General method for active balancing of combined shaking moment and torque fluctuations in planar linkages

Mech. Mach. TheoryVol. 25, No. 6, pp. 679--687, 1990 Printed in Great Britain. All fights reserved

0094-114X/90 $3.00 + 0.00 Copyright ~ 1990 Pergamon Press plc

GENERAL METHOD FOR ACTIVE COMBINED SHAKING MOMENT FLUCTUATIONS IN PLANAR

BALANCING OF AND TORQUE LINKAGES

I. S. KOCHEV Kettelerstr 8, 6084 Gernsheim (Rhein), B.R.D.

(Received 18 January 1989; receivedfor publication 3 November 1989)

Abatract--The paper describes a novel method of active balancing designed to eliminate completely the shaking moment, the driving torque or both simultaneously. Active balancing is achieved by a single additional member: a balancer disc revolving in a prescribed way related to the geometry and the mass characteristics of the original linkage. Nondreular gear drive or other means may provide the necessary rotation. The synthesis of the balancing device is based on simple general principles equally applicable for any planar mechanism.

1. I N T R O D U C T I O N

There is a substantial body of literature on the subject of shaking moment and driving torque balancing. Both passive and active methods have been extensively studied.

1.1. Passive balancing (mass redistribution) Berkof and Lowen [1] initialized a group of studies regarding the minimization of the r.m.s. shaking moment. Among the series of following publications, Haines [2] has to be specially mentioned for revealing the ultimate potentials of this concept. Torque fluctuations and their relationship to the mass distribution of links had been initially discussed by A. A. Sherwood, but the analysis took shape in later studies of Hockey [3]. In both cases, subject of examinations are 4R four-bars. From different point of view, the problem is also studied in Ref. [4] which conceptually follows Ref. [1], and is essentially general. 1.2. Active balancing Additional links, gears, dyads or more complex devices are introduced in this type of balancing, designed to counterbalance the inertia forces produced by the original mechanism. Among the profuse literature covering this vast area, we shall distinguish two papers of Kamenskii [5, 6], the investigations [7, 8] and the more recent studies [9, 10]. Kamenskii has formulated some valuable general ideas, whereas in Refs [7, 8] the real potentials of the method are revealed, if gears or dyads are introduced for the balancing of torques and shaking moments of crank-rocker *bars. Of course, the practical application of the proposed balancing techniques is highly questionable. As a rule, only partial balancing is achievable, and the price paid includes links with unrealistic configurations, complex balancing devices and extra additional masses. Moreover, such schemes affects unfavourably other dynamic characteristics. Thus, the shaking moment is reduced by 40% in Ref. [2], but on account of a greatly off-set coupler and considerable extra mass; in Refs [7, 8] shaking moment minimization results to 2.5 increase of the torque fluctuations, etc. Finally, the applicability of such methods depend largely on the mechanism under consideration, and active balancing is clearly unrealistic for linkages with rather complex structure [9, 10]. These shortcomings are generally bypassed in the author's paper [11], where the shaking moment is completely annulled by a prescribed nonuniform rotation of the crank, and adequate selection of its moment of inertia (MOI). The method is general, but the prerequisition of a common baseplate for the motor and linkage alike is essentially undesirable limitation. Another negative aspect of this scheme is the considerable rise of the torque fluctuation; about five times above its initial value. In recent years, Z. Huang and Z. D. Shi [12, 13] have made another serious 679

680

I. S.

bQt. wheel

KOCHEV

2/ [

D/q,~

% = AC

/ \

o.co 3\

\

%

=

D8

l--L--_

Fig. 1

advancement in the field. Their method is in general, active and ensures complete elimination of the fluctuating moments. The effect of balancing is achieved by one or three balancer discs revolving in a prescribed way related to the geometry and the mass parameters of the original linkage. Noncircular gear drive (Fig. l) or other means may provide the necessary rotation. The present study follows the same general concept and investigate the same balancing problems: (A) Full elimination of the shaking moment at constant input speed. (B) Full elimination of the driving torque at constant input speed. (C) Combined elimination of both by a prescribed crank rotation with specified fluctuation. In dealing with the first two items, there is no conceptual deviation from Ref. [12], however in both cases a valuable extention is provided concerning the control of the balancer fluctuations and the effect of balancing on other dynamic factors. In view of this and anticipating interest from the non-Chinese audience, the author considers such partial repetition as justified. In dealing with the general problem (C) however a different route is followed, which advantageously results to complete balancing by a single balancer instead of three, subsequent simplification of the whole device and considerable reduction of extra masses. The following sections primarily describe the procedure of synthesis of the balancing device capable to meet these functional requirements. Later studies substantiate that the method is general in two respects. First, it is applicable for any planar mechanism. Second, the procedure of synthesis is conceptually identical for any possible application. Numerical analyses given in Section 4, indicate that the proposed balancing technique demands moderate MOI for the crank and the wheel alike, leads to realistic configurations of the noncircular gears, and increases modestly the joint reactions. 2. GENERAL E X P R E S S I O N S

2. I. Balancing device The driver of the balancer disc could be any suitable shaft of the machine. We shall assume however, that the crank of the mechanism, rotating at angular speed ~o = 61 in positive (counterclockwise) direction provides the necessary motion. The scheme is shown in Fig. 1 and serves only illustrative purposes, that is, the four bar may be replaced by any other one DOF planar linkage. Basic parameters of the device are the mass moment of inertia JB of the balancer disc, its transmission function KB(O)=61B/61 (Fig. 1) and the geometry of the noncircular gears associated with this function. In the general case (C) [Section 1], an additional parameter is the flywheel MOI Jr which restricts the crankshaft fluctuations as required. In our further discussion, the flywheel is considered fixed to the crank 1, and therefore increases its initial MOI from Jt to Jr + JI.

Active balancing in planar linkages

681

Technical and feasibility requirements impose several restrictions on the transmission function Ks(0): (i) Technically, an external gearset provides the simplest method of driving the balancer disc. Therefore Ks(O)

oJs/w

=

=

On/O < 0.

(1)

(ii) Periodicity requirement demands the average ratio/~, of the transmission function KR(O) to be inverse of an integer. With regard to the previous provision, we therefore obtain R,~ = ~

[-Ks(0)] dO = 1/n;

n = 1, 2. . . . .

(2)

(iii) Smooth action of a noncircular gearset requires moderate fluctuation f of the transmission function. Therefore

f =IK,(O)l.~, IKs(O) Imi, ~
(3)

is bounded from above. There is no definite upper limit that can be placed upon f~t.. However, approximate analyses based on the relation between f and the extremal radii of the noncircular gears indicate that the allowable fluctuation fat falls in the range [2, 4]. Finally, let us note that in parametric form the gears polar radii are given by the expressions

R -- !

Im(o)l

d,

l

R,-- l +Ix,(O)I"

d,

(4)

where d marks the centre distance 00s (Fig. 1). 2.2. Angular momentum

The angular momentum of an active balanced mechanism may be expressed in the form /-to: =

[L(0) + Jsga(0)]o~,

(5)

where T..(0) defines the angular kinetic function of the original linkage [11]. Its general presentation with respect to the fixed origin 0 (Fig. 2) is given by Ref. [11]: T. = To- = ~ [J~O; + miGli(O) + m,u,G~(O) + m,v, G3,(O)],

(6)

Gu(O) = Xo, Y ; , - X~, Yo,, G2~(O) = O,p,~(O) + q2~(O), G~(O) = O~p2,(O) + q,(O), ply(O) = Xo, cos O~+ Yo~sin 0~,

(7)

qli(O) = X;;cos 0~+ Y~isin 0i, p2~(O) = -Xo~ sin 0~+ Y0~cos0i, q,.~(O) = - X~ sin 0; + Y~ cos 0;,

where mi, miu~, m~c~, J~ = J~ (noncentroidal!) designate the mass characteristics of link i, coordinates X0~, ¥0,, 0i are self explanatory (Fig. 2), and X~., F~, 0, are their derivatives w.r.t, the input angle 0 =01. The kinetic function of a force balanced mechanism is invariant with respect to the reference point. Also, note that the moment of inertia JI + JF of the crank 1 is an addend in the expression of T..(0) since 0~ = d01/d0t = I.

682

I.S. Kocl-mv

×

0

~

........

x

Fig. 2

2.3. Kinetic energy (K.E.) and generalized inertia The kinetic energy of an actively balanced mechanism is given by the expression K.E. = ½(J,(O) + Jn " K2n(O))to:,

(8)

where Jr(O) is the generalized inertia of the original linkage. For planar mechanisms it could be expressed in the general form [l l]

Jr(O) = ~. [JjO~2 + m,(X~ + Y6,2.) + 2(m,u,)O'~q:i(O) - 2(m,v,)O~q,(O)].

(9)

Clearly, the crank moment of inertia is an addend in this expression. 3. S Y N T H E S I S

OF T H E

BALANCING

DEVICE

This section integrates the procedures of synthesis of the balancing device for each of the functional requirements listed previously, and for a modified version of the general case (C).

(A ) Shaking Moment Balancing Shaking moment will vanish if the angular momentum (5) is a constant function, i.e. [7"..(0) + Js" KB(0)]to = const.

(10)

At constant speed to, a simplified condition follows

T:(O) + JB" KB(O) = Ca = const.,

(1 la)

and an alternative form is given by

.In" Kn(O) = Ca - T.(0).

(1 lb)

Obviously, precondition Kb(O)< 0 is satisfied if Ca < min T...

(12)

The unknown constant Ca may be expressed in term of the specified fluctuation (3). By eliminating Ja from the equations

- J , IgB(o)l=x = C o - max T.., -solgB(o)lm,, = Ca - min T.., obtained from expression (1 lb), and then using the result to solve for Ca, we have C~ = f " rain T: - max T. f-1 Note, that the solution just found holds inequality (12).

(13)

Active balancing in planar linkages

683

To make further progress, we now integrate equation (1 lb) and apply the periodicity provision (2). After replacing the constant (13) we readily obtain the necessary MOI of the balancer disc in the form G =

f(T. - rain T.) + max T.. - T. , (f- l).R~

where ~ labels the average value

lI "

= ~

7"..(o). dO.

(14)

(15)

Finally, using equation (I Ib) once more, we easily obtain the transmission function Ks(0) =

f(min T. - T:(0)) + T..(0)- max T.. ff - x)Js

(16)

Note that the M O I of the crank Jt + Jr is an addend in the expressions of Tz(0), ~, min T.. and max T...Equations (14) and (16), therefore indicate that a possible attachment of a flywheel will not affect the characteristicsJs and Ks(0) of the balancing device. Active balancing considered herein alters the driving torque M ° of the original mechanism. Derived from the equation Mi, = d(K.E.)/d0, equation (8) and the precondition co = const., the torque after balancing may be expressed in the form M,. = M°. + M~,

(17)

where M*

= J B K s ( O ) " K'a(O) " co2,

represents the generalized torque evoked by the balancing device. Substituting the derivative K's(O) obtained from equation (1 lb) we finally get M * = - K s ( O ) " T~(O). co2,

(18)

where prime throughout the paper indicates differentiation with respect to the input coordinate 0 =0,. (B) Driving Torque Balancing

Driving torque vanishes if the kinetic energy (8) is a constant function, i.e. 2K.E. = [Jz(0) + Js" K~(0)] co2 = const.

(19)

At constant speed co, this leads to the simpler condition Jg(O) + Js" K~(O) = Cb = const.,

(20a)

Js" K 2 ( O ) = Cb - J,(O).

(20b)

and its alternative form

The unknown constant Cb may be found by eliminating Js from equations Js" K~ max

= Cb -- rain Jr,

Js" K2 rain = Cb -- max Jg, and by substituting the prescribed fluctuation (3). This results to f2. max Js - rain Js Cb =

f2_

1

(21) '

and the useful auxiliary function F, (0) - Ch -- Js(O) = f2(max J~ - Js(O)) + Jr(O) - rain Jz

f2_ 1 MMT 2~ 6.--.G

(22)

684

I.S. KocH~V

To obtain the required MOI of the balancer disc, we now integrate the square root of equation (20b), utilize the provision (2), and readily find Jn = (l/R,o)2,

(23)

where

1 f~" x//-~

dO.

(24)

Finally, equations (20b) and (22) may be combined to yield the transmission function Ke(O) = -

~

,

(25)

where the negative sign complies with our general assertion (l). Note that the crank moment of inertia J~ + Jr is an addend in the expressions of Js(O), min Js and max Jg. Equation (22), therefore, indicates that a possible attachment of a flywheel will not affect function F~ (0), and clearly the basic design characteristics Jn and KB(0) defined via this function. Active balancing alters the shaking moment M ° of the original mechanism. From equation (5) and the precondition ca = const., it then follows that the shaking moment after balancing may be expressed in the form M, = M ° + M*,

(26)

where M* = - J n " K'B(O)" 092. Differentiating equation (20b) to obtain K'n(O), and substituting above, we finally have J~(O)

. ca2.

M* -- 2Kn(0)

(27)

(C) Combined Shaking Moment and Driving Torque Balancing Both moments will vanish if conditions (10) and (19) hold simultaneously. By eliminating the angular speed co from the latter it follows [T.(0) + Jn" Kn(0)]2 -- C[Jg(O) + Jn" K2n(0)],

(28)

where the unknown constant C is clearly positive. A modified form of the last equation is given by (C - Jn)Je" K2(O) - 2.7':(0). ,In" Ke(O) + C " Jg(O) - T~(O) -- 0.

(29)

By solving this quadratic equation for Kn(O), integrating the result, and applying the periodicity provision (2) we finally obtain " = "T- 12rt ~o'" [C. Je(JeJs(O) + T~(O) - C" J~(0))] 'a dO. [(C - Jn)" Ro, + T.]Jn Any triple of positive numbers C, arm,Jr which obeys this equation, which produces a negative transmission function Kn(O) implicitly defined by equation (29), and ensures allowable fluctuations of Kn(O) and ca(0), is clearly a feasible solution of the discussed problem. Generally speaking, infinite number of such solutions exist, and since the smaller moment of inertia Jn and Jr are desirable, the problem can be profitably studied from the point of view of optimization theory. However, such analyses are numerical in nature, and certainly will not prove the "existence problem" for linkages with other dimensions or different structures. With this in mind, we shall concentrate our further attention on a particular case, which leads to rather general conclusions. It is based on the assumption Js = C.

(30)

Active balancing in planar linkages

685

Equation (29) is then reduced to the linear form

J,(0)

2" JnKn(O) = Jn" T:(O--')- 7"=(0)=--Q(O),

(31)

from which directly follows that precondition KB(O)< 0 is fulfilled if Ja < min (Jz---~J"

(32)

The unknown MOI Ja may be found by integrating equation (31), and utilizing the periodicity provision (2). This gives

L

Ja = 2. R~, + 6 '

(33)

where

l

2 (0)

I2 = ~n

7"..(0)"dO.

(34)

Now, the transmission function Ka(O) is uniquely defined by equation (31) and its fluctuation is readily proved to be max[ Q [ f = mm-~-~"

(35)

Further, the speed fluctuation is easily derived from the basic condition (10), which by substituting JsKB(O) using equation (31), obtains the form co .F~ (0) = const., F~ =

1[

J,(0)+ ] Ja" T.---~ 7"..(0) .

(36) (37)

Finally, the coefficient of steadiness C, derived from equation (30, and based on the approximate average speed (comi,+ corn,x)/2 is given by rain F~ + max F2 6", -- 2(max F2 - rain F2)"

(38)

Characteristically, the results obtained in this subsection depend on the MOI of the crank Jr + Ji. As a reflection of this feature, we shall now prove a theorem of existence, which states that for a sufficiently large value of Jr, the combined shaking moment and driving torque balancing is always feasible. Proof. Jr is an addend in functions 7"..(0) and Jg(O). Therefore, as Jr approaches infinity we have the limit I2~ 1 according to equation (34), and the limit

r.. J,--' 2" R,o+ 1
686

I . S . KOCh, EV

This scheme is attractive at least in appearance, since unlike case (C), now the fluctuations of co and KB(O) are controlled independently by a proper selection of Jr and Ja. For example, Jr is obtained directly from T..(0) • co = const, as in Ref. [l l], and depends on the prescribed coefficient of steadiness Cs. The synthesis of this device follows already known principles, and its description is hardly necessary. Numerical assessment of this concept is given in the following section. 4.

NUMERICAL

ANALYSIS

The proposed variants of active balancing are illustrated by a full force balance four-bar shown in Fig. 1. Link's dimensions are given in Table 1 and follow the accepted values in Refs [7, I1]. Mass characteristics are based on the standard configurations [14] and material of unit density p -- 1 kg/m 3. Mass centre of the coupler is its midpoint. With regard to the masses mB of the balancing wheels, the latter are considered discs of thickness a~/2, as Ref. [14] specifies for the counterweights. Step size of A0 = 4 ° and transmission ratio R~, = 1 are used all over. For purposes of comparison, an average input speed of 1 rad/s, based on the exact formula COAV=

2"7~ ~2~ d 0

'

Jo co(0) is assumed. For the same reason, the first line of Table 1 displays all basic dynamic characteristics of the originalforce balancedlinkage. Useful additional data are the total mass m -- 3.35 kg of the original mechanism, and the flywheel MOI Jf = 6.41 kgm 2 corresponding to inertial motion and steadiness Cs ffi 10. Table 1 compiles all types of active balancing. Cases (A) and (C) are studied over a range of fluctuations f, mainly to explore its effect on the gear configurations. Active balancing affects all dynamic characteristics and column 7, 8 present their respective r.m.s, values. The reaction of the ground pivot A (Fig. 1) is excluded from this analysis, since it depends on the gear pivots locations, and the latter cannot be properly motivated within the context of this study. A brief inspection of Table 1 lead us to the following observations: • Shaking moment balancing triples the r.m.s, of the driving torque Mi, (problems A.1 and A.2). • Joint reactions as dependent on the crank motion, retain their original values for cases (A) and (B), and increase slightly in cases (C) and (D). • Combined balancing by a double wheel device [case (D)] demands considerable extra masses, and is clearly inefficient. Compare for example, the mass characteristics mB, JB and Jr associated with problems C.2 and D.I, which incidentally provide almost equal coefficient of steadiness (column 3). • By adequate selection of the fluctuation f, realistic configurations of the noncircular gears are easily attainable. This is evident from Fig. I which illustrates problems C.3, and from Fig. 3 which compares variants A. 1 and A.2. Of course, too many factors affect the allowable level of fluctuation, the type of balancing including, and certainly it cannot be specified in advance. T a b l e i. A c t i v e b a l a n c i n g o f a full force b a l a n c e d f o u r - b a r (Fig. 1) r.m.s. J o i n t reactions Type of balancing Prior to active b a l a n c i n g A.I: M,--0 A.2: M, ffi 0 B. 1: M ~ ffi 0 C.I: M , = M,, = 0 C.2: M~ = Mi~ = 0 C.3: M, ffi Mi, = 0 D.I: Ms = Mr, = 0 D o u b l e wheel 1

f -3 4 2 !.24 1.75 2.86 1.82 2

Cs to to to to

ffi const. =const. = const. = const. 22.8 9.9 6.5 10 3

Jr

JB

mB

r.m.s, M~

----49 20 12 48.97

-5.34 4.53 i.52 16.5 6.9 4.2 2 x 32.1

-4.10 3.77 2.18 7.21 4.64 3.63 2 x I0.0

1.84 --1.62 -----

4

5

6

7

r.m.s. M~,

C

D

0.47 1.561 1.563 ------

0.90 0.90 0.90 0.90 0.92 0.95 0.99 0.96

0.61 0.61 0.61 0.61 0.63 0.66 0.69 0.66 8

B 0.76 0.76 0.76 0.76 0.78 0.81 0.84 0.82

Active balancing in planar linkages

687

(b)

(a) Fig. 3 5. C O N C L U S I O N

The method of active balancing examined herein is universal, provides complete elimination o f the shaking m o m e n t and the driving torque, demands simple balancing device and is based on a general procedure of synthesis. These are considerable advantages over the existing proposals, where d u m m y links are often as numerous as the members o f the original mechanism. Another, not quite visible weakness o f these schemes comes from their general concept of balancing. Essentially, links are counterbalanced separately [9, 10], and since their shaking m o m e n t variations are not synchronized, the fluctuation o f the principal m o m e n t is always less than the sum of its components. By contrast, the method developed in Refs [12, 13] and further extended in this paper counterbalances the overall shaking moment, and thus achieves its goals with the highest efficiency possible. It is worth mentioning in this respect, that the ultimate potentials of the combined case (C) have not been entirely revealed. A synthesis based on the general condition (29) may move the already established limits in more favourable direction. REFERENCES 1. R. S. Berkof and G. G. Lowen, Trans. ASME JI Engng Ind. 93B, 53-60 (1971). 2. R. S. Haines, Ph.D. Thesis, University of Newcastle upon Tyne, England, Note pp. 156--157 0982). 3. B. A. Hockey, Mech. Mach. Theory 7, 335-346 0972). 4. R. S. Berkof, Mech. Mach. Theory 14, 61-73 (1979). 5. V. A. Kamenskii, J. Mech. 3, 303-322 0968). 6. V. A. Kamenskii, J. Mech. 3, 323-333 0968). 7. R. S. Berkof, Mech. Mach. Theory 8, 397-410 (1973). 8. J. L. Elliott and D. Tesar, Trans. ASME J! Engng Ind. 99B, 715-722 (1977). 9. C. Bagci, Trans. ASME JI Mech. Des. 104, 482-493 0982). 10. H. Zhen and L. Deyong, Design Engng Tech. Conf., ASME Paper No. 86-DET-167. I I. I. S. Kochev, Mech. Mach. Theory 25, 459-466 0990). 12. Z. Huang and Z. D. Shi, J. NE. Inst. Heavy Mach. (In Chinese). 11, 40-46 0986). 13. Z. Huang and Z. D. Shi, Proc. 7th IFToMM WId Congr., pp. 473-476 0987). 14. G. G. Lowen, F. R. Tepper and R. S. ikrkof, Mech. Mach. Theory 9, 299-323 (1974). A L L G E M E I N E M E T H O D E Z U M A K T I V E N A U S G L E I C H DES M O M E N T S D E R T R A G H E I T S K R , ~ F T E U N D DES T R E I B E N D E N MOMENTS EBENER GETRIEBE Zusammenfasung---Im Artikel wird eine universelle Methode zum aktiven Ausgleich ebener Getriebe dargestellt. Der Ausgleich erfolgt wit Hilfe eines einz/gen zu.~tzlichen Glieds: des ausgleichenden Schwungrads, das ein vorgeschriebenes ungleichmiBiges Drehen voUzieht, das dutch die geometrischen und Massencharakteristiken des Ausgansgetriebes bestimmt wird. Das ausgleichende Getriebe wird durch den Schenkel wittels nichtkreisf'6rmiger Zahnrider wit Au enverzahnung (Abb. 2) oder auch anders angetrieben. Durch die Methode werden die folgenden drei Aufgaben gel6st: (I) Vollstindiges Annulieren des Trigheitskrifte-oder des treibenden Moments bei gleichm~BigerBewegung des schenkels. (2) Gegenseitige Eliwinierung beider Momente bei ungleichmiBigem Drehen des Schenkels nach einem vorgeschriebenen Gesetz. Die Fluktuationen des Schenkelsund des ausgleichendenGetribes hingen von deren Inenionsmomenten ab und k6nnen leicht kontroUiert werden. Zahlenanalysen weisen darauf hin, dab die Anwendung der Methode zu gleichmi/3igen Inertionsmomenten dieser Glieder sowie zu realistischen Konfigurationen der nichtkreisf'6rmigen Zahrider und zu einer geringen Zunahme der Reaktionen in den Bindungen ffihn.