Dynamics of a ball on a vibrating plate

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Dynamics of a ball on a vibrating plate

Pierre Devillard

To cite this version:

Pierre Devillard. Dynamics of a ball on a vibrating plate. Journal de Physique I, EDP Sciences, 1994,

4 (7), pp.1003-1011. �10.1051/jp1:1994180�. �jpa-00246960�

Classification Physics ^Abstiacts

05.45 05.40 46.10

Dynamics of

_a

bail

_{on a}

vibrating plate

P. Devillard

Centre de

Physique Théorique

de Marseille, CNRS

Luminy,

Case ~07, CPT, 13288 Marseille Cedex 9, France

(Received 3

Februaiy

J994, received in final form 18 March J994, a<.<.epted 23 Maich 1994)

Abstract. We study both

analytically

and

numerically

the problem of _a partially ^{inelastic bail} on a platform vibrating with

frequency

_w. Two parameters ^control the dynamics, the restitution

coefficient _~, and the reduced acceleration r y/g, where _{y is} the maximum acceleration of the

motion of the plate and _g the acceleration of gravity. When _~ exceeds _some value,

simple

stable

fixed points no longer exist, but it is shown that

generic trajectories

are

periodic

with

penod

T~ which behaves _as T~ (l _~ )~~ for _~ close to1

Introduction.

In this paper, we revisit the

problem

of _one bail _{on a}

platform vtbrating

with

period

T

= 2 w/w and _maximum

amplitude

_a

[1-6],

which _was

recently

studied in reference

[6].

In the

following,

we show that ail the

periodic

motions, in _some parameter regions, mclude the _rest state, the _state where the bail _moves with the

plate.

The

period

between _{a rest state} and the _next rest state, scales _as

(1

_~

)~~,

_{~ is} the restitution coefficient.

l.

Description

of the model.

Instead of

taking

the _movement of the

platform

to be sinusoïdal, as m reference

[6],

_we take _it piecewise

parabohc,

with

period

T.

Namely,

the

height s(t

of the

platform

_in the

laboratory

frame _{is, on} each half

penod,

an arch of a

parabola,

whose

concavity

is

altematively

_up and clown

(except

at the very

beginning,

for t w

~

),

_as

depicted

in

figure

1. We have _: 2

~ ~~~ T

~ ~

T, ~~~~

s(t)

=

) (~ i ^~~

^~~ _~

3 T

(16)

3T

~)

^for

T<t<f' s(t)=)(T~~~~

2

~

(lc)

~~_ ne~.

~ ~ ~~n~ ₌

s(t)

_°~

2 ^'

1004 JOURNAL DE PHYSIQUE I N° 7

a

S(t)

~ t

-a

Fig. l Position _s(t of the vibrating platform as a function of time t. For _{t m}

~,

s(t )^is made of arches 4

of

parabola

with ~econd denvative ~~( equal to y or y.

dt~

y is an acceleration which relates to the

amplitude

_a of the motion

by

~~n2

a = -,

(2a)

and the reduced acceleration r will be defined

by

r

=

Y/g, (2b)

where g is the acceleration of

gravity.

From now on,

positions

and velocities of the bail will net be counted in the

laboratory

frame, but in the reference frame of the movmg

platform,

_so that the

position

^of the bail _is aise its

height

with respect to the

platform

and is

always positive.

In the reference frame of the

platform,

^{the force felt}

^by

^{a bail of mass one, when not in contact with the}

^platform,

^is

~~Î

_g. It takes

alternatively

the values _y g and _y g. We _assume that _we have _a

dt-

perfectly

hard

bail,

and that r is

larger

than 1. We denote

by

^{W+ and W~ the relative velocities}

of the bail with respect to the

platform respectively immediately

before and after _a bounce with the

platform.

The coefficient of restitution _~ is defined

by

W+

=

~W~ (3)

2.

Description

of the

dynamics.

We shall _{use two} alternative ways of

describing

the

dynamics.

One _is the standard way in the literature. The time ri where the Î-th bounce _occurs and

WI,

the relative

velocity

of the bail

just

after the

bounce,

_are recorded. A two dimensional map

(W/~

~, ri _~ =

f(W/,

_ri is obtained, where

fis

_a function. This, however, _is not convenient when the hall

performs

infinitely

_many

hops

^and comes to rest on the

platform.

In this _{case, as in} reference

[6],

_{we say} that the bail lands _{in an}

absorbing

_region. It is relaunched later, when the force felt

by

the bail

becomes

positive, namely,

when the

driving

acceleration, which is

equal

to

~~

,

overcomes

Î

gravity.

An alternative way is to

simply

record the

height

and

velocity

of the bail _at each _time the acceleration

changes sign

(at each half

period).

The acceleration felt

by

the bail will be

alternatively

_{y g} and _{y g.} In this

descrption,

_g will be set to one, the

length

of the half-

period

will be set to 1, and accelerations will be counted in units of _{y +} g. The acceleration

seen

by

the bail will thus

alternatively

take the values I and A

=

~

(4)

Time is denoted

by

t' _in this

description.

For any

integer

Î, with Î _w t' _w Î ₊ 1, the acceleration

felt

by

the bail will be A. For Î +1wt'mi +2, it will be -1 For every time

t'

equal

to an even integer ² n, the positions and velocities at this time will be denoted

by

x,, and u,,

respectively.

It will _{tum Dut to} be convenient to _use the variables

Yn ~ V,,,

(5a)

~~Î~

~l'

~

fi. _(5b)

We also introduce i~( and

v(,

the

positions

and velocities of the hall _{at times} t'=

2n+1

(1.e.

_every time the acceleration switches from

positive

to

negative). X(

and

Y( are defined in

analogy

to X,~ ^and Y,,. The

dynamics

is _now descnbed

by

a map :

(Xii,

_Y,,

Î (Xl, Yl ) Î

(Xii

+ i, Y,j

+

(6)

The motion of the

representative

points

(X,,,

_Y,, and

(X(,

_Y,[ in

phase

_space is restricted to the

cones (X,~ m

0,

_{Y,, w}

X,,

^and

(X(

m

0,

_{Y( w}

X( respectively.

The precise form of the

mapping

b and h is _a little

complicated

and _not very informative, it is

given

in the

appendix.

^One property ^{of h however is} important. ^For any

(X,[,

Y,() ^such that

Y( ~ l i~X(

(7a)

with ' =

~ ~

,

(7b)

~ln+

~

~n+

~ ~

(7C)

That _is to say, h is

identically

zero. This

corresponds

to the

absorbing

_region, the bail _{comes to} rest on the

platform.

3.

Scahng

of the

period

of the motion for

large

restitution coefficient.

In this

section,

the

period

^{of the}

periodic

motion for

large

restitution coefficient _is estimated.

Starting

from any initial

condition,

the bail

always

_{goes to} the rest state, moving

together

with the

plate.

The regime before the bail has reached the rest state for the first _time will be called

the transient regime. After _it has reached the rest state for the first

time,

its movement will then

be

penodic.

This _is called the permanent regime. The number of iterations between two rest states, the

period

of the motion _in this

parameter

region, is calculated _{as a} function of both the restitution coefficient and the _size of the

permanent

regime. The _size of the permanent regime

in tutu, is aise related _to the value of the restitution coefficient.

1006 JOURNAL DE PHYSIQUE I N° 7

Having

_set the

notations,

_{we now}

give

results about the

dynamics.

One issue is whether the hall

generically

lands in _an

absorbing region.

If this _occurs, then the

subsequent

motion will be

periodic.

It is

simpler

to discuss first _a situation where a bail does not

genencally

land in _an

absorbing region.

Let _us consider the

mapping (W/~

~, ri

~ ~) =

f(W/,

_ri

),

which relates times and velocities

just

after the

(Î

+

1)-th

^bounce of the bail will the

platform

to times and

velocities

just

after the Î-th bounce. There may exist fixed points ^{of the} map, i-e-

(W/~

=

WI

and _ri

~ ⁼

ri).

A fixed point ^with

W/~

=

WI

₌ 0

corresponds

to the hall

landing

in the

absorbing region.

We call this kind of fixed

point

a nuit

velocity simple

fixed

point.

A fixed

point

with

WI

_# 0 will be called _{a non-zero}

velocity simple

fixed

point.

^If such

a non-zero

velocity simple

fixed

point

exists, with

WI

=

Wo

#

0,

for

example,

and if it is

stable,

then there will be _a

neighbourhood

of this

point

for which the bail will be attracted _to it.

For

large Î, WI

tends to

Wo

^{and the bail will} never land in _an

absorbing region.

It is

possible

to

enumerate ail _non-zero

velocity simple

fixed

points,

for _a

given

value of _~ and l~ but the

analytic expressions

are

moderately comphcated

and not

particularly illuminating.

One result

is that for

(1+ ~)2

_{(1 +}

~2)-' ~2r-', ₍₈₎

no stable _non-zero

velocity simple

fixed

points

exist.

Although

the absence of such stable fixed

points

with

WI

_# 0 does not mean that _a

generic trajectory

fanas in _an

absorbing region,

other stable attractors, fixed

cycles,

strange attractors, etc... are not observed in the numencal

simulations for _~

larger

than 0.6 and

rlarger

than 2. Thus, we shall _assume that the

only

stable attractor, in this parameter

region,

is the rest state.

We _now calculate the

period

of the _motion

(from

_{one rest} state to

another),

_{as a} function of the restitution coefficient and of the size of the permanent

regime.

Later, we shall _see that the permanent regime

dynamics

is confined _{to a} region where W+

~

W,~~,

^where _W~~~

depends

_on

~, if the

dynamics

is described

by

_mapping

f.

If the

description

^{of the}

dynamics

^is done

using

m

j

^{and h}

^{(Eq. (6)),}

^{the permanent}

^{regime dynamics}

^{is confined to a region where}

,

X'~

₊ Y'~

~

Ro,

^where

Ro

is a number which

depends

_{on ~.}

From

equations (7a)

to

(7c),

we see that _a bail fanas _in the

absorbing

_region

if,

and

only

if, it

has net been absorbed before and

Y(w1-1.Xj.

The region

Y(w1-

_i~X,[, and

(Y((

_~

X(

defines _a

triangle

in

(X', Y') plane.

In

principle,

it suffices _to find the _{inverse iterates} of this

triangle

_via the

applications

b and h. We try to evaluate the _{area in}

(X', Y') plane

made

by

the

points

which _are going to land in the

absorbing

_region after _n

periods

of vibration of the

platform,

and _not before. More

precisely,

if _{one starts} with _a point ^of coordinates

(Xo, Yo),

it transforms under b into _a

point

^of coordinates

(Xi, Y()

and fanas _in the

absorbing

region if

Y(w1-1.Xj.

We say that the 0thorder

absorbing

zone, later denoted

by

mto is made

by

the

rectangle triangle

delimited

by

the

straight

fines:

Y(=1-1.Xj, yj

=

Xi

and

Y(

₌

Xi.

The n-th order

absorption

zone

A~,

is defined in _an

analogous

_way as

the points

X(, Y(

^{which land in the}

absorbing

_region after _n steps, ^{but have} not landed there

before _{; le-,} Y( w

-1.Xj

and Y( _~ ^l i~X[ for any s, 0

w s _~ n.

The _area

of,~o

_is

simply (1

_~

)~/(4

_~

).

^Since b and h _are defined in _a piecewise fashion, mi,, will be made _in

general

of many

patches.

The Jacobian of

b~',

_{taken at the value} (X,[,

Y(),

with

(X,,, Y,,)

₌ ^{b~ '}

(X(, Y(),

will be denoted

by 3(b~

^'

)(X(, Y().

It takes the value

3

(b~

^'

)(X(,

_Y,[

=

X(/X,,

,

if

Y(

~ k

(X( ), (9a)

where k

(Xi

)

=

)~ X(,

if X,[ _~

,~~Î _(9b)

A ₊

and

k(X(

=

X(~

+ A +

,

if

X(

_m

,~ _(9c)

3(b~ (X,[, Y( ₌ ~ ^~X,[/X~,

,

if

Y(

_~ ^k(X,[ )

(9d

We have also

3(h~

^'

= ~

~"',

where _m is defined in equation

(A.2fl

of the

appendix.

As _~

tends to 1, the _area of

Ao

^shrinks as

(1- ~)~

and the available

phase

_{space grows} like

RÎ,

where

Ro

^{is the size of the}

region

where the permanent

regime dynamics

takes

place.

We

only

_give here _a

simplified

backbone of the argument. ^The A,~ zones

gradually

_caver ail the available

phase

_{space as n} increase.

By definition,

_{the A,~ zones are}

disjoint. Otherwise,

if

two mt~~ and

~~t~~ with _{n # m}

overlapped,

the bail would land for the first time _in the

absorbing

region

after _n steps ^{and aise after} m steps, ^{which is}

impossible.

If A~~ ^is

mainly

built up

by points

located at a distance of order ₌ from the

origin

_in

(X', Y') plane,

the presence of the factor

X(/X,~

m the Jacobian

(Eq. (9a)) implies

that the _area

X(

_X,(

X(

~

Xi

of the _{~t,~ zone} shrinks

by

_a factor

1,~

₌

,

with X,[ z and _{X,~ z.} In the

~n ~n ~n

2

~0

regions

where the _main part ^{of the}

dynamics

takes

place,

_we have

X(

= X,~ _~. This _can be _seen

from equations

(A.2d)

to

(A.2g)

of the

appendix, giving

the precise from of the mapping h _{; we} have almost ail the _{time m}

= 0 in

equation (A.2fl.

The shrink factor

i,~

will thus be of order

Xi

-, which behaves _{as z~}

X~~

The _area of the

A,,

zone shrinks thus _as

=~'

_{Since every}

point belongs

to

only

one

Jt,~ zone, the number Î of iterations necessary to caver ail the accessible

phase

_space, where the permanent

regime

takes

place,

whose _{area is} of order

R(,

behaves _as

Î

~°

~

"(

_dz

_(l

~

)~~ R(. (10)

o (1 _~ z~

We

give

_a few words of

explanation

about this formula. One starts with

Ao,

whose _{area is}

already

of order (1 _~ )~. The

elementary portion

of

phase

_space, in

(X', Y') plane

limited

by

the two

diagonals

Y'

=

X',

Y'

=

X' and

by

the _two circles centered _at the

origin

and of radii _z and _{= +}

dz,

will have _to be covered

by A~

_zones. The _area of _one

particular gt,,

zone is of order (1 ~ )~ ^{z~ '} and the _area to be covered is 2

wz dz. It thus takes ~

"~

(~

^{iterations to}

(1 ~ ^Z~

caver this

elementary _portion

of

phase

_space. Since ail the accessible

phase

_space must

eventually

be covered,

equation (10)

follows. We _are thus left with the task of

estimating Ro.

The

typical

size of the region of

phase

_space where the permanent

regime

takes

place

has been

already

^{studied for} a model

slightly

different from _{ours, m} reference

[6]. Applying

their

method _{to our case,} the permanent

regime dynamics

will be confined _to the region

W+ ~ W~~~, with

W~up =

~~i

iii

+ ~

l~

^{+ 3 +}

1(1

^{+ 3 ~ )2}

+1(1 _~2)j (ii)

We have the

limiting

behaviour

W~~~

(l

_~ )~ ^' for _~

- l

(12)

This _means that _an upper bound _on the velocities _is of order

(1

_~ )~ ^'

Integration

^{of the}

equations ^{of motion between} successive bounces with the

platform

show that time between

1008 JOURNAL DE PHYSIQUE I N° 7

two consecutive bounces will be at most of order

(1-

_~ )~

,

and that maximum

heights

reached

by

the hall will be _{at most} of order (1 _~ )~^{~ The maximum values of the}

velocity

of the

hall,

taken at times where the acceleration

changes sign

will also be at most of order (1 _~ )~ ^' These _are

precisely

the values of

u,~ and

r(,

used in the

description

of the

dynamics by mappings

b and h. Thus _an upper bound _on

u,~ ^or

v(

is of order (1 _~ )~

'.

_{An upper bound} on.t.~~,

or,;,[

_is of order (1 _~ )~^~

By

definition

ofX(

and Y(

(Eqs. (5a)

and

(5b)),

the maximum

values ofX,( ^and Y( ^behave at most as

(1

_~ )~^' Thus, the

dynamics

_is confined _{to a} region of (X', Y')

plane

where X'~ X[~~, ^with X(~~

(l

_~ )~

'.

Since this _is

only

_{an upper} bound, _we checked

numerically

that the

typical

size

Ro

of the domain where the

dynamics

takes

place

in the permanent regime scales _as

(1 _~ )~ ^' for _~

- l We take _a hall at the

/p@', ^Y')

₌

^(0,

^{0 and}

integrate

the

dynamics.

We

study Ro,

the _maximum value of

,/X'~

₊ Y'~ reached

by

the hall. Since the values of

Ro strongly depend

_upon

A,

_averages have been taken _over

typically ^10~ trajectones,

with

different values of A in intervals

[A~,~,

A~~~

].

We did _not have to average on ~.

Figure

? shows

a

Log-log plot

of

Ro

as a function of (1 _~ )~^' (l A)~

"~,

for _~ between 0.5 and 0.9 and

(1 A between

10~~

_{and 10~} '

Ro

^behaves thus as (1 ~ )~^{' for} _{~ -} l.

iooo

Ro ~~

É~

ioo

1~o

,b~

io

b*

~o*

i

io ioo iooo

ji ~j-1/2 ₍₁ ~)-l

Fig.

2. Log-log

plot

of Ro ^the

typical

_size of the phase space region visited in the permanent regime (see text) _i>eisus (1 Ai ^"? (1 _~ )~

',

for values of _~ 0.5 (Q), _~ 0.6 ( + ), ~ = 0.8 (1), and

~ 0.9 ( _x ). Averages have been taken _over typically ^{102 values} of A. For example, for the point

centered at (1-A) 7.5

x10~~,

_and

q =0.6, _averages have been taken _over 100 vaiue~ of

(1 Ai _ranging from (1 A 5 _x 10~^~ _to (1 A)

= x 10~ ^~ A least square fit to the data gives a

slope

01. The uncertainty on the

slope

_is about 0.1

From

equation (10),

the number Î of iterations necessary to cover ail the accessible

phase

space behaves _as

Î

(l

_~

)~~. ₍₁₃₎

We

performed

numencal simulations,

starting

from _a hall at rest. We determined the first time

T~

^{for which the bail} cames to rest _on the

platform.

Because

T~

_very

strongly depends

_on

C results have been

averaged

_over

typically

10~ values of r in intervals

[r-

ôr,

r ₊

ôr],

with ôr

~ r. A

Log-log

of T~ versus (1 _~ )~^~ r~

',

for _two values of ris shown

in

figure

3. Least square fits _to the data give

slopes

1.002 _± 0.02 for r around 7.8 and 0.985 _± 0.03 for r around 15.8.

Moreover,

data _seem to

collapse

_{on a}

single

curve.

iooooo

ioooo

T~ r-i

iooo

ioo

io

io ioo iooo ioooo iooooo

(i v)-~

Fig.

3.

Log-log plot

of the ratio T~ ^r~ _~, where T~ is the

penod

of the movement of the bail and r the reduced acceleration

i,eisus (1-

~)~~

for _two values of n (%) 7.4w rw8.2 and (+) 15.4<

r _w 16.2. Each point

corresponds

to averages over

typically

¹⁰³ values of rand _one definite value of _~.

The straight lme is _a least square fit _to the data for 7.4 _w r

w 8.2 and has _a

slope

002.

4. Conclusion,

We have mtroduced _a

simplified

version of the

partially

inelastic

bouncing

hall _on a

vibrating platform

which enables the calculation of fixed points. ^For an acceleration

larger

than 2 g,

beyond

_some restitution coefficient _~

=

0.6,

ail

trajectones

_seem to end up to be

penodic.

However, the

period

_con be shown _to

diverge

as

(1- ~)~~

for _~

- l, ^and as r for

r

- oo.

JOURNAL DE PHYSIQUE T 4 N' 7 JULY 1994 17

1010 JOURNAL DE PHYSIQUE I N° 7

Acknowledgements.

We thank J. M. Luck for very useful discussions and the region P-A-C-A-

(Provence, Alpes,

Côte

d'Azur)

for generous allocations of computer time on the

Cray

YMP 2E of the I.M.T.

(Institut

méditerranéen de

technologie).

Appendix.

We

give

below the exact expressions for the mappings b and h which appear m

(6).

The

mapping

b is defined

by

X(= ~/X(+2(A+1)Y,,+A(A+1), _(A.la)

Y( _{= Y,~} + A

,

(A. ^lb

if

X,,

_{~ ,} A(A ₊ and _Y,~

~

l~

X,, (A. lc A ₊ '

or if

X,,

_{~ ,} ^A

(A

+ and Y,, ~ l

,fi, _(A.

là

and, otherwise,

by

Xl~ /) YÙ-(1(A+1)Yj-AX]j, (A.le)

~

Y(=Y,~+A+(1+~)~/(A+1)Y(-AXj. _(A.lf)

The mappmg h is defined, if X,[ # 0

by

_:

~ll

+ ^~

~n

+ ^~

~

~~'~~~

if

Y(

_~

iXj, (A.2b)

where

r =

~ ~

(A.2c

)

~

This

corresponds

to the

absorbing

_region,

Otherwise,

h is defined

by

X,~

~

= ~

"'X,(

,

(A.2d

Y,~_~ =

Y(

+

il

^{+ 2 ~}

^~~

⁺

^{"~) X(, (A.2e}

~

with

m =

Int

(Ln (

^{~ ~}

_{Q'j /Ln ~) (A.2f)}

2

and

Q'

y~j

= ~,

(A.2g)

n

« Int

» denotes the integer part ^of a real number.

References

[1] Fermi E., Phy.i, Rei,. 75 (1949) l169.

[2] Kowalik Z. J., Franaszek M. and Pieranski P.,

Fhys.

Rei>. A 37 (1988) 4016.

[3] Everson R. M., Physica 19D (1986) 355.

[4] Celaschi S. and Zimmerman R. L., Phys. Leit. A120 (1987) 447.

[5] Mehta A. and Luck J. M.. Fh_v.i. ^Rei,. Lent. 65 (1990) 393-396.

[6] Luck J. M. and Mehta A., Fhys. Rei,. E48 (1993) 3988.