042.345 [Ns/m]
Residual camber damping R.sub.r0x 0.358 [Nms/rad]
[0233] Unlike in the two earlier tests, there are two resonance
peaks in this third test corresponding to two modes of vibration of
the tire depicted in FIGS. 18 and 19.
[0234] Again in this case, the modes of vibration maintain the
sidewall and belt complex substantially rigid and may therefore be
described accurately by the concentrated-parameter model, in other
words the concentrated-parameter model is valid in the range of
frequencies between 0 and 100 Hz.
[0235] The frequency responses obtained with the loads of 4,611 N
and 6,302 N are not shown herein. The results, however, are similar
to those shown for the 2,914 N load.
[0236] FIG. 14 is a graph of the frequency response of the yaw
rotation of the hub obtained from the physical model and from the
finite-element model during the test D, with a normal load of 2,914
N bearing on the tire, whereas FIG. 15 is a graph of the frequency
response of the lateral displacement of the E point of the belt,
again in the test D.
[0237] The values of the concentrated parameters identified with
the test D are reported in the following Table IV.
4 TABLE IV Parameter Symbol Value identified Residual yaw stiffness
K.sub.r0z 14933 [Nm/rad] Residual yaw damping R.sub.r0z 14.318
[Nms/rad]
[0238] Again in the test D, there are two resonance peaks which
correspond to the two modes of vibration of the tire shown in FIGS.
20 and 21.
[0239] In this case, whereas the first mode of vibration (FIG. 20)
is comparable to a rigid mode, it is much more difficult to
describe the second one (FIG. 21) as a rigid mode. It seems
apparent that the belt as a whole becomes deformed in this second
mode of vibrating. The reconstructed frequency response does not
therefore accurately reproduce the second resonance peak. The
concentrated-parameter model in the test D is therefore only valid
in the range of frequencies from 0 to 70 Hz.
[0240] Again in the test D, the frequency responses obtained with
the loads of 4,611 N and 6,302 N are not shown herein. The results,
however, are similar to those shown for the 2,914 N load.
[0241] To evaluate a tire in relation to its road handling,
verification needs to be made that the total drift stiffness
K.sub.d of the tire and the total camber stiffness K.sub..gamma.
are within the following value ranges:
[0242] K.sub.d=500-2,000 [N/g]
[0243] K.sub..gamma.=40-3,500 [N/g]
[0244] K.sub.c=8,000-30,000 [N/g]
[0245] K.sub.b=150-400 [/g]
[0246] where g=degree.
[0247] The method according to the invention enables an evaluation
to be made of the steady state behaviour of the drifting tire. The
bidirectional brush model shown in FIG. 1 and illustrated above is
used.
[0248] Under drift conditions, the microinserts of the brush model
become deformed and a sideward force and a moment of torque act on
the beam that they are connected to. These forces and moments
result in a deformation of the beam that affects the configuration
of the microinserts. By means of successive iterations, the
deformation effectively assumed by the beams in relation to the
drift angle imposed on the hub is determined. At this point, the
total sideward force is determined, together with the total
self-aligning torque acting on the plate that the beams are
connected to. Under the total sideward force and self-aligning
torque, the plate snakes by an angle that depends on the overlying
structure, i.e. on the springs connecting it to the hub. On account
of this snaking, the deformation of the microinserts is altered. By
performing new iterations, the sideward force and self-aligning
torque that the tire summons up in reaction to the drift angle
imposed are determined.
[0249] The procedure is illustrated in the flow diagram of FIG.
22.
[0250] A drift angle .alpha. is applied to the physical model 1,
starting from a condition in which the beams 22 are in a
non-deformed configuration and the brush model has a null snaking
.sigma..sub.b (block 45). The sideward forces acting on the beams
through the effect of the drift angle .alpha. and depending on the
difference .alpha.-.sigma..sub.b and on the deformation of the
beams are determined (blocks 46a, 46b and 46c). The deformation of
the beams is determined (blocks 47a, 47b and 47c). A check is made
to see if the deformation is the same as that determined in the
previous step (blocks 48a, 48b and 48c). The procedure for
determining deformation of the beams is repeated until deformation
is verified to be equal to that found in the previous step. At this
point, the snaking of the plate, i.e. of the brush model, is
determined (block 49). A check is made to see if the snaking is
equal to that calculated in the previous step (block 50). The
procedure for determining snaking of the plate is repeated until
snaking is verified to be equal to that found in the previous step.
At this point, the sideward force and the self-aligning torque
acting on the hub due to the drift angle imposed are determined
(block 51). The procedure is repeated for the different values of
the drift angle .alpha. to produce drift, force and self-aligning
torque curves that enable steady state drift behaviour of the tire
to be evaluated.
[0251] The beams of the brush model do not all become deformed in
the same way since the pressure distribution acting on each beam is
different, as also are the sideward forces. In practice, the
sidewalls of the tire (shoulders) have greater stiffness than in
the central band of the tread.
[0252] To identify the flexural stiffness of each beam, a
distributed parameters model is used based on a model depicted in
FIG. 23 wherein the equivalent beams and springs (microinserts) of
the brush model are indicated with the same numerals as in FIG. 1.
A linear static analysis, corresponding to a sideways traction and
made by applying a sideward force to the contact surface of the
road, is then conducted on the finite-element model. The
accordingly stressed tire becomes deformed and the lateral
displacements are determined at circumferential sections thereof,
on a level with the external band. These circumferential sections
are divided into three groups corresponding to the central part of
the area of contact and to the two parts at either side. The mean
lateral displacement of each group is calculated over the full
length of the contact area. On determining the difference between
the generic deformation mean (mean lateral displacements over the
full length of the contact area) provided by the finite-element
model and the lateral deformation of the rigid ring of the
concentrated-parameter physical model, also subjected to the same
sideward force as applied to the finite-element model, a
"difference" deformation is obtained that must be offset by the
equivalent beam of flexural stiffness EJ (N*m.sup.2).
[0253] Knowing how the sideward force is divided over the three
parts of the contact area from the linear static analysis of
sideways traction on the finite-element model, a flexural stiffness
EJ is determined for each beam.
[0254] In FIG. 23 it may be seen that the sections closest to the
sidewall that the traction is exerted on become more deformed than
those further away (dashed lines). If a load of F/3 were to be
applied to each beam, there would be a monotone pattern of EJ when
moving from one side of the tire to the other, but this would be
contrary to what experience shows. Therefore, using the
finite-element model, a horizontal division of the load between the
various sections is also made.
[0255] The results of the identification procedure are reported in
the following Table V.
5 TABLE V EJ [N*m.sup.2] 1.sup.st load 2.sup.nd load 3.sup.rd load
1.sup.st beam 50 120 250 2.sup.nd beam 45 106 210 3.sup.rd beam 50
120 250
[0256] To describe as accurately as possible what happens in the
area of contact between tire and road, the actual pressure
distribution in this area must be determined.
[0257] Using the finite-element model, a static pressure
distribution is determined on a non-rolling tire with no drift. The
pressure distribution determined under a 55 range tire, quoted
earlier, is illustrated in FIG. 24, with a normal acting load of
2,914 N. Similar distributions were determined with normal loads of
4,611 N and 6,302 N.
[0258] It may be seen that two pressure peaks are presented in the
transverse direction and that these peaks are shifted outwardly
with respect to the centre of the area of contact. Furthermore the
distribution in the longitudinal direction is symmetrical. Finally
the ratio of the maximum value to the minimum value of the pressure
along the transverse direction increases as the normal load
increases.
[0259] To obtain the drift curves, the pressure values in those
points where the microinserts of the brush model are present must
be known beforehand. The contact was represented in the
concentrated-parameter physical model by means of a regular grid of
200.times.15 elements (200 elements longitudinally and 3.times.5
elements transversally). Accordingly, the pressure at the nodes of
this grid must be known, i.e. at 3,000 points. The finite-element
model provides the pressure at a much lower number of points
arranged in an irregular grid. A procedure for the interpolation of
the finite-element model data is used to move from the grid of
points provided by the finite-element model to that required by the
concentrated-parameter model. In the case in hand, the procedure
adopted is that of the "inverse distances". This interpolation
procedure requires the following inputs:
[0260] the coordinates of those points at which the pressure is
known;
[0261] the value of the pressure at these points;
[0262] the coordinates of the points for which the interpolated
pressure value is required.
[0263] The procedure provides the pressure value for the points
required.
[0264] Downstream of the identification stage, it is required that
any negative pressure values be equal to zero. Accordingly the
actual shape of the area of contact can be reproduced extremely
accurately as each microinsert of the brush model has an own length
and a known pressure distribution. More specifically, the length of
the contact of each microinsert changes upon variation of the
transversal position considered and each microinsert has an own
pressure value that depends on its position within the contact.
Therefore the curve of the contacts of each microinsert, which
identifies the position of the bottom ends of the microinserts when
the tire is drifting, varies from one microinsert to the next. When
the wheel is set rolling and drifting, and a beam and the five rows
of microinserts under it are considered, it may be seen that the
microinserts undergo a progressive, linearly increasing
deformation. The range of deformations is therefore triangular, in
the passage from entrance to exit of the footprint area, assuming
there is no slipping (FIG. 25). FIG. 26 illustrates the deformation
undergone by the beam due to the effect of the sideward forces
transmitted by the microinserts and the deformations undergone by
the microinserts. The deformations of the microinserts are given by
the relative displacements between the upper ends (attached to the
beam) and the lower ends (located on the line of contacts in the
event of adhesion). Pattern of the pressure in each microinsert is
illustrated in FIG. 27.
[0265] In order to determine the lateral and torsional stiffness
per unit of length of the microinsert, respectively {tilde over
(c)}.sub.p and {tilde over (k)}.sub.tor, the total stiffnesses of
the whole tread are taken and then the values found are divided by
the total length of the microinserts obtained as the sum of those
of the single microinserts: 17 { c ~ p = 2 a c py l tol k ~ tor = 2
a k tor l tot ( 1.43 )
[0266] where 2a is the length of the area of contact, l.sub.tot is
the sum of the lengths of the single microinserts, c.sub.py is the
stiffness per unit of length of the contact and k.sub.tor is the
torsional stiffness per unit of length of the contact.
[0267] These values are used to obtain the drift curves.
[0268] To determine the torsional contribution of the microinserts
in the yaw test with a non-rolling tire, the flexure of the inserts
present in the footprint area of the concentrated-parameter model
is identified, the sum is obtained of the contributions of each and
the value obtained subtracted from the torque per foot provided by
the finite-element model. The shape of the area of contact is
provided by the finite-element model. In this way, the purely
twisting torque arising following rotation about the Z axis of the
tire pressed to the ground is obtained.
[0269] The sideward force produced following deformation of each
microinsert and contributing to the flexure, assuming perfect
adhesion, is equal to the lateral stiffness of the microinsert
multiplied by the relative displacement in the plane of the contact
of the top end with respect to the bottom end of the microinsert.
Assuming a rotation about the centre of contact and perfect
adhesion, the deformation of a generic microinsert is equal to d
(x, y) * .alpha.', where .alpha.' is the rotation about the Z axis
effectively undergone by the tread and d is the distance of the
microinsert taken from the centre of the contact.
[0270] The lateral stiffness of the single microinsert is
determined starting from the stiffness per unit of length of the
contact: 18 c _ p = 2 a c py 4 a b dx dy ( 1.44 )
[0271] where a is the half-length of the contact, b is the
half-width of the contact and dx and dy are respectively the
longitudinal and transversal dimensions of each microinsert.
[0272] The bending torque due to each microinsert is calculated as
follows:
M.sub.f(x, y)=F(x, y).d(x, y) (1.45)
[0273] where F is the sideward force due to the flexure of the
microinsert and x and y are the coordinates as measured from the
centre of the contact.
[0274] The total bending torque created under the area of contact
for the rotation .alpha.' is calculated as follows: 19 M f = x y M
f = x y F ( x , y ) d ( x , y ) = x y c p _ ' d 2 ( x , y ) ( 1.46
)
[0275] By subtracting the total bending torque from the torque per
foot provided by the finite-element model, the pure twisting torque
is obtained:
M.sub.tors=C.sub.foot-M.sub.f (1.47)
[0276] If the twisting torque is divided by the rotation .alpha.'
and by the length of the area of contact 2a, the torsional
stiffness per unit of length of the contact is obtained: 20 K tor =
M tors 2 a ' ( 1.48 )
[0277] With the method according to the invention making allowance
for the deformability of the structure of the
concentrated-parameter model, of the local deformations of the
contact through the equivalent beams, of the lateral and torsional
stiffnesses of the microinserts, the drift curves for the range 55
tire were determined for three different loads.
[0278] To obtain calculated drift curves coinciding with the
experimental ones, the shape of the area of contact is
considered.
[0279] Bearing in mind the local deformation of the belt in
correspondence with the area of contact, a pear-shaped area of
contact is identified and a brush model with equivalent beams of
different lengths is used, as depicted in FIG. 28. The length of
the beams is obtained by performing a sensitivity analysis,
assuming that the central beam is equal in length to the
statistically measured contact area length and that the variation
of length of the external and internal beams is of equal modulus,
but opposite sign. The sensitivity analysis was conducted on the
range 55 tire.
[0280] It was found that the shape of the area of contact affects
the transversal pressure distribution: the pressure peak
corresponding to the outside of the curve is no longer equal to the
other one (the external pressure peak increases, whereas the
internal one decreases as the drift angle increases). It is
considered that the pressure distribution in the transverse
direction is always less symmetrical with a higher pressure peak in
correspondence with the longest equivalent beam and a lower
pressure peak in correspondence with the shortest equivalent
beam.
[0281] For an indication of the pattern of the distribution of
pressure from the finite-element model, the sideward forces arising
under the contact are calculated for different drift angles through
the concentrated-parameter physical model as described earlier.
This force distribution is applied to the nodes of the
finite-element model (non-rolling) in contact with the ground and
an extremely realistic indication of the pressure distribution in
the area of contact obtained. From the pressure distributions
obtained, a strong symmetry in the transverse direction results.
Illustrated in FIG. 29 is the pattern of pressure distribution
(N/mm.sup.2) for a vertical load of 2,914 N whenever a sideward
force and a self-aligning torque corresponding to a test with drift
of 6.degree. are applied to the finite-element model. On
application of these new data to the concentrated-parameter
physical model, calculated drift, sideward force and self-aligning
torque curves are obtained that are practically coincident with
those obtained by experimental means. FIGS. 30 and 31 illustrate
respectively the pattern of the sideward force (N) and the
self-aligning torque (Nm) in relation to the drift angle .alpha.
(.degree.) for a vertical load of 2,914 N. The calculated values
are marked with asterisks (*), whereas the experimental values are
marked with circles (o). Similar curves were obtained for normal
loads of 4,611 N and 6,302 N.
[0282] To determine the dynamic behaviour of the drifting tire
during the transient state, the tire is made roll by imposing laws
of motion on the hub that vary with time and are suitable for
numerically reproducing selected experimental tests commonly
carried out on tires.
[0283] Two experimental tests are conducted in the laboratory for
evaluating the dynamic behaviour of a drifting tire:
[0284] a first test called pendulum test, consisting in
simultaneously imposing a lateral displacement and a steering angle
on the hub of the tire;
[0285] a second test called drift test with yaw pattern, consisting
in directly imposing a drift angle equal to the yaw angle imparted
to the hub.
[0286] In both of the experimental tests, a wheel-road that
simulates the ground is used. The tire is blown up and pressed to
the wheel-road which is rotating at constant angular speed. The
axis of the tire is made oscillate, and a drift angle that varies
with time induced upon it. The oscillations are imposed on the
wheel with the two test arrangements described above.
[0287] FIG. 32 illustrates a cinematic diagram of a test machine
with the pendulum arrangement. By adopting a reference system
integral with the hub of the wheel, the longitudinal and transverse
components are defined of the velocity of the centre of the area of
footprint determined by the rotation velocity V imparted to the
tire by the wheel-road.
V.sub.T={dot over (.sigma.)}.l+V. sin (.sigma.)
V.sub.L=V. cos (.sigma.) (1.49)
[0288] from which the drift angle .alpha. is obtained for small
values of the angle of pendulum (steering) .sigma.: 21 . l + V V =
+ . V l ( 1.50 )
[0289] If a motion that is, for instance, sinusoidal is imposed on
the pendulum:
.sigma.=.sigma..sub.0. cos (.OMEGA..t) (1.51)
[0290] the drift angle becomes: 22 = 0 cos ( t ) - 0 V l sin ( t )
= 0 ( cos ( t ) + ) ( 1.52 )
[0291] Therefore, the drift angle .alpha. is equal to the steering
angle .sigma. imposed on the hub, with a phase difference of angle
.phi.. If .sigma. is constant, then .alpha. is always=.sigma..
[0292] FIG. 33 illustrates a cinematic diagram of a test machine
with the yaw arrangement. The longitudinal and transverse
components of the velocity of the centre of the hub are defined as
follows: 23 { V L = V cos ( ) V T = V sin ( ) (1.53)
[0293] The resultant drift angle for low snaking angles imposed on
the hub is: 24 = V sin ( ) V cos ( ) ( 1.54 )
[0294] Again in this case the steering angle imposed on the hub is
equal to the drift angle.
[0295] To determine the equations of motion of the tire under
dynamic conditions, suitable for reproducing the two experimental
tests described above, the absolute, right-handed trio of reference
axes shown in FIG. 1 is adopted. The horizontal X axis, orthogonal
to the axis of rotation of the hub, forms a null steering angle
.sigma.m. In the numerical simulations it is assumed that it is the
hub that moves while the wheel-road remains motionless. The
microinserts of the brush model enter the area of contact and their
bottom ends remain attached to the ground until slipping
occurs.
[0296] The independent variables of the physical model 1 with nine
degrees of freedom are again those indicated above (1.1). The
equations of motion governing the motion of the physical model are
again those in matrix form reported above (1.7). The matrix of mass
and that of stiffness are those reported above at (1.4) and (1.5).
The damping matrix is that given above at (1.6), wherein the terms
(5,3), (5,6) and (6,5) are equal to J.sub.y.omega.y (moment of
inertia and angular speed about the Y axis) to take gyroscopic
effects into account.
[0297] The vector of the forces comprises:
[0298] the force F.sub.by and the torque M.sub.bz transmitted by
the microinserts to the structure of the tire and a function of
time; the resulting sideward force F.sub.by is positive if
orientated similarly to the Y axis and the self-aligning torque
M.sub.bz is positive if orientated similarly to the Z axis;
[0299] the force F.sub.my and the torques M.sub.mx and M.sub.mz
transmitted by the test machine to the hub;
[0300] the torque M.sub.bx constraining the plate not to rotate
about the X axis because such a degree of freedom is not included
since a monodimensional brush model is used.
F={F.sub.my M.sub.mx M.sub.mz 0 0 0 F.sub.by M.sub.bx
M.sub.bz}.sup.T (1.55)
[0301] For the case under examination, in the
concentrated-parameter physical model in the transient state, the
yaw test is simulated by stating:
y.sub.m=.rho..sub.m=0
.sigma..sub.m=.sigma..sub.m(t) (1.56)
[0302] Then the three free coordinates of the hub are constrained
in the model whereas the roll of the brush model is not taken into
consideration since the contact is considered to be
monodimensional.
[0303] A partition is made between the generalized x.sub.l and
constrained x.sub.v degrees of freedom:
[0304] x={x.sub.l.sup.T x.sub.v.sup.T}.sup.T={{y.sub.c .rho..sub.c
.sigma..sub.c y.sub.b .sigma..sub.b}{y.sub.m .rho..sub.m
.sigma..sub.m .sigma..sub.b}}.sup.T (1.57)
[0305] and the matrices of mass, stiffness and damping are
accordingly rearranged by performing a division into four
submatrices so as to obtain two matrix equations from the equations
of motion: 25 { [ M ll ] x _ l + [ M lv ] x _ v + [ R ll ] x _ . l
+ [ R lv ] x . _ v + [ K ll ] x l + [ K lv ] x _ v = F _ l [ M vl ]
x _ l + [ M vv ] x _ v + [ R vl ] x _ . l + [ R vv ] x . _ v + [ K
vl ] x l + [ K vv ] x _ v = F _ v ( 1.58 )
[0306] In the first equation, F.sub.l represents the vector
containing the external active forces acting on the actual degrees
of freedom; in this case, the only external forces acting on the
x.sub.l are the sideward force F.sub.by and the self-aligning
torque M.sub.bz created under the contact and acting on the degrees
of freedom of the plate. These forces depend on the arrangement of
deformation of the inserts and on whether or not there is local
slipping; they are therefore, generally speaking, non-linear
functions of the degrees of freedom. In the second equation,
F.sub.v represents the vector of the generalized reactions applied
to the constrained degrees of freedom.
[0307] Remembering that the vector x.sub.v and its derivatives with
respect to time are vectors of known functions, the first equation
may be rewritten as follows:
[M.sub.ll]{umlaut over (x)}.sub.l+[R.sub.ll]{dot over
(x)}.sub.l+[K.sub.ll]x.sub.l=F.sub.l-[M.sub.lv]{umlaut over
(x)}.sub.v-[R.sub.lv]{dot over
(x)}.sub.v-[K.sub.lv]x.sub.v={circumflex over (F)} (1.59)
[0308] wherein the terms {circumflex over (F)}.sub.-- are all known
because they are the sum of the effective external forces and of
the equivalent forces due to the motion imparted to the
constraints.
[0309] The equation (1.59) therefore represents a system of "n"
equations, one for each unknown x.sub.l; by solving this equation,
the motion x.sub.l of the model can be determined.
[0310] Once the equations (1.59) have been integrated and the
values obtained for x.sub.l and their derivatives, the constraining
reactions F.sub.v may be obtained from the equations for the
constrained degrees of freedom.
[0311] To solve the non-linear equations (1.59) with explicit
numerical methods, the procedure is to invert the matrix of mass
[M.sub.ll] and, as the matrix is singular and cannot therefore be
inverted, the equation (1.59) is rewritten as a first order system.
A further partition is then made, followed by a change of
variables.
[0312] The selected partition of x.sub.l (1.57) is:
x.sub.l={y.sub.c .rho..sub.c .sigma..sub.c y.sub.b
.sigma..sub.b}.sup.T={{- y.sub.c .rho..sub.c .sigma..sub.c}{y.sub.b
.sigma..sub.b}}.sup.T={x.sub.c.- sup.T x.sub.b.sup.T}.sup.T
(1.60)
[0313] On rearranging the already rearranged matrices of mass,
stiffness and damping as described above, and performing a division
into four submatrices, the equation (1.59) becomes: 26 { [ M cc ] x
_ c + [ R cc ] x . _ c + [ R cb ] x _ . b + [ K cc ] x _ c + [ K cb
] x _ b = F _ ^ c [ R bc ] x _ . c + [ R bb ] x . _ b + [ K bc ] x
_ c + [ K bb ] x _ b = F ^ _ b ( 1.61 )
[0314] where {circumflex over (F)}.sub.c is a vector of three
elements containing the generalized forces acting directly on the
degrees of freedom of the belt and {circumflex over (F)}.sub.b is a
vector of two elements containing the generalized forces acting
directly on the degrees of freedom of the brush model (F.sub.by and
M.sub.bz).
[0315] Finally, to obtain a first order system, an auxiliary
identity is added to the system (1.61): 27 { [ M cc ] x _ c + [ R
cc ] x . _ c + [ R cb ] x _ . b + [ K cc ] x _ c + [ K cb ] x _ b =
F ^ _ c [ R bc ] x _ . c + [ R bb ] x . _ b + [ K bc ] x _ c + [ K
bb ] x _ b = F ^ _ b [ M cc ] x . _ c = [ M cc ] x . _ c ( 1.62
)
[0316] and the following change of variables is made:
z={{dot over (x)}.sub.c.sup.T x.sub.c.sup.T
x.sub.b.sup.T}.sup.T={{{dot over (y)}.sub.c {dot over
(.rho.)}.sub.c {dot over (.sigma.)}.sub.c}{y.sub.c .rho..sub.c
.sigma..sub.c}{y.sub.b .sigma..sub.b}}.sup.T (1.63)
[0317] If the matrices [B] and [C] are defined: 28 [ B ] = [ [ M cc
] [ 0 ] [ R cb ] [ 0 ] [ 0 ] [ R bb ] [ 0 ] [ M cc ] [ 0 ] ] [ C ]
= [ [ R cc ] [ K cc ] [ K cb ] [ R bc ] [ K bc ] [ K bb ] - [ M cc
] [ 0 ] [ 0 ] ] ( 1.64 )
[0318] the system (1.62) is synthetically expressed as:
[B].{dot over (z)}+[C].z={tilde over (F)} (1.65)
[0319] where {circumflex over (F)} is a vector of eight elements
comprised as follows:
{tilde over (F)}={{circumflex over (F)}.sub.c.sup.T{circumflex over
(F)}.sub.b.sup.T 0 0 0}.sup.T (1.66)
[0320] The matrix [B] is now invertible. These dynamic equations
are numerically integrated a Runge-Kutta step-by-step method of the
3.sup.rd order.
[0321] In the equations (1.65), the matrices [B] and [C] are
defined whilst the vector of the forces still needs to be
identified, in particular F.sub.by and M.sub.bz created under the
rolling tire in transient state.
[0322] As already stated earlier, to describe the behaviour of the
rolling tire in the transient state, a component of displacement
V*t in the X direction is imposed on the centre of the hub. In the
drift test with yaw arrangement, small variations of the degrees of
freedom are imposed on the physical model. Under these conditions,
the deformation modalities of the microinserts inside the area of
contact differ from those under steady state conditions. An insert
enters the area of contact in a generic position that depends on
the motion of the wheel and its deformation in the contact area
changes with time in a manner dictated by the pattern of the
model's degrees of freedom. The forces arising during the transient
state depend on the trajectory of the top and bottom ends of each
microinsert and are, as stated, influenced by the motion of the
wheel. The deformation of a generic i-th microinsert is defined as
follows:
Y.sub.i.sub..sub.--.sub.T(x.sub.i,
t)-Y.sub.i.sub..sub.--.sub.P(x.sub.i, t) (1.67)
[0323] where Y.sub.i-P is understood to be the absolute lateral
displacement of the top end of the i-th microinsert connected to
the plate and Y.sub.i-T is understood to be the absolute lateral
displacement of the bottom end interacting with the ground.
[0324] A procedure enabling an instant-by-instant evaluation of the
displacements of the top and bottom ends of all the microinserts in
the contact is now illustrated with reference to FIGS. 34 and 35.
The plate 21 of FIG. 1, connecting the belt to the microinserts in
a generic configuration, is illustrated in schematic form, seen
from above, in FIG. 34. Having earlier indicated the absolute
lateral displacement of the plate with y.sub.b and its absolute yaw
with .sigma..sub.b, the absolute lateral displacement of the top
end of the generic microinsert Y.sub.i-P is:
Y.sub.i.sub..sub.--.sub.P(x.sub.i,
t)=Y.sub.b(t)+(.alpha.-.xi..sub.l). sin (.sigma..sub.b) (1.68)
[0325] where .xi..sub.i is the abscissa indicating the generic
microinsert in the reference integral with the wheel (FIG. 34).
[0326] The position must now be identified of the bottom ends of
the microinserts Y.sub.i-T in order to determine their deformation
and accordingly obtain the contact forces.
[0327] A generic microinsert is taken at the generic instant of
time at which it enters the contact area: the top and bottom ends,
seen from above, are at the same point because the microinsert has
not yet been deformed. In the successive instants, the top end of
the generic microinsert has a longitudinal displacement component V
* t opposite to the feed direction of the tire while, at the same
time, due to the effect of the absolute lateral displacement of the
plate y.sub.b and of its yaw .sigma..sub.b, it also has a
transverse direction component. Simultaneously, the bottom end of
said microinsert, assuming perfect adhesion, remains motionless in
the absolute reference system whereas, in the reference system
integral with the wheel, it possesses a displacement component V *
t in the longitudinal direction. Having set a distance between two
adjacent microinserts of V * dt in correspondence with each step of
integration of the dynamic equations described above, a single
microinsert enters and exits from the contact (FIG. 35). Three
successive instants of time are taken. At the instant t, the
microinsert 1 has just entered under the contact and is in the
deformed configuration. An instant later, the top end of the
microinsert 1 has gone into 1' having a vertical coordinate .xi.=V
* dt, whereas the bottom end has remained in 1. Simultaneously, the
non-deformed microinsert 0 comes under the contact. The position
occupied by the bottom end of the microinsert 1" (which is the
third microinsert in the local reference) at the time t+2 * dt is
the same as occupied by the bottom end of the microinsert 1' an
instant earlier. In the same way, the position occupied by the
bottom end of the microinsert 2", the fourth in the local
reference, at the time t+2 * dt is the same as that occupied by the
microinsert 2' an instant earlier. The position occupied by the
bottom end of the generic microinsert at the time t is:
Y.sub.i.sub..sub.--.sub.T(x.sub.i,
t)=Y.sub.i.sub..sub.--T(x.sub.i-1, t-dt) (1.69)
[0328] The non-linearity of the brush model results from the
possibility of whether the microinserts slip or not. Two factors
have a fundamental role in this: the coefficient of friction at the
interface between the wheel and ground and the pressure
distribution. A side view is shown of the reference model, at the
bottom of FIG. 36. The pressure distribution is assumed to be
parabolic in the longitudinal direction, whereas no pattern need be
defined for the transverse direction since the brush model
considered is monodimensional.
[0329] The normal forces per unit of length that are discharged to
the ground are: 29 q z _ i = 3 F z 4 a { 1 - ( x i a ) 2 } x ( 1.70
)
[0330] where F.sub.z is the vertical load applied to the tire. The
maximum deformation possible for the generic microinsert is: 30 y i
_ max = 3 F z 4 c p x ( a 2 - x i 2 a 3 ) ( 1.71 )
[0331] where .mu. is the coefficient of friction between tire and
ground and c.sub.py is the stiffness per unit of length of the
brush model. When this maximum deformation is brought onto the line
x representing the top ends of the microinserts, the area within
which the bottom ends must fall is determined (seen from above, top
part of FIG. 36).
[0332] Knowing the position of the top end and the bottom end of
the generic microinsert, its deformation can be determined.
[0333] When the deformation of the microinserts and the stiffness
of the tread are known, it is possible to determine the forces
created under the contact area in the Y direction.
[0334] The resulting sideward force F.sub.by acting on the plate
is: 31 F by ( t ) = i = 1 n ( Y i _ T ( x i , t ) - Y i _ P ( x i ,
t ) c p x ( 1.72 )
[0335] If the single sideward forces created under each single
microinsert are integrated and multiplied by the respective arm,
the self-aligning torque M.sub.bz is found as follows: 32 M bz ( t
) = i = 1 n ( Y i _ T ( x i , t ) - Y i _ P ( x i , t ) c p x x i (
1.73 )
[0336] Some results obtained for a rolling tire from the procedure
described are illustrated below.
[0337] To simulate the behaviour of the tire in the transient
state, a test was conducted envisaging a step input of the steering
angle imparted to the hub. This test is particularly important as
it serves to evaluate the time taken by the tire to go into a
steady state and therefore, in practice, it gives its speed of
response.
[0338] In this test, a selected steering angle is imposed
instantaneously on the hub at a time t=0 and the dynamic behaviour
of the tire is obtained from the time history of the six free
degrees of freedom of the model, of the sideward force and of the
self-aligning torque.
[0339] The test was conducted on a range 55 tire with an applied
vertical load of 2,914 N and a steering angle of four degrees
imposed on the hub. The feed speed of the tire was 30 Km/h.
[0340] FIG. 37 shows the time pattern of the lateral displacement
of the belt y.sub.c with respect to the hub and of the absolute
lateral displacement of the plate y.sub.b.
[0341] In FIGS. 38, 39 and 40, the time pattern is shown
respectively of the snaking .sigma..sub.c of the belt, of the
snaking .sigma..sub.b of the plate and of the roll .rho..sub.c of
the belt.
[0342] FIGS. 41 and 42 illustrate the time pattern respectively of
the sideward force F.sub.by and of the self-aligning torque
M.sub.bz.
[0343] The pattern of the sideward force F.sub.by is used to
evaluate the length of relaxation which consists of the space
travelled by the tire before the sideward force reaches 63.2% of
its steady state value.
[0344] The behaviour of the tire in drifting in the transient state
is reproduced using, as the transversal drift force applied to the
vehicle F.sub.by, that given by the following equation: 33 V F . by
+ F by = F _ by ( ) ( 1.74 )
[0345] where .alpha. is the instantaneous drift angle, F.sub.by is
the steady state force and .delta. is the length of relaxation.
[0346] In order to determine the length of relaxation, the signal
of the sideward force F.sub.by is removed of its high frequency
harmonics. A low-pass filter was used with cut-off frequency 30 Hz
(FIG. 43). The filtered signal is comparable to the response of a
1.sup.st order system receiving a step signal as its input.
[0347] The general equation of motion of such a system is as
follows: 34 A F t + B F = F external ( 1.75 )
[0348] which, through the Laplace transform, may be written as
follows: 35 F F external = 1 A s + B = 1 B A B S + 1 = a s + 1 (
1.76 )
[0349] where .tau. is the time constant of the system and
represents the time taken by the system to reach 63.2% of its
steady state value. The (1.76) is the system transfer function. If
a step of amplitude x.sub.0 is provided as input, the system
response in the frequency domain is as follows: 36 F ( s ) = x 0 s
a s + 1 ( 1.77 )
[0350] and, on passing to the time domain, the following is
obtained: 37 F ( t ) = a x 0 ( 1 - - t ) ( 1.78 )
[0351] The term a * x.sub.0 is the steady state value of F.
[0352] When the equations (1.74) and (1.75) are compared, the
length of relaxation is obtained from the curve of the filtered
sideward force.
[0353] From the filtered signal of FIG. 43, the steady state value
of the sideward force is obtained so as to calculate 63.2% and,
from here, trace back to the time constant .tau.. For an immediate
evaluation of the steady state value, the same filtering operation
was also carried out on the self-aligning torque (FIG. 44).
[0354] In the case in question, the relaxation length was found to
be 0.23 meters.
[0355] FIGS. 45-47 illustrate the results obtained with the range
55 tire for three different loads and with different yaw angles
imposed on the hub with feeding speed of 100 Km/h.
[0356] Local deformability of the tire in the transient state is
also introduced into the brush model by means of the equivalent
beam described above. After determining the deformation curve, the
positions of the top ends of the microinserts are identified
whereas the position of the bottom ends is determined using the
procedure described above. On the basis of the data obtained, the
single elementary forces acting on each section of a beam can be
computed (FIG. 53).
[0357] The resulting sideward force and the self-aligning torque
are determined by integrating the forces q.sub.y distributed along
the beam. The constraining reactions F.sub.a and F.sub.b are
determined in order to give the deformation curve. 38 F y = - a a q
y * x ( 1.79 ) Mz = - a a q y * x * x ( 1.80 ) F a = - F 2 + M z 2
* a ( 1.81 ) F b = - F y 2 - M z 2 * a ( 1.82 )
[0358] Curvature of the beam is related to the bending torque
through the following equation:
EJ * y"(x)=+M.sub.f(x) (1.83)
[0359] This equation is resolved by determining the bending torque
in each section of the beam.
[0360] FIGS. 48, 49 and 50 illustrate the pattern respectively of
the sideward force, of the self-aligning torque, of the length of
relaxation at steady state in relation to the steering angle, with
a rigid and deformable contact, for the range 55 tire to which a
vertical load of 2,914 N is applied.
[0361] FIGS. 51 and 52 illustrate respectively the pattern of the
length of relaxation and of the time constant in relation to
velocity for the range 55 tire to which three vertical loads of
2,914 N, 4,611 N and 6,302 N were applied.
* * * * *