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THE MECHANICS OF ASYMMETRIC RETRACTION LOOPS USED IN FIXED APPLIANCE THERAPY N. E. WATERS Department of Dental Materials Science. UMDS, Guy’s Hospital. Medical School, London Bridge. London SEI 9RT. U.K. Abstmct,--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 digerent spring components are available to orthodontists for this purpose. a comparison of the merits of different designs has not been possible because analytic41 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 Uloops both with or without helices has been solved using the complementary (strain) energy method on the assumption that the eRect of a longitudinal tension or compression on the bending of a cantilever beam by lateral forces (the tie-bar elTect)is small. Unless the arms are of equal length it is then found that unequal couples 31 and N act at the posterior and anterior ends of the appliance, and in addition equal but oppositdy 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 gcncrally 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 stilTnes.ses (S and dQ/dy. respectively). the N/P and Q/P ratios. the self-righting ability (SRA)andd N/d/f, thechange in anti-tilt couple perdcgrcu:changein gable angle(theangleof the mesial arm with respect of a horizontal in the unactivatcd appliance).

In or&r

to move teeth the orthodontist

utilizes vari-

ous spring appliances and lessoften elastic mod&s

or loops which whtn activated are capable of storing the cncrgy required to supply the work necessary to tilt, translate or rot&c the teeth. In planning the treatment for a particular patient, the clinician has to decide which type of adpliance or appliances will bring about the desired tooth movement in the most etlicient way. For certain tooth movcmcnts, e.g. canine retraction as used in fixed ap@nce therapy (see Fig. I), there are a number of ditTetent designs from which the clinician can choose. Until fairly recently, however, the ncccssary criteria ofi which to base the selection of a particular component have been ill-defined as information on the characteristics of individual springs has not been available. Although a mechanical analysis cannot supply the clinician with al) the information required, there are in general various; characteristics of appliances which can only be properly evaluated in this way. These are the response of the appliance to activation which can produce forces, couples and torques on the teeth in more than one plane. Once these responsesto activation are known additional useful criteria such as the robustness or ability to withstand permanent distortion by excessive activation or by extraneous forces (e.g. food imparjtion) and also, for appliances which the patient for cleaning, the stability are removable or ability to w“:thstand dislodgement may be more readily evaluated. Rrceire6f injina//orm

? April

1990.

Space closing components which arc utilized in the cdgc-wise fixed appliance tcchniquc are of interest from the mechanical point of view in that they arc capable of applying a couple (M) as well as a rctraction force (I’) to a tooth. This is considcrcd to be a useful fcaturc of this type of spring for if the (M/t’) ratio has a specific valued, equal to the pcrpcndicular distance from the lint of action of the force P applied to the crown and the centre of resistance (the point within the root through which the resultant of all the resistive forces induced around the root by the force P acts), then thcorctically the tooth should move bodily (i.e. without tipping) (Smith and Burstone, 1984).

UNACTIVATED

ACT IVAT E D Fig. 1. Diagram to illustrate the method of use of one type of retraction loop, an open U-loop with helix.

1093

N.

109-a

E. WATERS

Retraction or closing loops. including the types to be considered in the present paper, have been examined by a number of workers. Chaconas er u!. (1974). using a simulated mouth model measured the force-displacement relationships for open and closing U-loops with and without helices, but although the importance of the IV/P ratio was stressed. the couples applied during activation were not recorded. Yang and Baldwin (1974) demonstrated that analytical solutions to the behaviour of closing springs could be obtained using a two-dimensional finite element method. The method was applied to a symmetrical open U-loop and a symmetrical Ricketts spring (Ricketts. 1974, 1976). both with non-gabled arms (i.e. for loops for which the angles x and /I are both zero in Fig. I) and reasonable agreement was obtained between theory and experiment for the force-deflection behaviour. More recently Greif et al. (1982) have utilized a three-dimensional finite element approach incorporating an iterative procedure to analyse the behaviour of a vertical loop under relatively large displacements. A study of the behaviour of T-loop retraction springs under large displacement conditions using a similar technique has been published by Faulkner or al. (1989). In the approach used by Koenig and Burstone and their co-workers (Koenig and Burstone, 1974; Burstone and Koenig, 1976; De Franc0 et ul.. 1976; Koenig cr 01.. 1980). generalized field equations were established for an arbitrary curved and twisted beam in space which were then solved by a standard finite ditlcrence matrix technique. Experimental confirmation of the predicted results was obtained for an open U-loop with non-gabled arms and for rectangular and T-loop components with a specially designed apparatus using transducers whose output was fed through A

Fig. 2. Jig used for measurements

on model springs.

an analog and digital converter and thence to a minicomputer. Whilst both these approaches are very powerful and have the potential capability of dealing with largescale deflections by iterative procedures, expressions for the required characteristics of a spring are not derived directly, and considerable expertise is required to obtain solutions for the simplest spring. The complementary (strain) energy approach (Waters, 1970, 1982) used in the present investigation, in contrast, has the basic merits of providing easily derived and accurate relationships in an explicit form, for small strain and small displacement conditions, which enables the factors controlling spring behaviour to be more readily assessed. Using this technique (Waters, 1982). expressions have been obtained for symmetrically disposed open and closed U-loops with and without helices incorporating gabled arms when used as either single or double loops. The present communication examines the mechanics of single loop components of the same type under the more normal situation in which the loop will be asymmetrically positioned. A summary of some of these results was presented at the Third Biomcchanics Conference in Davos, 1984 (Waters, 1986).

EXI’EWIMENTAL PROCEDURE

Experimental measurements were carried out on a number of enlarged model appliances and the results obtained compared with the results predicted by the derived analysis. The dimensions of the components of the models used were between ftve and ten times those of the dimensions of actual clinical-sized appliances. Confirmation of the correctness of the analysis in this way has been essential for elimination of errors in the derived equations and the computer programme. and also to validate the analysis itself which is derived on the basis of certain simplifying assumptions. For the purpose of the measurements it was assumed that the arms of the components when used clinically. are held in brackets which are in the same plane and whose slots are co-linear. The jig used for the measurements is shown diagrammatically in Fig. 2. In brief, one arm of the component is clamped vertically at A and the other arm is clamped perpendicular to a light symmetrical beam, BC. A light pointer DE fixed perpendicular to the centre of the beam serves to indicate when the lower end of the spring is vertically below A, so that the ends of the gabled arms are aligned as they would be in the clinical situation. Weights IV, and W, may be added to either end ofthe beam so that both a known vertical force and a known couple may be applied to the free end of the component. Provision is made for a light horizontal thread to be attached at one end to the centre of the beam at the point F and at the other end to the free end of a vertical cantilever beam G provided with calibrated strain gauges so that the tension in the

Asymmetric retraction loops thread could be measured. The cantilever rigidly attached

provided with horitontal.

machine

while keeping the thread horizontal. of the appliance

under

a given

vertical load P ivhen subjected to a couple M horizontal

(5) application chenko.

horizontal.

simulate the constraints tion component

rig was thus designed to

(I+/?).

(Timos-

the total rotation

produced

by the applied external couple M;

d 40,

(b’-=dQ

y. the deflection

in the clinical

in the direction

and at the point

of

of application

of the

external force Q;

imposed on an actual retrac-

when activated

theorem

forms:

and a

below the fixing point A and the beam

The experimental

of Castigliano’s

1955) in the following

d &o, (a) -= dM

forceQ such that the free arm at the point F

was vertically

strain energies

A vernier micro-

scope with a vertical traverse was used to measure the deflection

of the individual

to obtain the total strain energy UTOT of the appliance;

vertical and lateral traverses.

It was thus possible lo vary the tension in the thread

vertical

(4) the summation

beam was

to the table T of a milling

1095

situ-

ation and at the same time measure the forces and (‘)

couples induced’ at one of the brackets.

d Go, -= dP

x, the deflection

in the direction

and at the point

of

of application

of the

external force P. ANALYTICAL METHOD

This results in expressions of the form given in the Appendix.

The principles involved in the use ofcomplementary (strain)

energy

approach

uration

equations (AI)-(A3).

For any given contig-

it is necessary lo evaluate the deflection y, so

to the solution

of beam

that the only unknown

problems are discussed in most mechanical

engineer-

With the aid of the constructions

ing texts (e. g. Timoschenko. this approach metrical

1955). The application

to the small strain

U-loofl

retraction

solution

components

designs has beeri described previously The procedure metrically

involved

of

of sym-

of various

(Waters,

1982).

when the loop is not sym-

dispo$ed is the same, but requires an addi-

is the horizontal

5. the deflection y may be determined geometry

of the retraction

(6) the equations then s&cd

derived by steps S(a) and (b) arc

simultaneously

to find the unknowns

and Q in terms of I’. Explicit relationships and Q/P ratios arc thus obtained and (AS) in the Appendix);

the following

now involves

(7) the stilTness (P//x)

steps:

(I) the breakdown

of the appliance

into its basic

components; (2) the determination required

terms of the horizontal known

of the forces and

for the equilibrium couple

displacing

(Fig. 3 shows steps I and 2 completed

component;

in

force P. the un-

I+$ and the unknown

loop); (3) the determiination

couples

of each component vertical

details are

given in the Appendix;

tional step because the system now has two dcgrccs ol procedure

X.

from the general

loop. Further

static indcterminancy.

The

deflection

shown in Figs 4 and

unit displacement)

(A4)

(see equations

(i.e. the retraction

is obtained

M

for the M/P

force per

by substituting

for M

and Q in the expression derived from step 5(c). The above analysis ignores any tic-bar may occur in the arms II,

and

efrect that

I/, when a spring is

activated. An analysis which includes the tie-bar eflcct

force Q

but which is restricted to the simpler case of a sym-

for an open U-

metrical loop with arms grtbled at the same angle has been given

of the strain energy of each

in the literature

(Waters,

1982). Com-

parison of the results of the two analyses for the case where II, = II, = 5 mm, and the other parameters

-

M*rm,-

Ptt+ZQR

Fig. 3. Br akdvwn of an appliance into its basic cvmponcnk, together with the forces and cvuplcs required for the slaf ic quilibrium vfeach part (diagrammatic only). Arm If, shown gabled at an angle z, and arm H, at angle B to the horizontal.

are

1096

N. E. WATERS

Y f n++/q

+ 2Rp

Fig. 4. Geometrical construction for the determinalion of the deflection (y) necessaryto produce alignment in the ends of the arms of an unstrained open U-loop assuming the brackets into which the appliance is to be inserted are co-linear. Arms H, and H, gabled at II and fi. respectively.

Fig. 5. Additional terms 1o be added to Ihe cxprcssion for the de&lion y (.xeeFig. 4) if the anterior bracket is displaced a distance y,, parallel to. or tilted an angle y. with respexl to the original bracket axis.

those used for the data of Table 2 shows that the solutions gradually deviate as the retraction force and the gable angles increase. Typically, for the arms gabled at a=!= IO”and P= 100 g, theanti-tilt couple and the horizontal stitfnessare approximately 6% too large and 2 % too small. respectively, when calculated by the analysis presented here. EXPERIMENTAL RFSULTS AND DISCUSSION

The experimental and theoretical results obtained on nine model U-loops of diRering design are compared in Figs 6, 7 and 8. The dimensions of the components of these models together with the values for the stiffness characteristics of each spring dctermined by experiment and also by calculation from the theoretical analysis are presented in Table 1. From an initial examination of these results it was apparent that although the overall agreement was satisfactory, discrepancies between the predicted and experimental values in certain cases were considerable. Examination of the theoretical relationships for sensitivity to dimensional errors and also to the

0

-200 -200

0

200 CoUPLE

400

600

%pt/2sm

Fig. 6. Theoretical vs experimental couples N for nine Uloops of ditTering design (see Table I for details).

positional and alignment conditions indicated that the latter rather than the former were a major source of experimental error. This is demonstrated in Figs 6 and 7, where error bars corresponding to an error of + 2 mm in the positioning of the pointer connected to the loaded arm of a loop have been attached, for clarity, to only one predicted value of M or Q for each loop. In the majority of cases the experimental results are now seen to fall within the predicted range. As an absolute measure of the deflection x produced by a given retraction force P was not obtainable with the experimental arrangement used, the displacement relative to the first load applied (x-x,) has been plotted in Fig. 8 for both the experimental and theoretically predicted results. Although the agreement is good for small displacements an increasing discre-

Asymmetric

Table

Specimen NO.

Symbol used on graph

I

0

2 3 4 5 6 7 8 9

A

H,

Cl s ; A

n

Specimens respectively.

l-6:

of model retraction loops

H (cm)

R (cm)

(de&es)

(degfcxs)

El (g cmz)

8.63 8.63 8.63 6.07 3.41 3.45 3.68 4.25 4.4 I

5.27 3.66 1.7 7.83 7.83 7.95 1.82 7.58 6.44

5.25 5.25 5.25 5.25 5.25 5.25 4.43 5.62 4.79

1.43 1.43 1.43 I.43 1.43 1.43 1.5 1.92 0.88

3.85 3.85 3.85 13.6 13.6 3.7 2.2 I.2 4.5

13.6 13.6 13.6 3.85 3.85 3.0 7.75 2.9 0.3

9465 9465 9465 9465 9465 9465 9280 9600 9070

Specimens 7-9: open U-loop

dNldll gmmdegree-’

type

IO97

H2 (cm)

2. Characteristics

Loop

and stihsses

loops

(cm)

open U-loops.

Table

1. Dimensions

retraction

ID 2B 3B 48 IE 2E 3E 4E

34. I 9.0 26.3 5.6 51.5 - 17.5 42. I - 16.8

with helix. closed U-loop,

s (g 47

s (g c?

‘)

70.1 73.5 91.4 73.9 68.0 75.6 87.7 47.2 76.5

‘)

60.8 & 5.0 67.8 k 5.6 83.6 + 6.99 59.03k4.8 69.3 + 5.7 69.0 i 5.7 76.92 f 6.7 45.5 f 5.1 69.7 +6.6

and closed U-loop

with helix.

of different loops at the beginning and the end uf treatment

dQ/dy gmm-’

SRA

86.3 82.7 80.0 71.8 370 347 320 302

QIP

N:P

gmmdegrce-’

mm

48.5 41.9 38.8 35.1 97 8X XI 16

S g mm-’

gg-’

5.68 4.60 7.05 6.0 I 4.30 3.46 5.32 4.52

0.47 0.37 0.57 0.48 0.0 0.0 0.0 0.0

86. I 67.4 60.6 45.x 94.8 70.8 6R.0 4x.x

Coding: I, open and 2, closed U-loops; 3, open and 4, closed U-loops with hcliccs. B: hoginning (If, = 8 or I I mm, II, = 2 or 5 mm) and E: end of Ircatmcnl (If, = ? or 5 mm. f/,=Zor 5 mm). R= I.5 mm. ti=ZSBgcm’.

-40

-40

0

40

VERTICAL

80

FORCE

0

O.&Q

panty occurs as the displacements Taking

of the springs.

into account

of the experi-

the sensitivity

the agreement satisfactory.

errors it is considered that

t)etween theory and experiment

is very

1.2

r/mm

1.6

hptb

Fig. 8. Theoretical vs experimcnlal relative displacements on model U-loops (see Table I for details).

THEORETICAL

get larger, an effect

to changes in the geometry

mental results tb alignment

OI

OtFLECTlON

Fig. 7. Thcorckul vs experimental verrical rorces Q for nine U-loops of dlffcring design (see Table I for details).

ascribable

0.4

RFSULTS AND Dl!SCUSSlON

The derived theoretical

relationships

(see Appendix)

may be used to predict the characteristics sized

retraction

loops

providing

(I)

of clinically

that

the wire

behaves in a perfectly elastic fashion; and also (2) that

N. E. WATERS

1098

(5) for the same total span. closed loops with or without a helix are more flexible than their open loop counterparts. Effectively the longer the arms the smaller the induced couple acting to oppose the opening of the loop. In addition. for asymmetrical loops whose arms are not gabled: (6) unequal couples M and N of opposite senseact at the posterior and anterior ends of the component, respectively. Providing the anterior arm is equal in length or shorter than the posterior arm, the senseof N is such that it will oppose distal tipping of the crown of the tooth being retracted; (7) equal and oppositely directed vertical forces (Q) are induced at either end. The force on the tooth attached to the shorter arm (usually the anterior arm In common with U-loops with arms of equal length: attached to the canine) initially will be extrusive; (8) for a given loopconfiguration the magnitudes of (I) the retraction force P induced on activating a loop is proportional to the displacement (?I). If the Q, M. and N are proportional to the traction P, and hence also proportional to the amount of horizontal arms arc gabled, the displacement x is determined activation, (.x); after the arms are engaged in the brackets; (9) for any type of loop with a given span length (2) the horizontal stigness (S). is directly proportional to the flexural rigidity (Ef) of the wire used and which has fixed displacement. the magnitudes of N/P. for the same span is independent of the position of the M/P and Q/P arc dependent on the position of the loop along the span. This variation for closed loops loop and of any gabling of the arms; (3) on activating loops with gabled arms an initial with and without hclicesas retraction proceedsduring contraction occurs which, for symmetrical gabling. is the course of treatment for an average span of I3 mm is shown in Fig. IO; proportional to the gable angle; (4) the horizontal stitTncss(S) is decrcascd by in(IO) increasing the loop height ofany design subject creasing the loop height (ff) or by the addition of a to a given amount of activation increases the magnihelix to the loop. The variation in S with loop height tudcs of M. N and Q and hence also raises the ratios (II) for a closed loop with and without a helix is shown M/P. N/P, Q/P (WC Fig. 9 and Table 2); in Fig. 9; (I I) if. furthermore, the arms are gabled, M, N and the deformations of the loaded spring are such that the geometry does not change significantly. In addition to providing a better understanding of their behaviour in clinical use the results of the analysis allow the determination of the effects produced by changes in the dimensions on the stiffness(force/deflection), the antitilt couple/retraction force ratio (N/P) and vertical force/retraction force ratio (Q/P). A further benefit is that an unambiguous comparison of the characteristics of open or closed U-loop designs with or without helices may be obtained. Finally, the analysis allows the consequences of bracket misalignment to be assessed. The general conclusions of the analysis may be stated as follows.

400 -

U-K

% i

f

:a

A

300-

c m I, +OOt ii too -

Fig. 9. Predicted variation in the stitkss S, and in the ratios /V/P and Q/P with loop height (If) for a clinical-sized closed loop with and without a helix. (H, = 1 I mm, H, = 5 mm; R = I.5 mm; flexural rigidity E/=ZSOgcm-*.)

Asymmetricretractionloops r

I

I

I

I

I

I

1099

to the line ofaction of P. is difficult to achieve. Even for teeth with small roots ‘d’ is unlikely to be less than 8 mm, a value which can only be obtained by using a U-loop and helix with an exaptionally long leg length. It must therefore be concluded that this rquirement is rarely, if ever, complied with when these components are used clinically. In consequence. although the anti-tilt couple N tends to oppose tipping. some tipping does occur causing the appliance to deflect in the vertical plane. As soon as this happens the magnitude of N increases until the N/P ratio is sufficiently large to prevent further tipping occurring. In effect. the distal tipping of the crown causes the anterior arm to become gabled. The ability of a U-loop to resist tipping in this way is clearly a spring characteristic of some interest and may be termed its self-righting ability (SRA). Although it has been suggested(Waters. 1986) that the change in anti-lilt couple N per degree change in gable angle is a measure of the SRA. a more appropriate measure is clearly the increase in N per degree in angulation ( y) of the bracket on the tooth being retracted, which may be expressed mathematically as (d N/dy). This additional characteristic of a retraction component may be determined directly from the analysis presented (see Appendix). The value of the SRA of a particular design of spring will naturally vary according to its dimensions. A comparison of these values for the standard four designs of single loop configuration of the same ovcrall size under conditions appropriate to the initation of treatment is given in Table 2, where it may be seen that the SRA of open loops is slightly higher than that of closed loops, and is reduced if the loop contains a helix. These values enable estimates to be made of the amount of tilt likely to occur clinically. For example, if the N/P ratio for a tooth for pure translation is IO mm and an open loop without a helix is used having an N/P ratio of 5.7 mm and a SRA of 50 g mm degree- ’ then the angle of tilt (y) is given by:

1 I

-1

b

! B

Fig. IO. Prcdickd variation of M/P. N/P and Q/P as Irc~tmcnt proceeds for clinicd-sized closcd loops with and without a helix. (l/=8 mm; II, =S mm; H = I.5 mm:

EI=?H)gcm-‘.)

Q increase lincrurlyin magnitude with incrcasc in gable angle for LZ= b, or a = 0, /I = /I. Gabling the arms has a marked effect On the ratios N/P and Q/P as will be discussed further below. As has already been noted. for the efficient use of retraction components the retraction force should lie within the range considered to be optimal and the anti-tilt couple1ratio (N/P) should be such that the tooth being retracted does not tip. With regard to the first requirement it is apparent that in order to produce a reasonable amount of tooth movement betwteentwo consecutive visits of a patient to the clinic, an appliance with low stillness is required. This ensures that the forces are below the upper recommended force limit and reduces the likelihood that activation will cause permanent distortion by yiel’ding of the wire. Although this could be achieved by using a thineer wire with a lower flexural rigidity (El), any extraneous forces e.g. caused by food impaction, are more likely to cause distortion. It is for this reason. for exaMpIe. that when using an l8/8 stainless steel wire, a wi~reof 0.4 mm diameter is commonly used.As previoqsly noted an increase in flexibility may be obtained. however. by the use of closed rather than open loops, by increasing as far as is practical the height of the 107 or finally by the inclusion of a helix. It is clear that the second requirement, namely that the anti-tilt ratio (N/P) is equal to ‘d’, the perpendicular distance of the centre of resistance within the root

y=(lO-5.7)P/SO=O.O86P where P is the operative retraction force. Thus for P-IOOg the tilt y would be 8.6”. but for PalOg, y< I”. Thus the larger the applied retraction force the more detectable and clinically significant the tilt which will occur. This dependence of the angle of tilt upon the magnitude of the retraction force used, although elementary, has not previously been emphasized in the literature. It would appear to be an additional reason for ensuring that the force employed is within the range considered optimal and is not excessive. In all previous analyses it has been assumed that the brackets into which the appliance is inserted are co-linear (i.e. that y-0). Since the SRA (dN/dy) gives the couple acting on the tooth being retracted due solely to unit angular misalignment of the slot of the

N. & WATERS

1100

anterior bracket with respect to that of the posterior bracket. the effect of a finite angular misalignment (y) can be easily assessedby the present treatment. It has already been observed that the vertical force Q induced when the arms of the appliance are of unequal length is proportional to the retraction force P and will be extrusive on the tooth attached to the shorter arm, which will normally be the canine. Any extrusive movement of this tooth will cause a vertical displacement of this end of the spring and produce a vertical reaction tending to nullify Q, which once again, will depend on the design and geometry of the component. The stiffnessof the component to vertical displacement (without tilt occurring) may also be determined from the results of the analysis and is given by the change in Q for unit vertical displacement or (dQ/dy). Values of this additional spring characteristic are given in Table 2 for the conditions appropriate to the commencement and to the end of treatment, It may be observed that the vertical stiffness (dQ/dy) shows considerably less variation than the SRA with spring design, and as might be expected increases markedly as the span length decreases. On the assumption that the extrusive force Q on the canine is constant, and that the posterior end of the appliance is firmly anchored, an estimate of the possible vertical movement of the canine which might occur to nullify Q on a purely theoretical basis may be obtained from the spring characteristics, for

(Q/W’ y=(dQldy) Considering as before an open U-loop without a helix, then Q/P=O.5, and dQ/dy=80gmm-I. Thus for P= 100 g the displacemcut y=50/80=0.6 mm. Once again the lower the retraction force the less detectable and clinically significant the movement. The vertical stiffness (dQ/dy) also may be used to assessthe effect of a vertical disparity in the heights of the bracket slots at the start of treatment (see Fig. 5). One final spring characteristic of interest is dN/d,!?, which is effectively the increase in the couple N which may be obtained by gabling the anterior arm by one degree. For example, with a symmetrical open U-loop with the dimensions typical of the situation at rhe beginning of treatment (seeTable 2). dN/dfl= 34 g mm degree - ’ and N/P= 5.68. If the anterior arm is given a 10” gable angle, the couple N experienced by the tooth being retracted will be 5.68 P+ 340 g mm and the N/P ratio= 5.68 +340/P. As tooth movement occurs and P consequently decreases, the N/P ratio increases. Providing no distortion has occurred and root movement is not prevented by the presenceof adjacent teeth, and the distal arm is firmly anchored, the resultant large N/P ratio will produce ‘over correction’ and a final mesial tilt of IO” will result. together with an appropriateamount of Intrusion.

If, as will generally be the case, the arms are of unequal length so that a vertical force Q acts at the distal end of the bracket, then the true anti-tilt couple will be the algebraic sum of the couple N as calculated above together with a couple Qw/2, where w is the width of the bracket which is assumed to be centred on the axis of the tooth. This correction may be shown to be of more importance if the anterior arm is gabled when the arms are short. It may also be noted that dN/d/.l can be negative for reverse U-loops with and without a helix (see Table 2); in these cases although the anti-tilt couple N decreases if the anterior arm is gabled a relatively large intrusive force Q is generated by the gabling. It should be noted that some of thesecharacteristics will change considerably as the tooth moves. As an illustration of these changes, data are also given in Table 2 for the same springs under conditions appropriate to the end of treatment. This is not meant to imply, however, that all appliances should be left in to work out completely without adjustment.

Acknc~wled~um4nrs-The author gratefully acknowledges the

help of Mr P. J. Ward. MA (Onon) with the cxpcrimcntal work. REFEUENCFS

Burslone.C. J. and Koenig, H. A. (1976) Optimizing anlcrior and canine retraction. Am. J. Orrhodonr. 70, t-19. Chaconas. S. 1.. Caputo. A. A. and Hayashi. R. K. (1974). EfTccts of wire size. loop configuration, and gabling on canine-retraction springs. Am. J. Orrhodont. 65, 58-66. Dc France, J. C.. Koenig, H. A. and Buntone, C. J. (1976) Three dimensional large displaccmcnt analysis of orthodontic appliances. J. Binmechanics9, 793-801. Faulkner. M. G., Fuchshubcr, P.. Habcrstock, D. and Mioduchowski, A. (1989) A parametric study of the Corcc/ moment systemsproducedby T-loop retractionsprings. J. Biumeehanics22, 637-647. Grcif, R., Coltman. M., Gailus, M. and Shapiro, E. (1982) Force gcncration from orthodontic appliances. J. biomech. Engng 104,280-289. Koenig, H. A. and Burstonc, C. J. (1974) Analysis of generalized curved beams for orthodontic applications. 1. Biomechanics 7,429-W. Koenig. H. A., Vanderby. R.. Solonchc. D. J. and Burstone. C. J. (1980) Force systemslrom orthodontic appliances: an analytical and experimental comparison. J. blomech. Engng 102.29~300. Rickctts, R. M. (1974) Dcvclopmenl of retraction sections. Found. Orthodont. Res. News/err. 5.414%. Rickctts, R. M. (1976) Bioprogressive therapy as an answer to orthodontic needs. Part 2. Am. J. Orrhodonr. 70, X9-397. Smith, R. J. and Burstone. C. J. (1984) Mahanics of tooth movement. Am. J. Orrhodonr. 85,294-307. Timoschenko, S. (1955) Srrength of Mareriab. Part I (3rd Edn). Van Nostrand. Princeton. New Jcrxy. Waters. N. E. (1970) The mechanics of hngcr and retraction springs of removable orthodontic appliances. Arch. Urul Biol. 15. 349-363. Waters, N. E. (1982) The mcchania of retraction components used in fixed appliance therapy. In Biumechanics: Principles and Applirarions (Edited by Huiskcs, R., Van Campden, D. and Wijn, 1. de). pp. 445-450. Martinus NijhoR. The Hague.

Asymmetric retraction loops

Waters. N. E. (1986) The characteristics of asymmetrically disposed orthodontic retraction loops. In: Biomeckonics: Current lnferdixip~inury Research (Edited by Pert-en,S. M. and Schneider, E.), pp. 283-288. Martinus Nijhoff, The Hague. Yang, T. Y. and Bddwin. J. J. (1974) Analysis of space closing springs in orthodontics. J. Biomechanics 7. 21-28.

1101

-(M+QL)QH:+Q’H;,‘3]. Here L=H,+H,+ZkR. The total strain energy L’r,, is obtained by summing the strain energies of the individual components: Liror= C,,+

APPENDIX The displacement

non-symmetrical

characteristics open U-loop

of a

The model use-dfor the analysis is shown in Fig. 3. It is assumed that the, components are co-planar and that the arms H, and H, ate unqual in length and gabled at angles x and 4. respectively. When the appliance is activated (see Fig. I) by engagink the ends of the arms AB and EF into the slots in the brackets on the adjacent teeth and applying a lateral load P, a couple M and a vertical force Q will. in general. be induced at the end A. Initially for simplicity, the brackets are assumed to be co-linear, however. the analysis allows the effect df errors in bracket alignment to be examined provided the:bracket slots are in the same plane as the retraction component. The model in Fig. 3 has been shown with its components separated and theigabling exaggerated so that the forces and couples ncccssary to maintain each component in equilibrium when the appliance is activated may bc seen clearly. Unknown forces and couples are derived by considering the statical equilibrium of each component in sequence starting with the gabled atnn AB. The etTectof longitudinal forces on the bending of the components which are initially straight are ignored; the consequences of this simplifying assumption were considered in the Analytical Method Section. Taking each cdmponent in turn the bending moment is determined at a distance s from one end in the case of the straight components or R d0 from an end for the circular component where’ 0 is the angle subtended in moving along the arc from the and to the point under consideration. This gives:

uor;+ tJ,r.

Et (z+j?)=A,M+E,Q-C,P

(AlI

Ety=A,M+fl,Q-C,P

(AZ)

Et.x=

643)

-C,

M-C,Q+C,P.

Free these equations

kc, -4c*

M=

wz+a,-4y

1 1(A3 (44)

A,b-B,Az

A,B,-B,A,

A,b-A,b

A,C,-A,Cz

Q=

1

P+Et

A,b+B)-A,Y A,B,-A,&

[

’

Here A,=(tt,+H,+2tt+JnR) A,=[ti;:?-tf;/Z+tf,

(ZH+C’)

+I/, L+Rk(?ft+C)J 6, =A, 5,=[(tt:+tt:)/3+ff,L(L-tt,) +3 CRz/2+ff~fI~+ft(tt,+2R)’ +(tt+ZR)

CH,]

C,=(t/*+2R’+Ctt) C1=[tt

(tt,+kR)(C+tt+2R’/tt)]

CJ=[2ttJ/3+C(ttz+R’/2)+4ttR2] C=JnR where J = ively

G ,,=,%f+Q.s G,=Ps-(ICI

C,+Cce+

Application of Castigliano’s theorem as in steps 5(a), (b) and (c) in the Analytical Method Section then gives

+QI/,)

I or 3 for a U-loop or U-loop with helix, respect-

k = + I for an open loop. -

Gc,=(M+Qt/,-PI/-2QR)

1 for a closed loop.

From the above equations it follows that: -PRsinU-QR(I-cos6) G,,=Ps-(M+Qtt,+ZQR) G,,=Qr+(bf

+QH,

+?QR).

Q

The strain energy (U) of the individual components may then be shown to be: Ll

*’

U,=

=~(,~fzH,+MQt/;+Q’t/;13) 2Et -&(Ptt’,3-P(M+Qtt,)tt’ +(hf +Qtt,)’

ft)

u CD =~[(ht+Q(ft,+R)-Pft)‘JnR -4[bf+Q(ft,+R)-PftJPR*

-=K,+K, P

-&P’H’,:3-P(M+Q(tt,+?R)lft’

I

.

y=A’E’+B’A or y=ZR

+(M+Q(tt,+ZRI;‘H)

Ef p

Where K ,, K,, KJ. and K, are constants. For It, = It, and x=/3. K,=K,=O and hence Q-O. For ft,=H: and r=P=O. Kz=K,=K,=O and hence M/P = K ,. a constant independent of the horizontal force P. It remains to calculate the vertical deflection y from the geometry of the system. The necessarygeometrical construction for an open loop is shown in Fi8.4. Here y is the deflection necessary to engage the free ends of the arms It, and Hz into the bracket slots. If the bracket slots are colinear, then for an open U-loop the end A of the arm H, is deflected to the point A’ lying on the extension of H,. Thus:

+P’JnRJ/2+Q’JnRJ/2] um=

[

sinP+H,

sin(x+p).

In view of the assumptions made in the analysis. this may

N. E.

1102 be written with sufficient accuracy: y= H, (z+fl)+ZR/l

WATERS

(iv) the horizontal stiffness S. (ir the retraction fora developed per unit horizontal displacement)

Alternatively. for closed loops it may be shown that:

s-

y=H,(a+j?)-2Rg. Hence the vertical deflection for both open and closed loops may be written:

dQ [ dy

discrepancy yO and tilt 7 exist then from Fig. 5.

(i) the anti-tilt couple N = M +QL; (ii) the initial contraction on aligning the (gabled) arms xo=C(x+B)(4C,-A,Cs)

+Y,(A,G-AK,)IIQ~

(El):

x under the retraction load P + QC, - PC,)/G, where G is the tlexural rigidity

-GA,/Q,;

(vi) the change in anti-tilt couple per degree change in the gable angle g dh’ -

[ dS

1

=Gx[B,-LA,+(H,+2kr)(LA,-B)]/)80Q,;

(vii) the self-righting ability (SRA), i.e. the couple N in-

duced if the anterior bracket tilts about its antre by one degree

where Qr=A:-A,B,; (iii) the displacemen! x = -(MC,

= P/(x - xek

- 1=

y=H,(z+B)+2kR.

The remaining spring characteristics of interest are now readily derived. These are:

1

(v) the vertical stiffness (i.e. the vertical force Q developed per unit displaament vertically of the anterior bracket parallel to itself (Fig 5)

IF the bracket slots are not initially co-axial but a height y=H,(r+/l)+2kR+Ly+yo.

dP [ dx

[

dN dy

1

=Gn[-B,+L(B,+R,-R,L)]/l80Q,.