Gear Geometry and Applied Theory Episode 2 Part 5 pptx

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14.10 Nomenclature 403

parabolic function of transmission errors that is able to absorb the linear functions of

transmission errors caused by misalignments.

14.10 NOMENCLATURE

αn rack profile angle in normal section (Fig. 14.4.7)

αt rack profile angle in transverse section (Fig. 14.4.7)

βk (k = p, ρ) helix angle on pitch cylinder (k = p), on cylinder of

radius ρ (k = ρ) (Figs. 14.2.1 and 14.4.7)

λi (i = p, b, ρ) lead angle on the pitch cylinder (i = p), on the base cylinder

(i = b), and on the cylinder of radius ρ (Figs. 14.2.1, 14.4.5

and 14.4.7)

µ1 half of the angular width of the tooth space on the base circle of

gear 1 (Fig. 14.3.2)

θ, θ1, and θ2 surface parameter of the screw involute surface (Figs. 14.3.2

and 14.3.3)

φ, φ1, and φ2 angle of gear rotation (Figs. 14.4.1 and 14.5.1)

η2 half of the angular tooth thickness on pitch circle of gear 2

E shortest axes distance (Fig. 14.5.1)

F(12,n) normal component of contact force (Fig. 14.8.2)

H lead (Fig. 14.2.1)

l axial dimension of helical gear [Fig. 14.7.1(b)]

m12 gear ratio

mc gear contact ratio

N surface normal

n surface unit normal

pn circular pitch measured perpendicular to the direction of skew

teeth of the rack [Fig. 14.4.7(c)]

pt circular pitch in the cross section [Fig. 14.4.7(c)]

Pn and Pt diametral pitches that correspond to pn and pt

p = H/2π screw parameter

q orientation angle of straight contact lines on rack tooth surface

(Fig. 14.4.3)

rb radius of base cylinder (Fig. 14.4.4)

ro radius of operating pitch cylinder, axode

rpi radius of pitch cylinder i (Figs. 14.3.2 and 14.3.3)

s rack displacement (Fig. 14.4.1)

st tooth thickness on the pitch circle in the cross section

u surface parameter of a screw involute surface

wt space width measured on the pitch circle in cross section

X(12)

f , Y(12)

f , Z(12)

f components of contact force (Figs. 14.8.2 and 14.8.3)

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15 Modified Involute Gears

15.1 INTRODUCTION

Involute gears, spur and helical ones, are widely used in reducers, planetary gear trains,

transmissions, and many other industrial applications. The level of sophistication in the

design and manufacture of such gears (by hobbing, shaping, and grinding) is impressive.

The geometry, design, and manufacture of helical gears was the subject of research

presented in the works of Litvin et al. [1995, 1999, 2001a, 2003], Stosic [1998], and

Feng et al. [1999].

The advantage of involute gearing in comparison with cycloidal gearing is that the

change of center distance does not cause transmission errors. However, the practice

of design and the test of bearing contact and transmission errors show the need for

modification of involute gearing, particularly of helical gears. Figure 15.1.1 shows a 3D

model of a modified involute helical gear drive.

The existing design and manufacture of involute helical gears provide instantaneous

contact of tooth surfaces along a line. The instantaneous line of contact of conjugated

tooth surfaces is a straight line L0 that is the tangent to the helix on the base cylinder

(Fig. 15.1.2). The normals to the tooth surface at any point of line L0 are collinear and

they intersect in the process of meshing with the instantaneous axis of relative motion

that is the tangent to the pitch cylinders. The concept of pitch cylinders is discussed in

Section 15.2.

The involute gearing is sensitive to the following errors of assembly and manufacture:

(i) the change γ of the shaft angle, and (ii) the variation of the screw parameter (of one

of the mating gears). Angle γ is formed by the axes of the gears when they are crossed,

but not parallel, due to misalignment (see Fig. 15.4.4). Such errors cause discontinuous

linear functions of transmission errors which result in vibration and noise, and these

errors may also cause edge contact wherein meshing of a curve and a surface occurs

instead of surface-to-surface contact (see Section 15.9). In a misaligned gear drive, the

transmission function varies in each cycle of meshing (a cycle for each pair of meshing

teeth). Therefore the function of transmission errors is interrupted at the transfer of

meshing between two pairs of teeth [see Fig. 15.4.6(a)].

This chapter covers (i) computerized design, (ii) methods for generation, (iii) simu￾lation of meshing, and (iv) enhanced stress analysis of modified involute helical gears.

404

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15.1 Introduction 405

Figure 15.1.1: Modified involute helical gear

drive.

The approaches proposed for modification of conventional involute helical gears are

based on the following basic ideas:

(i) Line contact of tooth surfaces is substituted by instantaneous point contact.

(ii) The point contact of tooth surfaces is achieved by crowning of the pinion in the

profile and longitudinal directions. The tooth surface of the gear is a conventional

screw involute surface.

Contact lines L0

Base cylinder helix

Figure 15.1.2: Contact lines on an involute

helical tooth surface.