dFa () (Kac h() Kae )dz

(12)

dFx () dFt  cosdFr  sin 



dFy () dFt sin dFr cos



(13)

 dFz () dFa

The instantaneous forces can be expressed as follows:

ex

ex

ex

Fx   dFx , Fy    dFy , Fz   dFz

(14)

st

st

st

In addition, the resultant force is

F   (15)

2.4 Maximum and mean milling force or engagement angle in trochoidal machining

2.4.1 Maximum milling force or engagement angle

The maximum milling force and engagement angle are denoted as Ftro_max and atro_max. Because the engagement angle and milling force vary in a similar manner, it is only necessary to analyse the engagement angle. Fig. 6a shows that the line O1O2 intersects the circle O1 at point M. First, during the process of the cutter-circle rolling along the curve SE, the engagement angle will gradually increase and obtain a periodical extremum at point E. Then, the cutter-circle continues rolling along the arc EGH, and the engagement angle will gradually increase. When the cutter-circle passes through point M, the angle reaches its maximum (Fig. 6b); then, it gradually decreases.

Another particular case is when the point M is located within the cutter circle when the cutter circle is tangent to the point E. In this case, a maximum engagement angle occurs. In the subsequent rolling path of EGH, the engagement angle will decrease step by step.

2.4.2 Computing the variation curve diagram of milling force

For specific workpiece materials, if the milling parameters of the cutting tool are given, the method in Chapter 2.3 can be used to calculate the dynamic milling force for trochoidal machining. Corresponding to the EGH path in Fig. 6, Fig. 7 shows the dynamic milling force calculated under some parameters. The milling force first increases to the maximum gradually, and then decreases gradually. In Fig. 7, the two adjacent dots illustrate a rotation of the cutting tool. In this case, the cutting tool has

four edges, so there are four peak values between the two adjacent dots, respectively corresponding to the peak values obtained when the four edges cut in or out the workpiece. A series of peak points (F1, F2, … Fi, Fi+1 … Fn) for the first edge of the cutting tool during the machining can be connected to form a curve which is defined as the variation curve of milling force in this paper. The maximum of the series of peak points is defined as Ftro_max. In addition, the series of peak points can be averaged, thus to obtain an average value Ftro_ave, as below:

Ftro _ ave  (F1dl F2 dl Fi dl Fn dl) /(ndl) (F1  F2   Fi  Fn ) /(n)

(16)

Similarly, atro_max and atro_ave can be obtained from the variation curve of the engagement angle in Fig. 5. The four values, Ftro_max、Ftro_ave、atro_max and atro_ave, will be used in subsequent calculations and analysis.

2.5 Variation curve diagram of milling force and engagement angle

The large axial and small radial depths of cut are often adopted in cavity machining, which can provide improved use the cutting capacity of the cutting edges of the tool and allow more materials to be machined before the cutting tool becomes blunt. Usually, the radial depth of cut is smaller than 1/3 of the tool diameter, or even much smaller; while the axial depth of a cut may be larger than 1/3 of the tool diameter, or even much larger. In this context, carry out a calculation and comparative analysis for the variation curve of the trochoidal milling force and the variation curve of the engagement angle. Fig. 8 shows that for a cycle in trochoidal machining, we can first calculate each peak value of the milling force and then arrange them into a milling force curve. Second, we can calculate each changing engagement angle for trochoidal machining and then  arrange