© Strojni{ki vestnik 50(2004)12,594-597 © Journal of Mechanical Engineering 50(2004)12,594-597 ISSN 0039-2480 ISSN 0039-2480 UDK 621.914 UDC 621.914 Pregledni znanstveni ~lanek (1.02) Review scientific paper (1.02) Nov pristop k prera~unu aritmeti~nega srednjega odstopanja profila pri kopirnem frezanju A New Approach to Calculating the Arithmetical Mean Deviation of a Profile during Copy Milling Jozef Peterka Hrapavost površine, kot posledica frezanja s krogelnim frezalom, je v strokovni literaturi in univerzitetnih učbenikih le redko opisana. Problem je pogosto poenostavljen. Kar pomeni, da so uporabljena poenostavljena razmerja za teoretične izračune hrapavosti površine pri kopirnem frezanju s krogelnim frezalom: računski parameter Rz in največja višina valovitosti profila. V tem prispevku je predstavljen izpopolnjen izračun, ne samo parametra Rz, ampak tudi parametra Ra, aritmetičnega srednjega odstopanja profila. V prispevku so predstavljene nove enačbe za neposredni izračun aritmetičnega srednjega ostopanja prečnih in vzdolžnih profilov pri kopirnem frezanju ravne površine in poševne površine. © 2004 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: frezanje kopirno, finiširanje, hrapavost površin, oblike proste) Surface roughness as a result of milling with a cylindrical ball-end cutter is determined in the technical literature, as it is in university textbooks, relatively infrequently. The problem is usually simplified, which means simplified relations for the calculations of theoretical surface roughness during copy milling with cylindrical ball-end cutters are introduced: the calculation of parameter Rz, and the maximum height of the undulation profile. In this contribution an improved calculation, not only for parameter Rz , but also for parameter Ra, the arithmetical mean deviation of the profile, is presented. The paper presents new equations for the direct calculation of the arithmetical mean deviation of the transverse and longitudinal profiles during copy milling on a plane (face) surface and an oblique surface. © 2004 Journal of Mechanical Engineering. All rights reserved. (Keywords: copy milling, finishing, surface roughness, free form surfaces) 0 PREFACE Reference [1] describes various possibilities for calculating theoretical surfaces roughness. The surface roughness can be evaluated using different parameters. Theoretically, it is best to calculate ten points for the height of the irregularities, Rz. But this parameter is not mentioned in a drawing. This function fulfils the arithmetical mean deviation of the profile Ra. An alternative is an empirically determined equation between Ra and Rz, but this equation is assigned in the large enough interval [2] (see Equation (3), it depends on other parameters too, mainly on the cutting conditions) and its sum arithmetic value does not need to match the actual relation between Ra and Rz during the copy milling using copy tools. The copy-milling tools are mainly use in the CAD/ CAM systems branch [3], in order to manufacture oblique surfaces and free-form surfaces [4]. 1 ROUGHNESS ON THE PLANE SURFACE This section will show the calculation of roughness for two cases: for parameter Rz [5] and Ra, and for the transverse and longitudinal roughness. 1.1 The parameter Rz on the plane surface Fig. 1. shows the situation for the origin of theoretical transverse roughness during copy milling on a plane surface. Equation (1) is valid from Fig. 1: Rz = R- 1J4R2-a2@ a 2V e 8R (1), 2 jgnnatäüllMliBilrSO | | ^SsFÜWEIK | stran 594 Peterka J.: Novi pristop k prera~unu - A New Approach to Calculating 2ß Zi 3^\ z2 R \V/ /^ D J u/'Zl J \m&^^r fz J r\a »^>^^ **^W»4W> (^ v2 ae Vl x ¦ Rz Fig. 1. The origin of the transverse roughness during copy milling on a plane surface where R is the tool radius ae is the stepover (path interval), or the same equation for parameter Rt is published in [6] : DC Rt= 2 D2 (2), where Dc is the tool diameter ae is the stepover (path interval). Next, we can calculate the Ra parameter from its dependence on Rz parameter using different empirical equations. The reference example, according to [2], is shown in the following empirical equation (3) for conventional machining (turning, milling, drilling): triangle S1 S2V and to the area of the circular segments VV2S2 a V1S1V, Equation (5): Ar=ae-R-R2sinßcosß- R2arcß (5). Next, it is possible to express from Fig.1. the area of the profile valley A that is bounded by the points Z1V1Z0. The calculation of this area requires Equation (6): Ap=12 R2(arc2^-sin2pz) (6). In the next step we need to present the area of the profile peak A that is butted and bounded by the points Z2VZ1. This area we can calculate using the following Equation (7): Ra @ Rz (3-5) (3). Av =Ar ae- z -R2 (arc2(pz - sin 2q>z (7). The selection of the denominator value from the interval (3 to 5) depends on the methods of cutting and on the other experimental cutting parameters. The question is can we calculate directly the arithmetical mean deviation of the profile Ra using the tool radius R and the stepover (path interval) ae (or feed per tooth fz)? The answer is yes. 1.2 The direct calculation of Ra on the plane surface In this section we will show the derivation for the direct calculation of the arithmetical mean deviation of the profile during copy milling on a plane surface. From Fig.1 the angle b is: The definition of the mean line of the profile needs to compare the area of the profile valley and the profile peak, A = A. After the substitution and editing we obtain Equation (8): a ¦ z = a ¦ R - a ¦ R cos ß - R 2 arcß (8) 2 and from equation (8) we can find the position of the mean line of the profile (see the Fig.1) of surface roughness: z = R - 1 Rcos b - R2 arcb (9) 2ae or: b = arcsin ae 2R z =R (4). 1 R 1 - cos ß-----arcß | (10). The area of profile Ar (butted and bounded with points V2VV1) between two surfaces of the ball-shaped miller above stepover ae will be the area of the rectangle V1V2S2S1 reduced to the area of the From the definition of the arithmetical mean deviation of the profile Ra we can calculate this deviation as the width of the strip (of length ae) on which it is possible to transform the double area of the profile valley (or the double area of the profile I isfinHi(s)bJ][M]ifln;?n 04 stran 595 I^HSSTTIMIDC Peterka J.: Novi pristop k prera~unu - A New Approach to Calculating peak, or the area sum of the profile valley and the profile, depending on which is the better to calculate). For the area of the profile valley we need the angle jz, which from Fig.1. is: For the arithmetical mean deviation of the longitudinal profile we substitute the stepover, ae, with feed per cutter tooth, fz, and the new equation is (16): R-z 1 a R a cosz - sin 2cpz) (16), f z (pz = arccos cos ß + — arcß | (12) 2 a and the parameter R a we designate from the following condition (13): Ra-a = 2A (13), after the substitution and editing we obtain the equation: R2 Ra = —(arc2