Gear Geometry and Applied Theory Episode 2 Part 4 pdf

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CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36

13.8 Root’s Blower 373

Figure 13.8.3: For derivation of relation between the design parameters.

Figure 13.8.4: Applied coordinate systems.

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CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36

374 Cycloidal Gearing

Table 13.8.1: Properties of 2

Lobe number Convex Concave–Convex With singularities

2 0 < a

r < 0.5 0.5 < a

r < 0.9288 a

r > 0.9288

3 0 < a

r < 0.5 0.5 < a

r < 0.9670 a

r > 0.9670

which yield

xf = ρ sin(θ − φ) − a sin φ

y f = ρ cos(θ − φ) + a cos φ

r sin(θ − φ) − a sin θ = 0.

(13.8.5)

Equations of Dedendum Curve Σ2 of Rotor 2

Profile 2 is represented in S2 by the equations

r2 = M21r1, f (θ,φ) = 0, (13.8.6)

which yield

x2 = ρ sin(θ − 2φ) − a sin 2φ + 2r sin φ

y2 = ρ cos(θ − 2φ) + a cos 2φ − 2r cos φ

r sin(θ − φ) − a sin θ = 0.

(13.8.7)

Depending on the ratio a/r, profile 2 may be represented by (i) a convex curve, (ii) a

concave–convex curve, and (iii) a curve with singularities. The third case may be investi￾gated by considering the conditions of “nonundercutting” of 2 by 1 (see Section 6.3).

The first and second cases may be investigated by considering the relations between the

curvatures of conjugate shapes (see Section 8.3). The results of the investigations are

presented in Table 13.8.1.

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CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39

14 Involute Helical Gears with Parallel Axes

14.1 INTRODUCTION

Cycloidal gears (Chapter 13) and involute gears (Chapters 10, 11, 14, 15, and 16) have

different areas of application. This chapter covers involute gears with parallel axes,

whose design is based on the assumption that the gear tooth surfaces are in instanta￾neous contact along a line (line contact) in the case of aligned gear drives. Although the

influence of errors of alignment should be considered in the study of the real meshing

(see Chapters 15, 16, and 17), in this chapter we consider a preliminary study limited

to the theoretical study of meshing. This allows the reader to focus initially on the theo￾retical study of involute gears. However, we have to emphasize that the modern design

of helical gear drives is directed at observation of localized bearing contact (obtained

by tooth surfaces being in point contact instead of line contact), simulation of mesh￾ing of misaligned gear drives, and stress analysis (see Chapters 15, 16, and 17). The

nomenclature used in this chapter is presented in Section 14.10.

14.2 GENERAL CONSIDERATIONS

Helical gears that transform rotation between parallel axes in opposite directions are in

external meshing and are provided with screw tooth surfaces of opposite directions.

The axodes of nonstandard gears are two cylinders of radii ro1 and ro2 related as

ro2

ro1

= ω(1)

ω(2) = m12. (14.2.1)

These cylinders are called the operating pitch cylinders as well. Henceforth, we differen￾tiate standard and nonstandard helical gears. The operating pitch cylinders (the axodes)

coincide with the pitch cylinders in the case of standard helical gears, and they differ

from the pitch cylinders for nonstandard helical gears (see below). Axodes of standard

gears are the gear pitch cylinders. The line of tangency of the axodes is the instanta￾neous axis of rotation of the gears in relative motion. The cylinders of radii ro1 and ro2

roll over each other without sliding. The helices on the operating pitch cylinders are of

opposite direction but the magnitude of the lead angle (or the helix angle) is the same

for both helices.

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