# Rolling resistance modelling in the Celtic stone dynamics

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## Abstract

In the work there is presented a mathematical model of a rattleback lying on a horizontal plane, with special attention paid to modelling of the contact forces. Friction force modelling concerns a certain class of problems, where it is possible to assume fully developed sliding on the contact zone. The models are based on the integral expressions assuming the classical Coulomb friction law valid at each point of the contact. In order to obtain a higher simulation speed, a special class of approximations if integral functions is used, being some kind of generalization of Padé expansion. The rolling resistance is modelled as a result of special distortion of contact pressure distribution and takes into account the rolling direction and the elliptic shape of the contact. The work uses previous results of the authors, but more attention is paid to the rolling resistance model, which is extended and analysed in more detail. The models are validated experimentally and their various elements are tested.

## Keywords

Rolling resistance Celtic stone Contact forces## 1 Introduction

Modelling and numerical simulation of mechanical systems with frictional contacts in their general case require discretization of the deformable contacting bodies near the contact zone or at least discretization of the contact surface [1, 2, 3]. For the rolling contact one can use the well-known CONTACT software [2]. Space discretization and use of finite element methods lead, however, to high computational costs and long times of numerical simulations that are not always acceptable. It is the reason of the development of simplified models and algorithms [3, 4, 5].

There are special situations, where the local deformations of the bodies are negligible in comparison with their relative velocities in the contact area, and one can assume a fully developed sliding over the contact. In the case of planar contact one can assume relative motion of the contacting bodies in the contact zone, equivalent to the planar motion of a rigid body. Modelling of the resulting friction force and moment then leads to integral expressions over the contact area [6]. In order to obtain higher speeds of numerical simulations, scientists still searched for reduced and simplified models. Zhuravlev [7] applied Padé approximants in the construction of expressions for friction force and moment in the case of fully developed sliding over a circular contact area with Hertzian contact stress distribution and Coulomb friction law. These results were then developed and the models of mutually coupled friction forces and rolling resistance were obtained [8], where initially Hertzian contact stress distribution is distorted in a special way in order to take into account the rolling resistance [9]. An extension of this approach is presented in the work [10], where the generalized forms of approximations of the resulting friction force and moment as well as rolling resistance for elliptic contact shape are presented. Rolling friction is there modelled as a resistance against motion of the area of deformation over the body [11] and, ignoring the problem of rotation of the elliptical region, is consistent with the results obtained under the assumption of the linear hysteresis [12, 13, 14, 15]. Karapetyan [16] proposed a two-parametric model of friction forces for spherical contact between a ball and planar base being a result of deformations of the contacting bodies. This model joins three different kinds of friction (Coulomb, Contensou and rolling friction) and can be seen as a development of Zhuravlev model.

An example of a dynamical system, in which an essential role may be played by the particular elements of the contact models presented in the work [10], is the rattleback (known also as wobble stone or Celtic stone)—usually a semi-ellipsoidal top with a special mass distribution, moving on the plane base and possessing specific dynamical properties. The first scientific work on rattleback was published at the end of the 19th century [17] and it has been an object of interest till today [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], both from the point of view of dynamics and of modelling. During modelling and analysis of the Celtic stone, rolling without sliding is often assumed, leading to non-holonomic constraints [17, 21, 22, 24, 25, 26, 29, 31]. Some authors assume some kind of dissipation, but far from reality [18, 19, 20, 23]. More realistic modelling of friction is applied in the work [27]. Detailed modelling of contact between the Celtic stone and horizontal base, including rolling resistance and using results of the work [10], is presented in Ref. [34], where also experimental validation of particular parts of the model is performed. The same approach is used in modelling and numerical simulations of a billiard ball rolling and sliding on a table [35, 36].

In the present work the same experimental data is used as in Ref. [34], but deeper insight into the rolling resistance model is achieved—an extended expression for the rolling resistance additional elements and aspects are investigated.

## 2 Governing equations of the Celtic stone

Equations (1)–(2) define the motion of the mass centre \(C\) of the Celtic stone in the global coordinate system. Equation (3) represents the law of variation of the angular momentum expressed in the local reference frame. Equations (4)–(6) represent kinematic differential equations expressing relations between components of angular velocity and derivatives of angles defining orientation of the body. Equation (7) is the twice differentiated algebraic condition \(( \mathbf{r} _{C} + \mathbf{r} + \mathbf{k} ) \cdot \mathbf{n} = 0\) expressing assumption of permanent contact between the rattleback and plane \(\pi \). The set of Eqs. (1)–(6) is in fact a set of 13 scalar differential-algebraic equations of index 1. One can notice here 13 unknown scalar functions of time, but only 12 derivatives of these functions occur—there is no derivative of the function \(\hat{N} ( t ) \). Equations (1)–(6) can be solved algebraically with respect to these 13 unknown functions of time. One can obtain then the set of 12 ordinary differential equations with 12 unknown functions of time and additional algebraic expression for normal reaction \(\hat{N} ( t ) \).

Models (12)–(13) represent special case of approximations developed in [10] and tested in modelling of the Celtic stone in [34]. They are a minimal but sufficient form of more general models, allowing one to simulate the rattleback dynamics quickly and realistically. They are based on the assumption of fully developed sliding on the contact area and small enough local deformations resulting in relative motion in the vicinity of the contact being planar motion of rigid bodies. The resultant friction action is reduced to the friction force and moment acting at the centre of the finite contact area. Their models are based on the integral expressions under the assumption of the classical Coulomb friction law valid at each point of the contact zone. In order to obtain higher simulation speed, a special class of approximations is used, being some kind of generalization of the Padé expansion. In general, friction forces are coupled with rolling resistance via contact pressure distribution. A simplified model of pressure distribution is assumed, i.e. the Hertzian model, which is then modified (distorted) in a special way in order to take into account the rolling resistance. The parameters \(b _{i}\) (\(i=1,\dots,3\)) in models (12)–(13) have no direct physical meaning but depend on shape and size of the contact, as well as on the contact pressure distribution and model of friction. They can be identified based on the experimental data. In the case of model of friction force and moment (12)–(13), a circularly symmetric contact pressure distribution is assumed and there is no coupling with the rolling resistance (14). On the other hand, the rolling resistance model (13) assumes an elliptical shape of the contact patch. This is a simplified special case of the contact models chosen based on the work [34].

The parameters \(\varepsilon _{i}\) (\(i=1, 2\)) was initially introduced in order to eliminate singularities in the expressions (12)–(14) for vanishing sliding or rolling velocities. Their values was small enough to smooth set-valued friction and rolling models but not to change too significantly their properties. However, in this paper the model of rolling resistance is tested for higher values of the parameter \(\varepsilon _{2}\).

## 3 Numerical simulations and experimental validation

The rattleback used in the experimental investigations has the following parameters (found by direct length and mass measurements and numerical integrations on the corresponding volumes): \(a _{1}=110 \pm 0.01\) mm, \(a _{2}=25 \pm 0.01\) mm, \(a _{3}=25 \pm 0.01\) mm, \(m=(543.6 \pm 0.2)\times 10^{-3}\) kg, \(B _{1}=247.4 \pm 1~\text{kg}\,\text{mm}^{2}\), \(B _{2}=1358.9 \pm 1~\text{kg}\,\text{mm}^{2}\), \(B _{3}=1374.7 \pm 1~\text{kg}\,\text{mm}^{2}\), \(B _{12}=159.1 \pm 1~\text{kg}\,\text{mm}^{2}\), \(B _{13}=0 \pm 1~\text{kg}\,\text{mm}^{2}\), \(B _{23}=0 \pm 1~\text{kg}\,\text{mm}^{2}\), \(k _{1}=0 \pm 0.1\) mm, \(k _{2}=0 \pm 0.1\) mm, \(k _{3}=4.46 \pm 0.1\) mm. The vectors defining the positions of the measurement points \(\mathbf{r}_{OP_{1}}\), \(\mathbf{r}_{OP_{2}}\) and \(\mathbf{r}_{OP_{3}}\) (with origin at the point \(O\) and end at the corresponding measurement point \(P _{i}\)) possess the following coordinates in the local reference frame: \(r_{OP_{1}1}= -97 \pm 0.5\) mm, \(r_{OP_{1}2}= 0 \pm 0.5\) mm, \(r_{OP_{1}3}= 4 \pm 0.5\) mm, \(r_{OP_{2}1}= 97 \pm 0.5\) mm, \(r_{OP_{2}2}= 0 \pm 0.5\) mm, \(r_{OP_{2}3}= 4 \pm 0.5\) mm, \(r_{OP_{3}1}= 0 \pm 0.5\) mm, \(r_{OP_{3}2}= 0 \pm 0.5\) mm, \(r_{OP_{3}3}= 151 \pm 0.5\) mm. During the experiment a rubber plate is used as a base for the rattleback, so one can assume that the rolling and sliding body is rigid in comparison to the base. Moreover, one assumes a fixed value of the regularization parameter \(\varepsilon _{1}=10^{-5}\) m/s.

- 1.
Model A—the full model with all the parameters undergoing the process of estimation (with exception of the parameter \(\varepsilon _{1}\)). Note that \(b _{4}/ b _{5}= 0.240\) (result of identification), while the quotient of the corresponding axes’ lengths for the elliptical contact between the resting Celtic stone and the base according to the Hertz theory is equal to 0.148. In Figs. 5–7 there are presented three different experimental solutions used in the identification process compared with the corresponding simulation results obtained by the full model A.

- 2.
Model B—the parameter \(\varepsilon _{2}\) is equal to a small value of \(10^{-5}\) m/s and only play a role of regularization of the expression for rolling resistance moment. In this case as a result of identification one have \(b _{4}/ b _{5}= 0.227\).

- 3.
Model C—assuming \(b _{4}=1\)—resulting in direction of the rolling resistance perpendicular to the velocity \(\mathbf{v} _{A}\) of the contact point A on the plane \(\pi \).

- 4.
Model C

_{1}—assuming \(b _{4}=1\) and \(b _{4}/ b _{5}=0.148\) (see description of the model A). - 5.
Model D—assuming \(b _{5}=1\)—resulting in other kind of model of direction of the rolling resistance (chosen rather accidentally).

- 6.
Model D

_{1}—assuming \(b _{5}=1\) and \(b _{4}/ b _{5}=0.148\). - 7.
Model E—assuming \(b _{4}=b _{5}=1\)—the rolling resistance does not depend on the direction of rolling and elliptical shape of the contact is ignored.

- 8.
Model F—assuming \(b _{3} =b _{1}=0\)—i.e. no coupling between friction force and moment (finite size of the contact is ignored in the model of friction forces).

Values of the objective function \(F _{o}\) and the corresponding model’s parameters obtained during identification process

A | B | C | C | D | D | E | F | |
---|---|---|---|---|---|---|---|---|

\(F_{o}\) [mm | 1.25 | 1.52 | 1.48 | 1.73 | 2.05 | 2.13 | 3.55 | 3.47 |

| 0.231 | 0.231 | 0.233 | 0.238 | 0.234 | 0.241 | 0.235 | 0.189 |

| 0.482 | 0.481 | 0.485 | 0.738 | 0.494 | 0.725 | 0.407 | 0.476 |

\(b_{1}\) [mm] | 4.29 | 4.30 | 4.28 | 3.01 | 4.50 | 3.06 | 4.02 | |

\(b_{2}\) [mm] | 2.46 | 2.45 | 2.45 | 2.45 | 2.46 | 2.43 | 2.43 | 2.56 |

\(b_{3}\) [mm | 0.667 | 0.657 | 0.700 | 1.62 | 0.771 | 1.58 | 0.520 | |

\(b_{4}\) | 2.27 | 2.20 | | | 0.234 | | | 2.42 |

\(b_{5}\) | 9.46 | 9.71 | 4.23 | | | | | 10.2 |

\(\varepsilon _{2}\) [m/s] | 0.0530 | \(\mathit{10}^{-\mathit{5}}\) | 0.0209 | 0.0310 | 0.00490 | 0.00333 | 0.0756 | 0.0478 |

One can observe that the use of full model (12)–(14) is reasonable. The greater value of the parameter \(\varepsilon _{2}\) leads not only to better results but also makes the equations less stiff. An especially important element of the rolling resistance model is its dependence on the direction of rolling: the value of the resistance along the corresponding semi-axes of the elliptical contact as well as during rolling along intermediate directions.

## 4 Conclusions

The reported results have been obtained by the use the same experimental data as used in the previous work of the authors [34]. The extended model of the rolling resistance is used and new aspects of the rolling friction in the case of an elliptical contact are investigated. The model of the resultant friction force and moment is used in minimal but sufficient form (i.e. more details do not lead to noticeable improvement of the results) based on the results of Ref. [34].

First of all, it has been shown that the rolling resistance is smaller along the shorter semi-axis of the contact (compare model E with models A–D). It has only been confirmed that the vector of the rolling resistance moment is perpendicular to the direction of motion of the deformation region only in a certain approximation. The choice of an accidental model of direction of rolling resistance, like model D, gives much worse results than the model where the rolling resistance is always perpendicular to \(\mathbf{v} _{A}\) (model C). But extension of the model by addition of the parameter \(b _{4}\) to the set of parameters undergoing a process of identification (model B) leads to even better results. Another improvement of the rolling resistance model introduced in the present work, which allows one to obtain better agreement with experiment, is that of treatment of the parameter \(\varepsilon _{2}\) as an element not only playing a role of regularization of the model, but also a quantity undergoing the process of identification and allowed one to take any value (model A). Finally, we have confirmed that an important role in the modelling of the resultant friction force and moment is played by the coupling elements between them, i.e. the dependence of the friction force on the angular sliding velocity as well as the dependence of the friction moment on the linear sliding velocity (compare model F with models A–D).

The final conclusion is that the model here presented of the contact forces is very effective. In spite of its simplicity it allows for realistic and reliable numerical simulations of complex mechanical systems with frictional contacts, i.e. the rattleback having been the object of investigation presented in this work.

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