Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 123 132 Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/journals/dnc-default.aspx Mathematical Modelling and Simulation of thebifurcationalwobblestonedynamics Jan Awrejcewicz and Grzegorz Kudra Department of Automation, Biomechanic and Mechatronics, Lodz University of Technology, Lodz Poland Submission Info Communicated by Dima Volchenkov Received 20 August 2013 Accepted 11 March 2014 Available online 1 July 2014 Keywords Wobblestone Friction Rolling resistance Bifurcation Chaos Abstract The Celtic stone, sometimes also called wobbles tone or rattleback usually is a semi-ellipsoidal solid with a special mass distribution. Most celts lied on a flat and horizontal base, set in rotational motion around a vertical axis can rotate in only one direction. In this work the dynamics of the celt is simulated numerically, but the solid is forced untypically, i.e. it is situated on a harmonically vibrating base. Essential part of the model are approximate functions describing the contact forces, i.e. dry friction forces and rolling resistance. They are based on previous works of the authors, but some modifications of friction model are made, which can be described as a generalization of the earlier used Padé approximants. Periodic, quasiperiodic and chaotic dynamics of a harmonically forced rattleback is found and presented by the use of Poincaré maps and bifurcation diagrams. 2014 L&H Scientific Publishing, LLC. All rights reserved. 1 Introduction In physics and technology there are many situations, where one cannot assume, that friction force and torque, generated on some finite and very often rolling contact, are independent. The possible solution is to discretize the space [1, 2] and use, for example, the finite element method. This leads however to numerical complexity and time-consuming computations. Another approach to the problem is to construct some approximate and simplified models of the resulting friction force and torque, more suitable for relatively fast numerical simulations. In 1962, Contensou [3] proposed the integral model of friction force, assuming Coulomb friction law, Hertz stress distribution and fully developed sliding on circular contact. In the case of fully developed sliding of a planar object on flat surface, a group of researches, using analogy to the classical plasticity theory, tried to solve the problem by construction of the so-called limit surfaces [4, 5], which was subsequently approximated by the use of the ellipsoid [6]. In 1998, Zhuravlev [7] gave exact analytical solution to the integral model of Contensou and proposed its approximations based on the Padé expands. Then these results were extended to the higher-order Padé approximants [8], as well as the coupling between friction and rolling resistance was taken into account [9]. A different approach to the problem is presented in the work [10], where the coupled model of friction force and moment was obtained by the use of the Taylor expansions of the velocity pseudo potential. Later, the same group of authors [11] proposed the ellipse as an efficient approximation of the dependence between resulting friction force and moment, in the case of circular contact area and fully developed sliding. Corresponding author. Email address: jan.awrejcewicz@p.lodz.pl ISSN 2164 6376, eissn 2164 6414/$-see front materials 2014 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2014.06.002

124 Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 It is known that certain bodies of mass centre not coincident with their centroid and of principal centroidal axes of inertia not coincident with their geometric axes exhibit certain interesting dynamical behaviors. An example of such a rigid body is the so-called a rattle back top or wobble stone (a half-ellipsoid solid), which lied on a flat horizontal surface and set in rotational motion about the vertical axis (spin) can rotate in only one direction. The imposition of an initial velocity in the opposite direction leads to rapid cessation of rotation in this direction, and subsequently the stone starts its transverse vibrations and rotation in the opposite direction. On the other hand, the wobble stone is an example of mechanical system, where modelling of contact forces, including friction and rolling resistance, can occur important in realistic simulations. The first scientific work concerning dynamics of a wobbles tone was published by Walker [12] in 1896. Then problem has been attacked by Bondi [13], Magnus [14], Caughey [15], Kane and Levinson [16], Lindberg and Longman [17], and recently by Garcia and Hubbard [18]. In most cases the authors assumed dissipation-free rolling without slip.the lack of complete explanation, including the lack of an adequate mathematical model of dynamics of a wobble stone, was challenging also for leading Russian mechanicians. In 2002 Markeev [19] conducted the analysis of dynamics of a wobblestone, in the neighborhood of static and dynamic equilibrium positions on the assumption of absence of friction. However, the derived perturbation equations included only the local dynamics of the stone. On the other hand, they as well were verified in the experimental way. In 2006 Borisov, Kilin and Mamaev [20] performed a mathematical modeling of a heavy unbalanced ellipsoid rolling without slip on a horizontal surface pointing to some similar phenomena to those observed during motion of a wobble stone. In 2008 Zhuravlev and Klimov [21] presented the model of a wobble stone being the closest to reality, which introduced the possibility of slip of the contact point between the stone and horizontal surface, and the so-called Coulomb-Contensou-Zhuravlev friction model, based on the Padé approximants. In the work [22], for the first time, the complete set of hyperbolic tangent approximations of the coupled model of dry friction and rolling resistance for circular contact area, was presented and applied to the analysis of the Celtic stone dynamics. In the next paper [23], the approximate coupled models of friction and rolling resistance were developed for the elliptic shape of the contact patch, and applied in the Celtic stone mathematical modeling and simulation. It was shown, that the coupling between friction force and torque can play essential role in the simulation of this system. The paper [24] is a comprehensive and detailed study of the original approximations of coupled integral models of dry friction forces and rolling resistance for the elliptic contact area, based on the Padé approximants and their modifications and generalizations. In the present work we model and simulate numerically bifurcational dynamics of a celt, which is forced untypically, i.e. it is situated on a harmonically vibrating surface. In Section 2.1 we present differential and algebraic equations of motion of the celt as a rigid body. Section 2.2 is devoted to approximate modelling of contact forces, i.e. dry friction forces and rolling resistance. In Section 3, by the use of Poincaré maps and bifurcation diagrams, we present periodic, quasi-periodic and chaotic dynamics of a harmonically forced rattle back. 2 The governing equations 2.1 Differential and algebraic equations of motion of the celt The equations presented here rely on our earlier works concerning the Celtic stone modelling [22,23]. The rigid body of a semi-ellipsoid shape, with the geometry centre in the point O and with the mass centre in the point C (of the relative position determined by the use of vector k) lying on a rigid and flat surface π (parallel to the X 1 X 2 plane of the global coordinate system GX 1 X 2 X 3 ) with the contact in the point A is presented in Fig. 1. It is assumed that the surface is moving harmonically along the axis X 1, but it will be taken into account in the next section, during formulation of the contact model. The governing equations,written in the local co-ordinate system Cx 1 x 2 x 3 of central axes parallel to the geometric axes of the ellipsoid, have the following form

Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 125 Fig. 1 The Celtic stone. dv m dt + ω (mv) = mgn + ˆNn + ˆT s, dω B + ω (Bω) =(r k) ( ˆNn + ˆT s )+ ˆM s + ˆM r, (1) dt dn dt + ω n = 0, where m stands for mass of the body, while B denotes the tensor of inertia of the Celt at the mass centre B 1 B 12 B 13 B = B 12 B 2 B 23, B 13 B 23 B 3 where B 1,B 2 and B 3 are moments of inertia with respect to the axes x 1,x 2 and x 3, respectively. The symbols B 12,B 13 and B 23 denote the corresponding centrifugal moments. In Eq. (1) v and ω are the absolute velocity of the mass centre C and angular velocity of the body, respectively. The symbol ˆN denotes the magnitude of the normal reaction of the horizontal plane, n is the unit vector normal to the plane X 1 X 2, ˆT s (ignored in Fig. 1) is the friction force applied at the point A, ˆM s and ˆM r (ignored in Fig. 1) are the dry friction and the rolling resistance torques applied to the body. Vector r indicates the actual contact point position and the vector k determines the mass centre position. The notation da/dt stands for the derivative with respect to time of the vector a in the movable system Cx 1 x 2 x 3 Later in this work, we will denote the components of any vector u, along the axes of Cx 1 x 2 x 3 coordinate system, as u i (i = 1,2,3). For the components of this vector along axes of the global coordinate system GX 1 X 2 X 3, we will use notation u Xi (i = 1,2,3). The set of Eq. (1) consists of 9 first-order ordinary differential equations with 9 unknown differential variables: v i,ω i,n i (i = 1,2,3) and one algebraic variable ˆN. The missing relation is the following algebraic equation v A n = 0, (2) where v A =[v + ω (r k)], which follows the fact that the vector v A lies in the plane π. Equations (1) and (2) form now the differentialalgebraic equation set allowing to simulate the celt in the local coordinate system.

126 Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 Additional relations, necessary to solve the problem, are the following ellipsoid equation and the condition of tangent contact between the ellipsoid and the horizontal plane ϕ(r) = r2 1 a 2 + r2 2 1 a 2 + r2 3 2 a 2 1 = 0, 3 n = λ dϕ dr, where λ < 0 is a certain scalar multiplier and a 1,a 2 and a 3 are the semi-axes of the ellipsoid. Equation (3) result in the following relation between the components of the vectors r and n in the Cx 1 x 2 x 3 coordinate system: r 1 = a2 1 n 1 2λ, r 2 = a2 2 n 2 2λ, r 3 = a2 3 n 3 2λ, (4) where λ = 1 a 2 1 2 n2 1 + a2 2 n2 2 + a2 3 n2 3. In order to compute the absolute position of the mass centre C, one can use the following differential relation dr C + ω r dt C = v. (5) To determine completely the absolute angular position of the system, we use the following differential equation (3) where ψ is the precession angle. ψ = ω 1n 1 + ω 3 n 2 1 n 2, (6) 2 2.2 Friction forces and rolling resistance Following the works [23,24], we assume the elliptic contact between two bodies with the following properties: i) the contact is locally plane; ii) on its every element the Coulomb friction laws are applicable; iii) the friction coefficient is constant on contact area; iv) sliding is fully development the contact surface; v) deformation of the bodies do not affect the relative velocities in the contact plane. Then the components of the sliding friction occurring in the Eq. (1) read ˆT s = μ ˆN(T sx e x + T sy e y ), ˆM s = μ ˆNâM s n, where μ is dry friction coefficient, â is the length of major semi-axis of the contact area, the symbols e x and e y denote the unit vectors of axes x and y, respectively (see Fig. 2). The quantities T sx,t sy and M s are nondimensional components of the integral friction model. Taking into account the above mentioned assumptions, we can write v s cosϕ s ω s y T sx (v s,ω s,ϕ s )= σ(x,y) (vs cosϕ s ω s y) 2 +(v s sinϕ s + ω s x) dxdy, 2 T sy (v s,ω s,ϕ s )= M s (v s,ω s,ϕ s )= F F F v s sinϕ s + ω s x σ(x,y) dxdy, (8) (vs cosϕ s ω s y) 2 +(v s sinϕ s + ω s x) 2 ω s (x 2 + y 2 )+v s xsin ϕ s v s ycos ϕ s σ(x,y) (vs cosϕ s ω s y) 2 +(v s sinϕ s + ω s x) dxdy, 2 (7)

Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 127 Fig. 2 The contact area with the coordinate system and the velocities of sliding and rolling. where σ(x, y) is the non-dimensional contact pressure distribution over the non-dimensional contact area F presented in Fig. 2. The symbol v s denotes here the magnitude of linear sliding velocity at the centre A of the contact, while ω s is the angular sliding velocity. The angle φ s defines the direction of the linear sliding velocity v s. We assume the following model of non-dimensional contact pressure distribution over the area F: σ(x,y)=b 1 σ 0 ( x 2 + b 2 y 2 )(1 + d c x + b 1 d s y), (9) where 0 < b = ˆb/â 1 is dimensionless length of minor semi-axis of the elliptic contact (where ˆb is its real counterpart) and the function σ 0 (ρ ) defines the non-dimensional, circularly symmetric stress distribution over the unit circular area. The part 1 + d c x + b 1 d s y of the function (9) defines the deformation of stress distribution responsible for rolling resistance generation, where where d c = k r cosγ, d s = b r k r sinγ, (10) cosγ = sinγ = v rx v 2 rx + b 2 b 2 r b 1 b 1 r v ry v 2 rx + b 2 b 2 r v 2 ry v 2 ry,. In the Eq. (10), the parameter k r plays a role of the coefficient of the rolling resistance along the direction of the major axis of the contact, while b r k r defines the corresponding resistance along the minor axis. If only one of the bodies undergoes deformation, the symbols v rx and v ry denote the components of velocity of relative motion of the contact zone along the surface of the deformable body. The rolling resistance results from contact pressure deformation and reads ˆM r = ˆNâπS 3 (bd s e x d c e y ), (11) where S 3 belongs to the following family of parameters S i = ˆ 1 0 σ 0(ρ )ρ i dρ (12)

128 Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 Using the results of the work [24], we get the following approximation the integral model (8) T sx v2 s v sx + ω s (b Ts (4G(e)S 0 v sx ω s bd s (πs 3 v 2 sy + 4G(e)S 2ωs 2)) πs 3d c v sx v sy ) (v 3m Ts s + b m, Ts T s ω s 3m Ts ) m 1 Ts T sy v2 s v sy + ω s (b Ts (4H(e)S 0 v sy ω s + d c (πs 3 v 2 sx + 4H(e)S 2 ωs 2 )) + πbs 3 d s v sx v sy ) (v 3m Ts s + b m, (13) Ts T s ω s 3m Ts ) m 1 Ts M s 4b M s E(e)S 2 ωs 3 + πs 3 ((v 2 sx + b 2 v 2 sy)ω s + v 2 s(d c v sy bd s v sx )) (b m Ms M s ω s 3m Ms + v 3m, Ms s ) m 1 Ms fulfilling the derivatives of the integral components (with respect to v s for v s = 0 and with respect to ω s for ω s = 0) from zero to first order. The symbols v sx and v sy denote the components of velocity v s in the Axy coordinate system. The quantities S 0,S 2 and S 3 depend on the contact pressure distribution, while m Ts,m Ms,m Ts,m Ms,b Ts,b Ms, b Ts and b Ms are other constant parameters. The functions G(e) and H(e) are defined in the following way G(e) =(K(e) E(e))e 2, H(e)=(E(e)+(e 2 1)K(e))e 2, (14) where K(e) and E(e) are complete elliptic integrals of the first and second kind, and e = 1 b 2 is eccentricity of the contact. The approximation (13) is a generalization of the model presented in the work [23]. For m Ts = m Ms = 1 one obtains the Padé approximation, while for m Ts = m Ms = 2 one gets the model with square root in the numerator, which is able to describe the ellipsoid surface, like in the models proposed in the works [6,11]. In order to smooth the model and avoid the singularities, we use the following approximations of some elements of the functions (11) and (13) v s v 2 sx + v 2 sy + ε, d c k r v rx, (15) v 2 rx + b 2 b 2 r v 2 ry + ε R v ry d s b r k r, v 2 rx + b 2 b 2 r v 2 ry + ε R where ε and ε R are small numerical parameters. Then we can use classical methods of simulation of ordinary differential equations. The non-dimensional sliding and rolling velocities and the spinning velocity are computed as follows v s = v A v π, â dr v r = v s + /â, (16) dt ω s = ω n, where we have made an assumption that the rigid Celtic stone rolls over a deformable table. The term v π expresses the horizontal motion of the plane π along the axis X 1 : v π = q π ω π sin(ω π t)e X1, (17) where q π and ω π are constant parameters, and e X1 is the unit vector of the axis X 1. For more details see the references [23, 24]. Note, that using the presented equations, one often need to transform the vectors components between local and global coordinate systems.

3 Numerical simulations Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 129 Finally we solve numerically the set of algebraic-differential equations (1) and (2) and (5) and (6). It consists of 14 scalar equations with 14 unknown functions of time (v 1,v 2,v 3,ω 1,ω 2,ω 3,n 1,n 2,n 3, ˆN,r CX1,r CX2,r CX3,ψ). The vector r is function of vector n (see Eqs. (4)). Having the global position of the celt (n 1,n 2,n 3,ψ,r CX1,r CX2,r CX3 ), one can easy compute the position of the ellipsoid centre (r OX1,r OX2,r OX3 ). In the current section, using the model described in section 2, we present exemplary simulations of the wobble stone lying on vibrating base. We assume the following parameters of the celt and contact: a 1 = 110 mm, a 2 = a 3 = 25 mm, m = 543.6 10 3 kg, B 1 = 246.3kg mm 2, B 2 = 1358.9kg mm 2, B 3 = 1374.7kg mm 2, B 12 = 172.8 kg mm 2, B 13 = B 23 = 0, k 1 = k 2 = 0, k 3 = 4.29 mm, g = 9.81 m/s 2,â = 6.192 mm, b = 0.1813, μ = 0.2241, k r = 0.4345, b r = 1.196, S = 0.1498, S 2 = 0.1121, S 3 = 0.0589, b Ts = 0.5305, b Ms = 0.2476, m Ts = 0.8161, m Ms = 0.5366. The base is vibrating with the amplitude q π = 0.04 m. In Fig. 3 there is presented bifurcational diagram of the system with frequency of the base vibration ω π as a control parameter. One can observe rich bifurcational dynamics with many intervals of regular and irregular motion. Note that the Poincaré sections are observed as projection on ω 2 axis, so we observe here only a local dynamics of the celt (in the local coordinate system Cx 1 x 2 x 3 ). From that reason, as will be seen later, the observation of the recurrent motion on bifurcational diagram, not necessarily means the same behaviour in the case of global motion of the wobble stone. In Fig. 4 one can see an example of periodic motion for ω π = 14 rad/s, observed both in local coordinates of angular velocity (ω 1 ω 3 ) and global position of the geometric centre of the ellipsoid O(r OX1 r OX2 ). One can observe, that the global motion of the ellipsoid centre is limited. Figure 5 presents another example of periodic local motion for ω π = 15.6 rad/s, which repeats in the local coordinates (a), but global position of the point Fig. 3 The bifurcational diagram of celt with ω π as a control parameter. Fig. 4 The periodic solution for ω π = 14 rad/s.

130 Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 Fig. 5 The periodic solution for ω π = 15.6 rad/s: (a) and (b) trajectories, (c) Poincaré section. Fig. 6 The quasi-periodic solution for ω π = 17.0 rad/s: (a) and (b) trajectories, (c) and (d) Poincaré sections.

Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 131 Fig. 7 The chaotic solution for ω π = 19.0 rad/s: (a-b) trajectories, (c) Poincaré section. O wanders in an unlimited way (b). The corresponding Poincaré section is also presented (c). An example of quasi-periodic solution for ω π = 17.0 rad/s is presented in Fig. 6, where the trajectories in the local and global coordinates are shown (a-b), as well as the corresponding Poincaré sections (c-d). In this case, the global position of the body is bounded. The last example is chaotic motion for 19.0 rad/s, presented in Fig. 7 by the use of phase plots and Poincaré sections. Now the global motion of the wobble stone seems to be unlimited, but we did not prove that. 4 Conclusions In this paper, we have presented approximate models of coupled friction and rolling resistance in the case of elliptic contact, mainly based on our previous work [23]. We have added however some modifications, for example we have proposed a generalization of Padé approximation. This new approach, in certain sense, joins two different kinds of approximation: the one based on Padé approximates [7-9] and that using an approximation by the use of ellipsoid [11]. The presented models are suitable for fast numerical simulations of rigid bodies with frictional contacts, because they allow to avoid necessity of the use of the space-discretization methods. The models can be applied there where one needs such efficient simulations. As an example we can give the realistic simulations of reality in computer games. Another case of possible application is a plant model in predictive control systems, for example of a robot pushing and moving an object, which slides over some surface.

132 Jan Awrejcewicz, Grzegorz Kudra / Discontinuity, Nonlinearity, and Complexity 3(2) (2013) 123 132 We have also presented examples of rich bifurcational dynamics of a Celtic stone lying on harmonically moving base, including periodic, quasi-periodic and chaotic dynamics. This dynamical system can be used for testing different elements of the proposed approximate model of contact forces. Acknowledgments This paper was financially supported by the National Science Centre of Poland under the grant MAESTRO 2, No. 2012/04/A/ST8/00738, for years 2013-2016. References [1] Kalker, J.J. (1982), A fast algorithm for the simplified theory of rolling contact, Vehicle Systems Dynamics, 11(1), 1 13. [2] Kalker, J.J. (1990), Three-dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht. [3] Contensou, P. (1962), Couplage entre frottement de glissementet de pivotementdans la téorie de la toupe, Kreiselprobleme Gyrodynamics: IUTAM Symp., Calerina, 201 216. [4] Goyal, S., Ruina, A. and Papadopoulos, J. (1991), Planar sliding with dry friction. Part 1. Limit surface and moment function, Wear 143, 307 330. [5] Goyal, S., Ruina, A. and Papadopoulos, J. (1991), Planar sliding with dry friction. Part 2. Dynamics of motion, Wear 143, 331 352. [6] Howe, R. D. and Cutkosky, M. R. (1996), Practical force-motion models for sliding manipulation, The International Journal of Robotics Research, 15(6), 557 572. [7] Zhuravlev, V.P. (1998), The model of dry friction in the problem of the rolling of rigid bodies, Journal of Applied Mathematics and Mechanics, 62(5), 705 710. [8] Zhuravlev, V.P. and Kireenkov, A. A. (2005), Padé expansions in the two-dimensional model of Coulomb friction, Mechanics of Solids, 40(2), 1 10. [9] Kireenkov, A.A. (2008), Combined model of sliding and rolling friction in dynamics of bodies on a rough plane, Mechanics of Solids, 43(3), 412-425. [10] Leine, R.I. andglocker, Ch. (2003), A set-valued force law for spatial Coulomb-Contensou friction, European Journal of Mechanics A/Solids, 22(2), 193 216. [11] Möller, M., Leine, R.I. and Glocker, Ch. (2009), An efficient approximation of orthotropic set-valued force laws of normal cone type, Proceedings of the 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal. [12] Walker, GT. (1896), On a dynamical top, The Quarterly Journal of Pure and Applied Mathematics, 28, 175 184. [13] Bondi, S.H. (1986), The rigid body dynamics of unidirectional spin, Proceedings of the Royal Society of London. SeriesA, 405, 265 279. [14] Magnus, K. (1974), Zur Theorie der keltischenwackelsteine, Zeitschrift fürangewandte Mathematik und Mechanik, 5, 54 55. [15] Caughey, T.K. (1980), A mathematical model of the Rattleback, International Journal of Non-Linear Mechanics,15 (4-5), 293 302. [16] Kane, T.R. and Levinson, D.A. (1982), Realistic mathematical modeling of the rattleback, International Journal of Non-Linear Mechanics, 17(3), 175 186. [17] Lindberg, J. and Longman, R.W. (1983), On the dynamic behavior of the wobblestone, Acta Mechanica, 49(1-2), 81 94. [18] Garcia, A. and Hubbard, M. (1988), Spin reversal of the rattleback: theory and experiment, Proceedings of the Royal Society of London. Series A, 148, 165 197. [19] Markeev, A.P. (2002), On the dynamics of a solid on an absolutely rough plane, Regular and Chaotic Dynamics, 7(2), 153 160. [20] Borisov, A.V., Kilin, A.A. and Mamaev, I.S. (2006), New effects of rattlebacks, Doklady Physics, 51(5), 272 275. [21] Zhuravlev, V.Ph. and Klimov, D.M. (2008), Global motion of the celt, Mechanics of Solids, 43(3), 320 327. [22] Kudra, G. andawrejcewicz, J. (2011), Tangens hyperbolicus approximations of the spatial model of friction coupled with rolling resistance, International Journal of Bifurcation and Chaos, 21(10), 2905 2017. [23] Awrejcewicz, J. and Kudra, G. (2012), Celtic stone dynamics revisited using dry friction and rolling resistance, Shock and Vibration, 19, 1 9. [24] Kudra, G. and Awrejcewicz, J. (2013), Approximate modelling of resulting dry friction forces and rolling resistance for elliptic contact shape. European Journal of Mechanics A/Solids, 42, 358 375.