Irregular gears for cyclic speed variation

Mech. Mack. lTeeoryVol. 26. No. 2. pp. 171-183. 1991

0094-114X,91 $3.00+0.00

Printed in Great BnUun

Perlimon ~

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IRREGULAR GEARS FOR CYCLIC SPEED VARIATION'[" ADEL K. AL-SABEEH Mechanical Engineering Department. Kuwait University. P.O. Box 5969. 13060 Kuwait

(Receiced 28 April 1989; receired.for publication 28 March 1990)

Al~ract--Shock free motion of an irregular gear pair is generated using a specially designed pin and slot mechanism at the gear shifting points. The pin and slot takes over the motion at the running gear ratio, gradually shifting it and then releasing the motion at the other Bear ratio. The operation is partially like the pin and slot in the Geneva mechanism. The motion generated is a continuous unidirectional rotational motion with uniform ratio over most of the cycle. The design equations and constraints arc derived and applied to some examples.

INTRODUCTION With today's high-speed machinery the demand for a quick return class of mechanisms has been increasing. The speed of automatic machinery is becoming too high to permit an intermittent mechanism to come to a full stop. Furthermore, some mechanisms require a precise uniform advance speed with a quick return property such as fast photography, fast photocopying, automatic-feed machines.., etc. All of the above impose the need for mechanisms that possess the following characteristics: (a) (b) (c) (d) (e)

Cyclic speed variation. Well-balanced and suitable for high-speed operation. Unidirectional without dwells. Uniform speed over most of the cycle. Continuous, shock free and reliable operation.

The irregular gear pair poscsses most of the above requirements, Fig: I, however, because of the sudden change in speed (i.e. infinite acceleration) it involves shock and possible jamming and/or disengagement which makes it unreliable [i]. A better choice is the non-circular gear set Fig. 2 which can be designed to produce the required motion. However, it can readily be seen that the shifting angle may be as large as 90 ° which, relatively, leaves little for the uniform speed range, not to mention the manufacturing difficulties and the costs involved [2-6]. There are other mechanisms that produce such a motion such as eccentric gears, cams, four-bar linkages, Hooke joint, OIdham coupling . . . etc. Most have some limitations and/or disadvantages [6-9]. The mechanism under consideration is a modified irregular gear pair with a pin and slot shifting mechanism, Fig. 3. The motion produced, out of uniform angular velocity input, is a continuous unidirectional motion which confirms the above-stated characteristics.

Mechanism operation The operation of the mechanism may be described with the aid of Fig. 3. The figure shows the mechanism in its simplest combination, that is two basic wheels (i.e. two speeds per revolution). Any or both of the wheels can be of a higher order, say order n (n speeds per revolution), Fig. 4, as long as compatibility is maintained. Also, Fig. 3 shows the simplest pin and slot combination, that is the pin is located at the center of the leading tooth (on the pitch circle) and the effective slot is taken as a straight radial. tDuc to circumstances beyond the Publisher's control, this article appears in print without the author's corrections. MMT~ :--D

17[

172

A D E L K. A L - S A J ~ E H

..',L:

',.

",,

""

%

(b) 3- R, 3 I! nO

=m

I ! !

I I I I I I I

R I"

rr

I

Driver angle

01

2~

Speed ratio diagram Fig. I. (a) Irregular gear pair. (b) Speed ratio diagram.

It is worth mentioning that the location of the pin is not ideal, since it will produce high stresses on the leading tooth; furthermore, the resulting slot profile for this location, may jam the mechanism by back-rotation at the root of the slot due to the resultant sharp profile. However, for the theoretical analysis, the mechanism will be considered "as it is" and then will be modified to overcome those difficulties. Referring to Fig. 3(a), while the gears a and b are engaged with a ratio of Ri = b~/a,, the pin (e) on b, enters the slot on b2. The slot, from the edge of the slotted wheel up to point (d), is the involute of the running gear, to allow for full gear disengagement at (J').W h e n the pin is at (d), the motion is fully controlled by the pin and slot pair. The effectiveslot from (d) to (g) is straight and the radial which together with the pin will gradually shift the ratio to R2 = a2/bs. W h e n the pin is at (g), the pin and slot produces a ratio equal to that of the other gear pair to engage, namely b, and a:, Fig. 3(b). The other gear pair now takes over the motion and the remainder of the slot is cut so that the pin will not interfere with the motion, i.e.the involute of the meshing gears, Fig. 3(c). For two basic wheels, this operation is repeated similarly in shiftingback to R~. For higher order wheels the shirtings must maintain a repetitive continuous operation.

Irregular gears for cyclic speed variation

173

(b)

II

I

Shifting angle

3

I

II

I

0

=M

I

R

I

I

I I I I I

I I I

I 2/t Driver angle 01

Speed ratio diagram Fig. 2. (a) Non-circular gear set. (b) Speed ratio diagram. DESIGN E Q U A T I O N S Figure 5 shows a pair of basic wheels with the pitch radii a,. a2. b, and b, which are normalized with respect to the center distance O~ O2. The figure also shows the associated ratio curve as a function of the driving angle 0,. For some given motion requirement of R~ and R 2 it is desired to compute for the angles 0 . . 0,,, 0,~, 0 , . 02,, 022,023 and 024 in wheels I and 2. Also the normalized dimensions a,, b,, a, and b 2 on wheels I and 2 are to be determined so as to produce the desired ratios. From the gears ratios the following may readily be computed: f.D,

at

ca,

b~"

(I)

Also a:+b,=l.

(2)

From equations (I) and (2) the following may be obtained: I

b,=R,+l RI

(3)

a: = - Rl+l

(4)

0,, = R, 0~,,

(5)

Also from the given ratio

174

,aU3EL K. AL-SABEIEH

similarly for the other ratio R, I

(6)

al = R : + I R:

b:= R:+ I

(7)

0~3 = R,O:,.

(8)

To guarantee continuous cyclic motion for the basic wheel shown:

(a)

0~l + 01: + 01~ + 0~ = 2~r = 360".

(9)

O:l + 0:: + 0:3 + 0:4 = 2 ~ = 3 6 0 ' .

(10)

pin

b2

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i

/

I I

/

_

"-

v

/ /

\\:/ \I// "

/

, I

•

if:

II \\

pin

Dr|yen

/ bl

Driver

(b)

\\ \

'\\\ //I

/

Fig. 3---continuedopposite.

--

---

~

,

C

Irregular gears for cyclicspeed v~-iation {c)

~

175

.---~

//

// // //

f

\xx

/!

/

• \\~ ...7

//

//

\\

\\

Fig. 3. (a) Basic modified irregular Bear pair. before shift. (b) Basic modified irregular gear pair. during shih. (c) Basic modified irregular Bear pair. after shift.

Equations (5) and (8)-(10) contain eight unknowns namely 0 u. 01z, 0,.1,014,021, 02:, 0z~ and 024. The angles 01e, 014. 0z: and 0:4 are shifting angles and may be computed from the shifting requirements for a smooth shifting without shocks. TIlE StlIFTING MECtlANISM Consider the shifting mechanisms shown in Fig. 6. The smooth continuous shifting process requires the following conditions: (I) The pin and slot effectively takes over the motion at a position, say point c, at which the produced pin and slot ratio is exactly equal to the running gear ratio, namely, a,/ap. (2) The shifting mechanisms gradually shifts the ratio and leaves the motion at exactly the ratio of the gears to engage, namely b,/ap.

~.,,

"~_w~~ "~

/

Fig. 4. Second and fourth order compatible irregular Bear pair.

176

ADF.L K.. AL-S~a~'r'~!

(o}

Driver

Driven

I

a,

I

(b)

"

i

•

Shifting angle j

R I

~

l! ¢r

:

O d$

:

/

Rz I O I

I

012~

01 I

Driver

i

013

/

I

~ 014-~

angle 0t

Speed ratio diagram Fig. 5. (a) Schematic diagram for basic irregular gear pair. (b) Speed ratio diagram.

The second condition is already satisfied by the selected location of the pin. However; we need to determine the appropriate value of c, at which the resulting pin and slot ratio equals the running gear ratio a,/bp from which the associated values of 0p and 0, may be determined. For the crank glider mechanism, it is easily obtained that oJ..2p= c, ~, ap sin(~ - -

(II) n/2)'

as required by condition I, equation (! I) must be equal to a,/bp.

c, =--=a" R , apsin(p-~/2) bp

(12)

where R is the running ratio before shifting, and the wheel with the pin is considered the driver. At this particular position, p may be computed using the cosine law cos(p) = a ~ + c ~ - - I = - sin(]l - ~/2), 2ap c,

(13)

[rregularg~rs for cyclic ~

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variation

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177

s

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,,

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,,

~

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Fig. 6. Irregular gear mechanism equivalent to the sliding contact linkage during part of its cycle.

substitute equations (13) in (12) and solve for c,:

/!-~

c. = ~/(2/R) + ! "

(14)

Upon determining c. the angles 00 and 0. may be computed using the sine law: 0p = sin -'(c. s i n , ) .

(15)

0. = sin '(ap s i n , ) .

(16)

Now returning to the earlier discussion of the angles and applying the above equations to the first shift the following are obtained: =[

cl ,,

=

I-a~ '~(2/Ri)+ I'

~ I) cos_,(a, + c~~, 2a, c,

(17) (18)

'

012 = sin - '(c, sin,i ).

(19)

0:: = sin -I(a I sin,, ).

(20)

Similarly for the second shift; --a; (21)

,2=COS-t(a~+¢~-I)

7,,T~]

•

(22)

0 , = sin- '(a: sin,:).

(23)

0., = sin - '(cz sin,2).

(24)

Now we have eight equations, namely equations (5). (8). (9). (10). (19). (20). (23) and (24). with eight unknown angles which may be solved.

178

ADEY.K. AL-SABEEH

Example I

By way o f demonstration let us design an irregular gear with the ratios o f R, = I.S. R: = 0.6 for a center distance O~ O: = 100 ram. From equation equation equation equation

(6)--a~ = 62.5 mm, (4F--a: = 60.0 mm. (3)--b~ = 40.0 ram. ( 7 ) - - b , = 37.5 ram:

for the first shift equation equation equation equation

(17)--c~ = 51.1 mm, (18)---~')= 123.0 , (19)---0~, = 25.4 . (20]--07, = 31.6 ;

(a]

[b)

Fig. 7--continued oppo.site.

[r~gular gears for cyclicspeed variation

[79

(c)

O,

d

\

Driver

Driven

\

-,.. O.s

\

Fig. 7. (a) .~hcmatic diagram, just before shifting. (b) Schematic diagram for the sliding contact linkagcs, during shifting. (c) ~hcmalic diagram, just aftcr shiRing. It also shows thc modificd slot casc.

for thc second shift equation equation equation equation

(2l)----c2 52.9 mm, (22~--/~: = 122.6% (23)---0,4 = 30.3". (24)--0:4 = 26.5 °. =

Now solving equations (5) and (8)-(10) simultaneously for the remaining four angles, namely 0 . , 0 . . 0:, and 0:3. gives: 0,1 = 205.2 °, 0, 3 = 99.1 °,

0:j = 136.8°, 0u = 165.1°. The remainder of the slot which is defined by the involute of the engaged gears may be traced out graphically or may be computed using the following vector analysis.

The slot profile The slot profile is an important parameter in this mechanism and must be defined in algebraic form to be employed in the CNC milling process. Consider Fig. 7, when the slotted wheel is fixed with the radial slot along the x-axis and the pinned wheel (crank), rotating about it. The slot profile may be divided into three segments: (l) involute approaching slot; (ll) straight radial shifting slot and (III) involute exciting slot.

180

Am~. IC AL-S~o~e~

(I) Involute approaching slot: 0 > - 0 , . Referring to Fig. 7(a). for any given angle the following may be stated: = I0 + 0,I,

(25)

dp = 7( a,/bp) ,

(26)

x = cos 0 - ao cos( 10 t + 0p + ~ ) ,

(27)

Y = sin 0 +

(28)

av

sin( [ 01 + 0p + ~ ) .

(II) Straight radial shifting slot: - 0 , < 0 < 0. With reference to Fig. 7(b), for any given angle - 0 , < 0 < 0, the following may be obtained, from the cosine law: c = cosl01 +_ x/cos:101 - I + ap.

(29)

The two values o f c are real and represent the two possible geometries where we have the smaller value of c. namely: c = cosl01 - x/cosZl01 - I + ag.

(30)

From the sine law it follows that:

,(c sinl0 f~ d) = 0p - sin - \ ~ / .

(3 I)

x = cos 0 = ap cos(101 + 0p - ~ ) ,

(32)

y = sin 0 + a o sin(10 J + 00 - d)).

(33)

Then for this range:

Note that y throughout this range will be equal to zero for the given parameters. (111) lm,olute exciting slot: 0 > 0. Figure 7 (c) shows an arbitrary position where, 0 > 0 since the other gear set now engages and governs the motion, then:

dp = O(b,/ao),

(34)

and the profile for this range may be expressed as: X = COS 0 -- a o COS(O + ~ ) ,

(35)

y = sin 0 - a0 sin(0 + ~ ) .

(36)

The modified slot As stated earlier, the resulting slot would have a sharp point which introduces the possibility of back rotation. Furthermore, it was stated that the location o f the pin was not ideal due to the introduction o f extra stresses on the gear leading tooth. These two disadvantages may be dealt with simply by changing the crank length to be "a"" (actual crank) instead o f " a " (theoretical crank), only in the computation o f the x and y coordinates. Also the crank angle may be altered by a small amount ~, CW ( + ) or CCW ( - ) , again only in the x and y computation. The modified slot profile relations will be as follows.

(!) The modified approaching slot: 0 < - 0 , . x -- cos 0 - a~,cos(10 [ + 0p + ~ + ~),

(37)

y = sin0 +a~sin(101 + 0p+ ~ + ~).

(38)

(11) The modified shifting slot; -O, < 0 < O. x = cos 0 - a~ cos(101 + 0p - 0 + ~),

(39)

y = sin 0 + a~, sin([01 + 0p - ~ + ~).

(40)

Irregular gears for cyclic speed variation

181

(III) The modified exciting slot; 0 > O. x = cosO - a~ cos(O + ~ - ~ ) .

(41)

y = sinO - a~ sin(O + ~ - ,~).

(42)

Figure 8 shows some profiles for various value of a" and ~. In some design cases, the design requirements calls for specific R, for some given 0,,. The ratio R2 and the other design parameters are to be computed. This makes the closed form computation more complicated as compared to the case of the given R, and R:. To overcome this difficulty Fig. 9 is prepared as a design aid to determine R: for given R~ and 011 from which the above-presented computation may follow.

.o. . . . . . . . .

"..~ •

I . f.lr

. |.OQ6 '

//

t.o"

.|

a', t.o t.$

,'.

(

l[l'~. ,,,6..o

t

!

Fig. 8. Profiles for various values o f a" and ~.

ADEL K. ,4u.-SASEFJ.I

182

CAM-GEAR DESIGN CHART

3.2

2.8 2.6 2.4 2.2

R~

. . . .

2.0

\\

1.8 1.6 1.4 1.2 1.0 0.8

I

1

1

1

I

J

O 2

2

811 Fig. 9. Irregular gear design chart.

Example 2 Design an irregular gear set with the ratio o f Ri = 2.0 for 0,j = 180", the center distance Oi Ot = 100 ram. Referring to Fig. 9. 0,, = 180" would a p p r o x i m a t e l y be obtained when Example ! is repeated using R, = 2.0 and R 2 = 0 . 5 5 ( a s found from the figure) and the results are as follows: from equation equation equation equation

(6)--at = ( 4 ) - - a . -(3)---b t = (7)--h: --

64.5 66.7 33.3 35.5

mm,

mm. mm, ram.

For the first shift: from equation equation equation equation

(I 7)---c, = 54.0 ram, (18)--//, = 114.8 , (19)--0,, = 2 9 . 4 , (20)--0:: = 35.8".

For the second shift: from equation equation equation equation

(21)--c: -- 51.4 m m . (22)---//: = ! 15. I ~, (23)---0,4 = 3 7 . 1 , (24)--8:, = 27.8 ~-.

Irregular ~ars for cyclic speed variation

183

Solving equations (5) and (8)--(10) simultaneously yields: 011

=

180.0 °.

01j = 113-5 ~-. 0:, = 90.0 '~, 0,~ = 206.42. Note that 01, checks with the initial requirements. CONCLUSION In this paper it has been shown that an irregular gear pair may be modified to be a useful mechanical c o m p o n e n t that generates a cyclically varying speed ratio. T h e equations defining the wheels design parameters and the slot profile were developed and applied to some examples. The slot profile equations were in cartesian form so they can bc employed in the x and y C N C vertical milling machine. This work m a y be extended into analyzing the resulted m o t i o n and also developing the slot profile equations for an improved shifting motion. Acknowledgements--The author is grateful to the joint support of Kuwait University and The Kuwait Foundation for Advancement of Science (KFAS) through Grant No. EM-53: and to t h o r who assisted in this work.

REFERENCES I. N. P. Chironis, MechanL~m,~.Linkages and Mechanical Controls. McGraw-Hill, New York (1965). 2. N. P. Chironis, Gear Design and Application. McGraw-Hill, New York (1967). 3. Holmes. Ockford and Peters, Int. Fed. The,rv of Much. and Mech. Int. Syrup., pp 269-278 (1974).

4. 5. 6. 7. 8. 9.

H. C. Town. Des. Engng No*., 82 0970). T. Benett, ASME Paper No. 66-Mech-8 (1966). S. Rappaport Product Engng Match, 68 0960). L. R. Benford, Much. Des. 40(23), 151 -154 (1968). K. Mitome and K. Ishida, Bull, JSME I~;, 82 (1982). J. H. Bickford, Mechanism for Intermittent Motwn. Industrial Press, New York (1972).

UNREGELM~BIGE ZAHNR,~,DER FCR ZYKLISCHEN UBERSETZUNGSWECHSEL Zmammeafmmmg--Eswerden 2 Zahnr~iderpaare mitjc 2 verschicdenen Zahnr~idern lest aufjc ciner Wellc verwendet, um w~ihrcnd einer Umdrchung der Antricbswclle abwcchsclnd 2 verschicdcnc 0bcrsctzungcn zu errcichcn. Dic ZahnrJder trugcn nur abschninswcise Vcrzahnungcn. Dcr 0bcrsang yon eincr zur andc~c~ 0bcrsctzung crfolgt wie bcim Sternradsctricbc dutch cinen Treibcrbolzcn an cincm und ciner entsprechcnden Kurve am andcren Zahnrad des jcweiligcn Paares. Des Kurvenprofil, vorzugswciscals Nut aussebildet, wird so berechnet, dab ein ruckfreier Ubergang zwischen beiden Ubersctzunsen auftritt. Die Nut ist verzweigt, da die Auslaufkurve des Trciberbolzens in folge des Zahneingriffs eine Zykloide ist. Dutch Variation freier Parameter kann das gfinstisste Kurvenprofil ermittelt werden. Mit diesen Getrieben k6nnen elliptische oder andere unrunde, schwierig zu festigende Zahnfiiderpaare ersetzt werden.