2008
Dept. of Manufacturing Process & Automation Engineering Netaji Subhas Institute of technology Delhi University, New Delhi T.Sreekanth, Dr. S.K.Jha
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Surface Topography Generation of a diamond grinding wheel
Abstract: The paper presents the topographic model of an active surface of a diamond grinding wheel. The topography has been generated based on random tools and distribution of the grits. The main focus of this research has been grit geometry and its successful generation using matlab; so as to correlate to experimentally determined microtopography. Various models of grain geometry generated randomly are discussed and so is their distribution Keywords: simulation of topography, grit, interpolation
1. Introduction Computer simulation is a useful tool particularly in the field of manufacturing process. It allows us to shorten time of investigations and to reduce costs of changing or modification of existing technologies. The analysis of surface topographies after finishing processes is one of these methods. Computer generation of these structures makes forming of real surface easier. Surface quality is an integral contributor to overall product quality. The actual surface quality of finished product varies according to the chosen manufacturing process although the surface quality may be changed according to the product requirements. Amongst the machining processes grinding is most commonly used. In grinding, the surface quality depends on the grain/grit geometry, the kinematics of the grinding process and dynamics of the grinding system. The characterisation and measurement of grinding wheel surfaces is important owing to the dramatic influence that shape can have on grinding performance. Hence in this paper an attempt has been made to generate the topography by accurately modelling the abrasive particles, which are of special importance. We have concentrated on their geometry of the protruding parts because these parts of the grains are in contact with the workpiece. 2. Modelling of Abrasive particles In the past few decades there has been an increase in research on grinding due to its ever increasing application in manufacturing industry. Several statistical models have been developed to account for the random behaviour of the grinding process. In a model described by Baul [1], the emphasis was placed on grit heights and on spacing between grits along the wheel periphery. The peaks or asperities of the grinding wheel surface were represented by rods of varying height. The effect of variation in spacing between successive grits along the wheel periphery was not accounted for. The individual grit geometry was also neglected. In another model postulated by Yoshikawa [2], random numbers were used to determine the coordinates of the locations of the grains on the grinding wheel. A Monte Carlo simulation of the grinding process was conducted assuming a three dimensional –axial peripheral and radial distribution of the grain positions. McAdams [3], analyzed the profiles of abrasive surface of the grinding wheel by means of Markov chain theory. The Chapman-Kolmogorov equations together with recurrent event theory were used to deduce theoretical distributions for spacing between the active cutting points and lengths of lands on a worn grinding surface. Ghosh [4], in his M.S. thesis decribed the individual grits as cones of either the sharp cutting type or rounded edge grains. In most of the attempts to describe the geometry of the abrasive particles individual grits or grains have been considered only with single cutting edges. We take this into account by considering the model of each individual grain to consist of multiple cutting edges. Several models were made wherein each grain was modelled by successively displacing the volumes generated in either direction so as to form multiple cutting edges. Thus grains generated in this pattern resulted in a more accurate description of the surface topography. So essentially as shown in the picture each cone is displaced a certain distance δx and δz and this forms an individual grit.
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2.1. Modelling procedure using random tools For modelling individual grains, several programs were written in matlab which generated this geometry through random techniques. The method and code is briefly described below. The code written in matlab assumed that there is a surface function generated randomly using the symbolic toolbox from Mathworks wherein random functions were generated by using symbolic expressions in x and y. Such an approach imparts high degree of random base volume formation of the grain. The next procedure involved mainly generating the cutting edges on top of this surface. This was formulated by using 2 methods within matlab graphics. • Using linear interpolation • Using Delaunay triangulation Hence by this method described sufficient random nature with multiple cutting edges was imparted to the abrasive particle. While most parameters have been generated randomly the mean grit diameter has been assumed to be 90 from a finer mesh size of a TN 20 Cermet 100/80 grinding wheel. The grit height has been generated using normal distribution. The abrasive grain surface function can thus be summed up to be represented as Zs. The surfaces of grains are described by the function whose components determine the shape of grain fshape(x, y) and its cutting edge topography fctp(x, y). The elements of function are additive or multiplicative connecting (2.1): , , … … 2.1 an example of the randomly generated function[ fshape(x, y) and fctp(x, y ] is presented below. The models of the individual grits generated follow the discussion. |
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image generated using dealunay triangulation 3. Modelling of Grinding Wheel In the above discussion it has been shown how the individual grits have been generated now the topographic model is discussed. The random function shown above quite accurately generates various grit sizes and shapes which can be randomly patterned on the surface of the grinding wheel. The matlab code mainly involves a function to draw the grain which accepts a randomly generated number passed to it as a parameter. Depending upon the number passed to this volume render function it either generates a grit using linear interpolation or delaunay triangulation. Thus we see from the pictures shown above various grits can be successively patterned to create a topographic model of the diamond grinding wheel wherein it is carefully observed that both active cutting edges and blunt grits can be seen. For modelling the diamond grinding wheel the wheel can be divided into a number of discs as discussed by Ghosh [4], and then after the random generation of each grit it can be placed on sliced disc surface specified by polar coordinates. So these discs now successively patterned along the axis form the cylindrical wheel. Sieving determines the grit size of the abrasive grains. Since this method establishes the grit size the mean grit diameter and the height can be assumed to be more or less of the same order. The following assumptions made are briefly stated while generating the wheel topography: • The abrasive grains take the random shape as generated by the matlab code and also by deciding whether the volume rendering is through linear interpolation or through delaunay triangulation. • The grinding wheel is divided into a number of thinner discs or slices which follows a uniform distribution • The number of grits along each disc surface follows a gamma distribution. • The diameter of the grits is generated through a normal distribution. • From the discussion earlier the height of each grit is also normally distributed. Matlab also has powerful graphics capabilities amongst which the surf and griddata functions were extremely useful in generating the gird and the topography. The method of pattering the above grains was as stated using gamma distribution. A grid generated using matlab wherein the grain data within each cell formed on the grid consisted of no of grains.
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This was computed from the gamma distribution which generated the probability density function. Hence the entire topography was reduced to a simple matrix of the form
Thus the no of grains within each cell were computed by multiplying the prob density function with the interval value ij and each cell of the grid was selected by randomly generated integers i,j Nadolny & Balasaz [5]. This computed matrix could now be easily evaluated using matlab which has lot of surface plotting functions on matrices. A few of the models of the topography are generated below. Some of the models were interrupted during the generation procedure so as to compensate on processor power and time restraints hence only a part of the generated model is depicted in the pictures.
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4. Conclusions The model has been generated accurately focusing on the individual grain geometry. So far considered this is only one parameter which affects surface quality of the grinding process. The other parameters like presence of chatter and kinematics of grinding machine when taken into account would help produce a more accurate model of the grinding wheel topography.
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5. References [1] Baul, R.M. and Shilton, R., “mechanics of Metal Grinding with Particular Reference to Monte Carlo Simulation”, Proceedings of the Eighth International Machine Tool design and research Conference, 1997 [2] yoshikawa, H. and Sata, T., “Simulated grinding Process by Monte Carlo Method”, Annals of the CIRP, Vol 16,1998 [3] McAdams, H.T., “Markov chain Models of Grinding Profiles”, Trans. ASME, J. of Eng. For Ind., Vol 86, 1994 [4] S.Ghosh, “Modelling and Analysis of Grinding wheel process”, M.S. thesis Institute of Systems Research,1993 [5] K. Nadonly, B. Balasaz, “Modelling the surfaces of grinding wheels whose structure is zonally diversified”, Archives of Civil and Mechanical Engineering Vol 5, 2005
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