Gear Geometry and Applied Theory Episode 2 Part 1 pot

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CB672-10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:19

10.5 Meshing of Involute Gear with Rack-Cutter 283

Choosing ζ >ζmin, we have to limit ζ due to the possibility of tooth pointing (see

Problem 10.6.2).

(ii) N > Nmin. Then ζ ≤ 0, and the rack-cutter can be displaced toward the gear center,

or the setting can be conventional (ζ = 0).

Change of Gear Tooth Thickness and Dedendum Height

The displacement of the rack-cutter affects the gear tooth thickness and the dedendum

dimension. Henceforth, we consider the change of tooth thickness (space width) that is

measured along the gear pitch circle. The space width of the gear that is measured along

the pitch circle is equal to the tooth thickness of the rack-cutter that is measured along

I–I, the rack-cutter centrode. In the case of the conventional setting of the rack-cutter,

the nominal value of the gear space width is

w = sc = pc

2 = π

2P (10.5.10)

where sc is the tooth thickness of the rack-cutter on the middle-line a–a. When a non￾conventional setting of the rack-cutter is provided, the tooth thickness of the rack-cutter

on its centrode I–I is [Fig. 10.5.3(b)]

s ∗

c = sc − 2e tan αc = pc

2 − 2e tan αc . (10.5.11)

The gear space width is

w = s ∗

c = pc

2 − 2e tan αc . (10.5.12)

The radius of the dedendum circle is determined with the equation

rd = rp − b + e, (10.5.13)

and the dedendum height is (b − e). To keep the total height at the gear tooth at the

proper value it is necessary to change the radius of the addendum circle while preparing

the gear blank for cutting.

Problem 10.5.1

Consider a conventional setting of the rack-cutter (e = 0). The radius rG of the circle

where the initial point of the involute curve is located is represented by Eqs. (10.5.1)

and (10.5.2). Represent radius rG in terms of N, αc , and P; take a = 1/P.

Solution

rG = (N2 sin2 αc − 4N sin2 αc + 4) 1

2

2P sin αc

. (10.5.14)

Problem 10.5.2

Consider that radius rG is represented by Eq. (10.5.14). Derive an equation in

terms of N and αc when the initial point of the involute curve belongs to the base

circle.

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CB672-10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:19

284 Spur Involute Gears

Figure 10.5.3: Illustration of (a) generation of standard and nonstandard gears and (b) tooth thickness

of the rack-cutter.

Solution

N = 2

sin2 αc

.

Problem 10.5.3

Transform expression (10.5.8) by using Eq. (10.5.4). Represent ζ in terms of N

and αc .

Solution

ζ ≥

2 − N sin2 αc

2 .

Problem 10.5.4

A gear with tooth number N > Nmin is generated by a rack-cutter with the profile angle

αc ; the diametral pitch is P; the addendum of the rack-cutter is b = 1.25/P; a non￾conventional setting of the rack-cutter is used (e < 0). Represent in terms of N, αc , and

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CB672-10 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:19

10.6 Relations Between Tooth Thicknesses Measured on Various Circles 285

P the minimal radius of the gear dedendum circle with which undercutting might still

be avoided.

DIRECTIONS. Use Eq. (10.5.13) for rd . Expression (10.5.7) yields that undercutting may

still be avoided with

e = Nmin − N

P Nmin

= 2 − N sin2 αc

2P .

Solution

rd = N cos2 αc − 0.5

2P . (10.5.15)

Problem 10.5.5

Equation (10.5.15) determines the radius of the dedendum circle when the non￾conventional setting of the rack-cutter is applied. The radius of the dedendum circle

when a conventional setting of the rack-cutter is applied is represented by the equation

r ∗

d = rp − 1.25

P .

Determine N in terms of αc when: (i) rd > r ∗

d , (ii) rd = r ∗

d , and (iii) rd < r ∗

d .

Solution

(i) N <

2

sin2 αc

; (ii) N = 2

sin2 αc

; (iii) N >

2

sin2 αc

.

10.6 RELATIONS BETWEEN TOOTH THICKNESSES MEASURED

ON VARIOUS CIRCLES

Consider that the tooth thickness tp = 

AA on the pitch circle is given (Fig. 10.6.1). The

goal is to determine the tooth thickness tx = 

BB on the circle of given radius rx; tx must

be represented in terms of P, pressure angle αc , tooth number N, and radius rx.

The tooth half-thickness and the corresponding angle β (or βx) are related by the

following equations:

β =



AA

2rp

= tp

2rp

(10.6.1)

βx = tx

2rx

(10.6.2)

Figure 10.6.1 yields

βx = β + inv αc − inv αx (10.6.3)

where inv αc = tan αc − αc , inv αx = tan αx − αx, and

cos αx = rb

rx

= N cos αc

2P rx

. (10.6.4)