On the motion of the Oloid toy

On the motion of the Oloid toy

Alexander S. Kuleshov Mont Hubbard Dale L. Peterson Gilbert Gede [email protected]

Abstract Analysis and simulation are performed for the commercially available toy known as the Oloid. While rolling on the fixed horizontal plane the Oloid moves very swinging but smooth: it never falls over its edges. The trajectories of points of contact of the Oloid with the supporting plane are found analytically.

1 Introduction

Let us consider the motion of the Oloid on a fixed horizontal plane. The Oloid is a developable surface comprises of two circles of radius R whose planes of symmetry make a right angle between each other with the distance between the centers of the circles equals to their radius R. The resulting convex hull is called Oloid. The Oloid have been constructed for the first time by Paul Schatz [2,3]. The geometric properties of the surface of the Oloid have been discussed in the paper [1]. The Oloid is also used for technical applications. Special mixing-machines are constructed using such bodies [4]. In our paper we make the complete kinematical analysis of motion of this object on the horizontal plane. Further we briefly describe basic facts from Kinematics and Differential Geometry which we will use in our investigation. The Frenet - Serret formulas. Consider a particle which moves along a continuous differentiable curve in three - dimensional Euclidean Space 3. We can introduce the R following coordinate system: the origin of this system is in the moving particle, τ is the unit vector tangent to the curve, pointing in the direction of motion, ν is the derivative of τ with respect to the arc-length parameter of the curve, divided by its length and β is the cross product of τ and ν: β = [τ ν]. × Then the Frenet - Serret formulas for the derivatives of τ , ν and β are valid:

dτ /dt = ks˙ν, dν/dt = ks˙τ + æs ˙β, dβ/dt = æs ˙ν. − − The tangent τ , the normal ν and the binormal β unit vectors are known as the Frenet - Serret frame. The Poisson formulas. Let e1, e2, e3 is any moving coordinate system and let ω be the angular velocity of this system. Then for the derivatives of e1, e2 and e3 the following Poisson formulas are valid:

de /dt = [ω e ] , i = 1, 2, 3. i × i According to the Poisson formulas we can rewrite the Frenet - Serret formulas as follows: dτ /dt = [Ω τ ] , dν/dt = [Ω ν] , dβ/dt = [Ω β] , × × ×

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Figure 1: Kinematics of the moving lamina. where Ω = ks˙β + æs ˙τ is the angular velocity of the Frenet - Serret frame known also as the Darboux vector. In particular, for the plane curve we have æ = 0 and therefore

Ω = ks˙β. (1)

The law of composition of angular velocities. Consider three coordinate systems: the fixed system Oxyz and two moving systems Aξηζ and Cx1x2x3. Then the angular velocity of the system Cx1x2x3 with respect to Oxyz - system is the sum of the angular velocity of the Cx1x2x3 - system with respect to the Aξηζ - system and the angular velocity of the Aξηζ - system with respect to the Oxyz - system:

ωOxyz = ωOxyz + ωAξηζ . Cx1x2x3 Aξηζ Cx1x2x3

We will call the angular velocity ωAξηζ of the moving coordinate system Cx x x Cx1x2x3 1 2 3 with respect to another moving system Aξηζ as ”the relative angular velocity”. The Oxyz angular velocity ωAξηζ will called ”the transfer angular velocity” and the angular velocity ωOxyz will called ”the absolute angular velocity”. Cx1x2x3 Angular velocity of the moving lamina. Let us consider the following problem: the rigid lamina bounded by the convex curve of the given curvature k(s) is rolling without slipping on the fixed convex curve of the given curvature K(s) (we assume that the both curves have the same arc parametrization s = s(t)). Let us find the angular velocity Ω of the moving lamina. For this purpose we introduce three coordinate system: the fixed system Oxyz, con- nected with the fixed curve, the moving system Aξηζ rigidly connected with the moving lamina and the Frenet - Serret frame Cτνβ with the origin at the point of contact of two curves (Fig. 1). We will use the law of composition of angular velocities. The absolute angular velocity of the Frenet - Serret frame with respect to the fixed coordinate system Oxyz is Ks˙β (according to (1)). But this angular velocity is the vector sum of relative and transfer angular velocities. The relative angular velocity of the Frenet - Serret frame is its angular velocity with respect to the Aξηζ - system. This angular velocity is equal ks˙β (since in the moving frame Aξηζ the motion of the Frenet - Serret frame Cτνβ is the motion along the bound of the lamina). The transfer angular velocity of the Frenet - Serret frame is the angular velocity of the Aξηζ - system with respect to Oxyz - system, i.e. it is the angular velocity of the moving lamina Ω. Finally we have

Ω = (K k)s ˙β. − In particular when the lamina moves along the straight line, its angular velocity equals

Ω = ks˙β. −

276 On the motion of the Oloid toy

Two contact points. Let us consider the rolling of the body consisting of two similar symmetric laminas in perpendicular planes connected to each other along the common axis of symmetry.This body rolls without slipping on the fixed horizontal plane. Let G be the center of mass of the moving body and K1, K2 be two contact points of the body with the plane. From the rolling conditions for the two contact points follows that the angular velocity is always parallel to the line between the contact points i.e.

ω −−−→K K = 0. × 1 2 h i Now we introduce four coordinate systems. The first coordinate system is the fixed coordinate system Oxyz with the origin at any point O of the fixed plane. The Oz - axis is directed upward. The second coordinate system is Gx1x2x3 system with Gx2 axis directed along the common axis of symmetry of the two laminas. The Gx3 - axis is perpendicular to the plane of the first lamina and the Gx1 - axis is perpendicular to the plane of the second lamina. The third coordinate system is the Frenet - Serret frame K1τνβ for the bound of the first lamina: the τ - vector is the tangent vector to the bound of the lamina, the ν - vector is in the plane of the lamina and the β - vector coincides in the unit vector of Gx3 axis (i.e. β is perpendicular to the plane of lamina). The fourth coordinate system is also the Frenet - Serret frame. The origin of this system is also in K1, the first unit vector τ 1 of this system coincides with the vector τ of the system K1τνβ. The second vector ν1 of this system is in the plane of motion and the third vector β1 is perpendicular to the plane of motion (i.e. the vector β1 coincides with the vector ez of the system Oxyz). Therefore we can conclude that the coordinate system K1τ 1ν1beta1 is the Frenet - Serret frame for the curve which is obtained by the motion of the contact point K1 (i.e. it is a ”trace” of the point of contact on the plane). Let us try to find the angular velocity of the moving body. We will use the law of composition of angular velocities. Let us consider the systems Oxyz, K1τνβ and Gx1x2x3. The absolute angular velocity of Gx1x2x3 system with respect to Oxuz system is the sum of angular velocity of Gx1x2x3 with respect to K1τνβ and the angular velocity of K1τνβ with respect to Oxyz. But the angular velocity of Gx1x2x3 system with respect to K1τνβ system can be found very easy: it is equal ks˙β. So we need calculate now the angular − velocity of K1τνβ system with respect to Oxyz system. Let us denote by ϕ the angle between two unit vectors β = e3 and β1 = ez. Then we can represent the angular velocity of K1τνβ system with respect to Oxyz system is the sum of the angular velocity of K1τνβ system with respect to K1τ 1ν1β1 system and the angular velocity of K1τ 1ν1β1 system with respect to Oxyz system. But the angular velocity of K1τνβ with respect to K1τ 1ν1β1 system is equalϕ ˙τ and the angular velocity of K1τ 1ν1β1 system with respect to Oxyz system is equal Ks˙β1. Therefore finally we obtain: ω =ϕ ˙τ ks˙β + Ks˙β . − 1 Since β = ν sin ϕ + β cos ϕ, we have − 1 1 ω =ϕ ˙τ + ks˙ sin ϕν ks˙ cos ϕβ + Ks˙β . 1 1 − 1 1 But from the rolling conditions we already obtained that the vector ω is proportional to −−−→K1K2 vector i.e. ω is always in the plane of motion

ω = ω1τ 1 + ω2ν1.

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This means that K = k cos ϕ or ρ cos ϕ = r, (2) where ρ and r are radii of curvature of the curve on the fixed plane (trajectory of the point K1) and the bound of the first lamina. Natural equations of a curve. Let us consider the planar curve, defined by its parametric equations: r = r (s)= x(s)ex + y(s)ey, with the arc-length s as a parameter. We denote by α(s) the angle between the unit tangent vector τ to the given curve dr dx dy τ = = e + e ds ds x ds y and the unit vector ex of the Ox - axis. The initial value of α(s) at s = 0 can be chosen within the value divisible by 2π, for other points the angle α(s) is defined explicitly. Since τ (s) is the unit vector its projections on the Ox and Oy axes are cos α and sin α respectively. From the other side dx dy τ (s) = cos αe + sin αe = e + e , x y ds x ds y and therefore dx/ds = cos α, dy/ds = sin α. (3) Moreover, using the first Frenet – Serret formula we have dτ dα = ( sin αe + cos αe ) ν = k(s)ν, ds − x y ds and hence dα/ds = k(s). (4) This means that we can find the parametric equations of the curve if we know its curvature k(s). Thus we derived all the necessary facts for the investigation of the Oloid motion.

2 Motion of the Oloid

Coordinate frames and parametrization. We consider now the motion of Oloid. Let two circles of the same radius R in perpendicular planes be given such that each circle contains the center of the other. Then the convex hull of these circles is called Oloid. Let kA and kB be two circles of the same radius R in perpendicular planes Π1 and Π2 such that kA passes through the center MB of kB and kB passes through the center MA of kA. According to the previous theory let us introduce the moving coordinate system Gx1x2x3. The origin of this system will be at the midpoint G of MAMB (i.e. G is the center of mass of the system). The Gx3 - axis is perpendicular to the plane Π1 of the first circle, Gx1 - axis is perpendicular to the plane Π2 of the second circle and Gx2 axis is directed along the common axis of symmetry of two circles (Fig. 2). We will parametrize the first circle by the angle θ between the negative direction of Gx2 axis and the direction to the point of contact. Note that this parametrization is proportional to the arc-length parametrization s: s = Rθ. We introduce also the angle ψ for the parametrization of the second circle: let ψ be the angle between the positive

278 On the motion of the Oloid toy

Figure 2: The Oloid. direction of Gx2 axis and the direction to the point of contact B. Then the radius - vector of the point A can be written as follows: R −→GA = r = R sin θe + R cos θ e . 1 1 − 2 2   The radius-vector of the point B has the form: R −−→GB = r = + R cos ψ e + R sin ψe . 2 2 2 3   When Oloid rolling on a fixed plane the three vectors r r , r and r should situated 1 − 2 1′ 2′ in this plane. This condition can be written in the form:

cos ψ + cos θ cos ψ + cos θ = 0.

Thus we can express cos θ cos ψ = −1 + cos θ and obtain the radius – vector −−→GB = r2 in the θ – parametrization: R R cos θ R√1 + 2 cos θ −−→GB = r = e e . (4) 2 2 − 1 + cos θ 2 − 1 + cos θ 3   Note here the interesting fact: the length of the vector −−→AB

R cos2 θ R√1 + 2 cos θ −−→AB = −−→GB −→GA = R sin θe + R + e e − − 1 1 + cos θ 2 − 1 + cos θ 3   will be a constant AB = R√3. This feature is usually used in construction of Oloid. The expression (4) for the vector r2 should make a sense therefore we should have

O 1 + 2 cos θ. ≤ Therefore we have the following restrictions for the introduced parametrizations:

2π/3 θ 2π/3, 2π/3 ψ 2π/3. − ≤ ≤ − ≤ ≤ Trajectories of the points of contact. Let us obtain now equation for the plane of motion in the Gx1x2x3 coordinate system. Points A, B and the tangent vector to the first circle at A are always in this plane. Using this fact we can write equation for the fixed plane in the form:

2sin θX + 2 cos θY + 2√1 + 2 cos θZ + R (2 + cos θ) = 0. −

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The unit vector θ 1 θ √1 + 2 cos θ n = sin e + cos e + e . 2 1 θ 2 2 θ 3 − 2 cos 2 − ! 2 cos 2 is the normal vector to this plane. Therefore the angle between the plane of the first circle and the fixed plane is defined as follows:

√1 + 2 cos θ cos ϕ = (n e )= . · 3 θ 2 cos 2 The curvature of a circle at any point is equal to k = 1/R. Thus using (2) we can calculate the curvature of a curve drawn by the point of contact A on the fixed plane:

√1 + 2 cos θ K = k cos ϕ = θ . 2R cos 2 Having the expression for K we can find the parametric equations of the trajectory of the point A on the fixed plane. For this purpose let us introduce the fixed coordinate system Oxyz. The origin O of this system coincides with the point of contact of the first circle with the plane at θ = 0. The Ox - axis is tangent to the first circle, the Oz - axis is directed upwards. The Oy - axis forms the right triple with the Ox and Oz axes. Then we get:

dα dα dα √1 + 2 cos θ = = K (θ) , i.e. = . ds Rdθ dθ θ 2 cos 2 Integration of this equation gives us the following expression for α:

2 θ sin θ α = 2 arcsin sin arcsin 2 . √3 2 − √3 cos θ   2 ! Then θ 3 √3sin √3 (1 + 2 cos θ) 2 sin α = 2 (5 + 4 cos θ) , cos α = . θ 9 θ 9 cos 2 cos 2 Using these formulas together with (3), (4) we find after some trigonometric simplifi- cations

θ 2R√3 2 θ sin 2 θ xA (θ)= arcsin sin +arcsin +2sin √1+2 cos θ , 9 √3 2 √3 cos θ 2   2 ! ! 8R√3 θ 2R√3 θ 2π 2π y (θ)= sin2 ln cos , <θ< . A 9 2 − 9 2 − 3 3      These equations give the parametric representation for the trajectory of the point A on the fixed plane. We can use the similar ideas in finding the trajectory of point B. But we choose another way. Obviously we have the following vector relation:

−−→OB + −→OA + −−→AB = −→OA + ABeAB = −→OA + R√3eAB. We can represent this relation in the scalar form:

xB (θ)=xA (θ)+R√3 cos (α + γ) , yB (θ)=yA (θ)+R√3 sin (α + γ) .

280 On the motion of the Oloid toy

Here γ is the angle between the vector eAB and the tangent vector τ to the first circle at A. The tangent vector τ has the form τ = cos θe1 + sin θe2 and therefore its scalar product with −−→AB leads to:

θ −−→AB τ = AB (e τ )= AB cos γ = R√3 cos γ = R tan , · AB · 2   sin θ √1 + 2 cos θ cos γ = 2 , sin γ = . √ θ √ θ 3 cos 2 3 cos 2

Finally in the explicit form we have the following expressions for xB and yB:

θ θ 2R√3 2 θ sin 2 2sin 2 xB (θ)= arcsin sin +arcsin √1+2 cos θ , 9 √3 2 √3 cos θ − (1+cos θ)   2 ! !

7R√3 2R√3 2R√3 θ 2π 2π yB (θ)= + ln cos , <θ< . 9 9 cos2 θ − 9 2 − 3 3 2   

Figure 3: Trajectories of points A and B on the fixed plane Oxy.

These equations give the parametric representation for the trajectory of the point B on the fixed plane. Figure 3 shows the trajectory of point A (bottom curve) and B (upper curve) on the fixed plane Oxy.

References

[1] Dirnb¨ok H., Stachel H. (1997) The Development of the Oloid. J. Geometry Graphics 1:105-118.

[2] Schatz P. (1975) Rhythmusforschung und Technik. Verlag Freies Geistesleben, Stuttgart.

[3] Schatz P. (1933) Deutsches Reichspatent Nr. 589 452 in der allgemeinen Getriebeklasse.

[4] Bioengineering AG (Sagenrainsrt. 7, 8636 Wald, Switzerland).

281 Proceedings of XXXIX International Summer School–Conference APM 2011

Alexander S. Kuleshov, Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia Mont Hubbard, Department of Mechanical and Aerospace Engineering, University of California, Davis, USA Dale L. Peterson, Department of Mechanical and Aerospace Engineering, University of California, Davis, USA Gilbert Gede, Department of Mechanical and Aerospace Engineering, University of California, Davis, USA

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