DESIGN ET ANALYSE D’UN MÉCANISME DE · PDF fileFig. 1.Geneva mechanism with...

DESIGN AND ANALYSIS OF GENEVA MECHANISM WITH CURVED SLOTS

Jung-Fa HsiehDepartment of Mechanical Engineering, Far East University, Tainan, Taiwan

E-mail: [email protected]

Received November 2013, Accepted March 2014No. 13-CSME-185, E.I.C. Accession 3643

ABSTRACTA simple yet comprehensive method is proposed for the design of a Geneva indexing mechanism with curvedslots. In the proposed approach, conjugate surface theory is used to derive an analytical description of theprofile of the curved slots with and without an offset feature. Analytical formulae are then presented for thepressure angle of the Geneva mechanism and the principal curvatures of the curved slots. The effectivenessof an appropriate offset angle in eliminating the singular points and double-points on the curved slot profileis then demonstrated. Finally, a Geneva mechanism is fabricated in order to demonstrate the feasibility ofthe proposed approach.

Keywords: Geneva mechanism; offset; principal curvature.

DESIGN ET ANALYSE D’UN MÉCANISME DE MOUVEMENT ÀCROIX DE MALTE À RAINURES COURBES

RÉSUMÉUne méthode simple mais complète est proposée pour le design d’un mécanisme de mouvement à Croix deMalte d’indexation avec rainures courbes Dans l’approche proposée, la théorie des surfaces conjuguées estutilisée pour établir une description analytique du profil des rainures courbes avec et sans caractéristiques dedéviation. On présente des formules analytiques de l’angle de pression du mécanisme à Croix de Malte et lescourbures principales des rainures courbes. L’efficacité d’un angle de déviation approprié pour éliminer lespoints et doubles points sur le profil de la rainure courbe a été ainsi démontrée. Finalement, un mécanismede mouvement à Croix de Malte a été fabriqué pour démontrer la faisabilité de l’approche proposée.

Mots-clés : mécanisme de mouvement à Croix de Malte; déviation; courbure principale.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 4, 2014 557

NOMENCLATURE

(xyz)0 Configuration of frame (xyz)0 built in Geneva wheel0S Curved slot surfacerS Roller surface0n Unit outward normal of slot surfaceλ Roller heightr Roller radiusR Distance between z2 and zr axes (driving crank length)R1 Geneva wheel radiusa1 Distance between z2 and z0 axesα Half indexing angleψ Pressure angleθ1 Rotation angle of Geneva wheelθ2 Rotation angle of driving crankw1 Angular velocity of Geneva wheelε1 Angular acceleration of Geneva wheel

1. INTRODUCTION

Indexing mechanisms are of great practical importance in many applications, including weaving looms,precision measurement instruments, automated packaging and printing machinery, and so forth. One ofthe most common forms of indexing mechanism is the Geneva mechanism. The design and machining ofa conventional Geneva mechanism is generally straightforward since its structure consists primarily of nomore than a driving crank and a wheel with straight slots. However, in such mechanisms, a significant impactload is produced at the initial and final stages of the indexing operation as the roller enters and exits the slot,respectively. Consequently, conventional Geneva mechanisms are confined to low-speed applications, or tothose in which noise and vibration are of no importance.

Various approaches have been proposed for improving the kinematic characteristics and dynamic perfor-mance of Geneva mechanisms. For example, Fenton [1] deployed two Geneva mechanisms in series suchthat the second wheel accelerated and decelerated smoothly regardless of the motion status of the first mech-anism. Cheng and Lin [2] reduced the shock in a Geneva mechanism by using non-linear springs as dampingelements. Several studies have attempted to reduce the impact load through a redesign of the slot shape onthe Geneva wheel. For example, Fenton et al. [3] modified the conventional straight slot to a curved slot withan offset feature at its entrance. Lee [4] used a parametric polynomial expression to design and analyze thecurved slots of a Geneva mechanism. Figliolini and Angeles [5] performed the kinematic synthesis of bothconjugate Geneva mechanisms with curved slots and their pure-rolling cam-equivalents using the pressureangle as a figure of merit of the transmission force. Lee and Jan [6] presented a systematic method basedon envelope theory to determine the geometry of Geneva mechanisms with curved slots. Hsieh [7] utilizeda homogeneous coordinate transformation method to derive an analytical expression for the curved slots ona Geneva wheel and to generate the corresponding NC machining code. However, the problem of singularand double-points on the curved slot profile was not addressed.

Despite the contributions of the studies above, the literature lacks a systematic investigation into thedesign and machining of Geneva mechanisms with curved slots. Accordingly, the present study constructsa mathematical model of a Geneva mechanism with curved slots based on the input-output relationship ofthe mechanism and the dimensions of its links and then combines this model with the ability matrix of a3-axis CNC machine tool (i.e., the position and orientation of the spindle with respect to the table frame)in order to generate the NC code required to machine the designed Geneva wheel. In designing the curvedslot profile, the offset angle is carefully specified so as to eliminate the singular and double-points on the

558 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 4, 2014

Fig. 1. Geneva mechanism with curved slots.

slot profile without affecting the overall kinematic characteristics of the mechanism. The feasibility of theproposed methodology is demonstrated by fabricating a Geneva indexing mechanism.

2. SURFACE GEOMETRY

Referring to Fig. 1(a), the angular displacement of the driving crank takes place in the clockwise directionabout center 01. When the roller on the driving crank engages with the slot on the Geneva wheel and rotatesthrough an angle 2β , the Geneva wheel rotates about center 00 in the counterclockwise direction through anangle 2α (referred to as the indexing angle). Figure 1(b) shows that when the roller, with center 0r, entersthe curved axis 00x0, which rotates with the Geneva wheel, coincides with axis 0001, where θ1 = 0. Thus,from triangle 0r0001, it follows that

R1

Sβ=

RSα

=a1

S(π−α−β ), (1)

where a1 is the distance between the wheel axis z0, and the crank axis z1, R is the length of the driving crank,R1 is the radius of the Geneva wheel, and S denotes SINE.

In order to specify the slot profile in terms of the designed parameters using conjugate theory, it is firstnecessary to label the coordinate frames sequentially, starting with the Geneva wheel (marked as “0” in


Table 1. Kinematic parameters of Geneva mechanism.link bi θi ai αi λi1 b1 −θ1 a1 0◦ 02 0 −θ2 0 180◦ 0

Fig. 2. Roller location with respect to driving crank.

Fig. 1) and ending with the driving crank (marked as “2” in Fig. 1). Once frames (xyz)i (i = 0,1,2) havebeen assigned to link i according to the Denavit–Hartenberg (D-H) notation [8], the kinematic parameterscan be tabulated as shown in Table 1.

The configuration of frame (xyz)2 with respect to frame (xyz)0 is given by

0A2 = Π2i=1

i−1Ai =

C(θ2)+θ1) −S(θ2 +θ1) 0 a1Cθ1−S(θ2 +θ1) −C(θ2 +θ1) 0 −a1Sθ1

0 0 −1 b10 0 0 1

. (2)

Note that in Eq. (2), θ1 = θ1(θ2) represents the input-output relation of the Geneva mechanism.The configuration of the roller frame (xyz)r with respect to frame (xyz)2 (see Fig. 2) is given by the matrix

2Ar =

1 0 0 R0 −1 0 00 0 −1 00 0 0 1

. (3)

Similarly, the configuration of the roller frame (xyz)r with respect to frame (xyz)0 is expressed as

0Ar =0A2

2Ar =

C(θ2 +θ1) S(θ2 +θ1) 0 RC(θ1 +θ2)+a1Cθ1−S(θ2 +θ1) C(θ2 +θ1) 0 −RS(θ1 +θ2)−a1Sθ1

0 0 1 b10 0 0 1

. (4)

The surface equation, rS, and unit outward normal, rn, of the cylindrical roller can be expressed withrespect to frame (xyx)r as follows:

rS = [rCθ rSθ −u 1]T (0≤ u≤ λ ,0≤ θ ≤ 2π), (5)


rn =∂ rS∂u× ∂ rS

∂θ

/∣∣∣∣∂ rS∂u× ∂ rS

∂θ

∣∣∣∣= [Cθ Sθ 0 0]T , (6)

where θ is the polar angle, u is the height parameter of the cylindrical roller, and C denotes COSINE. Oncethe input-output relation of the Geneva mechanism has been defined, the conjugate points and slot profilescan be determined as

0nT · d0 Sdt

= (0Arrn)T d(0Ar

rS)dt

= 0, (7)

where 0n and 0S are the unit outward normal and surface equation of the slot with respect to frame (xyz)0,respectively. Equation (7) implies that for continuous contact to be maintained between the slot and theroller, the relative sliding velocity must be orthogonal to the common normal at the contact point. Imposingthe condition given in Eq. (7) on Eqs. (5) and (6), the conjugate point is obtained as

ES(θ)+ FC(θ) = 0, (8)

where E and F are defined as E = (R+a1Cθ2)dθ1/dθ2 +R and F =−a1Sθ2dθ1/dθ2, respectively.From Eq. (8), the value of parameter θ at the conjugate point can be derived as

θ =− tan−1 (F/E). (9)

Note that Eq. (9) yields two values corresponding to the two points of simultaneous contact betweenthe roller and the slot surface. The conjugate profile of the slots on the Geneva wheel can be obtainedby substituting Eq. (9) into Eq. (5), and then transforming rS to frame (xyz)0 via the matrix manipulation0S =0 Ar

rS. That is,

0S = [0Sx0Sy

0Sz 1]T =

rC(θ −θ1−θ2)+RC(θ1 +θ2)+a1Cθ1rS(θ −θ1−θ2)−RS(θ1 +θ2)−a1Sθ1

b1−u1

. (10)

For a curved slot profile with no undercut, a double-point occurs near the roller entry and exit pointsduring each cycle [6]. At this double-point, the roller may lose contact with the slot wall as the drivingcrank rotates, and thus the wheel is no longer properly controlled. Fenton et al. [3] showed that the double-point problem can be eliminated by adding an offset feature to the curved slot profile. Importantly, the offsetalters the shape of the slot, but has no effect on the kinematic characteristics of the mechanism. Figure 3presents a schematic illustration of the offset concept. As the roller engages with the slot at new point A,the original angle between line 00A and the center line 000′1 increases from α to α +∆α , where ∆α denotesthe offset angle and 0′1 is the new crank rotation center. The roller then exits the curved slot at point B. Inthe modified slot design, the axis distance 000′1 (denoted as a1) and crank length (denoted as R) are relatedas follows:

R1

sβ=

RS(α +∆α)

=a1

S(π−α−β −∆α). (11)

Thus, the geometric profile of the slot with an offset feature can be obtained by replacing a1 with a1 and Rwith R in Eqs. (9) and (10).

3. ANALYSIS OF SLOT CURVATURE

When machining the profile of the slots on the Geneva wheel, undercutting may occur even if the cutterhas the same form as the roller. Therefore, the principal curvatures of the slot surfaces must be carefully


Fig. 3. Geneva wheel with offset curved slot.

designed in order to prevent the occurrence of singular points and to enable the size of the slot profile to beproperly determined.

In accordance with the basic principles of differential geometry [9], the principal curvatures of the slot sur-faces can be evaluated using themathematical model of the slot profile developed in Section 2, i.e., 0S(θ2,u).That is,

K1,K2 = H±√

H2−K, (12)

where K1 and K2 are the principal curvatures of the slot profile, and H and K are defined respectively as

K =LN−M2

EG−F2 and H =2FM−EN−GL

2(EG−F2), (13)

where

L =0 n · ∂2 0S

∂u2 , M =0 n · ∂ 2 0S∂u∂θ2

, N =0 n · ∂2 0S

∂θ 22, E =

∂ 0S∂u· ∂

0S∂u

, F =∂ 0S∂u· ∂

0S∂θ2

,

and

G =∂ 0S∂θ2· ∂

0S∂θ2

are derived in Eqs. (A1–A6) in the Appendix.From Eqs. (12) and (13), it can be seen that the principal curvatures of the slot profile are related to both

the motion program of the Geneva wheel and the dimensions of the Geneva mechanism.

4. KINEMATICS

The performance of Geneva mechanisms can be improved by choosing a suitable motion program whichincorporates both zero velocity and zero acceleration at the beginning and end points of each engagementperiod. In the present study, the motion program is defined by the following modified sine curve:

θ1(θ2) =

h[

π

4+π

θ2−θdτ− 1

4(4+π)S(

4πθ2−θd

τ

)], 0≤ θ2−θd ≤ τ

8 ,

h[

24+π

+ π

4+π

θ2−θdτ− 9

4(4+π)S(

4π

3θ2−θd

τ+ π

3

)], π

8 ≤ θ2−θd ≤ 7τ

8 ,

h[

44+π

+ π

4+π

θ2−θdτ− 1

4(4+π)S(

4πθ2−θd

τ

)], 7τ

8 ≤ θ2−θd ≤ τ,

(14)


Fig. 4. Input-output relation of modified sine motion curve.

Fig. 5. Velocity and acceleration characteristics of Geneva mechanism with curved slots.

where θ1 is the Geneva wheel displacement, θ2 is the driving crank position, and θd is the dwell period. Inaccordance with this modified sine curve, the wheel displacement starts from a zero position and rises to atotal height of h over the period of the driving crank rotation, τ .

For illustration purposes, consider the case of a 4-station Geneva mechanism, i.e., N = 4. Let the dwellperiod of the Geneva wheel be specified as θ2 = 0◦ to θ2 = 137.34◦. Furthermore, let the indexing angleof the Geneva wheel as the driving crank rotates from 137.34◦ to 222.66◦ be equal to 90◦. Substitutingh = 90◦,τ = 85.32◦ and θd = 137.34◦ into Eq. (14), the input-output motion curve is found to have the formshown in Fig. 4. The corresponding velocity and acceleration characteristics are shown in Fig. 5.

Figure 6 shows the geometry relationship between the Geneva wheel and the locking disc at the momentthe driving roller enters the slot. To prevent interference between the wheel and the disc, parameters φ andξ relating to the circular arc of the locking disc must be carefully designed. From an inspection of Fig. 6, φ

and ξ can be obtained as


Fig. 6. Geometry relationship between Geneva wheel and locking disc.

Fig. 7. Curved slots with no offset feature.

cosξ =R2 + a2

1−R21

2Ra1, (15)

cosφ =R2

1 + a21−R2

L

2R1a1. (16)

When the roller has just entered the slot and the locking disc starts to unlock (or the roller has just exitedthe slot and the locking disc starts to lock), it is necessary to satisfy Eq. (15). Meanwhile, to preventinterference, the angular rotation ξ of the driving crank as the roller engages with the slot must be greaterthan that of the Geneva wheel, i.e., (360/N)−2φ , where N is the number of slots.

5. IMPLEMENTATION

To validate the design methodology presented in Sections 2–4, a Geneva mechanism was constructed withdesign parameters of α = 45◦,R = 120 mm, R1 = 115 mm, b1 = 2 mm, λ = 10 mm, r = 8 mm, and N = 4.From Eq. (1), parameters a1 and β were obtained as a1 = 169.564 mm and β = 42.66◦, respectively. The


Fig. 8. Curved slots with offset feature.

input-output relation of the designed mechanism was specified as shown in Fig. 5. Figure 7 shows the slotprofile obtained from Eqs. (9) and (10). It can be seen that the profile contains both singular points anddouble-points. As a result, the Geneva mechanism is unsuitable for practical applications.

Figures 8(a–c) illustrate the profiles of the curved slots given offset angles of ∆α = 3◦,5◦ and 70, re-spectively. (Note that in each figure, labels A and B indicate the entry point and exit point of the drivingroller, respectively.) It is noted that the shape of the slot and the relative location of the entry and exitpoints are both dependent on the magnitude of the offset. Figure 8(a) shows that given an offset angle of∆α = 3◦, the slot profile still contains singular points and double-points. However, given offset angles of∆α = 5◦ or ∆α = 7◦, the singular points and double-points both disappear, as shown in Figs. 8(b) and (c),respectively.

For the Geneva mechanism shown in Fig. 8(b), parameters a1 and R have values of a1 = 169.522 mmand R = 130.002 mm, respectively. Figure 9 shows the principal curvatures of the designed slot. It is notedthat one of the principal curvatures (i.e., K1) is equal to zero since the slot has a 2D profile. The resultsindicate that the designed slot has a minimum radius of curvature of 28.0662 mm. Thus, it is inferred thata cylindrical end miller with a diameter of 8 mm is required to machine an undercut-free slot profile. (Note


Fig. 9. Variation of principal curvatures of designed slots.

Fig. 10. Photographs of Geneva mechanism with curved slots.

that readers interested in the problem of generating the NC equations required to produce the correspondingGeneva wheel using a 3-axis machine tool are referred to [7].)

Substituting R1 = 115 mm, R = 130.002 mm, RL = 80 mm, and a1 = 169.522 mm into Eqs. (15) and (16),parameters φ and ξ are found to have values of φ = 24.203◦ and ξ = 42.659◦, respectively. The circular-arcangle of the Geneva wheel cutout is then obtained as (360/N)-2φ = 41.594◦. From Fig. 5, it can be seenthat when the driving roller enters the slot and the crank rotates through 42.659◦, the Geneva wheel rotatesthrough 45◦. In other words, interference does not occur between the locking disc and the Geneva wheel asthe crank rotates.

The feasibility of the proposed design methodology was confirmed by machining a Geneva wheel mecha-nism with the parameter settings described above. Figure 10 presents photographs of the machining processand the finished Geneva wheel mechanism, respectively. An experimental test showed that the indexingmechanism functioned satisfactorily. Thus, the basic feasibility of the proposed design methodology wasconfirmed.


6. CONCLUSIONS

This paper has presented a systematic approach for the design of a Geneva indexing mechanism with curvedslots. A kinematic model of the indexing mechanism has been derived utilizing the homogeneous coordinatetransformation method and conjugate surface theory. In addition, analytical expressions have been derivedfor the slot profile with and without an offset feature, respectively. Finally, analytical formulations havebeen presented for the principal curvatures of the slots on the driven wheel. It has been shown analyticallythat the use of an appropriate offset angle eliminates the singular points and double-points on the slot profile,and therefore improves the control of the indexing wheel. The feasibility of the proposed methodology hasbeen confirmed by designing and fabricating a mock-up Geneva wheel mechanism.

APPENDIX

E = 1, (A1)

F = 0, (A2)

G = (R2 +2RrCθ + r2)

(dθ1

dθ2+1)2

+(2a1rC(θ −θ2)+2a1RCθ2)

(dθ1

dθ2+1)(

dθ1

dθ2

)+ a2

1

(dθ1

dθ2

),

(A3)L = 0, (A4)

M = 0, (A5)

N =

(dθ1

dθ2+1)2

+ RCθ

(dθ1

dθ2+1)2

+ RSθd2θ1

dθ 22+ a1C(θ −θ2)

(dθ1

dθ2

)2

+ a1S(θ −θ2)d2θ1

dθ 22. (A6)

ACKNOWLEDGMENT

The author gratefully acknowledges the financial support provided to this study by the National ScienceCouncil of Taiwan under Grant No. NSC100-2221-E-269-009.

REFERENCES

1. Fenton, E.A., “Geneva mechanisms connected in series”, ASME Journal of Engineering for Industry, Vol. 97,pp. 603–608, 1975.

2. Cheng, C.Y. and Lin, Y., “Improving dynamic performance of the Geneva mechanism using non-linear springelements”, Mechanism and Machine Theory, Vol. 30, pp. 119–129, 1995.

3. Fenton, R.G., Zhang, Y. and Xu, J., “Development of a new Geneva mechanism with improved kinematic charac-teristics”, ASME Journal of Mechanical Design, Vol. 113, pp. 40–45, 1991.

4. Lee, H.P., “Design of a Geneva mechanism with curved slots using parametric polynomials”, Mechanism andMachine Theory, Vol. 33, pp. 321–329, 1998.

5. Figliolini, G. and Angeles, J., “Synthesis of conjugate Geneva mechanisms with curved slots”, Mechanism andMachine Theory, Vol. 37, pp. 1043–1061, 2002.

6. Lee, J.J. and Jan, B.H., “Design of Geneva mechanisms with curved slots for non-undercutting manufacturing”,Mechanism and Machine Theory, Vol. 44, pp. 1192–1200, 2009.

7. Hsieh, J.F., “Application of homogenous transformation matrix to the design and machining of a Geneva mecha-nism with curved slots”, Proc. Instn. Mech. Engrs, Part C: J. Mechanical Engineering Science, Vol. 221, pp. 1435–1443, 2007.

8. Paul, R.P., Robot Manipulators-Mathematics, Programming and Control, MIT, Cambridge, MA, 1982.9. Kreyszing, E., Differential Geometry, Dover Publications, New York, 1991.


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