Asymmetric Plastic Spur Gears

7/28/2019 Asymmetric Plastic Spur Gears

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By J.L. Moya, A.S. Machado, J.A.Velsquez, R. Goytisolo, A.E.Hernndez, J.E. Fernndez, andJ.M. Sierra

A Study inAsmmerc Plasc Spr Gears

This paper presents a

theoretical analysis of a

procedure to determine the

Lewis Factor, which can play

a major role in the fracture

of asymmetric plastic gear

teeth.


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The basic weakness of plastic spur gear teeth is

tooth fracture brought on by the accumulation of

stress at the root of the tooth and by the geometr y

of the tooth. Tooth width and height play a major

role in failure, as does the Lewis Factor, which has

a direct effect on the expression to calculate tooth

strength. This study describes a theoretical analy-

sis of a procedure to determine the Lewis Factor

for asymmetric teeth.

IntroductIonThe most common failure of well-lubricated metal-

lic gears is caused by pitting at the flank of the

tooth. Volumetric fatigue is of secondary impor-

tance. However, the most frequent failure in

plastic spur gears teeth is tooth fracture, when

stress originating at the root of the tooth and tooth

geometry play decisive roles [1]. The Lewis Factor

is decisive for calculating the bending strength of

gears, and most traditional texts in the literature

on gears employ graphs and tables to calculate it

[2-11]. However, since asymmetric plastic teeth

are still somewhat recent inventions, their Lewis

factor values are nowhere to be found in the

above-cited papers despite them being a sine qua

non to calculate bending strength.

This paper analyzes the procedure to calcu-

late the Lewis factor for asymmetric teeth and

describes the values of this parameter in terms

of the coefficient of asymmetry and teeth number

[12-13].

SymmetrIc GearSThere is a range of procedures for calculat-

ing the Lewis Factor (LF) for symmetric gears.

According to Black [14], the starting point to

calculate this factor is the constant strength

parabola in the tooth, as shown in fig. 1,

which establishes that the LF is determined

by the expression: ; t and h are

obtained from the figure and pis diametral pitch.

Faires [15] proposes the procedure illustrated in fig.

2. It can be deduced from the figure that:

and , where the expression, is

termed the LF (y) Other standards such as DIN

3990 [16] calculate the most critical section by

Load angle in asymmetric gear tooth.

Outside angle for asymmetric spur gear drive for

coast side profile.

Pressure angle for asymmetric spur gear drive for

driving side profile.

Pressure angle for asymmetric spur gear drive for

coast side profile.

Load angle in spur and helical gear teeth

Pressure angle for spur and helical gear teeth.

Angle at top of the tooth.

Critical radius of curvature at the

root of the asymmetric tooth.

Radius of curvature at the root of the tooth for driv-

ing involute profile.

Radius of curvature at the root of the tooth for coast

involute profile.

Inverse shift profile coefficient.

Face width

Coefficient of asymmetry.

Radial clearance.

Outside circle diameter.

Base circle diameter for asymmetric spur gear drive

for coast side profile.

Pitch diameter.

Eccentricity.

Geometric factors.

Tooth height coefficient.

Distance from critical section to intersection of the

tooth centerline and the line of action for load at tip

of tooth, in the asymmetric tooth.

Distance from critical section to intersection of the

tooth centerline and the line of action for load at tipof tooth and the curvature radius of the root trochoid

for spur and helical gear teeth.

Module.

Bending moment produced by the horizontal compo-

nent of transmitted force.

Normal resultant force in the tooth.

Tangential force.

Vertical force.

Top land thickness.

Critical tooth thickness in the asymmetric tooth.

Critical tooth thickness for driving side profile.

Critical tooth thickness for coast side profile.

Tooth thickness on pi tch diameter.

Tooth thickness at critical section for

spur and helical gear teeth.

Variable angle to determine Lewis factor.

Section module.

Shift profile coeff icient.

Lewis factor for asymmetric gear teeth.

Lewis factor for spur and helical gear teeth.

Lewis factor for asymmetric teeth from the regression

equations obtained by the Statgraphics software.

Numbers of teeth.

nomenclature


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extending a line at an angle of 30 from the tooth axis of sym-

metry, tangent to the tooth-root profile in points a-a.

aSymmetrIc GearSThe authors calculations of the LF were based on figs. 4, 5, and

6. The stress produced at the root of the tooth due to flexion can

be calculated using the following expression:

Multiplying and dividing by the module:

From the expression (2), is designated as:

then the expression (2) can be rewritten as follows:

(4

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is the LF for asymmetric gear teeth.

Observe the difference between the expression (3) and the

expression (5) for symmetric gear teeth [16]:

(5)

After considering the DIN 3990 standard [16], the algorithm

proposed by Gonzlez [17], and the characteristics of asym-

metric teeth, the authors propose the algorithm in fig. 5 to

calculate . The concept of the asymmetry coeff icient was

incorporated to cater for asymmetry. It is the relation between

the driving side profile and the coast side profile angles:

. Fig. 4 illustrates how the LF varies for different

numbers of teeth and asymmetry coefficients, and its differing

values are described in table 1.

Using Statgraphics software, the authors found the regres-

sion equations to obtain the values of the LF for asymmetric

teeth depending on the number of teeth and the asymmetry

coefficients. Three expressions were established consider

the range of number of teeth so that values were precisely c

culated:

For (with 98,9% reliability level and abso-

lute error of 0,0053) = -0,0315949 + 0,0269414Z+

0,405287C - 0,000462649Z2 - 0,103118C2

Fig. 1. Determination of the Lewis Factor, according to Bla



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For (with 99,13% reliability level and abso-

lute error of 0,00058) = -0,0543935 + 0,011263Z+

0,0685655C - 0,0000845454Z2 + 0,0404041C2

For (with 99,13% reliability level and absolute

error of 0,00058) = -0,0599137 + 0,00834091Z+

0,028519C - 0,0000413636Z2 + 0,0581197C2

Table 2 provides a comparison of LF values for asymmet

teeth calculated using the algorithm developed by the aut

Fig. 2. Determination of the Lewis Factor for symmetric

gears.

Fig. 3. Asymmetric tooth profiles.

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C Values o Lewis actor according to the number o teeth Z and the coefcients o asymmetry CZ=10 Z=12 Z=15 Z=20 Z=30 Z=40 Z=50 Z=60 Z=70 Z=80 Z=90 Z=100

1,00 0,201 0,245 0,289 0,320 0,358 0,389 0,408 0,421 0,429 0,429 0,442 0,446

1,05 0,228 0,267 0,308 0,351 0,392 0,396 0,415 0,435 0,437 0,438 0,450 0,454

1,10 0,232 0,272 0,314 0,354 0,399 0,403 0,422 0,442 0,444 0,445 0,458 0,462

1,15 0,242 0,277 0,320 0,358 0,406 0,410 0,430 0,450 0,452 0,453 0,466 0,470

1,20 0,237 0,282 0,326 0,371 0,413 0,417 0,438 0,458 0,460 0,461 0,474 0,478

1,25 0,247 0,288 0,332 0,378 0,421 0,424 0,446 0,466 0,468 0,469 0,482 0,487

1,30 0,252 0,294 0,339 0,385 0,429 0,432 0,454 0,475 0,477 0,478 0,490 0,496

1,35 0,257 0,300 0,345 0,392 0,437 0,440 0,463 0,483 0,486 0,487 0,498 0,505

1,40 0,263 0,306 0,352 0,400 0,445 0,448 0,472 0,492 0,495 0,496 0,510 0,515

1,45 0,268 0,312 0,359 0,408 0,454 0,457 0,481 0,502 0,504 0,505 0,520 0,525

1,50 0,274 0,319 0,366 0,416 0,463 0,466 0,491 0,512 0,514 0,515 0,530 0,535

Z=30 Z=40 Z=50 Z=90

C

=1,

00

C

=1,

35

C

=1,

50

C

=1,

00

C

=1,

35

C

=1,

50

C

=1,

00

C

=1,

35

C

=1,

50

C

=1,

00

C

=1,

35

C

=1,

50

0,358 0,437 0,463 0,389 0,446 0,471 0,408 0,463 0,491 0,442 0,498 0,530

0,378 0,435 0,451 0,370 0,427 0,454 0,406 0,463 0,491 0,445 0,503 0,532

Table 1: Values of the Lewis Factor for asymmetric teeth.

Table 2: Comparison of Lewis Factor values for asymmetric teeth calculated using the algorithm developed by author an

the regression equations from the Statgraphics software.

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and the regression equations from the

Statgraphics software.

concluSIonSNew expressions for calculating the Lewis

Factor are obtained. The principal failures

and the calculation methods are analyzed

and compared with the Finite Elements

Method [18]. The limitations of present cal-

culation expressions and new coefficients

and formulas for the particular case of asym-

metric gears are offered. All information was

obtained by employing the most advanced

methods of graphic design, geometric mod-

eling and simulation. The positive influence

of asymmetric profiles combined with tooth

profile modification on the bending strength

of teeth is demonstrated.

referenceS:1) K. Cavdar, F. Karpat, F.C. Babalik,

Computer aided analysis of bend-

ing strength of involute spur gears

with asymmetric profile Journal of

Mechanical Design 127 (3) (2005)

477-484.

2) W. Lewis, Investigation of strength of

gear teeth, Proceedings of Engineers

Club, Philadelphia, P.A. (1892) 16-23.

3) E. Buckingham, Manual of gear desi

Editorial Industrial Press Inc., N

York, 1971.

4) D.W. Dudley, Gear Handbook: Desig

Manufacture and Application of Gea

Fig. 4: Variation of the Lewis Factor according to the number of teeth and the

asymmetry coefficient.

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Fig. 5. Algorithm to determine Lewis Factor values (YF)



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Editorial Continental, S.A., Mxico,

1980 (in Spanish).

5) G. Henriot, Trait Thorique et Pratique

des Engrenajes, Villars, Paris, 1991 (in

French).

6) V.N. Kudriatzev, Design Machine

Elements, Mashinostroienie, Leningrad,

1980 (in Russian).

7) F.L. Litvin, Gear Theory, Nauka, Moscow,

1968 (in Russian).

8) H. Merrit, Gear Engineering, Pi tman

Publishing, England, 1990.

9) D. Reshetov, Machine Elements, La

Habana, Pueblo y Educacin, 1985 (in

Spanish).

10) M.F. Spotts, T. E. Shoup, Mechanical

Design, McGraw-Hill, New York, 1999.

11) K. Zirpke, Zahnrader, Veb Fachbuch

Verlag, Leipzig 1980 (in German).

12) A.L. Kapelevich, Geometry and Design

of Involute Spur Gears with Asymmetric

Teeth, Mechanism and Machine Theory

35 (2000) 117130.

13) J.I. Pedrero, A. Rueda, A. Fuentes,

Determination of the ISO Tooth Form

Factor for Involute Spur and Helical

Gears, Mechanism and Machine

Theory 34 (1999) 89103.

14) P. Black, Machine Design, McGraw-Hill,

New York, 1948.

15) V.M. Faires, Machine Design, Limusa,

1998 (in Spanish).

16) DIN 3990, Calculation of Load Capacity

of Cylindrical Gears: Calculation of

Tooth Strength, 1987-12 (in German).

17) G. Gonzlez, P. Frechilla, R. Garca, T

Finite Element Method as alternat

to design gears, Ingeniera Mecn

1 (2001) (in Spanish).

18) J. Moya, A. Machado, J. Velzquez,

Hernndez, J. Fernndez, The influen

of tooth geometry on the strength

plastic spur gears. (submitted to t

Mechanism and Machine Theory).

Fig. 6. Stress at the root of the

asymmetric gear tooth.

about the authors:

acknowledgements:

J.L. Moya and A.S. Machado are in the Department of Applied Mechanicsat Central University of Las Villas [www.uclv.edu.cu] in Cuba. J.A. Velsquezis on the mechanical and electrical engineering faculty at the UniversidadVeracruzana [www.uv.mx] in Mxico, and R. Goytisolo is on the mechani-cal engineering faculty at the University of Cienfuegos [www.ucf.edu.cu] in Cuba. A.E. Hernndez, J.E. Fernndez, and J.M. Sierra are in theDepartment of Mechanical and Civil Engineering at the University of Oviedo[www.uniovi.es] in Spain.

A. Machado was the recipient of a postdoctoral fellowship from the StateSecretary of Education and Universities, Ministry of Education and Science,Spain. This study was financially supported by research project MAT2003-06153

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Asymmetric Plastic Spur Gears