The rolling of a disk on a horizontal plane and

International Journal of Mathematical Education in Science & Technology Vol. 00, No. 00, Month? 200?, 1–13

The rolling of a disk on a horizontal plane and Euler’s disk Grant Keady, University of Western Australia () The rolling of a disk (or hoop, etc.) on a horizontal plane – without dissipation – is a classical problem. Many second-year undergraduate students of the author’s generation, studied the stability of the motion in which the disk rolls steadily in a straight line. The problem is treated in the text book [1], the author used as an undergraduate 40 years ago. Current mathematical software, e.g. Matlab, makes work on the problem accessible to present-day second-year engineering mathematics students. This paper is partly a composite of assignment and exam questions used in 2006. There are research topics surrounding the problem. A catalyst for some recent work on the problem is the toy marketed under the name ‘Euler’s disk’.

1

Introduction

This paper is submitted as a case-study, suitable for teaching use and involving quite easy uses of Computer Algebra (CA). The extra item of interest is the value of CA, for the lecturer, in further – not for teaching – investigation of the problem. In particular, CA helped us with the observation that the system of differential equations (d.e.s) – denoted (D) in the following – is integrable. This proved to be a rediscovery: see [2]. Some of this material was developed for my second-year engineering mathematics notes where the following sequence of topics are amongst those treated: • • • •

determinants and eigenvalues; linear systems of constant coefficient d.e.s, exponential matrices; numerical solution of d.e. initial-value problems (using Matlab); classification of stability of equilibria for nonlinear systems.

The class uses Matlab, including its Symbolic Toolbox: a textbook is [3], a recommended reading book is [4]. The matlab m-files associated with the story are made available to the class. For readers of this journal, the easiest route through to the materials and research-related code is via the link at http://www.maths.uwa.edu.au/~keady/papers.html The rolling disk problem served to provide several problems for the assignment sheets, sometimes in the ‘Additional Problems’ section where some harder,

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not-directly-assessed problems are appropriate. I also devoted one lecture to the problem, very much emphasising the items listed above, and, actually not even mentioning material in §5. Up until now, with my engineering mathematics class I have treated only simple, perhaps excessively-simple, and, therefore, perhaps unimpressive, examples, both in connection with numerical solution of o.d.e.s and in connection with the classification of equilibria. In particular, the lead-in example is the nonlinear (rigid-bar) pendulum. This is used as an example of numerical solution and also as an example of classifying equilibria as stable or unstable. The second-year engineering students not only treat examples but also consider the general setting. In this the d.e. system is written dy = f (y) , dt and we define equilibrium points as vectors ye where f (ye ) = 0. We assess stability by linearizing in the usual way, setting up a problem for the remainder ρ where y = ye + ρ, which is dρ = Jρ where J = Df (ye ) , dt and J is the Jacobian of f evaluated at ye . One classifies the stability of an equilibrium, of course, by determining the signs of the real parts of the eigenvalues of J. If all the real parts are negative, the equilibrium is stable: if all the real parts are nonpositive, with at least one of the real parts zero, we will describe the equilibrium as neutrally stable. If any real part is positive, the equilibrium is unstable. The history of how I came to choose to study the rolling disk problem is as follows. In August 2005, Prof Keith Moffatt stopped at the University of Western Australia en route to an IUTAM Committee meeting in Adelaide. His colloquium was entitled “The paradoxical behaviour of spinning bodies” and amongst other things treated • the tippe-top problem, and • the final stages of motion of the Euler disk: see §6. During a ‘Workshop’ series several of us participated in prior to his visit, we studied these problems. The standard Jacobian approach to study stability was one of the tools used. My efforts on the rolling disk problem were partly because I knew all along that parts of it would be simple enough to develop into an example for my second-year notes. Although I would not expect modernday students to be able to do the hand calculations that those of my generation could do, the CA Systems (CAS)

The rolling of a disk

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• give them a tool whereby they can bypass the deficiencies in their skills in hand-calculation, and • enable related numerics and graphics to be completed easily and accurately. Returning to the particular problem of the rolling disk, of course, one thing led to another. • The numerical solution of the initial value problem (which I also had done for the tippe-top) is easy with Matlab (or Maple or Mathematica). I was surprised to notice that there is an exactly-solvable second-order linear system of d.e.s lurking in the full fourth-order nonlinear system describing the rolling disk. (The system is given in equations (D0 ), (D1 ), (D2 ), (D3 ) below.) The ready availability of CAS, i.e. the ease with which it is possible to ask any CAS to dsolve is the key factor in all this. This is described in §5. It proved to be a rediscovery, with the particular form we noted having been noted in 1903-04: see [2]; see also [5]. There are deeper uses (e.g. to establish that the system is integrable: see §5), but the use made here is as a check on the numerical solutions of §4. • The commercial Euler disk is sold with a convex base. The problem of a disk rolling inside a sphere has also been studied: for references, see [6]. The equilibria, and their stability, can be studied in a similar way to rolling on a flat base: see [6]. However, this system of d.e.s is 6th order, and may well not be integrable. The variety of behaviour it exhibits may well be richer than the case of rolling on the plane, but little seems to be known of this.

2 2.1

The rolling of a disk on a horizontal table Motion of a rolling disk: 4th order system

The first examples concerning the stability of equilibria should be of low order, no higher than 2nd. The nonlinear pendulum is a 2nd-order system: the logistic equation is a 1st-order d.e. It seems educationally worthwhile to give one example which is larger than this, partly to show that Matlab can help with engineering sums, or, more accurately for this example, ‘simple’ mechanics sums. We won’t set up the d.e.s here, but take them from mechanics texts: see p402 of [1] for example. We denote the radius of the disk by a, and the acceleration due to gravity by g. Following [1], let θ be the inclination of the plane of the disk to the vertical. Also following [1], we introduce a rectangular coordinate system moving with the disk and with its origin at the centre of the disk. The first coordinate axes, and associated direction i, is in the direction of the line joining the the centre of the disk to the point of contact. k is normal to the plane of the disk. The

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vector j so that ijk is an orthogonal triad is therefore in the plane of the disk and horizontal. The angular velocity of the disk is ω = ω1 i + ω2 j + ω3 k. From the equations in [1], we obtain 3 scalar equations each linear in each of (the angular velocities) ω˙ 1 , ω˙ 2 and ω˙ 3 . Solving for ω˙ 1 , ω˙ 2 and ω˙ 3 , we have the following set of four equations for the four unknowns θ, ω1 , ω2 , ω3 . In these, γ = 2/3 for a solid disk and γ = 1/2 for a hoop. θ˙ = −ω2

(2.0)

and ω˙ 1 = −ω2 (tan(θ)ω1 + 2ω3 ) ω˙ 2 =

(1 −

γ) tan(θ)ω12

+ 2ω1 ω3 − 2γ(g/a) sin(θ) (1 + γ)

ω˙ 3 = −γω2 ω1

(2.1) (2.2) (2.3)

We denote by (D) the system of four d.e.s (2.0), (2.1), (2.2), (2.3).

Conservation of energy. The system is autonomous. Invariance of equations under translations in time is associated with the conservation of energy. Hence dE/dt = 0 where E is defined by 2.1.1

g (1 − γ)ω12 + (1 + γ)ω22 + 2ω32 = 4γE − 4γ cos(θ) a

2.1.2 The position of the centroid. System (D) can be solved numerically. The motion of the centroid (xO , yO , zO ) can also be obtained by integrating a larger system of d.e.s:

ω1 φ˙ = − cos(θ)

(2.4)

x˙O = aω3 sin(φ) + aω2 cos(φ) cos(θ),

(2.5)

y˙O = −aω3 cos(φ) + aω2 sin(φ) cos(θ).

(2.6)

(zO = a cos(θ) does not require any further integration.)


3 3.1

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The equilibria and their stability The equilibria in general

It is straightforward to solve (with or without Matlab’s solve) the equations obtained by putting θ˙ = 0, (so ω2 = 0, ω˙ 1 = 0, ω˙ 3 = 0). We have found it convenient, e.g. in [6], to split these into cases: • rolling in a straight line • spinning about a vertical diameter • the general case, rolling in a circle, and the sub-case where the centroid remains fixed. We begin, though, with the most trivial of the equilibria 0 = ω1 = ω2 = ω3 with θ = 0: this equilibrium is unstable. We note in passing that 0 = ω1 = ω2 = ω3 with θ = π is also an equilibrium of system (D), and it is a stable. Clearly, if we are really intending to model the rolling above a table we need to add other constraints, for example that the reaction at the point of contact should point upward. The issue arises also in [6] concerning rolling inside a sphere.

3.2

The Jacobian, in general

Setting ω2 = 0 in the expression for the Jacobian J, required for the analysis of the stability of these equilibria, leads to matrices (which obviously have their trace 0 and determinant 0) of the form 

Jgen

0  0 =  j31 0

0 0 j32 0

 −1 0 j23 0  . 0 j34  j43 0

This is rank 2. To solve the d.e.s ρ˙ = Jgen ρ one begins with finding the characteristic polynomial pJgen (λ) = λ2 (λ2 − j34 j43 − j32 j23 + j31 ). With a similar question having been set on an assignment, we set, as the first question on the 2006 exam paper: 2006 Exam, Q1. A matrix J (which arose in the ‘rolling disk’ application in lectures) is specified and studied in the following matlab transcript, the last line being the output of its characteristic polynomial.

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syms j31 j23 j32 j34 j43 x J= sym([0,0,-1,0;0,0,j23,0;j31,j32,0,j34;0,0,j43,0]) charPolJ=factor(poly(J,x)) charPolJ = x^2*(x^2-j34*j43-j32*j23+j31) (a) Given that at least one of j32 or j34 is nonzero, (i) what is the rank of J? (ii) What is det(J)? (b) What is the condition on j31 which ensures that there is a positive eigenvalue of J? (c) If J is such that it has a nonzero eigenvalue, is it diagonalizable?

Not only can the eigenvalues be found explicitly, but there is a simple formula for the exponential matrix. Deriving this is assisted by the observation that 3 Jgen = λ2 Jgen . The following is a (largely unseen) exam question based on this: 2006 Exam, Q2. (a) Consider any matrix J for which J 3 = λ2 J for some nonzero λ. Verify that the exponential matrix of Jt is given by expm(tJ) = (I −

J2 cosh(λt) sinh(λt) ) + J2 +J . λ2 λ2 λ

(b) Let u be an eigenvector of matrix A with Au = λu. (i) Show that u is also an eigenvector of expm(At) and find its corresponding eigenvalue. (ii) Given that, for any matrix, • the sum of the eigenvalues equals the trace, • the product of the eigenvalues equals the determinant, show that det(expm(At)) = exp(trace(A)t) (iii) For the matrix J of Q1, what is det(expm(Jt))? (c) Let J0 be the matrix of Q1 with the further restriction that, while it remains rank 2, the only eigenvalue is 0. Is matrix J0 diagonalizable? Given that J03 = 0, use the exponential series to write down a formula for expm(J0 t).

In any event, the solution of the linearized equations associated with perturbing an equilibrium solution is easy, after the nonzero eigenvalues are obtained. Although caution is needed, some aspects of the solution can be represented by the trajectories in the (θ, ω2 )-plane. We will see, in this plane, either ellipses or hyperbolae according to whether λ2 < 0 or λ2 > 0. We also have det(expm(Jt) = exp(trace(Jt)) = 1 .


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The equilibria with θ = 0. The easiest solution has θ = 0 and ω1 = 0 = ω2 and ω3 a constant, and this corresponds to the hoop/disk rolling in a straight line. Perturbing about this, the remainders ρ satisfy ρ˙ = Jρ with   J = −g

0 0

2γ a 1+γ

0

0 −1 0 −2ω3 2ω3 0 1+γ 0 0

 0 0  0 0

The eigenvalues are 0 (with algebraic and geometric multiplicity 2) and ±λstLine where

λ2stLine

2 2( gγ a − 2ω3 ) . = 1+γ

We set this as an assignment question, following from an earlier calculation of the eigenvalues of the Jacobian: Assignment Question: Consider the d.e.s describing the rolling of a disk on a horizontal plane. (a) Verify that θ = ω2 = ω1 = 0 and ω3 =constant is an equilibrium solution of these d.e.s. (b) Use the Matlab code given at the links for this assignment to calculate the Jacobian at this equilibrium. Verify that the form of the Jacobian is that given in the previous assignment. (c) Show that the vertically-upright straight-line equilibrium motion of part (a) is neutrally stable only if the angular-velocity ω3 satisfies ω32 > gγ/(2a).

(For readers of this paper, the matlab code and related materials are available from the link at the ‘papers’ page , as specified in the Introduction.) The other, ‘spinning’, equilibrium, in which θ = 0 and ω3 = 0 = ω2 can be analyzed similarly.

3.2.1 The modelling of the ‘table’ is inadequate for some unstable solutions. The nonlinear pendulum

Briefly, if we let the initial rolling speed go to zero, and have an initially nearly vertical hoop, it will ‘fall over’. Setting ω1 = 0 and ω3 = 0 in the equations gives the nonlinear pendulum d.e. for (θ, ω2 ). For more details, see [6].

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Numerical solution

Some Matlab code to generate numerical solutions was provided to the students. This code is available to readers of this paper via the URL given in the introduction. The lead-in example, omitted here (but see [6]) concerns really tiny oscillations. The matlab code also plots out the trajectory in the (θ, ω2 )-plane: it is very close to a small, thin, ellipse. These oscillations are so small that the linear stability analysis, the eigenvalues of J, can be used to check the frequency. An example with a much larger perturbation gives the path of the centre of the disk shown in Figure 1.

Figure 1. A more interesting trajectory. The upper plot is the trajectory in the (θ, ω2 )-plane. The lower plot is the path of the centre (xO (t), yO (t)).


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It is also easy to produce 3-dimensional plots of the trajectory of the centre of the disk. The Matlab plots allow one to interactively move the viewpoint. We leave it to readers to run the code if they wish to see the 3-dimensional plots. There are numerous other examples. Those showing (θ, ω2 )-planes include [7] Figs 3,4 pp296-297.

5

Legendre functions and the rolling disk

On dividing each of the equations, for ω˙ 1 , and for ω˙ 3 , by θ˙ = ω2 , a homogeneous second-order linear system is obtained for functions ω1 (θ) and ω3 (θ). Matlab’s Symbolic Toolbox (or Maple - which is the package in the Symbolic Toolbox) solves this in terms of conical Legendre functions P 1 +iλ (sin(θ)). 2 There are many equivalent ways to present this. For example V = ω1 cos(θ)/ω3 considered as a function of s = sin(θ) satisfies the Riccati equation dV C γV 2 γV 2 = − =2− . 2 ds A 1−s 1 − s2 Again, Matlab’s Symbolic Toolbox (or Maple) solves this in terms of conical Legendre functions. After noting this, and deciding that it was worth pursuing, we continued to study the problem, and its literature, eventually reaching the reference [7], which uses the conical Legendre functions. The history is treated in [7]. Several authors, independently, noted it – using hypergeometric functions – in the nineteenth century: see, for example, [5]. The Legendre function form is in a 1904 paper [2]. The problem is old, and the observation not all that deep, so we were not surprised that it proved to be a rediscovery. The readily available code for calculating the conical Legendre functions numerically proved to be rather slow. Hence we chose for most of our checks on the numerics to proceed as follows. The check consists of looking at V against sin(θ), where we have this from two different approaches: (i) from the numerical solution of the 4th-order system (D); (ii) from the Legendre d.e., and the Riccati form above. A more important use of the observation it to conclude that the system is integrable, and to find some implications for its global behaviour. The observation that, besides the energy E, there are two other conserved quantities – associated with the conical Legendre solutions – has many implications. [8] state: “ ... for the rolling disk Vierkandt ( [5], 1892) showed something very

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interesting: On an appropriate symmetry-reduced space, ... all orbits of the system are periodic.” Perhaps this means that all solutions of the system (D) are periodic. A similar statement occurs in the first paragraph of [7], where the ‘quasi-periodic’ motions mentioned there possibly refer to the 7-th order system including φ, xO and yO as well as the other variables.

6

The Euler disk: Introduction and flat table

Euler’s disk, named after Leonhard Euler, is a circular disk that spins, without slipping, on a surface. A common example is a coin spinning on a table. A spinning disk ultimately comes to rest; and it does so quite abruptly, the final stage of motion being accompanied by a whirring sound of rapidly increasing frequency. There is a commercial ‘Euler disk’, a toy metal disk, about 3.75cm in radius, and about 400gm in weight. Dissipation is essential for the explanation of the phenomena involved, and there are possibly unresolved puzzles. For references see [6] and http://www.eulersdisk.com/pubs.html The equations we have in §2.1 are for non-dissipative rolling. We begin by ˙ In this section we are concerned with θ near recalling that ω1 = − cos(θ)φ. π/2, θ = π/2 − α, so ω1 ∼ −αφ˙ in the final stages of an Euler-disk motion, α is small, φ˙ is large, while ω1 is small. We first consider the simplest non-dissipative approximation possibly relevant to the Euler disk. In the simplest model, as the disk rolls, • the point P of rolling contact describes a circle with a constant angular velocity φ˙ in our notation; • the centroid is fixed; • the motion would persist forever. Stated in terms of the variables in the earlier part of this article, we have ˙ and the equation that says ω˙ 2 = 0 gives ω2 = 0 = ω3 , ω1 = − cos(θ)φ, ω12 =

g 2γ cos(θ) a1−γ

g 2γ or φ˙ 2 = a (1 − γ) cos(θ)

(6.0)

Now when θ is near π/2, θ = π/2 − α with α small, and we are dealing with a disk, we have 4g φ˙ 2 ∼ aα as in [9], equation (1).

(6.1)


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Approximate models giving a first impression of what might be happenning have the centroid slowly falling vertically and energy slowly being dissipated. The expression for the energy, in the ω2 = 0 = ω3 equilibrium case, for a disk, is 3 3 E = mga cos(θ) ∼ mgaα, 2 2

(6.2)

as in [9], equation (2). An analysis supposing that energy is slowly dissipated is given in [9] and it leads to the ‘finite stopping time’. I’m uncertain if there is more to be gained from the study without dissipation, but will look at the non-dissipative rolling disk more anyway! For example, it is easy to calculate the natural frequency for the very small perturbations on the steady rolling above, and q it has the same asymptotic depeng ˙ with Const a numeric constant. dence on α as φ, namely varying as Const aα Without dissipation, periodic perturbations of the steady rolling also continue. Let’s describe these periodic perturbations as ‘rocking’ (as one can imagine them as rocking about a diameter).

6.1

Motion with θ near π/2

[10] p154 notes that one can do much more than merely a linear stability analysis, and that there are significant simplifications to the d.e.s describing the nonlinear motions, when we take θ = π/2 − α near π/2. We approximate cos(θ) ∼ α and sin(θ) ∼ 1. The system of 4 equations becomes α˙ = ω2 ω˙ 1 = −ω2

ω

1

+ 2δω3

α 2 2γ g (1 − γ) ω12 + ω1 ω3 − ω˙ 2 = (1 + γ) α (1 + γ) (1 + γ) a ω˙ 3 = −γω2 ω1 where, here, δ = 1, and the parameter is introduced because of a forthcoming approximation. Once again we can divide the second and fourth equations by the first to obtain a linear system: ω dω1 1 =− + 2δω3 dα α

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dω3 = −γω1 dα This is recognized (by Matlab or Maple) as Bessel’s equation. For further details, including those of the further restriction (effectively δ = 0), realistic for Euler’s disk, in [10], see [6]. There is a closely associated autonomous system of d.e.s, presented in a model form, in an (unseen) exam question for the students: 2006 Exam, Q5. The following autonomous system of d.e.s arises in connection with certain sorts of non-dissipative motions of a rolling disk: α˙ = f1 = ω ω˙ = f2 =

(1 − α)(1 + α + 2α2 ) α3

The function f2 is a decreasing function of α on α > 0. (a) Find the equilibrium solution or solutions. (b) Using Matlab one finds that the jacobian of f = [f1 , f2 ]T with respect to y = [α, ω] is 0 1 2 J = Df = − 3+α α4 0 Use this to classify any equilibrium point as stable, neutrally stable or unstable. (c) In Figure 2 is given the ‘phase-portrait’ of this system. (i) Mark the equilibrium point with a *. (ii) Sketch, with attention to the main qualitative features, but without too much concern about detailed numerics, the general shape of the trajectory of the solution of of the initial-value problem where the solution starts with α(0) = y1 (0) = 0.8 and ω(0) = y2 (0) = 0.2.

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Conclusion

Realistic problems become accessible to students with limited mathematical manipulation skills, or limited stamina, on using mathematical software on appropriate problems. Furthermore, real problems often have a research element for the lecturer to investigate, if the lecturer so desires.

References [1] [2] [3]

Synge, J.L. and Griffith, B.A. Principles of Mechanics. (McGraw-Hill 3rd ed: 1959). Gallop, E.A. On the rise of a spinning top. Trans. Cambridge Philos. Soc. 19 (1904) 356-373. Strang, G. Linear algebra and its applications. (Brooks/Cole, 3rd ed: 1988; 4th ed: 2006).


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[4]

Harman, T.L., Dabney, J. and Richert, N. Advanced Engineering Mathematics Using Matlab (PWS 2nd edition: 1999). [5] Vierkandt, A. Uber gleitende und rollende Bewungen, Monatshefte der Math. und Phys, 3, 31-54, 1892. [6] Keady, G. The rolling of a disk on a horizontal surface and inside a sphere. School of Mathematics and Statistics, University of Western Australia, Research Report (also available electronically). 2007, in preparation. [7] O’Reilly, O.M. The dynamics of rolling disks and sliding disks. Nonlinear Dynamics 10 (1996) 287-305. [8] Bloch, A.M., Marsden, J.E. and Zenkov, D.V. Nonholonomic dynamics. Notices Amer. Math. Soc. 52 (2005), no. 3, 324–333. [9] Moffatt, H.K. Euler’s disk and its finite-time singularity. Nature 404 (2000) 833. [10] Thomson, W.T. Introduction to Space Dynamics. (Dover: 1986 reprinting of Wiley 1963).

Figure 2. You are asked in Q5(c) to show, on this phase plot, (i) any equilibrium solution, (ii) a solution to an initial value problem.