Two-dimensional matrix-based non-complex axis-of-rotation error modeling

.•2D matrix-based non-complex spindle error calculations and their bounds are shown.•Unidirectional resonances are improperly perceived as retrograde errors.•mth spectral components can produce effects at the m + 1 and m − 1 frequencies.•Vector-matrix integrals and discrete data calculations are given for determining the vector amplitudes of the rotating model.•An equation is given for determining 3D part errors on radially varying surfaces.

This work applies to the evaluation of a spindle as an integral part of a machine whose task is the measurement or modifying the shape or other parameter of a workpiece. This includes the co-influences of the spindle, its drive units and structural loop of the machine. First, the matrix-based two-dimensional simplified solution free of complex numbers is described. Second, bounds that encompass all possible radial error motions are determined from the matrix-based model. Third, it is shown that excited unidirectional resonances are improperly perceived as retrograde errors during evaluation due to the calculation and removal of reference ball eccentricity effects. Fourth, using a Fourier multi-dimensional spatial spectrum model, the stator referenced spindle error can result in a particular mth spectral component producing an effect at both the m + 1 and m − 1 spatial frequencies on the workpiece if the tool is mounted on the rotor. Fifth, it is shown how to directly calculate the (+ and −) mth characteristic rotating vectors. Sixth, a quick consideration will be given to estimating 3D part errors on radially varying surfaces such as but not limited to cones and spheres.