Dynamical behaviour of an oscillator sustained by an intermittent mechanism

Journal of Sound and Vibration (1983) 88(l), 27-36

DYNAMICAL SUSTAINED

BEHAVIOUR

OF AN OSCILLATOR

BY AN INTERMITTENT

MECHANISM

F. BADRAKHAN College of Engineering and Petroleum, Kuwait University, Kuwait (Received 9 March 1982, and in revised form 22 June 1982)

A new method for studying the dynamical behaviour of an oscillator sustained by an intermittent mechanism is presented. Graham escapement is considered as a typical model for such a mechanism. A new form given to the equation of motion permits the study of the natural frequency of the system, the establishment of a first integral in the case of a free motion and the analysis of effects of external harmonic or random perturbations.

1. INTRODUCTION For a linear oscillator sustained by an intermittent mechanism (a ratchet mechanism, an intermittent gearing or a Geneva wheel), the mechanism effect may be represented by periodic pulses. The length (or duration) of a pulse depends on the mechanism and it is predefined by design requirements. Figure 1 shows the theoretical shape of the pulse (force or torque) during one period for a ratchet mechanism or an intermittent gearing (Figure l(a)) and that for a Geneva wheel acting during one-sixth or one-quarter of a period (Figures l(b) and (c), respectively). Knowing the time function defining the periodic pulses, one may study the dynamical behaviour of the oscillator if this time function is replaced by its Fourier series.

(a)

(b)

ICI

Figure 1. &ape of the pulse transmitted per cycle by (a) a ratchet mechanism or an intermittent (b) and (c) a Geneva wheel.

gearing,

The effect of any intermittent mechanism may be interpreted by periodic pulses too; but the shape of the pulse is not always a simple one especially if this shape is determined experimentally only and has no analytical function. The Fourier series becomes difficult to obtain and the classical method cannot be applied in this case. The same difficulty arises if the oscillator is non-linear. The purpose of this paper is to present a new approach for the study of the dynamical behaviour of an oscillator sustained by any intermittent mechanism. The Graham escapement mechanism will be considered as a model for the study, but the method presented is general. It can be applied to the case of any intermittent mechanism even if the pulse is determined experimentally and even if the oscillator is non-linear. 27 @ 1983Academic Press Inc. (London) Limited 0022-460X/83/090027+10 $03.00/O

28

F. BADRAKHAN

The Graham escapement is known as an intermittent motion mechanism for producing rotary or rectilinear motion by steps. In this type of mechanism, a toothed wheel (Figure 2), to which a torque is applied, is allowed to rotate in discrete steps by the action of a pendulum or a spring. Because of this action, the mechanism can be used as a timing device and, as such, it finds its widest application in mechanical clocks and watches. A second application is its use as a governor to control displacement, torque or velocity [l]. The other principal component of Graham escapement is constituted by the anchor arms, called pallets, which make contact with the teeth of the escape (toothed) wheel.

Figure

2. Escapement

mechanism

(the oscillators

are not shown).

A complete study of the actual behaviour of escapement and that of design and performance characteristics is beyond the scope of this paper (see references [2] and [3], for example). The aim here is to reconsider the study of the dynamical behaviour of an oscillator sustained by this type of mechanism. The intention is to give a new approximate method by considering the system (mechanism+oscillator) as a whole and taking into account the non-linearity of its motion. The method used is simpler and more appropriate than any other known method. It is related to the “trace” method suggested for the study of systems with hysteresis [4,5]. It allows the study of free oscillations, the determination of a first integral of motion in the presence of viscous and Coulomb damping and the study of the influence of harmonic or random external perturbations. 2. THE (TORQUE-DISPLACEMENT)-DIAGRAM Denote by 0 the angle of displacement of the oscillator (pendulum or spiral spring), of motion, J its mass moment of inertia and by % the total torque, including the effect of escapement. To analyze the escapement phenomenon, two cases are considered, as follows. Case 1, e < 0 (Figure 3(a)): for A > 19> 19,, the oscillator is free and the escapement components are maintained fixed by appropriate abutments; for til > 8 > &, the escapement is released; a resistant torque Cd < 0 results; this torque is supposed to be constant; for 8 = 02, an impulsive motion starts; an impulsive torque Ci >O is applied and is supposed to be constant; for 13= 8 3, the impulsive motion ceases, the escapement components are immobilized and the oscillator is free again. Case 2, 4 > 0 (Figure 3(b)): similar phenomena to those of case 1 occur; specifically, for -A < 6J< 03, the oscillator is free; for & < 8 < 04, a resistant torque -C& results; for o4 < 8 < 13r,an impulsive torque Cl is applied. If the escapement is considered to be kinematically and dynamically symmetrical, which is the case in general, then tY4= -&, 19~= -8, and CL = Cd, Cl = Ci.

A the amplitude

INTERMITTENT

MECHANISM

Figure 3. Torque-displacement

OSCILLATOR

29

diagram. (a) 4 < 0; (b) 4 > 0.

The angle r = & +A =A --el is the rest angle, while d = e1 -& = 04- O3 is called the clearance (or release) angle and i = & -& = @I-e4 is the impulse angle. In practice, usually d
Figure 4. Escapement loop: loop elements and limit cycle

30

F. BADRAKHAN

counterclockwise in accordance with the fact that a certain amount of energy, expressed by the area of the loop, is given to the oscillator during each cycle. In a steady motion, this energy is to balance the amount of energy dissipated by damping due to Coulomb friction and/or to viscous effects. The energy given per cycle by the escapement mechanism is expressed by W = Cd(el -eJ + Ci(ez -e3) - C& (e, - e,) + Cl (e, -19~),or,in thecaseofasymmetricalescapement, W=2Ci(e2-es)-2Cd(e1--e2)=2Cii-2Cdd. The equations of the loop segments in the (torque-displacement)-plane are the following:

ko:

‘g=ke, ABeBe %=kke-c,, e1 > 8 > e2 % = ke + Ci, e,>e>-8, ’ -e,>e>-A .V=ke, I

e>o:

%=ke, %=ke+c,,

-Ace<-e1

%=ke-Ci, vz=kke, I

-e2
-el
(1, 1’)

:

3. THE EQUATION OF MOTION The equation of motion of the oscillator sustained by escapement has the general form 8;+2h~+Fsgn~+f(e,~)=E(t),

(2)

where A is the constant of viscous damping, F is the torque exerted by Coulomb friction, %Y =f(@, 6) is the torque due to the escapement, represented by the loop in the (torquedisplacement)-plane and E(t) is the external perturbation. E(f) may be a deterministic or a random function of time and is equal to zero in the case of a free motion. The function f depends on 8 and on sgn 6 (the sign of the velocity d). It can be written in the form f =f+ with

f =f1f=f+

for 4 < 0 ford>0

(equation (1)) (equation(l’)) I ’

One can write also (as in refernece [4]), f(e, 6) = g(6) - h (e, e’), with g = $(f_ +f+) (an odd function of e), and h =$(f_-f+) =-h(e) sgn 4 = -h, (an even function of 0). In particular, the curve %’= g(8) may be called the truce of the escapement loop (Figures 4(b) and (c)). With these considerations, the equation of motion (2) may be written in the form 8i+2hi+[F-h(e)]sgni+g(e)=E(t).

(3)

This equation is non-linear with h(e) and g(8). It can be shown [4] that the left-hand side of equation (3) may be replaced by an approximate expression preserving the non-linearity of g(e). The approximate equation leads to a solution more accurate than that obtained by the first harmonic method of Krylov-Bogolioubov. It is obvious that any non-linearity originally present in the stiffness of the oscillator may be added to that of g(B) and the equation of motion (3) remains of the same form. In the case of a harmonic motion, the torque V = 2hi is represented in the (torquedisplacement)-plane by an ellipse (8) defined by %’= 2AwdA* -e’, or (%‘/2AoA)*+ (e/A)* = 1, where w is the radian frequency. The torque 5%’ = F sgn i is represented by a rectangle (9) defined by bO+%=F, 1 O
-AsesA Aze>-A

I’

by a loop (Figure 4(b)).

INTERMITTENT

MECHANISM

31

OSCILLATOR

The main idea of the approximate method suggested in this paper is to replace the rectangle and the loop by two ellipses (Z?i) and (g2) respectively (see reference [4]). The ellipse (8i) replacing the rectangle is defined by (%/B1wA)* + (8/A)’ = 1, where Bi is such that rB1wA2 = area of rectangle =4FA, and the ellipse (g2) replacing the loop of escapement is defined by (%‘/B2~f31)2+(6/8~)2 = 1, where B2 is such that rB2w& = area of loop = 2Ci(8i + 0,) - 2C, (f3, - &). Ellipses (8) and (‘8r) are described clockwise (energy dissipated), while the ellipse (SYZ) is described counterclockwise (energy supplied). Finally, the equation of motion (3) may be approximated by Bi+[2A +Bl(A)-B&h,

&)]i+g(@)=E(t).

The quantity

between brackets is zero if the energy supplied by the escapement per cycle is exactly equal to that dissipated by damping. In practice, this condition is always assumed, although it is impossible to be realized. Therefore, it is logical to assume here that a certain difference always exists and the equation of motion is rather of the form B+B(A,

6, e,)i+g(e)

4. THE NATURAL

=15(t).

(4)

FREQUENCY

The natural frequency of the undamped oscillator alone is w. = & But this frequency is perturbed by the presence of escapement. The actual value may be calculated from the undamped equation

with G(0) =

8 J

k’+g(B)=O,

or 8’=2[G(A)-G(8)]

(5,6)

g(x) dx. The period of motion is deduced from equation (6) to be

0

T=

J

(7)

A[G(A)-G(B)]-1’2d0.

0

The expressions for the function G(8) are the following: G(8) = $kO* G(B)=qk8’-~(Ci-Cd)(6-62)

G(0) =$k6* +&Ci - GM0 + 62) G(8)=~ke2-~(Ci-Cd)(81-e,) G(e)=3ke2+3(Ci-C,)(e,-e,)

for --13~d19S&, for &Si3S19i, for -f9158S-62, forBis@
with k = @i, 8 = (Ci - Cd)lo& and To = 27r/wo. Equation

(8)

(7) may then be written in the

form

T2wo II

002

[A'

- 8*]-“*

which, after manipulation, T = To+;

d8 +

J

81 02

J01 A

[A2--e2-(A-8)S]-1’2dt?+

[A2-8*]-*‘*de

1 ,

yields

[sin-’ (?)

-sin’

(2)

+sin-’ (s)

-sin-’ ($$$)I.

(9)

It may be noted that the perturbation of the period, expressed by the bracket, becomes zero when 6 = 0, and this result is easily predictable.

32

F. BADRAKHAN 5.

THE LIMIT CYCLE

Since the trace of the loop, defined by % = g(8), is piecewise linear, the limit cycle of the oscillator in the reduced phase plane (f3,i/w) is constituted by arcs of circles (Figure 4(d)). This limit cycle may be used to determine the period with escapement by graphical or energetic considerations. In fact, the period may be expressed in the form T = (4/w&a

(10)

+P +Y).

The common radius of arcs cx and /3 is A while that of arc y is determined by expressing the conservation of energy at f31 and 02. This radius is found to be R = [A* +S2/4]1’2 and the value of the perturbation is given by T - To = (4/oo)(7 - y’), with cos y =

It can be verified easily that this result is similar to that obtained analytically by direct integration. But this graphical method has the great advantage of applicability to the case of an escapement loop determined experimentally. The trace of such a loop may be approximated by a piecewise linear one. 6 LAGRANGIAN FORMULATION AND FIRST INTEGRAL Generally, in the design of the system (oscillator + escapement), all external perturbations are neglected and the motion is assumed to be steady: the effect of damping is supposed to be balanced by the impulse given by the escape wheel itself, using the potential energy of another external oscillator. In the case of a free motion the equation of motion becomes f!j+2h~+[F-h(e)]sgn8+g(e)=O. It can be shown (reference [6]) that a Lagrangian dynamical system in the form

(11) function may be defined for this

9=e2*‘{$ti2-[[HI(e)-H1(A)]sgn8-G(8)},

(12)

whereh1(8)=F-h(B),H1(8)=5hl(e)deandG(e)=jg(e)de.Equation(ll)isobtained then by the Lagrange equation d/dt(XZ/ab) - &%‘/a0= 0, it being taken into account that (i) a/ae(sgn 6) = d/di(sgn 4) = S(0, 4) (Dirac impulse), (ii) d/dfjlhl =sgn 4, (iii) 8S(O, ti) = 0, because S(0, 4) is null except for e’= 0 and therefore the product is zero, and (iv) [H1(8)-H1(A)]S(O, 6) = 0 because 4 = 0 for 8 =A. Realizing that p = az/ad = ie**’ rse*=p e-2Arjsgn

tj=sgnp,

one finds that the Hamiltonian function is then given by %=ppe-Z=$p’e

-‘*’ + e2”‘{[H1(e) --H,(A)]

sgn p + G(B)},

(13)

which leads to the two canonical equations B = asflap = p e-2ht,

0 = aX/ae = -e2*‘[h1(e) sgn p

+g(e)].

(14)

Reducing the dynamical study of the oscillator sustained by escapement to a problem of analytical dynamics is advantageous by itself. But the most important feature of the

INTERMITTENT

MECHANISM

OSCILLATOR

33

Lagrangian formulation resides in the possibility of writing a first integral of the free unperturbed motion. It can be shown [6] that the gauge-variance (invariance up to an exact differential) of the Lagrangian function leads, by virtue of Noether’s generalized theorem, to a first integral obtained if a system of Killing’s generalized partial differential equations has a solution. In this case, a first integral exists only if the function f* = hl(e> sgn d 4 g(8) is linear, or piecewise linear of the form f* = 20 (00 + C), and the integral is then given by (see reference

(15)

[6])

1 = e2Ar[b&j+ b (C/D)B + 4’ + 2F.+(0)] = constant,

(16)

F,(B)=H1(8)sgn~+G(B)+K,

(17)

being of the form F,(8) = (00 + C)*. The limitation of existence of a first integral by the linearity of the function fJf3) may be overcome since any function may be approximated by a piecewise linear one. It is clear that in this problem C and D depend on the sign of 8. They are defined in the following manner. For -A d 8 s -81 and e1 6 8
C=Fd%,

D=a,

K = F2/2k.

For --& s @5 -19~: bOJf,=k@+F+Cd,

C = (C, +F)/&,

D = Jk/2,

K = (C, + Fi2/2k,

e
C= (CL-F)/JG,

D = Jk/2,

K = (Ci -F)‘/2k.

For --02 s 8 s 02: f.+=k@+(F-Ci)sgnd,

C = [(F - Ci)/Jk]

Sgn4,

D = dk/2,

K = (F - Cj)2/2k.

For02sBcBl: B>O+f*=k0+F_Ci, &cOrSf*=ke-F-C,, In particular,

C=(F_Ci)/J2k, C=-(F+Cd)/&,

D=Jk/2, D=Jk/2,

K =(F_Ci)*/2k, K=(F+C&2k.

if Aj is an amplitude (for which 4 = 0), one can use the first integral to write

e2*“+“*)[H1(Ai+l) sgn 6 + G(Ai+I) + K] = e’^‘[Hl(Ai) sgn k + G(Ai) + K], where sgn d designates the sign of the velocity h along the path from Ai to A+ 1. This

equation leads to Ti =$ln

ITI

sgn 4 + G(Ai) f K

Hl(Ai+1) sgn d + G(Ai+,) +K I *

(18)

In general, the period of motion 7’i is variable by virtue of the difference between the energy supplied and the energy dissipated. If the Coulomb friction torque F and the viscous damping h are estimated accurately, equation (18) may be used for design purposes, in the determination of the clearance angle and of the impulses C’, and Cl, by writing Ti = Ti+l = constant. The first integral defined by equation (16) may be used also in the study of stability of the system by Liapunov’s second method.

F. BADRAKHAN

34

7. RESPONSE TO AN EXTERNAL

HARMONIC

PERTURBATION

In the presence of an external harmonic perturbation, the equation of motion (2) becomes of the form ., or $+B(A)i +f(e) = E cos wt, 8 +fi(& t9) = E cos wt, (19920)

with B(A) = 2h + B1(A) - B2(e1, e2). A first-harmonic approximation of the solution may be obtained by Bogolioubov-Krylov’s method. It is of the form 8 = A cos Cc,with CL= wt + 4, and must meet the condition 8 = -Au sin I,?,equivalent to A cos $ -A+ sin 4 = 0. Substituting into equation (19) and replacing all the terms in the equations obtained by their averages over one period of $, the steady state solution (A = 0, c#~= 0) is defined by [C(A)-w2A]2+[S(A)]2=E2,

tan 4 = S(A)/[C(A)

S(A) =L~ ozm/~(A, 4) sin CLd9, I

- @'A],

C(A)=L ~ ozwf~(A, Ilr) ~0s 9 d$. I

It is important to note that S(A) is related only to the coefficient of ti in equation (20): S(A) = -oAB(A). Knowing the expression of B(A), one has S(A)=-~AWA-(~F/T)+(~A/TT~~)[C~(~~+~~)-C’~(~~-~~)]. C(A)

is related to the function g(f3) in the equation of motion, 271

C(A)

where K,(A)

=K,(A)A

=’~ o I

g(ti)cosG

W,

is the equivalent stiffness of the system. Thus, C(A)

= kA - (2/r)(Ci

- C,){[l - (eJA)2]“2

-[l

- (e,/A)‘]“‘}.

The stability of the steady state depends on the non-existence of vertical tangents to the resonance curve A = A(w). According to a classical known result (see reference [7], for instance), one can write stabilitye[S(A)/A]

+ [dS(A)/dA]

< 0.

In the case considered here, this condition of stability becomes stability e - 2bo - (4F/rA)

+ (4/7re:)[Ci(e1 + e,) - Cd(e, - &)I < 0.

8. INFLUENCE OF AN EXTERNAL

RANDOM

PERTURBATION

Since in mechanical timing devices one uses escapement mechanisms to change the energy of an external oscillator into indication of time, it is important to study the influence of an external random perturbation on the period of the oscillator sustained by escapement. It will be assumed that the perturbation is a white noise with normal distribution. The power spectral density (PSD) is constant: S(w) = S = constant. In this case, the equation of motion (4) may be rewritten in the form f9+Bi+g(e)=N(t),

(21)

where N(t) is the white noise signal. It is known [8] that if g(8) is an odd function of 0 (which is the case here), one can write a Fokker-Planck equation giving the joint distribution of displacement and velocity.

INTERMITTENT

MECHANISM

35

OSCILLATOR

The equation is of the form dp/dt = -a/ae[pu]

+ ~/Lw{[Bv + g(e)]p} + da*p/av*,

(22)

where u = 4 is the velocity and p =p(& v) is the joint distribution density. For the case of a steady state motion (ap/at= 0),equation (22) may be arranged in the form (sa/av-a/ae)[~p+(~s/B)ap/av]+a/a~Cpg(e)+(~s/B)ap/ae ]=0.

Since the solution of the Fokker-Planck solution of the form p =p(8, v) =p(B)p(v) vanish. Thus p(e) = Cl exp

(23)

equation is unique [9], one can try to find a for which the two brackets of equation (33)

[-WV/&l,

p(v) = C2 exp [-v*/2~:],

where G(B) = I,”g(x) dx = potential energy (defined by equations (8)), and ~2 = d/B = velocity variance = E[t?] (expected value ofi 6*). This form of p =p(O, v) is similar to that of the Maxwell-Boltzmann distribution; C1 and C2 are two constants of integration which can be determined by normalization conditions. In particular, the condition In the same manC21TZ exp [--u*/2~~] dv = 1 yields p(v) = (l/(+,J%) exp [-v*/2~:]. ner, the condition C1 Jr,” exp [-G(e)/az] de = 1 gives Cr = 1/Ice, with ~~=~JZli(erf~+(erf~--erf$T

exp[s2~~@] + l-erf (

0,

u=--,

WO

i&3=&,

,=Ciecd

k

’

erfx=-

2) 2

-7.r

exp [ (eyg:)‘]],

(24)

X I0

ept2 dt (error function).

In order to determine the influence of the external random perturbation on the period of the system (mechanism+oscillator), one can use the result obtained by Rice [lo] giving the expected number of times, per second, that the signal B = e(t) crosses the level 8 =a with a positive slope: N,’ = Jot”u du [p(8, v)]+~. It gives N,’ = (cU,lI,&) exp [-G(a)/c+t]. Th e interest here lies in the frequency corresponding to the level a = 0: N,C = CT&&.

(25)

Substituting equation (24) into equation (25) and taking into account that No = oo/27r, one obtains

No T,’ --z----_erf N; To (26)

It can be noticed that if 8r + f3*(isochronous escapement), No+ + No. Also if A + 0, u + w and NC + N,, since erf co = 1. Equation (26) may be used in design in order to limit the errors due to external random perturbations if the mechanism is subjected to such perturbations.

36

F. BADRAKHAN 9. CONCLUSION

The method of analysis presented in this paper, which is based on a new form given to the equation of motion can be applied to cases of an oscillator sustained by an intermittent mechanism of any type. Its great advantage is its applicability to cases where the diagram of the torque communicated by the mechanism is determined experimentally. The problem is reduced then to the approximation of the curve %‘= g(0) obtained experimentally by a piecewise linear or by a polynomial function. In particular, the first integral obtained in a closed form may be very useful for design purposes. It helps in the choice of the required angles 81 and &. Once these angles are known, the perturbation of the oscillator natural period may be determined and the influence of external perturbations on the stability and on the period may be examined and ascertained.

ACKNOWLEDGMENTS

The author wishes to thank Professor Raymond Chaleat from Besancon University (France) who suggested the subject of this paper.

REFERENCES I, H. H. MABIE and F. W. OCVIRK 1978 Mechanisms and Dynamics of Machinery, 36-38, New York: John Wiley & Sons, Inc., third edition. 2. J. HAAG 1962 Oscillatory Motions. Belmont, California: Wadsworth. 3. R. CHALEAT 1959 Annales Francaises de Chronome’trie 3-4, 139-264. Theorie g&ret-ale de l’echappement a ancre. 4. F. BADRAKHAN 1977 International Journal of Non-Linear Mechanics 12, 1-12 Etude approchee d’un systeme a hysteresis. 5. F. BADRAKHAN 1974 Annales Francaises de Chronometrie Bl-12. Inflence des excitations aleatoires sur un oscillateur entretenu par Cchappement. 6. F. BADRAKHAN 1982 Journal of Sound and Vibration 82, 227-234. Lagrangian formulation and first integrals of piecewise linear dissipative systems. 7. C. M. HARRIS and C. E. CREDE (editors) 1976 Shock and Vibration Handbook 4, 30-31. New York: McGraw-Hill Book Company, second edition. 8. T. K. CAUGHEY 1963 Journal of the Acoustical Society of America 35,1683-1692. Derivation and application of the Fokker-Planck equation to discrete dynamic systems subjected to white random excitation. 9. A. H. GRAY 1964 Ph.D. Thesis, California Institute of Technology. Stability and related problems in randomly excited systems. 10. S. 0. RICE (published by N. Wax) 1954 in Selected Papers on Noise and Stochastic Processes. 133-194. London: Dover Publications. Mathematical analysis of random noise.