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The input torque in linkages
Mechanism and Machine TheoryVOl. 14, pp, 61-73 © Pergamon Press Ltd., 1979. Printed in Great Britain
0094-114X/79/0101-0061/502.00/0
The Input Torque in Linkages Richard S. Berkoft
Received 11 August 1977 Abstract This paper reviews the various principles which have been used to obtain an analytic expression for the input torque of linkages. The simplest formulation appears to be a concise kinetic energy derivative, from which a simple expression is derived for the input torque of an inline force-balanced 4-bar linkage. All mass effects due to linear and rotary inertia components are included. The relationship between input torque, shaking moment, and moment of ground bearing forces then becomes apparent. Torque balancing is of interest, so the least-square minimization of the input torque by internal mass redistribution is discussed. 1. Inroduction
A Utah-SPEEDmachine must have energy supplied to it to perform its job. This energy is usually provided by a motor which transmits a torque to the machine through an input or drive shaft. For simple rotating machinery, this input torque is used to overcome frictional forces and to do useful work, and is often constant with respect to time, as is the speed of the device. However, many machines have non-rotating parts as well as non-constant work requirements. Such mechanisms may typically be employed in packaging, materials handling, forming, sewing, power generating and other high-rate machines. These devices are generally constructed so that the input torque and input speed vary with time, although usually in a periodic way. By the nature of their functions, however, it is often required that the input angular velocity, and consequently the input torque, have certain prescribed maximum fluctuations. If we examine only the inertial effects of a single degree of freedom mechanism, then the input torque Mm and the input angular velocity ~1 are related to the total kinetic energy T by 1 dT
where all variables are functions of time t. Note that we are concerned here with rigid-body linkages without friction, gravity, or external loads. Thus, if the total kinetic energy of a linkage is made constant, the resulting "isoenergetic" mechanism can operate at any constant speed with zero input torque (aside from friction and possible work requirements). Furthermore, because of the complex nature of the motion in a mechanism which contains non-rotating parts, virtually the only way to minimize speed and torque fluctuations is to minimize the total energy variations. Since a motor is generally sized for the peak operating conditions, it follows that the smoothing of torque requirements is economically rewarding. Not only can a smaller motor do the job, but reduced noise and increased life for certain components will probably result. It is the purpose of this paper to review torque and speed smoothing techniques as applied to mechanisms, and then to discuss the input torque of the 4-bar linkage in more detail. tManager, Advanced Technology, Gulf and Western Advanced Development and Engineering Center, Swarthmore, PA 19081, U.S.A.
61
62 2. State of the Art
Traditionally, the flywheel has been used to alternately store and discharge energy in order to smooth out the power cycle. Mahig[1,2] and Mahig and Kakad[3] have shown how to minimize oscillations in mechanisms by tuning the flywheel, i.e. by optimizing the choice of flywheel inertia and connecting shaft stiffness, the results are general in that a Fourier analysis is made of the resisting moment, which includes inertia, friction and work. Damping is also used in the later papers. Other energy storage systems have been proposed in the literature. The use of a spring equivalent to a flywheel to minimize velocity fluctuations was examined by Genova[4]. Skreiner [5] used a spring attached to a moving link and the mechansim frame to decrease speed fluctuations. Spring balancing has also been shown by Genova[6], Hain and Becker[7], Harmening[8], Sheino [9], and Van Sickle and Goodman[10]. The more general problem of the synthesis of force-generating components (e.g. springs and dashpots) which are attached to a mechanism to produce a desired time response was investigated by Carson[Ill, Gardocki and Carson[12], and Halter, Smith and Carson[13]. Matthew and Tesar[14, 15] synthesized spring parameters to provide balancing properties. The latter paper deals specifically with the torque smoothing problem. Benedict and Tesar [16] optimized the input torque variation with both balancing springs and cam subsystems. These cam subsystems, described more fully by Benedict, Matthew and Tesar[17], were shown to smooth out the torque curve almost to perfection. Torque compensation by means of cams was also described by Sarring[18]. Energy compensating cam devices have been designed by Artobolevskii[19], Dizioglu [20], and Meyer zur Capellen [21, 22] to make the total kinetic energy of the mechanism system virtually constant with respect to time. The idea of using the minimization of input torque fluctuation as one of the criteria of kinematic synthesis of linkages was discussed by Chen[23]. Smoothing by kinematic synthesis techniques was presented by Sherwood[24], Liniecki[25], Dizioglu[26] and Hilpert[27]. In a more comprehensive manner, Conte, George, Mayne and Sadler[28] and Dresig and Sch6nfeld [29, 30] used nonlinear programming to synthesize a linkage with optimum dynamic properties, such as minimum input torque fluctuation. Elliott and Tesar[31] showed that the driving torque can be specified at a finite number of positions as one constraint during the mass synthesis of 4-bar linkages. The minimization of kinetic energy fluctuation by the internal redistribution of masses in the mechanism has been investigated by Sherwood[32,33], Sherwood and Hockey[M], Hockey[35, 36] and Hilpert[37]. Hockey[38] also minimized the input torque fluctuations of mechanisms subject to external loads by means of proper choice of link masses. A more general kinetic energy minimization (although without reference to the input torque) was shown by Johnson [39]. Ogawa and Funabashi [40] similarly reduced input torque fluctuation by proper selection of link masses and moments of inertia, addition of link dyads, and the attachment of unbalanced planetary mechanisms. Kulitzscher[41] smoothed mechanism operation by mass distribution and additional linkages. Another approach, shown by Liniecki[42], was to supply a variable transmission ratio device which automatically compensates for any variations in order to produce a constant input speed. A comparative study was presented by Offt[43], who experimentally compared 4-bar linkages without balancing, with balancing using a coupler mass redistribution, and with smoothing using a flywheel. Note that techniques which completely force and moment balance linkages (as in [44]) do not eliminate variations in the input torque. Although the shaking moment can be made to vanish, the input torque and moment of the ground bearing forces remain equal and opposite to each other (although their magnitudes can be changed). There are two traditional types of dynamic analyses and associated syntheses which involve the input torque: (1) For a given mechanism motion, determine the input torque or driving forces required (Wittenbauer I).
63 (2) For a given input torque, determine the motion (time response) of a specific mechanism (Wittenbauer II). This paper is limited to discussing the first type of dynamic problem, although the analytic tools used are applicable to all dynamic analyses. The determination of the input torque requires a force analysis. There are a number of ways to formulate an equation for the input torque, namely: (1) Newton's second law, i.e. freebody diagrams and equilibrium equations; (2) Virtual work; (3) Lagrange's equation; (4) Other energy formulations. An overall discussion of these types of analysis was given by Starr[45]. In addition to the techniques used in the above referenced torque balancing papers, the following are some papers which use these formulations for dynamic problems. Classical force analysis using freebody diagrams and equilibrium equations have been illustrated by authors such as Marshall and Waldron[46], Mabie and Ocvirk[47] (complex numbers), Bonham[48], Lowen, Tepper and Berkof[49], Yang[50] and Bagci [51, 52] (screw calculus), Sadler[53], and Smith and Maunder[54]. The use of influence coefficients or "transfer functions in dynamic systems was described by Benedict and Tesar [55] and Rehwald [56]. The classical use of an equivalent moment of inertia, reduced to the input link, was illustrated by Zhukovsky [57], Artobolevskii and Loshchinin [58], Timoshenko and Young [59], and Meyer zur Capellen[21]. Virtual work formulations can be used to find the input torque, as presented by Skreiner and Roberts [60], Paul and Krajcinovic [61], and Yokoyama and Ogawa [62]. A typical input/output formulation is shown by Hunka[63]. Energy formulations have also been popular. Quinn [64] showed the use of kinetic energy in a dynamic problem. Lagrangian techniques have been used in papers by Chace [65], Chace and Bayazitoglu [66], Meyer zur Capellen [21, 67], and Uicker [68]. 3. Input Torque FotmulaUon$ Three techniques are well known for formulating an expression for the input torque of an arbitrary linkage: classical force analysis, virtual work, and Lagrange's equation. One other technique utilizes a simple kinetic energy formulation, and is generally unknown and unused. It is the purpose of this section to discuss this kinetic energy approach. As verification, the virtual work and Lagrange formulations are shown to be equivalent. A. Kinetic energy.
A simple derivation of eqn (1) can be obtained by equating the infinitesimal change of total kinetic energy T to the work done by the input torque Mm, where the input angle is (b~ d T = Mm d~j. This can be rewritten, for continuous functions, as dT Mm
- d~l
--
dT dt dt d ~ l '
or
•
(1)
Let us consider an arbitrary n-bar planar linkage, with input torque Mm acting on link 1, and only D'Alembert forces and D'Alembert moments acting on other moving links (see Fig. 1). The D'Alembert force Fo, acts through, and the D'Alembert moment Mol acts about, the MMT VoL/4, No. 1--E.
64
yl
O
Figure 1. Arbitrary n-bar planar linkage. center of mass S~ of any link i. The vectorial position of the center of mass S~ is given by (2)
rs~ = xli + Yij,
where xi and y~ are the cartesian coordinates of link i, taken with respect to an arbitrary inertial origin 0. The angular position $i of each link i, with respect to the x-axis, is also defined. The total kinetic energy, KE, T~ of any rigid link i is the sum of the translational K E and the rotational K E of the link 1
2
1
2 '2
(3)
Ti = ~ mii'si + ~ mik i~)i ,
where m; is the mass and k~ is the radius of gyration of link i. Also, the velocity and acceleration of the center of mass S~ are denoted by fsi and/~si, while the angular velocity and angular acceleration of each link are given by 6i and 6~, respectively. The total K E of the entire n-bar linkage is just the sum of the K E ' s of the n - 1 individual moving links (where the ground link is n) 1 n-l
n-1
(4) Referring to eqn (1), the input torque can be obtained by differentiating eqn (4) n-I
_ 1 ~_~ rn,(fsfs, + k,2t~,6,). Mm - 61 i=l
(5)
A version of this equation with cartesian components of velocity, xi and yi, and acceleration, J/i and 9~, can be obtained by noting that eqn (4) can also be written as
1 n-I T = ~ ~ mi(xi 2 + #i2 + k/26/2).
(6)
Applying eqn (1), we obtain n--I
M.N =
4,,
X
+.,
+
(7)
This is a general equation for the input torque of any single degree-of-freedom planar rigid-body linkage. B. Virtual work The Principle of Virtual Work for an n-bar planar rigid-body linkage is written as
65 n-I
n-1
~'~ F, .v, + ~ Mi" to, = 0, i=1
(8)
i=1
where, for each link i: F;, M~ is external force dr moment; v~ is velocity of point of application of Fi; to~ is angular velocity of link acted upon by M~. Disregarding friction and external work requirements, only the input torque and inertia effects enter the formulation. The D'Alembert force and D'Alembert moment of any link i are given by FDi = - m i f s~ = - mi(.fiii + Yij),
(9) Mo, = - miki2 dpi = - miki2 ~i k,
while the input torque is (I0)
Mm = MiNk.
Since reaction and bearing forces do no work, the Principle of Virtual work, eqn (8), becomes a-I
Mm" 4~, + ~ (Fo, "i's, + M o , " dp,)=0.
(Ib
FDi "rsl = -- rrli(.iciJii + YtYi),
(12)
i=1
Note that
SO that, with eqns (9) and (10), eqn (11) can be rewritten in scalar form as 1 n-I
MIN : ~ ~ lni(fgiXi -1"Yiyi "t- ki2~t)i~),), which just coincides with eqn (7). C. L a g r a n g e ' s equation
Lagrange's equation for a single degree-of-freedom planar rigid-body linkage can be written as
d 0(_~) OT
M m = "-~
Od~l'
(13)
where Mm is the generalized force acting on the system, and Ot is the chosen generalized coordinate. Equation (13) will now be shown to reduce to eqn (1), which is a much simpler formulation. ~Because 4h is the only independent coordinate of a complex n-bar linkage, the velocity of the center of mass and the angular velocity of any link can be expressed as functions of ~ and
6,
fSi "~jrli(l~l)l~l,
~i ----f2i(l~l)l~l •
(14)
Therefore, from eqn (4), we can express the total K E as T = jr(~bl)~l2, where jr is a very complex function, but which can be written in terms of $1 exclusively.
(15)
66
Thus, from eqn (15)
d (O~): ~t(2#~')= 2'~l+ 2d-d~f dt aT
~16)
al
=
Also
dgb,_(k, Of
d r = 0f d 6 , + 0f dt 0~2 dt 0~ dt
0~'
(17)
since f = f(dh) and OrlOn1= O. Substituting eqns (16) and (17) into (13) gives df Mm = 2f~, +~-~ d,.
(18)
Equation (15) can be rewritten as T
I &2, so that d [ = 1 dT dt ~12 dt
2~T. ~13
Upon substitution into eqn (18), we obtain MIN =-
1
dT
~1 dt '
which is identical to eqn (1). Thus, for a single degree-of-freedom lin.kage, eqn (1) is a reduced form of Lagrange's equation. The full eqn (13) would also lead to the expression for Mm given by eqn (7), but the use of eqn (13) is an extremely long, complex derivation.
4. Input Torque and Shaking Moment Components of Inline Force Balanced 4-Bar Linkage The simple kinetic energy formulation shown in Section 3A enables us to find meaningful expressions for the two components of the shaking moment. These components, the input torque and the moment of the ground bearing forces, are found for an inline force-balanced 4-bar linkage. "Inline" refers to links which can have any shape as long as the link centers of mass lie on the straight lines between the pivots. D'Alembert forces and D'Alembert moments cause ground reactions opposing the shaking force and the shaking moment. Referring to Fig. 2, we define: M m = M4t is input torque, or torque of motor acting on link 1; F4t, F43 are ground bearing forces acting at pivots on links 1 and 3; M6lo = OAo x F4t + OA3 x F43 is moment of ground bearing forces about reference origin 0. The shaking force, or total force transmitted by the mechanism to the ground, can be written in alternate forms (see eqn (8)) 3
Fs = - [, = ~ Fo, = - F41 F43, -
-
i=l
where L is the total linear momentum of the linkage.
(19)
67
t ' - / ~v7"../i-~',
0
/ ,.~) i
---
A3
% ,
-4
Fa3/ ~
~F41
oI
x
Figure 2. Inline force-balanced 4-bar linkage. The shaking moment, or total moment transmitted to the ground by the mechanism, is given by 3
Mslo = - HIo = ~ (rs, x Fo, + Mo,) = - Mm - Malo,
(20)
i=1
where HIo is the total angular momentum of the linkage with respect to reference origin O. Upon force balancing, the shaking force vanishes and the shaking moment reduces to a pure torque, which is independent of reference point or origin O. Note that, although the shaking force is zero, the ground bearing forces are not. They are equal in magnitude and opposite in direction, and produce a pure torque. The total angular momentum of the inline force-balanced 4-bar linkage (also independent of reference point) can be written in scalar form as (refer to [69]) 3
(21)
H=~'.K,6i, i=1
where (see Fig. 2) KI
=
ml(kt2 + rl2+ alrO,
-
K2 = - m2(k: 2 + r22 --- a2r2),
(22)
K3 = - m3(k32 + r32 + a3r3),
and, with eqn (20) 3
M s = ~'. Ki~,.
(23)
i=l
(Refer to [44]). Expressions will now be derived for Mm and Me, showing that they too are concise functions of the coefficients Ki. The simple relationship between the input torque and the total kinetic energy T, shown in the preceding section, is used. the key equation, derived subsequently, is T=-~.=
K,6, 2
Thus, from eqn (li, the input torque becomes
(24)
68 and, from eqns (20) and (25), the moment of the ground bearing forces is written as
i=l
~-[1-1
tl
(26)
"
Derivation of kinetic energy expression The kinetic energy expression, eqn (24), is derived as follows. First, eqn (6) is rewritten for the 4-bar linkage 1 3 T = ~ .~'~ mi(Jci 2 + ~1i2 + kiwi2).
(27)
Since .~12+ ~12 -----r12~l 2,
(28)
3C32+ y32 ----r32~32,
it22 + i¢22 = r22(b22+ al2(b22 + 2alr2 cos (4,2 - 4~,)~,~2,
then eqn (27) becomes 1 T = ~2 {m i(kl 2 + r,2)gbl 2 + ms(ks 2 + r32)~s 2
+ m2[(k22 + r22)~22 + a12~12 + 2air2 cos (~2 - ~1)(~1(J)2]}
which can be written in shortened form as 1 3
T = - ~ .~". Kit~l 2 + Z,
(29)
where the Ki are defined by eqn (22), and the remainder Z is given by 1
;2
1
'2
1
•
Z = - ~ mlrlalq)l - ~ m3r3as4~3 + ~ m2r2a24~2 z 1
2" 2
+ ~ m2[al 4), + 2atr2 cos (4~2- 4~1)~1~2].
(30)
The force balancing relations for an inline 4-bar linkage are (refer to [70]) m l r l = mE(az- r2) ~ ,
(31) a3 m3r3 = m2r2--. a2
Substituting eqns (31) into (30) gives Z = ~m2r2 [a,Ed~l2 + a22~22 - as2~s: + 2a,a2 cos (4,2 - 4~1)~1~2].
(32)
The velocity expression in the brackets is identically zero, which can be shown with the aid of the loop equations al cos ~b~+ a2 COS ~2 -- as cos ~3 + a4, ] al sin ~bl + a2 sin 4,2 = as sin ~bs.
(33)
69
Differentiating each expression in eqn (33), squaring both sides of each, and then adding both equations together results in a12~, 2 + a22t~22- a32~32+ 2a,a2 cos (4~2- tb~)~,~2 = O.
(34)
Thus, eqn (32) becomes Z = O,
(35)
and eqn (29) reduces to eqn (24).
5. Least-Square Minimization of Input Torque by Internal Mass Rearrangement The input torque of a force-balanced 4-bar linkage cannot be made to vanish by internal mass rearrangements. However, it may be desirable to minimize this input torque without adding external masses. A least-square technique can be used to determine which linkages have optimum mass distribution from an input torque point of view. In this section, the least square technique is shown for inline force-balanced 4-bar linkages. The method works best for this simplified case because of the form of the input torque expression, i.e. the coefficients Ki can be adjusted. The theory is the same which has already been used for shaking moment optimization (see [69]), since the form of the expression for the input torque is identical to that of the shaking moment. Thus, eqn (25) is a linear combination of linearly independent terms of the form 3
(36)
MIN = ~ Kin'l, /=l
where
in which the qbi are functions of the mechanism input variable thl and depend only upon the fixed lengths. The coefficients K~ depend only upon the mass distribution of the individual links. In order to make the periodic function of eqn (36) deviate least from zero in the root-mean-square sense, it is sufficient to satisfy 21r
E =
~0
M e d4~1= min.
(38)
The required relationships between the coefficients K~ are obtained by taking the partial derivatives of E with respect to certain Kj, one at a time, and setting the results equal to zero, i.e. OE = 2 f2,~ MIN OMIN d~, = 0. OKj Jo -~j
(39)
Reference [69] shows that a nontrivial solution Can only be obtained by assuming that one of the coefficients of eqn (36), say Kp (p = 1, 2, or 3), is fixed. Under these circumstances, eqn (38) produces two equations in two unknowns, since the partial derivative OMm/OKp vanishes. Substitution of eqn 06) into (39) gives the set of equations
fo2CC(.31~=lKif~i)~jd~l~O
(j = 1, 2, 3; j # p;
After rearrangement, these expressions become
Kp fixed).
(40)
70
Ki
dPicI)i dbl := - Kp
dp, qbi dthl.
1411
i:1 t :p
If one defines 6t~i = #i~ =
qb~qbj d61 =
~s
d~h,,
(42)
then eqn (41) results in 3
q=1,2,3;
E Ki#0 = - Kw#pj
j~p).
(43)
i=l
There are three sets of eqn (43), depending upon whether p = 1, 2, or 3. Each set has two equations in two unknowns. The solution of each leads to six optimum coefficient ratios of the type Ki[optimum
--
~ip =- Kp[ti×ed --
# ip~ jj 2
-- ~'~ij# jp
(i,j,p = 1,2,3;
i#j~p),
(44)
where the optimum coefficients Ki are found uniquely for a fixed Kp, and determine the best associated mass distribution for the linkage. Since the #ij terms of eqn (42) are functions of the given linkage, i.e. link lengths, angles, and the input angular velocity and acceleration, then there are certain "best" mass distributions for that linkage. Although all of the ratios G must be evaluated, some are applicable while others are not. Thus, the minimum RMS input torque can be achieved by an internal mass rearrangement which follows the above ratios. When a linkage is being designed, these ratios can be used as a guide for evaluating link parameters or actually synthesizing the links.
Physical pendulum coupler link If the coupler link has the characteristics of a physical pendulum link (see [44]), i.e. k f = r2(a2- r2),
(45)
then K2 = 0 from eqn (22), and eqn (25) reduces to
M,,,
=
-
~46)
- K,;;,
In this case, only ~31 and f13 exist, and are given by
K3[°ptimum ,-,~31
fo 2~ (~1(~3~r~(~1 d(~)1 -
~31=
Kt[axed
~,3-
K3l~xed
533 f02~ *"'32/6~2d~ l|,~/
-
#,,
41
,
(47)
(48)
While these ratios are found mathematically by assuming that each of the coefficients K~ are held constant in turn, there is no implication that one of the link mass distributions must remain unchanged for optimization. Generally speaking, only one optimum coefficient ratio will be able to be realized for a given linkage.
71
The actual design procedure for matching actual and ideal coefficient ratios is beyond the scope of this paper, but is similar to the technique shown in [71].
Constant input angular velocity Perhaps the most practical application of this theory is for the case in which the input angular velocity is constant. Such a linkage may be driven by a motor directly or through a gearbox. The motor, together with the input crank and its counterwe!ght, may have enough of a flywheel effect to provide a nearly constant angular velocity. Of course a flywheel can also be utilized. This constant velocity may also be desired by the designer, since it is the easiest case to handle for timing purposes. For this case, the RMS input torque can be minimized by redistributing rink masses as follows. Since ~t = 0 for this case, apt vanishes, and eqn (36) reduces to a two term expression 2.
(49) (~1
(~1
In this case, only ~32 and ~23 exist, and are given by
K3[ optimum_ ~:32 =
K21nxe~
-
°032 - O03---~=
K21optimum_ O023 ~:23= K3llixed --O02-"~ =
(50) o "rr (~353) 2 d ~ l
fo2rr(~2~2)(~3~3) d ~ l (51)
fo~ ($2,~2)2 d,h By evaluating the integrals of eqns (50) and (51), ratios of K2 and K3 can be obtained which minimize the input torque fluctuation in a least square sense. One last trivial case comes about if the linkage has constant input angular velocity as well as a physical pendulum coupler link. The input torque expression reduces to M,N -- - K 3 &
(52)
01 In terms of mass distribution, the input torque of this linkage can be optimized only by minimizing coefficient K3. Referring to eqn (22), K3 can be minimized only by optimizing the mass distribution of the output link. Thus, for an existing linkage, i.e. given link lengths, angles, input angular velocity and acceleration, certain optimum mass distribution ratios can be obtained. For the inline forcebalanced 4-bar linkage, therefore, the RMS input torque can be minimized by redistributing the link masses according to these ratios• • References 1. J. Mahig, Effects of higher harmonics on optimal damping and tuning for a tandem compressor. ASME Paper No. 74-DET-18, Mech. Conf. New York, New York (1974). 2. J. Mahig, Minimization of mechanism oscillations through flywheel tuning. Trans. ASME, J. Engng Ind. 93B(1), 120--124(Feb. 1971). 3. J. Mahig and Y. P. Kakad, Theoretical reduction of a tandem compressor's shaft oscillation through tuning and damping, ASME Paper No. 72-Mech-56,Mech: Conf. San Francisco, California (1972). 4. P. J. Geneva, Synthesis of spring equivalent to flywheel for minimal coetficient of fluctuation. ASME Paper No. 68:Mech-65, Mech. Conf. Atlanta, Georgia (1968). 5. M. Skreiner, Dynamic analysis used to complete the design of a mechanism. Prec., Appl. Mech. Conf., Oklahoma State University, Paper No. 12 (1969); also J. Mechanisms SOL 105-109 (1970). 6. P. I. Geneva, Synthesis of spring balancing mechanisms (Russian). "Prec. 2nd Int. Congr. Theory of Machines and Mechanisms 3, 39-51. Zakopane, Poland (1969).
72 7. K. Hain and R. Becker, Die Anwendung von Federn in Lenkergetrieben, insbesondere zur Beeinflussung de~ Drehmomentenverlaufes. Maschinenbautechnik 6(4), 232-240 (1957). 8. W. A. Harmening, Static mass balancing with a torsion spring and four-bar linkage. ASME Paper No. 74-DET-29, Mech. Conf. New York, New York (1974). 9. L. S. Sheino, The balancing of four-bar mechanisms. Vestnik Mashinostroeniya and Russian Engng J. 46(6), 203-208 (1%6). 10. R. C. Van Sickle and T. P. Goodman, Spring actuated linkage analysis to increase speed. Prod. Engng 24, 152-158 (1953). I 1. W. L. Carson, Force system synthesis for a desired mechanism time response. Proc. 3rd Appl. Mech. Conf. Paper No. 16. Oklahoma State University (1973). 12. M. J. Gardocki and W. L, Carson, Force system synthesis using a weighted velocity squared error criteria. Proc. 3rd Appl. Mech. Conf. Paper No. 41. Oklahoma State University (1973). 13. J. M. Halter, R. E. Smith, and W. L. Carson, Dynamic time response synthesis of mechanisms. ASME Paper No. 72-Mech-69, Mech. Conf. San Francisco, California (1972). 14. G. K. Matthew and D. Tesar, Synthesis of spring parameters to balance general forcing functions in planar mechanims. Trans. ASME, J. Engng Ind. 99B(2), 347-352 (1977). 15. G. K. Matthew and D. Tesar, Synthesis of spring parameters to satisfy specified energy levels in planar mechanisms. Trans. ASME, J. Engng Ind. 99B(2), 341-346 (1977). 16. C. E. Benedict and D. Tesar, Optimal torque balance for a complex stamping and indexing machine. ASME Paper No. 70-Mech-82, Mech. Conf. Columbus, Ohio (1970). 17. C. E. Benedict, G. K. Matthew and D. Tesar, Torque balancing of machines by sub-unit cam systems. Proc. 2nd Appl. Mech. Conf. Paper No. 15. Oklahoma State University (1971). 18. E. J. Sarring, Torque compensation for cam systems. Trans. 7th Conf. on Mechanisms. pp. 179-185. Purdue University (1%2). 19. 1.1. Artobolevskii, Das dynamische Laufkriterium bei Getrieben. Maschinenbautechnik. 7. 663-667 (1958). 20. B. Dizioglu, Getriebelehre, Band 3. Dynamik., 199 pp. Vieweg, Braunschweig (1%6). 21. W. Meyer zur Capellen, Die Bewegung periodischer Getriebe unter Einfluss von Kraft- und Massenwirkungen. lndustrie-Anzeiger 86(9, 17). 135-140 and 282-289 (1964). 22. W. Meyer zur Capellen, Dynamische Ausgleichprobleme bei ebenen und spharischen Kurbeltrieben. Trans., Int. Conf. Mech. and Machines. I, 3-22. Varna, Bulgaria (1%5). 23. F. Y. Chert, An analytical method for synthesizing the four-bar crank-rocker mechanism. Trans. ASME. J. Engng Ind. 91B(I), 45-54 (1%9). 24. A. A. Sherwood, The dynamic synthesis of a mechanism with time-dependent output. J. Mechanisms 3(I), 35--40 (1%8). 25. A. Liniecki, Synthesis of slider-crank mechanism with consideration of dynamic effects. J. Mechanisms 5(3), 337-349 (1970). 26. B. Dizioglu, Synthese der periodischen Getriebe mit vorgeschriebener dynamischer wirkung. Maschinebautechnik 5(! 1), 565-574 (1956). 27. H. Hilpert, Zur Getriebeauswahl bei totlagenkonstruktionen. Maschinenbautechnik 17(3), 159-162 (1%8). 28. F. L. Conte, G. R. George, R. W. Mayne and J. P. Sadler, Optimum mechanism design combining kinematic and dynamic force considerations, Trans. ASME, J. Engng Ind. 97B(2), 662-670 (1975). 29. H. Dresig and S. Schrnfeld, Rechnergestiitzte Optimierung der Antriebs- und Gestellkraftgrrssen ebener Koppelgetriebe--Teil I (Computer-aided optimization of the input torque and frame forces in planar linkages--Part I). Mechanism and Machine Theory 11(6), 363-370 (1976). 30. H. Dresig and S. Sch6nfeld, RechnergestOtzte Optimierung der Antriebs- und Gestellkraftgrfssen ebener Koppelgetriebe--Teil II (Computer-aided optimization of the input torque and frame forces in planar linkages--Part II). Mechanism and Machine Theory 11(6), 371-379 (1976). 31. J. L. Elliott and D. Tesar, The theory of torque, shaking force, and shaking moment balancing of four link mechanisms. Trans. ASME, J. Engng Ind. 99B(3), 715-722 (1977). 32. A. A. Sherwood, The dynamics of the harmonic space slider-crank mechanism. J. Mechanisms I(2), 203-208 (1%6). 33. A. A. Sherwood, The optimum distribution of mass in the coupler of a plane four-bar linkage. J. Mechanisms 1(3, 4), 229-234 (1966). 34. A. A. Sherwood and B. A. Hockey, The optimisation of mass distribution in mechanisms using dynamically similar systems. J. Mechanisms 4(3), 243-260 (1%9). 35. B. A. Hockey, An improved technique for reducing the fluctuation of kinetic energy in plane mechanisms. ,L Mechanisms 6(4), 405--418 (1971). 36. B. A, Hockey, The method of dynamically similar systems applied to the distribution of mass in spatial mechanisms. J. Mechanisms 5(2), 16%180 (1970). 37. H. Hilpert, Zur Ungl¢iclfformigkeit des Antriebsdrehmomentenverlaufs in Gelenkgetrieben. Wiss. Zeitschrift der TU KarI-Marx-Stadt 7(3), 83-89 (1%5). 38. B. A. Hockey, The minimization of the fluctuation of input-shaft torque in plane mechanisms. Mechanism and Machine theory 7(3), 335-346 (1972). 39. W. E. Johnson, A method of balancing reciprocating machines. Trans. ASME, J. Appl. Mech. 57, A81--86 (1935); (discussion) 58, A34-35 (1936). 40. K. Ogawa and H. Funabashi, On the balancing of the fluctuating input torques caused by inertia forces in the crank-and-rocker mechanisms. Trans. ASME, J. Engng Ind. 91B(1), 97-102 (1%9). 41 P. Kulitzscher, Leistungsausgleich von Koppelgetrieben durch Ver~derung der Massenverteilung oder Zusatzkoppelgetriebe (Performance balancing of linkages by redistribution of mass or linkage addition). Maschinenbautechnik, 19(11), 562-568 (1970). 42. A. Liniecki, The equation of a variable transmission ratio for a mechanical system with fluctuating load. J. Mechanisms 4(2) 13%144 (1%9). 43. C. Offt, E xperimentelle Untersuchungen an Koppelgetrieben mit Leistungsausgleich (Experimental investigation of linkages with balancing of power). Mechanism and Machine Theory 9(2), 23%246 (1974). 44. R. S. Berkof, Complete force and moment balancing of inline four-bar linkages. Mechanism and Machine Theory 8(3), 397-410 (1973).
73
45. P. J. Start, Methods of time response, Proc. 3rd AppL Mech. Conf. Paper No. 12. Oklahoma State University (1973). 46. W. E. Marshall and P. Waldron, Inertia torques in four-bar mechanisms. J. Mech. Engng Sci. 9(4), 309-317 (1967). 47. H. H. Mahie and F. W. Ocvirk, Mechanisms and Dynamics of Machinery, 2nd Edn. Wiley, New York (1963). 48. W. E. Bonham, Calculating the response of a four-bar linkage. ASME Paper No. 70-Mech-69, Mech. Conf. Columbus, Ohio (1970). 49. G. G. Lowen, F. R. Tepper and R. S. Berkof, The quantitative influence of complete force balancing on the forces and moments of certain families of four-bar linkages. Mechanism and Machine Theory 9(3, 4), 299-323 (1974) and Erratum, 10(4), 361-363 (1975). 50. A. T. Yang, Inertia force analysis of spatial mechanisms. Trans. ASME, J. Engng Ind., 93B(1), 27-33 (1971). 51. C. Bagci, Screw calculus in dynamics of plane mechanisms. Communications 3rd World Congr. Theory of Machines and Mechanisms. Vol. H, Paper H-5, 57-72. Kupari, Yugoslavia (1971). 52. C. Bagci, Static force and torque analysis using 3 x 3 screw matrix, and transmission criteria for space mechanisms. Trans. ASME, J. Engng Ind. 93B(1), 90--101 (1971). 53. J. P. Sadler, Dynamic analysis of mechanisms including coulomb friction via equilibrium equations. Proc. 3rd Appl. Mech. Conf. Oklahoma State University, Paper No. 13 (1973). 54. M. R. Smith and L. Maunder, Inertia forces in a four-bar linkage. J. Mech. Engng Sci. 9(3), 218-225 (1967). 55. C. E. Benedict and D. Tesar, The use of influence concepts in treating the dynamics of machine problems. Proc. 3rd Appl. Mech. Conf. Paper No. 43. Oklahoma State University (1973). 56. W. Rehwald, Kinetische Bewegungsanalyse ebener, ungleichf6rmig i~bersetzenderKurbelgetriebe. VDI-Forschungsheft 503 (1964). 57. N. E. Zhukovsky, The dynamic problems of a kineticchain reduced to problems of a lever(Russian).Complete Works I, Gostekhizdat (1937). 58. I. I. Artobolevskiiand V. S. Loshchinin, Investigationand finding of the laws of distributionof inertialforces in machine assemblies.J. Mechanisms 5(4),467--494(1970). 59. S. Timoshenko and D. H. Young, Advanced Dynamics. McGraw Hill,N e w York (1948). 60. M. Skreiner and B. W. Roberts, Dynamics of hydraulicallyactuated linkage mechanisms. Mech. Chem. Engng Trans., No. 2, pp. 207-215 (1967). 61. B. Paul and D. Krajcinovic,Computer analysisof machines with planar motion, Part 2: Dynamics. Trans. ASME, 92E, J. Appl. Mech. 37(3),703-712 (1970). 62. Y. Yokoyama and K. Ogawa, Dynamic characteristicof planar six-barlinkage.Bull.JSME 14(78),1352-1362 (1971). 63. J. Hunka, Analysis of the fluctuationof input torque on the four-bar crank-roc,k, er linkage. Aust. J. Appl. Sci. 14, 222-230 (1963). 64. B. E. Quinn, Energy method for determining dynamic characteristicsof mechanisms. J. Appl. Mech. 16, 283-288 (1949). 65. M. A. Chace, Analysis of the time-dependence of multi-freedom mechanical systems in relativecoordinates. Trans. ASME, J. Engng Ind. 89B(I), 119-125 0967). 66. M. A. Chace and Y. O. Bayazitoglu,Development and applicationof a generalizedd'Alembert force for multi-freedom mechanical systems. Trans. ASME, J. Engng Ind.,93B(I),317-327 0971). 67. W. Meyer zur Capellen,Torsionalvibrationsin the shaftsof linkagemechanisms. Trans. ASME, J. Engng Ind. 89B(I), 126--136(1967). 68. J. J. Uicker, Jr.,Dynamic behavior of spatiallinkages.Trans. ASME, J. Engng Ind. 91B(1),251-265 (1969). 69. R. S. Berkof and G. G. Lowen, Theory of shaking moment optimizationof force-balancedfour-barlinkages. Trans. ASME, J. Engng Ind. 93B(I),53-60 (1971). 70. R. S. Berkof and G. G. Lowen, A new method for completelyforcebalancing simple linkages.Trans. ASME, J. Engng Ind. 91B(1),21-26 (1969). 71. G. G. Lowen and R. S. Berkof, Determination of force-balancedfour-bar linkages with optimum shaking moment characteristics.Trans. ASME, J. Engng Ind. 93B(I),39--46(1971). ANTRIEBSMOMEN~ VON KOPPELGETRIE~N R. S. Berkof Kurzfassuns - Viele Maschinen enthalten Get~lebe, deren Antriebsmoment und Antriebsgeschwindigkeit zeitlich variieren, normalerweise in periodischer Form. Es wird meist gefordert, dab die Antriebswinkelgeschwlndigkeit und infolgedessen das Antriebsmoment in gewissen vo~geschrlebenen Gxenzen zu halten sind. Da eln Motor Gblicherweise fttw die hSchste Belastung gew~hlt wird, is~ es vo~teilhaft, die Drehmomentenspitzen zu gl~tten. DeE Vorteil ist nicht nux ein schw~cherer Motor, sondern auch ein ger~uschloserer Lauf und eine erhShte Lebensdauer gewisser Maschlnenteile. Es ist dee Zweck dleser Abhandlung, einen [~berbllck Gber die Methoden fdr Antriebsmomerit- und Geschwindlgkeitsausgleich bei Getrleben zu geben. Eine tiefer gehende Diskussion Gber das Antriebsd2ekmoment von Viergelenkgetrieben folgt. Die Abhandlung gibt einen Sberblick Gber die verschiedenen angewendeten Grundlagen fGr analytlsche AusdrGcke fOm das Antriebsmoment yon Koppelgetrleben. Die einfachste Form scheint ein ku~zer Differentialquotient dee kinetischen EneEgie zu sein. Von ihm Mann ein elnfacheE Ausdruck f~r das AntEiebsmoment eines 4glledrigen k.w~fteausgeglichehen roppelgetriebes abgeleltet weEden. Alle Tr~heitswiEkungen aus Translation und Rotation slnd de.Tin enthalten! es folgt daraus die Beziehung zwischen AntEiebsmoment, RGttelmoment und Moment dee Geetellkr~fte. Da dee Momentenausgleich InteressieEt, wiEd mittels des Minimumver~ahzens des kleinsten Quadrates das AntEiebsmoment dutch Massenumverteilung minimiert.
0094-114X/79/0101-0061/502.00/0
The Input Torque in Linkages Richard S. Berkoft
Received 11 August 1977 Abstract This paper reviews the various principles which have been used to obtain an analytic expression for the input torque of linkages. The simplest formulation appears to be a concise kinetic energy derivative, from which a simple expression is derived for the input torque of an inline force-balanced 4-bar linkage. All mass effects due to linear and rotary inertia components are included. The relationship between input torque, shaking moment, and moment of ground bearing forces then becomes apparent. Torque balancing is of interest, so the least-square minimization of the input torque by internal mass redistribution is discussed. 1. Inroduction
A Utah-SPEEDmachine must have energy supplied to it to perform its job. This energy is usually provided by a motor which transmits a torque to the machine through an input or drive shaft. For simple rotating machinery, this input torque is used to overcome frictional forces and to do useful work, and is often constant with respect to time, as is the speed of the device. However, many machines have non-rotating parts as well as non-constant work requirements. Such mechanisms may typically be employed in packaging, materials handling, forming, sewing, power generating and other high-rate machines. These devices are generally constructed so that the input torque and input speed vary with time, although usually in a periodic way. By the nature of their functions, however, it is often required that the input angular velocity, and consequently the input torque, have certain prescribed maximum fluctuations. If we examine only the inertial effects of a single degree of freedom mechanism, then the input torque Mm and the input angular velocity ~1 are related to the total kinetic energy T by 1 dT
where all variables are functions of time t. Note that we are concerned here with rigid-body linkages without friction, gravity, or external loads. Thus, if the total kinetic energy of a linkage is made constant, the resulting "isoenergetic" mechanism can operate at any constant speed with zero input torque (aside from friction and possible work requirements). Furthermore, because of the complex nature of the motion in a mechanism which contains non-rotating parts, virtually the only way to minimize speed and torque fluctuations is to minimize the total energy variations. Since a motor is generally sized for the peak operating conditions, it follows that the smoothing of torque requirements is economically rewarding. Not only can a smaller motor do the job, but reduced noise and increased life for certain components will probably result. It is the purpose of this paper to review torque and speed smoothing techniques as applied to mechanisms, and then to discuss the input torque of the 4-bar linkage in more detail. tManager, Advanced Technology, Gulf and Western Advanced Development and Engineering Center, Swarthmore, PA 19081, U.S.A.
61
62 2. State of the Art
Traditionally, the flywheel has been used to alternately store and discharge energy in order to smooth out the power cycle. Mahig[1,2] and Mahig and Kakad[3] have shown how to minimize oscillations in mechanisms by tuning the flywheel, i.e. by optimizing the choice of flywheel inertia and connecting shaft stiffness, the results are general in that a Fourier analysis is made of the resisting moment, which includes inertia, friction and work. Damping is also used in the later papers. Other energy storage systems have been proposed in the literature. The use of a spring equivalent to a flywheel to minimize velocity fluctuations was examined by Genova[4]. Skreiner [5] used a spring attached to a moving link and the mechansim frame to decrease speed fluctuations. Spring balancing has also been shown by Genova[6], Hain and Becker[7], Harmening[8], Sheino [9], and Van Sickle and Goodman[10]. The more general problem of the synthesis of force-generating components (e.g. springs and dashpots) which are attached to a mechanism to produce a desired time response was investigated by Carson[Ill, Gardocki and Carson[12], and Halter, Smith and Carson[13]. Matthew and Tesar[14, 15] synthesized spring parameters to provide balancing properties. The latter paper deals specifically with the torque smoothing problem. Benedict and Tesar [16] optimized the input torque variation with both balancing springs and cam subsystems. These cam subsystems, described more fully by Benedict, Matthew and Tesar[17], were shown to smooth out the torque curve almost to perfection. Torque compensation by means of cams was also described by Sarring[18]. Energy compensating cam devices have been designed by Artobolevskii[19], Dizioglu [20], and Meyer zur Capellen [21, 22] to make the total kinetic energy of the mechanism system virtually constant with respect to time. The idea of using the minimization of input torque fluctuation as one of the criteria of kinematic synthesis of linkages was discussed by Chen[23]. Smoothing by kinematic synthesis techniques was presented by Sherwood[24], Liniecki[25], Dizioglu[26] and Hilpert[27]. In a more comprehensive manner, Conte, George, Mayne and Sadler[28] and Dresig and Sch6nfeld [29, 30] used nonlinear programming to synthesize a linkage with optimum dynamic properties, such as minimum input torque fluctuation. Elliott and Tesar[31] showed that the driving torque can be specified at a finite number of positions as one constraint during the mass synthesis of 4-bar linkages. The minimization of kinetic energy fluctuation by the internal redistribution of masses in the mechanism has been investigated by Sherwood[32,33], Sherwood and Hockey[M], Hockey[35, 36] and Hilpert[37]. Hockey[38] also minimized the input torque fluctuations of mechanisms subject to external loads by means of proper choice of link masses. A more general kinetic energy minimization (although without reference to the input torque) was shown by Johnson [39]. Ogawa and Funabashi [40] similarly reduced input torque fluctuation by proper selection of link masses and moments of inertia, addition of link dyads, and the attachment of unbalanced planetary mechanisms. Kulitzscher[41] smoothed mechanism operation by mass distribution and additional linkages. Another approach, shown by Liniecki[42], was to supply a variable transmission ratio device which automatically compensates for any variations in order to produce a constant input speed. A comparative study was presented by Offt[43], who experimentally compared 4-bar linkages without balancing, with balancing using a coupler mass redistribution, and with smoothing using a flywheel. Note that techniques which completely force and moment balance linkages (as in [44]) do not eliminate variations in the input torque. Although the shaking moment can be made to vanish, the input torque and moment of the ground bearing forces remain equal and opposite to each other (although their magnitudes can be changed). There are two traditional types of dynamic analyses and associated syntheses which involve the input torque: (1) For a given mechanism motion, determine the input torque or driving forces required (Wittenbauer I).
63 (2) For a given input torque, determine the motion (time response) of a specific mechanism (Wittenbauer II). This paper is limited to discussing the first type of dynamic problem, although the analytic tools used are applicable to all dynamic analyses. The determination of the input torque requires a force analysis. There are a number of ways to formulate an equation for the input torque, namely: (1) Newton's second law, i.e. freebody diagrams and equilibrium equations; (2) Virtual work; (3) Lagrange's equation; (4) Other energy formulations. An overall discussion of these types of analysis was given by Starr[45]. In addition to the techniques used in the above referenced torque balancing papers, the following are some papers which use these formulations for dynamic problems. Classical force analysis using freebody diagrams and equilibrium equations have been illustrated by authors such as Marshall and Waldron[46], Mabie and Ocvirk[47] (complex numbers), Bonham[48], Lowen, Tepper and Berkof[49], Yang[50] and Bagci [51, 52] (screw calculus), Sadler[53], and Smith and Maunder[54]. The use of influence coefficients or "transfer functions in dynamic systems was described by Benedict and Tesar [55] and Rehwald [56]. The classical use of an equivalent moment of inertia, reduced to the input link, was illustrated by Zhukovsky [57], Artobolevskii and Loshchinin [58], Timoshenko and Young [59], and Meyer zur Capellen[21]. Virtual work formulations can be used to find the input torque, as presented by Skreiner and Roberts [60], Paul and Krajcinovic [61], and Yokoyama and Ogawa [62]. A typical input/output formulation is shown by Hunka[63]. Energy formulations have also been popular. Quinn [64] showed the use of kinetic energy in a dynamic problem. Lagrangian techniques have been used in papers by Chace [65], Chace and Bayazitoglu [66], Meyer zur Capellen [21, 67], and Uicker [68]. 3. Input Torque FotmulaUon$ Three techniques are well known for formulating an expression for the input torque of an arbitrary linkage: classical force analysis, virtual work, and Lagrange's equation. One other technique utilizes a simple kinetic energy formulation, and is generally unknown and unused. It is the purpose of this section to discuss this kinetic energy approach. As verification, the virtual work and Lagrange formulations are shown to be equivalent. A. Kinetic energy.
A simple derivation of eqn (1) can be obtained by equating the infinitesimal change of total kinetic energy T to the work done by the input torque Mm, where the input angle is (b~ d T = Mm d~j. This can be rewritten, for continuous functions, as dT Mm
- d~l
--
dT dt dt d ~ l '
or
•
(1)
Let us consider an arbitrary n-bar planar linkage, with input torque Mm acting on link 1, and only D'Alembert forces and D'Alembert moments acting on other moving links (see Fig. 1). The D'Alembert force Fo, acts through, and the D'Alembert moment Mol acts about, the MMT VoL/4, No. 1--E.
64
yl
O
Figure 1. Arbitrary n-bar planar linkage. center of mass S~ of any link i. The vectorial position of the center of mass S~ is given by (2)
rs~ = xli + Yij,
where xi and y~ are the cartesian coordinates of link i, taken with respect to an arbitrary inertial origin 0. The angular position $i of each link i, with respect to the x-axis, is also defined. The total kinetic energy, KE, T~ of any rigid link i is the sum of the translational K E and the rotational K E of the link 1
2
1
2 '2
(3)
Ti = ~ mii'si + ~ mik i~)i ,
where m; is the mass and k~ is the radius of gyration of link i. Also, the velocity and acceleration of the center of mass S~ are denoted by fsi and/~si, while the angular velocity and angular acceleration of each link are given by 6i and 6~, respectively. The total K E of the entire n-bar linkage is just the sum of the K E ' s of the n - 1 individual moving links (where the ground link is n) 1 n-l
n-1
(4) Referring to eqn (1), the input torque can be obtained by differentiating eqn (4) n-I
_ 1 ~_~ rn,(fsfs, + k,2t~,6,). Mm - 61 i=l
(5)
A version of this equation with cartesian components of velocity, xi and yi, and acceleration, J/i and 9~, can be obtained by noting that eqn (4) can also be written as
1 n-I T = ~ ~ mi(xi 2 + #i2 + k/26/2).
(6)
Applying eqn (1), we obtain n--I
M.N =
4,,
X
+.,
+
(7)
This is a general equation for the input torque of any single degree-of-freedom planar rigid-body linkage. B. Virtual work The Principle of Virtual Work for an n-bar planar rigid-body linkage is written as
65 n-I
n-1
~'~ F, .v, + ~ Mi" to, = 0, i=1
(8)
i=1
where, for each link i: F;, M~ is external force dr moment; v~ is velocity of point of application of Fi; to~ is angular velocity of link acted upon by M~. Disregarding friction and external work requirements, only the input torque and inertia effects enter the formulation. The D'Alembert force and D'Alembert moment of any link i are given by FDi = - m i f s~ = - mi(.fiii + Yij),
(9) Mo, = - miki2 dpi = - miki2 ~i k,
while the input torque is (I0)
Mm = MiNk.
Since reaction and bearing forces do no work, the Principle of Virtual work, eqn (8), becomes a-I
Mm" 4~, + ~ (Fo, "i's, + M o , " dp,)=0.
(Ib
FDi "rsl = -- rrli(.iciJii + YtYi),
(12)
i=1
Note that
SO that, with eqns (9) and (10), eqn (11) can be rewritten in scalar form as 1 n-I
MIN : ~ ~ lni(fgiXi -1"Yiyi "t- ki2~t)i~),), which just coincides with eqn (7). C. L a g r a n g e ' s equation
Lagrange's equation for a single degree-of-freedom planar rigid-body linkage can be written as
d 0(_~) OT
M m = "-~
Od~l'
(13)
where Mm is the generalized force acting on the system, and Ot is the chosen generalized coordinate. Equation (13) will now be shown to reduce to eqn (1), which is a much simpler formulation. ~Because 4h is the only independent coordinate of a complex n-bar linkage, the velocity of the center of mass and the angular velocity of any link can be expressed as functions of ~ and
6,
fSi "~jrli(l~l)l~l,
~i ----f2i(l~l)l~l •
(14)
Therefore, from eqn (4), we can express the total K E as T = jr(~bl)~l2, where jr is a very complex function, but which can be written in terms of $1 exclusively.
(15)
66
Thus, from eqn (15)
d (O~): ~t(2#~')= 2'~l+ 2d-d~f dt aT
~16)
al
=
Also
dgb,_(k, Of
d r = 0f d 6 , + 0f dt 0~2 dt 0~ dt
0~'
(17)
since f = f(dh) and OrlOn1= O. Substituting eqns (16) and (17) into (13) gives df Mm = 2f~, +~-~ d,.
(18)
Equation (15) can be rewritten as T
I &2, so that d [ = 1 dT dt ~12 dt
2~T. ~13
Upon substitution into eqn (18), we obtain MIN =-
1
dT
~1 dt '
which is identical to eqn (1). Thus, for a single degree-of-freedom lin.kage, eqn (1) is a reduced form of Lagrange's equation. The full eqn (13) would also lead to the expression for Mm given by eqn (7), but the use of eqn (13) is an extremely long, complex derivation.
4. Input Torque and Shaking Moment Components of Inline Force Balanced 4-Bar Linkage The simple kinetic energy formulation shown in Section 3A enables us to find meaningful expressions for the two components of the shaking moment. These components, the input torque and the moment of the ground bearing forces, are found for an inline force-balanced 4-bar linkage. "Inline" refers to links which can have any shape as long as the link centers of mass lie on the straight lines between the pivots. D'Alembert forces and D'Alembert moments cause ground reactions opposing the shaking force and the shaking moment. Referring to Fig. 2, we define: M m = M4t is input torque, or torque of motor acting on link 1; F4t, F43 are ground bearing forces acting at pivots on links 1 and 3; M6lo = OAo x F4t + OA3 x F43 is moment of ground bearing forces about reference origin 0. The shaking force, or total force transmitted by the mechanism to the ground, can be written in alternate forms (see eqn (8)) 3
Fs = - [, = ~ Fo, = - F41 F43, -
-
i=l
where L is the total linear momentum of the linkage.
(19)
67
t ' - / ~v7"../i-~',
0
/ ,.~) i
---
A3
% ,
-4
Fa3/ ~
~F41
oI
x
Figure 2. Inline force-balanced 4-bar linkage. The shaking moment, or total moment transmitted to the ground by the mechanism, is given by 3
Mslo = - HIo = ~ (rs, x Fo, + Mo,) = - Mm - Malo,
(20)
i=1
where HIo is the total angular momentum of the linkage with respect to reference origin O. Upon force balancing, the shaking force vanishes and the shaking moment reduces to a pure torque, which is independent of reference point or origin O. Note that, although the shaking force is zero, the ground bearing forces are not. They are equal in magnitude and opposite in direction, and produce a pure torque. The total angular momentum of the inline force-balanced 4-bar linkage (also independent of reference point) can be written in scalar form as (refer to [69]) 3
(21)
H=~'.K,6i, i=1
where (see Fig. 2) KI
=
ml(kt2 + rl2+ alrO,
-
K2 = - m2(k: 2 + r22 --- a2r2),
(22)
K3 = - m3(k32 + r32 + a3r3),
and, with eqn (20) 3
M s = ~'. Ki~,.
(23)
i=l
(Refer to [44]). Expressions will now be derived for Mm and Me, showing that they too are concise functions of the coefficients Ki. The simple relationship between the input torque and the total kinetic energy T, shown in the preceding section, is used. the key equation, derived subsequently, is T=-~.=
K,6, 2
Thus, from eqn (li, the input torque becomes
(24)
68 and, from eqns (20) and (25), the moment of the ground bearing forces is written as
i=l
~-[1-1
tl
(26)
"
Derivation of kinetic energy expression The kinetic energy expression, eqn (24), is derived as follows. First, eqn (6) is rewritten for the 4-bar linkage 1 3 T = ~ .~'~ mi(Jci 2 + ~1i2 + kiwi2).
(27)
Since .~12+ ~12 -----r12~l 2,
(28)
3C32+ y32 ----r32~32,
it22 + i¢22 = r22(b22+ al2(b22 + 2alr2 cos (4,2 - 4~,)~,~2,
then eqn (27) becomes 1 T = ~2 {m i(kl 2 + r,2)gbl 2 + ms(ks 2 + r32)~s 2
+ m2[(k22 + r22)~22 + a12~12 + 2air2 cos (~2 - ~1)(~1(J)2]}
which can be written in shortened form as 1 3
T = - ~ .~". Kit~l 2 + Z,
(29)
where the Ki are defined by eqn (22), and the remainder Z is given by 1
;2
1
'2
1
•
Z = - ~ mlrlalq)l - ~ m3r3as4~3 + ~ m2r2a24~2 z 1
2" 2
+ ~ m2[al 4), + 2atr2 cos (4~2- 4~1)~1~2].
(30)
The force balancing relations for an inline 4-bar linkage are (refer to [70]) m l r l = mE(az- r2) ~ ,
(31) a3 m3r3 = m2r2--. a2
Substituting eqns (31) into (30) gives Z = ~m2r2 [a,Ed~l2 + a22~22 - as2~s: + 2a,a2 cos (4,2 - 4~1)~1~2].
(32)
The velocity expression in the brackets is identically zero, which can be shown with the aid of the loop equations al cos ~b~+ a2 COS ~2 -- as cos ~3 + a4, ] al sin ~bl + a2 sin 4,2 = as sin ~bs.
(33)
69
Differentiating each expression in eqn (33), squaring both sides of each, and then adding both equations together results in a12~, 2 + a22t~22- a32~32+ 2a,a2 cos (4~2- tb~)~,~2 = O.
(34)
Thus, eqn (32) becomes Z = O,
(35)
and eqn (29) reduces to eqn (24).
5. Least-Square Minimization of Input Torque by Internal Mass Rearrangement The input torque of a force-balanced 4-bar linkage cannot be made to vanish by internal mass rearrangements. However, it may be desirable to minimize this input torque without adding external masses. A least-square technique can be used to determine which linkages have optimum mass distribution from an input torque point of view. In this section, the least square technique is shown for inline force-balanced 4-bar linkages. The method works best for this simplified case because of the form of the input torque expression, i.e. the coefficients Ki can be adjusted. The theory is the same which has already been used for shaking moment optimization (see [69]), since the form of the expression for the input torque is identical to that of the shaking moment. Thus, eqn (25) is a linear combination of linearly independent terms of the form 3
(36)
MIN = ~ Kin'l, /=l
where
in which the qbi are functions of the mechanism input variable thl and depend only upon the fixed lengths. The coefficients K~ depend only upon the mass distribution of the individual links. In order to make the periodic function of eqn (36) deviate least from zero in the root-mean-square sense, it is sufficient to satisfy 21r
E =
~0
M e d4~1= min.
(38)
The required relationships between the coefficients K~ are obtained by taking the partial derivatives of E with respect to certain Kj, one at a time, and setting the results equal to zero, i.e. OE = 2 f2,~ MIN OMIN d~, = 0. OKj Jo -~j
(39)
Reference [69] shows that a nontrivial solution Can only be obtained by assuming that one of the coefficients of eqn (36), say Kp (p = 1, 2, or 3), is fixed. Under these circumstances, eqn (38) produces two equations in two unknowns, since the partial derivative OMm/OKp vanishes. Substitution of eqn 06) into (39) gives the set of equations
fo2CC(.31~=lKif~i)~jd~l~O
(j = 1, 2, 3; j # p;
After rearrangement, these expressions become
Kp fixed).
(40)
70
Ki
dPicI)i dbl := - Kp
dp, qbi dthl.
1411
i:1 t :p
If one defines 6t~i = #i~ =
qb~qbj d61 =
~s
d~h,,
(42)
then eqn (41) results in 3
q=1,2,3;
E Ki#0 = - Kw#pj
j~p).
(43)
i=l
There are three sets of eqn (43), depending upon whether p = 1, 2, or 3. Each set has two equations in two unknowns. The solution of each leads to six optimum coefficient ratios of the type Ki[optimum
--
~ip =- Kp[ti×ed --
# ip~ jj 2
-- ~'~ij# jp
(i,j,p = 1,2,3;
i#j~p),
(44)
where the optimum coefficients Ki are found uniquely for a fixed Kp, and determine the best associated mass distribution for the linkage. Since the #ij terms of eqn (42) are functions of the given linkage, i.e. link lengths, angles, and the input angular velocity and acceleration, then there are certain "best" mass distributions for that linkage. Although all of the ratios G must be evaluated, some are applicable while others are not. Thus, the minimum RMS input torque can be achieved by an internal mass rearrangement which follows the above ratios. When a linkage is being designed, these ratios can be used as a guide for evaluating link parameters or actually synthesizing the links.
Physical pendulum coupler link If the coupler link has the characteristics of a physical pendulum link (see [44]), i.e. k f = r2(a2- r2),
(45)
then K2 = 0 from eqn (22), and eqn (25) reduces to
M,,,
=
-
~46)
- K,;;,
In this case, only ~31 and f13 exist, and are given by
K3[°ptimum ,-,~31
fo 2~ (~1(~3~r~(~1 d(~)1 -
~31=
Kt[axed
~,3-
K3l~xed
533 f02~ *"'32/6~2d~ l|,~/
-
#,,
41
,
(47)
(48)
While these ratios are found mathematically by assuming that each of the coefficients K~ are held constant in turn, there is no implication that one of the link mass distributions must remain unchanged for optimization. Generally speaking, only one optimum coefficient ratio will be able to be realized for a given linkage.
71
The actual design procedure for matching actual and ideal coefficient ratios is beyond the scope of this paper, but is similar to the technique shown in [71].
Constant input angular velocity Perhaps the most practical application of this theory is for the case in which the input angular velocity is constant. Such a linkage may be driven by a motor directly or through a gearbox. The motor, together with the input crank and its counterwe!ght, may have enough of a flywheel effect to provide a nearly constant angular velocity. Of course a flywheel can also be utilized. This constant velocity may also be desired by the designer, since it is the easiest case to handle for timing purposes. For this case, the RMS input torque can be minimized by redistributing rink masses as follows. Since ~t = 0 for this case, apt vanishes, and eqn (36) reduces to a two term expression 2.
(49) (~1
(~1
In this case, only ~32 and ~23 exist, and are given by
K3[ optimum_ ~:32 =
K21nxe~
-
°032 - O03---~=
K21optimum_ O023 ~:23= K3llixed --O02-"~ =
(50) o "rr (~353) 2 d ~ l
fo2rr(~2~2)(~3~3) d ~ l (51)
fo~ ($2,~2)2 d,h By evaluating the integrals of eqns (50) and (51), ratios of K2 and K3 can be obtained which minimize the input torque fluctuation in a least square sense. One last trivial case comes about if the linkage has constant input angular velocity as well as a physical pendulum coupler link. The input torque expression reduces to M,N -- - K 3 &
(52)
01 In terms of mass distribution, the input torque of this linkage can be optimized only by minimizing coefficient K3. Referring to eqn (22), K3 can be minimized only by optimizing the mass distribution of the output link. Thus, for an existing linkage, i.e. given link lengths, angles, input angular velocity and acceleration, certain optimum mass distribution ratios can be obtained. For the inline forcebalanced 4-bar linkage, therefore, the RMS input torque can be minimized by redistributing the link masses according to these ratios• • References 1. J. Mahig, Effects of higher harmonics on optimal damping and tuning for a tandem compressor. ASME Paper No. 74-DET-18, Mech. Conf. New York, New York (1974). 2. J. Mahig, Minimization of mechanism oscillations through flywheel tuning. Trans. ASME, J. Engng Ind. 93B(1), 120--124(Feb. 1971). 3. J. Mahig and Y. P. Kakad, Theoretical reduction of a tandem compressor's shaft oscillation through tuning and damping, ASME Paper No. 72-Mech-56,Mech: Conf. San Francisco, California (1972). 4. P. J. Geneva, Synthesis of spring equivalent to flywheel for minimal coetficient of fluctuation. ASME Paper No. 68:Mech-65, Mech. Conf. Atlanta, Georgia (1968). 5. M. Skreiner, Dynamic analysis used to complete the design of a mechanism. Prec., Appl. Mech. Conf., Oklahoma State University, Paper No. 12 (1969); also J. Mechanisms SOL 105-109 (1970). 6. P. I. Geneva, Synthesis of spring balancing mechanisms (Russian). "Prec. 2nd Int. Congr. Theory of Machines and Mechanisms 3, 39-51. Zakopane, Poland (1969).
72 7. K. Hain and R. Becker, Die Anwendung von Federn in Lenkergetrieben, insbesondere zur Beeinflussung de~ Drehmomentenverlaufes. Maschinenbautechnik 6(4), 232-240 (1957). 8. W. A. Harmening, Static mass balancing with a torsion spring and four-bar linkage. ASME Paper No. 74-DET-29, Mech. Conf. New York, New York (1974). 9. L. S. Sheino, The balancing of four-bar mechanisms. Vestnik Mashinostroeniya and Russian Engng J. 46(6), 203-208 (1%6). 10. R. C. Van Sickle and T. P. Goodman, Spring actuated linkage analysis to increase speed. Prod. Engng 24, 152-158 (1953). I 1. W. L. Carson, Force system synthesis for a desired mechanism time response. Proc. 3rd Appl. Mech. Conf. Paper No. 16. Oklahoma State University (1973). 12. M. J. Gardocki and W. L, Carson, Force system synthesis using a weighted velocity squared error criteria. Proc. 3rd Appl. Mech. Conf. Paper No. 41. Oklahoma State University (1973). 13. J. M. Halter, R. E. Smith, and W. L. Carson, Dynamic time response synthesis of mechanisms. ASME Paper No. 72-Mech-69, Mech. Conf. San Francisco, California (1972). 14. G. K. Matthew and D. Tesar, Synthesis of spring parameters to balance general forcing functions in planar mechanims. Trans. ASME, J. Engng Ind. 99B(2), 347-352 (1977). 15. G. K. Matthew and D. Tesar, Synthesis of spring parameters to satisfy specified energy levels in planar mechanisms. Trans. ASME, J. Engng Ind. 99B(2), 341-346 (1977). 16. C. E. Benedict and D. Tesar, Optimal torque balance for a complex stamping and indexing machine. ASME Paper No. 70-Mech-82, Mech. Conf. Columbus, Ohio (1970). 17. C. E. Benedict, G. K. Matthew and D. Tesar, Torque balancing of machines by sub-unit cam systems. Proc. 2nd Appl. Mech. Conf. Paper No. 15. Oklahoma State University (1971). 18. E. J. Sarring, Torque compensation for cam systems. Trans. 7th Conf. on Mechanisms. pp. 179-185. Purdue University (1%2). 19. 1.1. Artobolevskii, Das dynamische Laufkriterium bei Getrieben. Maschinenbautechnik. 7. 663-667 (1958). 20. B. Dizioglu, Getriebelehre, Band 3. Dynamik., 199 pp. Vieweg, Braunschweig (1%6). 21. W. Meyer zur Capellen, Die Bewegung periodischer Getriebe unter Einfluss von Kraft- und Massenwirkungen. lndustrie-Anzeiger 86(9, 17). 135-140 and 282-289 (1964). 22. W. Meyer zur Capellen, Dynamische Ausgleichprobleme bei ebenen und spharischen Kurbeltrieben. Trans., Int. Conf. Mech. and Machines. I, 3-22. Varna, Bulgaria (1%5). 23. F. Y. Chert, An analytical method for synthesizing the four-bar crank-rocker mechanism. Trans. ASME. J. Engng Ind. 91B(I), 45-54 (1%9). 24. A. A. Sherwood, The dynamic synthesis of a mechanism with time-dependent output. J. Mechanisms 3(I), 35--40 (1%8). 25. A. Liniecki, Synthesis of slider-crank mechanism with consideration of dynamic effects. J. Mechanisms 5(3), 337-349 (1970). 26. B. Dizioglu, Synthese der periodischen Getriebe mit vorgeschriebener dynamischer wirkung. Maschinebautechnik 5(! 1), 565-574 (1956). 27. H. Hilpert, Zur Getriebeauswahl bei totlagenkonstruktionen. Maschinenbautechnik 17(3), 159-162 (1%8). 28. F. L. Conte, G. R. George, R. W. Mayne and J. P. Sadler, Optimum mechanism design combining kinematic and dynamic force considerations, Trans. ASME, J. Engng Ind. 97B(2), 662-670 (1975). 29. H. Dresig and S. Schrnfeld, Rechnergestiitzte Optimierung der Antriebs- und Gestellkraftgrrssen ebener Koppelgetriebe--Teil I (Computer-aided optimization of the input torque and frame forces in planar linkages--Part I). Mechanism and Machine Theory 11(6), 363-370 (1976). 30. H. Dresig and S. Sch6nfeld, RechnergestOtzte Optimierung der Antriebs- und Gestellkraftgrfssen ebener Koppelgetriebe--Teil II (Computer-aided optimization of the input torque and frame forces in planar linkages--Part II). Mechanism and Machine Theory 11(6), 371-379 (1976). 31. J. L. Elliott and D. Tesar, The theory of torque, shaking force, and shaking moment balancing of four link mechanisms. Trans. ASME, J. Engng Ind. 99B(3), 715-722 (1977). 32. A. A. Sherwood, The dynamics of the harmonic space slider-crank mechanism. J. Mechanisms I(2), 203-208 (1%6). 33. A. A. Sherwood, The optimum distribution of mass in the coupler of a plane four-bar linkage. J. Mechanisms 1(3, 4), 229-234 (1966). 34. A. A. Sherwood and B. A. Hockey, The optimisation of mass distribution in mechanisms using dynamically similar systems. J. Mechanisms 4(3), 243-260 (1%9). 35. B. A. Hockey, An improved technique for reducing the fluctuation of kinetic energy in plane mechanisms. ,L Mechanisms 6(4), 405--418 (1971). 36. B. A, Hockey, The method of dynamically similar systems applied to the distribution of mass in spatial mechanisms. J. Mechanisms 5(2), 16%180 (1970). 37. H. Hilpert, Zur Ungl¢iclfformigkeit des Antriebsdrehmomentenverlaufs in Gelenkgetrieben. Wiss. Zeitschrift der TU KarI-Marx-Stadt 7(3), 83-89 (1%5). 38. B. A. Hockey, The minimization of the fluctuation of input-shaft torque in plane mechanisms. Mechanism and Machine theory 7(3), 335-346 (1972). 39. W. E. Johnson, A method of balancing reciprocating machines. Trans. ASME, J. Appl. Mech. 57, A81--86 (1935); (discussion) 58, A34-35 (1936). 40. K. Ogawa and H. Funabashi, On the balancing of the fluctuating input torques caused by inertia forces in the crank-and-rocker mechanisms. Trans. ASME, J. Engng Ind. 91B(1), 97-102 (1%9). 41 P. Kulitzscher, Leistungsausgleich von Koppelgetrieben durch Ver~derung der Massenverteilung oder Zusatzkoppelgetriebe (Performance balancing of linkages by redistribution of mass or linkage addition). Maschinenbautechnik, 19(11), 562-568 (1970). 42. A. Liniecki, The equation of a variable transmission ratio for a mechanical system with fluctuating load. J. Mechanisms 4(2) 13%144 (1%9). 43. C. Offt, E xperimentelle Untersuchungen an Koppelgetrieben mit Leistungsausgleich (Experimental investigation of linkages with balancing of power). Mechanism and Machine Theory 9(2), 23%246 (1974). 44. R. S. Berkof, Complete force and moment balancing of inline four-bar linkages. Mechanism and Machine Theory 8(3), 397-410 (1973).
73
45. P. J. Start, Methods of time response, Proc. 3rd AppL Mech. Conf. Paper No. 12. Oklahoma State University (1973). 46. W. E. Marshall and P. Waldron, Inertia torques in four-bar mechanisms. J. Mech. Engng Sci. 9(4), 309-317 (1967). 47. H. H. Mahie and F. W. Ocvirk, Mechanisms and Dynamics of Machinery, 2nd Edn. Wiley, New York (1963). 48. W. E. Bonham, Calculating the response of a four-bar linkage. ASME Paper No. 70-Mech-69, Mech. Conf. Columbus, Ohio (1970). 49. G. G. Lowen, F. R. Tepper and R. S. Berkof, The quantitative influence of complete force balancing on the forces and moments of certain families of four-bar linkages. Mechanism and Machine Theory 9(3, 4), 299-323 (1974) and Erratum, 10(4), 361-363 (1975). 50. A. T. Yang, Inertia force analysis of spatial mechanisms. Trans. ASME, J. Engng Ind., 93B(1), 27-33 (1971). 51. C. Bagci, Screw calculus in dynamics of plane mechanisms. Communications 3rd World Congr. Theory of Machines and Mechanisms. Vol. H, Paper H-5, 57-72. Kupari, Yugoslavia (1971). 52. C. Bagci, Static force and torque analysis using 3 x 3 screw matrix, and transmission criteria for space mechanisms. Trans. ASME, J. Engng Ind. 93B(1), 90--101 (1971). 53. J. P. Sadler, Dynamic analysis of mechanisms including coulomb friction via equilibrium equations. Proc. 3rd Appl. Mech. Conf. Oklahoma State University, Paper No. 13 (1973). 54. M. R. Smith and L. Maunder, Inertia forces in a four-bar linkage. J. Mech. Engng Sci. 9(3), 218-225 (1967). 55. C. E. Benedict and D. Tesar, The use of influence concepts in treating the dynamics of machine problems. Proc. 3rd Appl. Mech. Conf. Paper No. 43. Oklahoma State University (1973). 56. W. Rehwald, Kinetische Bewegungsanalyse ebener, ungleichf6rmig i~bersetzenderKurbelgetriebe. VDI-Forschungsheft 503 (1964). 57. N. E. Zhukovsky, The dynamic problems of a kineticchain reduced to problems of a lever(Russian).Complete Works I, Gostekhizdat (1937). 58. I. I. Artobolevskiiand V. S. Loshchinin, Investigationand finding of the laws of distributionof inertialforces in machine assemblies.J. Mechanisms 5(4),467--494(1970). 59. S. Timoshenko and D. H. Young, Advanced Dynamics. McGraw Hill,N e w York (1948). 60. M. Skreiner and B. W. Roberts, Dynamics of hydraulicallyactuated linkage mechanisms. Mech. Chem. Engng Trans., No. 2, pp. 207-215 (1967). 61. B. Paul and D. Krajcinovic,Computer analysisof machines with planar motion, Part 2: Dynamics. Trans. ASME, 92E, J. Appl. Mech. 37(3),703-712 (1970). 62. Y. Yokoyama and K. Ogawa, Dynamic characteristicof planar six-barlinkage.Bull.JSME 14(78),1352-1362 (1971). 63. J. Hunka, Analysis of the fluctuationof input torque on the four-bar crank-roc,k, er linkage. Aust. J. Appl. Sci. 14, 222-230 (1963). 64. B. E. Quinn, Energy method for determining dynamic characteristicsof mechanisms. J. Appl. Mech. 16, 283-288 (1949). 65. M. A. Chace, Analysis of the time-dependence of multi-freedom mechanical systems in relativecoordinates. Trans. ASME, J. Engng Ind. 89B(I), 119-125 0967). 66. M. A. Chace and Y. O. Bayazitoglu,Development and applicationof a generalizedd'Alembert force for multi-freedom mechanical systems. Trans. ASME, J. Engng Ind.,93B(I),317-327 0971). 67. W. Meyer zur Capellen,Torsionalvibrationsin the shaftsof linkagemechanisms. Trans. ASME, J. Engng Ind. 89B(I), 126--136(1967). 68. J. J. Uicker, Jr.,Dynamic behavior of spatiallinkages.Trans. ASME, J. Engng Ind. 91B(1),251-265 (1969). 69. R. S. Berkof and G. G. Lowen, Theory of shaking moment optimizationof force-balancedfour-barlinkages. Trans. ASME, J. Engng Ind. 93B(I),53-60 (1971). 70. R. S. Berkof and G. G. Lowen, A new method for completelyforcebalancing simple linkages.Trans. ASME, J. Engng Ind. 91B(1),21-26 (1969). 71. G. G. Lowen and R. S. Berkof, Determination of force-balancedfour-bar linkages with optimum shaking moment characteristics.Trans. ASME, J. Engng Ind. 93B(I),39--46(1971). ANTRIEBSMOMEN~ VON KOPPELGETRIE~N R. S. Berkof Kurzfassuns - Viele Maschinen enthalten Get~lebe, deren Antriebsmoment und Antriebsgeschwindigkeit zeitlich variieren, normalerweise in periodischer Form. Es wird meist gefordert, dab die Antriebswinkelgeschwlndigkeit und infolgedessen das Antriebsmoment in gewissen vo~geschrlebenen Gxenzen zu halten sind. Da eln Motor Gblicherweise fttw die hSchste Belastung gew~hlt wird, is~ es vo~teilhaft, die Drehmomentenspitzen zu gl~tten. DeE Vorteil ist nicht nux ein schw~cherer Motor, sondern auch ein ger~uschloserer Lauf und eine erhShte Lebensdauer gewisser Maschlnenteile. Es ist dee Zweck dleser Abhandlung, einen [~berbllck Gber die Methoden fdr Antriebsmomerit- und Geschwindlgkeitsausgleich bei Getrleben zu geben. Eine tiefer gehende Diskussion Gber das Antriebsd2ekmoment von Viergelenkgetrieben folgt. Die Abhandlung gibt einen Sberblick Gber die verschiedenen angewendeten Grundlagen fGr analytlsche AusdrGcke fOm das Antriebsmoment yon Koppelgetrleben. Die einfachste Form scheint ein ku~zer Differentialquotient dee kinetischen EneEgie zu sein. Von ihm Mann ein elnfacheE Ausdruck f~r das AntEiebsmoment eines 4glledrigen k.w~fteausgeglichehen roppelgetriebes abgeleltet weEden. Alle Tr~heitswiEkungen aus Translation und Rotation slnd de.Tin enthalten! es folgt daraus die Beziehung zwischen AntEiebsmoment, RGttelmoment und Moment dee Geetellkr~fte. Da dee Momentenausgleich InteressieEt, wiEd mittels des Minimumver~ahzens des kleinsten Quadrates das AntEiebsmoment dutch Massenumverteilung minimiert.