Gear Geometry and Applied Theory Episode 3 Part 2 pptx

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CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28

19.13 Prospects of New Developments 613

where Ehg and Ewg are the center distances between the hob and the worm-gear and

between the worm and the worm-gear, respectively; r = r ph − r pw; r ph and r pw

are the radii of pitch cylinders of the hob and the worm, respectively; γ = λw − λh;

λw and λh are the lead angles of the worm and the hob, respectively.

For instance, in the case of an involute worm-gear drive the hob and the worm are

two involute helicoids. In the case of K worm-gear drives (see Section 19.7), the hob

and the worm are generated by a cone with the same profile angle.

Figure 19.13.2 shows the output of TCA for a K worm-gear drive wherein the worm￾gear has been generated by an oversized hob [Seol & Litvin, 1996]. The path of contact

is oriented across the worm-gear surface and is located around the center of the worm￾gear surface [Fig. 19.13.2(a)]. The function of transmission errors is of a parabolic type

[Fig. 19.13.2(b)].

For some cases of misalignment, an oversized hob that is too small fails to provide a

continuous function of transmission errors. In the opinion of the authors of this book,

localization of the bearing contact by double crowning of the worm is the approach

with much greater potential.

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CB672-20 CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:11

20 Double-Enveloping Worm-Gear Drives

20.1 INTRODUCTION

The invention of the double-enveloping worm-gear drive is a breathtaking story with

two dramatic characters, Friedrich Wilhelm Lorenz and Samuel I. Cone, each acting in

distant parts of the world – one in Germany and the other in the United States [Litvin,

1998]. The double-enveloping worm-gear drive was invented by both Cone and Lorenz

independently, and we have to credit them both for it [Litvin, 1998]. The invention of

Samuel I. Cone in the United States has been applied by a company that bears the name

of the inventor, known by the name Cone Drive.

The invented gear drive is a significant achievement. The special shape of the worm

increases the number of teeth that are simultaneously in mesh and improves the con￾ditions of force transmission. The conditions of lubrication and the efficiency of the

invented drive (in comparison with a worm-gear drive with a cylindrical worm) are

substantially better due to the special shape of lines of contact between the worm and

gear surfaces (see below).

The theory of double-enveloping worm-gear drives has been the subject of intensive

research by many scientists. This chapter is based on the work by Litvin [1994]. We

consider in this chapter the Cone double-enveloping worm-gear drive.

20.2 GENERATION OF WORM AND WORM-GEAR SURFACES

Worm Generation

The worm surface is generated by a straight-lined blade (Fig. 20.2.1). The blade per￾forms rotational motion about axis Ob with the angular velocity Ω(b) = dΨb/dt, while

the worm rotates about its axis with the angular velocity Ω(1) = dΨ1/dt; ψb and

ψ1 are the angles of rotation of the blade and the worm in the process for gener￾ation (Fig. 20.2.2). The shortest distance between the axes of rotation of the blade

and the worm is Ec . The generating lines of the blade in the process of generation

keep the direction of tangents to the circle of radius Ro. The directions of rotation

shown in Figs. 20.2.1 and 20.2.2 correspond to the case of generation of a right-hand

worm.

614

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20.2 Generation of Worm and Worm-Gear Surfaces 615

Figure 20.2.1: Worm generation.

Worm-Gear Generation

The generation of the worm-gear is based on simulation of meshing of the worm and

the worm-gear in the process of worm-gear generation. A hob identical to the generated

worm is in mesh with the worm-gear being generated on the cutting machine. The axes

of rotation of the hob and the worm-gear are crossed; the shortest distance E between

the axes is the same as in the designed worm-gear drive; the ratio m21 between the

angular velocities of the hob (worm) and the worm-gear is also the same. Here,

m21 = ω(2)

ω(1) = N1

N2

(20.2.1)

where N1 and N2 are the numbers of worm threads and gear teeth.

Figure 20.2.2: Coordinate systems ap￾plied for worm generation.