O2 C

is set as b=[bx，by，0], and

bx  =(R2–r)*cosφ

by  =(R2–r)*sinφ

According to the vector dot product, cos∠E1CO1=a·b/(|a|·|b|)

2 2

=（ax*bx+ay*by）/[(ax2+bx2)+( ay   +by  )] (7)

∠E1CO1 can be acquired using the inverse cosine in the range between 0 and π.

Calculating the engagement angle   ∠DCE1

Obviously, the engagement angle satisfies the following geometric formula:

atro=∠DCE1=2π–∠E1CO1–∠DCO1 (8)

When the angle φ increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation.

(3) Determining processing limitations (Fig. 4)

First, the current offset circle (centred by O2) should intersect the previous offset circle (centred by O1). The limitations thereof should be studied; that is, determine whether the current offset circle is internally tangent or externally tangent with the previous offset circle (see Fig. 4a). The distance of O1 and O2 should meet the following condition:

R1–R2<L<R1+R2 (9)

Second, after the cutter travels through the previous offset circle and the current circle, no residual material should remain in their middle area. According to the geometrical relationship, the limitation should be as follows: O1, O2, and the cutter centre are in a straight line, and the tool circle is internally tangent with the current offset circle and externally tangent with the previous offset circle (see Fig. 4b). Therefore,

S=(L –R1)+R2<2*r

That is, the distance of O1 and O2  should  satisfy

L < R1–R2+2*r (10)

In most cases, the cutting width S is less than r to avoid large cutting loads and tool fatigue; therefore, the following inequality can be obtained.

L < R1–R2+r (11)

(4) Calculating and drawing the variation diagram of the engagement angle

A trochoidal path normally incorporates a few trip cycles. Each cycle is combined with one connected curve and one circle arc. By calculating the engagement angle for a

series of positions in a trip cycle, a variation curve diagram of the engagement angle can be drawn.

Fig. 5 includes several typical examples. The horizontal coordinate axis is the rolling length, and the vertical axis is the engagement angle (in degrees). When the ratios of r, R1, and R2 vary, similar results can be obtained.

Based on the curve diagram analysis, two conclusions can be drawn: (1) Increasing the centre distance of two neighbouring circles can increase the maximum value of the engagement angle of the trochoidal path. (2) Assuming that the cutter radius (r) and the trochoidal circle spacing (L) are certain, the maximum engagement angle will decrease with an increasing trochoidal circle size.

2.3 Modelling of milling force in trochoidal machining

Altintas et al. (1991) presented an efficient simulation system for milling mechanics by using tool path generation algorithms based on a solid modeler; they also experimentally validated their models. According to the mechanistic force model, the differential forces are defined in the tangential, radial, and axial directions as in Eq. (12). Then, they are transformed into x, y, and z coordinates as in Eq. (13)

dFt () (Ktch() Kte )dz



dFr () (Krc h() Kre )dz