Geometry Ontemporary

8/10/2019 Geometry Ontemporary

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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 woulddepend 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 wireincreases 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 anddecreases range by a factor of two.

More generally, for a round cantilever beam, the strength of the beam changes asthe 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 changesdirectly 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).
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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. AsTable 9-2indicates, 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 ismade, 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 itsspringiness 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 torsionquite differently from bending: springiness and range in torsion increaseproportionally with length, while torsional strength is not affected by length.

Changing from a cantilever to a supported beam, though it complicates themathematics, does not affect the big picture: as beam length increases, there are

proportional decreases in strength but exponential increases in springiness andrange.

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
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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 thecantilever spring, effectively increasing its length to obtain more desirable mechanical

properties.

Before beginning to discuss control of root position, it is necessary to understandsome basic physical terms that must be used in the discussion:

Forcea load applied to an object that will tend to move it to a differentposition 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).

Center of resistancea point at which resistance to movement can be

concentrated for mathematical analysis. For an object in free space, thecenter 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 toothroot embedded in bone, the center of resistance will be determined by the


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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., abouthalfway between the root apex and the crest of the alveolar bone;Figure 9-

17).

Momenta 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 offorce 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 doesnot 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 (seeFigure 9-17). Not only is the tooth displaced in the

direction of the force, it also rotates around the center of resistancethus thetooth tips as it moves.

Coupletwo 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 (CR) 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 alsorotate around CR(i.e., the center of rotation and center of resistance are

identical) because a moment is created by applying a force at a distance fromCR. 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 (seeFigure 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 ofresistance, while the combination of a force and a couple can change theway an object rotates while it is being moved (Figure 9-18).

Center of rotationthe 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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