The Geometrical Model of Calculation of Metal Removal and Roughness in Magnetical-Abrasive Machining

The geometrical model of interaction between abrasive particle and the surface in magnetic-abrasive machining is offered. The theoretical research of metal removal is made. The dependences for determination of metal removal and maximum depth of particle introduction are received.

The scheme of single grain material removal in MAM On the basis of the adopted design scheme and the above assumptions, the amount of material removed (1) where k c -chips coefficient, which is equal to the actual area of removed metal considering elasticplastic deformation of machined material to geometric area, L a chips length. The depth of grain forcing into the metal [2,3,6] where F n -normal force, H v -BHN. Normal force is equal to P magnetic pressure multiplied by the grain cross sectional area: ( La chip length in mm is determined by n inductor rotation frequency (rpm), t п (с) polishing time and the width of inductor and workpiece contact zone. Supposing that the magnets are disposed along the generatrix of the inductor, and the width of contact zone is equal to B core width (Fig. 3), the total chip length for one grain is determined by: where n -the frequency of rotation(rev / min).  Fig.3 The scheme of workpiece machining.
To determine the amount of grains involved in cutting we denote the diameter of magnetic abrasive particles З а , and core area where n М -core number. Then the number of working magnetic-abrasive particles is determined by the formula: Supposing, that n a cutting abrasive particles are attached to one magnetic particle we get the total amount of cutting grains: The total amount of removed material (7) To consider the dynamic phenomena during machining we offer the additional coefficient k d . It allows to consider the deformation in the tool and workpiece contact zone. This is due to the fact that during the extensive flat surface machining cutting force causes a considerable deformation of both the tool and the magnetic-abrasive powder. And the metal removal is reduced. The coefficient k Д is always less than 1, but it approaches 1 when the surface significantly smaller than core width B is processed. This coefficient depends on a large number of factors. Therefore for it empiric dependence is offered. Over time, the cutting properties of grains go down because of their bluntness and destruction. So, we propose to use one more empirical coefficient k I (τ) in the formula . To consider the process of blunting the following empirical dependence is used. (9) After preliminary cutting marks on the prepared surface look like Fig.3.
Since the profile is periodic, the surface roughness parameters are actually determined by the area We assume that in this area the mark profile is set by the equation b ax y = .
Then the surface roughness parameters is determined by coefficients a and b. The mark height corresponds to R max parameter. Herewith Ra parameter is determined by the formula During processing from workpiece surface having the profile of Fig.3, allowance is removed. Herewith, the volume of the material that is removed can be calculated taking into account the expression (9). Metal is removed from mark tops. Herewith, after some time they take the form shown in Fig. 4.

Figure4-Mark profile during machining.
Since the profile of original mark is described by degree dependence, parameters S z и b p are related by: Dimensions Δ and b p are determined by metal removal process.
where S -feed in mm / rev, n -inductor rotational speed in rev /min. The volume of removed metal in the first pass On the other hand, this amount is determined by dependence (10) where Δ is calculated with the formula (16). Equation (14) can be solved for b Р only numerically, excluding trivial solution b Р = 0. Fig. 5showsthe dynamics of roughness parameter changes Ra from pass to pass when feeding S = 0.05mm / rev. The program for performing calculations is designed in VBA in MS Excel. The calculation time on modern computers is less than 1 second. Results and discussion. The proposed program allows to calculate the metal removal and the surface roughness in magnetic-abrasive machining, depending on the chosen parameter of machining. Conclusions. The dependences derived from modeling can help to accomplish some vital engineering tasks dealing with the workpiece finishing process. They can be used to determine optimal sizes of abrasive powder portions participating in the process of magnetic abrasive machining, an appropriate technical rate setting, the most effective trajectory and operation modes. Using the obtained results a production manager can assess a potential effect of some process related parameters (e.g. an inductor radius, the radius of the powder portion, the machine table feed, the rate of the inductor rotation, etc.) on the production efficiency. The model presented above is of great scientific value because it can be used for parameter prediction for machining sophisticated flat workpieces when machining conditions are not constant and have to be readjusted due to the peculiarities of technical characteristics of the magnetic abrasive machining process.