Numerical study of a model of vibro-transporter

Volume 135, number 2

PHYSICS LETTERS A

13 February 1989

NUMERICAL STUDY OF A MODEL OF VIBRO-TRANSPORTER M.-O. HONGLER, P. CARTIER and P. FLURY Instilut de Microtechnique, Departement de Mecanique, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland Received 12 October 1988; accepted for publication 5 December 1988 Communicated by A.R. Bishop

We discuss the transport properties of a simple model of vibratory feeder commonly used in automated assembly chains. The dynamics is described by a set of coupled, non-linear and strongly dissipative mappings. Periodic verus chaotic solutions are clearly exhibited and their influence on the transport rate ofthe system is analyzed.

1. Introduction of the model One of the difficulties in the realization of automatic assembly lines is to convey parts to the ad hoc locations in the chain. A solution which is commonly adopted, is to use vibratorytransporters (also called vibratory feeders). Basically, a vibratory feeder is constituted by an oscillating track on which the parts to be conveyed are disposed. When the track is set into motion, the mobile lying on it is itself set into movement. Since the pioneering work of Redford and Boothroyd [1], theoretical and experimental aspects of vibro-transportation have been abundantly studied (a selection of articles is given in ref. [2]). This important activity of research clearly reflects the difficulties which the constructors of feeders have to deal with. Schematically, the device is represented in figs. 1 and 2 where the notations to be used are introduced, The reference frame xOy is mobile and attached to the track. In actual applications, the vibratory transporter is either a bowl or a linear track. Here, we shall restrict our discussion to the linear case for which the centripetal and Coriolis accelerations are absent (the dynamics for the bowl shape case presents, in its essence, identical features as locally it reduces to the linear case fig. 2). In view of fig. 2, the general equations of the motion have the form 106

/

-~

Th

Bowl Suspension

--

spring Electromagnet

Fig. 1. Vibratory feeder.

m5~(1) = maw2 sin (wt) mg sin (a) + F, 2 my(t)=mbw sin(wt+y)—mgcos(a)+N, —

(1 a) (lb)

where dots denote the derivatives with respect to the time, F and N stand respectively for the friction and the constraint forces, a is the slope ofthe track, g the gravitational acceleration and y the phase shift between the parallel and perpendicular components of the excitation force. Depending on the external parameters, various types of motion exist and a detailed analysis of the possible periodic motions is given in ref. [3]. Here, we shall confine our attention to the jumping regimes (i.e. we shall select the parameters such that aco2 > gcos (a)). In these regimes which are the most

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~-u(t)I1~=W~,

jv(t)Ir~=~n.

(6)

N

\

Using eqs. (4)—(6), we can reactualize the initial conditions each time an impact has occurred. Hence the direct integration ofthe free flight equations (2) yields the set of non-linear mappings [21

‘~ mg

0

2b

=sin(r~~1)—sin(;),

(7)

—cos(r~)—Y’~]

(8)

,

=R11{—,i[cos(r~+1 +y)—cos(r~+y)] —ktg(a) (r~+~ —t~)+~~}

~J~+i

2a

(9)

.

Fig. 2. Modelization of a vibratory feeder.

common situations (typical values are w=27rX50 s’ and b = 50—100 ~tm) the dynamics between the impacts simply reduces to free flight equations, namely [2]

The transport rate itself can be calculated with the mean velocity W~ (in the parallel direction) attamed between successive impacts; thus we obtain

w,,=

1

tn+iTn

JC Tn

ü(r)=sin(r)—k,

av(r’)d, a~’

(2a) ~

i3(t)=,~sin(r+y)_ktg(a)

(t~+1—t~)

(2b)

,

where r, u(r), v(r), kand ,~are dimensionless quantities defined by 2, ~=a/b, twt, k=gcos(ct)/bw u=y/b, v—x/a. (3) The dynamics at the nth impact time r= r,, is specified in the form

—

~

~—; [sin(r~+1+y)—sin(t~+y)].

(10)

The dynamics of the model is now completely characterized by eqs. (7)—(10).

2. Discussion of the solutions Let us now discuss the different solutions of the

=

—R

~

(4a)

model. First of all, one has to remark that eqs. (7) and (8) can be discussed independently of eq. (9).

V(t)IT,,,Tn_~~

(4b)

Eqs. (7) and (8) are precisely the mapping recently studied in the context of a bouncing ball on vibrating tables [6,7]. Let us briefly recall the main results. The mappings eqs. (7) and (8) exhibit a Feigenbaum cascade of bifurcations (here, the control parameteris k) [4,5,9,10]. From eqs. (7) and (8), the period-one solutions (characterized by r~ = + (27tn ) r, reiN) are immediately found in the form:

1 ~-u(r)

at

v(r)J~+~=—R11

-~--

where the coefficients of the perpendicular and parallel restitution are denoted respectively by R ~ and R11 and e is an infinitesimal quantity which relates times just before and after the impact time t t~. Obviously we have 0
(5)

2irrkR1 1+R1

(ha)

Now, let us introduce the notations 107

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ro=Acos(itrk~~).

(llb)

k

One R ~ is fixed and hence the mobile to convey selected, the unique control parameter which remains is k. The stability intervals for the period-one solutions are obtained by a standard linearization procedure. The result reads [5—8] ki,r>k>k

13 February 1989

0.636

(12)

2,r,

where k1 ,. and

k2r

read

0.498

(13a)

kir~~

_______

and

0.318

k2r4~2r2(~

)2

(i+R2)2]~2

(l3b) When k is decreased below k2~a stable period-two orbit is found. This behavior is observed until a new critical value, say k3r, is reached; there, a new period doubling occurs and so on until ~ below which the chaotic regime is attained [6—10]. A sketch of the situation is summarized in fig. 3. The succession of the critical values ku,. approach an accumulation point koo,r according to the sequence [7,9,10]:

0295

chaos

...

lim k

= 0.46992...,

i,r 4~r ~~+2.r

7

1,2,3

~7+l,r

( )

Using eq. (13), the transport rate in the simplest periodic regime takes the form / 1 —R 1 +R ~ Wfl=W=7trk(\llcos(Y) 1+R~—tg(a) lR,)

(

— i sin ~‘)

[

— ~ 2k 2 r2

21~2 (l—R±\ 1 +R ) j ~.

(15)

While it is obvious to obtain eq. (15), the estimation of the transport rate is far less trivial in the chaotic regimes. Indeed, the calculations would require the knowledge of the invariant measure governing the statistics ofthe t~, the cI3~and the ~ Unfortunately, such a knowledge is presently not available and we have to explore the behavior of the transport rate numerically. In actual feeder, the situation coefficients R ~ and 108

chaos

0212

penod-one

0.205 Fig. 3. Scenario ofthe dynamicalbehavior. R1 = 1/3.

R~are relatively small. Typical values are in the range of R.1 and R11=0.3333 [1]. In this case, we have a strongly dissipative dynamical system (observe that the Jacobian of the mapping of eqs. (7), (8) equals R1) and the simplification which reduces the imphicit eqs. (7) and (8) to an explicit mapping (the so called standard mapwe[4,5]) available [8]. In the present paper, choseistonot iterate eqs. (7), (8) with R = 1 / 3 and R 11 = 0.2. With these values, eqs. (13) predict the simple periodic motion in the intervals r=l: r=2:

k21 =0.498
(l6a) (l6b)

r=3: k2,3=0.205
(16c) (1 6d)

Our numerical calculations enable us to observe

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1.0

0.8 >~

=

--I

C,

-.

0.6

/

2

a 0.4

0.2

0.0~--~•

.

-

2

4

6

8

Phase

Fig. 4. Iteration of the mapping eqs. (7) and (8). k= 0.43; R

1 = 1/3; 1500 interations.

Mean of W as a function of K.

~

Psiinit=.9424

a 3

~

gamma=o.o alpha =0.0

0

~

0~4

0~6

Fig. 5. Mean transport rate as a function of the excitation parameter k (the increment on For each value ofk, we perform 600 iterations of the mapping.

K

k is 0.0025). R1 =

1/3; a=0.0;

1.0

y= 0.0.

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3

2

0~.1.

0~2

0~3

K

04

Fig. 6. Mean transport rate as a function ofthe excitation parameter k (the increment on k is 0.0025). R~= 1/3; a=0.2; y=O.O.

(for r= 1 and r= 2) the Feigenbaum cascade of bifurcations. In particular, in the case r= 1, we observe that the k,1 0.47 and then the chaotic solutions occur for k,i <0.47. A typical representation of the phase space is sketched in fig. 4. The chaotic nature of the motion is explicitly visible, Concerning the transport properties of the system, we have sketched in figs. 5 and 6 the mean transport rate = (1/N) ~,, W~for a selection of kranging from 0.05 to 0.98 (step 0.0025), ,~=4,y=O and 0.2; a=0 and 0.5. The period-one regions eqs. (l6a)— (1 6b) (for r= 1, 2, 3, and 4) are easily identified (they are drawn in fig. 5). The period-one regime leads to high transport rates . The decrease of the transport in these regimes characterized by eqs. (16), follows directly from eq. (15). Remark the linear dependence obtained when y= 0. Except in the regions determined by eqs. (16) and in the rather narrow intervals defined by k2r
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high occur, the transport rate exhibits a (positive definite) random looking behavior which implies a relatively low seems to be relatively independent of the external parameter k; i.e. no net tendency emerges. Let us further devote special attention to the fine structure of the curve in chaotic regions. We can ask whether this fine structure remains unchanged under (reasonably small) perturbations and how it is affected by the precision used to iterate the mapping eqs. (7) and (8) (remember, eq. (7) is implicit and has therefore to be solved by successive increments). To answer these questions, using identical t0, !P~Dand ~ we have calculated are indeed dependent on the

Volume 135, number 2

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/

4

2

00,0

~

0~2

0~3

0~4

‘

K

Fig. 7. Mean of < W> over a selection of nine different values ofthe increments used to solve eq. (7). For each value of k (the increment on k is 0.0025), we perform 400 iterations.

choice ofthe increments. In fig. 7, we show the mean (> obtained from these different calculations. The fine structure of << W>> exhibited in fig. 7 therefore is more robust under small perturbations than the fine structure of fig. 5. Observe how the details drawn in fig. 5 have been smoothed in fig. 7. In industrial realizations, it is not expected that the external parameter k remains precisely fixed at one single value. In fact, the system is permanently subject to external noise and then is likely to explore a range of values of k. The sources of the external noise are of course multiple; let us mention that in practice the excitations are never perfectly sinusoidal and their amplitudes can be altered by the environment (thermal dilatation, time dependent characteristics of the vibrators, etc.). On the other hand, the geometry ofthe conveyed parts itselfplays an important role; in particular the restitution coefficients R1 and R11 will be dependent not only on the constituting material but also on the shape of the mobiles. Think for instance ofthe bouncing of a cube;

its restitution coefficients

R

1 and R11 are certainly quite different for an impact on the edge or on- the face. The differences between these behaviors could be introduced phenomenologically by taking into account different values of R1 and R11. Incidently, the influence of external noise can be partially appreciated from the influence of the choice of different increments in solving eq. (7). Indeed, note that small precision errors in the resolutions of eq. (7) affect the dynamics in very much the same manner as the introduction of random errors after each impact in the sequence of r,,. Hence, the noise does not significantly affect the simple periodic solutions and in the chaotic domain has to be estimated by taking the mean over different realizations of the noise (as performed to obtain <> in fig. 7) and over different values of the control k.

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3. Conclusions and perspectives We have analyzed numerically the dynamical behavior and the transport properties of a modelization of a vibratory conveyor used in automated assembly chains. Our study is performed in regimes for which the parts arejumping over the tracks. This regime is always selected when high transport rates are required. Contrary to most studies performed on this topic (note the exception of ref. [11]), our discussion is not confined to the periodic regimes. Here, we have carefully taken into account the presence of chaotic solutions the existence of which is inherent to the non-linearity of the dynamics. In view of the numerical results, we can roughly isolate two types of transport regimes, namely: a high transport rate attained when the solutions of the equations of the motion are periodic and a lower (although positive) transport rate which occurs when the motion is chaotic. In this last regime, the transport rate depends erratically on the external control k. However, this pseudo-random variable besides its strictly positive nature, does not exhibit a net tendency to increase or decrease as k varies and hence the actual transport rate (which can be estimated by taking the mean of < W> for different values of k) will roughly be independent of k. In ref. [11], where a slightly different dynamical model is considered, a similar qualitative observation is reported. In actual realizations, except the simplest periodic regime tunable for a relative wide range of external parameters eq. (1 6a), the other periodic regimes (characterized by narrow bands of the external control given by eqs. (16c) and (16d)) are unlikely to be observed. Indeed, the ubiquitous presence of external noise prevents the possibility of stable motions in regimes characterized by these narrow gaps of k [6]. Finally, a direct qualitative observation on actual feeders in operation, reveals that the progression of the conveyed mobiles is manifestly erratic. This ob-

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servation, together with the relative independence of of k (a property required for proper operating conditions), indicate that the chaotic dynamics, far from being a complication to be omitted from the description, is in fact important in the discussion of vibro-transportation.

Acknowledgement Professor C.W. Burckhardt is warmly thanked for his interest. We are indebted to Professor J. Figour for drawing our attention to the problem of vibrotransportation. Dr. C.V. Quach Thi is thanked for her careful reading of the manuscript.

References [1] A.H. Redford and G. Boothroyd, Proc. Inst. Mech. Eng.

182

(1967—68) 135. [2] M.-O. Hongler and J. Figour, Periodic versus chaotic motion

in vibratory feeders, Helv. Phys. Acta, to be published. Taniguchi, M. Sakata, Y. Suzuki and Y. Osanai, Bull. JSME 6 (1963) 37. [4] A.J. Lichtenberg and M.A. Lseberman, Regular and stochastic motion, Appl. Math. Sci. 38 (Springer, Berlin, 1983). [5] J. Guckenheimer and P.J. Holmes, Non-linear oscillations, dynamical systems and vector fields, Appl. Mech. Sci. 42

131 0.

(Springer, Berlin, 1983). [6] N.B. Mello, Y.M. Chos and A.M. Albano, J. Phys.Tufillaro, (Paris) 47T.M. (1986) 173. [71N.B. Tufillaro and A.M. Albano, Am. J. Phys. 54 (1986) [8] C.N. Bapat, S. Sankar and N. Popplewell, J. Sound Vjb. 108 (1986) 1477.

[9] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25. [10] P. Collet and J.-P. Eckmann, Iterated map on the interval as dynamical systems (Birckhaüser, Basel, 1980). [Il] Ya.F. Vaynkof and S.V. Inosov, Mech. Sci. Maschinovedeniye 5 (1976) 1.