Moment to force ratios and the center of rotation

The purpose of this study was to investigate the relationship between moment to force (M/F) ratios and the centers of rotation by use of the finite element method (FEM). A three-dimensional FEM model was developed for the upper right central incisor on the basis of average anatomic dimensions. The center of resistance and centers of rotation were determined for varying M/F ratios applied at the midpoint of the crown. The center of resistance was located at 0.24 times the root length measured apical to the level of alveolar crest. The centers of rotation varied with the M/F ratios following a curve of hyperbola. The M/F ratio was -9.53 for root movement (Co at the incisal edge), -8.39 for translation, and -6.52 for tipping around the apex. It was found that even a small difference in the M/F ratios produced clinically significant changes in the centers of rotation.