Determination of the Mechanical Efficiency of the Gears-PETRESCU

7/31/2019 Determination of the Mechanical Efficiency of the Gears-PETRESCU

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1

Determination of the Mechanical Efficiency of

the Gears

F.I. Petrescu and R.V. Petrescu

1Polytechnic University of Bucharest, Romania, e-mail:

[email protected]

2Polytechnic University of Bucharest, Romania, e-mail:

[email protected]

Abstract. The paper presents an original method to determine the efficiency of the gear. The original-

ity of this method relies on the eliminated friction modulus. The paper is analyzing the influence of a

few parameters concerning gear efficiency. These parameters are: z1 - the number of teeth for the

primary wheel of gear; z2 - the number of teeth of the secondary wheel of gear; 0 - the normal pres-

sure angle on the divided circle; - the inclination angle. With the relations presented in this paper, one

can synthesize the gears mechanisms.

Key words: Gear, efficiency, tooth, pressure angle, wheel

1 Introduction

In this paper the authors present an original method to calculating the efficiency of

the gear.

The originality consists in the way of determination of the gears efficiency

because one hasnt used the friction forces of couple (this new way eliminates the

classical method). One eliminates the necessity of determining the friction coeffi-

cients by different experimental methods as well. The efficiency determined by

the new method is the same like the classical efficiency, namely the mechanical

efficiency of the gear.

Some mechanisms work by pulses and are transmitting the movement from an

element to another by pulses and not by friction. Gears work practically only bypulses. The component of slip or friction is practically the loss. Because of this the

mechanical efficacy becomes practically the mechanical efficiency of gear.

The paper is analyzing the influence of a few parameters concerning gear effi-

ciency. With the relations presented in this paper, one can synthesize the gears

mechanisms. Today, the gears are present every where in the mechanicals world.


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2 F.I. Petrescu and R.V. Petrescu

2 Determining the Momentary Mechanical Efficiency

The calculating relations are the next (1-20), (see the fig. 1):

+===

+===

12211112112

11

sincos

sincos

vvvvvvv

FFFFFFFmmm

(1)

With: - the motive force (the driving force);mF F - the transmitted force (the

useful force); F - the slide force (the lost force); - the velocity of element 1,

or the speed of wheel 1 (the driving wheel); - the velocity of element 2, or thespeed of wheel 2 (the driven wheel); - the relative speed of the wheel 1 in rela-

tion with the wheel 2 (this is a sliding speed).

1v

2v

12v

The consumed power (in this case the driving power) takes the form (2). The

useful power (the transmitted power from the profile 1 to the profile 2) will be

written in the relation (3). The lost power will be written in the form (4).

1vFPP mmc = (2) (3)12

12 cos == vFvFPP mu

(4)12

112 sin == vFvFP m

The momentary efficiency of couple will be calculated directly with the next

relation:

=

== 12

1

1

2

1 coscos

im

m

mc

ui

vF

vF

P

P

P

P(5)

The momentary losing coefficient will be written in the form (6):

=+=+=

== 1sincossin

sin1

2

1

2

1

2

1

1

2

1

ii

m

m

m

i

vF

vF

P

P(6)

One can easily see that the sum of the momentary efficiency and the momen-

tary losing coefficient is 1.


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Determination of the Mechanical Efficiency of the Gears - EUCOMES 2010 3

P

K1

n

n

t

t

1

1

rb1

rp1

Fm

F

F

v1

v12

v2

1

1

0

2 2002 Victoria PETRESCUThe Copyright-Law

Of March, 01, 1989

U.S. Copyright Office

Library of Congress

Washington, DC 20559-6000

202-707-3000

O1

Fig. 1 The forces of the gear

Now one can determine the geometrical elements of the gear. These elements

will be used in determining of the couple efficiency, .

3 The Geometrical Elements of the Gear

One can determine the next geometrical elements of the external gear (for the

right teeth, =0): The radius of the basic circle of wheel 1 (of the driving wheel)(7); the radius of the outside circle of wheel 1 (8); the maximum pressure angle of

the gear (9):

011 cos2

1= zmrb (7) )2(

2)2(

2

1111 +=+= z

mmzmra (8)

2

cos

)2(2

1

cos2

1

cos1

01

1

01

1

11 +

=

+

==

z

z

zm

zm

r

r

a

bM

(9)

And now one determines the same parameters for the wheel 2: the radius ofbasic circle (10) and the radius of the outside circle (11).

022 cos2

1= zmrb (10) )2(

222 += z

mra (11)

Now one can determine the minimum pressure angle of the external gear (12):


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)cos/(]44sinsin)[( 012022

20211 +++= zzzzztg m (12)

Now one can determine, for the external gear, the minimum (12) and the

maximum (9) pressure angle for the right teeth. For the external gear with bended

teeth (0) one uses the relations (13, 14 and 15):

cos

0tgtg t = (13)2

cos

cos

cos

cos1

1

1

+

=

z

z t

M (15)

t

ttm

z

zzzztg

cos

cos]4

cos4

cos

sin

cos

sin)[(

1

22

222211

+++= (14)

For the internal gear with bended teeth (0) one uses the relations (13 with16, 17-A or with 18, 19-B).

A. When the driving wheel 1, has external teeth:

t

ttm

z

zzzztg

cos

cos]4

cos4

cos

sin

cos

sin)[(

1

22

222211

++= (16)

2cos

cos

cos

cos1

1

1

+

=

z

z t

M (17)

B. When the driving wheel 1, have internal teeth:

t

ttM

z

zzzztg

cos

cos]4

cos4

cos

sin

cos

sin)[(

1

22

222211

+++= (18)

2cos

cos

cos

cos1

1

1

=

z

z t

m (19)


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4 Determination of the Efficiency

The efficiency of the gear will be calculated through the integration of momentary

efficiency on all sections of gearing movement, namely from the minimum pres-

sure angle to the maximum pressure angle; the relation (20).

5.0)(4

)2sin()2sin(]

2

)2sin()2sin([

2

1

])2sin(2

1[

2

1cos

11 2

+

=+

=

=+

=

=

=

mM

mMmM

a

i

M

m

M

m

M

m

dd

(20)

5 Determining of Gearing Efficiency in Function of the

Contact Ratio

One calculates the efficiency of a geared transmission, having in view the fact that

at one moment there are several couples of teeth in contact, and not just one.

The start model has got four pairs of teeth in contact (4 couples) concomi-

tantly. The first couple of teeth in contact has the contact point i, defined by the

ray ri1, and the pressure angle i1; the forces which act at this point are: the motorforce Fmi, perpendicular to the position vector ri1 at i and the force transmitted

from the wheel 1 to the wheel 2 through the point i, Fi, parallel to the path of ac-

tion and with the sense from the wheel 1 to the wheel 2, the transmitted force be-

ing practically the projection of the motor force on the path of action; the defined

velocities are similar to the forces (having in view the original kinematics, or the

precise kinematics adopted); the same parameters will be defined for the next

three points of contact, j, k, l (Fig. 2).

i

O1

O2

K1

K2

j

A

rb1

rb2

i

j

kl

ri1rj1

rl1

rk1

Fl, vl

Fml, vml Fi, vi

Fmi, vmi

Fig. 2 Four pairs of teeth in contact concomitantly


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For starting we write the relations between the velocities (21):

111

111

111

111

coscos

coscos

coscos

coscos

===

===

===

===

blllmll

bkkkmkk

bjjjmjj

biiimii

rrvv

rrvv

rrvv

rrvv

(21)

Fromrelations (21), one obtains the equality of the tangential velocities (22),

and makes explicit the motor velocities (23):

11 ==== blkji rvvvv (22)

l

b

ml

k

b

mk

j

b

mj

i

b

mi

rv

rv

rv

rv

cos;

cos;

cos;

cos

11111111 =

=

=

= (23)

The forcestransmitted concomitantly at the four points must be the same (24):

FFFFF lkji ==== (24)

The motor forces are (25):

l

ml

k

mk

j

mj

i

mi

FF

FF

FF

FF

cos;

cos;

cos;

cos==== (25)

The momentary efficiency can be written in the form (26).

lkji

lkji

l

b

k

b

j

b

i

b

b

mlmlmkmkmjmjmimi

llkkjjii

mc

ui

tgtgtgtg

rFrFrFrF

rF

vFvFvFvF

vFvFvFvF

P

P

P

P

2222

2222

2

11

2

11

2

11

2

11

11

4

4

cos

1

cos

1

cos

1

cos

1

4

coscoscoscos

4

++++=

+++=

=

+

+

+

=

=+++

+++===

(26)

Relations (27) and (28) are auxiliary:


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11

111111

11

111111

11

111111

11111111

23

23);(

22

22);(

22);(

;;;

ztgtg

zriKlKtgtgriKlK

ztgtg

zriKkKtgtgriKkK

ztgtg

zriKjKtgtgriKjK

tgrlKtgrkKtgrjKtgriK

ilbilb

ikbikb

ijbijb

lbkbjbib

+=

==

+=

==

+=

==

====

(27)

111

23;

22;

2

z

tgtg

z

tgtg

z

tgtg ilikij

=

=

= (28)

One keeps relations (28), with the sign plus (+) for the gearing where the drive

wheel 1 has external teeth (at the external or internal gearing) and with the sign (-)

for the gearing where the drive wheel 1, has internal teeth (the drive wheel is a

ring, only forthe internal gearing). The relation of the momentary efficiency (26)

uses the auxiliary relations (28) and takes the form (29).

)1(2

)12()1(3

21

1

)1(23

)12()1(21

1

2

)1(4

6

)12()1(41

1

)1(2

2)1(4

1

1

)3210(2

2)3210(4

44

4

)2

3()2

2()2

(4

4

4

4

12

1

112122

1

2

1

2

1

12

1

2

12

1

1

2

1

2

1

2

111

2

2

1

22

1

2222

2

1

22

2

1

2

1

2

1

2

2222

++=

=

++

=

=

++=

=

++

=

=++++++++

=

=++++

=

=++++

=

==

z

tg

ztg

zEtg

zEEtg

EE

zE

tgEEE

zEtg

izE

tgizE

tg

ztg

ztg

z

tg

z

tg

z

tgtg

tgtgtgtg

E

i

i

E

i

i

ii

iiii

lkji

i

(29)


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In expression (29) one starts with relation (26) where four pairs are in contact

concomitantly, but then one generalizes the expression, replacing the 4 figure

(four pairs) withE couples, replacing figure 4 with the E variable, which repre-

sents the wholenumber of the contact ratio +1, and after restrictingthe sums ex-

pressions, we replace the variable E with the contact ratio 12, as well.The mechanical efficiency offers more advantages than the momentary effi-

ciency, and will be calculated approximately,by replacing in relation (29) the

pressure angle 1, with the normal pressure angle 0 the relation taking the form

(30); where 12 represents the contact ratioof the gearing, and it will be calculatedwith expression (31) for the external gearing, and with relation (32) for the inter-

nal gearing.

)1(2

)12()1(3

21

1

12

1

012122

1

2

0

2

++=

z

tg

ztg

m (30)

0

02120

22

210

22

1..

12cos2

sin)(44sin44sin

++++++=

zzzzzzea (31)

0

00

22

0

22

..

12 cos2

sin)(44sin44sin

++++

=

eiiieeiazzzzzz

(32)

6 Conclusions

The best efficiency can be obtained with the internal gearing when the drive wheel

1 is the ring; the minimum efficiency will be obtained when the drive wheel 1 of

the internal gearing has external teeth. For the external gearing, the best efficiency

is obtained when the bigger wheel is the drive wheel; when one decreases the

normal angle0, the contact ratio increases and the efficiency increases as well.

References

1. Petrescu, R.V., Petrescu, F.I., Popescu, N.: Determining Gear Efficiency. Gear Solutions

magazine, 19-28, March (2007)

Determination of the Mechanical Efficiency of the Gears-PETRESCU

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