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44 Silviu Berbinschi et al
⎛x⎞ ⎛X⎞ ⎜ ⎟ ⎜ ⎟
⎜
y
⎟
= 1 1 ⋅
⎜
Y
⎟
⎜z⎟ ⎜Z ⎟
⎝ ⎠ ⎝ ⎠
T
(5) ω ( ϕ )
⎛X⎞ ⎜ ⎟
where
⎜
Y
⎟
is the matrix of the current point in space XYZ, and
⎜Z⎟ ⎝ ⎠
⎛1 0 0 ⎞
⎜ ⎟
= ⎜ ⎟
⎜ ⎟
(6) ω1( ϕ1) 0 cos( ϕ1) sin(
ϕ1)
0 −sin(
ϕ ) cos(
ϕ )
⎝ 1 1 ⎠
is the rotation transformation matrix, around X axis, with angle ϕ1
(counterclockwise rotation).
The assembly of equations (3) and (6), with the respect of rolling condition
(1), determine the relative motion,
⎛ξ⎞ ⎛1 0 0 ⎞ ⎛X⎞ ⎛ 0 ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟
= ⎜ ⎟⋅
⎜
Y
⎟
−⎜ −λ
⎟
⎜ ⎟ ⎜0 −sin 1 cos ⎟ ⎜
1 Z ⎟ ⎜−R⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ rp ⎠
(7) η 0 cos( ϕ1) sin(
ϕ1)
ζ ( ϕ ) ( ϕ )
while the Σ profile, belongs to the profile’s ordered curl, associated with the C1
centrode, in form,
⎛ 0 ⎞
⎜ ⎟
Σ= ⎜ ⎟
⎜ ⎟
⎝ ⎠
(8) Y( u)
Z ( u)
with u variable parameter, describe a profile’s family in the rack-gear space:
(9) ( )
( u,
1 ) = 0;
( , 1) ( ) cos( 1) ( ) sin(
1)
( , ) ( ) sin( ) ( ) cos(
)
ξ ϕ
u ϕ1
Y u Z u Rrp
1
u Y u Z u R
;
Σ η ϕ = ϕ − ϕ + ⋅ ϕ
ζ ϕ = ϕ + ϕ − .
1 1 1
The enveloping of the profile family (9) represents the rack-gear tool’s
profile.
rp