MILLING OF COMPLEX SURFACES

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Fiabilitate si Durabilitate - Fiability & Durability No 1/ 2018 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 167

MILLING OF COMPLEX SURFACES

Assoc. Prof. PhD. Eng.Evstati LEFTEROV1, Assist. Prof. PhD. Eng. Tanya AVRAMOVA2

1

Technical University of Varna, Bulgaria [email protected] 2

Technical University of Varna, Bulgaria [email protected]

Abstract: In the article are considered principle kinematic schemes of cutting for perform the machining of recessed and convex rotating surfaces. The mathematical dependencies determining the trajectory character described by the relative motion of the respective kinematic cutting scheme are presented. Of particular importance in performing of various principle kinematic schemes of cutting is the direction of motion of the tool relative to its rotation.

Key words: kinematical cutting schemes, milling, face milling, trajectory, cutting edge of a tool

1. Introduction

Milling tools can perform processing on kinematic schemes classified by Acad. D. I. Granovsky [1] from IV to VIII group. Fig. 1 shows the basic schemes of milling. There is clearly a need for analyzing of principle kinematic schemes of cutting (PKSC) described by the motions different from one rotational and one rectilinear motion, which allows processing only planar surfaces.

Fig.1 Milling schemes

1-milling; 2-duplex milling; 3-form milling; 4-hollow milling; 5-slot milling; 6-lathe milling; 7-thread milling; 8-grooving; 9-milling with big feed; 10-face milling; 11- oblique surface

milling; 12-helical milling; 13- circular interpolation; 14-trochoidal milling

2. Kinematic schemes of cutting

To perform the machining of recessed and convex rotating surfaces, PKSC with two uniform rotational motions can be used [2] (fig. 2 a)÷d)) A and B operating in the coordinate plane YOZ of the spatial coordinate XYZ system (fig. 2 a)). The following two variants are of interest to the practice [3, 6]:

 Variant I – the two equal motions A and B have the same direction and ratio between their angular velocities i =0,003

B A

 

 [3]. Furthermore, motion A belongs to the tool and the uniform rotational motion B of the workpiece. The two motions in the given direction can be imitated by the rolling of a circle on another one making contact.

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Fig.2 PKSC and trajectories of the relative work motion – Variant I

If point A lies on a circle with a radius r, then the trajectory described by it (fig. 2 b)) when rolling on a circle with a radius R will be the desired relative trajectory of a point of the cutting edge of the tool.

The nature of this trajectory is determined by dependencies:

 



-K.cos ;z=(R-r).sin +K.sin r).cos

-(R =

y , (1)

where: θ – the angle at which the center of the moving circle is moved relative to its initial position; α – the angle of displacement of point A with respect to the direction of the Y axis and fixing its position.

From fig. 2 b) it is found that: θ+α=ψ; α= ψ- θ. From the condition for rolling without sliding R.θ=r.θ or ψ=(R/r).θ. In this case, for angle α is valid the expression:

  . r r) -(R = (2) Then the equation of the trajectory of the relative work motion (TRWM) of a point of the cutting edge of the tool gets the type:

) . r r -R K.cos( -r).cos -(R = y   ) . r r -R K.sin( r).sin -(R = z   , (3) where: K – the distance from point A to the center of the moving circle

At K=r – the trajectory is hypocycloid (fig.2 b));

At К<r – the trajectory is shortened hypocycloid (fig.2 c)); At К>r – the trajectory is extended hypocycloid (fig.2 d)).

 Variant II – both regular and rotating motions A and B have opposite directions and the ratio between the angular velocities i=0,005 [3]. Motion B belongs to the tool, and motion A of the workpiece (fig. 3 a)). Both motions can be imitated as a rolling of two circles without sliding in the direction indicated and external contact.

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Fig.3 PKSC and trajectories of the relative work motion – Variant II

If point A lies on a circle with a radius r, then the trajectory described by it when rolling on a circle with a radius R will be the desired trajectory of the relative work motion of a point on the cutting edge of the tool. The nature of this trajectory is determined in the following way:     

r).cos -K.cos ;z=(R r).sin -K.sin (R

=

y (4)

The angles θ, α and the distance K have the same meaning as in the first variant. From fig.3 b) it was found that α = ψ + θ or ψ = α- θ. From the condition for rolling without sliding: R. θ=r. ψ, from which follows that   .

r r) (R

= .

The TRWM equation takes the form:

) . r r R K.cos( -r).cos (R = y     ) . r r R K.sin( r).sin (R = z     (5) At K=r – the trajectory is hypocycloid (fig.3 b));

At К>r – the trajectory is shortened hypocycloid (fig.3 d)); At К<r – the trajectory is extended hypocycloid (fig.3 c)). Using this variant helps to machining rotational surfaces.

At present, most machining of CNC machines is performed according to the kinematic scheme shown in fig. 4 a) [5], which is practically 701 PKSC from the classification of G.I. Granovsky [1]. This PKSC is composed of two uniform rotational motions A and B and a rectilinear uniform motion C. Depending on who performs these motions, the tool or the workpiece, the following two variants are of practical interest:

 Variant I – the uniform rotational motion B belongs to the tool, and the motions A and C to the workpiece. Moreover, A and B have opposite directions. In this case, the uniform rotational motion B determines the main motion of the cutting, A - the feeding, and C is an

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auxiliary one. The ratio of peripheral speeds is recommended 0,005 V V = B A   [2], because there is no kinematic connection at the CNC machines.

Fig.4 701 PKSC and two variants of trajectories of relative working motion of a point on the cutting edge

When considering 701 PKSC [1] establishes that it is a combination of 501 PKSC [1] and one rectilinear uniform motion acting on the X axis of the spatial coordinate system XYZ. In this case, the equation of the relative working motion of the point on the cutting edge of the tool will be identical to that of the extended epicycloid (where К>r), described in a coordinate plane YOZ, and the coordinate of the point of this trajectory, obtained by the action of a uniform rectilinear motion С. If α is denoted as angle of inclination of the screw line, the mathematically sought equation of the relative trajectory will look like this:

 .tg -r. = x ) . r r R K.cos( -r).cos (R = y     (6) ) . r r R K.sin( r).sin (R = z    

This is a spatial epicycloid equation described by a screw line on the surface of a cylinder at an angle α relative to X axis of the XY coordinate system (fig. 4 b)). With this trajectory, screw grooves can be milled on a cylindrical surface if it is satisfied the ratio ε = 0,005.

 Variant II – motions A and B have the same direction. The rotational A and rectilinear C uniform motions belong to the tool and the uniform rotational motion B of the workpiece. The peculiarity of this variant consists in the fact that neither rotational motions A and B are not identified with the motion of cutting. The latter is the result of the two elementary rotational motions A and B. In this case C is a feeding motion. At К<r and

2

B

A 

 

the equation of the trajectory of the relative working motion at a point of the cutting edge of the tool is a screw shortened hypocycloid with an elliptical outline (fig.4 c)):

r r R ; .tg . -r = x '   ' 

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Fiabilitate si Durabilitate - Fiability & Durability No 1/ 2018 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 171 ) . r r R K.cos( -r).cos (R = y     (7) ) . r r R K.sin( r).sin (R = z    

One of the possibilities of this variant allows development of methods of processing of polyhedral parts with a practically precise shape.

3. Direction of milling and incision characteristics

Of particular importance in performing of various PKSC is the direction of motion of the tool relative to its rotation [4]. There are two ways of working – forward and backward (fig.5).

Fig.5 Milling type

a) climb (down) milling; b) conventional (up) milling

The choice of the diameter of the milling tool is usually made on the basis of the workpiece parameters, the gauge of the track to be machined and the available machine power. From this point of view, there are three types of milling:

 when the workpiece width is bigger than or equal to the diameter of the milling cutter. In this case, thin chips (fig. 6 a)) are removed during incision and exit of the milling cutter;

 when the diameter of the milling cutter is relatively bigger than the width of the workpiece (fig. 6b);

 when the diameter of the milling cutter is considerably bigger than the treated area of the workpiece and the axis of the milling cutter is outside the width of the workpiece (fig. 6 c)).

Fig.6 Milling type according to the dimensions of the workpiece

In the case of milling, the critical moment for the operation of the individual inserts is the impact stress at incision, of which depends the cross-section of the cut metal layer. The positioning of the inserts is shown in fig. 7 a). In terms of attacking the workpiece blank, the two ways shown in fig. 7 a) directly affect the geometric parameters of the tool [6, 7, 8]. Fig. 7 b) shows the various steps of entering and exiting the tool in the workpiece.

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Fig.7 Position of the cutting edges when entering and exiting the work area

The above mentioned features must be considered in all cases when working on certain PKSC.

4. Conclusions

4.1. The analysis of PKSC provides specific information on the shaping of individual surfaces of the workpiece, and allows to recommend the most appropriate strategy for production.

4.2. The design of the shape and the number of cutting parts of the tool are indissolubly linked to the selected PKSC.

4.3. Choosing PKSC with more than two elementary motions significantly increases the productivity of the milling process.

4.4. When building a particular method of treatment should be determined geometric parameters of the tool in accordance with the nature of the trajectory of the relative work motion. References [1] Грановски Г. И., Кинематика резания, Maшгиз, Москва, 1948г., p.150 [2] Грановский Г.И., Грановский В.Г., Резание металлов, Высшая школа, Москва, 1985, p.304 [3] Лефтеров Е., Оптимални методи и средства за механична обработка, ИЦ на РУ „Ангел Кънчев―, Русе, 2013 [4] Лефтеров Е., Режещи инструменти, Лазаров Дизайн ЕООД, Варна, 2017, p.482 [5] Симеонов Н. Г., Лефтеров Е. Л., Ефективност и приложимост на високоскоростното фрезоване, сп. МТТ, Печатна база на ТУ-Варна, 2014, ISSN 1312-0859, p. 69-75 [6] Симеонов Н. Г., Лефтеров Е. Л., Материал и геометрия на режещи инструменти за високоскоростно фрезоване, сп. МТТ, Печатна база на ТУ-Варна, 2014, ISSN 1312-0859, p. 64-69

[7] Rao V.S., Rao P.V.M., Modelling of tooth trajectory and process geometry in peripheral milling of curved surfaces, International Journal of Machine Tools & Manufacture, Volume 45, Issue 6, Elsevier, May 2005, p. 617-630

[8] Wei Zh., Wang M., Ma R., Wang L., Modeling of process geometry in peripheral milling of curved surfaces, Journal of Materials Processing Technology, Volume 210, Issue 5, Elsevier, 2010, p.799-806