geared transmissions design

7/31/2019 Geared Transmissions Design

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TECHNICAL UNIVERSITY OF CLUJ-NAPOCAACTA TECHNICA NAPOCENSIS

International Conference on Engineering Graphics and Design

12-13 June 2009

GEARED TRANSMISSIONS DESIGN

Relly Victoria PETRESCU, Florian Ion PETRESCU

Abstract: In this paper one presents shortly an original method to obtain the efficiency of the geared

transmissions in function of the cover grade of gearing. With the presented relations one can make thedynamic synthesis of geared transmissions, having in view increasing the efficiency of gearingmechanism in works.

Key words: Efficiency, Force, Geared Transmission, Cover Grade, Dynamic, Tangential Velocity,Motor Velocity, Momentary Efficiency, Mechanical Efficiency, Wheel, Tooth, Four Pair in Contact.

1. INTRODUCTION

In this paper one makes a brief presentationof an original method to obtain the efficiency of

the geared transmissions in function of thecover grade. With the presented relations one

can make the dynamic synthesis of the geared

transmissions having in view increasing the

efficiency of gearing mechanisms in work.

2. DETERMINING THE GEARING

EFFICIENCY IN FUNCTION OF THE

COVER GRADE

One calculates the efficiency of a geared

transmission, having in view the deed that atone moment there are more couples of teeth in

contact, and not just one.The start model has got four pairs of teeth in

contact (4 couple) concomitantly. The firstcouple of teeth in contact has the contact point

i, defined by the ray ri1, and the pressure angle

i1; the forces which action in this point are: the

motor force Fmi, perpendicular in i on the

position vector ri1 and the transmitted forcefrom the wheel 1 at the wheel 2 by the point i,

Fi, parallel with the gearing line and with the

sense from the wheel 1 to the wheel 2, the

transmitted force being practically the

projection of the motor force at the gearingline; the defined velocities are similar with the

forces (having in view the original kinematics,or the precisely kinematics adopted); the same

parameters will be defined for the next threepoints of contact, j, k, l (see the picture number

one).For starting one writes the relations between

the velocities (see the system 1):

111

111

111

111

coscos

coscos

coscos

coscos

===

===

===

===

blllmll

bkkkmkk

bjjjmjj

biiimii

rrvv

rrvv

rrvv

rrvv

(1)

From the relations (1), one obtains theequality of the tangential velocities (2), and one

explicit the motor velocities (see the relations3):

11 ==== blkji rvvvv (2)

l

bml

k

bmk

j

bmj

i

bmi

rv

rv

rv

rv

cos;

cos

;cos

;cos

1111

1111

=

=

=

=

(3)

The transmitted forces concomitantly in the

four points must be the same (see the relation4):


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FFFFF lkji ==== (4)

The motor forces are (5):

l

ml

k

mk

j

mj

i

mi

FF

FF

FF

FF

cos;

cos

;cos

;cos

==

==

(5)

i

O1

O2

K1

K2

j

A

rb1

rb2

i

j

kl

ri1rj1

rl1

rk1

Fl, vl

Fml, vml Fi, vi

Fmi, vmi

Fig. 1 Four pairs of teeth in contact concomitantly.

The momentary efficiency can be written in

the form (6):

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

++++=

=

+++

=

+

+

+

=+++

+++

===

(6)

The relations (7) and (8) are auxiliary

relations:

1

1

111

111

1

1

111

111

1

1

111

111

1111

1111

23

23

);(

22

22

);(

2

2

);(

;

;;

ztgtg

zriKlK

tgtgriKlK

ztgtg

zriKkK

tgtgriKkK

ztgtg

zriKjK

tgtgriKjK

tgrlKtgrkK

tgrjKtgriK

il

b

ilb

ik

b

ikb

ij

b

ijb

lbkb

jbib

+=

=

=

+=

=

=

+=

=

=

==

==

(7)

11

1

23;

22

;2

ztgtg

ztg

tgztgtg

ili

kij

=

=

=

=(8)

One keeps the relations (8), with the signplus (+) for the gearing where the drive wheel

1, has external teeth (at the external or internalgearing), and with the sign (-) for the gearing

where the drive wheel 1, has internal teeth (the

drive wheel is a ring, only at the internalgearing).The relation of the momentary efficiency

(6), use the auxiliary relations (8) and takes theform (9):

It start in the expression (9) with the relation

(6) where one has four pairs in contactconcomitantly, but then one generalizes the

expression exchanging the 4 figure (four pairs)with E couples (exchanging figure 4 with the E

variable, which represents the whole of cover

grade +1), and after while one restricts the


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sums expressions, one exchanges the variable E

with the cover grade 12, as well.

)1(2

)12(

)1(3

216

)1(2

3

)12()1(215

2)1(4

6)12()1(

414

)1(2

2

)1(4

13

)3210(22)3

210(4

442

)2

3()2

2(

)2

(41

6

1

5

1

4

1

3

1

2

4

1

4

4

4

12

1

112

122

1

2

1

2

1

1

2

1

2

1

2

1

1

2

1

2

1

2

11

1

2

2

1

22

1

2

222

2

1

22

2

1

2

1

2

1

2

2222

++=

++=

++=

++=

++++

+++++=

++

+++=

======

=++++

=

=

=

z

tg

ztgA

z

Etg

z

EEtgA

EEzEtgEEE

zEtgA

izE

tg

izE

tgA

ztg

ztgA

ztg

ztg

ztgtgA

AAAAAA

tgtgtgtg

E

i

i

E

i

i

i

i

ii

ii

lkji

i

(9)

The mechanical efficiency is more

interesting than the momentary efficiency, andwill be calculated approximately, exchanging

the pressure angle 1, with the normal pressureangle 0, in the relation (9) which takes the

form (10); where 12 represents the cover gradeof the gearing, and it will be calculated with the

expression (11) for the external gearing, and

with relation (12) for the internal gearing as

well.

)1(2

)12(

)1(3

211

1

1

12

1

012

122

1

2

0

2

++=

=

z

tg

ztgB

Bm

(10)

0

021

0

20

22

2

0

10

22

1..

12

cos2

sin)(

cos2

44sin

cos2

44sin

+

+++

+

++=

zz

zz

zzea

(11)

0

0

0

0

22

0

0

22

..

12

cos2

sin)(

cos2

44sin

cos2

44sin

+

++=

ie

ii

eeia

zz

zz

zz

(12)

The calculations results one has beencentralized in the table 1.

3. CONCLUSION

The best efficiency can be obtained with the

internal gearing when the drive wheel 1 is thering; the minimum efficiency will be obtained

when the drive wheel 1 of the internal gearing

has external teeth. At the external gearing, thebetter efficiency can be obtained when the

greater wheel is the drive wheel; when one

decreases the normal angle 0, the covergrade and the efficiency increase. The

efficiency increase too when the number of

teeth of the drive wheel, 1, increase as well(when one increases z1), see the table number 1.


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4. REFERENCES

[1] Petrescu, R.V., Petrescu, F.I., Popescu, N.,Determining Gear Efficiency, In Gear

Solutions magazine, March 2007, USA, pp.19-28.

[2] Petrescu, F.I., Grecu, B., Comnescu, A.,

Petrescu, R.V., Sinteza dinamic laangrenajele cu roi dinate cu axe paralele.

In al treilea Seminar Naional de MecanismeCraiova 2008, SNM08, Proceedings, ISBN

978-973-746-910-6, 582 pag, Ed. SITECH,

Craiova, p. 319-324, 2008.

[3] Petrescu, F.I., Grecu, B., Comnescu, A.,Petrescu, R.V., Geared Transmissions

Dynamic Synthesis, In NEW TRENDS INMECHANISMS, Ed. Academica

Greifswald, 2008, ISBN 978-3-9402-37-10-1.

[4] Petrescu, F.I., Grecu, B., Comnescu, A.,

Petrescu, R.V., Determining the Efficiencyof Geared Transmissions, Proceedings of

33Rd

International Conference onAutomotive Engineering Off-Road

Vehicles, ORV 2008, Military TechnicalAcademy, Bucharest, Romania, 2008, ISBN

978-973-640-149-7.

Table 1Determining the efficiency of the geared transmissions

z1 0

[grad]

z2 12ae

12ae

21ae

12ai

12ai

21ai

42 20 126 1.799463 0.844769 0.871197 1.920902 0.83866 0.89538

46 19 138 1.875212 0.856744 0.882524 2.004616 0.850911 0.905915

52 18 156 1.964633 0.869323 0.893686 2.099177 0.864094 0.915635

58 17 174 2.062449 0.880938 0.904223 2.205675 0.87608 0.925023

65 16 195 2.173287 0.892173 0.914292 2.326635 0.887672 0.933877

74 15 222 2.301654

0.903305 0.9239632.465121

0.899241 0.94210385 14 255 2.449656 0.914052 0.933147 2.624685 0.910419 0.949774

98 13 294 2.620345 0.924221 0.941781 2.810001 0.920962 0.956932

115 12 345 2.822316 0.934095 0.949936 3.027795 0.931246 0.963487

137 11 411 3.062854 0.943503 0.957556 3.286764 0.941054 0.969476

165 10 495 3.351511 0.952271 0.964586 3.599006 0.950177 0.974938

204 9 510 3.687451 0.960703 0.970155 4.020382 0.958606 0.980652

257 8 514 4.097102 0.968417 0.975013 4.577099 0.966232 0.98587

336 7 672 4.666475 0.975348 0.980651 5.214391 0.973645 0.989299

457 6 914 5.427328 0.981528 0.98562 6.067023 0.980251 0.992231

657 5 1314 6.495028 0.986917 0.989899 7.264006 0.986011 0.994672

DESIGNUL TRANSMISIILOR (ANGRENAJELOR) CU ROI DINATE

Rezumat:Lucrarea prezint pe scurt o metod original pentru obinerea eficienei (randamentului) angrenajelordinate n funcie de gradul de acoperire (al angrenajului respectiv). Cu relaiile prezentate se poate face sinteza

dinamic a unui angrenaj cu roi dinate, avnd n vedere creterea eficienei (randamentului) angrenajului nfuncionare.

Petrescu Relly Victoria, PhD.Eng., Lecturer at Polytechnic University of Bucharest, GDGI Department (Department

of Descriptive Geometry and Engineering Graphics), [email protected], 0214029136;Petrescu Florian Ion, PhD. Eng. Assistant Professor at Polytechnic University of Bucharest, TMR Department(Theory of Mechanisms and Robots Department), [email protected], 0214029632.

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