Scholarly article on topic 'The Mathematical Model of Generating Kinematic for the Worm Face Gear with Modified Geometry'

The Mathematical Model of Generating Kinematic for the Worm Face Gear with Modified Geometry Academic research paper on "Mechanical engineering"

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Procedia Technology
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{"Worm face gear" / "Spiroid gearing" / "Modified geometry" / "Mathematical model" / Modeling}

Abstract of research paper on Mechanical engineering, author of scientific article — Claudiu-Ioan Boantă, Vasile Boloş

Abstract This paper aims to present the mathematical model for the generating kinematic and numerical simulation of worm face gear with simplified geometry. Through simplification we mean that the thread or the tooth profile is trapezoidal in cross section and tooth's flanks angles are symmetrical, in this case 20°. It is highlight the way haw are generated the tooth flanks of the worm face gear by the method of generating kinematics, where is considered that the worm face gears can be replaced, in terms of kinematics, by a gear composed by rack and a worm wheel. The method was exemplified for a worm face gear with Archimedean worm heaving following characteristics: i12 = 47, A = 56mm, axial module 2.5.Matrix-vector method was used in order to create the model, the results of calculations performed in MATHCAD wererepresented in INVENTOR 2012.

Academic research paper on topic "The Mathematical Model of Generating Kinematic for the Worm Face Gear with Modified Geometry"

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Procedia Technology 12 (2014) 442 - 447

200The 7th International Conference Interdisciplinary in Engineering (INTER-ENG 2013)

The mathematical model of generating kinematic for the worm face

gear with modified geometry

Claudiu-Ioan Boantaa*, Vasile Bolosb

aUniversitatea Tehnica din Cluj Napoca, B-dul Muncii nr 103-195, Cluj Napoca 400641, Romania bUniversitatea Petru Maior, Str. N Iorga nr 1, Tirgu Mure§ 540088, Romania

Abstract

This paper aims to present the mathematical model for the generating kinematic and numerical simulation of worm face gear with simplified geometry. Through simplification we mean that the thread or the tooth profile is trapezoidal in cross section and tooth's flanks angles are symmetrical, in this case 20°. It is highlight the way haw are generated the tooth flanks of the worm face gear by the method of generating kinematics, where is considered that the worm face gears can be replaced, in terms of kinematics, by a gear composed by rack and a worm wheel. The method was exemplified for a worm face gear with Archimedean worm heaving following characteristics: i12 = 47, A = 56 mm, axial module 2.5.

Matrix-vector method was used in order to create the model, the results of calculations performed in MATHCAD were represented in INVENTOR 2012.

©2013The Authors.Published byElsevierLtd.

Selection andpeer-review underresponsibilityofthePetru Maior University of Tirgu Mures. Keywords: Worm face gear; Spiroid gearing; Modified geometry; Mathematical model; Modeling

1. Introduction

The worm face gear made up by conical worm geared with a taper worm wheel, respectively a cylindrical worm gear with a flat worm face wheel, specifically is the fact that the conical or cylindrical worm has different pressure

* Corresponding author. Tel.: +4-072-753-3661; fax: +4-036-581-5577. E-mail address: claudiu_boanta@yahoo.com

2212-0173 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Petru Maior University of Tirgu Mures. doi:10.1016/j.protcy.2013.12.512

angles on the two sides, respectively teeth height is slightly higher (2.75 x module) than usual worm gears (2.25 x module) (Fig. 1), this makes the output and operating conditions of the gear to be different, for the two directions of rotation.

Fig. 1. a) spiroid worm reference profile; b) spiroid worm simplified profile; c) usual cylindrical worm profile.

This geometry creates the technological difficulties, the worm wheel machining tools must be made with almost identical geometry. Pressure angle of 100 has the effect of reducing of the functional relief angle for the mill that process the face worm wheels, which affects the roughness of the flanks obtained.

To reduce the difficulties mentioned above and to ease the implementation of the face worm gear was proposed a modified worm face gear [1], which is characterized by having equal pressure angles of flanks(14,50 [1] or may be 200) and teeth height identical to that of the usual worm gear by 2,25 x module (Fig. 1).

This paper present a mathematical model of the modified worm face gear, characterized in that the worm has a symmetric pressure flanks. The mathematical model developed here, is based on the mathematical model presented in the paper [2, 3], which relates to a worm face gear of an Archimedean type. The mathematical model is built as matrix-vector system and ensure the determination of the coordinates for the points of the surfaces of worm and worm wheel flanks.

The domain of worm face gear is currently a topic of interest for many studies of theoretical, technological and experimental character [4 - 12].

2. Definition of the helical surfaces of the worm

To determine the coordinates of the worm flanks, respectively the flanks of the worm face wheel, the following reference systems are used: fixed reference system OFXFYFZF, reference system related to the wrapped worm OjXjYjZj and the system O0X0Y0Z0 related to the generating curve of the wrapped worm, O2X2Y2Z2 system related to the worm wheel.

Further it will be used the following notation: A - Axial distance

B - a parameter that defines the position of the reference system related to the wrapped worm related to the fixed reference system

a - flank pressure angle h - helical parameter of the flank i12 - transmission ratio

k - indicator of the flank (k = 1 right, k = 2 left) M - current point

N-l - the normal raised to the surface index 1 p - wrapped worm step

pk - position parameter along to a curve (k = 1 or k = 2)

u1, u2, v - rotational motion parameters between reference systems

Xj- position vector in the reference system index 1

w1, w2 - angular velocity of the worm, respectively of the worm wheel

Fig. 2. a) Worm; b) Detail of the worm tooth.

Based on the notation from the Fig. 2 and using elements defined in [2] and [13], expressions of the coordinates of the points of the wrapped worm will be:

^it —

— (r0 + pk x cos a) sin v (To + Pk x cos a) cos v (do + Pk x si-n o)(2k — 3) + h x v 1

x\k Vlk zlk 1

= Xlk(pk,v~)

Similar to those presented in [3] the report Hk is noted as follow:

Hk=—-—

r0+pkxcos a

The normal to the flanks is:

~(2k - 3) sin a x sin v — Hk x cos a x cos v nlxk

(2k- 3) sin a x cos v — Hk x cos a x sin v nlyk

— cos a nlzk

= Nlk(pk,v)

Expressions (1) and (3) define completely and unequivocally the helicoids of the modified wrapped cylindrical worm gear.

3. Definition of the tooth flanks surfaces of the face worm wheel

To define the face worm wheel tooth flank, it was considered the way of kinematic generating, which means that the worm face gear can be considered, from the kinematic point of view as formed by a rack and a face wheel. The

movements of the elements are: translational motion of the worm by speed hw1 and the rotation movement with angular velocity w2 (Fig. 3).

To determine the coordinates of the flanks for the worm face wheel are used the following reference systems: reference system related to worm face wheel OjXjYjZj and the system O0X0Y0Z0.

Fig. 3. Relative position of the worm and worm face wheel.

Based on the elements presented in [2], by customization (taking into account the symmetry of the flanks for conjugate worm) following expressions are obtained:

cos u2 x xik — sin U2 X zlk -{B- hu-i) x sin u2 4 A x cos u2 x2k

— sin U2 X xlk — COS U2 X zlk -{B — hui) x cos u2 + A x sin u2 V2k

Vik z2k

1 1

The normal vector to the tooth flank of the wheel surface point is:

nlxk X cos u2 ~ nlzk X Sln u2 —nlxk x sin U2 — nlzk X COS U2

nlyk 0

n2xk n2yk n2zk 0

= N2k(p, V)

Relations (4) and (5) define completely the face worm wheel tooth flank surface. 4. Numerical modeling

= X2k(p,v,u2) (4)

For numerical verification of the mathematical model described above it was considered a gear heaving following geometrical characteristics: Number of threads for worm z1=1, Number of teeth for gear z2=47, Axial distance A=56mm, Pressure angles a=200, Axial module ma=2,5mm, Reference range of worm r0=18,2325.

Using Mathcad it was performed the determination of the coordinates for the points of left and right flank of the worm, respectively of the conjugate face worm wheel were then represented in INVENTOR 2012 (Fig. 4 and Fig.

Fig. 4. Coordinates for the points of the flanks of the worm.

______

MX 1 ttIZ

v\ \ v V '' \ I 1 A// m/ni SA!/ / T^l "l / ///

V \ [} —7 -

\ * ' \ \ V \ vv \ \ \r a i a \ HJl /1 1/

V - WxTT

Fig. 5. Detail - the space between two teeth.

The numerical model achieved, allows the determination of the points family which form the flanks of the worm, respectively the flanks of the wheel, that for the each case make it possible to modeling of the geometric configuration of the case further studied by transferring such data in various environments graphical specific (INVENTOR, CATIA etc.).This opportunity is especially useful for designers to optimize each specific application of this gear.

5. Conclusions

The face worm gear with modified geometry it is one embodiment that can be considered for industrial application, combining functional advantages of mutual positions of the worm and worm wheel by the reduction of the technical difficulties specific to execution of the face worm gears.

In order to apply this kind of gears at industrial scale, involves the performing of mathematical and numerical study and experimental determinations related to the performance, efficiency, noise, vibration etc.

The study provides coordinates determination for the flanks of the worm and the face worm wheel and provide for the appropriate graphical modeling environments that will optimize the constructive nature.

From this point of view the work corresponding to the requirements underlying the development of this category of gear.

References

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[2] Bolos V. Angrenaje melcate spiroide. Danturarea rotilor plane, Tg Mures: Editura Universitatii "Petru Maior"; 1999.

[3] Gyenge C. Contributii asupra imbunatatirii preciziei frezelor melc duplex pentru executarea angrenajului melcat duplex, Cluj Napoca: Institutul Politehnic; 1979.

[4] Abadjieva E, Abadjiev V. Geometric Synthesis and Computer Modelling of Conic Linear Helicoids, in The 4th International Conference on Power Transmissions, Sinaia; 2012.

[5] Balajti Z. New Modelling of Computer Aided Design of Worms, in Manuf. and Ind. Eng; 2012.

[6] Bolos V, Gavrila I. Research on the Possibility to Determine the Tooth Flank Surface of Spiroid Plane Wheels, MTeM, Cluj-Napoca; 2007.

[7] Bucur B. Cercetari privind geometria §i tehnologia de danturare a angrenajelor melcate frontale cu conicitate inversa, Cluj-Napoca: Universitatea Tehnica; 2012.

[8] Dudas I. General Mathematical Model for Investigation of Cylindrical and Conical Worms and Hobs, in MTeM Cluj-Napoca; 2011.

[9] Dudas I. The Theory and Practice of Worm Gear Drives, London: Penton Press; 2000.

[10] Dudas I, Bodzas S. Spiroid Csiga Matematikai, Geometriai Modellezese es Gyors Prototipus Gyartasa, in Multidiszciplinaris tudomanyok, Miskolc; 2011.

[11] Napau ID. Contributii privind modelarea, simularea si experimentarea angrenajelor melc-roata plana cu contact localizat, Cluj Napoca: Universitatea Tehnica; 2005.

[12] Nieszporek T, Lyczko K. Generation of the Worm Helical Surface, in Proceedings in Manufacturing Systems; 2010.

[13] Bolos V, Boanta C. The Mathematical and numerical model of the spiroid gearing, in Inter-Ing, Tg. Mures; 2007.