Model of radial deformations of protector of vehicle tire

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.


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.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 elementprotector are needed to be evaluated individually.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.

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 http://dx.doi.org/10.5755/j01.mech.17.1.199 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.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.

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.

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 0.5 ν = .
In the works, analyzing pneumatic tire, rather wide limits of rubber elasticity modulus 20 2 − = r E MPa are prescribed [1,2].For the analysis of simplified car tire model in the work [2] 18 = r E 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 mm, and height of not ribbed one is Breaker 2 layers Cord 2 layers Breaker 4 layers Cord 6 layers we would get , 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.In order to correct the results obtained, operational conditions of the protector were examined in greater detail.

Porous composite model
The simplest model evaluates filling of the ribbed part where is protector filling coefficient, and is area of continuous part in the selected by pattern recurrence protector part; is area of protector part selected; 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 ( ) where ; is i-th part of band width in the protector width.bi k Elasticity modulus of the whole protector is calculated from equation 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.

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 where A is area of prism base: δ is height of the prism; ∆δ is deformations of the prism; α and η are parameters of form change under compression.
As α and η are interconnected, system of equa- tions is proposed in the work [8], obtained from energy minimum condition ( ) where a and b are dimensions of prism base.
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, 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.

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] ( ) ( )  Steel -rubber 0.9 5 Rubber 0.9 6 Steel -rubber 0.9 7 Viscose -rubber 0.9 8 Sealing layer 0.9 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 ( = ν 0.5)steel ( = ν 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.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 [ ] Elasticity modulus and Poisson's ratios of the components are taken from the works [1 -3].It was as-sumed, 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° and -20°), it was assumed, that shear forces appearing in these layers counterbalanced and were not transferred to other layers.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 less.With the variation of rubber elasticity modulus in the range 2 -20 MPa, 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 : with variation tenfold, varies 9.97 of time (Fig. 7).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 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 (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).

z E
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.The experiment confirmed, that linear dependence of pressure and deformation remains till pressures, exceeding the range of operational pressures (tests were performed till 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.

1. q =
According to simulation data, with an assumption, that for rubber 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: where is a member of stiffness matrix with respect to deformations constriction.* z C 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.

Comparative determination of pressure
It is rather complicated thing to compare the models investigated, because they present different characteristics -elasticity modulus or deformation sponding to the value of radial loading .To compare the results generalized characteristic, describing relation between loading and strain -stiffness , was used where is protector stiffness, is loading of radial tire.
1 C z R For determination of value 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 where is radial stiffness of tire, R C 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: where is width of the tire protector.

B
The assumptions permit finding pressure in the contact zone.Tire loading was calculated according to the tire nominal loading (Table 5).The pressure in the contact is calculated for loadings (0.25 -1.25)For all tires investigated nonlinear znom R znom R ( ) Z q R 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 Nonlinear dependence (Fig. 12) was noticed, which is related to nonlinear dependence of and the length of contact with the road, from this follows nonlinear 1 ( ) 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 (Fig. 13), as well. 1 ( ) C R 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 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.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 Fig. 16 Scheme of road unevenness 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 785 0 4 .= π .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: r .4550 = Δ , here r is radius of the sphere.
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 Δ .Dent radius is equal to .Elastic resolution of such task is known [15].Dent radius c is equal c ( ) where ; , ν is elasticity modulus and Poisson's ratios of material of the sphere and plane ( MPa, ).The average pressure in contact ( ( )) Value F is selected such way, that pressure , acting at tire and road contact when the tire is loaded with , is equal to the average pressure in the contact sphere -protector , with the evaluation of coefficient of filling with spheres in the contact.Taking into account, that according to elastic resolution: and with entering condition , we would get Calculations confirm that condition r .4550 < Δ (protector does not fill all space among inequalities) for an inequality of 5 mm height is fulfilled for pressures in tireroad 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 z R C − remains similar and is related with nonlinear z R q − dependence.

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.

Fig. 2 2 c 2 k 3 ,
Fig. 2 Vehicle quarter model: -length of the tire and road contact; m k L 1, c 1 and k 1 -mass, rigidity and damping factor of part of the tire protector; , -tire rigidity and damping factor; c 2 c

Fig. 3
Elements of the radial tire: 1a -protector part with a pattern, 1b -continuous protector part, 2b -breaker, 2b -breaker, 3 -radial ply, 4 -inner sealing layer, 5 -fixing ring, 6 -sidewall belts compression the protector was loaded up to the loading, corresponding by pressure to nominal radial loading 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.znom R The obtained experimental results are generalized in Fig.

Fig. 4
Fig. 4 Protector band deformations (1 -experimental points, 2 -generalized line) and shear modulus of composite matrix and filling; m ϕ are volume parts of filling and matrix; f ν and m ν are Poisson's ratios of composite, filling, and matrix; 1 -direction along fiber; 2 -direction across fiber.

Fig. 6
Fig. 6 Calculation schemes of breaker layers: a) with rubber layer excluded, b) with rubber layer not excluded

Fig. 7
Fig. 7 Elasticity modulus of the tire band in directions X and Z: 1 -a model in direction X, 2 -a model in direction Z, 3 -b model in direction Z, 4 -b model in direction Z (a model -8 layers, b model -7 layers) Fig. 8 Dependence of elasticity modulus of tire band on rubber elasticity modulus ( 2 0. z = σ MPa): 1 -7 plies, 2 -8 plies

C
Thus the member of stiffness matrix of tire pro-tector tested would be equal to 27 MPa.

Fig. 14 Fig. 15
Fig. 14 Dependence of stiffness of ribbed part of the protector of tire A on protector's height: 1 -new tire, 2 -wear 50 %, 3 -wear out

Fig. 18
Fig. 17 Scheme of interaction of unevenness -protector

Table 4
Dividing into layers of tire 175/70R13 bands