geneva mech

8/12/2019 Geneva Mech

1/5

to minimize contact stress and torsional vibration

Fig. I-Four-station Geneva mechanismin pos it ion of maximum acceleration

,I

T t ~ T b - - 21__ L

echanismseneva

1 0 1

How to design

By Ray C Johnson Senior De.lgn EngineerEastman Kodak Co Rochester N Y

I NCREASED demands fo r high-speed Genevawheel operation have focused attention on twoundesirable characteristics inherent in thismechanism. These characteristics, found to somedegree in many intermittent-motion mechanismsare 1) high contact stress between the dr ive p inand the wheel slot and. 2 vibratory motion of thedriven inertia load. Both of t hese facto rs ad versely affect performance and life.Chief effect of high contact stress is to causeex.cessive drive-pin wear with premature failureot the Geneva drive. On the other hand vibratoryInotion often interferes with the operation of thetnechanism or causes excessive wear on GenevaParts such as the locking device.This article outlines a simple method fo r minitnizing drive-pin contact s tress, and gives a procedure for reducing undesirable vibratory motion.Minimizing Contact Stress

A four-station Geneva mechanism which interlUittently rotates an inertia load JL against aMarch 22 1956

frictional retarding moment M is sho wn in FigLichtwitz has analyzed the motion of Geneva mechanisms and shows that, for a given number ofstations, the angular motion of the Geneva wheelis a function of the angular position and velocityof the drive-pin shaft. Angular motion of theGeneva wheel is also shown to be independent ofit s diameter D T he me ch an is m i n Fig 1 is shownin it s position of maximum wheel acceleration andthe following discussion is concerned with theforces and stresses obtained in this ins tantaneousposition.Since both the distance from the drive-pin axisto the drive-shaft axis and the distance from thedrive- shaf t axi s to the Geneva-wheel axis are directly proportional to the wheel diameter } thetorque arm of the pin contact force with respectto the wheel axis is also directly proportional tothe diameter. This torque arm is d eno te d by kDin Fig 1 where k is a constan t of proportionality. Because wheel acceleration is maximumin the position shown the drive pi n exerts atorque F rna.1 kD on the Geneva wheel. Summing

.Otto Llehtwltz.- Meebantsms tor Intermi ttent Mot ion, } fAcnlNtDBSTGN Dee. 19ri1. Pages 134-148.

1 7


2/5

(4 )Again from proportional design considerations

pin length-and therefore wheel thickness t :depend upon the diameter d of the drive PinLonger pins are permissible with larger diameters'Because Equa;tion 4 has already shown thatpin diameter is proportional to the wheel diameterit follows that wheel thickness t can also besumed proport ional to wheel diameter. Denoting

must be an optimum wheel diameter that tn.i .mizes the pin-slot contact stress.In the derivation of optimum wheel dialllet

proportional design techniques are used in thtGeneva wheel layout. These techniques will reeQ enize the desirabil ity of having pin diametersl arge as practicable. As for other principal dilrteassions, pin diameter d can be assumed to be d i r e t ~ ~proportional to wheel diameter. Denoting the eo yostant of proportionality by k n

(3)

. . . (1 )

. . . 2

11 - I l lE 1

k ~ F 1 tdwhere

J a J d amQx M fkD

Since D is in the denominator of Equation 1, thepin force at the condition of maximum angularaccelaration obviously depends on the Genevawheel diameter. Also, D effectively exists in thenumerator, since i t is one of the fac tors thattermines the value of Geneva wheel intertia Jo

Drive pin contact stress is given by the wellknown Hertz equation as

torques about the wheel axis 0 and solving forFmalll

k , = 0.798 ~ (Table l G e n e v a Design Constants

Hasie Geneva Propertiesn amax/w 2 av/w 2

S u g g e s t e d Design Constants--=:k k k 3 k ks

A study of Equation 2 shows that minimumcontact stress is obtained when both the drivingpin diameter d and the Geneva wheel thickness tare as large as practical. However, the Genevawheel diameter D cannot be specified with certainty at this point. Equation 1, fo r example, showsthat small wheel diameters cause high pin forcesbecause D is in the denominator. Furthermore,large wheel diameters, because of high wheel i n e r ~ti a J also cause high pin forces. Obviously there

3 4 46'4 110 246 17 346 22 547 27 338 3 1 3 89 35 16

10 38 30

31.446.4092.2291.3500.92840.6998

. 0.55910.4648

1. 7321.0000.72650.57740.48160.41420.36400.3249

0.15460.23840.29300.33110.35920.38070.39770.4112

0.160.150.140.140.140.130.120.10

0.160.150.140.140.140.130.120.10

0.700.350.240.170.130.100.080.08

0.003530.005060.00516 SSS0.005730.005750.005840.00540

k - Ratio of distance between Geneva centerand d riv e p in to Geneva wheel diameterk 1 to 1C5 =Constantsl =: ; Length of Geneva wheel shaft, in.

M J=Retarding moment acting on Genevawheel, lb-in.n Number of slots in Geneva wheelR = Locking radius on Geneva wheel, in.

Sma:\: =: ; Contact compressive stress betweeneva drive pin and wheel slot at maximumangUlar acceleration of Geneva wheel, ps i

T =: ; Torque, lb-in.t =: ; Thickness o f Geneva wheel, in.

W =: ; Energy available fOl free torsionalbrations when Geneva drive pin leaveswheel slot, in.-Ib

W =: ; Specific weight of Geneva wheel material,lb pel ell in .

Angle shown in Fig. 1 for Geneva mechanism in position of maximum angularacceleration, deg.

P l =: ; Poisson s ratio for Geneva drive p in materialP2 = Poisson s r at io for Geneva wheel slot ma

terialw= Angular velocity of Geneva drive shaft.

md per secI = Elas tic rota t ional deformation of Geneva

wheel shaft when drive pin leaves wheelslot, rad

amax --.: Maximum angular accelerat ion o feva wheel, rad per sec2all = Angular acceleration of Geneva wheel a t

instant when dr ive p in l eaves th e wheelslot, rad per sec2

b = Radius of Geneva wheel shaf t, in.c = Torsional stiffness of Geneva wheel shaft,lb-in. per radD =:; Diameter of Geneva wheel, in.d = Diameter of Geneva drive pin, in.E l =Modulus of e l s ~ i c i t y of Geneva drive p inmaterial, ps iE =: ; Modulus of elasticity of Geneva wheels lot mater ia l, ps iE =: ; Shearing modulus of e last ic ity of Geneva

wheel shaft material, ps iIn = Natural frequency of free torsional vi

brations for l oad ine rt ia , cpsF maZ' = Force between the Geneva drive pin and

wheel slot a t instant when th e angularacceleration o f Geneva wheel is amum, lbg =Acce ler at ion due t o gravity, 386 in. persec2

J 0 :::;: Mass moment of inertia of Geneva wheelabout axis o f rota tion, Ib-sec2-in.

J L =Mass momen t of inertia of load aboutaxis of rotation, Ib-sec2'in.

J =:; Polar moment of inertia of cross sectional area of Geneva wheel shaft, in .4

Nomenclature


3/5

. 7 )

~ w ~ )g +JL ama +M,F ma == k

GENEV ME H NISMS

When Equation 6 is substituted in Equation 1,

This relationship, together with Equations 4 and 5.are ~ u b s t i t u t e d in Equat ion 2, and the expression

where kG is a constant of proportionality and wis the specific weight of the material Incidentally,locking radius R Fig 1, is al so r ela ted to th ewheel diameter by th e constant of proportionality ,. Therefore

. 6

. . . (5 J

g

stant of proportionality by k tbiS con_ 3 Dt - proportional design assumption also proThe weans for relating the Geneva inertia Jade s a . . \l J th wheel diameter D From basIc mec ames,to moment of inertia of th e Geneva wheelthe al to its mass times it s r ad ius of gyration. equIS red The mass of the wheel is directly propursqua 1 to i ts specif ic weight and to it s volume.t i O ~ h e r m o r e under the propor tional design as.Fu ption, the volume is proportional to DB andSLlIllsquare of the radius of gyration is proportionaltheD2 Hence, from this line of reasoning t he mass: O J J l ~ n t of inertia of a proportionally designedGeneva wheel can be expressed as

kGwDG :::=

Example 1- Design for Minimum StressA fou r- st at ion Geneva mechani sm intermit

tently rotates a solid steel 4-in. diameter cylinder, 1 4-Jn. thick. A constant frictional momentof 2 lb-in. opposes rotation of th e cylinder anddrive-shaft speed is 600 rpm.

First steps ar e to calculate th e load inertiaand the Geneva wheel acceleration. The momentof inertia of the l oad i s

1 arc entered in Equations 4, 5, and 7. Resultsare t F 0.16(4.63) = 0.68-in., d : 0.15(4.53)= 0.68-in., and R =: 0.35(4.53) =: 1.59 in. Foroetorque arm kD =: 0.2384 (4.53) = 1.08 In.

From Equation 1,(:...0_ 0_0_70_4__+_0_ 0_0_46:...):...(:...2:...1.:...30_0:...)_+_2_F rl l lU = 1.08

R L 2 1rR,}w ( R22 J L ;;:: M L = =

2 4g. . (2)2 (0.283) (2J2= --4C-:-:( 8 6 )--:( 2 )= 0.0046 Ib-sec2-in.

=232 IbFor the steel drive pin and Geneva Wheel, E :;;:30,000,000 psi and . =: 0.3. Equation 3 thereforegives k 1 = 3240.Finally from Equation 2 contac t s tress for th eootimum Geneva design is

where M Land R L are the mass and radius ofthe load, respectively. From Table I, maximumacceleration of th e Geneva wheel is

[ 600 (2. .) J2ama:t == 5.409 1 2 =5.409 60== 21,300 rad per sec2

Contact stresses will now be determined fo rth e optimum Geneva wheel and fo r wheelssmaller and larger than optimum. In each case,t he design proportions of Table 1 ar e used.

Optimum \Vheel: From Equation 9, th e massmOment of inertia of the optimum GenevaWheel is

J0 :::: -. .-2 [0.0046 + 2 J21,300:;:: 0.00704 Ib-sec2-in.

For a four-stat ion Geneva wheel with th e valueof from Table I , Equation 6 gives

JG = 0.00606 D ( 0.283 ) =0.00704. 386Therefore, D = 4.53 in.

The value of D and the cons tants from Table

j 232Sm,. = 3240 (0.68) (0.68)

= 72,600 psi\Vheel Too Small: Here, th e Geneva wheel

diameter is arbitradly set at 2 in. Then: D :;;:2.00 in., d = 0.30-in., and t = 0.30-in. InertiaJ is again calcu la ted from Equation 6 usingk:l from Table 1. In this case, J0 = 0.000119Ibsec2in. From Equation 1 as before, Fma:r; =215 lb. From Equation 2 contact s tress is r l l a ~=: 158,200 psi for the design where the Genevawheel is too small. This con ta ct s tr es s is 218pe r cent of the stress of 72,600 psi calcu la ted forthe optimum wheel.This case shows the possible hazard of lettinga desire fo r minimum inertia inf luence designof high-speed mechanisms.

\Vlteel Too Large: Assume that the Genevawheel diameter is arbitrarily se t at 6 in. Whencar ri ed through as before, the analysis givesF rlla = 499 lb, and 8ma = 80,400 psi. This contact stress is 111 pe r cent of the optimum stressat 72,600 ps i. Moreover, pin force is deoidedly larger than th e optimum F mov whichin i tself is a disadvantage.

March 22. 1966 109


4/5

(12)

IU )

Polar moment of inertia J is9 =

MACHINE DESJGI l

. .bJ = ..2

= -J

Lichtwitz has shown in his analysis that thGeneva wheel is decelera ting at the instant thdrive pin leaves the wheel slot. Deceleration eis given in b ~ e Because the Geneva ishaft, Fig. 2), is decelerating the inert ia load it transmits a torque at t he instant the drive leaves the wheel slot. The magnitude of this Shafttorque neglecting frictional forces is given byT JLu l .. 10

When a shaft of length I transmits a reasonabl,torque it undergoes elastic rotational deformationand the angle of deformation is

Minimizing Vibration

wishes and then work ou t the design by sEquation 9. lHow these foregoing ideas may he apPlied .actual design i s demonstra ted hy Example 1.

Therefore, from Equation 10, the elastic rotationaldeformation at the instant the pin leaves theslot is

where b is the shaft radius.The shaft twist 8 Fig. 2) at the beginningof the rest period represents stored potentialenergy which is capable of initiating torsionalvibrations of the inertia load. These vibrationspersist during the rest period of the Geneva wheeland have an initial amplitude given by Equation 11For many applications vibrations of even smallmagnitude cannot be tolerated. Large vibrationson the other hand often cause exorbitant wear onparts such as the Geneva locking device. Equstions 10 11 and 12 reveal a means for reducingthe amplitude of such free torsional vibrations.For example from Equation 10 it is obvioUSthat shaft torque can be reduced by decreasingthe inertia of the load. Decreasing the load inertiaJL is also desirable from a p n ~ s t r s s standpoint.

From Eqllation 11, the length of the Genevawheel shaft I should also be as smali as possible,and the shaft should be a stiff material with largeshearing modulus EFinally, from Equation 12, the polar momentof inertia of the shaft J. should be as large aspractical, implying the use of a large_diametershaft.For a shaft made of a single material but consisting of various diameter cross sections thetorsional deflection equation corresponding to

8 )

(9 ) MJ h2 Bmll

8 , r = k , I k ;D + JL Gmu + MVkk k , l D

To minimize the contact stress a partial derivativeof 8 with respect to D is taken in Equation 8and equated to zero. This derivative expressionafter simplification by use of Equation 6, can berearranged to give

for maximum contact stress becomes

Fig 2-Geneva mechanism at in-slant drive pin s leaving slot

Equation 9 is th e crux of the analysi s to thispoint. shows that minimum pin contact stressis obtained when the mass moment of inertia ofthe Geneva wheel has a particular value that de-pends on 1 inertia of the load, 2 frictionalforces and 3 speed of operation. For mostpractical problems, the term M G m ~ in Equation 9will be negligible with respect to JL and can beIgnored,

Accurate layouts were made by the author forGeneva wheels that have three to ten stationsnsing a set of proportions that seemed to conformwell with good practice. The proportionality con-stants for these designs are given in Table 1.Use of these suggested constants wlll simplifythe design procedure. The calculated value forJ0 f rom Equat ion 9 can he placed in Equat ion 6and the desired wheel diameter D found at onceby extracting kr from Table 1. However a de-signer can choose different proportions. if he

110


5/5

From basic mechanics, th e energy of free torsionalvibrations is

Tbe torsional stiffness c of the shaft is found fromgquation 13 :

GENEVA MECHANISMS

Summary of Geneva Design

Application of these analytical methods is demonstrated in xample2

For a Geneva mechanism having a given numberof sta tions, the following design steps are recommended:1. Using Equations 10 through 16, select aGeneva wheel shaft that will ~ v satisfactoryvibratory characteristics. Equation 13 is the mostimportant equation in this group and should beused at the start of th e analysis for choosing anacceptable shaft size.

2. Calculate th e mass moment of inertia of theGeneva wheel shaft and, if appreciable, include itwith th e load inertia, JL3. Use Equation 9 to calculate the optimum moment of inertia fo r the Geneval wheel, Jo

4. Using Table 1 and Equat ion 6, calculate theoptimum diameter D of th e Geneva wheel. Detel -mine th e optimum Geneva wheel physical dimensions by using Equations 4, 5 and 7., The pin stress at th e condition of maximumangular acceleration may then be calcula ted byEquation 8.

. (16)

ation 11 isj lquhO ~ ~ + ~ + ~ + ...) (13), s: iT b b3

nsidering that th e Geneva wheel is clamped durthe rest period, the natural frequency for free,ng oDal vibrations of the load inertia JL istors2 I_c_ .. 1 4 )/ .. ::::: 2 iT ., JL

C6W = ....Equations 11 through 16 enable a designer to calculate the amplitude, frequency and energy of theload vibrations and demonstrate that, throughcareful selection of components, the magnitude ofvibrational motion can be held to an acceptablevalue.

Example 2 - Design for Acceptable VibrationDesign data and calculated values from x-

ample 1 are used in this example. Also, a constant-diameter shaft connects the Geneva wheelwith the load. A shaft design that gives exorbitant vibratory motion will be considered first.ExorbItant Vibratory l\lotlon: Here, th e Gen

eva-wheel shaft diameter 2b is chosen as -in.Such a small diameter might be selec ted withthe idea of keeping inertias low. The shaft issteel and is 5 in. long.From Equation 10, shaft torque is

[ 6006(02'lT) JL au =0.0046 [1.000]= 18.15 b-in.From Equation 12.

J == 1T2b = : : )

= 0.00194-in. 4Hence. ft.:om Equation 11,

1 8 . 1 ~6 .= 7.=; ; :=;:;;;;=: ;-0.00194 12,000.000)=O.o0390-rad

Therefore; a t th e periphery of the 4-in. diameterload cylinder, the amplitude of free vibrationswill be 26 = 2(0.0039) = 0.0078-in or 0.0156-in.total v ibra tory motion. This vibrat iona l mot ionis exorbitant.

AcCeI}table Vibratory Motion: Assume thatth e wheel shaft diameter 2b i s inc reased to 1in. and that the s haft length Z is reduced to 3in. Then, f rom Equat ion 12, J = O.0982-inch4Because the initial shaft torque is still 18.15Ib-in.. Equation 11 gives

1 8 . 1 ~ (3 )6 = ===-=-==:-:-,=0.0982 12.000.000)

= O.0000462-radTherefore, at the periphery of the load cylinderth e amplitude of free vibrations ha s been reduced to 2(6) = 2(0.0000462) O.0001-in., andthe total vibratory motion is now less than0.0002-in,Reduction of vibration, by a factor of 85 toI, ha s been achieved with comparatively li tt le

increase in inert ia .The mass moment of inertia of the I-in. by3 in . shaft is only 0.000216 Ib-sec2-in., which

is less than 5 per cent of the load inertia of0.0046 Ib-sec2-in.A comparison of the vibratory character is ticsis given below for the two designs. Frequenciesand energies shown were obtained d irectly fromEquations 14.15 and 16.

SbaU Doub1 VibraUouShaUDlam. Leo,rtb Ampllwde . E n e l p lu.) (10.) (10.) c p a ) In.-Ib)

0.0156 '0 9 _1 , 0.0002 14.70 0.00042March 22, 1956 111

geneva mech

Documents