Gear Geometry and Applied Theory Episode 3 Part 1 potx

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

19.7 Geometry and Generation of K Worms 583

Figure 19.7.2: Coordinate systems applied for generation of K worms.

1 is represented as the family of lines of contact of surfaces c and 1 by the following

equations:

r1(uc , θc , ψ) = M1oMoc rc (uc , θc ) (19.7.1)

Nc (θc ) · v

(c1)

c (uc , θc ) = f (uc , θc ) = 0. (19.7.2)

Equation (19.7.1) represents the family of tool surfaces; (uc , θc ) are the Gaussian

coordinates of the tool surface, and ψ is the angle of rotation in the screw motion.

Equation (19.7.2) is the equation of meshing. Vectors Nc and v

(c1)

c are represented

in Sc and indicate the normal to c and the relative (sliding) velocity, respectively. It

is proven below [see Eq. (19.7.8)] that Eq. (19.7.2) does not contain parameter ψ.

Equations (19.7.1) and (19.7.2) considered simultaneously represent the surface of the

worm in terms of three related parameters (uc , θc , ψ).

For further derivations we will consider that the surface side I of a right-hand worm

is generated. The cone surface is represented by the equations (Fig. 19.7.3)

rc = uc cos αc (cos θc ic + sin θc jc ) + (uc sin αc − a) kc . (19.7.3)

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

584 Worm-Gear Drives with Cylindrical Worms

Figure 19.7.3: Generating cone surface.

Here, uc determines the location of a current point on the cone generatrix; “a” deter￾mines the location of the cone apex.

The unit normal to the cone surface is determined as

nc = Nc

|Nc |

, Nc = ∂rc

∂uc

× ∂rc

∂θc

, (19.7.4)

which yields

nc = [− sin αc cos θc − sin αc sin θc cos αc]

T. (19.7.5)

The relative velocity is represented as the velocity in screw motion (Fig. 19.7.4)

v(c1)

c = −ωc × rc − Rc × ωc − p ωc (19.7.6)

where Rc = −Ec ic is the position vector of point O

1 of the line of action of ω. Equation

(19.7.6) yields

v(c1)

c = ω











− sin γc zc + cos γc yc

− cos γc (xc + Ec ) − p sin γc

sin γc (xc + Ec ) − p cos γc









 . (19.7.7)

The equation of meshing of the grinding surface with the worm surface after elimi￾nation of (−ω sin γc cos θc ) is represented as

nc · v(c1)

c = f (uc , θc ) = a sin αc − (Ec sin αc cot γc + p sin αc ) tan θc

− (Ec − p cot γc ) cos αc

cos θc

− uc = 0 (19.7.8)

where uc > 0. Equation (19.7.8) with the given value of uc provides two solutions for

θc and determines two curves, I and II in the plane (uc , θc ) (Fig. 19.7.5). Only curve I is

the real contact line in the space of parameters (uc , θc ).

P1: JsY

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

19.7 Geometry and Generation of K Worms 585

Figure 19.7.4: Installment of grinding cone: (a)

illustration of installment parameter Ec ; (b) illust￾ration of installment parameter γc .

Figure 19.7.5: Line of contact between generating cone and K worm surface: representation in plane

of parameters.