A GRAPHICAL-ANALYTICAL TECHNIQUE FOR THE SYNTHESIS OF NON­CIRCULAR GEARS IN PATH-GENERATING GEARED FIVE-BAR MECHANISMS

D. Mundo, G. GattiDepartment of Mechanical Engineering - University of Calabria

87036 Arcavacata di Rende (CS) - Italy2 Department of Mechanical Engineering - National Cheng Kung University, Tainan

Contact: d.mundo@unical.it

Received July 2008, Accepted September 2008No. 08-CSME-22, E.I.C. Accession 3059

ABSTRACTThe paper presents a graphical-analytical technique for the synthesis of non-circular gears in path­

generating geared five-bar mechanisms. In such mechanisms, a two degree-of-freedom (dot) five-barlinkage is integrated by a pair of non-circular gears to precisely guide a coupler point along a prescribedplanar trajectory.The synthesis method proposed here considers the most general case, where the prescribed trajectoryconsists of a series of open curves. Each segment of the prescribed path identifies a phase of themechanism motion that is referred to as requested motion branch. In each of these phases, for anyprescribed position of the coupler point, the inverse kinematic analysis of the linkage and the Aronhold­Kennedy theorem are used to identify the actual configuration of the system and locate the instantaneouscentre of the relative motion between the two cranks of the linkage. The regions of the gear's centrodes,corresponding to the requested motion branches, are thus synthesized. These regions are connected toeach other by using proper polynomial functions, as to guarantee a continuous and cyclic motion of themechanism. An example is illustrated where the requested coupler point trajectory consists of a series ofstraight line segments.

TECHNIQUE GRAPHIQUE-ANALYTIQUE POUR LA SYNTHESE DE ROUES NONCIRCULAIRES DANS LES MECANISMES COUPLES ACINQ BARRES DE

GENERATION DE TRAJECTOIRE

RESUMECet article presente une technique graphique-analytique pour la synthese de roues non circulaires dans

les mecanismes couples it cinq barres de generation de trajectoire. Dans de tels mecanismes, unecommande it cinq barres it deux degres de liberte est integree par une paire de roues non circulaires pourguider de fa~on precise un bout de coupleur Ie long d'une trajectoire planaire prescrite.La methode de synthese proposee ici envisage Ie cas Ie plus general en vertu duquel la trajectoire prescriteconsiste en une serie de courbes ouvertes. Chaque segment de la trajectoire prescrit reconnait une phasedu mouvement mecanique it laquelle on fait reference comme etant une branche de mouvement requise.Dans chacune de ces phases, pour toute position prescrite du bout du coupleur, f'analyse cinematiqueinverse de fa commande et Ie theoreme Aronhold-Kennedy sont utilises pour determiner la configurationreelle du systeme et localiser Ie centre instantane du mouvement relatif entre les deux manivelles de lacommande. Les regions des centrodes des roues correspondant aux branches de mouvement sont doncsynthetisees. Elles sont reliees entre elles en utilisant les fonctions polynomiales adequates de sorte itgarantir un mouvement continu et cyclique du mecanisme. II y a un exemple illustre montrant ou latrajectoire du bout du coupleur requis consiste en une serie de segments de lignes droites.

Transactions a/the CSME Ide fa SCGM Vol. 32, No. 3-4, 2008 487

1. INTRODUCTION

Geared five-bar mechanisms (GFBM) represent an effective solution for problems of rigid bodyguidance, path generation or function generation, as demonstrated by the wide literature that deals withsuch mechanical systems. Potential applications of these mechanisms, such as sewing and stampingmachines, steering mechanisms and straight-line generators, were discussed by Reuleaux [1].Freudenstein and Primrose [2] analyzed the kinematic properties of the motion of a point belonging to thecoupler of a GFBM. Roth and Freudenstein [3] proposed a numerical method for the synthesis of aGFBM for path generation tasks defined by nine precision points, while a GFBM was proposed by Oleksaand Tesar [4] as function generating mechanism. More recently, continuation methods have beenproposed by Starns and Flugrad [5] to synthesize a GFBM for a path generation task defined by sevenprecision positions, while Nokleby and Podhorodesky [6] proposed a method for the optimal synthesis ofGFBM, based on a quasi-Newton optimization routine.

A study on the kinematic properties of GFBM with various gear ratios and phase angles was proposedby Zhang et al. [7], who proposed an algorithm for the optimal synthesis of a symmetric GFBM as a pathgenerating mechanism. In the last paper, the authors mentioned the idea of enhancing the versatility ofsuch mechanisms, in terms of displacement or velocity profile of the coupler point, by coupling the twocranks through a variable gear ratio function.

The integration of five-bar linkages with non-circular gears was proposed by Mundo et al. [8] tosynthesize a mechanism capable of precisely moving the coupler according to a motion defined by twogeneralized coordinates, as in path generation In that paper, the synthesis of the non-circular gears isbased on a velocity analysis of the linkage. Since the shape of the gears is dependent on the geometriccharacteristics of the linkage, an optimization is also proposed to reduce the gears' non-circularity, thusenhancing the feasibility of gear manufacture and the dynamic behaviour of the output noncircular gear.

In this paper, a different approach, based on a graphical construction, is proposed for the synthesis ofnon-circular gears in path-generating geared five-bar mechanisms. The prescribed trajectory of thecoupler point is supposed to be defined by n open curves. Each curve corresponds to a phase of themechanism motion that is referred to as requested motion branch. For any prescribed position of thecoupler point, the actual configuration of the system is identified through the inverse kinematic analysisof the linkage, so that the primary centres of instantaneous rotation of the linkage are located. A graphicaltechnique, based on Aronhold-Kennedy theorem [9], is then used to locate the secondary instantaneouscentres, including the instant centre of relative motion between the two cranks of the linkage. In the actualconfiguration of the system, this centre defines the tangency point between the pitch curves of the non­circular gears. By rigidly rotating this point along with each crank, the regions of the gear's centrodescorresponding to the requested motion branches, referred to as requested profiles, are synthesized. Inorder to guarantee a continuous and cyclic motion transmission from the input crank to the output crankof the mechanism, two consecutive requested profiles are connected to each other by a proper connectionprofile, defined by a fifth order polynomial functions. This way, n more phases of the mechanism motion,referred to as connection motion branches, are defined.

The outline of the paper is as follows. In Sections 2, 3 and 4, the methodology for the synthesis ofGFBMs with non-circular gears is described as consisting of three phases: an inverse kinematic analysisof the linkage, which enables locating the primary instant centres; the identification of the secondaryinstant centres and the simultaneous synthesis of the gear's requested profiles; a forward kinematicanalysis of the GFBM for evaluating the boundary conditions that enables the definition of eachconnection profile. In section 5, an example of the proposed technique is illustrated, where a GFBM withnon-circular gears is synthesized to precisely move the coupler point along a prescribed trajectory, whichis defined by three straight line segments.

Transactions ofthe CSME Ide La SCGM VoL. 32, No. 3-4, 2008 488

2. INVERSE KINEMATICS OF THE FIVE-BAR LINKAGE AND LOCATION OF THEPRIMARY INSTANT CENTRES

In figure 1, a five-bar linkage is schematically represented. It is a 2-dof mechanical system, i.e. twoindependent inputs must be specified to fully define its configuration. For this reason, such a mechanismcan be used to precisely guide the coupler point M along a prescribed path. At any instant of the motion,in fact, coordinates XM and YM of point M can be imposed according to the requested trajectory. By meansof the inverse kinematic model of the linkage, the configuration of the system can be identified, thusenabling the location of the primary instant centres of the mechanism (i.e. the instant centres of therelative motion between two adjacent links, which can be located by inspection of the linkage).

With reference to figure 1, the geometry of the mechanism is defined by angles 0 and /3, by thecoordinates of point A, and by the lengths of the links (Ll>oo, L6). In the same figure, the actual orientationei (i=1, .. ,4) of each link, considered to be positive in the counter-clockwise direction and measured withreference to the horizontal axis, is represented. The coordinates of point E are defined as

X E = X A + L s cos 0

YE = YA + Ls sin 0

For a given position (XM, YM) of the coupler point M, the values of angles e/ and e2are given as

(la)

(lb)

(2)

(3)

where the ± signs in Eq. (2) correspond to the two possible assembly configurations of dyad ABM.

Figure 1 - Scheme of the five-bar linkage used for exact path generation

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4, 2008 489

Once angles 8/ and 82 are known, the coordinate values of points Band C can be determined as:

X 8 = X A + L I cos 8 1

Y 8 = Y A + L I sin BI

Xc =xB+LzcosBz

Yc = Y B + L2 sin B2

The orientation 84 and 83 of links L4 and L3 respectively can be now determined as

8 -I [Yc - YA - Ls sin £5)4 = tan ±If

Xc - XA- Lscos £5

where

-I[ L4z + (xc - XA- Lscos £5y + (Yc - YA- Lssin b"y - L/ JIf = cos

2L4~(XC -XA-Ls cos£5)Z +(Yc - YA -Lssin£5y

and

8 _1[YC-YA-LsSin£5-L4sin84]= tan3 xc-xA-Lscos£5-L4cosB4

(4a)

(4b)

(Sa)

(5b)

(6)

(7)

(8)

The double solution for Eqs. (2) and (6) is due to the fact that the mechanism's dyads can beassembled in different configurations. Generally speaking, this problem may be avoided by one of themethods developed for the branch and order problems. In this specific case, the selection of the actualsolution is straightforward. Link 4, in fact, is to be guided, through non-circular gears, by link 1.Therefore, both cranks are requested to rotate according to monotonic functions, so that a motioninversion of the gears does not occur. The selection of the proper solution for Eqs. (1) and (5) is thenstraightforward.

The coordinates of point D can be finally determined as

X D = X E + L4 cos B4

YD =YE+L4sinB4

(9a)

(9b)

As a result of the inverse kinematics of the linkage, the primary centres C 1S, CZb C32 , C43, C14 arelocated in correspondence of points A, B, C, D and E respectively, as shown in figure 2.

3. SYNTHESIS OF THE GEARS' PITCH CURVES: REQUESTED PROFILES

In order to reduce the mobility of the system represented in figure 1, a pair of geared bodies is to besynthesized: the rotation of link 4 is thus made dependent on the rotation of link 1. The synthesis of thenon-circular pitch curves is performed in two phases, as discussed earlier. In this section, the synthesis ofthe gears' requested 'profiles is described, while the connection profiles will be dealt with in the nextsection.In order to synthesize the requested profiles, the secondary instant centres, including C 14, must be locatedfor any position of the coupler point along the prescribed trajectory. Such a task can be accomplished bymeans of a graphical technique, which is based on the Aronhold-Kennedy (AK) theorem. The AK

Transactions ofthe CSME /de fa SCGM Vol. 32, No. 3-4,2008 490

theorem can be enunciated as follows: the three instant centres Cij, Ci/o and Cpo shared by three rigidbodies i, j and k in relative motion to one another, all lie on the same straight line.In figure 2, the primary instant centres are represented as deriving from the inverse kinematic analysisdescribed in the previous section. In the same figure, the unit vectors t and D, which identify the tangentand the normal direction to the prescribed trajectory, are represented as well. The instant centre C2S ,

which is the instant centre of the coupler's absolute motion, lies along a line that is normal to thetrajectory of point M. According to the AK theorem, C 2S must be collinear with points C I2 and CIS.

Therefore, it is located at the intersection between two lines: the first line passes through point M and hasthe same direction of the unit vector D, the second line is defined by points C 12 and CIS, as shown in figure2.

,t'

Figure 2 - Location of the primary and secondary instant centres of rotation

By applying the AK theorem, the instant centres C3S and C13 can be located simultaneously at theintersection between two pairs of straight lines, respectively the lines defined by points C 2S - C 23 and C 4S ­

C 34 for the first centre, by points CIS - C 3S and C 12 - C 23 for the second centre. Finally, C I4, which is theinstant centre of the relative motion between the two cranks of the linkage, is located by intersecting thelines crossing points C 34 - C 13 and C 4S - CIS. As point M moves along the prescribed trajectory, theconfiguration of the linkage changes as well as the location of the instant centres. The curves 0"1 and 0"2

traced by the instant centre CI4 on links I and 4, while the mechanism moves, represent the centrodes ofthe relative motion between the two cranks. Therefore, by definition, 0"1 and 0"2 are the requested profilesof the non-circular pitch curves. It is to be noticed that a necessary condition for the GFBM to be feasibleis that C I4 is located between CIS and C 4S , so that undercutting does not occur between O"j and 0"2.

Therefore, during the synthesis process an inspection is needed and not in any case a GFBM can besynthesised.

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4, 2008 491

4. SYNTHESIS OF THE GEARS' PITCH CURVES: CONNECTION PROFILES

In order to guarantee a regular working of the GFBM, a continuous and cyclic motion transmissionfrom the driving gear to the driven gear must be achieved. For this purpose, the results of the inversekinematic analysis described in section 2 are used to derive the requested input-output (I/O) relationship,that is the function that the gears are demanded to perform during the requested motion branches. Such arelationship defines the driven gear's rotation (output) as a function of the driving gear's rotation (input),that is

(10)

where¢J;1l =8t -81,0 and ¢Joul = 84 -84,0 are the angular posItIOns of the input and output gears

respectively, while 81,0 and 84,0 are the orientations of links 1 and 4 in the initial configuration of thelinkage.

In figure 3 two consecutive segments (solid line) of the I/O function are schematically represented. Inthe same figure, points PI and P2 represent the last instant of the first requested motion branch and theinitial instant of the second requested motion branch respectively.

t/Jouf

¢ P2.. auf,2f----------..

I/

t/J auf) f--__----::;P"""1

¢ in, 1 t/Jin,2

4>.,m"----------:-.1....-----,.-'------------7

Figure 3 - Relationship between the input and the output gear during requested (solid line) andconnection (dashed line) motion branches

To guarantee a continuous and cyclic motion transmission from the driving crank to the driven crank,the two requested profiles must be connected to each other by a connection profile, which isschematically represented in figure 3 with a dashed line. In order to define the I/O relationship in theconnection phase, guaranteeing for at least a C2 continuity of the gear I/O relationship during the phase ofthe mechanism motion between points PI and P2, the following fifth-order polynomial function is used:

(11)

where ¢J;1l I ~ ¢Jill ~ ¢Jill 2 and the over-bar denotes the function f in the connection phase., ,

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4,2008 492

The unknown coefficients Cj (j=O, .. ,5) in Eq. (11) are determined by imposing the following boundaryconditions:

l(¢in,i ) = ¢oul,i

1.- I (d. ) d¢oUIoUI'fJin,i =~

'fJm Pi

(12)

(13)

(14)

where i = 1,2 correspond to the final and initial phase of one requested motion branch and itsconsecutive one respectively, while the second member in the above equations is evaluated by means ofan instantaneous velocity and acceleration analysis of the mechanism at the extremes of the tworequested motion branches [10].

Eqs. (11) to (14) guarantee a C2 continuity of the gear I/O relationship during the phase of themechanism motion between points PI and Pl.

Once such a relationship is defined, the gears' connection profiles can be derived as follows. Theinstantaneous value of the gear ratio is evaluated as

-Cd. ) - dioul _ 2· d. 3 . d. 2 4 . d. 3 5 . d. 4 . h d. < d. < d.r 'fJin - dd.. - C1 + C2 'fJin + C3 'fJin + C4 'fJin + Cs 'fJin ' WIt 'fJin,1 - 'fJin - 'fJin,2If/m

(15)

where, again, the over-bar denotes the gear ratio function in the connection phase.From the condition of pure rolling between the two pitch curves, the gear ratio function can be also

expressed as:

(16)

where R;n and R oul are the variable radii of the driving (input) and driven (output) pitch curvesrespectively, and must obey the following geometric condition:

(17)

where /), is the distance between the axes of rotation of the two gears, that is the distance between thefixed hinges of the five-bar linkage.

Once the instantaneous gear ratio function is formulated according to the expression of Eq. (15), thelinear system formed by Eqs. (16) and (17) can be solved analytically, enabling, this way, the synthesis ofthe pitch curves during the connection motion branch.

5. EXAMPLE

In order to illustrate the synthesis methodology described in the previous sections, a task of pathgeneration is to be defined. In practical engineering applications, a typical task is to guide a coupier pointalong a straight-line. In this paper, a prescribed path, consisting of three straight-line segments, is

Transactions ofthe CSME Ide La SCGM VoL. 32, No. 3-4, 2008 493

generated as follows. At first, by using the atlas of coupler curves of GFBM proposed Zhang (1984), afive-bar linkage is selected so that the coupler point is moved along a triangle-like trajectory. Figure 4-ashows the selected linkage, whose geometric parameters are listed in Table 1, as well as the path that thecoupler point describes if the two cranks are operated by a pair of circular gears. From the generated paththree segments of straight-line are derived, as shown in figure 4-b.

Paramo XA I YA I L I I L2 I L3 I L4 I Ls I L6 fJl in I fJ4 in I .B I 5Units Dimensionless DegreesValue 0.0 I 0.0 I 0.8 I 4.2 I 4.2 I 0.8 I 3.0 I 4.2 o I 160 I 0 I 0

Table 1. Design parameters for the linkage represented in figure 4-a

1.2 1.4 1.6 1.8X

0.8

4.8 ·:7f··.. · ..r··· ..r··:;1~·· ..;<·..[· ··4.6 : >.4( j :...\[ ..

4.4 ( ( .. · ..t ·( ·(..·t; .;: : : : : :

4.2 ...../.~ .......:....... , ....... ;....... :...... ~ .......

(~ i : i i !

~ 3: :\:ttt·••••TLCI•••.. rt •••. , , , .

I , I • • ,

:: ••••·.l··••••l······j·····l~·l/···l ••••••••1 '----'-_'----'------''---'------l_-'--'-OS O.S 1.S 2.S 3.S

x

S .•••••. ,..••..•. ;.... .. ..

, , , . ,4 ...... ,..... ..·~·...... ' ....•..l....·..!

> :"',:1: '::, !,:,'

1 ....... ~........ ; .... .. .~ ,.

0.... ..: -r.......!....... j"....

a) b)

Figure 4 - Five-bar linkage selected for path generation (a) and requested trajectory of the coupler point(b)

Transactions ofthe CSME /de la SCGM Vol. 32, No. 3-4,2008 494

·90

·270

UJQ)

~

~ ·180~

4 --I---------T.-.-..

3 -....................•........

> 2 ·-------·i·--······+······· ··r--------!-----······:··········

: ···~············1······· ·i·~···,·········;·········"·)·····.1------~····1··········~-·········[·-···----+--·-··/· .

:' :'-2 '-----'----'-------'------:'---'---~--'

-2 -1

Figure 5 - Requested profiles of the non-circular pitch curves

By using the methodology described in sections 2 and 3, the requested profiles, shown in figure 5 aresynthesized. In order for these profiles to be connected to each other, the gears' I/O relationship is derivedand made continuous through the interpolation procedure described in section 4. Figures 6-a and 6-b showthe entire I/O relationship and the non-circular pitch curves respectively. In both figures, solid and dashedlines refer to the requested and to the connection phases of the mechanism motion respectively.

Figure 7 shows the CAD assembly model of the combined mechanism, used for a kinematic simulation,where the driving and the driven crank of the linkage are rigidly connected to the input and to the outputnon-circular gear respectively. The coupler point, whose trajectory is represented in the same figure,precisely follows the requested path.

o~."" "

'--

"'~''',"

"

""'~'"·360 '-------'------'-----'-----""~,

o 90 180 270 360

<l>in (degrees)

a)

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4, 2008 495

2,...----,..------.------r-----r----,---,----..,, , I • I •, , I , , ,, , , . , ,

1.5 :::::::::t~~~~<:ts:~::::::::l?=r:~~~:~~~~t::::::::::,J" : : : \. /': : : \/: : : ~ / : : : \

0.5 .- - i l • - - - - ~---- - --- - --- ~ - - --- - - - - --~- -- - t-/- --- ~ --- ------ --~- ---- -- ---- -~ ---i{ -- -- ---

>- 0

-0.5

-1

-1.5

-2 '----__-'--__-'-__--'-__--'-__--' .1...-_-----'

-2 -1 2 3 5

x

b)

Figure 6 - Gears' I/O relationship (a) and non-circular pitch curves (b). Solid and dashed lines refer to therequested and to the connection phases of the mechanism motion respectively

Figure 7 - CAD assembly model of the GFBM with non-circular gears

CONCLUSION

The main contribution of the paper is the presentation of a graphical-analytical technique for thesynthesis of non-circular gears in path-generating GFBMs. Such mechanisms represent an effectivesolution in problems ofprecise path generation.

A case study is presented where a GFBM with non-circular gears precisely guides a coupler point alonga trajectory consisting of three segments of straight-line, which form a triangle-like closed curve.

Transactions ofthe CSME Ide La SCGM Vol. 32, No. 3-4,2008 496

The authors believe that the idea illustrated in this paper will contribute to the diffusion of gearedlinkages, since the proposed graphical-analytical technique can be easily implemented in parametricComputer Aided Design environment.

REFERENCES

[1] Reuleaux F., 1963, The kinematics of machinery. Dover, New York, (reprint of 1876 edn).[2] Freudenstein F. and Primrose E. J. F., 1963, Geared five-bar motion. Trans. ASME, 85E (1. App.

Mech.),30:161-75.[3] Roth B. and Freudenstein F., 1963, Synthesis of path-generating mechanisms by numerical methods.

Trans. ASME, J.Engng. Ind., 30: 298-305.[4] 01eksa S. A. and Tesar D., 1963, Multiply separated position design of the geared five-bar function

generator. Trans. ASME (1. Engng. Ind.), 298-305[5] Starns G. and Flugrad D.R. Five-bar path generation synthesis by continuation methods, ASME

Journal of Mechanical Design, 115: 988-994, 1993.[6] Nokleby S.B. and Podhorodesky R.P., 2001, Optimization-based synthesis of Grashof geared five-bar

mechanisms, ASME Journal of Mechanical Design, 123: 529-534.[7] Zhang C., Norton R.L. and Hammonds T., 1984, Optimization of parameters for specified path

generation using an atlas of coupler curves of geared five-bar linkages, Mech. Mach. Theory, 19: 459­466.

[8] Mundo D., Gatti G. and Dooner D.B., 2008, Optimized five-bar linkages and non-circular gears forexact path generation. Mech. Mach. Theory, available on line at www.sciencedirect.com.doi: 10.1016/j.mechrnachtheory.2008.04.011

[9] Uicker U., Pennock, G.R., and Shigley J.E., 2003, Theory of Machines and Mechanisms - ThirdEdition, Oxford University Press, Inc.

[10] Zhou, H., and Cheung, E. H. M. Analysis and optimal synthesis of hybrid five-bar linkages,Mechatronics, 11 (3): 283-300,2001

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4, 2008 497

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4,2008 498