The mechanics of asymmetric retraction loops used in fixed appliance therapy

In order to move a tooth bodily without tipping, both a force P and an anti-tilt couple N of a suitable constant magnitude are required. Although many different spring components are available to orthodontists for this purpose, a comparison of the merits of different designs has not been possible because analytical solutions to the deformation behaviour have only been generally available for the atypical case in which the loop is centrally positioned. The asymmetric case, with two redundancies, for open and closed U-loops both with or without helices has been solved using the complementary (strain) energy method on the assumption that the effect of a longitudinal tension or compression on the bending of a cantilever beam by lateral forces (the tie-bar effect) is small. Unless the arms are of equal length it is then found that unequal couples M and N act at the posterior and anterior ends of the appliance, and in addition equal but oppositely directed forces Q are induced at either end. Check measurements were carried out on enlarged model components using a special jig utilizing a mixed dead-load and strain-gauge system. Agreement with theory was generally within the known errors of measurement. The main characteristics of these retraction components in the clinical situation can consequently be predicted. The characteristics examined are the lateral and vertical stiffnesses (S and dQ/dy, respectively), the N/P and Q/P ratios, the self-righting ability (SRA) and dN/d beta, the change in anti-tilt couple per degree change in gable angle (the angle of the mesial arm with respect of a horizontal in the unactivated appliance).