Model of radial deformations of protector of vehicle tire/ Automobilio padangos protektoriaus radialiniu deformaciju modelis.
Sapragonas, J. ; Darguzis, A.
1. Introduction
For stability of a vehicle, the problem of interacttion of tire
with road is one of the most complicated tasks. For analysis of the tire
and road contact, it is necessary to evaluate tire deflection features.
It is rather complex task due to some uncertainty of tire operational
conditions and sophisticated structure of the tire itself.
Problem's complexity is emphasized by the absence of common opinion
about tire shape till now. The geometric tire model was described by
toroid [1] in earlier works, and the band model was used later [2].
Modern studies of contact zones of broad tires [3] would conform more to
the ring model of limited flexibility.
Variation of dimensional position of a wheel due to kinematic
properties of the suspension [4] causes tire deflection in transversal,
longitudinal and vertical directions, and protector angular deflections
with respect to the plane of the wheel (Fig. 1). Increasing loadings
cause deflection of elements of suspensions and steering-wheel
mechanism, what can change the tire dimensional position.
For the formation of corrected vehicle stability models, tire
deflection characteristics are needed [5]. Experimental data are
supplied rarely and they are typical to the concrete tire.
[FIGURE 1 OMITTED]
Tire characteristics need to be corrected when solving tasks of
dynamics of the tire itself. Operation analysis of high speed vehicles
suspensions [6] demonstrates that there appears an additional vibration
source, which is related to natural tire vibrations by the authors of
the works quoted. For this purpose in vehicle's quarter model (Fig.
2) deformations of an additional element--protector are needed to be
evaluated individually.
[FIGURE 2 OMITTED]
Purpose of this work is to compare deformability of tires of
different types, thus permitting to use for modeling tire deflection
characteristics of the same tire group. Common regularities of composite
mechanics were used in the work and possibilities of the application of
some assumptions were analyzed.
2. Subjects of investigation
Modern tire has complex structure (Fig. 3). For the analysis of
tire deflection features, structure of the tire is simplified, dividing
the tire into a ring and sides. The ring, consisting of protector,
breaker, plies of cord and internal ply, is described as layered
structure [1, 2]. It is assumed in the simpler models, that its band is
absolutely rigid to stress but ideally flexible, though investigations
of its features [7] confirm, that the band is not absolutely flexible.
The sides in many models are described as a band of cord fiber or a
band, in which rubber is used just for ensuring tightness [1].
Mechanical features of rubber were not taken into account, so the sides
are calculated as single cord threads. Such model is applicable for the
description of static tire deflection, when mass of the tire elements,
their inertia moments and damping elements may not be evaluated. With
these assumptions applied, deformations of the tire elements could be
expressed by elasticity characteristics of their materials. Other tire
elements are simplified too. Experience of the application of
computational models shows that these models do not specify tire
characteristics in full, and results are to be corrected according to
experimental data.
[FIGURE 3 OMITTED]
The radial tires of three types with different structure were
analyzed in the work: tires of a motorcar (175/70R13; 195/50R15), light
truck (185/75 R14C) and lorry (12.00R20). Characteristics of the tire
elements are presented in Table 1.
3. Investigations of protector deformations
Influence of the ring and side on tire deformation properties is
analyzed in work [2] in more details. Deformation characteristics of
separate tire elements in this work are defined after disintegration of
them into monolayers, consisting of rubber or rubber, reinforced by
fiber oriented to single-direction. Investigations done on interaction
of the tire and the road [4, 6] confirmed, that while evaluating tire
compensation function it is necessary to evaluate deformation features
of the band, with entering an additional element into quarter model of
the vehicle (Fig. 2). In order to determine deformation features of this
element the tire is divided by elements into a band of ring shape,
consisting of a protector, breaker, cord and protective layers,
connected flexibly with toroidal sides, composed of rubber reinforced
with cord. The protector model consists of two layers: ribbed and not
ribbed ones, connected consequently.
3.1. Protector deformations under compression
Protector part with a pattern and continuous protector part of the
protector are isotropic materials and conditionally can be analyzed as a
material close to the rubber by their features. As modeling their
behavior, we used equations applied for rubber parts, evaluating, that
Poisson's ratio is v = 0.5.
In the works, analyzing pneumatic tire, rather wide limits of
rubber elasticity modulus [E.sub.r] = 2-20 MPa are prescribed [1, 2].
For the analysis of simplified car tire model in the work [2] [E.sub.r]
= 18 MPa there was used, what is related to objective of reaching
adequate deformation characteristics for the whole tire. In order to
correct initial characteristics, an experiment was performed to define
trodden down band of a truck tire, coincident to protector of the tire
12.00R20, with wide part 313 mm. Height of the ribbed part of the
protector is [[delta].sub.1a] = 17.8 mm, and height of not ribbed one is
[[delta].sub.1b] = 3.9 mm. Under compression the protector was loaded up
to the loading, corresponding by pressure to nominal radial loading
[R.sub.znom] for tire of type 12.00R20. The universal
extension--compression machine Amsler was used for the investigation.
Deformations were estimated by an indicator of 0.01 mm accuracy
measuring variation of the distances between machine plates. Friction
between polished machine steel plates and the protector was reduced by
lubrication.
The obtained experimental results are generalized in Fig. 4.
Dependence of loading-deformation achieved was very close to linear
(correlation coefficient [r.sub.xy] = 0.947 ), what confirmed the
possibility of application of linear elastic material description as
investigating protector deformations. According to experimental results
protector elasticity modulus defined is 7.84 MPa, with the evaluation of
ribbed protector part by filling coefficient, we would get [E.sub.r] =
6.16 MPa, that is considerably less, than in the work [2]. Besides, it
is necessary to take into account, that softer rubber is used for
motorcar tires.
[FIGURE 4 OMITTED]
In order to correct the results obtained, operational conditions of
the protector were examined in greater detail.
3.2. Porous composite model
The simplest model evaluates filling of the ribbed part
E = [k.sub.u][E.sub.r] (1)
where [k.sub.u] is protector filling coefficient, and [k.sub.u] =
[A.sub.i] / [A.sub.p]; [A.sub.i] is area of continuous part in the
selected by pattern recurrence protector part; [A.sub.p] is area of
protector part selected; [E.sub.r] is elasticity modulus of protector
rubber.
As protector pattern in different its parts is not the same,
filling coefficient for every protector band is calculated individually.
Then, equivalent elasticity modulus of the protector is equal to
[E.sub.1] = [E.sub.g] [summation][k.sub.ui][k.sub.bi]/1 - [k.sub.a]
(1 - [summation][k.sub.ui][k.sub.bi]) (2)
where [k.sub.a] = [[delta].sub.1a] /[[delta].sub.1];
[[delta].sub.1] = [[delta].sub.1a] + [[delta].sub.1b]; [k.sub.bi] is
i-th part of band width in the protector width.
Elasticity modulus of the whole protector is calculated from
equation
1/[E.sub.1] = ([[delta].sub.1a]/[E.sub.1a] +
[[delta].sub.1b]/[E.sub.1b]) 1/[[delta].sub.1] (3)
Results, obtained for different tires, are presented in Table 2.
This model reflects operational conditions of the protector
incorrectly. Deformations of ribbed part of the protector may be
restricted by contact with road, continuous breaker. Besides,
deformations of ribbed part of the protector are running not exactly by
ideal compression conditions, because with restriction of ends
deformations, element's form changes.
3.3. Model of rubber prism under compression
Deformations of ribbed part of the protector may be evaluated by
calculation methods of rubber prism shock absorbers. In this way,
protector element is imagined as rubber prism, deformed with more or
less constricted end deformations, subject to conditions. For this
purpose deformation equations of rubber shock absorbers are used in work
[8]. If deformations of both ends of rubber shock absorber of prism
shape are constricted, relation between loading and deformations is
expressed by the equation
[R.sub.z] = 1/3 [beta]EA(1/[[lambda].sup.2] - [lambda]) (4)
where [lambda] = 1 - [DELTA][delta]/[delta], [beta] = 4/3(1 +
[[eta].sup.2] - [eta])[(1 - 2/[alpha][delta] th
[alpha][delta]/2).sup.-1]
A is area of prism base: [delta] is height of the prism;
[DELTA][delta] is deformations of the prism; [alpha] and [eta] are
parameters of form change under compression.
As [alpha] and [eta] are interconnected, system of equations is
proposed in the work [8], obtained from energy minimum condition
[[alpha].sup.2] = 1/[b.sup.2] 48(1 + [[eta].sup.2] -
[eta])/[[eta].sup.2](a/b) + [(1 - [eta]).sup.2] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where a and b are dimensions of prism base.
Equations (5) and (6) were solved by approximation method.
Such model of ribbed part of the protector was applied in two
cases: when end deformations are not constricted and when they are fully
constricted. If deformations of shock absorber ends are not constricted,
[beta] = 1. With deformations constricted, rigidity of separate bands of
ribbed part of the protector increases up to 4.5 times (Table 3).
The second model of protector deformations permits the influence of
the evaluation of protector deformations constriction in circular (X)
and radial (Y) directions on radial deformations in the case of absolute
constriction of deformations. Under real conditions the constrictions of
deformations of ribbed part of the protector ends are not absolute. In
contact with road deformations of ribbed part of the protector are
constricted by friction forces between the road and the tire,
deformations of not ribbed part of the protector are constricted by
layers of breaker and cord, reinforced with fiber. If breaker layers are
not rigid (especially in transversal direction), conditions of
deformations of ribbed part of the protector are closer to the ones of
shock absorber with freely deforming ends, and if the deformation is
constricted, we will have to evaluate the deformation constriction.
In order to estimate influence of the latter constriction more
exactly, deformation characteristics of tire abstracted ring, consisting
of protector, breaker, cord and protective layer (Fig. 3), were
determined.
3.4. Influence of deformations constriction of the protector on
stiffness
Deformation characteristics of separate tire elements were defined
dividing them into monolayers (Table 1.), consisting of rubber or
rubber, reinforced with single-oriented fiber.
Band of the tire 175/70R13 designed for motorcars by Nokian company
was simulated. Its structure in more details is presented in Table 4.
Elasticity characteristics of a mono-ayer are defined according to
equations of composite mechanics [9, 10]
[E.sub.c1] = [E.sub.f][[phi].sub.f] + [E.sub.m](1 - [[phi].sub.f])
(7)
[E.sub.c2] = [E.sub.m] [((1 - [[phi].sub.f]) +
[E.sub.m]/[E.sub.f2][[ph].sub.f]).sup.-1] (8)
[G.sub.c12] = [G.sub.m] [((1 - [[phi].sub.f]) +
[G.sub.m]/[G.sub.f12][[phi]/sub.f]).sup.-1] [G.sub.f] (9)
[V.sub.c12] = [[phi].sub.f][v.sub.f12] + [[phi].sub.m][v.sub.m]
(10)
where [E.sub.c], [G.sub.c], [E.sub.m], [G.sub.m] are elasticity and
shear modulus of composite matrix and filling; pf and pm are volume
parts of filling and matrix; [[phi].sub.f] and [v.sub.m] are
Poisson's ratios of composite, filling, and matrix; 1--direction
along fiber; 2--direction across fiber.
The given equations should be corrected, so as they do not evaluate
the difference of Poisson's ratios of components [10]. In our case,
for pair--rubber (v = 0.5) steel (v = 0.25), having even the largest
difference of Poisson's ratios due to a big difference of
elasticity modulus of the components, the correction does not change the
value of composite elasticity modulus, so while defining characteristics
of the layers, differences of Poisson's ratios are not taken into
account.
[FIGURE 5 OMITTED]
Features of the layers of cord and breaker in longitudinal and
transversal directions are defined by using expressions of state of
plane stress for anisotropic monolayer [9, 2]: with the application to
axes presented in Fig. 5
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Elasticity modulus and Poisson's ratios of the components are
taken from the works [1-3]. It was assumed, that fiber of nylon and
viscose is anisotropic, elasticity modulus along and across the fiber
are different.
While investigating tire band structure, the problem of dividing
into layers appeared. There is the rubber layer between separate bands
of breaker, reinforced with steel wire. Therefore the band was examined
additionally separating the rubber layer between the reinforced layers
(Fig.6, a; Table 4) and using simplified separation into two layers
(Fig.6, b).
While investigating deformation features of the whole band, the
model used in layered composites was applied firstly. Theoretical
elasticity modulus of the band, as the composite has been calculated
with the evaluation of elasticity module of the layers connected in
parallel (directions X and Y) or consequently (direction Z).
So as reinforcement angles of adjacent breaker layers differ just
by a sign (+ 20[degrees] and -20[degrees]), it was assumed, that shear
forces appearing in these layers counterbalanced and were not
transferred to other layers.
[FIGURE 6 OMITTED]
It was found out, that while understanding the large difference
between elasticity modulus of reinforcement fiber and rubber, the rubber
elasticity modulus does not have a considerable influence upon
elasticity modulus of the layer in the direction of reinforcement. With
the variation of elasticity modulus of the rubber in the range 2-20 MPa,
band's elasticity modulus in direction Y (direction of cord fiber)
varies by 3.7% only. The way of layers exclusion (a and b) in this case
does not have any influence too. Herewith it was found out, that in the
direction of breaker fiber (X) and in direction Z fiber's
elasticity modulus depends more distinctly on the rubber elasticity
modulus. Elasticity modulus in direction X depends on [E.sub.r] less.
With the variation of rubber elasticity modulus in the range 2-20 MPa,
Ex differs by 3.7% for a model (Fig. 6, a) and by 3.5% for b model (Fig.
6, b). Especially distinct is the dependence of [E.sub.x] : with
[E.sub.r] variation tenfold, [E.sub.z] varies 9.97 of time (Fig. 7).
[FIGURE 7 OMITTED]
So as deformational features of the band in direction Z differ
considerably from other authors' ones, an assumption was done, that
more rigid cord and breaker layers might constrict deformations of the
protector and with the state of constricted deformations, Poisson's
ratio, close to 0.5, might have considerable influence on
protector's deformation. Therefore protector deformation under
compression was investigated additionally together with assumptions,
reflecting interaction of separate layers in excluded tire band more
correctly. For this purpose two cases were discussed. In the first case
an assumption was applied, that when the band is deformed in direction
Z, deformations of all layers in directions X and Y are the same
[[epsilon].sub.x1] = [[epsilon].sub.x2] = ... = [[epsilon].sub.xn],
[[epsilon].sub.y1] = [[epsilon].sub.y2] = ... = [[epsilon].sub.yn] (12)
Sum of unitary axial forces appearing in separate layers in
directions X and Y is equal to 0
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Numerical experiment has shown, that elasticity modulus of rubber
in direction Z increases greatly, therefore elasticity modulus of the
whole band in direction Z (Fig. 8) increases. Numerical experiment
confirms, that the models, projecting an equal layers deformation, are
especially sensitive to components with Poisson's ratio, close to
0.5. Conditional rubber elasticity modulus when structure of the layers
matches the tire 175/70R13 subject to a layer is equal 86.3 MPa. Because
of this reason the model with abstracted rubber interlayer predicts
unreal big stiffness.
[FIGURE 8 OMITTED]
In the second case we estimate friction between the protector and
the road and assumed, that protector's deformation is constricted,
till shear force appearing in directions X and Y does not exceed the
friction force. So as a control experiment was intended, in
computational model we assume, that friction forces are acting in
another band side, too--in contact with the sealing layer. Numerical
experiment confirmed that friction has an influence on value [E.sub.z]
(Fig. 9). With friction coefficient increasing, band stiffness in
transversal direction increases quickly. Abstracted additional rubber
layers have less influence in this case (Fig. 9).
In order to correct the results obtained, an attempt to deform a
band cut out from the tire in direction Z was done. The experiment was
performed with a band of tire 175/70R13 by Nokian company applied for
the simulation.
[FIGURE 9 OMITTED]
An universal tension-compression machine and equipment, used to
investigate protector band, was applied for the research. Generalized
experimental data are presented in Fig. 10.
[FIGURE 10 OMITTED]
The experiment confirmed, that linear dependence of pressure and
deformation remains till pressures, exceeding the range of operational
pressures (tests were performed till q = 1.5 MPa). The calculated
elasticity modulus of the whole band is equal 27 MPa, what corresponds
to computational model of simplified structure (7 layers), evaluating
friction between the plates and the band.
According to simulation data, with an assumption, that for rubber
[E.sub.r] = 7.84 MPa, calculated elasticity modulus of ribbed part of
the protector is equal to 5.72 MPa, and one of not ribbed part is equal
to 7.84 MPa. Cord is evaluated in the calculation of radial loadings in
simplified tire models. Therefore breaker layers may be considered
conditionally participating in radial deformation of the protector, as
well. Thus by calculation expressions for layered composites with
consequently located layers we would get:
1/[C.sup.*.sub.prz] = [[delta].sub.1]/[C.sup.*.sub.1z] +
[[delta].sub.2]/[C.sup.*.sub.2z] + [[delta].sub.3]/[C.sup.*.sub.3z] (14)
where [C.sup.*.sub.i] is a member of stiffness matrix with respect
to deformations constriction.
Thus the member of stiffness matrix of tire protector tested
[C.sup.*.sub.z] would be equal to 27 MPa.
For simulation of the real protector operation loadings and
operational conditions of the protector should be corrected by
approaching the computational model to real conditions.
4. Correction of protector loading
4.1. Comparative determination of pressure
It is rather complicated thing to compare the models investigated,
because they present different characteristics--elasticity modulus
[E.sub.1] or deformation [DELTA][delta], corresponding to the value of
radial loading [R.sub.z]. To compare the results generalized
characteristic, describing relation between loading and
strain--stiffness [C.sub.1], was used
[DELTA][delta] = [C.sub.1] [R.sub.z] (15)
where [C.sub.1] is protector stiffness, [R.sub.z] is loading of
radial tire.
For determination of value [C.sub.1] by the first or the second
model, we have to evaluate the fact that area of the contact with road
and volume of the deformed part of the protector depends on loading. For
finding the area of contact with road, we used these assumptions:
* the contact with road is of elliptic shape, restricted by tire
area and length of the contact with the road;
* layers under the protector are rigid enough, their length does
not change with the tire being deformed;
* dependence of the tire radial deflection and radial loading are
linear;
* pressure in the contact area distributes evenly.
The second and third assumptions permit finding out length of the
contact with the road
[L.sub.k]/2 = r arccos((1 - [C.sub.R][R.sub.z]) / r) (16)
where [C.sub.R] is radial stiffness of tire, r is free radius of
tire (Table 5).
By expression of an ellipse area we get the contact area and
comparative pressure in the contact:
q = 4[R.sub.z]/[pi]BL (17)
where B is width of the tire protector.
The assumptions permit finding pressure in the contact zone. Tire
loading was calculated according to the tire nominal loading
[R.sub.znom] (Table 5). The pressure in the contact is calculated for
loadings (0.25-1.25) [R.sub.znom] For all tires investigated nonlinear
q([R.sub.z]) dependence (Fig. 11) was obtained. We notice, that tires
operate in different pressure sections and operational conditions of the
protector for motorcars and trucks differ by 1.5-1.7 time, the ones of
motorcars and lorry--up to 3.2 times (Fig. 11). Therefore we compared
protector's stiffness using not comparative pressure, but ratio
[R.sup.*.sub.z] = [R.sub.z]/[R.sub.znom].
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
Nonlinear dependence [C.sub.1a] ([R.sub.Z]) (Fig. 12) was noticed,
which is related to nonlinear dependence of [R.sub.z] and the length of
contact with the road, from this follows nonlinear q([R.sub.Z])
dependence, reducing protector's deformation in larger loadings
zone. That is the particularity of tire contact with the road what
explains that the tire type has no essential impact on the nature of
dependence [C.sub.1a] ([R.sub.z]) (Fig. 13), as well.
Thickness of ribbed part has an influence on protector's
stiffness. While evaluating the influence of this part by protector
filling coefficient or by the model of rubber prism deformations,
results differ not so much, if constrictions of the prisms ends are not
taken into account.
The influence of protector's wear on its stiffness was defined
by the model of short rubber prisms deformation. For the calculations of
protector's stiffness, the influence of constriction of prism ends
is not taken into account. For the calculation of protector's
stiffness of the tire of A type protector's height is 8.1 mm for a
new tire, 1.6 mm for worn one, besides, intermediate wear level (3.2 mm)
is projected. Protector's wear changes the dependence C - [R.sub.z]
very slightly (Fig. 14), but has considerable influence on the stiffness
value itself. The influence is not monotonous --stiffness especially
varies in the beginning of wear (Fig. 15). For the tires investigated
wear of the protector may increase protector's stiffness for
nominal loading up to 1.3 times.
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
4.2. Model of protector interaction with road covering texture
The third deformation model of the protector was used in order to
evaluate particularity of the tire interaction with road texture. The
road texture is unevenness of road covering. For blacktop height of
these inequalities is 1-5 mm [13-14]. An assumption was done, that the
inequalities are bumps of semisphere shape located regularly (Fig. 16).
This assumption permits modeling of interaction of a single bump with
the protector with the assumption that a bump acts on continuous part of
the protector. Coefficient of filling with spheres, required for
recalculation of comparative pressure is equal to [pi]/4 = 0.785. From
geometric conditions it is found out, that gaps between the spheres will
be filled when impact of a sphere into protector is equal to: [DELTA] =
0.455r , here r is radius of the sphere.
[FIGURE 16 OMITTED]
To investigate the interaction resolution of elastic deformations
of the known sphere--plane contact was used. The interaction scheme is
presented in Fig. 17. Shape inequality of semisphere with radius r,
force F presses on protector, which is on rigid base. Therefore
inequality squeezes in the protector by depth [DELTA]. Dent radius is
equal to c. Elastic resolution of such task is known [15]. Dent radius c
is equal
c = 0.721 [cube root of 2Fr([k.sub.1] + [k.sub.2])] (18)
where [k.sub.i] = 1 - [v.sup.2.sub.i] / [E.sub.i]; [E.sub.i],
[v.sub.i] is elasticity modulus and
Poisson's ratios of material of the sphere and plane
([E.sub.1] = 18 MPa, [v.sub.1] = 0.5 , [E.sub.2] = 6 GPa, [v.sub.2] =
0.3 ).
The average pressure in contact
[q.sub.0] = 0.918 [[cube root of F/(2r([k.sub.1] +
[k.sub.2])).sup.2]] (19)
[FIGURE 17 OMITTED]
[FIGURE 18 OMITTED]
Value F is selected such way, that pressure q, acting at tire and
road contact when the tire is loaded with [R.sub.z], is equal to the
average pressure in the contact sphere--protector [q.sub.0], with the
evaluation of coefficient of filling with spheres in the contact. Taking
into account, that according to elastic resolution:
[DELTA] = 0.8255 [[cube root of [F.sup.2]/r([k.sub.1] +
[k.sub.2]).sup.2]] (20)
[q.sub.0] = F/[pi][c.sup.2] (21)
and with entering condition [q.sub.0] = q , we would get
[DELTA] = 5.552 [q.sup.2]r[([k.sub.1] + [k.sub.2]).sup.2] (22)
Calculations confirm that condition [DELTA] < 0.455r (protector
does not fill all space among inequalities) for an inequality of 5 mm
height is fulfilled for pressures in tire road contact.
This effect could have influence on the sliding [16] and on the
investigation of contact between tire and road as well in the aspect of
noise [17] and safety [18].
Protector rigidity values of separate models are presented in Fig.
18. The model evaluating interaction with road texture (r = 5 mm)
prescribes less stiffness values, but dependence C - [R.sub.z] remains
similar and is related with nonlinear q - [R.sub.z] dependence.
5. Conclusions
1. Factors, having the largest influence in the evaluation of
stiffness of tire protector in radial direction, i.e. dependence of
pressure of tire -road contact on radial loading and breaker and cord
influence on radial deformation of the protector, were determined.
Evaluation model of protector deformations has no essential influence
and a simple model, evaluating filling coefficient of the protector
pattern may be used.
2. Evaluation of deformations of ribbed part of the protector by
filling coefficients of the protector pattern is adequate to the model
of deformation of rubber prisms.
3. Numerical simulation and experiments has confirmed that
deformations of the protector band constrict breaker and cord layers,
therefore stiffness of the protector band in direction Z is evaluated
with stiffness of the whole band in directions X and Y defined. When
elasticity modulus of the rubber is [E.sub.r] = 6 MPa, for a motorcar
tire is obtained [E.sub.1z] = 27 MPa.
4. While evaluating ring stiffness of the protector, nonlinear
[DELTA][delta] - [R.sub.z] dependence has been obtained, so as due to
tire deflection particularity dependence of relative pressure onto road
q on radial loading is nonlinear.
5. It is recommended to evaluate protector stiffness of the tire in
relative coordinates [R.sub.z] / [R.sub.znom], because in this case the
tire type has less influence.
Received October 04, 2010
Accepted January 28, 2011
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J. Sapragonas, Kaunas University of Technology, Kqstucio 27, 44312
Kaunas, Lithuania, E-mail:
[email protected]
A. Darguzis, Kaunas University of Technology, Kqstucio 27, 44312
Kaunas, Lithuania, E-mail:
[email protected]
Table 1
Data of tire constitutive parts
Tire Tire elements
angle of fiber,
thickness, mm [degrees]
A 175/70R13 [[delta].sub.1a] = 8.0 --
[[delta].sub.1b] = 2.1 --
Breaker 3 layers [[delta].sub.2] = 2.7 [[theta].sub.2] =
Cord 1 layer 20[degrees]
[[delta].sub.3] = 0.9 [[theta].sub.3] =
90[degrees]
[[delta].sub.4] = 0.9 --
B 195/50R15 [[delta].sub.1a] = 8.1 --
[[delta].sub.1b] = 2.1 --
Breaker 2 layers [[delta].sub.2] = 3.6 [[theta].sub.2] =
Cord 2 layers 20[degrees]
[[delta].sub.3] = 0.9 [[theta].sub.3] =
90[degrees]
[[delta].sub.4] = 1.0 --
C 185/75R14C [[delta].sub.1a] = 12.0 --
[[delta].sub.1b] = 3.0 --
Breaker 2 layers [[delta].sub.2] = 3.2 [[theta].sub.2] =
Cord 2 layers 20[degrees]
[[delta].sub.3] = 1.8 [[theta].sub.3] =
90[degrees]
[[delta].sub.4] = 1.0 --
D 12.00R20 [[delta].sub.1a] = 17.8
[[delta].sub.1b] = 3.9
Breaker 4 layers [[delta].sub.2] = 6.4 [[theta].sub.2] =
Cord 6 layers 20[degrees]
[[delta].sub.3] = 7.2 [[theta].sub.3] =
90[degrees]
[[delta].sub.4] = 1.0 --
Note: [theta]--angle of cord with respect to rolling direction,
[[delta].sub.1a]--protector part with a pattern, [[delta].sub.1b]--
continuous protector part, [[delta].sub.2]--breaker,
[[delta].sub.3]--cord, [[delta].sub.4]--sealing layer.
Table 2
Elasticity modulus of the protector, obtained with
evaluation of filling
Ribbed part Continuous part Protector
Tire type [E.sub.1a], MPa [E.sub.1b], MPa [E.sub.1], MPa
A 5.72 7.84 6.16
B 5.88 7.84 6.22
C 7.08 7.84 6.51
D 5.33 7.84 5.37
Table 3
Values of stiffening coefficient of protector elements for
individual protector bands
Tire Band Length Width [beta] for
type number a, mm b, mm band
A 1 39 29 3.83
2 32 29.6 3.67
B 1 37 29 3.75
2 31 29 3.67
3 239 24 4.58
C 1 33 20 2.62
2 36 18 2.08
D 1 116 42 3.33
2 90 41 3.17
Table 4
Dividing into layers of tire 175/70R13 bands
Layer Layer description Thickness, mm
No.
1 Protector with pattern 2.9
2 Protector without pattern 2.1
3 Nylon--rubber 0.8
4 Steel--rubber 0.9
5 Rubber 0.9
6 Steel--rubber 0.9
7 Viscose--rubber 0.9
8 Sealing layer 0.9
Table 5
Exploitational data of tires investigated
Exploitational data A B C D
Free radius r, mm 288 293 323 561
Static radius [r.sub.st], mm 262 267 288 520
Nominal loading [R.sub.z nom], N 4.7 4.7 8.3 36.8
Protector width B, mm 177 201 184 313
