c Allerton Press, Inc., 2016. ISSN 0025-6544, Mechanics of Solids, 2016, Vol. 51, No. 3, pp. 298–307.  c A.E. Belkin, V.K. Semenov, 2016, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2016, No. 3, pp. 71–82. Original Russian Text 

Theoretical and Experimental Analysis of the Contact between a Solid-Rubber Tire and a Chassis Dynamometer A. E. Belkin* and V. K. Semenov Bauman Moscow State Technical University, ul. 2-ya Baumanskaya 5, Moscow, 105005 Russia Received December 10, 2015

Abstract—We consider the problem of modeling the test where a solid-rubber tire runs on a chassis dynamometer for determining the tire rolling resistance characteristics. We state the problem of free steady-state rolling of the tire along the test drum with the energy scattering in the rubber in the course of cyclic deformation taken into account. The viscoelastic behavior of the rubber is described ¨ by the Bergstrom–Boyce model whose numerical parameters are experimentally determined from the results of compression tests with specimens. The finite element method is used to obtain the solution of the three-dimensional viscoelasticity problem. To estimate the adequacy of the constructed model, we compare the numerical results with the results obtained in the solid-rubber tire tests on the Hasbach stand from the values of the rolling resistance forces for various loads on the tire. DOI: 10.3103/S0025654416030067 Keywords: solid-rubber tire, chassis dynamometer, contact, steady rolling, rubber viscoelasticity, energy dissipation, rolling resistance.

1. INTRODUCTION One most informative method for experimental studies of the working characteristics and wear life of tires is the test on a tire rolling stand with a chassis dynamometer. As the tire rolls on chassis dynamometer, one can, in particular, determine the rolling resistance forces and the tire self-heating temperature, i.e., the characteristics depending on the energy scattering in the rubber in the course of cyclic deformation. Calculating these characteristics is a complex mathematical modeling problem, where it is very important to tune the model so that it can adequately reproduce the test results. In this paper, we solve the problem of free steady rolling of a solid-rubber tire on the chassis dynamometer with energy dissipation taken into account and verify the reliability of the proposed model compared with the experimental results. The problem of rolling contact for a viscoelastic thick-wall cylinder with a rigid core (of the rubbed roller) was solved in [1–4] by the finite element method (FEM). The contact along the normal to the drum surface without taking into account the tangential forces was studied in [1]. The general statement of the one-sided contact conditions for a rolling cylinder with the friction forces taken into account in the contact spot was considered in [2, 3]. In these papers, the plane deformations of the rubber massive roller were investigated. The viscoelasticity relations with exponential relaxation kernel containing one relaxation time typical of the material were used for the rubber. A more general approach to the description of viscous effects in elastomeric materials under cyclic loads was used in [4], where the law of state with internal variables of the deformation process determining the variations in the viscous strain components was stated. The computational aspects of the contact problem of the tire rolling were discussed in [5–7]. 2. STATEMENT OF THE PROBLEM Figure 1 outlines the solid-rubber tire and the problem of tire rolling on the chassis dynamometer. The tire consists of an undeformable rim of the wheel (position 1 ) and a solid rubber vulcanized on it *

e-mail: [email protected]

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Fig. 1.

(position 2 ). An absolutely rigid drum rotating at a constant angular velocity ωd sets the tire in motion with angular velocity ω. The parameter determining the contact of the two bodies is the approach of their axes u0 ; in this case, the force interaction factors, i.e., the force F1 pressing the tire to the drum and the pulling force F2 , as well as the torque M applied to the drum axis, are the unknown variables. The tests show that, under maximum working loads, the ratio of the tire compression u0 to the solid rubber thickness does not exceed 8%; i.e., the rubber deformations are small. This fact permits solving the problem in a geometrically linear statement. We state the equations in the Cartesian coordinate system x1 , x2 , x3 with the unit vectors e1 , e2 , and e3 (Fig. 1) and, if necessary, use the cylindrical coordinates r, ϕ, x3 with the unit vectors e1 , i2 , and i3 = e3 in the radial, circular, and axial directions, respectively. In steady-state rolling, the velocity vector of an arbitrary point of the tire is calculated by the formula v = ωri + ω

∂u , ∂ϕ

where u = ui ei is the point displacement vector due to the tire deformation; from now on, the sums are taken over repeated indices from 1 to 3. In the Cartesian coordinates, we have v = ω(−x2 + ∂ϕ u1 )e1 + ω(x1 + ∂ϕ u2 )e2 + ω∂ϕ u3 e3 , where ∂ϕ ≡ ∂/∂ϕ = x1 ∂/∂x2 − x2 ∂/∂x1 is the operator of differentiation with respect to the angular coordinate. MECHANICS OF SOLIDS

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The strains εij and the stresses σij at any material point of the tire are periodic functions of time, εij (t + T ) = εij (t),

(2.1)

σij (t + T ) = σij (t),

and the period it the time of the wheel rim revolution T = 2π/ω. To describe the energy losses in the rubber, we use the viscoelastic elastomer deformation model ¨ and Boyce in [8, 9]. According to this model, the viscous strain rate of elastomers proposed by Bergstrom obeys the relations of the theory of flow with strengthening, and the elongation of the polymer chain is taken as the strengthening parameter. In the triaxial stress state in the case of small strains, the ¨ Bergstrom–Boyce model is described by the equations v , σij = KΘδij + 2G ij +Sij v Sij

∗

= 2G (ij −

vij

d dt

(2.2)

vij ),

(2.3)

v = f (λvch , σuv )Sij ,

f (λvch m, σuv ) =

(2.4)

B(σuv )m , (λvch − 1 + ε0 )n

(2.5)

where the functions with the superscript v correspond to the viscous structure of the material, K is the modulus of bulk compression, Θ = εii is the bulk strain, δij is the Kronecker symbol, G and G∗ are the equilibrium and relaxation shear moduli, ij and vij are the strain deviator components v S v )1/2 is the stress intensity in the viscous structure, and their viscous components, σuv = ( 32 Sij ij  √ v v v v 2 2 2 λch = (λ11 ) + (λ22 ) + (λ33 ) / 3 is the multiplicity of averaged viscous elongation of the macromolecular chain of the elastomer [8], m, n, and B are the strain law parameters, and ε0 is a small constant of deformation, which is added to describe the creep rate at zero strain. The proposed model was experimentally verified is tests for determining the cyclic compression of rubber specimens [10] and polyurethane specimens [11]. In the steady rolling of the tire at slow velocities (the maximum velocity of the transport vehicle is 70 km/hour), one can neglect the forces of inertia and consider the process as a static one. The analysis of the rolling tire is based on the virtual work principle equation written out at an arbitrary time,  σij δεij dV = δWc , (2.6) V

where δWc is the work of forces in the contact region Ωc . We expand the desired contact load pc in the components in the axial, circular, and normal directions with respect to the drum, pc = pt1 td1 + pt2 td2 + pt3 nd , where td1 = e3 , td2 , and nd are the unit vectors in these directions. Consider the conditions of contact between the tire and the chassis dynamometer. The equation of the drum slanting surface in the possible contact region can be written as (see Fig. 1) x1d = R0 +

x22d − u0 . 2Rd

The “unit” (in the approximation of the theory of slanted surfaces) vectors are determined by the formulas x2 x2 e1 + e2 , nd = −e1 + e2 . td2 = Rd Rd The spatial coordinates of points xi + ui of the outer surface of the deformed tire must satisfy the condition of impenetrability into the depth of the drum, x1 + u1 ≤ R0 +

(x2 + u2 )2 − u0 , 2Rd

where x1 = R0 − x22 /(2R0 ) is the equation of the outer surface of the tire in unstrained state. MECHANICS OF SOLIDS

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By introducing the function of penetration of contacting bodies g = u0 −

x22 (x2 + u2 )2 − + u1 , 2R0 2Rd

we can rewrite the impenetrability condition as the requirement that this function be nonpositive; i.e., g ≤ 0. By linearizing the function g with respect to small displacements of the tire points, we obtain f = f ◦ − un ,

(2.7)

where g◦ = u0 − x22 /(2R0 ) − x22 /(2Rd ) is the initial penetration that would occur in the case of a permeable contact surface and un = nd · u is the displacement of the tire point along the normal to the supporting surface. Let Ωc be the desired contact region. Then the normal contact conditions are stated as g◦ − un = 0, g◦ − un < 0,

pn ≥ 0 inside Ωc , pn = 0 outside Ωc .

(2.8)

To state the tangential contact conditions, we introduce the function of tire slip velocity along the drum, ξ = ωd Rd td2 − v (r = R0 ). Its projections onto the drum axes td1 and td2 have the form ξt1 = −ω∂ϕ u3 ,

        x2 2 1 x2 2 1 x2 x2 + −ω ∂ϕ u1 − ω 1 − ∂ϕ u2 . ξt2 = ωd Rd − ωR0 1 − 2 R0 Rd Rd 2 Rd

adhesion region Ωst In the general case of tire rolling, the contact region Ωc is divided into the c and sl the sliding region Ωc . In the adhesion region, the total tangential stress pt = (pt1 )2 + (pt2 )2 remains smaller than the limit stress calculated by the Coulomb law, and there is no sliding; i.e., the following conditions are satisfied:  pt < μpn , ξt = (ξt1 )2 + (ξt2 )2 = 0, where μ is the sliding friction coefficient. In the sliding region, the tangential stress has the limit value pt = μpn , and by the Coulomb friction law we have     ξt1 ξt2 , pt2 = μpn . ξt = 0, pt1 = μpn ξt ξt The conditions ξt1 = 0 and ξt2 = 0 imposed on the sliding velocity in the adhesion region can be stated not only in velocities but also in displacements. In the contact region, we have |x2 /Rd | < 0.05, and hence we can set td2 ≈ e2 . After coming into contact, the corresponding material points of the tire and the chassis dynamometer cover the same path, and their spatial coordinates coincide on this path. The adhesion conditions in the circular and axial directions can be represented as gt◦ − u2 = 0,

u3 (ϕ0 ) − u3 = 0,

(2.9)

g◦ ,

the function of initial “penetration” in the where, by analogy with the the initial penetration function circular direction is introduced as well,   ωd ◦ Rd − R0 (ϕ − ϕ0 ) + u2 (ϕ0 ). (2.10) gt = ω In the expressions (2.9) and (2.10), ϕ0 stands for the angular coordinate at the beginning of contact, which depends on x3 . Under the normal conditions of testing the tire on the chassis dynamometer, sliding is practically absent. We assume that the adhesion condition (2.9) is satisfied in the entire contact region Ωc . We solve the contact problem by using the penalty method where we treat the work δWc of contact forces in Eq. (2.6) as a variation of the penalty function. To satisfy the normal contact conditions (2.8), we determine the relationship between the desired pressure pn = kn (g◦ − un ) (kn is the penalty coefficient) and the penetration function (2.7) of the contacting bodies. MECHANICS OF SOLIDS

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The pressure work is represented as  δWpn = kn (g◦ − un )δun dΩ,

δun = nd · δu.

(2.11)

Ωc

The same procedure is used to compose the expression for the virtual work of the tangential forces in the contact. We assume that these forces are proportional to the discrepancies under the adhesion conditions (2.9), pt1 = kt (u30 − u3 ) and pt2 = kt (gt◦ − u2 ), where kt is the penalty coefficient in the case where the adhesion conditions are satisfied, u30 = u3 (ϕ0 ). Then the work of tangential forces becomes  (2.12) δWpt = [kt (gt◦ − u2 )δu2 + kt (u30 − u3 )δu3 ] dΩ. Ωc

The sum of works (2.11) and (2.12) determines the function δWc , i.e., the right-hand side of the variational equation(2.6). 3. ON THE NUMERICAL SOLUTION OF THE VISCOELASTICITY PROBLEM In the problem under study, the strains and stresses in the tire decay very rapidly as the distance from the contact region increases. We restrict the domain of significant stresses and strains by the angles ϕ− and ϕ+ under the assumption that σij = 0 and εij = 0. The periodicity conditions (2.1) are replaced by the attenuation conditions for the stress–strain state components as the boundary of the computation domain is approached, σij → 0,

εij → 0

as ϕ → ϕ± .

(3.1)

To compute the work of internal forces, we write out the viscoelasticity relations (2.2)–(2.3) in the different form σij =

e σij

∗

t

− 2G

˙ vij dt˜,

e σij = KΘδij + 2(G + G∗ ) ij ,

(3.2)

−∞ e are the instantaneous stresses determined by the elasticity relations and the ˙ v are the viscous where σij ij strain rates determined by formulas (2.4) and (2.5); the dot denotes differentiation with respect to time. For the steady rolling, the integration over the time in (3.2) is replaces by the integration over the angular coordinates ϕ = ωt, and the lower limit of integration t˜ = −∞ is replaced by ϕ = ϕ− with conditions (3.1) taken into account,

σij =

e σij

∗ −1

ϕ

− 2G ω

˙ vij dϕ. ˜

(3.3)

ϕ−

The initial conditions of integration are set to be zero, σij (ϕ− ) = 0, εij (ϕ− ) = 0. After several transformations, we can write out the virtual work principle equation as   e σij

∗ −1

ϕ

− 2G ω

V

 =

˙ vij

 dϕ˜ δεij dV

ϕ−

[kn (g◦ − un )δun + kt (gt◦ − u2 )δu2 + kt (u30 − u3 )δu3 ] dΩ,

(3.4)

Ωc

where V is understood as the computation domain of the tire bounded by the radial cross-sections ϕ = ϕ− and ϕ = ϕ+ . MECHANICS OF SOLIDS

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Fig. 2.

When solving Eqs. (3.4), the stresses (3.3) must be determined by an iteration method, because the viscous strain rates ˙ vij depend on the unknown stresses. At each iteration step, the viscous strains vij (ϕ) = ω −1

ϕ v f (λvch , σuv )Sij dϕ˜

(3.5)

ϕ−

are calculated by numerical integration by the Runge–Kutta method. In this paper, Eq. (3.4) was solved by using the finite element method (FEM). The bulk eight-node elements shaped as parallelepipeds are used with the three-linear approximation of the displacements. The chosen representation of the displacements by a C 0 -smooth function leads to discontinuous strains and stresses on the interface between neighboring finite elements (FE). To calculate the integral (3.5) at the current iteration step, we approximate the strains obtained at the preceding iteration step. The algorithm is illustrated in Fig. 2, where the outlined tire region is divided into FE. Consider an arbitrary row of elements arranged in the circular direction over which the strain rates are integrated numerically. Between same-name Gaussian points (i.e., points with the same local coordinates) of two and εkij , we introduce the adjacent FE with numbers k − 1 and k at which the strains are equal to εk−1 ij linear approximation + α(εkij − εk−1 εαij = εk−1 ij ij ),

α ∈ [0, 1].

Under the assumption that the viscous strains (vij )k−1 of the (k − 1)st element are known, we successively compute the coefficients of the Runge–Kutta method   1 v v k−1 k−1 2 v v k−1 1 Δϕk 0.5 , εrs , + krs kij = ˙ ij ((rs ) , εrs ), kij = ˙ ij (rs ) 2ω     (3.6) 3 v v k−1 2 Δϕk 0.5 4 v v k−1 3 Δϕk k , εrs , kij = ˙ ij (rs ) , εrs , + krs + krs kij = ˙ ij (rs ) 2ω ω MECHANICS OF SOLIDS

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where Δϕk is the integration step (Fig. 2). To compute the coefficients (3.6), we determine the viscous strain rates by formulas (2.4) and (2.5). Finally, the deformations of the viscous link at the corresponding Gaussian point of the kth element are calculated as Δϕk 1 2 3 4 (vij )k = (vij )k−1 + (kij + 2kij + 2kij + kij ). 6ω Further, by substituting the obtained strains into the expressions (2.3) and (2.2), we redetermine the stresses. The above-described process of integration starts from the fictitious (which is not used to solve the problem) element in the row, for which the stresses and strains are set to be zero. To complete the description of the theory of calculations, we note that, in the case of free steady rolling of the tire, its rotational velocity ω is determined from the prescribed drum rotation velocity ωd starting from the condition that there is no tractive effort or braking torque applied to the tire axis. 4. RESULTS OF THEORETICAL AND EXPERIMENTAL STUDIES The object of our study is a solid-rubber tire of standard size 630 × 170 with the following geometric parameters: R0 = 312.8 mm is the outer radius, Ri = 272.8 mm is the inner radius, H0 = 149.0 mm is the width of the tire tread, and Hi = 162.6 is the width of the rubber base. ¨ Prior to analyzing the tire, we determined the values of the parameters of the Bergstrom–Boyce model for the tire rubber. From the results of cyclic pulsation compression tests with short cylindrical rubber specimens on the Instron ElectrodPuls 1000 stand, we determined the following parameter values: G ≈ 3.9 MPa is the equilibrium shear modulus, G∗ ≈ 3.2 MPa is the relaxation shear modulus, and m ≈ n ≈ 1 are the exponents in the flow law (2.4), (2.5). We assume that the coefficient of proportionality between the viscous strain and stress rates in the law (2.4), (2.5) depends on the material loading frequency, i.e., on the tire rotational velocity. This decision is justified in detail in [10]. We set B/ω ≈ 0.147 MPa−(m+1) in the calculations. The modulus of bulk compression was equal to 108 MPa. The tire computational sector was separated by the angles ϕ− = −π/3 and ϕ+ = π/3. To satisfy the contact conditions, we took the penalty coefficients kn = kt = 2000 MPa/mm. All above-listed values of the material model parameters are approximate and must be refined when tuning the model. In what follows, we present the results of modeling of the tire rolling over the drum on the test stand Hasbach whose radius is Rd = 1000 mm. Figure 3 shows the distribution of the normal pressure pn (curve 1 ) and circular adhesion forces pt2 (curve 2 ) in the contact spot in rolling at the velocity 60 km/h (ωd = 16.7 rad/s) under the loading of 11.5 kN. The figure also presents the computational (curve 3 ) and experimental (curve 4 ) pressure diagrams for the immovable tire pressed by the same force to the plane supporting surface. All pressure diagrams were constructed for the circular cross-section passing through the center of the contact spot, the stresses pn and pt2 are given in MPa, and the coordinate x2 , in mm. A comparison of curves 3 and 4 show that the computational results agree well with the experimental results for the immovable tire. Note that the immovable tire was tested on the equipment produced by the XSENSOR Technology Corporation and the sensor IX500:256.256.16 was used. Compared with the solution for the immovable tire, the maximal pressure in the solution for the tire in rolling is displaced towards the contact input. The circular adhesion forces at the contact input and output have values commensurable with the normal pressure, which shows that sliding is possible in these regions. The sliding regions taken into account in the calculations can affect the distribution of friction forces and, as a consequence, the value of the calculated rolling resistance force. But in this version of the solution, the normal pressure forces, which determine the tire stress state, cannot change significantly. Therefore, it is expedient to express the rolling resistance force FR in terms of the power of the energy scattered in the rubber,  σij ε˙ij dV V . FR = ωd Rd The authors tested the tire over the drum stand to obtain the dependence of the rolling resistance force on the velocity and load. The tire rolled on the drum in the free rolling mode. Figure 4 shows the obtained graphs of the resistance force FR (N) versus the rolling velocity Rd ωd (km/hour) for various MECHANICS OF SOLIDS

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Fig. 3.

Fig. 4.

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Fig. 5.

forces F1 pressing the tire to the drum. The values of the force F1 in kN are indicated near each curve. Note that these results correspond to the “cold” state of the tire. One can see that as the velocity increases from 30 to 70 km/hour, the resistance force increases by less than 10%. This result agrees well with the experimental data for rubber specimens, and the hysteresis losses are practically independent of the load frequency. This fact permits reducing the number of the tire tests, because, in the velocity range under study, the resistance force is independent of the rolling velocity. We can average the measurement results over the velocities and construct the graph of the rolling resistance force versus the force pressing the tire of the drum. Figure 5 shows this dependence. Here the forces FR are still measured in Newtons, and the forces F1 , in kN. One can see that the experimental results fit well to a straight line (graph 1 in Fig. 5). The same figure shows the results of mathematical modeling (graph 2 ). There is a qualitative correspondence between the computational and experimental results. The experimental value of the tire rolling resistance coefficient determined as the ratio FR /F1 is equal to 0.017–0.019; its computational values are somewhat lower, 0.015–0.018. The computations show that, for a fixed load on the tire, the rolling resistance forces decreases as the chassis dynamometer radius increases. It is of interest in practice to determine the correspondence between the values of the rolling resistance along the drum FR and along a plane surface of the road FR∞ . The latter case is illustrated by graph 3 in Fig. 5. The ratio FR∞ /FR can be used as a correction coefficient for determining the characteristics of the tire rolling on the plane from the results of the tire tests on the stand with the chassis dynamometer. Comparing graphs 2 and 3 in Fig. 5, one can obtain the correction coefficient FR∞ /FR ≈ 0.75 for the drum radius of 1000 mm. 5. CONCLUSION The obtained results for the rolling resistance testify the qualitative reliability of computations of the ¨ energy dissipation in the solid-rubber tire according to the Bergstrom–Boyce model used to describe the viscoelastic cyclic deformations of the rubber. This model can be used to estimate the power of heat production in solid-rubber tires in the analysis of their self-heating. REFERENCES 1. C. N. Bapat and R. C. Baira, “Finite Plane Strain Deformations of Nonlinear Viscoelastic Rubber-Covered Rolls,” Int. J. Numer. Meth. Engng 20, 1911–1927 (1984). 2. J. T. Olden and T. L. Lin, “On the General Rolling Contact Problem for Finite Deformations of a Viscoelastic Cylinder,” Comput. Meth. Appl. Mech. Engng 57 (3), 1911–1927 (1984). 3. J. T. Olden, T. L. Lin, and J. M. Bass, “A Finite Element Analysis of the General Rolling Contact Problem for a Viscoelastic Rubber Cylinder,” Tire Sci. Technol. 16 (1), 18–43 (1989). MECHANICS OF SOLIDS

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4. P. Le Tallec and C. Rahier, “Numerical Models of Steady Rolling for Nonlinear Viscoelastic Structures in Finite Deformation,” Int. J. Numer. Meth. Engng 37, 1159–1186 (1994). 5. U. Nackenhorst, “The ALE-Formulation of Bodies in Rolling Contact: Theoretical Foundations and Finite Element Approach,” Comput. Meth. Appl. Mech. Engng 193 (39–41), 4299–4322 (2004). 6. P. Wriggers, “Finite Element Algorithms for Contact Problems,” Arch. Comput. Meth. Engng 2 (4), 1–49 (1995). 7. Guangdi Hu and P. Wriggers, “On the Adaptive Finite Element Method of Steady-State Rolling Contact for Hyperelasticity in Finite Deformations,” Comput. Meth. Appl. Mech. Engng 191 (13–14), 1333–1348 (2002). ¨ and M. C. Boyce, “Constitutive Modeling of the Large Strain Time-Dependent Behavior of 8. J. S. Bergstrom Elastomers,” J. Mech. Phys. Solids 46 (5), 931–954 (1998). ¨ and M. C. Boyce, “Mechanical Behavior of Particle Filled Elastomers,” Rubber Chem. 9. J. S. Bergstrom Techn. 72 (4), 633–656 (1999). 10. V. K. Semenov and A. E. Belkin, “Mathematical Model of Viscoelastic Behavior of Rubber under Cyclic Loading,” Izv. Vyssh. Uchebn. Zaved. Mashinostr., No. 2, 46–51 (2014). 11. A. E. Belkin, I. Z. Dashtiev, and V. K. Semenov, “Mathematical Model of Viscoelastic Behavior of Polyurethane in Compression with Moderately High Strain Rates,” Vestink MGTU im. Baumana. Ser. Mashinostr., No. 6, 44–58 (2014).

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