Stochastic and Regular Motions in System with 'Yawn'

Copyright © IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995

STOCHASTIC AND REGULAR MOTIONS IN SYSTEM WITH 'YAWN' Alexander Yu. Pogromsky Laboratory of Control ofLarge-Scale Systems, Institute ofProblems ofMechanical Engineering, 6l , BoLshoy, VD, St. Petersburg, 199178, RUSSIA

Abstract

The complex oscillations appearing in the system with clearance between the drive and driven sections of mechanical transmission are considered and the mathematical model is proposed for certain assumptions. It is shown that the model possesses a resemblance to the Ulam map and has regular and irregular behavior that is typical for the motions of the driven section in system with 'yawn'. Key words. Conservative system, dissipative system, chaos, strange attractor, bifurcation.

1. In the present consideration the 'yawn' is described by the kinematical model, the energetics of the system is not studied. 2. This paper regards only periodical reference motions of the drive section.

1. Introduction and Problem Statement.

In automatic control systems, mechanical connection between an executive motor and a controlled object may be accompanied with a 'yawn'. Constructively, the 'yawn' is a clearance between drive and driven sections of the mechanical transmission (fig. 1). It is known that 'yawn' has a significant influence on the quality of servo systems, particularly on the high-accuracy systems. In [1-2] various aspects of this phenomena were considered with respect to automatic control systems, and mathematical models were proposed. However, these models assume absolute nonelastic impacts between drive and driven sections. The goal of the present consideration is the scrutiny of the oscillations in the 'yawn' when the impacts between the sections are assumed to be elastic, the friction is absent and the reference signal is sinusoidal.

Assume that the motion of the drive section is sinusoidal : (Ll) .9 = Acos qJ, where A is the magnitude and qJ plays the role of time. Then, for the n + 1th impact it can be written: vn +l = -av n - A(l + a)sin qJn (1.2)

where v n' Vn+l denote the speed of the driven section before and after impact and a is coefficient of restitution, 0 < a < 1. The next moment of the impact can be found as a solution of the following transcendent equation: ACOSqJn+l - AcosqJn

±B= v n+l (qJn+l -

qJJ,

0 .3)

where B is the value of the clearance between the sections and among the set of the solutions of thi s equation only the least is acceptable.

~.

Equations (1.2), (1.3) detennine the motion of the system with clearance between two sections of mechanical transmission when the friction is absent and the drive section moves sinusiodally.

,,- ' Fig. I.

Jacobi matrix of the map (1.2), (1.3) can be obtained as:

Before the description of the proposed model, it is necessary to remark the following :

(lA)

385

where jl

j. ,

=

=-a,

j2

=-A(l + a)cos
a(

all" '

~n+l where p = q lB .

j = -aAsin IPn - avn + A(l + a)(IPn+l - IPn)cos IPn

A sin IPn+l - A(l + a)sin IPn - avn It is seen that 4

fI det J; =

a 2n

A sin CfJI + VI Asin CfJn+1+ V,,+I

(2.2)

1

=
.

Jacobi matrix of this system is obtained as:

in

(1.5)

;=1 Thus, if the denominator of this expression is not equal to zero, one can find n, such that the multiplication (1.5) is less than unit. It implies the dissipative behavior of the system (1.2), (1.3) (when a < 1).

aa

=[ _

1-

(a~n +f3sincpj

Moreover, detJ n conservative.

=a

f3P~~:~n ].

(2.3)

(a~n +f3sincpj

and for a

=1

this system is

We will first study the map (2.2) in its conservative fonn (a 1) and for & 0. In this case map (2.2) possesses a resemblance to the Ulam map studied in [4].

=

The system (1.2), (1.3) demonstrates complex behavior, particularly for the coefficients of restitution which are close to 1. For such coefficients the dynamics of the map (1.2), (1.3) can be irregular for a long time (thousands of the impacts) before the convergence to the simple attractor is occurred. This erratic behavior is not chaotic, because the trajectories settle to an equilibrium sooner or later, but the transient time can be significantly large and the trajectories look like chaotic in usual sense. That is why to analyze the oscillations in system with 'yawn' it is reasonable to simplify the model (1.2), (1.3) such that the simplified model has similar behavior, including regular and irregular. The next section of the paper deals with a description of how the irregular motions of (1.2), (1.3) appear and disappear.

=

For ~» /3, a= 1 , system (2.2) can be described by the following system of differential equations:

d~(n) = ,8sin
(2.4)

where n plays the role of time. Excluding n and separating variables in written for & = 0: ~

2. Simplified Model of System with 'Yawn'.

= Ce - fJcos rp.

(~.4)

it can be (2.5)

Here C is an integral constant.

Assume that the value of clearance B is sufficiently large, such that B I A > > 1. This assumption leads to the consideration of the following map:

Expression (2.5) determines curves which the invariant phase trajectories of system (2.2) asymptotically try to attain when ;»,8.

= all" +qsin
Then denoting ~ n rewritten as

= V niB,

(2.1)

The phase portrait of system (2.2) for considering case

the map (21) can be

(&= 0, a= 1) is implemented in figure 2 (,8= 1). One can notice that (2.5) gives good quantitative description of the invariant curves. Moreover, it can be seen that there is a chaotic motion in the area of small

speeds, i.e. there exists a minimal constant Co for which the behavior of the map (2.2) is regular and determined by (2.5). As ~ decreases, the invariant solutions destroy, forming stochastic area.

* To obtain this map from (1.2),(1.3) it shoul be assumed that V in (2.1) is positive. Generally it is not valid, however, motions of such kind are rather typical for a system with 'yawn'. See [3] for more fonnal consideration in similar model. 386

{:'~~~~n(l~al' {:"~~:='mc~al

(2.7)

The first of them is a node (focus) and the second is a

10-

a) / /3~ > 1 map (2.2) has only one saddle. When attractor - strange attractor which is implemented in figure 4 for the following parameters: -8

a=0.8, Fig.2.

When the condition

Stochastic motions in considered system turns out to be sensitive to the variation of parameter & . In the conservative case for & =f:. 0 system (2 .2) has two equilibria, corresponding periodic motion of the driven section with minimal period:

{

~' =-~ cp'

=0

& '

{

~" =-~ cp"

/3=1, &=0.

COS CPn

=0

(2.8)

is valid, it can be obtained from (2 .3) that

SpJn

= a + 1.

(2 .9)

In other words, if system (2.2) possesses an equilibrium (2.7) for which condition (2.8) is satisfied, then one of its multiplicators is equal to 1. Lines cP = 7r / 2 and

(2.6)

cP = 37r / 2 determine two areas:

= 7r

The first of them is a center, the second is a saddle. If these equilibria do not belong to the stochastic area, then the phase portrait of the system is similar to the one implemented in figure I and the way of intermission does not change. As these points 'penetrate' to the stochastic area, the separatrix motions through the saddle around the center fonn the stochastic channel. Figure 3 illustrates this phenomena for

7r 37r O
(2.10)

=-0.06· 27r, /3 = 0.8 . It

should be noticed that the dynamics of the exploring system in conservative case bears a remarkable resemblance to the Fermi model of stochastic acceleration studied in [3-6]. &

·1.5

Fig. 4. In the area <1:>+, both multiplicators are less than unit, Fig.3.

and in the area <1:>-, one of them is greater than unit. Since the greatest of multiplicators is equal to I on the

In the dissi pative case (a < 1), system (2.2) has the following equilibria which correspond to 27r-periodic motion of the driven section as well as (2.6):

lines qJ = 7r / 2 and cp = 3 7r / 2, the saddle-node bifurcation can occur on these lines. From (2.7), the condition of this bifurcation is as follows :

387

unstable motions of the strange attractor. It can be seen

1- a

- - =1. /3&

in figure 5 that as I~ increases, the saddle point must

(2.11)

get into the strange attractor area. When I~ is equal to some threshold value, the strange attractor contacts the separatrix, that results ip. the crisis of the strange attractor. This bifurcation is similar to the bifurcation of the transformation of a limit cycle to a homoclinic loop. A similar crisis was considered in [7]. However, in that case after the crisis strange attractor merges with the other strange attractor and as a result, the motion can be possible either in the area of the first or the second 'former' strange attractors. The considered case is not the same: after the crisis the revival of chaotic motions is possible provided that the explored scenario is realized in the inverse order. Considered crisis results in that the chaotic motions get into the attractive basin of the focus, which becomes the sole attractor of the system. This case is illustrated in figure 6 for the following parameters: a=0.8, {3=0.8, &=-0.06·27r.. It can be seen that the center considered in figure 3 is transformed into the focus and the stochastic channel with separatrix motions turns into the separatrix which goes to the focus. One can notice from the figure that for these parameters the chaotic motions are preserved yet and the chaotic behavior can be retained for a long time. Moreover, the presence of disturbances usual for a real system can result in that the stable motion will not be observed for the time acceptable for an experimenter. As

Assume that this condition is satisfied for some value of & . Let us consider what will happen when the absolute value of & increases. After the birth of the nonrobust saddle-node equilibrium when I ~ increases it forks to the saddle (in the area

<1>-) and the stable node (in the area <1>+). Therefore, the system now has two attractors: the node and the strange attractor. Expressions (2.7) determine in an implicit form subsequent movement of the saddle and the node in the phase space as I~ changes. Figure 5 shows the phase portrait of the system for

a = 0.8, /3 = 0.8, &= -0.04 · 27r. Curves a and b are the lines along which the equilibria 1 and 2 moves as I~ increases. Excluding & from (2.7), it is not difficult to obtain that these equilibria moves along sinusoid:

;' = ~sin rp' , rp' E <1>+, 1- a

;" = ~ sin rp", rp" 1- a

I~

(2.12) E

<1> - .

further increases, the saddle penetrates deeper and

deeper into the area of the destroyed strange attractor, and regular dynamics begins to prevail over the chaotic one (fig. 7). This figure shows the phase portrait of the (2.2) for the following parameters: map

a = 0.8, /3= 0.8, &= -0.1· 27r. It is seen that the trajectories which are close to the separatrix go to the focus through the saddle and the domain of the irregular motions is sharply shortened with comparison to the previous figure . ~

Fig. 5. As the node point moves along section a of the curve (212), when the condition

{SpJn(;n

= ;', rpn = rp,)}2 -4a= 0

(2.13)

is valid, the node is transformed into the focus . The presence of the saddle equilibrium after the saddlenode bifurcation ensures the existence of the separatrix, separating the stable node (focus) motions from the

Fig. 6.

388

[5] Zaslavsky G.M., Sagdeev R.Z. , Usikov D.A. Chemikov AA "S/abyi chaos i Jevasiregu/yarnye struktury," Moscow-Nauka, 1991, [in Russiuan] [6] Moon F.e. Chaotic vibrations, New York: Willey and Sons, 1988 [7] Afraimovich V.S., Rabinovich M.I. , Ugodnikov AD . "Kriticheskie tochki i fazovye perehody v parametricheski vozbuzdaemom angarmonicheskom oscillyatore so stohasticheskim povedeniem", Pisma v ZETF, 1983, v.38, N2, pp.64-67 [in Russian)

.;;.

Fig. 7. 3. Conclusion In conclusion we have demonstrated the behavior of the simplified model of the system with ya'M1 under the following assumptions: • The impacts between the sections of the mechanical transmission are considered to be elastic. • The friction and external forces do not affect the motion of the driven section. • The drive section moves sinusoidal. • The value of the clearance between two sections of the transmission is sufficiently large. It has been sho'M1 that these assumptions lead to the consideration of the map possessing the complex behavior. Under certain conditions, it describes the regular and irregular motions which are typical for the motions in the system with ya'M1.

It is necessary to say that explored model is very simplified to be utilized in the synthesis of the servosystems but it possesses the complex behavior that can be observed in experimentation [see 6 and reference therein] . Author believes that studying this model can help design more competitive control algorithms for servosystems with ya'M1. References [1] Kblypalo E.A "Nelineinye systemy avtomaticheskogo upravlenia ", Leningrad-Energia, 1967 [in Russian] [2] Mikerov AG. "0 vybore parametrov nezestkosty mehanicheskoi peredachi po pokazatelyam kachestva sledyachei systemy" Elementy, ustroistva i programnye sredstva in!ormazionnopreobrazovatelnyh system, 1989, Ryazan, p.102-111.

[in Russian] [3] Zaslavsky G.M., Chirikov B.v. "0 mechanizme uskorenia Fermi", Doldady Academ. Nauk USSR, 1964, v.159, N2, [in Russian] [4] Lichtenberg AJ. , Liberrnan M.A. Regular and stochastic motion.- Springer-Verlag, New York, 1982.

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