The influence of the rolling resistance model on tippe top inversion A. A. Kilin, E. N. Pivovarova Abstract In this paper, we analyze the effect which the choice of a fric- tion model has on tippe top inversion in the case where the resulting action of all dissipative forces is described not only by the force applied at the contact point, but also by the additional rolling resis- tance torque. We show that, depending on the friction model used, the system admits different first integrals. In particular, we give ex- amples of friction models where the Jellett integral, the Lagrange integral or the area integral is preserved. We examine in detail the case where the action of all dissipative forces reduces to the horizontal rolling resistance torque. For this case we find permanent rotations of the system and analyze their linear stability. Also, we show that for this friction model no inver- sion is observed. arXiv:2002.06335v2 [math.DS] 20 Feb 2020 Introduction In this paper we analyze the dynamics of the tippe top on a smooth plane in the presence of forces and torques of rolling resistance. Tippe top inver- sion has attracted the attention of researchers for the last decades [1–12]. Reference [1] discusses the development of a spherical prototype of the top which a capable of performing various modes of motion (in particular, complete or partial inversion) by changing the mass-geometric characteris- tics. Many studies from the last century [2–7] and from the early part of this century [8–12] gave mathematical explanations of tippe top inversion. They investigated the stability of steady-state (dissipation-free) solutions, analyzed parameter values, and found cases where tippe top inversion is possible. A detailed analysis of the dynamics of an axisymmetric top is possible since the system has the Jellett integral. In his “Treatise on the Theory 1 of Friction” [13] Jellett pointed out that some quantity (named later after him) in the problem of the motion of a body of revolution on a plane remains unchanged when adding an arbitrary friction force applied at the point of contact. There exist various generalizations and analogs of the Jellett integral. As an example, we mention the integral found in [21] for a rigid spherical shell with internal body. Most studies of tippe top dynamics use the classical sliding friction model, which is proportional to the velocity of the point of contact of the tippe top with the plane. However, various friction models have been proposed recently for a more accurate analysis which provides not only a qualitative, but also a quantitative description of the system dynamics. For example, a comparative analysis of the most frequently used friction models is made in Ref. [18]. The authors of Ref. [20] present a model of viscous rolling friction which gives a fairly accurate description of the rolling motion of spherical bodies on a horizontal plane. Also, this friction model explains qualitatively some dynamical effects, in particular, retrograde motion of a rolling disk at the final stage [19]. In this paper we address the problem of the influence of the friction model on tippe top inversion. In particular, we examine the situation where the resultant action of all dissipative forces is described not only by the force applied at the point of contact, but also by an additional rolling resistance torque. It turns out that, depending on the chosen friction model, the system admits different first integrals: the Jellett integral, the Lagrange integral or the area integral. In this work we also carry out a qualitative analysis of tippe top dy- namics in the case where the action of dissipative forces reduces to the horizontal rolling resistance torque. We show that, in this friction model, no tippe top inversion is observed. 2 1 Equations of motion and conservation laws 1.1 Formulation of the problem Consider the motion of a heavy unbalanced ball of radius R and mass m with axisymmetric mass distribution which rolls with slipping on a hori- zontal plane under the action of gravity (Fig. 1). The system is acted upon by different resistance forces, which depend on the type of coating of the contacting surfaces, air resistance etc. As is well known, this system of forces generally reduces to the resultant of resistance forces, F , and to the resistance torque M f . We assume that in this case the motion of the ball is subject to the following assumptions: – the ball contacts the surface at one point P ; – the resultant of resistance forces, F , is applied to the point of contact; – the ball is acted upon by the principal rolling resistance torque M f , which includes torque r F , but may not be equal to it in the general case. × Figure 1 To describe the motion of the ball, we introduce two coordinate systems: – a fixed (inertial) coordinate system Oxyz with origin on the support- ing plane and with the axis Oz directed vertically upwards. – a moving coordinate system ox1x2x3 attached to the ball, with origin at the center of mass of the system and with the axis ox3 directed along the symmetry axis of the ball. In what follows, unless otherwise specified, all vectors will be referred to the moving axes ox1x2x3. 3 We assume that the center of mass of the system is displaced relative to the geometric center of the ball along its symmetry axis by distance a and is given by the vector a = (0, 0,a). Let us denote the projections of the unit vectors directed along the fixed axes Oxyz onto the axes of the moving coordinate system ox1x2x3 as follows: α = (α1, α2, α3), β = (β1,β2,β3), γ = (γ1,γ2,γ3). The orthogonal matrix Q SO(3) whose columns are the coordinates of the vectors α, β, γ specifies∈ the orientation of the body in space. Remark. By definition, the vectors α, β and γ satisfy the relations (α, α)=1, (β, β)=1, (γ, γ)=1, (α, β)=0, (β, γ)=0, (γ, α)=0. Let v be the velocity of the center of mass of the ball, and let ω be its angular velocity, both defined in the coordinate system ox1x2x3. Then the evolution of the orientation and of the position of the ball is given by the kinematic relations γ˙ = γ ω, α˙ = α ω, β˙ = β ω, × × × (1) x˙ = (v, α), y˙ = (v, β), where x and y are the coordinates of the center of mass o of the ball in the fixed coordinate system Oxyz. The coordinate z of the center of mass is uniquely defined from the condition that the ball move without loss of contact with the plane z + (r, γ)=0, (2) where the radius vector of the contact point r in the axes ox1x2x3 can be represented in the form r = Rγ a. (3) − − 4 1.2 Equations of motion Differentiating Eq. (2) taking (3) and the relationz ˙ = (v, γ) into account, we obtain a (holonomic) constraint equation in the form f = (v + ω r, γ) = (vp, γ)=0, (4) × where vp is the velocity of the ball at the point of contact with the plane. We write the Newton – Euler equations for changing the linear and an- gular momenta of the ball in the form ∂f mv˙ + mω v = N + F mgγ, × ∂v − (5) ∂f Iω˙ + ω Iω = N + M f , × ∂ω where I = diag(i1,i1,i3) is the central tensor of inertia of the ball, g is the free-fall acceleration, F is the friction force applied at the point of contact, ∂f ∂f M f is the rolling resistance torque, N ∂v and N ∂ω are the force and the torque of the reaction of the supporting plane, respectively. Let us express N from the first equation of (5) and from the derivative df of the constraint equation (4) with respect to time =0 dt · N = m(v, γ) (F , γ)+ mg. (6) − If the surfaces of the ball and the plane are homogeneous (but not necessarily isotropic), then the force F and the torque M f depend only on the variables (v, ω, γ). In this case, the system of equations describing the change of variables (v, ω, γ) closes in itself and takes the form · mv˙ + ω mv + m(ω, r γ) γ = F k, × × · Jω˙ + ω Iω + m((ω, (r γ) ) g)r γ = M f r F , (7) × × − × − × ⊥ γ˙ + ω γ =0, × where J = I + m(r γ) (r γ)1), F = F (F , γ)γ is the horizontal × ⊗ × k − component of force F , and F ⊥ = (F , γ)γ is its vertical component. In addition, these equations must be restricted to the submanifold given by the constraint equation (4) and the geometric relation (v + ω r, γ)=0, γ2 =1. (8) × 1)Symbol ⊗ denotes tensor multiplication of vectors, which associates the matrix A with components Ai,j = aibj to the pair of vectors a, b. 5 Since these functions are first integrals of the system (7), this restriction is satisfied uniquely. Remark. In principle, one can consider the trajectories of the system (7) on other level sets of the integrals (8), but they have no explicit physical interpretation. Thus, Eqs. (1) and (7) completely describe the motion of the tippe top on a smooth plane with friction. 1.3 Laws of resistance and cases of existence of addi- tional integrals In the case of rolling on an absolutely smooth plane (F = 0, M f = 0) the system (7) admits, in addition to the integrals (8), three more integrals of motion: – the energy integral 1 1 = (ω, Iω)+ m(v, v)+ mgaγ , (9) E 2 2 3 – the Lagrange integral F = i3ω3, (10) – the Jellett integral G = (Jω, r).