20
Force systems from an ideal arch Charles J. Burstone, D.D.S., MS.,* and Herbert A. Koenig, M.S.M.E., Ph.D.** Farmington, Conx T he force systems delivered from commonly used orthodontic appli- ances 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. Two-tooth segments When an arch wire is placed in the mouth a complicated set of forces is produced at each tooth. Reduction of this complex system into less complicated basic units would offer a simpler approach to understanding and solving many of the clinical problems that exist. The smallest basic unit that one could study is the two-tooth segment of an arch, An example of a two-tooth segment could *Head, Department of Orthodontics, School of Dental Medicine, University of Connecticut. **Associate Professor, Department of Orthodontics, School of Dental Medicine, University of Connecticut. 270

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Page 1: 6 geometrias

Force systems from an ideal arch

Charles J. Burstone, D.D.S., MS.,* and

Herbert A. Koenig, M.S.M.E., Ph.D.**

Farmington, Conx

T he force systems delivered from commonly used orthodontic appli- ances 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.

Two-tooth segments

When an arch wire is placed in the mouth a complicated set of forces is produced at each tooth. Reduction of this complex system into less complicated basic units would offer a simpler approach to understanding and solving many of the clinical problems that exist. The smallest basic unit that one could study is the two-tooth segment of an arch, An example of a two-tooth segment could

*Head, Department of Orthodontics, School of Dental Medicine, University of Connecticut. **Associate Professor, Department of Orthodontics, School of Dental Medicine, University of Connecticut.

270

Page 2: 6 geometrias

Volume 65 Number 3 Force systems from ideal arch 271

A B

Fig. 1. Positive forces. A, Anterior and lateral forces are positive (+). B, Buccal, labial, and mesial forces are positive (t).

be a straight wire placed between a canine and a premolar only. If the wire is not passive in the brackets, a force system will be produced on the canine and premolar in isolation from the rest of the arch. By summing a series of two-tooth force systems, the force system can be found for each tooth along the arch. Thus, the two-attachment segment is the basic unit for understanding forces in a complete arch. Even though the determination of the force system on a given tooth may be more complicated than merely summing two attachment forces, two-tooth analyses offer a basic building block for understanding the force systems from an orthodontic appliance.

Sign conventions

It is important to adopt a universal sign convention for forces and moments which will be applicable for dentistry and orthodontics. The convention is as follows :

Anterior forces are positive (+), posterior forces are negative (-) , lateral forces are positive (+), and medial forces are negative (-) (Fig. 1, A). Forces acting in a mesial direction are positive (+) ; those acting in a distal direction are negative (-). Buceal forces are positive (+) ; lingual forces are negative (-) (Fig. 1, B). Extrusive forces are positive (+), and intrusive forces are negative (-9 (Fig. 2, A).

Moments (couples) tending to produce mesial, labial, or buccal crown move- ments are positive (+), and moments tending to produce distal or lingual crown movements are negative (-) (Fig. 2, B and C).

The same convention is used for groups of teeth (a segment or an entire arch) or for establishing signs of orthopedic effects on the maxilla or mandible.

Force systems acting on the wire and the teeth

Some force systems on a wire can be fully determined if one force is known. For example, let us determine the force systems acting on the two-tooth segment

Page 3: 6 geometrias

272 Burstone and Koenig Am. J. Orlhod. March 1974

B

C Fig. 2. Positive forces and moments. A, Extrusive forces are positive (t). B and C, Moments that move crowns mesially, buccally, or labially are positive (t).

shown in Fig. 3. A straight wire is placed between a premolar edgewise bracket and a hook on the canine. The wire is tied into the premolar bracket, and then a force gauge is used to measure the force needed to lift the wire occlusally to the hook. The force gauge records +lOO Gm., which is an activation force since it is exerted on the wire.

The equilibrium diagram in Fig. 3, A shows two unknowns, Py and M. These unknowns can be found using the laws governing equilibrium. The wire must be in equilibrium, once it is placed between the two attachments. If it were not, the wire would accelerate the patient out. of the orthodontist’s office. By the laws of statics, we know that the force system on the wire at the premolar bracket is -100 Gm. and +700 Cm. - mm. (Fig. 3, B). The force system is therefore stati- cally determinate with the clinical measurement of a single force by the force gauge.

It is well known that for every action there is an equal and opposite reaction.

Page 4: 6 geometrias

Force systems from idecrl wch 273

100 gm

-b n I M . . . : :. . . :,

3*+;:.:-. : .;., : .:I:. ;::::: - :.:::::. :. :,:: :. - _ _ ;.. :.j:>::-:.: . . . . . _

?[-l’ y “???i :

fif

,..‘:. :: ::: :: ., . . :.::., ” 1:”

..:.::; .,

I -7mm- I

0 1 . FY

1OOgm , 700gm-mm

-- t n ‘.-,- I

I -- I ----=--

1 I c---7mmW I I

0 6

100 gm

0 C

100 gm

700 gm-mm

n ..::,::.:.. ‘. ,... . . . . :. ;. . . . . .: . . . . . . . . . . ., .,., ‘..;.‘..:.;.

m

.;.,..,..,. :.::;;,.:.,. ..I, ,y: . . . . . . .y . . . ...... :.:.;,;:: .,. 1.;. . . . . . . ;: ‘;:,.. :.:.:.., I

100 gm

Fig. 3. A wire is placed in the premolar bracket and lifted to a hook on the canine. A, A magnitude of 100 Gm. force is measured at the canine hook. B, Activation forces acting on the wire. C, Deactivation forces acting on the teeth.

For example, a rifle is tired. d force propels the bullet forward while an equal force pushes at the shoulder of the marksman. In a similar way, the forces and moments acting on the wire are balanced by equal and opposite forces acting 011

the teeth. It should be clearly noted that, in order to find the forces acting on the wire at the premolar, it was necessary to place the wire in equilibrium. The force system which is found by the laws of statics and shown in Fig. 3, B is t,hc activcltimt force system. It consists of the forces or moments that the orthodontist exerts on the wire to place it into the attachments or the forces and

moments exerted by the attachments on the wire. In order to find the forces and moments acting on the teeth, it is necessary to reverse the signs of all forces and

Page 5: 6 geometrias

274 Burstone a?ld Koenig Am. J. Orthod. March 1974

Fig. 4. Wire-attachment geometry is defined by the interbracket distance (L) and the

angles of the brackets at positions A and B (e, and es).

moments (Fig. 3, C). Thus, the premolar has acting on it a force of +lOO, a couple of -700 Gm. - mm. and ,the canine, a force of -100 Gm. The equal and opposite forces acting on the teeth comprise the deactivation force system. It is important not to confuse the activation and deactivation force systems. In order to determine the unknowns in the orthodontic force system, the following pro- cedure is followed. First, the wire is placed in equilibrium. All activation un- knowns are determined. Once these unknowns are found, their directions are then reversed using the same magnitudes which give the moments and forces (deactivation force system) acting on the teeth.

If we would like to know how a tooth will move (its center of rotation), it is necessary to know the moment to force (M/F) ratio at the attachment. (It is also required that one replace the M/F ratio at the bracket with an equivalent system at the center of resistance of the tooth.) For now, however, let us concern ourselves only with M/F at the bracket in Fig. 3, C. At the premolar bracket the M/F ratio is -7/l, and at the canine hook the ratio is 0 with only a -100 Gm. force acting. The effects on the premolar are extrusion (t-) , crown-distal and root mesial (-) ; and on the canine, intrusion (-) .

The M/F ratio determines the center of rotation of a tooth, group of teeth, or bones. In the above example, the laws of statics were sufficient to solve for the unknowns, so that the complete force system including the M/F ratio was known. Unfortunately, in most two-attachment segments, the solution is statically indeterminate.

In order to solve more complex and statically indeterminate clinical problems and to throw light on the workings of all arch wires, a computer program based on the linear beam theory was developed. To simplify the descriptions of the force systems and to better develop the needed fundamental concepts, only force systems in one plane of space were considered, and effects within the bracket slot were disregarded. The wires studied were 0.016 inch high-temper wire (400,000 p.s.i. yield strength). The force systems which are described are initial force systems only. (As teeth move under the influence of the forces, the force system will change.)

Wire-attachment geometry

The force system produced by a straight wire placed between two attachments can be determined only if the wire-attachment geometry is accurately defined.

Page 6: 6 geometrias

Volume 65 Ntcmbw 3

Force systems from ideal arch 275

BASIC TWO-TOOTH GEOMETRIES

CLASS: 1 IL m Ip Y m

8, 1.0 0.5 0 -0.5 -0.75 - 1.0

6

Fig. 5. The six basic geometries based on the ratio 8Jes. Classes are independent of

interbracket distance. Position A is the canine; position B, the premolar.

Fig. 4 shows a two-tooth segment. The interbracket axis (L) connects the centers of the two attachments. @A and 8,( are the angles of the attachments (brackets) measured with respect to the interbracket axis. 8~ is always the smaller of the two angles. The geometry of a two attachment-wire segment can be fully defined if one knows eA, en, and L.

Six basic two-tooth geometries will be presented in this article. These geometries, which form six classes, are shown in Fig, 5. The ratio of @A/% defines each of the six classes.

Thus, e&I% = 1 describes two brackets that are angled the same amount and are in the same direction with respect to the interbracket axis (Class I) ; 8J8B = 0.5 describes two brackets in which one is angled one half of the other and both are angled in the same direction (Class II) ; eA/eR = -0.5 describes two brackets in which one is angled one half of the other and both brackets are angled in opposite directions from each other (Class IV).

In Fig. 5 the reader will observe a gradually changing @A as eR is kept constant. Since, by convention, @A is always smaller than f&, the six classes can be used to describe any two-tooth segments, regardless of the teeth involved. A consistent sign convention is used to describe the angle of the brackets. The sign of the angle is the same sign as the moment required to place the wire into the bracket. Thus, the Class I bracket arrangement shown for the lower left quadrant has premolar and canine brackets with positive (+) angles.

Let us now proceed to describe the relative force system produced in each class of geometry, followed by a specific description of the actual forces, moments, and moment-to-force ratios acting on the wire.

Relative force systems

The state of the art is such that even a general understanding of the relative magnitudes of forces and moments and their directions is a major advancement.

Table I gives for each of the six classes (1) the actual force system (given at the yield strength) developed in the wire for both a 7 mm. and a 21 mm. interbracket distance; (2) the ratio of the moment produced at A with respect

Page 7: 6 geometrias

276 Burstone and Koenig Am J Orthod. March 1974

Table 1. Force systems by class

CLASS I II Iu m P lzr

to B; and (3) the force system that acts on the teeth. It should be noted that, with the exception of the forces and moment labeled “force system on teeth,” all other forces and moments act on the wires. Each class will be separately described, followed by a comparative description of the general force systems produced. All descriptions refer to a straight wire connecting the lower left canine (bracket A) with the lower left premolar or first molar (bracket B) as shown in Fig. 5.

In Class I geometries, two equal and positive moments would act at position A and position B. Since the two moments are equal, the ratio MA/MB = +I. Although the magnitude of the moments may vary, depending upon the amount of activation and the interbracket distance, the ratio of MA to Me always remains +l in Class I.

In addition to the moment, two vertical forces are also produced-a positive force at position A and a negative force at position B. Force A equals force B. In all of the geometries considered in this work, force A equals force B for equilibrium.

If one would like to predict how the teeth might potentially move, it is necessary to know the force system acting at the brackets. The forces and moments acting on the wire are reversed; thus, two equal moments act on the canine and the premolar. Both of them are negative, which would tend to move the canine and premolar crowns back and roots forward. In most clinical situations the mesial movement of the canine root is an undesirable side effect. In addition, the vertical forces produce intrusion at the canine (position A) and extrusion at the premolar (position B) .

Page 8: 6 geometrias

Force systems front ideal arch 277

1 w1 W2

w2 Mg(gm-mm) 848 1860 -

WI w2

MA(gm-mm) 048 1860

FA km) 242.3 531.4

e,(degre.s) 2.0 4.4

M/Fl A LLLL

Fig. 6. Class I geometry-7 mm. interbracket distance.

The Class II geometry is characterized by QA having a magnitude of one half of &. Two positive moments are created at the wire at positions A and B. The magnitude of the moment at A is 0.8 of the moment at B. A positive force is found at A, and a negative force at B.

In Class III geometries the interbracket axis cuts across the two brackets, SO

that the ratio of eA to 81~ is 0; thus, a wire placed in the canine bracket (A) would cross the premolar bracket (B) at the center of the slot. What is the relative force system produced in this geometry?

AS in Classes I and II, two moments are produced, both positive. However, the moment at position A is one half the moment at position B. Although the actual magnitudes of the moments may vary, depending upon the activation and interbracket distance, the ratio MA/MB is a constant of 0.5. Thus, whenever the orthodontist observes this geometry, he anticipates two moments in the same direction, one moment being one half of the other.

In addition, vertical forces are produced on the wire-positive at position A and negative at position B. It should be noted that the relative magnitude of the vertical forces with the same interbracket distance is considerably less than would be found in geometries I and II.

Reversing the forces to obtain the deactivation force system, one would find acting on the canine and the premolar, two negative moments which tend to move roots forward and crowns back. The moment on the canine would be one half the moment on the premolar. The vertical forces produce an intrusive force on the canine and an extrusive force on the premolar.

Page 9: 6 geometrias

278 Burstone avid Koenig Am. J. Orthod. March 1974

35 1.0 8 L= 21mm

B

.5

.4

.3

E E

2

; 1

i= v 5 MA c

k -. 1 n ii -_ 2

2 -.3

-. 4

-. 5

POSITION - mm -

Fig. 7. Class I geometry-21 mm. interbracket distance.

In geometry IV the ratio between o,J@,~ is -0.5; in other words, the canine bracket is angled one half the premolar bracket in relation to the interbracket axis.

In this geometry, a positive moment is found at position B, but no moment whatsoever is found at position A. Only a single force operates at position A, with an equal and opposite force at B. Since no moment is acting in position A, the ratio of lQ’MII is equal to 0.

Geometry V could describe a clinical situation in which the canine root is forward of the crown and the posterior segments, including the premolar, are tipped into an extraction site. The angle of tip of the canine to the premolar QA/@( is -0.75. The canine bracket is angled 0.75 of the premolar bracket. In this example, the moment at A is negative and its magnitude is two fifths of the positive moment at B. The ratio of M,/M, is -0.4. Equal and opposite verti- cal forces act at positions A and B. Note that the direction of the moment at the canine is opposite to geometries I, II, and III.

The deactivation force system in geometry V has moments acting to move the root back and the crown forward on the canine (positive) and to move the root forward and the crown back on the premolar (negative), and the vertical forces tend to intrude the canine (negative) and to extrude the premolar (positive).

Class VI has premolar and canine brackets equally tipped into an extraction

Page 10: 6 geometrias

Volume 65 Number 3

Force systems from ideal arch 279

e,; 0.5 L =7mm 8

B

z :02-

F v -.03 - !t i -.04-

,” TO5 -

5 -.06-

-.07-

:oa -

Fig. 8. Class II geometry-7 mm. interbracket distance.

site. The ratio of 8*/8n is -1.0. The force system acting on the wire is composed of equal and opposite moments (negative at A and positive at B) . No vertical forces are present. The ratio of Ma/MB is -1.0. Thus, equal and opposite couples would be produced on the teeth, tending to move the canine root back and crown forward (positive) and the premolar crown back and root forward (negative).

The six clinical classes actually represent a continuum of possible force systems that can place a wire between two brackets into equilibrium. The clinician will find it useful, even without knowing the exact magnitudes, to know the direction of the moments, their relative magnitudes, and the direction of the vertical forces. As the ratio @*/en changes from one geometry to the next, the force system on the wire, and hence on the teeth, will radically change. Note that the moment acting on the canine in Class I will tend to move the root forward ; in Class IV no moment is present ; in Class V a moment tends to move the canine root back ; and in Class VI a larger moment tends to move the root back.

The vertical forces FA and FR are equal in each case. However, their relative magnitudes decrease as the ratio of @A/@r becomes smaller. Since

F

A = &IA + MR

L

it can be readily seen that the magnitude of FA and Fn becomes increasingly less. For example, in geometry I the vertical forces are 531 grams for a 7 mm. inter- bracket distance. In geometry III the magnitude of the force is reduced to 398

Page 11: 6 geometrias

280 Am. J. Orthod. March 1974

2 f!L = 0.5 L=21 mm e

B

I

F

1 B

MB WIRE POSITION -mm ./ I

is -. 5

5 ~6

-.7

I6 I8 20 24

/;

i w1 w2 w

MAlw-md 1482

Fig. 9. Class II geometry-21 mm. interbracket distance.

grams; in geometry V, 160 grams; and, finally, in Geometry VI no vertical forces are present (Table I).

The relative force systems that have been described hold true, provided the geometry as defined by eA/er, is present for each of the six classes. It is possible, by interpolation, to determine the relative force systems for other geometries. For example, a eA/eR ratio of 0.75 would give an MA/MB ratio of +0.9. It should be noted further that the relative force systems are independent of the amount of activation of the wire.

Of particular clinical interest is the fact that very small changes in geometry can radically alter the force system. The difference between canine bracket angles of -0.5 and -0.75 degrees with respect to the premolar is not a great difference, and yet the force systems can vary greatly. The reader is encouraged to look at the actual force values for a 7 mm. and a 21 mm. segment, so that he can better visualize the changing force systems of the different geometries (Table I).

The diagram of the various classes presented in Fig. 5 shows the interbracket axes parallel to the occlusal plane. In many clinical situations the interbracket axis may not be parallel to the occlusal plane; however, the force system in each class is determined by the 8*/8r ratio as discussed previously.

Actual moment-to-force ratios

In the section entitled “Relative Force Systems,” we only considered the ratio of MA/MB and the relative magnitude of the vertical forces. In order to determine the moment-to-force ratios acting on each bracket, it is necessary to know the interbracket distance (L) in addition to the ratio eA/eB. The data

Page 12: 6 geometrias

.12-

Force systems from ideal arch 281

A SLzo

‘8

L =7mm

FA F8 WIRE POSITION - mm

MA c j”8

Fig. 10. Class III geometry-7 mm. interbracket distance.

that follow, which describe the actual force and moment values for the six classes, have been obtained by the computer program previously described. For convenience, data are plotted and pertinent parameters are placed in the inset tables in Figs. 6 to 15.

The reader is referred to Fig. 6, which uses a format that all other graphs will follow. In Fig. 6 the force system is given for a Class I geometry (0,/e, = 1) with a 7 mm. interbracket distance. Positions A and B could represent the lower left canine and premolar positions, respectively. F, and MA, therefore, act on the wire at the canine bracket. The vertical axis measures the deflection of the wire in millimeters, and the horizontal axis denotes wire position (the 0 position is at the canine bracket and ‘7 mm. is at the premolar bracket). The elastic curve of a wire placed between bracket A and bracket B is denoted by curves W, and W,. Wire, (W,) connects brackets angled at 2.0 degrees; wire* (W,) connects brackets angled at 4.4 degrees. The directions of the forces and moments are illustrated on the graph; the actual values of the moments, forces, angular relation of the bracket to a straight wire, and the moment-to-force ratios are given in the inset tables. Note that a table on the left describes the force system at position A and one on the right, at position B.

At positions A and B, a moment of 848 Gm.-mm. acts on the wire. Force* is 242 Gm. and Forcer is -242 Gm. It should be appreciated that relatively large

Page 13: 6 geometrias

282 Bwrstone and Koenig

OA -z 0 8

L =21mm

8

Am. J. Orthod. March.1974

E E .02 -

5 F u .04 - !Y kl n 2 .06 - 2

fl,(d.gr..s) 6.6 20

Mh]B + +-

.08 -

.I -

Fig. 11. Class III geometry-21 mm. interbracket distance.

forces are produced with only a 2 degree rotation of brackets A and B with a 0.016 inch wire. The moment-to-force ratios are 3.5/l and 3.5/l. Wire, placed between brackets rotated 4 degrees produces proportionately greater moments and forces; however, its moment-to-force ratio is the same as that of wire,.

Fig. 7 describes a Class I geometry in which the interbracket distance is 21 mm. One could visualize this geometry as a wire connecting brackets on a lower left canine and first molar. As in Fig. 6, moment, force, angular deflection, and moment-to-force ratios are given at positions A and B. Classes I through VI are described graphically in Figs. 6 to 15 and should be referred to in the following generalizations.

4 FA Q3

WIRE POSITION - mm

The moment-to-force ratio is constant for any given class and interbracket distance, regardless of the amount of deflection. Class I-7 mm. geometries have M/F ratios of 3.5/l at A; Class II-7 mm. geometries, 3.1/l at A; and so on (Figs. 6, 7, 8, 10, 11, etc.). Thus, the M/F ratio is independent of the amount of activa- tion of a wire required to engage it in the brackets for any given class and inter- bracket distance. The M/F ratios for a 7 mm. interbracket distance are given for position A and B in lines 4 and 5 of Table II. Note that, at position A, the ratio is largest in a Class I geometry (3.5/l) ; becomes smaller in Classes II and III ; is 0 in Class IV ; and negatively increases from Classes IV to VI. At position B the M/F ratio is smallest in Class I (3.5/l) and increases to its largest value in

Page 14: 6 geometrias

Force systems fwm idenl nrch 283

WIRE POSITION- mm

Fig. 12. Class IV geometry-7 mm. interbracket distance.

Class V (12/-l). A similar gradient is shown for positions A and B with a 21 mm. interbracket distance in lines 6 and 7 of Table II. It is evident that, for a given class, the M/F ratio increases proportionately with the interbracket distance.

The moment-to-force ratios for any two-tooth segment in a clinical context can be found by the following procedure :

1. Measure the magnitude of angles 8 A and eII and the interhracket distance (L).

2. Select the class which most closely describes the two-tooth geometry ( @A/h).

3. Determine the M/F ratio at position A, using the formula

M [I FA = kl

in which k is a constant given in line 3 of Table II. Interpolation of k values between classes is possible.

4. Determine the M/F ratio at position B, using the formula

[f-J, = * [$I* . z It can be readily seen that the ratios of (M/F) A and (IV/F) B are proportional

to MA/MB, since FA equals F,. The MA/MI3 ratios are given in Table II, line 2.

Page 15: 6 geometrias

284 Burstone and Koewig Am. J. Orthod. March 1974

MA c

-0.5

E

= 0 -1.0

5

Iii 8 -1.5

2 3

-2.c

MA (

- .l

E

$ -.2

ki ti 2 -.3

ii z

WIRE POSITION -mm

Fig. 13. Class IV geometry-21 mm. interbracket distance.

e, i - 0.75 8B

L=7mm

FA WIRE POSITION - mm

FB

r’l 2 3 1 5 6 / ‘M,

Fig. 14. Class V geometry-7 mm. interbracket distance.

Page 16: 6 geometrias

Porte systems from ideal arch 285

!!L z-1.0 Lz21 mm

% %I r POSITION - mm

fig. 15. Class VI geometry-21 mm. interbracket distance.

The reader should be able to determine the M/F ratios given in Table II by using the above method.

Actual moment and force values

The clinician may want to estimate the actual moment and force magnitudes for each of the six classes. This is accomplished by the following steps:

1. Measure the magnitude of angles ed and eB and the interbracket distance (L) .

2. Select the class which most closely describes the two-tooth geometry (@A/%).

3. Determine the magnitude of the moment at position B (MB), using the formula

where k is a constant given in line 8 of Table II. Interpolation of k values is possible.

4. Determine the magnitude of the moment at position A (MA) by using the appropriate 8A/& constant for each class

e)A = k%s The 8~ constants are given in line 1 of Table II.

Page 17: 6 geometrias

286 Burstone and Koenig Am. J. Orthod. March 1974

Table II. Moment-to-force ratios for two-tooth segments

I Class I II IZI I IV v ) VI

1.0

1.0

0.50

3.5

1

3.5

-1

10.5

0.5 0 -0.5 -0.75 -1.0

0.8 0.5 0.0 -0.4 -1.0

0.44 0.33 0 -0.66 Undefined

3.1 2.3

r 1

0 -4.6 Undefined

Undefined

Undefined

Undefined

987

(M/F), = K

L

1

3.9 4.7 7 11.6

-1 -1 -i -1

0

M [ 1 FJJ L = 7 mm.

-13.8

1

M [ 1 -F* L = 21 mm.

9.3

-i-

7.0

-1 1

10.5 M [ I E‘, L = 21 mm.

-1

11.6 14 21 35

-1 -1 -1 -1

LMB = K 2,960. 2,467. 1,974. 1,480. 1,234.

5. Determine the magnitude of Fn and Fn, using the formula

MA + MB FA = L = -F,

Yield characteristics of two-tooth segments

The clinician is interested in knowing the greatest moment, force, or deflec- tion that he can obtain from a wire without producing permanent deformation. Thus, a knowledge of the yield characteristics of a two-tooth segment can be most useful. The yield characteristics for the six classes are given in Table III.

Using a 0.016 inch round wire with a yield strength of 400,000 p.s.i., the following would be expected. The moment at position A is largest in Class I (1,860 Gm.-mm.), becomes smaller in Class II and Class III, drops to zero in Class IV, and negatively increases in Classes V and VI, becoming -1,860 Gm.- mm. in the latter. The moment at position B is always 1,860 Cm.-mm., regardless of class, The moment values given above are independent of the interbracket distance. Lines 1 and 2 of Table III can be used to estimate the moments at yield

Page 18: 6 geometrias

Volume 65 Number 3

Force systems from ideal arch 287

Table III. Yield characteristics of two-tooth segments

Class

I 1 II ( III 1 IV 1 v I VI M, (Gm.-mm.) 1,860 1,482 930 0 -740 -1,860 MB (Gm.-mm.) 1,860 1,860 1,860 1,860 1,860 1,860 M, + MB = K (Gm.-mm.) 3,720.O 3,342.0 2,790.o $860.0 1,120.o 0 FA (Gm.) (L = 7 mm.) 531.4 477.4 398.0 265.7 160 0 F, (Gm.) (L = 21 mm.) 177.0 160.0 133.0 88.6 53.1 0 8, (degrees) (L = 7 mm.) 4.4 2.6 0 -4.4 -7.9 -13.2 8, (degrees) (L = 7 mm.) 4.4 5.3 6.6 8.8 10.5 13.2 8, (degrees) (L = 21 mm.) 13.1 8.0 0 -13.3 -23.7 -40.0 8, (degrees) (L = 21 mm.) 13.1 16.0 20.0 26.6 31.5 40.0

for 0.016 inch wire. Interpolation of the table is possible between listed classes. The maximum force at A and B is length (1,) dependent, since

F =MA+& A

L

FA = -Fs

MA t Mn is constant for each class. Line 3 of Table III gives the MA + MIS constant. The constant divided by the interbracket distance (L) gives the yield force at A or B. Yield forces at A for ‘7 mm. and 21 mm. interbracket distances are given in lines 4 and 5 of Table III. Note that the forces at yield are greatest in Class I and then diminish until no force is present in Class VI. The force at yield is inversely proportional to the interbracket distance for each class.

The angular deflection at yield for ‘7 mm. and 21 mm. wire length is given in Table III. QA has its largest positive value in Class I, is less in Class II, 0 in Class III, and negatively increases to Class VI. en is smallest in Class I and increases between Class I and Class VI.

It should be remembered that all of the yield values refer to a 0.016 inch round wire with a yield strength of 400,000 p.s.i. If the yield strength is reduced or raised, the yield values will vary proportionately.

Discussion

Perhaps no concept has held back the development of clinical orthodontics more than the idea of the ideal arch. The dogma of the ideal arch states that if a wire is bent into the shape in which one would like the brackets to be found at the end of treatment, the teeth will move to that position on the ideal arch and thus produce the desired occlusion. There is some validity to the ideal arch concept if one considers a very rigid wire which acts as a mold and the teeth are slowly displaced through the thickness of the periodontal ligament by inter- mittent ligation. Today, however, most orthodontists use highly flexible wires, and as one increases the flexibility of wire, a complicated force system comes into play which commonly produces undesirable side effects not wanted during

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288 Burstone and Koenig Am. J. Orthod. March1974

A

~ ~ . . : ,+: ,>: : . . , ‘. . : .

6 C Fig. 16. “Reading” of a wire. A, Wire is placed in molar bracket. To reach the wire, the canine must intrude (--) and the root must go back (+) according to the ideal arch principle. B, Incorrect force system based on “ideal” arch principle, C, Correct force system for depicted Class IV geometry.

treatment. The straight wires running between two brackets described in this article are nothing more than segments of an ideal arch, and it has been clearly demonstrated that these force systems are not under the control of the ortho- dontist. If one makes an ideal wire, in a sense the arch wire is doing the thinking for the clinician. It would be only by sheer chance that the desired force system would be produced.

Traditionally, the orthodontist predicts how a tooth will move by a so-called “reading” of the arch wire. “Reading” means placing the wire in one bracket and determining the linear and angular change required of the other bracket in order for it to approach the line of the wire. Fig. 16, A shows a Class IV geometry involving the lower left canine and first molar. According to the ideal arch principle, the canine should intrude (-) and the root should move back (+) (Fig. 16, B). The actual force system for a Class IV geometry has only an in- trusive force (-) and no moment acting on the canine (Fig. 16, C). If each of the six classes is reviewed by placing a straight wire in one bracket and tradi- tionally “reading” the other bracket to the wire, it becomes obvious that the

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Force systems from ideal nrch 289

prediction of force systems by this method is incorrect. The correct method to read a wire is to lay it across the centers of the two brackets and determine t,he QA/@13 ratio and the interbracket distance as outlined in this article. It should be apparent that very small changes in the eA/err ratio and the inter- bracket distance can readily alter the force system. This is particularly the cast in progressing from Classes IV through VI.

This article has concerned itself only with the initial force systems acting on the teeth from a,n orthodontic wire. As a flexible wire continues to deactivate, the force system will change. This is an interesting story in itself, since teeth will “wiggle back and forth” before they reach a given position at the end of an adjust,ment. The significance of the initial force system is that it is the force system of the greatest magnitude and the one t,hat is most likely to be active during the time the arch wire is in place. After a given number of weeks and months, a wire is usually removed, so the force system acting at the point of

zero deflection of the wire is theoretical at best. In a later publication it is our intent to describe the changing force system as a wire approaches zero deflection. If one considers the “round-trip ride” of the teeth produced by changing M/F ratios during deactivation, the over-all concept of the ideal arch becomes even more untenable.

To demonstrate the basic force systems produced, a 0.016 inch wire of high yield strength has been used. As yield strengths vary, the force systems will vary proportionately, as described by elementary engineering formulas. It should also be remembered that, for other cross sections, the load-deflection rate will vary as the fourth power of the diameter and the yield forces and moments as the third power of the diameter. The yield rotations at points A and B vary inversely as the diameter of the wire.

Thus, using the formulations described, the force system from any two- attachment segments can be determined in any plane of space where a straight wire is placed into irregular attachments. More important, the basic theory of determining the force systems from orthodontic wire is given, so that the clinician might avoid obvious errors leading to highly undesirable side effects.

Although it is now possible to develop nomograph tables, so that force values can be predicted from an ideal wire, it is obvious that such wires usually will not give the force system desired. The solution to the problem lies in creative spring design, namely, designing a wire that delivers the proper force system and has a geometry that is not so critical but that the clinician is assured that that force system will actually he delivered to the teeth. A description of the scientific basis of orthodontic spring design is beyond the scope of this article.

REFERENCES

1. Burstone, C. J.: The biomechanics of tooth movement. In Kraus, B. S., and Riedel, R. A. (editors) : Vistas in orthodontics, Philadelphia, 1962, Lea 6; Febiger, pp. 197-214.

2. Burstone, C. J.: The application of continuous forces to orthodontics, Angle Orthod. 31: 1-14, 1961.

3. Burstone, C. J.: Biomechanics of the orthodontic appliance. In Graber, T. M. (editor) : Current Orthodontic Concepts and Techniques, Philadelphia, 1969, W. B. Saunders Company, vol. I, pp. 160-178.