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The radius of the previous offset circle is set as R1, the radius of the current offset circle is set as R2, and the tool radius is set as r. Obviously, |CD|=r，|CE1|=r，
|O1D|=|O1S|= R1, |O2E|= R2, R1=rtro1+r, R2=rtro2+r.
To avoid a tedious calculation at the circular intersection, a geometric modelling method for the engagement angle in trochoidal machining is proposed as follows. It mainly includes two sections (Figs.3 b-c).
(1) Geometric modelling on the first section SE
Calculating angle ∠E1CO1
Fig. 3b shows that the cutter rolls along the curve SE and that the tool centre trajectory is S’E’. The coordinates of the cutter centre C(xc,yc) can be computed using
the tangency point E1 and vector
E1C .
The length of CO1 is
CO1
In addition, the length of E1O1 is
E1O1
Obviously, from the geometric relationship, ∠E1CO1 can be calculated by the following formula:
cos E1CO1
( CO1
CE1
E1O1
) /(2 * CO1 * CE1 )
(2)
( CO1
r 2
E1O1
) /(2 * CO1 * r)
Calculating the angle ∠DCO1
∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0<∠DCO1<π. It can be obtained using the cosine theorem:
cos∠DCO1=(|CO1|2+r2–R12)/(2*|CO1|*r) (3)
Calculating the engagement angle ∠DCE1
Obviously, the engagement angle ∠DCE1 satisfies the following geometric formula:
atro=∠DCE1=2π–∠E1CO1–∠DCO1 (4)
When SE1 increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation.
(2) Geometric modelling of the second section EGHE
Establishing the trajectory equation for tool centre
The moving trajectory of the cutter centre C is the arc centred by O2 (xO2, yO2) with radius R2-r. Therefore, the following equations for the variation in C(xc, yc) may be established.
xc=xO2 +(R2–r)*cosφ
yc=yO2 + (R2–r)*sinφ (5)
Calculating the angle ∠DCO1
Obviously, ∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0<
∠DCO1<π.
From the geometric relationship, |CD|=r，|O1D|=R1 are known, and the length of CO1 is h2=( xc–0)2+( yc–0)2,
According to the cosine theorem,
cos∠DCO1=(h2+r2–R12)/(2*h*r) (6)
The angle ∠DCO1 can be obtained by the inverse cosine.
Calculating the angle ∠E1CO1
The angle ∠E1CO1 may be calculated as the vector dot product of
O2 C
and
CO1 .
However, because ∠E1CO1 is likely to be larger than π (see Fig. 3c), the relationship of the vector cross product is required to identify whether ∠E1CO1 is greater than or less than π.
The vector quantity
CO1 is set as a=[ax，ay，0], and
ax = –xO2 – (R2–r)*cosφ
ay = –yO2 – (R2–r)*sinφ
The vector quantity
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