rotating disks

Rotating bouncing disks, tossing pizza dough, and the behavior of ultrasonic motors

Kuang-Chen Liu, James Friend, and Leslie YeoMicroNanophysics Research Laboratory, Department of Mechanical Engineering, Monash University, Victoria 3800, Australia

Received 4 May 2009; revised manuscript received 1 August 2009; published 1 October 2009

Pizza tossing and certain forms of standing-wave ultrasonic motors SWUMs share a similar process forconverting reciprocating input into continuous rotary motion. We show that the key features of this motionconversion process such as collision, separation and friction coupling are captured by the dynamics of a diskbouncing on a vibrating platform. The model shows that the linear or helical hand motions commonly used bypizza chefs and dough-toss performers for single tosses maximize energy efficiency and the doughs airbornerotational speed; on the other hand, the semielliptical hand motions used for multiple tosses make it easier tomaintain dough rotation at the maximum speed. The systems bifurcation diagram and basins of attraction alsoprovide a physical basis for understanding the peculiar behavior of SWUMs and provide a means to designthem. The model is able to explain the apparently chaotic oscillations that occur in SWUMs and predict theobserved trends in steady-state speed and stall torque as preload is increased.

DOI: 10.1103/PhysRevE.80.046201 PACS numbers: 05.45.a, 45.20.dc, 46.40.f, 47.20.Ky

I. INTRODUCTION

Like many nonlinear dynamical systems, the seeminglysimple equations that govern a ball bouncing on a vibratingplatform under a constant gravitational field describe verycomplex behavior. A variety of interesting phenomena ac-company the bouncing ball system, including noise-sensitivehysteresis loops 1, period doubling route to chaos, andeventually periodic orbits known as the sticking solution,or complete chattering 2,3. Due to its physical simplicityand rich nonlinear behavior, the system has found a varietyof applications, from the dynamics of two-dimensionalgranular gases 4, high energy ball milling 5, to its use asa pedagogical demonstration of chaos 6.

Regardless of the various complex bouncing dynamicsthat it displays, the response of the traditional bouncing ballsystem is ultimately an oscillatory one: the mean verticalvelocity is zero, and, as long as the collision is not perfectlyelastic, the displacement of the ball is bounded. However,through a simple extension of the system by adding a rota-tional degree of freedom and an angular component to theplatform vibration, a different set of phenomena is made pos-sible: stick-slip rotation while the disk and platform are incontact, impulsive frictional torque imparted at each colli-sion, and the potential for nonzero mean angular velocity andunbounded rotary motion.

This process of motion transferthe conversion of recip-rocating motion into continuous rotary motion in the modi-fied systemis intimately related to pizza tossing and thestator-rotor interaction in a class of standing-wave ultrasonicmotors SWUMs 79. As illustrated in Fig. 1, the systemmay be represented as a disk or a pizza dough with angular and linear x displacements, bouncing on an oscillating plat-form or a pair of dough-tossing hands with a combined an-gular and linear oscillation s described by

t = sint + and st = A sint . 1

Figure 2 shows a family of potential trajectories traced by adough-tossing hand or a SWUM stator as the amplitude ratio

L=ae /A=tan L and phase lag are varied ae is the ef-fective friction contact radius.

Of the many different hand trajectories that may be usedto toss a pizza, we observe that distinct hand motions areused in two different dough-tossing modes. In the first mode10, the dough begins each cycle at rest relative to the hand:the dough is launched, caught upon its descent, allowed tocome to rest, and the process is repeated. Since each tossbegins with the same initial conditions, the process can beconsidered as a chain of single tosses. In this case, we ob-serve that a linear trajectory resembling Fig. 2c is em-ployed. In the second mode 11, the dough is not allowed tocome to rest after each collision, and thus the rotation of thedough is maintained over multiple tosses. In this case, thetossing motion traces a semielliptical trajectory resemblingFig. 2b.

One of our key goals in this paper is to investigate whythe particular hand motions are adopted by dough-tossingperformers for the two dough-tossing modes. Do these hand

x

s

sx

platform/hand trajectory

bouncing disk/pizza dough

datum

FIG. 1. Color online The displacement variables of thebouncing-disk system and pizza tossing. The trajectories traced by apoint on the oscillating platform and a dough-tossing hand are givenby Eqs. 1, which describe a family of closed curves on the surfaceof a cylinder.

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motions provide any advantages in terms of the effort re-quired, the rotary speed reached by the dough, and ease ofhandling? Noting that the operation of SWUMs can be seenas a continuous sequence of multiple pizza tosses, answers tothe above questions about pizza tossingoriginally con-ceived as merely a pedagogical toolwill also help us betterunderstand the underlying motion transfer process inSWUMs: the generation of continuous rotation from an os-cillatory input.

A number of attempts have been made to model SWUMsover the years 7,12,13, however, these models have so farassumed that the vertical motion of the rotor is negligible.This assumption may be valid when the rotor preload domi-nates over the inertial force of the stator acceleration and therotor operates in the nonbouncing regime. More precisely, interms of the dimensionless forcing parameter =A2 /g,where g is the rotor acceleration due to preload, the rotor willremain in contact with the stator when 1 /. However,most motors operate with 1 /. For example, our SWUM8 operating at 70 kHz with the rotor under gravitationalacceleration has a lower estimate of 6 if we assume astator vibration of merely 1 nm; Tsujinos SWUM 14 witha stator vibration of 5 m at 55 kHz and a rotor under apreload acceleration of 9103 m /s2 has 70. As will beshown, the bouncing-disk model reproduces key qualitativefeatures of the motors dynamics, including the seeminglychaotic oscillations of the rotors transient speed curves andthe effect of preload on steady-state speed and stall torque.

The structure of the current paper is as follows: in Sec. IIwe describe details of our bouncing-disk model; in Sec. IIIwe use our model to investigate the optimal hand motion forsingle tosses; in Sec. IV we investigate the best way to main-tain dough rotation over multiple tosses and its implicationfor SWUMs by considering how various bouncing-disk or-bits e.g., periodic, chaotic, or chattering and their basins ofattraction affect the motion transfer process; and in Sec. V,we compare our models predictions with results from a pro-totype motor, showing that the bouncing-disk model is ableto account for important SWUM characteristics that couldnot be explained by existing models.

Note that we neglect dough deformation in this paper forthe following reasons: 1 our focus is on pizza tossing as amethod for imparting rotary motion rather than dough shap-ing, 2 the typical rate of dough deformation is low 3%increase in diameter per toss, see Appendix, and 3 pizzatossing is used as a pedagogical tool for understanding therotor dynamics in SWUMs. Dough plasticity only has a mi-nor effect on the dynamics and will not be considered further

here. Although we assume that the rotor of SWUM is underconstant preload, our model would still be valid for sprungrotors if the spring with stiffness k operates purely in com-pression and the springs mean displacement l0 is muchgreater than the rotors peak-to-peak displacement ampli-tude. Under such conditions, the spring force is effectivelyconstant and the rotors acceleration due to the spring pre-load is given by g=kl0 /m.

In a previous publication 15, we have briefly reportedthe application of the bouncing-disk model to pizza tossing,and its potential implications for SWUMs. In this paper, weprovide the full details for the modeling of single and mul-tiple pizza tosses, and we make an in-depth comparison be-tween our theoretical predictions and the experimental re-sults of SWUMs.

II. THE BOUNCING-DISK MODEL

A. Equations of motion

As an extension of the traditional bouncing ball problem,the vertical component of the bouncing-disk system has thesame assumptions and governing equations as previous work3: collisions have zero duration and the coefficient of res-titution is independent of the impact velocity. For the rotarycomponent, we assume that the contact pressure is uniformand that the frictional torque can be modeled using Coulombfriction with a constant coefficient of friction acting at aneffective contact radius ae we assume that the kinetic andstatic coefficients of friction are equal. The resulting gov-erning equations for the motion of the disk are

mag2 = Tf +

n=1

H nt tn , 2a

and

mx = mg + N + n=1

F nt tn , 2b

where N is the normal contact force, Tf is the frictionaltorque, F n and H n are the linear and angular impulse at timetn from the disks nth collision with the platform, is theDirac delta function, m is the disks mass, and ag is theradius of gyration. The platform displacements s , pre-scribed by Eqs. 1 enters Eqs. 2 through the followingforcing terms: N, Tf, F n, and H n; their functional forms aredescribed below.

During the contact phase, the normal contact force N as-sumes a value such that x=s. However, in the absence ofadhesion forces N must remain non-negative; thus the rotorand the stator separates if N falls to zero when sg. Thefunctional form of N is then

N = 0, s g or x smg + s , s g and x = s . 3

Depending on the normal contact force N, the relativeangular velocity and the angular acceleration of thestator, the torque acting on the rotor can arise from either

a

0.5

b

0.25

c

0

d

L 0.05

e

L 0.25

f

L 0.45

FIG. 2. Color online Potential dough-tossing motions andSWUM stator trajectories: ac the effect of varying phase lag with L= /4, and df the effect of varying amplitude ratio L=tanL with = /4.

LIU, FRIEND, AND YEO PHYSICAL REVIEW E 80, 046201 2009

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static or kinetic friction. When the sliding speed is zero =0 and the angular acceleration is within or enteringthe static friction envelope Senv= : env where env=aeN /mag

2, the frictional torque is static and it exactlybalances the rotors inertia so that =. If the sliding speedis nonzero 0, or the angular acceleration is outsideor exiting the static friction envelope, then the frictionaltorque is kinetic and it acts to oppose the sliding motion. Theabove description is partially summarized by the followingequation:

Tf =mag

2 , = and env aeN , = and env

aeN , = and env

aeN sgn , ,

4

where sgn is the signum function.The axial impulse of the nth collision F n is given by

F n = tn

tn+

Ftdt = mx+ x , 5

where the superscripts ,+ are used to denote, respectively,a quantity before and after the collision. Since we are mod-eling the collision with a speed-independent coefficient ofrestitution , the relationship between the preimpact andpostimpact relative vertical velocity is

s x+ = s x . 6

Substituting Eq. 6 into Eq. 5, we obtain the expression forthe nth axial impulse

F n = mx+ x = m1 + s x . 7

The determination of the frictional impulse H n requiresconsideration of the available impulse H n,a and the maxi-mum possible impulse H n,p. The available impulse H n,a isthe torsional impulse that would be transmitted if slidingfriction were present over the whole collision

H n,a = tn

tn+

ae sgn Ftdt = ae sgn F n.

8

Note that the frictional torque may only be present over afraction of the collision, vanishing when the relative angularvelocity between the rotor and the stator is reduced to zerostatic friction is ignored since the collision is short. If a capis not imposed on H n, scenarios where the relative rotation ofthe rotor reverses direction after impact would occur. Themaximum transferable impulse is thus limited to the amountthat would result in + =0, which is

H n,p = mag2 , 9

and the frictional angular impulse is given by

H n =H n,a, if H n,a H n,pH n,p, if H n,p H n,a

. 10B. Method of solution

Except for gravity, the forces experienced by the bouncingdisk are all associated with some form of discontinuity: theimpulsive forces F n ,H n cause sharp changes to the disksvelocity, the contact force can only support compressionalloads N0, and the frictional torque can switch betweenstatic and kinetic friction and has a jump discontinuity whenrelative sliding velocity changes sign aeN sgn .The result of these discontinuities is that the disk describedby Eq. 2 experiences four distinct phases: 1 parabolicflight, 2 impact, 3 sticking contact, and 4 sliding con-tact, each of which may be solved analytically. The follow-ing paragraph outlines how the four phases are sequenced asa solution to Eq. 2 and the whole process is summarized inFig. 3.

During parabolic flight, gravity is the only acting force sothat xt=xn+vnt tn

g2 t tn

2 and t= n. The phaseends when the disk collides with the platform, where thetime of collision is determined by solving for tn+1 such thatxtn+1=stn+1

xn + vntn+1 tn g2tn+1 tn2 = A sintn+1 . 11

After determining the postimpact velocities through Eqs. 6and 10, if the rebound speed x s is zero and the contactforce is positive N0, the system enters one of two contactphases, otherwise the disk is relaunched into parabolic flight.The two contact phases differ in terms of whether the frictionis static or kinetic. If the disk satisfies the condition de-scribed in Eq. 4 for static friction, then = ; otherwise, thesystem undergoes sliding contact and the angular rotation ofthe disk is determined by solving

reboundspeed iszero

contactforce fallsbelowzero

True

TrueFalse

False

Separation 1

parabolicflight

Collision

Attachment

C A

S

impact2

3

stickingcontactTrue

slidingcontact

4Falsefrictionaltorque isbelow thestatic limit

relativeangularspeed iszero

&

(a)

FIG. 3. Color online Model simulation flow chart. The inseta shows a potential trajectory of the bouncing disk dashed lineand the oscillating platform solid line for one cycle of simulation,starting with collision c, attachment a, and ending with separa-tion s.

ROTATING BOUNCING DISKS, TOSSING PIZZA DOUGH, PHYSICAL REVIEW E 80, 046201 2009

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mag2 = aeN sgn . 12

The two contact phases switch between each other accordingto the following conditions: stick changes to slip if the plat-forms angular acceleration exceeds the static friction limit,and slip changes to stick if the sliding speed reaches zero.Both contact phases end when N falls to zero and the disk islaunched into parabolic flight.

III. SINGLE TOSSES OF PIZZA DOUGH

A. Performance measures and parameters

We simulate our bouncing-disk model for a single launchcycle to investigate what the best hand motion is for singletosses as and L are varied. To compare the different tossingmotions, the following performance measures are used: 1the airborne rotational speed f reached by the pizza dough,2 the energy efficiency or ratio of the rotational kineticenergy gained by the dough over the total energy input due tothe tossing motion, and 3 the speed ratio of the doughsairborne angular velocity f over the maximum rotary toss-ing speed .

Noting that the amplitude ratio L=tan L=ae /A, andthat the arc length of the linear dough-tossing trajectory c=A2+ ae2, we can express A and as functions of Land c

ae,A = csin L,cos L . 13

Specifying A , in terms of c ,L has the following ad-vantages: by keeping c constant, the stroke length of thedough-tossing motion remains bounded as we vary the am-plitude ratio L, and by varying L between 0 and /2 allpossible L from 0 to are explored.

The full specification of a single toss in our bouncing-diskmodel can be determined by the following set of independentparameters: , L, c, , ae, ag, , and g. Initial conditions

and the coefficient of restitution are not part of the specifi-cation because we are only concerned with a single launchcycle and the dough always begins each toss at rest with t0=3T /4=3 /2 i.e., the lowest point of the tossing motion.The parameter values we use in this investigation are shownin Table I. The coefficient of friction between skin anddough is estimated to be =0.60.1 through the inclinedplane test a baked bread begins to slip on a lightly flouredhand at a slope of 305; the gravitational accelerationg=9.8 m /s2. The rest of the parameters are based on a videoof pizza tossing recorded at a local pizza shop 10: , ae, ag,and c are estimated by visual inspection; L and , however,are determined from the estimated vertical and angularlaunch speeds: xsep=3.10.3 m /s and sep=141.5 rad /s.Assuming that there is no slip between the dough and thehands,

xsep = c cos L and sep = c sin L/ae,

and L are thus given by

2 = xsepc2 + sepae

c2 14

and

L = tan1aesep

xsep .

B. Results and discussion

The effects of varying the amplitude ratio L and thephase lag are shown in Figs. 4a4c. For clarity, we onlyshow the results for 0 since f, , and all possesssome form of translation symmetry: for f and , f+=f, and for , f+= f. We have also limited thedomain of L to 0,cos1g /c2 rather than the full rangeof 0, /2 because the vertical force exerted by the tossing

TABLE I. Full specification of a single toss

rad

Lrad

c

m

rad/saem

agm

gm/s

0 0.550.14 0.250.05 155 0.1350.015 0.110.01 0.60.1 9.8

a

0 84

3 8

2

0.00

20

10

0

10

20

L rad

rota

rysp

eed

fr

ads

n 012345678

b

0 84

3 8

2

0.00

0

0.1

0.2

0.3

0.4

0.5

L rad

ener

gyef

fici

ency

n 0

1

2

3

45

6

7

c

0 84

3 8

2

0.00

1.

0.5

0

0.5

1.

L rad

spee

dra

tio

n 012345678

d

0 84

3 8

2

0.00

0

0.1

0.2

L rad

phas

e s

ep2 L cos

1gc2

e

0 84

3 8

2

0.00

0

5

10

15

20

25

L rad

rota

rysp

eed

fr

ads

increasing

FIG. 4. Color online The performance of different pizza tossing techniquesas measured by a dough rotation speed f, b energyefficiency , and c dough-to-hand speed ratio when the amplitude ratio L is varied at a phase lag of =n /8. d f is the doughrotation speed at the phase of separation sep, which is a function of L with a range of sin1g /c2 , /4. e The effect of on f; themaximum f occur at greater L as is increased 0.5, 0.6 and 0.7 for the three curves.


046201-4

motion is insufficient for overcoming gravity when Lcos1g /c2, which means the dough never leaves thehand.

Though our model is simple, the essential physics of thetorque transfer process in pizza tossing are included. Thepredicted dough rotation speed is 145 rad /s for the singletoss specified by Table I, which is in good agreement withthe average airborne angular speed of 141.5 rad /s in thefirst four pizza tosses of the video footage 10. The effectsof the uncertainties in the input parameters to our model areshown in Table II.

From Figs. 4a4c, we can see that the optimal param-eters are =0 or , and L=0.323 L=1.6, where themaximum airborne speed f=22 rad /s and energy effi-ciency =0.42 are reached; the speed ratio at these points=0.97 have a negligible difference from max=0.99 atL ,= 0.132 ,0. The optimal hand motions predicted byour bouncing-disk model are linear trajectories with =0 asshown in Fig. 2c, which are precisely the motions we ob-serve in actual pizza tossing. We note, however, that there isa significant difference between the optimal L=0.323 pre-dicted by our model and the observed L=0.1750.045. We will first explain why L ,= 0.323 ,0 is optimal in our model before returning todiscuss this discrepancy.

When the contact force and the coefficient of friction aresufficiently large, the airborne rotation speed of the doughwill be the same as the rotation speed of the tossing motionat the point of separation when s falls below g. In terms ofthe nondimensional phase =t, the point of maximum handrotation speed max and the point of separation sep are,respectively, given by

max = + n, where n = 0,1. . .

and sep = sin11/ . 15

The amplitude ratio tan L=ae /A affects three things:the point of separation sep see Fig. 4d, the maximumhand rotation speed , and the normal contact force N=mg+ x; the phase lag, on the other hand, controls the pointat which the maximum hand rotation speed occurs max . Ifwe let L /2, we increase the hand rotation speed ,however the dough will be sliding at sep due to reducednormal contact force; if we let L0, there will be littlerotary hand motion compare the three columns in Fig. 5.The optimal amplitude ratio L=0.323 thus results fromthe compromise between the goal of minimizing slip and thegoal of maximizing . The optimal phase lag =0, on theother hand, results from the balance between the torquetransmitted at the beginning 0 and the end sep of the

contact phase. If 0, max is shifted away from sep to theleft and thus the torque is reduced as we approach separation;if 0, max is shifted toward sep on the right but therewould initially be reverse torque at 0.

Our model suggests that the linear hand motions observedin the single toss video =0 are chosen to maximize doughrotation and energy efficiency. However, the observed ampli-tude ratio L= 0.1750.045 is much lower than the op-timal value of L=0.323 predicted by our model, whichmay be caused by other factors that influence the choice ofthe amplitude ratio L. For example, a dough-toss performermay be aiming for a particular toss height which requires asmaller L see Fig. 5 for the effect of L on toss height.Furthermore, Fig. 4e shows that the optimal L shifts to-ward zero as decreases. Since the coefficient of frictiondepends on the amount of dry flour present on the handsafactor difficult to controla dough toss performer may thuschoose a low L to minimize slip.

IV. MULTIPLE TOSSES OF PIZZA DOUGH AND SWUMBEHAVIOR

In this section we investigate the underlying motion trans-fer process that applies to both SWUMs and multiple pizzatosses. We will thus refer to both the pizza dough and theSWUM rotor as the disk, and both the dough-tossing handand the SWUM stator as the platform.

Most of the angular momentum transfer between the plat-form and the disk occurs during impact. Much informationabout the motion transfer process can thus be gained bystudying the next collision map of the bouncing-disk sys-tem: the location of attracting orbits will affect the steady-state rotation speed, and the size of their basins of attractionswill affect the sensitivity of SWUMs to perturbations and theease at which multiple pizza tosses may be executed. Thefollowing state variables are used in our next collision map:the relative axial collision velocity wA= x s, the relative an-gular collision velocity wT= , and the phase at impactimp=timp. The nondimensionalized form of the map hasfour parameters: , , as previously defined, and an angular

TABLE II. Effects of input parameter uncertainties

frad/s

Uncertainties rad/s due to

c ae ag L

14 3 5 2 0.01 0.01 3

a b c

s,x

ms

0 75 1 1 25 1 5

0.60.4

00.4

sep

0 75 1 1 25 1 5

0.60.2

00.2

sep

0 75 1 1 25 1 5

0.60.1

0

0.1sep

,

rad

s

0.75 1. 1.25 1.5

0.6

20

0

20 12

0.75 1. 1.25 1.5

0.6

20

0

201

2

0.75 1. 1.25 1.5

0.6

20

0

20 12

phase 2

FIG. 5. Color online Details of various hand motions dashedcurves and the resulting dough trajectories solid curves. Two tra-jectories with different phase lags 1=0 and 2= /2 are shownfor each of the following amplitude ratios L: a 0.132, b0.323, and c 0.426. The effect of choosing a suboptimal L,such as slip at high L, and low hand rotation rate at low L,can be seen by comparing the trajectories in a and c with theoptimal trajectory with L=0.323 in b.


046201-5

forcing parameter T=ag22 /aeg that can be interpreted

as the ratio of the inertial torque to the frictional torque dueto the rotor preload. In the next two sections we will discussthe implications of the systems bifurcation plots and basinsof attraction on pizza tossing and SWUMs.

A. Bifurcation diagram

The use of Coulomb friction in our bouncing-disk modelmeans that the angular component of the disk is affected byits vertical motion but not vice versa. The governing equa-tion for the vertical component of our bouncing-disk systemis thus identical to the traditional bouncing ball system. Weplot the bifurcation diagram for =0.5 by following theperiod 1 attractor 16 at =0.335, which undergoes a perioddoubling route to chaos as is increased. The resulting bi-furcation diagram for imp, shown in Fig. 6a, is consistentwith Tufillaros 2.

Also shown in Fig. 6 is the evolution of the chatteringorbit, which coexists with the periodic orbit as a separateattractor in the phase space. A key feature of the chatteringorbit is the occurrence of rapid collisions with decaying im-pact velocity and flight duration. In the idealized bouncing-disk system, it is possible for the disk to come to rest on theplatform after completing an infinite number of collisions infinite time; to avoid this computational supertask 17, weartificially set wA to zero if the impact speed is smallwA /A106 or if there are 20 consecutive short dura-tion flights /2104. Although the complexity of thechattering orbits grows as is increased, they still appear tobe eventually periodic. For 0.52, the chattering orbitsfollow a pattern similar to those of single pizza tosses: fol-lowing a long toss, the dough comes to rest on the hand

during the chattering phase and the process is repeated. Onthe other hand, the periodic orbits better match the dynamicsof the multiple pizza tosses, and will thus be the focus of ourdiscussion.

Since the frictional angular impulse H n acts always toreduce the relative angular impact speed wT between thedisk and the platform, wT will eventually be zero for anyperiod 1 orbit. That is,

p1,ss = timp = cosimp + , 16

where p1,ss is a period 1 steady-state dough rotational ve-locity, and timp is the platform rotation velocity at impact.The maximum steady-state rotational speed p1,ss= willbe achieved if =imp. In Fig. 6a, we can see that period1 orbits occur for 1 /30.486, and the phase of impactimp /2 starts at 0 and shifts to 0.130 at the first point ofbifurcation, thus the stator motion needed to achieve themaximum rotation velocity can vary from a linear to asemielliptical trajectory depending on .

For example, a linear trajectory with =0 would give themaximum rotation speed p1,ss= when =1 /3 and de-crease to p1,ss= cos0.26 when =0.486, as shown inthe bifurcation diagram of / for ,T= 0,100 inFig. 6c. In contrast, a semielliptical trajectory with =0.26 gives a low rotary speed when =1 /3 and reachesthe maximum speed when =0.486, as shown in Fig. 6d.

In the period 2 region 0.4860.531, Eq. 16 nolonger applies as the steady-state orbit alternates betweentwo collision states: the large-wA branch with a decreasingimp, and the small-wA branch with an increasing imp seeFig. 6b. Since timp of the large-wA branch has a greaterinfluence on the disks steady-state rotational speed, as impshifts toward zero on the large-wA branch, the average ro-tational speed rises for Fig. 6c and falls for Fig. 6d untilreaching the local extremum corresponding to the period 4bifurcation.

As the disk visits an increasing number of points in thephase space post period 4 bifurcation, and into the chaoticand chattering regime, the rotational dynamics of the disk isincreasingly affected by T. For a large T, the rotary inertiaof the disk dominates over the frictional torque, and the diskrotates at an average speed with small perturbations due toangular impulses H n transferred at different times imp. For asmall T, however, the frictional torque dominates and therotary speed of the disk oscillates wildly as the disk tracks timp more closely. The effect of T on this oscillation ofthe speed can be seen by comparing Fig. 6c where T=100 and Fig. 6d where T=1. Note that the bifurcationdiagram may appear to have three branches in what shouldbe a period 4 region of Fig. 6d. However, this is caused bythe one-dimensional projection of orbits that exist in three-dimensional space imp,wA ,wT.

Past SWUM researchers 7,13 have assumed that the ro-tor can be considered stationary relative to the stator, leadingto the conclusion that contact always occurs as the statorreaches its maximum vertical displacement /2, andthat an elliptical trajectorysuch as shown in Fig. 2a, with

0.35 0.45 0.550.40.2

00.20.4

a bc dea

imp2

0.35 0.45 0.551.5

1.

0.5

0.a bc deb

wA

A

Axial forcing parameter

0.35 0.45 0.55

0.

0.5

1.

0.0

a bc dec

0.35 0.45 0.551.0.50.0.51.a bc ded

Axial forcing parameter

FIG. 6. Color online Bifurcation diagrams =0.5 showingperiod doubling routes to chaos dark gray and the evolution of thechattering mode light gray as is increased for: a the phase ofimpact imp, b the relative vertical collision speed, cd thenormalized angular collision speed when c ,T= 0,100 andd ,T= 0.26 ,1. Note that L is not fixed. The basins ofattraction at ae are shown in Fig. 7. The axial forcing parametern at the first five bifurcation points are 0.486, 0.531, 0.5402,0.5421, and 0.5425. The ratio nn1n+1n of the distance between thefirst few bifurcation points are 4.891, 4.842, and 4.750, which ap-proaches the Feigenbaum constant 4.669


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=max = /2is the optimal stator motion. We haveshown, however, that when the vertical rotor motion is takeninto account, the maximum steady-state rotational speed isachieved by choosing the right phase lag =imp for agiven in the period 1 region 0.3330.486. Contraryto the assumption of SWUM researchers, neither imp is nec-essarily /2 nor is elliptical motion the best stator trajectory.

B. Basins of attraction

We can see from the previous section that in order toachieve the maximum dough rotation speed, one shouldchoose a hand motion within the period 1 regime 0.333

0.486 for multiple pizza tosses. We can further com-pare the relative ease at which a hand motion maintainsdough rotation over multiple tosses by examining the effectof on the basins of attraction; a larger basin of attractionimplies that the orbit is more tolerant of variability in thehand motion. In Fig. 7, the basin of attraction of the period 1orbit at =0.34 only covers 11.6% of the phase spaceshown; as is increased, the basin of attraction widens to39.1% of the phase space shown at =0.486. We can thussee that the use of semielliptical hand motion =0.26with 0.486 has the advantage of being easier for main-taining the period 1 orbit required for maximum dough rota-tion speed.

The wide basins of attraction of the chattering orbits makethem far easier to perform than period 1 multiple tosses. Thefact that the wA=0 axis is deep within the basins of chatter-ing orbits shows that period 1 orbits cannot be attained for adough initially at rest if the tossing motion purely followsEq. 1. This explains why dough-toss performers do notstart multiple tosses with semielliptical hand motion rightfrom the beginning, but instead employ a linear hand motionfor the first few tosses.

Although it is unlikely that dough-toss performers willneed to consult Fig. 6 and 7 to avoid chaotic and chatteringorbits, the bifurcation diagrams and the basins of attractionprovides SWUM researchers useful insights into the physicalbehavior of SWUMs. For example, SWUMs should be de-signed to operate away from the chaotic regime e.g., =0.55 in Fig. 7d because the extended region of the phasespace visited by the disk reduces the rotor speed, and thefractal basin boundary of the attractors makes the motor be-havior unpredictable. The maximum rotor speed can beachieved in the period 1 regime, though complex controlmay be needed to actively adjust the stator motion to accom-modate the evolving stator-rotor dynamics, since the basin of

attraction for period 1 orbits do not cover SWUMs usualinitial condition. For 0.561, we see a sudden widening ofthe phase space explored by the orbits; the strange attractorof the chaotic orbits merges with the chattering orbit and thebasin of attraction appears to fill the entire phase space seeFig. 6d. Although these orbits are eventually periodic, theydiffer from the low period chattering orbits in Figs.7a7c: they have a longer period, and they follow theshadow of the chaotic strange attractor. SWUMs operated inthis regime will not reach the maximum rotor speed, how-ever, the wide basin of attraction suggests that they will op-erate most consistently. In Sec. V, our simulation results sug-gests that our prototype SWUM predominantly operates inthis regime.

V. DYNAMICS OF SWUMS

The rotor and stator of our SWUM are, respectively, a20-mm-diameter steel ball and a pretwisted beam that isglued to an axially poled piezoelectric transducer see Fig. 8for details of the geometry. By applying a sinusoidal elec-trical input to the transducer at a resonance frequency of thestator assembly, the stator tip is excited to vibrate with suf-ficient amplitude in the form described by Eq. 1 to inducerapid rotor rotation. Since the systems resonance frequen-cies are fixed by its geometry and material properties, themotor has two adjustable experimental parameters: the am-plitude of the input voltage, altered via the use of a signalgenerator; and the rotor preload, altered by adjusting the dis-tance between the steel ball and a magnet placed above it.

colli

sion

velo

city

wA

A

0.5 0 0.55.

2.5

0a

0.5 0 0.55.

2.5

0b

0.5 0 0.55.

2.5

0c

0.5 0 0.55.

2.5

0d

0.5 0 0.55.

2.5

0e

phase of collision 2

chattering orbit

chaoticorbit

chattering orbit

FIG. 7. Color online The basins of attraction for the following values of : a 0.34, b 0.486, c 0.50, d 0.55, and e 0.60. The whiteregions are the basins of attraction of the chattering orbits marked by black s in ac, red dots in d, and black dots in e. The blackregions are the basins of attraction of the attractors that undergo the period doubling route to chaos marked by white in ac andyellow dots in d.

FIG. 8. Geometric details of our SWUM not to scale. Note thatthis SWUM Ref. 8 is a simple prototype with preload providedby gravity.


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A. Model parameters

In order to make direct comparisons between the SWUMand our model, we need to know the correspondence be-tween the experimental conditions and the model parameters.A total of nine input parameters ag, ae, , g, , , A, , are required to perform dimensional simulations, however,only four ag, ae, , g are accurately known: ae and agdepend on the design geometry of the SWUM, is set by thesignal generator at a resonance frequency, and g can be ad-justed and inferred from the measured weight of the motorsetup. We thus compare the motor and the model indirectlyby estimating the remaining parameters , , A, , andusing them as default values about which we study the sen-sitivity of the system to variations in the five parameters.

The default values of the parameters used in our simula-tions are shown in Table III; justifications for the values weselected for , , A, , and are detailed below. We wereunable to measure A, , and during the motors operationbecause the rotor blocks the access our instrumentation re-quires to the stator tip; the stator motion is thus estimated bymeasuring its vibration without the rotor, which shows thatA107 m and 30 rad /s. Assuming that the presenceof the rotor decreases A more than , we chose A=108 m as the default parameter while keeping =30 rad /s. The phase lag is assumed to be close to that ofthe free vibration of the stator, and thus we chose =0.

We estimated the coefficient of restitution by dropping a20 mm steel ball from a height of 4.5 cm onto an aluminumbeam sitting on a laboratory bench in a configuration similarto the motor; the resulting rebound height of 0.90.5 cmimplies =0.40.1 at an impact speed of 0.95 m/s. Thecoefficient of friction is estimated by measuring the rate atwhich the rotor decelerates brake when power input to themotor is switched off 18; in the absence of stator vibration,Eq. 2b implies that

= mag

2brakeaeN

. 17

The deceleration profiles were obtained for a range of pre-loads see Fig. 9 giving us an average of 0.1 with astandard deviation of 0.06. Although reference values of between steel and aluminum under dry sliding conditions isabout three times the results of our measurements in the0.30.4 range 19, the difference may be explained by thefact that the stator and rotor are under line contact, whichreduces the true contact area and correspondingly the coeffi-cient of friction 20.

B. Comparison with experimental results

In Fig. 10, transient speed curves predicted by our modelat the default parameter values with A varied across its ex-pected range, are shown next to measurements from threeseparate trials of our SWUM. The two sets of curves arequalitatively very similar: not only does our model predictthe speed transients curves to be of the observed formss1et/tc, it also accounts for the presence of what ap-pears to be a high frequency, random oscillation that existingSWUM models have been unable to explain. In our model,the oscillation originates from the vertical rotor motion thatis undergoing long period chattering possibly with transientchaos, and is strongly affected by A: when A is small, col-lisions are more frequent and occur at lower speeds, leading

TABLE III. Default values for the dimensional parameters

Unknown Known

radA

m

rad/sae

mmag

mm

kHzg

m /s2

0.1 0.5 0 108 30 1.25 6.3 70 10

0 2 4 6 80

10

20

30

0.1

time s

roto

rsp

eed

rad

s

44.3t7.77

80 160 240 3200

0.1

0.2

0.3

preload mNm

FIG. 9. Color online Determining the coefficient of frictionbetween the rotor and the stator. A curve of the form =braket tc is fitted to the deceleration profile when the motor is switchedoff; is then inferred from brake through Eq. 17. The right handfigure shows the measured at various preloads; in accordancewith Coulomb friction, is relatively independent of preload andhas an average of 0.10 and standard deviation of 0.06.

a

0 2 4 60

10

20

30

rotorspeed

rads

23.91et0.854

d

0 2 4 60

10

20

30

rotorspeed

rads

18.31et0.788

b

0 2 4 60

10

20

30

rotorspeed

rads

20.81et0.743

e

0 2 4 60

10

20

30

rotorspeed

rads

16.1et0.924

c

0 2 4 60

10

20

30

rotorspeed

rads

21.91et0.755

f

0 2 4 60

10

20

30

rotorspeed

rads

16.21et0.804

time s time s

FIG. 10. Color online Transient speed curves each fitted withan equation of the form ss1et/tc. ac Simulation results atfull preload for the following values of A: a 109 m, b 108 m,and c 107 m; all other parameters take on the default values inTable III. df Measured behavior of our SWUM at full preloadand the same voltage input 0.55 V peak-to-peak on three separatetrials.


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to smoother transient speed curves; when A is large, infre-quent high speed collisions results in large oscillations inrotor speed.

Quantitative comparisons are avoided since accurate val-ues of five parameters are unknown; in fact our sensitivitystudy in the next section shows that the five parameters maybe adjusted for a perfect match. We merely comment thatreasonable values were chosen for our estimates and the re-sulting predictions are of the same order of magnitude as themeasurements.

We thus focus on qualitative comparisons, and have foundgood agreement between the observed and predicted trendsin steady-state speed and stall torque when preload is in-creased. In our model, the effects of preload are apparent intwo ways: explicitly, through the parameter g, and implicitly,by modifying the stator motion. Except for the general ex-pectation that the stator vibration amplitude will decreasewhen g is increased, A and and are unknown functionsof g. We conjecture that A is more significantly affected by gthan and , and thus in determining the effect of preloadin our model, we varied g between 0.1 and 10 m /s2 at threedifferent axial vibration amplitudes, starting with A=107 m at low preloads and reducing to 108 and 109 min the presence of preload the default values were used for and .

Our experimental study 8 of SWUMs showed that for alarge range of preloads between 80 and 280 mNm, the stalltorque is proportional to preload, while the steady-statespeed is relatively independent of preload. We can see fromFig. 11, where we overlay the simulation results with theexperimental data, that similar trends are predicted by ourbouncing-disk model when preload is in the 80280 mNmrange. Note also that the response of our model at the defaultfrequency =70 kHz and the second operating frequency

=184 kHz are both consistent with experimental observa-tion, adding further support that our model has correctly cap-tured the stator-rotor momentum transfer process.

However, there are some features of the SWUM that oursimulations fail to reproduce: for the =70 kHz case, thereis a sharp fall in steady-state speed at low and high preloads,and the plateau in the stall torque when preload is greaterthan 200 mNm; for the =184 kHz case, there is a sharprise in steady-state speed at high preloads. The discrepanciesat high preload may be caused by the following: first, would be reduced as the preload is increased due to the sup-pression of stator axial vibration and therefore torsional vi-bration through the coupling mechanism within the twistedstator structurethis could explain the drop in steady-statespeed at high preload when =70 kHz; and second, a quali-tative change in the motors dynamics can occur at high pre-load if A is reduced such that 1 /, in which case thestator acceleration would be insufficient for the stator-rotorseparation to occur, and the rotor would no longer bebouncing. The differences at low preload may be causedby the assumption that g is constant in our model, which maybe valid when the axial motion of the rotor is small relativeto its distance to the magnet used to adjust the preload, butfails as the magnet is drawn close to the rotor to achieve alow preload.

C. Predicted behavior of SWUMs

When the transient speed curves are of the form ss1et/tc, we can compare simulated and observed results bytheir steady-state speed ss and the stall torque ss / tc. Note,however, that this transient response occurs in our bouncing-disk model when the vertical motion of the rotor undergoeslong period chattering that traces the shadow of a chaoticstrange attractor; for different parameters, the bouncing-disksystem may undergo periodic or low period chattering orbitsthat give rise to qualitatively very different speed trajecto-ries. Figure 12 shows an example of the speed trajectory thatresults from a low period chattering orbit: the periodic natureof the of orbit causes the angular momentum transfer to oc-cur in a regular pattern, resulting in the linear rise in rota-tional speed. The angular impulse is limited by the relativeangular speed of collision; thus when the rotor reaches therotary speed of the stator at impact, there is an abrupt tran-sition from the apparent constant acceleration to steady-staterotation. Note also that the speed trajectory appears to formmultiple lines as the rotor cycles through collisions with dif-ferent impact speeds.

So far, piecewise linear transient speed curves of this formhave not yet been observed in SWUMs. We suspect that realmotors operate in a region of the parameter space where lowperiod chattering orbits are uncommon, and that they may bereproduced experimentally under the right conditions. Thisleads us to the results of our sensitivity study of the fiveparameters with estimated default values, where we foundthat the coefficient of restitution appears to have a strongeffect on the presence of low period chattering orbits.

We varied each of the parameters , , A, , and individually over their expected range while keeping all

a b

0 80 160 240 3200

10

20

30

preload mN

stea

dyst

ate

spee

d

0 80 160 240 3200

0.01

0.02

0.03

0.04

preload mN

stal

ltor

que

mN

m

c d

0 80 160 240 3200

10

20

30

preload mN

stea

dyst

ate

spee

d

0 80 160 240 3200

0.01

0.02

0.03

0.04

preload mN

stal

ltor

que

mN

m

FIG. 11. Color online Effect of preload on steady-state speedand stall torque at two of the motors operating frequencies: ab=70 kHz with clockwise rotor rotation, and cd =184 kHz with anticlockwise rotor rotation. The measured resultsare marked by crosses while the simulation results are markedby dots and joined to guide the eye. Points joined by solid,dashed, and dotted lines are, respectively, simulations performed atA=109, 108, and 107 m.


046201-9

other parameters at their default values. The predicted effectof each parameter on the steady-state speed and stall torqueare shown in Fig. 13. Most of our simulations yielded tran-sient speed curves of the form ss1et/tc, however, when0.5, low period chattering becomes the prevalent behav-ior of our model. With our estimated to be 0.40.1 at 0.95m/s and theories predicting to rise as collision speed isdecreased 21, this may explain why low period chatteringorbits have not been observed experimentally and suggestswe should reduce in attempting to reproduce these orbits inSWUMs.

While the chief purpose of our sensitivity study is to de-termine the likely effect of errors in our estimated param-eters, it also provides predictions that may be used for futurecomparisons with SWUMs and suggests which parametersare important for improving the motor. From Fig. 13, we cansee that an error in some of the parameters will have agreater significance than the others; for example, the steady-state speed is most sensitive to , , and , while the stalltorque is most sensitive to , , and . Note that and appear to be the only two parameters that significantly affectboth steady-state speed and stall torque; their effects are,however, limited to reducing steady-state speed and stalltorque from the maximum values that are set, respectively,by and .

We have seen from Fig. 10 that A has a strong effect onthe oscillation of the speed-time curve, thus it may seemcurious that the steady-state speed and stall torque are notaffected significantly by A even when it is varied by twoorders of magnitude. A possible explanation for this is thatthe increased angular impulse H caused by larger collisionspeed wA is offset by the longer interval between collisionsT

H 1 + wA and T wA/g

implies H /Ttorque g1 + .

Although the above argument also seems to explain the in-dependence of stall torque to , and the linear relationshipbetween and the stall torque, it should be noted that thissimplistic argument neglects the fact that collisions occurover a range of phases and speeds, and ultimately fails tocorrectly predict the effect of .

Consideration of how the collisions of the orbit are dis-tributed in the state space phase space is needed for ex-plaining the effects of and . The results for correspondsto a cosine curve because the collisions are more concen-trated at 0, thus the momentum transfer occurs predomi-nantly when the angular velocity of the stator is = cos. The collisions are less concentrated at =0when is increased, thus leading to the decrease in bothsteady-state speed and stall torque.

VI. CONCLUSIONS

Pizza tossing and SWUMs share a common mechanismfor converting reciprocating input into continuous rotary mo-tion, and the physical behavior of both may be representedby the convenient adaptation of the bouncing ball model2,3 to include rotation. Key features of the motion transferprocess such as impact, separation and stick-slip frictionaltorque are captured by our bouncing-disk model, giving us abetter understanding of SWUMs and pizza tossing. Whilethere are many factors influencing the choice of tossing mo-tion that we did not consider in detail, our bouncing-diskmodel shows that aspects of the tossing motion employed inthe two basic pizza tossing modes do optimize certain per-formance measures: the linear trajectory used in single tossesmaximizes rotation speed and efficiency, while the semiellip-tical motion used in multiple tosses maximizes the ease ofmaintaining the dough in the period 1 orbit that is required

a b

0 30 60 900

10

20

30

time s

rotorspeedrad

s

21.51et18.30.5 0 0.5

150100500

2

w Amms

c d

0 6 12 18 240

1

2

3

10

time s

rotorspeedrad

s

0.5 0 0.53020100

100

2

w Amms

FIG. 12. Color online Comparison of the simulated transientspeed curves when the rotors vertical displacement undergoes longperiod and low period chattering orbits. The default model param-eters in Table III are used except for: ab A=107 m and g=0.5 m /s2 in the long period chattering case, and de A=108 m and g=1 m /s2 in the low period chattering case.

a b c d e

stea

dystat

esp

eed

0 0.2 0.40

0.2

0.4

0.6

0.8

1

0

0.06

0.12

0.18

0. 0.5 1.0

0.2

0.4

0.6

0.8

1

0

0.02

0.04

0.06

periodic

orbit

0 2 1

0.5

0

0.5

1

0.06

0.03

0

0.03

0.06

rad

109 108 1070

0.2

0.4

0.6

0.8

1

0

0.02

0.04

0.06

A m

3 30 3000.

0.2

0.4

0.6

0.8

1.

0

0.02

0.04

0.06

rads

stal

ltor

que

mN

m

FIG. 13. Color online Effect of varying a , b , c , d A, and e about the default parameter values. In each plot, thesteady-state speed predictions are denoted by crosses with scales on the left hand frame, while the stall torque predictions are denotedby dots with scales on the right hand frame. Note that the rotor speed is nondimensionalized with respect to .


046201-10

for maximum steady-state speed. We applied our bouncing-disk model to our SWUM, using estimated parameters, andfound that it reproduces the oscillations that have been ob-served in the motors transient speed curves and have notbeen explained by existing models. Additionally, the pre-dicted effect of preload on the steady-state speed and stalltorque agrees with experimental observation.

APPENDIX: NEGLECTING DOUGH DEFORMATION

In the bouncing-disk model, the effects of dough defor-mation during the impact and contact phase of pizza tossingmay be modeled with an appropriate choice of friction coef-ficient and coefficient of restitution . What is not modeledis the change in dough geometry, which causes changes tothe moment of inertia and aerodynamic forces. However, aswe will show below with data extracted from videos of pizzatossing, the change per toss in dough diameter is relativelysmall, thus the essential character of the process is the sameas the bouncing-disk system.

A frame by frame diameter estimate of the dough in themultiple-toss video 11 starting from the end of the second

toss at t=2.365 s is shown in Fig. 14; the diameter wasmeasured in arbitrary units, fitted with a least-squares bestfit, and finally the diameter was normalized with respect tothe best fit diameter d0 at t=2.365 s. The start of each toss ismarked with an arrow. From n=3 to n=12 the diameter grewby 30%, which is an average of 3% per toss.

1 P. Pieraski, Phys. Rev. A 37, 1782 1988.2 N. B. Tufillaro, Phys. Rev. E 50, 4509 1994.3 J. M. Luck and A. Mehta, Phys. Rev. E 48, 3988 1993.4 J.-C. Gminard and C. Laroche, Phys. Rev. E 68, 031305

2003.5 K. Szymanski and Y. Labaye, Phys. Rev. E 59, 2863 1999.6 N. B. Tufillaro and A. M. Albano, Am. J. Phys. 54, 939

1986.7 K. Nakamura, M. Kurosawa, and S. Ueha, IEEE Trans. Ultra-

son. Ferroelectr. Freq. Control 40, 395 1993.8 D. Wajchman, K.-C. Liu, J. Friend, and L. Yeo, IEEE Trans.

Ultrason. Ferroelectr. Freq. Control 55, 832 2008.9 B. Watson, J. Friend, and L. Yeo, Journal of Micromechanics

and Microengineering 19, 022001 2009.10 Video footage of single pizza tosses performed by Noah El-

hage, captured on 3 October, 2007, camera operated by Kuang-Chen Liu.

11 Online videos of multiple-toss, accessed 29 April, 2008 http://www.throwdough.com/trick_basictoss.htm and http://www.youtube.com/watch?vfAW0aDWaiSI.

12 K. Nakamura, M. Kurosawa, and S. Ueha, IEEE Trans. Ultra-son. Ferroelectr. Freq. Control 38, 188 1991.

13 J. Guo, S. Gong, H. Guo, X. Liu, and K. Ji, IEEE Trans.Ultrason. Ferroelectr. Freq. Control 51, 387 2004.

14 J. Tsujino, Smart Mater. Struct. 7, 345 1998.15 K.-C. Liu, J. Friend, and L. Yeo, EPL 85, 60002 2009.16 H. L. Nusse and J. A. Yorke, Dynamics: Numerical Explora-

tions Springer-Verlag, Berlin, 1994.17 J. P. Laraudogoitia, in The Stanford Encyclopedia of Philoso-

phy, edited by E. N. Zalta Stanford University, Palo Alto, CA,2008.

18 K. Nakamura, M. Kurosawa, H. Kurebayashi, and S. Ueha,IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 4811991.

19 CRC Handbook of Engineering Tables, edited by R. C. DorfCRC Press, Boca Raton, FL, 2004.

20 shppE.-S. Yoon, R. A. Singh, H.-J. Oh, and H. Kong, Wear259, 1424 2005.

21 G. Kuwabara and K. Kono, Jpn. J. Appl. Phys. 26, 12301987.

3 4 5 6 7 8 90.0

0.5

1.0

1.5

Time s

Est

imat

eddi

amet

erd

d0

3 45 6

7 89 10

11 12

FIG. 14. Color online Frame by frame estimate of dough di-ameter from the video of the mutliple-toss Ref. 11. The equationfor the least-squares best fit is d /d0=1+0.0499t2.365. The startof the nth toss is marked with an arrow.


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