Gear Geometry and Applied Theory Episode 2 Part 6 ppsx

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15.8 Undercutting and Pointing 433

The derivation of line L is based on the following considerations:

(i) Equation (15.8.1) yields

∂rc

∂uc

duc

dt +

∂rc

∂θc

dθc

dt = −v(cσ) c . (15.8.3)

Here, ∂rc/∂uc , ∂rc/∂θc , and v

(cσ) c are three-dimensional vectors represented in sys￾tem Sc of the pinion rack-cutter.

(ii) Equation (15.8.2) yields

∂ f

∂uc

duc

dt +

∂ f

∂θc

dθc

dt = − ∂ f

∂ψσ

dψσ

dt . (15.8.4)

(iii) Equations (15.8.3) and (15.8.4) represent a system of four linear equations in two

unknowns: duc/dt and dθc/dt. This system has a certain solution for the unknowns

if matrix

A =











∂rc

∂uc

∂rc

∂θc

−v

(cσ) c

∂ f

∂uc

∂ f

∂θc

− ∂ f

∂ψσ

dψσ

dt











(15.8.5)

has the rank r = 2. This yields

1 =





























∂xc

∂uc

∂xc

∂θc

−v(cσ) xc

∂yc

∂uc

∂yc

∂θc

−v(cσ)

yc

∂ f

∂uc

∂ f

∂θc

− ∂ f

∂ψσ

dψσ

dt





























= 0 (15.8.6)

2 =





























∂xc

∂uc

∂xc

∂θc

−v(cσ) xc

∂zc

∂uc

∂zc

∂θc

−v(cσ) zc

∂ f

∂uc

∂ f

∂θc

− ∂ f

∂ψσ

dψσ

dt





























= 0 (15.8.7)

3 =





























∂yc

∂uc

∂yc

∂θc

−v(cσ)

yc

∂zc

∂uc

∂zc

∂θc

−v(cσ) zc

∂ f

∂uc

∂ f

∂θc

− ∂ f

∂ψσ

dψσ

dt





























= 0 (15.8.8)

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434 Modified Involute Gears

4 =





























∂xc

∂uc

∂xc

∂θc

−v(cσ) xc

∂yc

∂uc

∂yc

∂θc

−v(cσ)

yc

∂zc

∂uc

∂zc

∂θc

−v(cσ) zc





























= 0. (15.8.9)

Equation (15.8.9) yields the equation of meshing f (uc , θc , ψσ ) = 0 and is not ap￾plied for investigation of singularities. The requirement that determinants 1, 2,

and 3 must be equal to zero simultaneously may be represented as

2

1 + 2

2 + 2

3 = 0. (15.8.10)

Equation (15.8.10) enables us to obtain for determination of singularities the fol￾lowing function:

F (uc , θc , ψσ ) = 0 (15.8.11)

NOTE. In most cases, it is sufficient for derivation of function F = 0 to use instead

of (15.8.10) only one of the three following equations:

1 = 0, 2 = 0, 3 = 0. (15.8.12)

An exceptional case, when application of (15.8.10) is required, is discussed in

Section 6.3.

Singularities of the pinion may be avoided by limitation by line L of the rack-cutter

surface c that generates the pinion. The determination of L [Fig. 15.8.1(a)] is based

on the following procedure:

(1) Using equation of meshing f (uc , θc , ψσ ) = 0, we may obtain in the plane of pa￾rameters (uc , θc ) the family of contact lines of the rack-cutter and the pinion. Each

contact line is determined for a fixed parameter of motion ψσ .

(2) The sought-for limiting line L is determined in the space of parameters (uc , θc ) by

simultaneous consideration of equations f = 0 and F = 0 [Fig. 15.8.1(a)]. Then,

we can obtain the limiting line L on the surface of the rack-cutter [Fig. 15.8.1(b)].

The limiting line L on the rack-cutter surface is formed by regular points of the rack￾cutter, but these points will generate singular points on the pinion tooth surface.

Limitations of the rack-cutter surface by L enable us to avoid singular points on the

pinion tooth surface. Singular points on the pinion tooth surface can be obtained by

coordinate transformation of line L on rack-cutter surface c to surface σ .

Pointing

Pointing of the pinion means that the width of the topland becomes equal to zero.

Figure 15.8.2(a) shows the cross sections of the pinion and the pinion rack-cutter. Point

Ac of the rack-cutter generates the limiting point Aσ of the pinion when singularity of

the pinion is still avoided. Point Bc of the rack-cutter generates point Bσ of the pinion

profile. Parameter sa indicates the chosen width of the pinion topland. Parameter αt

indicates the pressure angle at point Q. Parameters h1 and h2 indicate the limitation of

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15.9 Stress Analysis 435

(mm)

(mm)

Figure 15.8.1: Contact lines Lcσ and limiting line L: (a) in plane (uc , θc ); (b) on surface c .

location of limiting points Ac and Bc of the rack-cutter profiles. Figure 15.8.2(b) shows

functions h1(N1) and h2(N1) (N1 is the pinion tooth number) obtained for the following

data: αd = 25◦, β = 30◦, parabola coefficient of pinion rack-cutter ac = 0.002 mm−1,

sa = 0.3 m, parameter s12 = 1.0 [see Eq. (15.2.3)], and module m = 1 mm.

15.9 STRESS ANALYSIS

This section covers stress analysis and investigation of formation of bearing contact

of contacting surfaces. The performed stress analysis is based on the finite element

method [Zienkiewicz & Taylor, 2000] and application of a general computer program

[Hibbit, Karlsson & Sirensen, Inc., 1998]. An enhanced approach for application of

finite element analysis is presented in Section 9.5.