More precise superfinishing by means of statistical modeling

Abstract

Superfinishing is the final stage in the manufacture highprecision parts that take the form of solids of rev� olution. The main goal of superfinishing is the increase the precision of the part's crosssectional profile. It is assessed in terms of noncircularity—in particular, oval distortion, faceting, and undulation of the surface. Pro� duction experience shows that faceting and undulation may be effectively reduced in most cases, whereas it is much more difficult to reduce oval distortion. In centerless superfinishing, stable reduction of oval distortion is impossible for the following reasons. (1) Centerless basing tends to retain the shape of the blank from preceding operations. Therefore, much of the resultant error consists of basing error, with par� tial copying of the initial shape errors and the forma� tion of new errors. (2) In a batch of blanks, the shape fluctuations are stochastic, and it is not always possible to identify one or two dominant harmonics of the profile. Therefore, setup of the machine tool on the basis of determinate models only ensures precision in the first approxima� tion. Thus, further improvement in the precision of cen� terless superfinishing entails creating statistical models of shaping for batches of blanks and developing meth� ods of machinetool setup on that basis. The mathematical model of basing in centerless superfinishing was considered in (1). However, certain simplifications in the formulation of the problem introduced considerable error in the calculation. In addition, random initial phases and correlations between the amplitudes of the harmonics significantly complicate the problem and call for other methods of solution. In the present work, we adopt a statistical modeling method—the MonteCarlo method (2). In machine� tool setup so as to minimize the basing error, this method involves simulation of the stochastic input data (the shape fluctuations of the blanks); repeated implementation of the analytical basing model; and the derivation of probabilistic characteristics whose numerical values agree with the results of determinate solution. In the end, we obtain a series of particular values of the basing error, whose statistical analysis reveals the influence of machinetool setup on the machining precision for the batch of blanks. The precision of the crosssectional profile is assessed in terms of the noncircularity, which indicates how much the actual profile differs from some basic circle. Most often, the basic circle adopted is the mean circle corresponding to the periphery of the part's cross section according to the leastsquares method (3). Then the blank's cross section with shape fluctua� tions may be described by a trigonometric polynomial in polar coordinates (4)