Reduction of delivery fluctuation and optimum tooth profile of spur gear rotary pumps

M S 331 Mech. Mach. Theory Vol. 23, No. 2, pp. 141-146, 1988

0094-114X/88

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OF DELIVERY FLUCTUATION AND TOOTH PROFILE OF SPUR GEAR ROTARY PUMPS

T H . C O S T O P O U L O S , A. K A N A R A C H O S a n d E. P A N T A Z I S National Technical University of Athens, Odos 28 Octovriou 42, Athens, Greece

(Received in revisedform 17 February 1987) Abstract--The optimum tooth profile of spur gear pumps is calculated in the present paper, and a method of reducing delivery fluctuation is also proposed. Reduction of delivery fluctuation may be achieved through a delivery penalty or through a pump size penalty. Also given is a way of calculating pump delivery through the gear's cross-sectional area which corresponds to a given rack profile.

I. INTRODUCTION Previous analyses (see Refs[l-4]) have examined several configurations of gear pumps without trappings. These analyses are generally of the form in which a given function for the rack profile or for the path of contact is used to evaluate the remaining characteristics of the pump. In the present paper, however, the circular pitch and the number of gear teeth are used to find the rack profile that gives a pump with maximum delivery. Calculations are presented for the path of contact and the gear tooth profile, through which the optimal gear area and therefore the maximal delivery of the pump is determined. During the search for the optimum rack profile, two conditions have been used[3]. First, rack and gear must be kept moving in the same direction; and second, intersection of the mating profiles must be avoided. Mathematical investigation shows that spur gear pumps with no delivery fluctuation cannot exist. Researchers[l-4] have proposed, however, that one way to eliminate the delivery fluctuation of a hydraulic pump would be to use helical tooth gears. But the present paper recommends another approach. Reduction of delivery fluctuation in spur gear pumps can be achieved through a delivery penalty, by reducing the tooth height while keeping the same circular pitch; or through a pump size penalty, by changing the circular pitch and tooth height without altering the delivery.

sponds to the point B (X~, Y~) of the path of contact and to the point C(Xg, Yg) of the gear tooth profile. It is obvious from Fig. 1 that point A of the rack profile, moved a distance K, and point B of the gear tooth profile, rotated through the angle {0}, will coincide at point C of the path of contact. Hence,

Ypc = Y

and

Xp~ = X + K.

(1)

The law of gearing states that the common normal to the mating profiles at the point of contact must pass through the pitch point P(0, 0). Thus, for {~} < 0 in the clockwise direction, ~ - , tan{~ } = - 1,

(2)

where tan{s} = YwlXw.

(3)

From equations (1)-(3) it follows that

K = - [ Y * d Y / d X + X].

(4)

When the rack moves a distance K, the gear will rotate through an angle {0 }, which means that

K = R* {0 }

(5)

and the corresponding point C(Xs, Yg) of the gear profile is given by the following relations:

X g = ( X + K ) , c o s { O } - - ( Y + Ro),sin{O}

(6)

Yg = (X + K) • sin{0 } + ( Y + Ro)* cos{ 0 } -- Ro, where the pinion pitch radius is given by

Ro = p * Z/2{n }.

(7)

2. PATH OF CONTACT AND GEAR TOOTH PROFILE 3. GEAR AREA

Figure 1 shows the rack profile Y = F(X) in the coordinate system (X, Y). The origin is at pitch point P(0, 0), the X-axis coincides with the rack pitch line, and the positive X-axis shows the direction of the rack's motion. In this case, the gear has a clockwise rotation. Point A(X, Y) of the rack profile corre-

From Fig. 1 we find that

n~(~) = x~ + (no + r~) ~ = ( Y ' d Y / d X ) 2 + (Ro + y)2,

0 =4~ +4~, 141

(8) (9)

TH. COSTOPOULOSet al.

142

Y Path of contact

Y =F'(X) track)

/ --,9

"

/

X

a

R(*~".qL "

0,

)

Fig. 1. Rack, gear and path of contact of mating profiles.

tan ~b~ = X~/(Ro + Y~)

4. OPTIMUM RACK PROFILE

= ( x + K)I(Y + Ro).

(10)

Using the above equations and differentiating equation (9) with respect to X, we find that

ddp/dX = - [(d Y/dX) 2 + Y. (d 2r/d2X) + I]/R o + [(Y + Ro)" y.d2y/d2X + Ro'(dY/dX) 2]

/[(Y + Ro) 2 + (Y.dY/dX)2].

(l 1)

The area of a section of the gear between two points on the gear tooth profile corresponding to angles {$}1 and {4}2 may be calculated by using E12 = ~

R2(~b)'d~b,

(12)

I

which, through equations (8) and (11), becomes Ej2

___l2Jx,

[(Y+Ro)'y'd2y/d2X

+ Ro.(dY/dX)21.dX

l ~ x2

2Ro Jx, [(Y'dr/dX)2 + (g + R°)21 x [(dY/dX) 2 + y.d2y/d2X + 11dX, (13) where Xl and X2 are the ordinates (in the X-axis) of the points on the rack profile corresponding to the points on the gear profile that determined angles {4~}i and {4}2, respectively. Furthermore, it is clear from Fig. 1 that when {q5} changes from 0 to 2{n } the area evaluated by equation (13) is the whole area of the gear, Eg, which may be used in calculating the pump delivery, as will be done later in this paper.

All previous analyses of hydraulic pumps start with the assumption of a known path of contact or a known rack profile[I-4]. At best, researchers assume a family of curves for the rack profile, from which, by various methods, they select and propose the one rack profile that, according to them, will give the best results: the highest delivery rate. The present paper does not begin with these restrictive assumptions. In this analysis, the only data used are, as mentioned before, the number of gear teeth and the circular pitch. Other restrictions on the final design are those that hold for all mating profiles: that is to say, the rack and gear must move in the same direction and the profiles must not intersect. The first of these restrictions may be expressed (refer to Fig. 1),

(dY/dX)2 + y.d2y/d2X + 1/>0,

(14)

whereas the second may be expressed (see Ref.[3]),

- R o <<.Y'[(dY/dX)2 + y'd2y/d2X + l] <~Ro.

(15)

The zero value of equation (14) leads to a rack profile of circles with radii equal to p/4 and centers located on the rack pitch line. These circles represent the upper limit of all possible rack profiles; this limit, however, cannot be reached because of the restriction described by equation (15). Furthermore, rack profiles of the form represented in Fig. 2 are prevented by the restriction imposed by equation (14). The following further restrictions ensure a symmetrical profile and continuous contact: Y=0

at

X=-n.p/4

Reduction of delivery fluctuation and optimum tooth profile

143

Y

Optimum

X 3

1

Fig. 2. Prohibited rack profile.

Fig. 4. Trial curves and the optimum rack profile.

and

dY/dX=O at X = - ( n - 1 ) . p / 4 , with

n=1,3,5,7 ......

(16)

The restrictions expressed in equations (15) and (16) will lead to a U-shaped rack profile fitting beneath the circles generated by equation (14) (as illustrated by profile 2 in Fig. 3) with the following characteristics: (1) continuous contact--no trapping; (2) no intersection; (3) optimum size and shape (see the relevant material which follows). The performance index of the profile is a resultant of the delivery per revolution of the gear and the unit width of the gear, and it is defined as

v = 2 ( { n } R ~ - Eg). As an alternative, the following ratio has been selected as the performace index to be maximized:

DR

=

({re} R~ - E,)/{n}

R~.

(17)

Because we are interested in a symmetrical shape, the search for the optimum profile is restricted to the interval of the X-axis between the points 0 and -p/4.

5. THE

METHOD

Study of known U-shaped curves (such as the sinusoidal) shows that as Y(X = 0) increases, the performance index, DR, increases. The optimum rack

-

profile must also give the maximum DR. Toward this end, our search should begin with a U-shaped curve having low values of Y. With this in mind, we have constructed the following algorithm. As illustrated in Fig. 4, the interval X(0, -p/4) has been divided into N equal spaces; and at all points of division, a value of Y is sought such that all points of the U-shaped curve will obey the constraints defined by equations (14) and (15). The upper curve, all points of which will also be regulated by those constraints, is the optimum rack profile. This is so because if the value of Y for any point on the curve were to increase, to the extent of the increment in Y, the new point would not obey the given restrictions. Furthermore, the calculated rack profile is smoothed by using proper smoothing techniques (see Fig. 5). Observation of the so constructed optimum shows that, for the interval of X(0, -p/4), its upper part resembles a circular arc whose center is at the pitch point. The rest of the profile for this interval is a curve of an unknown function. However, this optimum rack profile may be found analytically, as follows. For the interval of X under consideration, a Ushaped rack profile has a path of contact such as the curve (a) of Fig. 6. In such a path of contact there will always be an upper part with dY/dX > 0, for which, due to equation (14), the upper bound is a circular arc whose center is the pitch point and whose radius is Ymax;and a lower part with dY/dX < 0, for

2

x

I

I

I

I

I

[

to Fig. 3. U-shape curves for the rack profile.

Fig. 5. Optimum rack profile.

,,I

X

TH. COSTOPOULOSet al.

144

)

~

from which RcrlRo may be calculated for various numbers of gear teeth, Z. For example, if Z = 2, then Rer/Ro = 0.7654; or equivalently, R¢,/(p/4) = 0.9745, which gives DR = 0.5881. Since a circular rack profile with its center at the pitch point is difficult to construct, we propose a more easily constructed profile that, like the other, is a circular arc, but whose center is at the point (X = 0 , Y = - R e + Ymax) and whose radius is

(O, Ra)

(x¢,,Y~,)

R e _-_-

X

2 + (P/4)z]/eYmax, [Ymax

where (see Fig. 7) Ym,x= Y(X = 0).

P Fig. 6. Optimum path of contact.

which, due to equation (15), the lower bound is a circular arc whose center is at the point (0, Ro) and whose radius is the pitch radius, Ro. The above mentioned bounds for the path of contact determine the optimal path of contact for a design subject to constraints (14) and (15), which obviously gives the optimum rack profile for the given Y~x. Any other path of contact located inside the bounds will give a rack profile with a circular pitch larger than that of the optimum rack profile. The point of intersection for the above mentioned bounds, (X~, Y~), may be derived from the following relations: -2 X 2 r - ~ Y2 r - - R cr

and X2cr+(Ycr-Ro)2=R2

Table I. Characteristics of the pump with the proposed tooth profile

Ym~/P

(DR)~

2 3 4 5 6 7 8 9 10

0.22224 0.23139 0.23584 0.23897 0.24061 0.24183 0.24283 0.24370 0.24446

0.576935 0.500734 0.437218 0.386642 0.345621 0.312171 0.284500 0.261279 0.241533

Table 1 gives the ratio Ym~/P ratio and the pertinent values of (DR)rex for various values of Z (the n u m b e r of gear teeth) for the proposed circular tooth rack profile. At X = - p / 4 matching of the curves of the symmetrical profile is secured due to the way they are constructed. The construction of the rack profile, furthermore, is very simple.

o.

6. DELIVERY FLUCTUATION

From this one obtains Xcr = - R , , ( 1

Z

- R~:r/4R2o)'/2

and

The delivery rate of the pump (see Fig. 8) is given by

dV/d{O} = R~ - R:o -- [ ( Y * d Y / d X ) 2 + y2], Yc, = R~r)ZRo.

Following Ref. [4], in which the rack profile is calculated from a given path of contact, we find

where Y = Y(X) is the rack profile. For a spur gear pump with no delivery fluctuation, the following would need to hold true:

dZV/d:O = 0 K=K~+K2=f

[l+(Y~/Xr~) × (d Ypc/dX~)] dX~ = p/4,

where K~ =

;?

[1 + (Y~/X~)(dY~/dX~)] dX~ = 0

and

K2 =

(18)

or equivalently

y . d Y / d X . [ ( d Y / d X ) 2 + Y.d2Y/d2X + l] = 0,

/

[1 + (Y~/X~).(dY~/dX~)I dX~ =p/4. r

The last equation gives (1 - R:cr/4R2) ~'~:"Rcr/Ro = sin(n/2Z),

Fig. 7. Proposed circular tooth rack profile.

(19)

Reduction of delivery fluctuation and optimum tooth profile

145

LO Z=2 Z=4

sf/y /

0.8

,/.~//

Z=10 0.6

A

B

NK 0.4

® 0.2

0

Fig. 8. Rotary pump.

'

0;2

'

OJ4

'

0J6

DR/(DR)ma=

'

0;8

'

1.0

Fig. 10. Characteristics of fluctuation reduction. which means that at least one of the three factors would have to be precisely zero. Is this possible? Firstly, ( d Y / d X ) 2 + Y*d2Y/d2X + 1 cannot be zero because it would lead to a circular rack profile, which has no physical meaning, as indicated by equation (14). Secondly, either Y = 0 or d Y / d X = 0 leads to the trivial case of a line rack, which is of no interest. It is obvious then that spur gear pumps of nonfluctuating delivery cannot exist. According to equation (19) dV/d{O } = R ~ - R2o- r 2,

where (20)

r 2 = ( y . d Y / d X ) 2 + yz,

r being the distance from the pitch point to the point of contact of the mating profiles. It is clear from Fig. 1 that, for a single revolution of the gear, r becomes greatest at the points X = 0, - p / 2 and - p , whereas r becomes zero at the points X = - p / 4 , and - 3p/4. Therefore,

pump, having the optimum rack profile, will supply maximum delivery, V~=~; but it will also have maximum delivery fluctuation, r~=~. Any othor pump, with lower tooth height but with the same circular pitch, will give less delivery, VI, but will produce less delivery fluctuation, r~, too. We are seeking a new pump. It will be similar to the pump with the low degree of fluctuation: the similarity expressed by the ratio r/rl > 1; but it will also give us Vm~: the same delivery, that is, supplied by the pump with the optimum rack profile. This new pump will have a circular pitch larger than that of the optimum pump. Figure 10 has been constructed using the above conditions. It plots r2/r2m~ vs DR/(DR)m~ foe the new pump, where r ~ and (DR)m~ are given in Table 1. Typical gears are shown in Figs 11 and 12.

(dV/d{O})m~ = Re -- R2o

and 2 (dV/d{O})=. = R~ - R2o - rm~x.

The delivery rate for a pump with the proposed rack profile is presented qualitatively in Fig. 9. This

~dV/d8

2 R~2_RI-r~;=

L

90 °

180 °

270 °

360 °

-p/4 -p/2 -3p/4 -p

Fig. 9. Delivery fluctuation. MM.T. 23/2--E

0

X Fig. 11. Gear profile for Z = 4.

146

TH. CosToPouu~s et al. calculated r and D R for the new pump. Furthermore, D R = Vm~/2[n] (Ro, s ~ + r) 2, from which we can find the radius Ro, New for the pitch circle of the new pump. And so the new circular pitch is calculated by PN,w = 2{It} * Ro.N,w/Z. Thus, all the necessary quantities have been calculated for a new pump with the same delivery, Vm,x, but with a delivery fluctuation, r 2, half that of the original optimum pump. Finally, the tip radius of the new pump may be found as follows: Rk.New = (Ro + rm~,)/(0.71) m = 1.187(R o + rmax). 8. CONCLUSIONS

Fig. 12. Gear profile for Z = 10.

7. EXAMPLE

For a gear with Z = 2 , Table 1 gives Ym~ = rm~ = 0.22224"p, where p is the circular pitch such that Ro = p * Z / 2 { n } , which may be defined by the designer. The radius Re of the rack circular tooth profile is given by Re = [Ym~ 2 + (P/4)2]/2Ym~.

The performance index, 0.576935, may be read from Table ! or calculated by using (DR)n,

=

(V=,U2)l{n} (Ro + Ym,,) 2,

where V~,~ is the delivery to be transferred. We are looking for a pump with this same delivery, V=,~, but with a fluctuation of delivery r 2 such that ri/r2=~ = 0.5. From Fig. 10 we find that, for the curve Z = 2, DR/(DR)=~,=0.71. In this way, we have

Although for spur gear pumps non-fluctuating delivery is impossible, reduction of the fluctuation may be achieved by applying the optimality criterion given in equation (17) and using Fig. 10 in conjunction with the similarity condition. It should be noted that Fig. 10 holds for any given delivery. As illustrated in our example, reduction of the delivery fluctuation may be achieved by selecting a larger pump with a lower tooth height and a larger circular pitch.

REFERENCES

1. S. Togashi and H. Iyoi, The synthesis of tooth profile shapes and helical gears of high hydraulic performance for rotary type pumps. Mech. Mach. Theory 8 (1973). 2. H. Iyoi, M. Oka and T. Iyoi, Determination of an improved gear surface of helical gear for pumping action. Mech. Mach. Theory 12 (1977). 3. H. Iyoi and S. Ishimura, X-Theory in gear geometry. Trans. ASME J. Mech. Transmiss. Automat. Design 105 (1983). 4. K. Mitome and K. Seki, A new continuous contact-low noise gear pump. Trans. ASME J. Mech Transmiss. Automat. Design 105 (1983).

Zusamt~ffutug--In dem vorliegendem Aufsatz wird die optimale form der geraden Z~hnen von Zahnradpumpen ermittelt, und es wird auch eine Methode vorgeschlagen, mit der eine Reduktion der Schwankungen des F6rderstromes m6glich wird. Diese Reduktion kann entweder fiber eine Reduktion des F6rderstromes oder fiber eine Vergr6sserung der BaumaBe erfolgen. Schliesslichwird eine analytische Beziehung flit die Berechnung des F6rderstroms als Funktion der Zahnnform bzw. des erzeugenden Werkzeuges gegeben