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  Geometry: Size and Shape Changes related to size and shape are independent of the material. In other words, decreasing the diameter of a steel beam by 50%would reduce its strength to a specific percentage of what it had been previously (the exact reduction would depend on how the beam was supported, as we discuss below). Decreasing the diameter of a similarly supported TMA beam by 50%would reduce its strength by exactly the same percentage as the steel beam. But keep in mind that the  performance of a beam, whether beneath a highway bridge or between two teeth in an orthodontic appliance, is determined by the combination of material properties and geometric factors. Cantilever Beams When a round wire is used as a finger spring, doubling the diameter of the wire increases its strength eight times (i.e., the larger wire can resist eight times as much force before permanently deforming or can deliver eight times as much force). Doubling the diameter, however, decreases springiness by a factor of 16 and decreases range by a factor of two. More generally, for a round cantilever beam, the strength of the beam changes as the third power of the ratio of the larger to the smaller beam; springiness changes as the fourth power of the ratio of the smaller to the larger; and range changes directly as the ratio of the smaller to the larger (Figure 9-12). FIGURE 9-12 Changing the diameter (d) of a beam, no matter how it is supported, greatly affects its properties. As the figures below the drawing indicate, doubling the diameter of a cantilever beam makes it 8 times as strong, but it is then only as springy and has half the range. More generally, when beams of any type made from two sizes of wire are compared, strength changes as a cubic function of the ratio of the two cross-sections; springiness changes as the fourth power of the ratios; range changes as a direct proportion (but the precise ratios are different from those for cantilever beams).    As the diameter of a wire decreases, its strength decreases so rapidly that a point is reached at which the strength is no longer adequate for orthodontic purposes. As the diameter increases, its stiffness increases so rapidly that a point is reached at which the wire is simply too stiff to be useful. These upper and lower limits establish the wire sizes useful in orthodontics. The phenomenon is the same for any material, but the useful sizes vary considerably from one material to another. As Table 9-2 indicates, useful steel wires are considerably smaller than the gold wires they replaced. The titanium wires are much springier than steel wires of equal sizes but not as strong. Their useful sizes therefore are larger than steel and quite close to the sizes for gold. Geometry: Length and Attachment Changing the length of a beam, whatever its size or the material from which it is made, also dramatically affects its properties (Figure 9-13). If the length of a cantilever beam is doubled, its bending strength is cut in half, but its springiness increases eight times and its range four times. More generally, when the length of a cantilever beam increases, its strength decreases proportionately, while its springiness increases as the cubic function of the ratio of the length and its range increases as the square of the ratio of the length. Length changes affect torsion quite differently from bending: springiness and range in torsion increase  proportionally with length, while torsional strength is not affected by length. Changing from a cantilever to a supported beam, though it complicates the mathematics, does not affect the big picture: as beam length increases, there are  proportional decreases in strength but exponential increases in springiness and range. FIGURE 9-13 Changing either the length of a beam or the way in which it is attached dramatically affects its properties. Doubling the length of a cantilever beam cuts its strength in half but makes it 8 times as springy and gives it 4 times the range. More generally, strength varies inversely with length, whereas springiness varies as a cubic function of the length ratios and range as a second power function. Supporting a beam on both ends makes it much stronger but also much less springy than supporting it on only one end. Note that if a beam is  rigidly attached on both ends, it is twice as strong but only one-fourth as springy as a beam of the same material and length that can slide over the abutments. For this reason, the elastic properties of an orthodontic archwire are affected by whether it is tied tightly or held loosely in a bracket. A removable appliance incorporating a cantilever spring for initial tipping of a maxillary canine toward a premolar extraction site. Note that a helix has been bent into the base of the cantilever spring, effectively increasing its length to obtain more desirable mechanical properties. Before beginning to discuss control of root position, it is necessary to understand some basic physical terms that must be used in the discussion:    ã  Force  —  a load applied to an object that will tend to move it to a different  position in space. Force, though rigidly defined in units of Newtons (mass × the acceleration of gravity), is usually measured clinically in weight units of grams or ounces. In this context, for all practical purposes, 1.0 N = 100 gm (the actual value is between 97 and 98 gm).    ã Center of resistance  —  a point at which resistance to movement can be concentrated for mathematical analysis. For an object in free space, the center of resistance is the same as the center of mass. If the object is partially restrained, as is the case for a fence post extending into the earth or a tooth root embedded in bone, the center of resistance will be determined by the  nature of the external constraints. The center of resistance for a tooth is at the approximate midpoint of the embedded portion of the root (i.e., about halfway between the root apex and the crest of the alveolar bone; Figure 9-17).    ã  Moment   —  a measure of the tendency to rotate an object around some point. A moment is generated by a force acting at a distance. Quantitatively, it is the product of the force times the perpendicular distance from the point of force application to the center of resistance and thus is measured in units of gram-millimeter (or equivalent). If the line of action of an applied force does not pass through the center of resistance, a moment is necessarily created.  Not only will the force tend to translate the object, moving it to a different  position, it also will tend to rotate the object around the center of resistance. This, of course, is precisely the situation when a force is applied to the crown of a tooth (see Figure 9-17). Not only is the tooth displaced in the direction of the force, it also rotates around the center of resistance  —  thus the tooth tips as it moves.    ã Couple  —  two forces equal in magnitude and opposite in direction. The result of applying two forces in this way is    FIGURE 9-17 The center of resistance (C R  ) for any tooth is at the approximate midpoint of the embedded portion of the root. If a single force is applied to the crown of a tooth, the tooth will not only translate but also rotate around C R   (i.e., the center of rotation and center of resistance are identical) because a moment is created by applying a force at a distance from C R  . The perpendicular distance from the point of force application to the center of resistance is the moment arm. Pressure in the periodontal ligament will be greatest at the alveolar crest and opposite the root apex (see Figure 8-9). a pure moment, since the translatory effect of the forces cancels out. A couple will produce pure rotation, spinning the object around its center of resistance, while the combination of a force and a couple can change the way an object rotates while it is being moved (Figure 9-18). ã Center of rotation  —  the point around which rotation actually occurs when an object is being moved. When two forces are applied simultaneously to an object, the center of rotation can be controlled and made to have any desired location. The application of a force and a couple to the crown of a tooth, in fact, is the mechanism by which bodily movement of a tooth, or even greater movement of the root than the crown, can be produced.
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