 Llow: Ftro max=Fstr; Calculate a proper centre spacing of trochoidal circles to   make

the maximum milling force equal to the milling force Fstr in the contour-parallel cutting; this spacing is defined as Llow.

 Lmid:  Ftro  ave   =Fstr;  Calculate  a  proper  centre  spacing  of  trochoidal  circles  to

make the average milling force equal to the milling force astr in the contour-parallel cutting; this spacing is defined as Lmid.

 Lhigh: Ftro ave =1.5Fstr;

Four areas can be delineated according to the three values above (as shown in   Fig.

10):

Area 1: The machining is safe, and the cutting tool is subject to relatively small forces and obtains a longer service life, but the milling efficiency is reduced.

Area 2: The machining is relatively safe, the tool wear is reduced, and the tool load is greater than that of the machining mode in contour-parallel cutting.

Area 3: The cutting tool load is larger, and the possibility of fatigue for cutting tool increases. However, the machining efficiency will also be improved.

Area 4: The machining efficiency is higher, but the cutting tool load is larger; moreover, the tool can be easily damaged, and tool wear is more serious.

Obviously, in general, Area 2 or Area 3 should be chosen in high-speed cavity milling. Area 3 can be chosen to improve the milling efficiency if the strength of the cutting tool permits.

To obtain better control, we can set the values of t1 and t2 (for example t1=10% and t2= 50%) and calculate a proper centre spacing of neighbouring trochoidal-circles to make the average milling force in trochoidal milling satisfy

Ftro_ave≤Fstr * (1+t1) (17)

or the maximum engagement angle satisfy

Ftro_max≤Fstr* (1+t2) (18)

Sometimes the engagement angle a (astr、atro、atro_max) can be used as a substitute for the milling force (Fstr、Ftro、Ftro_max) to simplify the calculation.

(4) According to the geometry features of the cavity, corner, and narrow slot, compute the trochoidal path in the material gathering area. The corresponding algorithm will be shown in Part 4.

4. Trochoidal machining for cavity

4.1 Geometry description

Contour-parallel tool paths represent a common high-speed milling method for cavity moulds. In this paper, the contour-parallel cutting approach adopts the milling mode from inside to outside. The path elements involved are mainly arcs and linear segments. The following analysis will focus on the line-line corner type, and other corner types composed of complex curves can be deduced accordingly.

Fig. 11 shows the tool path of contour-parallel cutting. Assuming that the current trajectory is U and V, the trajectory formula can be expressed as

U:   A1 x B1 y C1  F1t1   0

V: A2 x B2 y C2  F2t1   0

(19)

where F1 and F2 refer to the offset director and take on the value of 1 or -1; the coefficients of x and y are normalized.

The previous trajectory with the offset distance d is

PU:

A1 x B1 y C1  F1 (t1  d ) 0

PV:

A2 x B2 y C2  F2 (t1  d ) 0

(20)

The bisector of two geometric curves is the trajectory in which each point is equidistant to both curves. As mentioned by the paper of Persson (1978), the bisector of the corner may be a straight line, parabola, or hyperbola. Calculating the bisector is an important step in establishing the trochoidal path. Let the curves U and V be offset by an amount equal to h. The offset curves intersect at a point M, which is a point on  the