Comparative analysis for low-mass and low-inertia dynamic balancing of mechanisms

V. van der WijkUniversity of Twente,

Faculty of Engineering Technology,

Volkert van der Wijk,

P.O. Box 217, 7500 AE,

Enschede, Netherlands

B. DemeulenaereAtlas Copco Airpower NV,

Bram Demeulenaere,

Boomsesteenweg 957, B-2610,

Wilrijk, Belgium

C. GosselinLaval University,

Department of Mechanical Engineering,

Clement Gosselin, Pavillon Adrien-Pouliot,

1065 Avenue de la medecine,

Quebec, Quebec G1V 0A6, Canada

J. L. HerderUniversity of Twente,

Faculty of Engineering Technology,

Just L. Herder, P.O. Box 217, 7500 AE,

Enschede, Netherlands

Comparative Analysisfor Low-Mass and Low-InertiaDynamic Balancing ofMechanismsDynamic balance is an important feature of high speed mechanisms and robotics thatneed to minimize vibrations of the base. The main disadvantage of dynamic balancing,however, is that it is accompanied with a considerable increase in mass and inertia. Aim-ing at low-mass and low-inertia dynamic balancing, in this article the relative impor-tance of the balance parameters of common balancing principles is analyzed and thebalancing principles are compared. To do this, the evaluation of a balanced rotatablelink is found to be representative for a large group of balanced mechanisms. Therefore, arotatable link is balanced with duplicate mechanisms (DM), with a countermass (CM)and a separate counter-rotation (SCR), and with a counter-rotary countermass (CRCM).The equations for the total mass and the inertia are derived and compared analyticallywhile the balancing principles are compared numerically. The results show that the DM-balanced link is the best compromise for low mass and low inertia but requires a consid-erable space. For the CRCM-balanced link and the SCR-balanced link that are morecompact, there is a trade-off between mass and inertia for which the CRCM-balancedlink is the better of the two. [DOI: 10.1115/1.4006744]

1 Introduction

Dynamic balance, i.e., shaking force and shaking moment bal-ance, is an important feature of machines and mechanisms thathave to run at high speeds with minimum vibrations of the baseand, in addition, of free floating mechanisms such as space manip-ulators to maintain position and orientation. Advantages ofdynamically balanced mechanisms include increased accuracy [1]and reduced cycle times [2]; reduced noise, wear, and fatigue [3];and improved ergonomics [4]. Since the base of a dynamicallybalanced machine does not vibrate, heavy supports and rigidfloors are not needed. Balanced machines therefore can havesmaller footprints, which increase the capacity of a factory floor.Since balanced machines do not have disturbing effects to thebuilding and surroundings [5], they also can be placed on leveledfloors allowing factories to be built vertically up.

The main disadvantage of dynamic balancing is that often aconsiderable amount of mass and inertia is added [6]. For movingvehicles, space manipulators, robot end-effector tools, and mate-rial and transport costs, a low mass is important. For low drivingtorques and low driving power, low inertia is important.

Especially for high speed manipulators and robotics, the focuson low-mass and low-inertia dynamic balance solutions is essen-tial. So far, however, little work has been done from this perspec-tive. Serial and parallel manipulators are mainly balanced bydirect application of known solutions of the balancing of linkages[7,8], showing a dramatic increase in mass and inertia [9]. On onehand, the balancing of manipulators is more complicated becauseof their many degrees-of-freedom (DoF). On the other hand, sincethe kinematics of manipulators is not as determining as it often isfor linkages, there is an increased freedom for the design focusedon dynamics rather than kinematics. For linkages, approaches intothis direction have been proposed for optimal input torque balanc-ing [10].

With the focus on robotics, in Refs. [6,11] various balancingprinciples are analyzed and compared with respect to their massincrease and inertia increase. From these studies, general guide-lines for low-mass and low-inertia dynamic balancing werederived [12]. The main results include that mechanism links haveto be positioned such that they counter-rotate with one another,CMs should be solely at links connecting the base, and the designspace for balancing elements should be as large as possible. Thereis a trade-off between the addition of mass and the addition ofinertia, which needs to be made for the intended application.

Synthesis following the guidelines has resulted in various ad-vantageous balance solutions. Spatial Delta-like robots can beforce balanced with masses near the base and one leg being a spa-tial pantograph [13]. A dynamically balanced redundant planar 4-RRR parallel manipulator was designed with just four counter-masses near the base [14]. Dynamically balanced 2-DoF graspermechanisms [15] were derived from linkage architectures that areinherently balanced having no countermasses at all.

The comparison of balancing principles regarding additionalmass and additional inertia in Refs. [6,11] was done by applyingthem to a double pendulum (dyad). This was found to represent aconsiderable part of all balanced mechanisms from the literaturesince often they can be regarded as composed of balanced doublependula. A fair comparison was aimed at by selecting a specificset of values for the linkage and balance parameters and for thedesign of the balancing parts. The relative influence of these link-age and balance parameters, however, was not studied in detail.

This article is aimed at investigating the relative importance ofthe balance parameters to gain detailed understanding on howwithin the design process of a specific balance architecture thetotal mass and inertia can be affected. To do this, the balancingprinciples of Refs. [6,11] are applied to a rotatable link, analyzedwithin the parameters, and compared.

After the fundamentals, the relevance of a balanced rotatablelink is discussed. Subsequently, the equations for the total massand the inertia of the DM-balanced link, the SCR-balanced link,and the CRCM-balanced link are derived. They are comparedboth analytically and numerically. The influence of the trade-off

Contributed by the Mechanisms and Robotics Committee of ASME for publica-tion in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received November12, 2011; final manuscript received April 25, 2012; published online June 7, 2012.Assoc. Editor: Sundar Krishnamurty.

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between mass and inertia is considered for which the optimal pa-rameter values of the balanced links are selected and compared.

2 Fundamentals

The conditions for which mechanisms are dynamically bal-anced can be derived from the linear momentum and the angularmomentum equations [6,16]. If the linear momentum of a mecha-nism is conserved, then the resultant force on the base is zero andthe mechanism is shaking force balanced. If the angular momen-tum is conserved, then the resultant moment on the base is zeroand the mechanism is shaking moment balanced. Both force bal-ance and moment balance are necessary to have a (completely)dynamically balanced mechanism.

The total mass of a mechanism is defined as the sum of themass values of all the moving elements. The inertia of a mecha-nism is defined as the reduced inertia in Ref. [17]. This is the iner-tia of the mechanism reduced to the input angles which can bederived from the kinetic energy equations.

For selecting the optimal parameter values of the balanced linksfor the trade-off between low mass and low inertia, the mass–inertia factor [6] is used. This factor is the weighted sum of the ra-tio of the total mass and the ratio of the inertia before and afterbalancing. The weight factors determine the relative importanceof low mass and low inertia for the intended purpose.

3 Rotatable Link for Comparison

Various authors have investigated the balancing of mechanismsby considering the balancing of a double pendulum, includingRefs. [6,18–20]. Figure 1(a) shows a double pendulum balancedwith CRCM from Ref. [6]. This mechanism can be regarded as acomposition of two CRCM-balanced rotatable links as illustratedin Fig. 1(b). The resulting mass and inertia of the upper link are aconstant part of the lower link. The influence of the balance pa-rameters on the total mass and inertia of the double pendulumtherefore can be investigated by considering a single rotatablelink. This also holds for double pendula balanced with separatecounter-rotations.

In addition, with the target to balance mechanisms solely withbalance elements at the links connecting the base, as in the 4-RRRrobot in Ref. [14], the results of a detailed analysis of the influenceof the balance parameters of a single balanced rotatable link corre-spond to those of the complete balanced robot.

4 Principle of Duplicate Mechanisms

Figure 2 shows a link l modeled with a lumped mass m withinertia I that rotates about pivot O and is dynamically balanced

with axial and mirror duplicate links l*. This is named the dupli-cate mechanisms principle [3]. The three duplicate links are iden-tical to the initial link and can be coupled by equal gears at theirpivots to move synchronously. The horizontal mirror duplicate atO0 moves synchronously with the initial link at O in oppositedirection. Then the horizontal shaking force and the shakingmoment are balanced. With two vertical mirror duplicates rotatingsynchronously about O00 and O0 0 0 also the vertical shaking force isbalanced.

The reduced inertia can be obtained from the equation of the ki-netic energy [17], which writes

TO ¼1

2Iredh

_h2

(1)

The kinetic energy of the initial link can be written with

TO ¼1

2ðI þ ml2Þ _h

2(2)

Since the four links are equal, the kinetic energy of the balancedsystem writes

TO ¼1

2ð4I þ 4ml2Þ _h

2(3)

in which the reduced inertia is

Iredh ¼ 4I þ 4ml2 (4)

The total mass of the balanced mechanism then is written as

mtot ¼ mþ 3m� ¼ 4m (5)

5 Principle of Separate Counter-Rotations

The most common way to balance a linkage is with counter-masses and separate counter-rotations for the moment balance[21,22]. For the rotatable link, this principle is shown in Fig. 3.For force balance, a countermass m* is added at distance l* fromO. To balance the moment, an additional rotatable element withmass m�cr and inertia I�cr is placed on the base and is driven by agear attached to the link at O with which it counter-rotates withrespect to the link.

To derive the balance conditions and the equations for the totalmass and total inertia, the positions of the two masses m and m*can be written in vector notation [x,y,z]T as

r ¼l cos h

l sin h

0

264

375 r� ¼

�l� cos h

�l� sin h

0

264

375Fig. 1 A double pendulum balanced with CRCM, shown in (a)

[6], can be regarded for detailed analysis of the parameters astwo individually balanced single links as shown in (b)

Fig. 2 DM-balanced link with axial and mirror duplicates of theinitial link l, drawn to scale

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With the derivatives of these position vectors, the linear momen-tum of the SCR-balanced link is written as

pO ¼ð�mlþ m�l�Þ _h sin h

ðml� m�l�Þ _h cos h

0

264

375 (6)

Since the separate counter-rotation is pivoted to the base, its CoMis stationary with respect to the base for which its linear momen-tum with respect to the base is zero and mCR does not appear inthe equations. The linear momentum is conserved for all motionwhen pO¼ 0, which holds for the force balance condition

ml� m�l� ¼ 0 (7)

For a planar linkage, i.e., a linkage in which all elements movewithin a single plane and in which all are mass symmetric withrespect to this plane (which is when one principal axis of inertia isaligned with the axis normal to the plane), the angular momentumcan be written with a single equation. For the force balanced link,the angular momentum about the z-axis can be written as

hO;z ¼ ðI þ I� þ ml2 þ m�l�2Þ _hþ I�cr_h�

(8)

The angular velocity of the SCR _h�

depends on _h with transmis-sion ratio k being

_h� ¼ k _h (9)

with which h0,z can be written as

hO;z ¼ ðI þ I� þ ml2 þ m�l�2 þ kI�crÞ _h (10)

For the SCR-balanced link, the transmission ratio is written as

k ¼ � dO

dSCRwith dO and dSCR being the gear diameters of the gear

at the link at O and the gear at the counter-rotating element. Theangular momentum is conserved for all motion for the momentbalance condition

I þ I� þ ml2 þ m�l�2 þ kI�cr ¼ 0 (11)

Equivalent to the DM-balanced link, the inertia equation for theSCR-balanced link can be derived from the kinetic energy whichwrites

TO ¼1

2ðI þ I� þ ml2 þ m�l�2Þ _h

2 þ 1

2I�cr

_h�2

(12)

With the kinematic relation of Eq. (9), this results in

TO ¼1

2ðI þ I� þ ml2 þ m�l�2 þ k2I�crÞ _h

2(13)

in which the reduced inertia equation is

Iredh ¼ I þ I� þ ml2 þ m�l�2 þ k2I�cr (14)

The equation for the total mass is written as

mtot ¼ mþ m� þ m�cr (15)

6 Counter-Rotary Countermass Principle

Figure 4 shows three possible CRCM configurations for balanc-ing a rotatable link. Equivalent to the SCR-balanced link, a massm* is placed at a distance l* from O for force balance. For themoment balance, in Fig. 4(a) the inertia I* of this mass is drivenwith transmission ratio k by a chain (or belt) attached around agear (or pulley) at O, which is fixed to the base [23]. When link lis moved, I* will rotate in opposite direction (k is negative) forwhich the countermass m* is a counter-rotary countermass. InFig. 4(b), the CRCM is driven by a pair of external gears [16], andFig. 4(c) shows a configuration by using internal gears [24]. Foreach configuration, k can be obtained from the gear diameters as

k ¼ 1� dO

dCRCMwith diameters dO and dCRCM of the fixed gear at O

and the CRCM gear, respectively. The latter gear is fixed to theCRCM but its diameter can differ from the diameter of theCRCM.

The force balance condition for this configuration is equal tothe force balance condition of the SCR-balanced link (Eq. (7)).The angular momentum about the z-axis is written as

hO;z ¼ ðI þ ml2 þ m�l�2Þ _hþ I� _h�

¼ ðI þ ml2 þ m�l�2 þ kI�Þ _h (16)

in which the kinematic relation of Eq. (9) was substituted. Theangular momentum is conserved for all motion for the momentbalance condition

I þ ml2 þ m�l�2 þ kI� ¼ 0 (17)

The kinetic energy of the CRCM-balanced link can be writtenwith

TO ¼1

2ðI þ ml2 þ m�l�2Þ _h

2 þ 1

2I� _h�2

(18)

With the kinematic relation of Eq. (9), this results in

TO ¼1

2ðI þ ml2 þ m�l�2 þ k2I�Þ _h

2(19)

Fig. 4 CRCM-balanced rotatable link by using: (a) gears withchain [23], (b) external gears [16], and (c) internal gears [24]

Fig. 3 Balanced rotatable link with a countermass and a sepa-rate counter-rotation

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in which the reduced inertia equation is

Iredh ¼ I þ ml2 þ m�l�2 þ k2I� (20)

The total mass of the CRCM-balanced link is calculated with

mtot ¼ mþ m� (21)

7 Relations of the Parameters

By comparing the moment balance condition, mass equation,and inertia equation of both CRCM- and SCR-balanced links, twodifferences are noted. Due to the extra counter-rotating elementfor the SCR-balanced link, an extra parameter appears in eachequation. Furthermore, the balancing moment for the CRCM-balanced link is coming from I*, while for the SCR-balanced linkit is coming from the additional I�cr .

To evaluate the influence of these differences in a practical situa-tion and to find the optimal set of parameter values for which theaddition of mass and the addition of inertia are minimal, the designof the masses must be chosen. For maximum clarity, the mass andthe inertia elements in the configurations of Figs. 3 and 4 are mod-eled as solid discs of thickness t and radius Ri. Herewith, the num-ber of parameters remains low. The mass and the inertia of thesediscs are related by

mi ¼ qptR2i

Ii ¼1

2miR

2i

9=;) m2

i ¼ 2qptIi (22)

with material density q. Collecting all equations for the SCR-balanced link results in the following set of equations:

Force balance: ml� m�l� ¼ 0 (7)

Moment balance: I þ ml2 þ I� þ m�l�2 þ kI�cr ¼ 0 (11)

Inertia: Iredh ¼ I þ ml2 þ I� þ m�l�2 þ k2I�cr (14)

Total mass: mtot ¼ mþ m� þ m�cr (15)

Design relations: m�2 ¼ 2qptI�

m�2cr ¼ 2qptI�cr (22)

With m, I, l, t, and q being known, there are six equations andeight unknowns (m*, I*, l*, mtot, Ired

h , k, I�cr , and m�cr). Two balanceparameters (one of the original linkage and one of the additionalelement) must be chosen for which the others are determined. Bymaking combinations, various relations can be obtained in whichone balance parameter depends on a single other balance parame-ter including the following:

Iredh ðm�;m�crÞ ¼ I þ ml2 þ m�2

2qptþ m2l2

m�

þ 2qpt

m�2cr

I þ ml2 þ m�2

2qptþ m2l2

m�

� �2

(23)

Iredh ðk;m�Þ ¼ ð1� kÞ I þ ml2 þ m�2

2qptþ m2l2

m�

� �(24)

kðm�;m�crÞ ¼ �2qpt

m�2cr

I þ ml2 þ m�2

2qptþ m2l2

m�

� �(25)

With Eq. (15), the total mass–inertia relation Iredh ðmtotÞ is obtained

from Eq. (23) and the total mass-transmission ratio relation k(mtot)is obtained from Eq. (25).

For the CRCM-balanced link, there are in total five equations,one less than for the SCR-balanced link because of the counter-rotating element. The complete set of equations is as follows:

Force balance: ml� m�l� ¼ 0 (7)

Moment balance: I þ ml2 þ m�l�2 þ kI� ¼ 0 (17)

Inertia: Iredh ¼ I þ ml2 þ m�l�2 þ k2I� (20)

Total mass: mtot ¼ mþ m� (21)

Design relation: m�2 ¼ 2qptI� (22)

in which there are six unknowns (m*, I*, l*, mtot, Iredh , and k).

Therefore, one of them can be chosen independently. With combi-nations from this set of equations, the following relations areobtained:

Iredh ðm�Þ ¼ I þ ml2 þ m2l2

m�þ 2qpt

m�2I þ ml2 þ m2l2

m�

� �2

(26)

Iredh ðl�Þ ¼ I þ ml2 þ mll� þ 2qptl�2

m2l2I þ ml2 þ mll�� �2

(27)

kðm�Þ ¼ � 2qpt

m�2I þ ml2 þ m2l2

m�

� �(28)

kðl�Þ ¼ � 2qptl�2

m2l2I þ ml2 þ mll�� �

(29)

With Eq. (21), from Eqs. (26) and (28) the total mass–inertia rela-tion and the total mass-transmission ratio relation are obtained,respectively. The dependency of Ired

h and k on length l* followsfrom Eqs. (27) and (29).

8 Numerical Comparison

For a numerical comparison, the parametric values of Table 1 arechosen. For these values, Fig. 5 shows the dependency of the bal-ance parameters I*, l*, and k on the countermass m* of the CRCMprinciple. This figure can be useful as a design chart. For example,when k¼� 4 it can be found that m*¼ 1.65 kg, I*¼ 0.0056 kg m2,and l*¼ 0.046 m, as indicated by the dotted lines.

Figure 5 shows that for an increasing value of countermass m*,the length l* and the transmission ratio k decrease while the inertia

Table 1 Parameter values

m¼ 0.3 (kg) l¼ 0.25 (m) q¼ 7800 (kg m�3)I¼ 184 (kg mm2) t¼ 0.01 (m)

Fig. 5 Design chart for the CRCM-balanced link showing thedependency of I*, l*, and k on countermass m*

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I* increases. Since mass m is constant, it follows from Eq. (21)that these characteristics also hold for the relation of the totalmass mtot with I*, l*, and k.

For the SCR-balanced link, Fig. 6 shows the relation betweenthe transmission ratio k with the masses m* and m�cr . In each direc-tion, the surface has equivalent behavior as the curve of k dependson m* for the CRCM-balanced link. Since the equations are equal,the relations of I* with m*, I�cr with m�cr , and l* with m* for theSCR-balanced link are equal to the curves of the CRCM-balancedlink.

For a realistic visualization of the balancing principles, fork¼� 4 the SCR- and CRCM-balanced links are drawn to scale inFigs. 7 and 8, respectively. For the same parameters also the DM-balanced link in Fig. 2 is drawn to scale.

Figure 9 shows the relation between the reduced inertia Iredh and

the total mass mtot for the three balancing principles for k¼� 4.For the SCR-balanced link, the curves are shown for a range ofvalues of m�cr . The result of the DM-balanced link is a point sinceit has a single solution.

Figure 9 shows that for the CRCM-balanced link a decreasingtotal mass leads to an increasing inertia and vice versa. The mini-mum values are the horizontal asymptote with value Ih¼ Iþml2

and the vertical asymptote for mtot¼m. For each curve of theSCR-balanced link, there exists a minimum for the inertia. Thisminimum can be obtained from the derivative of Eq. (24) and

holds for m�SCR;minðIred

h Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqptm2l23

p. This minimum is independent

of the transmission ratio. The minimum values for the total massare determined by the vertical asymptotes mtot ¼ mþ m�cr.

From Figs. 5 and 9, it is observed that for the CRCM-balancedlink for increasing l* and k and for decreasing I*, the inertiaincreases and vice versa. The least inertia is obtained for an

infinitely large disc as CRCM at an infinitely small distance l*with an infinitely low transmission ratio. The minimum total massis obtained with an infinitely small mass at an infinitely large dis-tance l*. This means that for the CRCM-balanced link there is atrade-off between the total mass and the reduced inertia.

For the SCR-balanced link, a small transmission ratio and alarge m�cr together with the optimal m* lead to the smallest inertia.However, for m�cr being smaller there is a trade-off between thetotal mass and the reduced inertia too.

The total mass–inertia relation of the CRCM-balanced link inFig. 9 is in any situation below the curves of the SCR-balancedlink. This means that for equal total mass or for equal reducedinertia, the CRCM-balanced link has a lower reduced inertia or alower total mass, respectively. The DM-balanced link, however,shows the lowest total mass and the reduced inertia combination.

Table 2 shows some numerical results for CRCM- and SCR-balanced links for various transmission ratios. With Eqs. (28) and(25), it can be shown that for any k the total mass of the CRCM-balanced link is smaller than the total mass of the SCR-balanced

Fig. 6 Relation of the transmission ratio k with respect to themasses m* and m�cr for the SCR-balanced link showing similarbehavior as compared to Fig. 5

Fig. 7 SCR-balanced link drawn to scale with k 5 24

Fig. 8 CRCM-balanced link drawn to scale with k 5 24

Fig. 9 Relation between the total mass and the inertia of theCRCM-, SCR-, and DM-balanced links. For the CRCM-balancedlink, the curve is below the curves of the SCR-balanced link.The DM-balanced link has the lowest value.

Table 2 The results of the total mass and inertia with respectto the transmission ratio (with m�SCR ¼ m�

SCR;minðI redhÞ) with equal

inertia for k 5 2 23.6

SCR CRCM

Transmission ratio mtot (kg) Iredh ðkgm2Þ mtot (kg) Ired

h ðkgm2Þ

k¼�1 5.01 0.053 3.48 0.042k¼�4 3.21 0.133 1.95 0.112k¼�16 2.31 0.452 1.18 0.431k¼�23.6 2.15 0.653 1.04 0.653

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link. For the inertia however, there is a break-even point. Form�SCR ¼ m�

SCR;minðIredh Þ

, the break-even point is at the lowest k, in

this case being k¼� 23.6 as shown in Fig. 10. For a smaller orlarger m�SCR, the break-even point exists for higher transmissionratios. For transmission ratios lower than the break-even point, theinertia of the CRCM-balanced link is smaller, while for highertransmission ratios the SCR-balanced link has a lower inertia. Fora low inertia, a low transmission ratio is needed, which is inadvantage of the CRCM-balanced link.

8.1 Influence of Trade-Off Between Mass and Inertia. Toselect the optimal parameter values and the best balancing princi-ple for a particular application, the relative importance of the totalmass and the reduced inertia has to be chosen. This is done withthe mass–inertia factor, which is written as [6]

l ¼ wM � mþ wh � Ih (30)

where wM and wh are, respectively, the weight factors for the massand the inertia, and m and Ih are, respectively, the dimensionlessnumbers for the increase in total mass and the increase in thereduced inertia. These ratios are calculated with

m ¼ mtot

motot

(31)

Ih ¼Iredh

Ired;oh

(32)

where motot and Ired;o

h are, respectively, the total mass and thereduced inertia before balancing, and mtot and Ired

h are, respec-tively, the total mass and the reduced inertia after balancing. Theoptimal parameters are found for the lowest l, and the best bal-ancing principle is the principle with the lowest minimal value l.The results for both weights being equal to one is defined as thecharacteristic mass–inertia factor lc.

The values for the total mass and the reduced inertia of the linkbefore balancing are mo

tot ¼ 0:3000 and Ired;oh ¼ 18;934ðkg mm2Þ.

For the characteristic mass–inertia factor (wM¼wh¼ 1), theresults for the balanced links are shown in Fig. 11.

The minimum lc is found for the DM-balanced link. The mini-mal lc for the SCR-balanced link is for all m�cr higher than that ofthe CRCM-balanced link.

When a low mass is more important than a low inertia, then theweight for the total mass can be chosen higher, for example,wM¼ 2. The resulting mass–inertia values are shown in Fig. 12.

With respect to the values of lc in Fig. 11, for each curve the min-imum mass–inertia value is found for a lower total mass.

When a low inertia is more important than a low mass, then theweight for the inertia can be chosen higher, e.g., wh¼ 2. Figure 13shows the resulting mass–inertia values. With respect to the val-ues of lc in Fig. 11, for each curve the minimum mass–inertiavalue is found for a higher total mass. For both situations inFigs. 12 and 13, the CRCM-balanced link shows to be advanta-geous for low mass and low inertia with respect to the SCR-balanced link with the DM-balanced link being the best of thethree.

9 Influence of q and t

The choice for the material density q and the thickness t withwhich the countermasses and the counter-rotating elements weremodeled affects the results. Since these two parameters appearsolely as the product qt, investigation can be done regarding thisproduct.

For the product qt being 15 times smaller than the product ofthe parameters from Table 1, the characteristic mass–inertia

Fig. 10 The reduced inertia with respect to the transmissionratio for CRCM- and SCR-balanced links. A break-even pointexists and is at the lowest k for m�SCR ¼ m�

SCR;minðI redhÞ.

Fig. 11 Characteristic mass–inertia curves (wM 5 wh 5 1). Thevalue is the lowest for the DM-balanced link. The values of theSCR-balanced link are, for all m�cr , higher than those of theCRCM-balanced link.

Fig. 12 Mass–inertia values for a low total mass being twice asimportant than a low inertia (with wM 5 2 and wh 5 1)

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values are shown in Fig. 14. This could be the material being 15times lighter or the discs being 15 times thinner. Figure 14 showsthat for these values, the CRCM-balanced link has the lowest min-imum mass–inertia value, while the DM-balanced link shows alower value than the SCR-balanced link. When the value qt isreduced further, also the SCR-balanced link eventually will havelower values than the DM-balanced link. When the value qt isincreased, the results will be similar to Fig. 11 with the CRCM-balanced link having lower values than the SCR-balanced link.

For qt being 15 times smaller, Fig. 15 shows that the positionof the break-even point between the CRCM- and the SCR-balanced link for the inertia with respect to the transmission ratiochanged. For the SCR-balanced link with m�

SCR;minðIredhÞ, the break-

even point has become k¼� 13.7 which is lower than in Fig. 10.

10 Discussion

For the analysis and comparative study, the countermasses andcounter-rotating elements were modeled as solid discs. If thecounter-rotating elements would be designed for instance as ring-shaped elements, by which the inertia is higher for equal mass,this would be advantageous for both CRCM- and SCR-balancedlinks. A higher inertia of the counter-rotating elements implies a

lower transmission ratio and a lower reduced inertia of the mecha-nism. Or for equal inertia the total mass is lower. Since also a lowvalue for qt implies a higher inertia for equal mass, this effect hasbeen shown.

From the results it can be summarized that for a low mass addi-tion, the (counter-rotary) countermass should be far away fromthe center of rotation. For a low inertia addition the counter-rotations should rotate slowly for which they need to have a lowtransmission ratio. For a low inertia addition, the (counter-rotary)countermass should be located close to the center of rotation.Since this is disadvantageous for low mass addition, there is atrade-off between the addition of mass and the addition of inertia.

A low transmission ratio is practically applicable for all threeCRCM configurations of Fig. 4. Having the countermass far awayfrom the origin for low mass addition requires considerable space.The DM-balanced link also requires a considerable space, how-ever, this may be the reason it has a relatively low mass and iner-tia. It has the advantageous features of a low transmission ratio(�1) and of the “countermasses” being placed relatively far fromthe origin with respect to the CRCM- and the SCR-balanced links.

The influence of mass and inertia of the transmission mecha-nisms, i.e., the gears, pulleys, or other transmission solutions, hasnot been considered in this article. A specific choice from themany possible designs of such a transmission would have reducedthe clarity of the investigation significantly. Since all three balanc-ing principles need a transmission, it is likely that its influence oneach principle is comparable. When desired, mass and inertia ofthe transmissions can be included in the evaluation by extendingthe mass–inertia factor (Eq. (30)).

Another possible extension of the mass–inertia factor (Eq. (30))is to independently weight the mass and inertia of the movinglinks and the mass and inertia of the base and the elements on thebase. This can be of specific interest for machines and mecha-nisms for which the importance of the mass and inertia of the baseis different from the importance of the moving mass, for instancewhen the base is fixed.

11 Conclusions

In this article, the relative importance of the balance parametersof a CRCM-balanced link and a SCR-balanced link were ana-lyzed. Together with a DM-balanced link, they were comparedboth analytically and numerically. With the mass–inertia factor,optimal balance parameter values were selected, taking intoaccount the trade-off between a low mass and a low inertia. It was

Fig. 13 Mass–inertia values for a low inertia being twice as im-portant than a low total mass (with wM 5 1 and wh 5 2)

Fig. 14 Curves for the characteristic mass–inertia factor(wM 5 wh 5 1) for qt /15. The minimum mass–inertia value for theCRCM-balanced link is lower than that of both DM- and SCR-balanced links.

Fig. 15 By reducing the product qt (here qt/15), the transmis-sion ratio the transmission ratio for which the minimum break-even point for the inertia exists also reduces

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found that the evaluation of a balanced rotatable link is the repre-sentative for a large group of balanced mechanisms.

The results show that for both CRCM- and SCR-balanced links,there is a trade-off between low mass and low inertia. For lowmass addition, the (counter-rotary) countermass has to be placedfar away from the center of rotation. For a low inertia addition,the transmission ratios have to be low and the (counter-rotary)countermass has to be placed close to the center of rotation.

For the SCR-balanced link, there is an optimum for the inertia.The total mass–inertia relation of the CRCM-balanced link isalways lower than that of the SCR-balanced link for which theCRCM-balanced link is favorable for low mass and low inertiadynamic balancing. For counter-rotating elements having a rela-tively low mass and high inertia, e.g., with a ring-shaped design,the CRCM- and the SCR-balanced links become better than theDM-balanced link for low mass and low inertia dynamicbalancing.

For equal transmission ratios of CRCM- and SCR-balancedlinks, there is a break-even point for the inertia. For low transmis-sion ratios, the CRCM-balanced link shows to have the leastinertia.

NomenclatureI ¼ inertia

Ired ¼ reduced inertiaI ¼ ratio of inertia before and after balancing

m ¼ massm ¼ ratio of mass before and after balancingl ¼ link lengthd ¼ gear diameterk ¼ transmission ratioh ¼ absolute angle of link with respect to reference framer ¼ mass position vector

(.)* ¼ balance propertypO ¼ linear momentum about the originhO ¼ angular momentum about the origin

T ¼ mechanism’s kinetic energyl ¼ mass–inertia factorw ¼ weight factor

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