Rattleback Research Articles

Visualizing the Movement of the Celtic Stone as High School Students Activity

The Celtic Stone (also known as a rattleback) is a semi-ellipsoidal shaped solid object which when spun rotates on its axis in a preferred direction. If spun in the opposite direction, it goes to the stop and reverses its spin to the preferred direction. As the movement of the stone is multidirectional it is a challenge to perform quantitative measurements of its motion characteristics. The work presents the experimental set-up and the procedure for collecting date of a rattleback motion as the high school students activity. Proposals for specific student actions and their benefits for learning were described in connection with the results obtained. Some preliminary results of video measurements performed with a specific metal rattleback and meant to visualize its motion are reported. Attempts to compare the results with predictions based on the numerical model of the situation are undertaken. The advantages and disadvantages of the measurement system are presented and discussed.

Celtic Stone Dynamics on a Plane with Viscous Friction

Celtic Stone Dynamics on a Plane with Viscous Friction

Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion

We survey recent results on the theory of dynamical chaos from the point of view of topological dynamics. We present the concept of three types of dynamics: conservative, dissi-pative, and mixed dynamics, and also show several simple examples of attractors and repellers of all three types. Their similarities and differences with other known types of attractors and repellers (maximal and Milnor ones) are discussed. We also present elements of the qualitative theory of mixed dynamics of reversible systems. As examples of such systems we consider three nonholonomic models of rigid body motion: the Suslov top, rubber disk, and Celtic stone. It is shown that they exhibit mixed dynamics of different nature; in particular, the mixed dynamics observed in the model of rubber disk is extremely difficult to distinguish from the conservative one.

On the Region of Existence of a Discrete Lorenz Attractor in the Nonholonomic Model of a Celtic Stone

In this work, we consider the problem of existence of discrete Lorenz attractors in the nonholonomic model of a Celtic stone. To this end, the main local and global bifurcations leading to the appearance and destruction of the attractors are studied in two-parameter families of such models of certain types. The corresponding bifurcation diagram, in which the region of existence of a discrete Lorenz attractor is shown and its boundaries are described, is obtained on the plane of the governing parameters (the angle of dynamic asymmetry of the stone–the total-energy value). The similarities and differences in the scenarios of the appearance of a discrete Lorenz attractor in the nonholonomic model of a Celtic stone and attractor in the classical Lorenz model are found.

Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone

We study dynamic properties of a Celtic stone moving along a plane. We consider two-parameter families of the corresponding nonholonomic models in which bifurcations leading to changing the types of stable motions of the stone, as well as the chaotic-dynamics onset are analyzed. It shown that the multistability phenomena are observed in such models when stable regimes various types (regular and chaotic) can coexist in the phase space of the system. We also show that chaotic dynamics of the nonholonomic model of a Celtic stone can be rather diverse. In this model, in the corresponding parameter regions, one can observe both spiral strange attractors various types, including the so-called discrete Shilnikov attractors, and mixed dynamics, when an attractor and a repeller intersect and almost coincide. A new scenario of instantaneous transition to the mixed dynamics as a result of the reversible bifurcation of merging of the stable and unstable limit cycles is found.

Complex Dynamics in Generalizations of the Chaplygin Sleigh

The article considers Chaplygin sleigh on a plane in potential well, assuming that an external potential force is supplied at the mass center. Two particular cases are studied in some detail, namely, a one-dimensional potential valley and a potential with rotational symmetry; in both cases the models reduce to four-dimensional differential equations conserving mechanical energy. Assuming the potential functions quadratic, various behaviors are observed numerically depending of the energy, from those characteristic to conservative dynamics (regularity islands and chaotic sea) to strange attractors. This is another example of nonholonomic system manifesting these phenomena (similar to those for Celtic stone or Chaplygin top), which reflects a fundamental nature of these systems occupying intermediate position between conservative and dissipative dynamics.

Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

On some approximations of the resultant contact forces and their applications in rigid body dynamics

The work presents the possible applications and effectiveness of certain class of models of the resultant friction force and rolling resistance. The friction models are based on the integral model constructed under assumption of fully developed sliding on the plane contact area of general shape and any pressure distribution. Then the integral model of friction force and moment are approximated based on Padé approximants and their generalizations. These models are expected to be computationally effective in numerical simulations of rigid bodies with frictional contacts, such like billiard balls, Thompson top, the wobble stone and many others. In the present work two different examples of application of the developed contact models are presented and tested: a) a billiard ball rolling and sliding on the plane horizontal table; b) a full ellipsoid of revolution in contact with plane and horizontal base.

Application and experimental validation of new computational models of friction forces and rolling resistance

In the process of modelling of the Celtic stone rotating and rolling on a plane surface, different versions of simplified models of contact forces between two contacting bodies are used. The considered contact models take into account coupled dry friction force and torque, as well as rolling resistance. They were developed using Pade approximations, their modifications, and polynomial functions. Before the use of these models, some kinds of regularisations have been made, allowing to avoid singularities in the differential equations. The models were tested both numerically and experimentally, giving some practical guesses of the most essential elements of contact modelling in the Celtic stone numerical simulations. Since the tested contact models do not require the space discretisation, they can find application in relatively fast numerical simulations of rigid bodies with friction contacts.

Mathematical Modelling and Simulation of the Bifurcational Wobblestone Dynamics

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 aflat 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 Pade approximants. Periodic, quasiperiodic and chaotic dynamics of a harmonically forced rattleback is found and presented by the use of Poincare maps and bifurcation diagrams.

Richness of chaotic dynamics in nonholonomic models of a celtic stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.

Approximate modelling of resulting dry friction forces and rolling resistance for elliptic contact shape

There are many examples of mechanical systems with non-point friction contacts (billiard ball, Thompson top, wobblestone, electric polishing machine, the wobblestone, the Celtic stone), where the assumption of one-dimensional dry friction model do not necessarily lead to satisfactory accuracy of the numerical simulation. Moreover the rolling resistance often plays an important role in such systems. The paper is devoted to the problem of developing an approximate coupled model of resulting dry friction force and moment as well as rolling resistance, suitable for fast numerical simulation of rigid bodies with friction contacts, i.e. allowing to avoid the space discretization. An integral model of dry friction components is built under assumption of classical Coulomb friction law and fully developed sliding on the contact area of general shape and arbitrary contact pressure distribution. Then the special model of stress distribution over the elliptic contact area is developed, being a kind of generalization of Hertzian normal stress distribution, with addition of special distortion related to the rolling resistance. Finally some original approximate models of friction force and moment are proposed, based on Pade approximants and their generalizations as well as in the form of piecewise polynomial functions.

Об аттракторах лоренцевского типа в модели кельтского камня

Об аттракторах лоренцевского типа в модели кельтского камня

Феномены нелинейной динамики диссипативных систем в неголономной механике «кельтского камня»

Феномены нелинейной динамики диссипативных систем в неголономной механике «кельтского камня»

О некоторых новых аспектах хаотической динамики «кельтского камня»

О некоторых новых аспектах хаотической динамики «кельтского камня»

Celtic Stone Dynamics Revisited Using Dry Friction and Rolling Resistance

The integral model of dry friction components is built with assumption of classical Coulomb friction law and with specially developed model of normal stress distribution coupled with rolling resistance for elliptic contact shape. In order to avoid a necessity of numerical integration over the contact area at each the numerical simulation step, few versions of approximate model are developed and then tested numerically. In the numerical experiments the simulation results of the Celtic stone with the friction forces modelled by the use of approximants of different complexity (from no coupling between friction force and torque to the second order Padé approximation) are compared to results obtained from model with friction approximated in the form of piecewise polynomial functions (based on the Taylor series with hertzian stress distribution). The coefficients of the corresponding approximate models are found by the use of optimization methods, like as in identification process using the real experiment data.

Celtic stone dynamics revisited using dry friction and rolling resistance

The integral model of dry friction components is built with assumption of classical Coulomb friction law and with specially developed model of normal stress distribution coupled with rolling resistance for elliptic contact shape. In order to avoid a necessity of numerical integration over the contact area at each the numerical simulation step, few versions of approx- imate model are developed and then tested numerically. In the numerical experiments the simulation results of the Celtic stone with the friction forces modelled by the use of approximants of different complexity (from no coupling between friction force and torque to the second order Pade approximation) are compared to results obtained from model with friction approximated in the form of piecewise polynomial functions (based on the Taylor series with hertzian stress distribution). The coefficients of the corresponding approximate models are found by the use of optimization methods, like as in identification process using the real experiment data.

TANGENS HYPERBOLICUS APPROXIMATIONS OF THE SPATIAL MODEL OF FRICTION COUPLED WITH ROLLING RESISTANCE

In this paper, for the first time, the complete set of Tangens hyperbolicus approximations of model of dry friction coupled with rolling resistance for circular contact area between interacting bodies is proposed. The developed approximations are compared with corresponding Padé approximants of the first and second order well known from the literature and with the numerical solution of the exact integral model as well. It is shown that Tangens hyperbolicus approximants are closest to the exact solution. Then the approximated models are applied to the celtic stone dynamics, however with the significant simplifying assumption of circular contact between stone and the table, presenting differences between them again. Certain specific approximations and regularizations of the friction and rolling resistance models enabling and facilitating their application to the real problem are shown. The analysis of the response dependence on initial conditions is performed by the use of a special kind of diagram.

On stability when a singular point is non-isolated

Assuming that m zero roots correspond to the non-isolated singular point of the equations of disturbed motion, the problem of stability when some of the remaining roots are pure imaginary, while others have a negative real part in inessentially singular cases, is considered. This situation is typical for non-holonomic systems. The stability and instability theorems obtained are used to study the stability of the rotations of a celtic stone on the boundary of the domain of stability.

The use of the method of averaging to study non-linear oscillations of the celtic stone

The approximate solution of the equations of the perturbed motion of a celtic stone near its position of equilibrium, obtained in /1/ by retaining in these equations terms of the second order with respect to the perturbations and averaging ★★ is used for a qualitative and quantitative explanation of the following effect established by numerical integration of the complete equations /2/. If the celtic stone is rotated about the vertical axis in a specified direction, then after a fairly short time it ceases to rotate, begins to oscillate about the horizontal axis, and then resumes its rotation in the opposite direction. For some of the models of the Celtic stone the change in the direction of rotation may occur more than once.