Herbert A. Koenig

Force systems from an ideal arch

T he force systems delivered from commonly used orthodontic appliances are relatively unknown. It is little wonder that unpredictable and many times undesirable tooth movement is produced during treatment. In the more sophisticated orthodontic appliances, the force system is produced totally or in part by placing a wire with a given configuration into a series of attachments (brackets, tubes, etc.) on the teeth. In an attempt to determine the force system, orthodontists in the past have used force gauges to measure the amount of force required to seat an arch wire in a bracket. IJnfortunately, this bit of information is inadequate to describe the force system completely in most clinical applications, since the situation is statically indeterminate; in other words, there are too many unknowns to calculate the forces from an appliance using the laws of statics. Clinically, such measurements represent little more than pseudoscience, since they incompletely describe the physical realities and, hence, will not predict the biologic response and the nature of the tooth movement to be expected. The purpose of this article is threefold: (1) to describe the force system which is produced when a straight wire is placed in a nonaligned bracket produced by a malocclusion ; (2) to develop the terminology and the approach to solve and describe force systems from all appliances; and (3) to offer a scientific basis for developing the orthodontic appliances of the future. To reach these objectives, the simplest clinical situation will be considered-the placing of a straight wire in two attachments on two teeth.

Optimizing anterior and canine retraction.

Vertical loops or modified vertical loops are basically frictionless springs which are used for canine and anterior tooth retraction. The design and selection of a proper loop or retraction spring should be based on a number of scientific criteria. Foremost among these would be a sufficiently high moment-to-force ratio so that root apices are not displaced mesially or anteriorly. A retraction spring with zero angulation of its horizontal-occlusal arms delivers a moment when activated to produce a force. The ratio of this moment and force is constant throughout the elastic range of activation of the spring. The higher the moment-to-force ratio, the greater is the clinicians control over the apices of the anterior teeth. An analysis of design factors demonstrates that the higher the loop occluso-gingivally, the shorter its horizontal length occlusally, and the greater the gingival horizontal length as in a T loop; these are significant factors in increasing the moment-to-force ratio. The placement of helices is a useful design consideration but the main effect is in reducing the load-deflection rate. By keeping these design factors in mind, the clinician can build into his retraction springs, without the placement of any gable bend, the largest possible moment-to-force ratio so as to optimize his tooth movement. Although it may be possible to design retraction springs to deliver an adequate moment-to-force ratio for controlled tipping around the apex of an incisor or a canine, translatory movements are not possible, considering the intraoral limitations on spring height. This can be overcome by the placement of gable bends or angulation in a vertical loop or retraction spring. Unfortunately, with the typically used high-load-deflection-rate vertical loops, activation to achieve the desired moment-to-force ratio is too critical, exacting, and changeable with small displaced movements of the tooth. This can be partly overcome by utilizing designs that have not only the highest possible moment-to-force ratio during pure horizontal activation of their arms but low-load deflection rates as well. Because of the low load-deflection rate, moment-to-force ratios are relatively more constant if a gable bend (angulation) is placed. The science of spring design as applied to the problems of canine and anterior tooth retraction in this article allows the clinician to optimize the design of his retraction springs. More important, with properly designed springs, it allows him to estimate with relative accuracy the force systems produced and to avoid undesirable side effects which might not have been apparent from superficial observation.

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.

Three-dimensional large displacement analysis of orthodontic appliances

Abstract An incremental technique for predicting the nonlinear load-displacement characteristics of orthodontic appliances is developed. The analysis is based upon a three-dimensional finite-difference formulation which permits the treatment of appliances that have arbitrary configurations and non-uniform material and cross-sectional properties. The effects of intermediate loading and attached springs may also be studied. The results show the activated configurations and the associated force-couple systems for three appliances that are activated to the yield point. The incremental analysis results are compared to those of the linear solution for each case.

Precision adjustment of the transpalatal lingual arch: Computer arch form predetermination

Correction of widths and axial inclinations with a transpalatal lingual arch has the advantage of distributing forces across the arch, thus minimizing undesirable side effects. The proper shape of an arch to deliver the required force systems for both unilateral and bilateral width change was determined by using an analytical approach which enabled the deactivated shape of the arch to be established and drawn by computer. The transpalatal lingual arch is extremely sensitive to shape in producing a force system. The critical nature of this shape requires the clinician to understand the over-all pattern of properly forming the arch for various applications and the clinical procedures for evaluation before final insertion. The deactivated shape and the force systems for representative applications are presented; these differ significantly from the concept of the ideally shaped wire.

Analysis of generalized curved beams for orthodontic applications

Abstract The generalized field equations for an arbitrary curved and twisted beam in space are presented and discussed. A numerical algorithm is generated from these equations yielding a computer code which is capable of determining the static, dynamic and free vibration characteristics of a non-uniform, non-homogeneous, anisotropic beam which has arbitrary curvature and arbitrary twist. The static portion of this analysis is shown to be directly applicable to the study of force systems delivered by simple and complex orthodontic appliances. Several orthodontic configurations are studied using the subject model. Clinical considerations are discovered as a result of this study.

Force systems from an ideal arch--large deflection considerations.

A sophisticated mathematical simulation is presented which allows for the consideration of large activations in orthodontic appliances and their effect upon the resulting force systems which are delivered to teeth. Effects of bracket/wire interaction are studied using this new tool. Previous studies of force systems from an ideal arch were redone with the new analysis in which the wire was either rigidly restrained or free to slide. The restraint of the wire produced large mesio-distal forces and increased the magnitude of the moments on each bracket. If the wire is free to slide, both large deflection and small deflection solutions give similar results. The relative force system M1/M2 fundamentally held true with large deflections and restraint; however, some differences were noted. The significance of allowing wire to slide in the bracket is discussed.

Dynamical finite element analysis for elastic waves in beams and plates

Abstract A special method of analysis, hereafter referred to as “Direct Analysis”, is described and applied to the solution of the problem of traveling flexural waves in beams and plates, for which shear correction and rotatory inertia are considered. Finite beams and plates are considered so that the influence of reflected waves is included. The proper boundary conditions for these problems, several input stress pulses, as well as the effect of the length of the bounded medium (beam or plate) on the magnitude of the stress are considered. Implications to design of structures are discussed. The characteristic features of the Direct Analysis are then presented.

Prediction of edge stresses in layered media using the surface integral-finite element technique

Abstract A method for predicting the state of stress in a finite body consisting of two isotropic material layers, including a characterization of the singular stress state at the intersection of the interface with a stress-free boundary, is presented. The prediction of the “free edge stress” is accomplished using elastic stress intensity parameters. Arbitrary two-dimensional geometries with mechanical and thermal loading and plane strain or plane stress behavior are addressed. The approach consists of coupling the finite element solution of two unattached layers to a singular integral representation of distributed dislocations along the interface of two semi-infinite layers. Coupling of the two methods occurs along the interface, where displacement compatibility is enforced, and at the finite boundary, where correction forces are defined to eliminate the tractions from the semi-infinite dislocation solution. The stress intensity factors at the free edge interface are derived from the singular integral solution.

Non-linear formulation for elastic rods in three-space

Abstract Six one-dimensional deformation measures are defined to describe the state of deformation of a naturally curved and twisted rod experiencing large displacements and rotations. In terms of these deformation measures six exact kinematic equations and six equations of equilibrium are formulated. A general elastic deformation functional of the deformation measures is proposed as the necessary link between the two sets of equations.