Gear Geometry and Applied Theory Episode 3 Part 7 pptx

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26.4 Generation of a Surface with Optimal Approximation 763

Figure 26.4.6: Determination of maximal

deviations along line Lgk.

MINIMIZATION OF DEVIATIONS δi, j . Consider that deviations δi,j(i = 1,..., n; j =

1,..., m) of g with respect to p have been determined at the (n, m) grid points. The

minimization of deviations can be obtained by corrections of previously obtained func￾tion β(1)(θp). The correction of angle β is equivalent to the correction of the angle that

is formed by the principal directions on surfaces t and g . The correction of angle β

can be achieved by turning of the tool about the common normal to surfaces t and

p at their instantaneous point of tangency Mk.

The minimization of deviations δi,j is based on the following procedure:

Step 1: Consider the characteristic Lgk, the line of contact between surfaces t and

g , that passes through current point Mk of mean line Lm on surface p (Fig. 26.4.6).

Determine the deviations δk between t and p along line Lgk and find out the maximal

deviations designated as δ

(1)

kmax and δ

(2)

kmax. Points of Lgk where the deviations are maximal

are designated as N(1)

k and N(2)

k . These points are determined in regions I and II of surface

g with line Lm as the border. The simultaneous consideration of maximal deviations

in both regions enables us to minimize the deviations for the whole surface g .

Note. The deviations of t from p along Lgk are simultaneously the deviations of

g from p along Lgk because Lgk is the line of tangency of t and g .

Step 2: The minimization of deviations is accomplished by correction of angle βk that

is determined at point Mk (Fig. 26.4.6). The minimization of deviations is performed lo￾cally, for a piece k of surface g with the characteristic Lgk. The process of minimization

is a computerized iterative process based on the following considerations:

(i) The objective function is represented as

Fk = min

δ

(1)

kmax + δ

(2)

kmax (26.4.46)

with the constraint δi,j ≥ 0.

(ii) The variable of the objective function is βk. Then, considering the angle

β(2)

k = β(1)

k + βk (26.4.47)

and using the equation of meshing with βk, we can determine the new characteristic,

the piece of envelope (k)

g , and the new deviations. The applied iterations provide

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764 Generation of Surfaces by CNC Machines

the sought-for objective function. The final correction of angle βk we designate

as β(opt)

k .

Note 1. The new contact line L(2)

gk (determined with β(2)

k ) slightly differs from the real

contact line because the derivative dβ(1)

k /ds but not dβ(2)

k /ds is used for determination

L(2)

gk. However, L(2)

gk is very close to the real contact line.

Step 3: The discussed procedure must be performed for the set of pieces of surfaces

g with the characteristic Lgk for each surface piece.

Recall that the deviations for the whole surface must satisfy the inequality δi,j ≥ 0.

The procedure of optimization is illustrated with the flowchart in Fig. 26.4.7.

Curvatures of Ground Surface Σg

The direct determination of curvatures of g by using surface g equations is a com￾plicated problem. The solution to this problem can be substantially simplified using the

following conditions proposed by the authors: (i) the normal curvatures and surface

torsions (geodesic torsions) of surfaces p and g are equal along line Lm, respectively;

and (ii) the normal curvatures and surface torsions of surfaces t and g are equal

along line Lg . This enables us to derive four equations that represent the principal cur￾vatures of surface g in terms of normal curvatures and surface torsions of p and t.

However, only three of these equations are independent (see below).

Further derivations are based on the following equations:

kn = kI cos2 q + kII sin2 q = 1

2

(kI + kII ) +

1

2

(kI − kII ) cos 2q (26.4.48)

t = 0.5(kII − kI ) sin 2q. (26.4.49)

Here, kI and kII are the surface principal curvatures and angle q is formed by unit

vectors eI and e and is measured counterclockwise from eI and e; eI is the principal

direction with principal curvature kI ; e is the unit vector for the direction where the

normal curvature is considered; t is the surface torsion for the direction represented by e.

Equation (26.4.48) is known as the Euler equation. Equation (26.4.49) is known in

differential geometry as the Bonnet–Germain equation (see Chapter 7).

The determination of the principal curvatures and principal directions for g is based

on the following computational procedure (see Section 7.9):

Step 1: Determination of k(1)

n and t(1) for surface g at the direction determined by

the tangent to Lm. The determination is based on Eqs. (26.4.48) and (26.4.49) applied

to surface p. Recall that p and g have the same values of k(1)

n and t(1) along the

previously mentioned direction.

Step 2: Determination of k(2)

n and t(2). The designations k(2)

n and t(2) indicate the

normal curvatures of g and the surface torsion along the tangent to Lg . Recall that

k(2)

n and t(2) are the same for t and g along Lg . We determine k(2)

n and t(2) for surface

t using Eqs. (26.4.48) and (26.4.49), respectively.

Step 3: We consider at this stage of computation that for surface g the following

are known: k(1)

n and t(1), and k(2)

n and t(2) for two directions with tangents τ 1 and τ 2

that form the known angle µ (Fig. 26.4.8). Our goal is to determine angle q1 (or q2)

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26.4 Generation of a Surface with Optimal Approximation 765

Figure 26.4.7: Flowchart for optimization.

for the principal direction e

(g)

I and the principal curvatures k(g)

I and k(g)

II (Fig. 26.4.8).

Using Eqs. (26.4.48) and (26.4.49), we can prove that k(i)

n and t(i) (i = 1, 2) given for

two directions represented by τ 1 and τ 2 are related with the following equation:

t(1) + t(2)

k(2)

n − k(1)

n

= cotµ (26.4.50)