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T h e a s ic s o f O r t h o d o n t i c M e c h a n i c sS t ev
e n J L i n d a u e r
The b i o l og i c c a s c a de o f e v e n t s t ha t u l t i m
a t e l y r e s u l t s i n bone r e m ode l i nga n d o r t h o d
o n t i c to o t h m o v e m e n t b e g in s w i t h t h e m e c h
a n i c a l a c t iv a t io n o fa n o r t hodont i c a pp l i a nc
e . The f o r c e s y s t e m s pr oduc e d by o r t hodont i c a p
-p l ia nc e s , c ons i s ti ng o f bo t h f o r c e s a nd m om e
nt s , d i s p la c e t e e t h i n a m a n ne rt ha t i s bo t h p
r e d i c t a b l e a nd c on t r o l la b l e . B y v a r y i ng t
he r a t i o o f m om e n t t of o r c e a p p l i e d t o t e e t
h , t h e t y p e o f t o o t h m o v e m e n t e x p e r i e n c e
d c a n b er e gu l a t e d by t he o r t hodont i s t . O r t hod
ont i c a pp l i a nc e s obe y the l a w s o fphy s i c s a nd c a
n be a c t i v a t e d t o ge ne r a t e t he de s i r e d f o r c
e s y s t e m s t oa c h i e v e p r e de t e r m i ne d t r e a t
m e n t goa l s f o r i nd i v idua l pa t i e n t s . L i k e w i
s e ,a ny o r t hodont i c a pp l i a nc e c a n be a na l y z e d
t o de f i ne t he m e c ha n i c a l f o r c es y s t e m s i t p
r oduc e s . U nde r s t a nd i ng t he b i om e c ha n i c a l p r
i nc i p l e s unde r l y -i ng o r t hodo nt i c a pp l i a nc e a
c t i v a t i ons is e s s e n t ia l f o r e x e c u t i ng e f f
ic i e n t a nds uc ce s sf ul o r t hodont i c t r e a t m e nt .
Se m i n O r t hod 2 0 0 1 ; 7 : 2 - 1 5 .) C o p y r i g h t 2 0
01 b y W B S a u n d e r s C o m p a n y
he physical concepts that form the founda-tion of ortho donti c
mechanics are the key
to understanding how orthodontic applianceswork and are critical
for designing new appli-ances. The principles are not unique to
orth-odontics but are basic to the science of staticmechanics.
Physical laws of statics may be ap-plied to explain the force
systems developed byorthodontic appliance activations. Simple
me-chanical principles can help deduce how teethwill be displaced
as a result of application ofthese force systems. It is the
biologic reaction tothese perturbations that ultimately results
inorthodontic tooth movement. To control toothmovement with the
objective of achieving pre-dictable results based on predetermined
treat-ment goals the basic mechanics underlyingorthodontic
appliance activations must be thor-oughly understood.
From the Department of Orthodontics, School of Dentistry,
Vir-ginia C ommo nwealth University, Richmo nd, V A.Supported in
part by the Medical College of V b~ nia OrthodonticEducation and
Research Founda tion.Ad vss correspondence to Steven J. Lindau er,
D MD , M DSc,Department of Orthodontics, School of Dentistu,
Virginia Comm on-wealth University, Richmond, VA
23298-0566.Copyright 2001 by W.B. Saunders C ompany
1073-8746/01/0701-0002535.00/0doi: 10.1053/sodo . 2001.21053
P r e d ic t in g H o w T e e t h W i ll M o v ein Respo nse to
ppl icat ion ofOr t h o d o n t i c F or ce S y s t e m
sOrthodontic appliances are used to produceforce systems that will
displace teeth and initiatea biologic cascade allowing teeth to
move. Al-though many treatment modalities and philoso-phies
advocate differen t appliances the forcesystems they produce can be
dissected into thesame basic physical compone nts: forces and
mo-ments. Forces alone can be used to move teethand will often
produce moments causing theteeth to also rotate tip and torqu e. By
usingappliances to control rotation practitioners gainmore control
over how teeth move. Varying theratio of mom ent to force applied
allows the orth-odontist to vary the type of tooth
movementeffected.F o r c e sForces are vectors having both
direction andmagni tude . To move a toot h predictably a forceneeds
to be applied in the desired directionwith the desired magnitude
and at the correctposition on the tooth. Changing the
directionmagnitude or point of force application willaffect the
quality of the toot h d isplacement thatwill occur. In oth er words
if mesial move men t ofa tooth is indicated a force to push or pull
the
2 Seminars in Orthodontics, Vol 7 , No 1 (March ), 2001: pp
2-15
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Basics o f Mechanics
t o o t h i n a m e s i a l d i r e c t i o n m u s t b e a p p
l i e d . T h ef o r c e c a n a c t a n y w h e r e a l o n g i ts
l i n e o f a c t io n s op u s h i n g i s e q u i v a l e n t t o
p u l l i n g . I n c r e a s i n g t h em a g n i t u d e o f f o
r c e w i ll i n c r e a s e t h e a m o u n t o ft o o t h d i s p
l a c e m e n t i n it i al ly . H o w e v e r , i t i s u n -c l e
a r h o w f o r c e m a g n i t u d e i s r e l a t e d t o t h e r
a t eo f t o o t h m o v e m e n t , w h i c h i s a b i o l o g i
c a l l y c o n -t r o l l e d p h e n o m e n o n . 1,9
I n t ui t iv e l y, t h e p o i n t o f f o r c e a p p l i c a
t i o n a l s oh a s a n i n f l u e n c e o n t h e q u a l i ty o
f t o o t h m o v e -m e n t . T h e r e i s o n l y o n e p o i n
t o n a t o o t ht h r o u g h w h i c h a f o r c e c a n b e a p
p l i e d t h a t w i llm o v e t h e t o o t h i n t h e d i r e c
t i o n o f t h e f o r cew i t h o u t t i p p i n g o r r o t a t
i n g i t . T h i s p o i n t is th ec e n t e r o f r e s i s ta n
c e a n d a f o r c e a c t i n g t h r o u g h i tw i ll c a u s e
p u r e t r a n s l a t i o n o f t h e t o o t h a s s h o w ni n
F i g u r e 1 .
I f a t o o t h w e r e t r u l y a f r e e b o d y f l o a t i
n g i ns p a c e , t h e c e n t e r o f r e s i st a n c e w o u l
d b e c o i n c i -d e n t w i t h t h e t o o t h s c e n t e r o
f m a s s o r g r a vi ty .H o w e v e r , t e e t h a r e r e s t
r a i n e d b y s u r r o u n d i n gt is su e s, m o s t n o t a b
l y t h e p e r i o d o n t a l l i g a m e n t( P D L ) . T h e l
o c a t i o n o f t h e c e n t e r o f re s i s t an c e ,t h e r
e f o r e , d e p e n d s o n t h e s iz e a n d s h a p e o f t h
et o o t h a s w e l l a s o n t h e q u a l i t y a n d l e v e l
o f t h es u p p o r t i n g s t r u c tu r e s . I n a h e a l t h
y t o o t h w i t h a ni n t a c t P D L , t h e c e n t e r o f r
es i s t a n c e is p r e s u m e dt o b e s o m e w h e r e b e t
w e e n 1/3 a n d 1/2 t h e d i s t a n c ef r o m t h e a l v e o
l a r c r es t t o t h e r o o t a p e x . 3 F o r am a x i l l a r
y c e n t r a l i n c i s o r , t hi s is a p p r o x i m a t e l
y1 0 m m a p i c a l t o t h e l e v el o f p l a c e m e n t o f a
no r t h o d o n t i c b r a c k e t . T h e c e n t e r o f re s i
s t a n c e isl o c a t e d m o r e a p i c a l ly f o r a p e r i
o d o n t a l l y c o m -
igure 1 . C e n t e r o f r e s i s ta n c e , s h o w n t h r o
u g h o u t a s as o l id b l a ck d o t . A f o r c e a c t i n g
t h r o u g h t h e c e n t e r o fr e s i s t a n c e r e s u lt s
i n p u r e t r a n s l a t i o n o f a t o o t h .
B
igure 2 . T h e c e n t e r o f r e s is t a n c e in a t o o t
h w i t h f u llp e r i o d o n t a l s u p p o r t ( A ) . T h e c
e n t e r o f re s i s ta n c e ism o r e a p i c a l i n a p e r i
o d o n t a l l y c o m p r o m i s e d t o o t h(B) .p r o m i s e
d t o o t h w i t h l os s o f a t t a c h m e n t a s s h o w ni n
F i g u r e 2 . + 6
F o r c e s , as v e c t or s , c a n b e c o m b i n e d o r d
i -v i d e d m a t h e m a t i c a l l y . T w o o r m o r e f o r
c e s a c t in ga t a s in g l e p o i n t c a n b e a d d e d u s
i n g s i m p l et r i g o n o m e t r y o r v e c t o r a d d i t
i o n a n d r e p r e -s e n t e d a s a s in g l e f o r c e a t t
h a t p o i n t . F o r e x a m -p l e , s e p a r a t e d i s ta l
ly a n d e x t r u si v e l y d i r e c t e df o r c e s c an b e c
o m b i n e d i n t o o n e d i s t a l -e x t r u si v ef o r c e
a s s h o w n i n F i g u r e 3 . S i m i l a r ly , a s i n g l ed
i s t al - e x tr u s i ve f o r c e c a n b e r e s o l v e d m a
t h e m a t -i c al l y i n t o i ts d i s ta l a n d e x t r u s
iv e c o m p o n e n t s .oments
W h e n a f o r c e i s a p p l i e d a t an y p o i n t o t h e
r t h a nt h r o u g h t h e c e n t e r o f r e s i st a n c e , i
n a d d i t i o n t o
igure 3 . F o r c e s a r e v ec t o rs a n d c a n b e c o m b
i n e d o rr e s o l v e d m a t h e m a t i c a l l y .
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8/13/2019 2001 Lindauer
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4 S t e v e n J L i n d a u e r
m o v i n g t h e c e n t e r o f re s i s t a n c e i n t h e d
i r e c t i o no f t h e f o r c e, a m o m e n t i s c r e a t e d
. A m o m e n t i sd e f i n e d a s a t e n d e n c y t o r o t a
t e a n d m a y r e f e r t or o t a t i o n , t i p p i n g , o r
t o r q u e i n o r t h o d o n t i c t e r -m i n o l o g y . I f
a d i s t a l f o r c e i s a p p l i e d b u c c a l t ot h e c e
n t e r o f r es i s t an c e , t h e c e n t e r o f r e s i s t
an c eo f t h e t o o t h w i l l m o v e d i s t a l ly a n d t h
e t o o t h w i l lr o t a t e m e s i o b u c c a l l y F i g 4 A
) . I f a d i s t a l f o r c e i sa p p l i e d c o r o n a l t o
t h e c e n t e r o f r e s i s ta n c e , a sw h e n a n e l a s
ti c c h a i n i s s t r e t c h e d o v e r t h eb r a c k e t o n
a p r e m o l a r , t h e c e n t e r o f r e s i s ta n c ew i l l
m o v e d i s t a l ly a n d t h e c r o w n o f t h e t o o t h w
i l lt i p d i s t a l l y F i g 4 B ) . I f a n i n t r u s i v e
f o r c e i s a p -p l i e d f a c ia l to t h e c e n t e r o f re
s i s ta n c e , t h e c e n t e ro f r e s i s ta n c e w i ll i n
t r u d e a n d t h e c r o w n o f t h et o o t h w i l l t o r q
u e f a c i a l l y F i g 4 C ) .
A f o r c e a p p l i e d d i r e c t l y t h r o u g h t h e c
e n t e r o fr e s i s ta n c e w i l l c a u s e p u r e t r a n s
l a t i o n o f t h et o o t h i n t h e d i r e c t i o n o f t h
e f o r c e w i th n o r o t a -t i on . I f th e s a m e f o r c e
i s a p p l i e d a w a y f r o m t h ec e n t e r o f r e s i s ta
n c e , t o w a r d t h e c r o w n f o r e x a m -p l e , t h e t
o o t h w i l l m o v e i n t h e d i r e c t i o n o f t h ef o r
c e a n d t h e r e w i l l b e a m o m e n t c r e a t e d t o t i
pt h e c r o w n i n t h e d i r e c t i o n o f t h e fo r c e F i
g 5 A ) .I n c r e a s i n g t h e m a g n i t u d e o f t h e f o
r c e F i g 5 B )o r a p p l y i n g t h e s a m e f o r c e e v e
n f u r t h e r f r o mt h e c e n t e r o f r e s i s t a n c e F
i g 5 C ) w i ll i n c r e a s e t h et e n d e n c y f o r ro t a
t io n . T h e r e t o r e , t h e m a g n i t u d eo f a m o m e n
t M ) is e q u a l to t h e m a g n i t u d e o ft h e a p p l i e
d f o r c e F ) t i m e s t h e d i s t a n c e d ) o ft h a t f o
r c e f r o m t h e c e n t e r o f r e s i s t a n c e , M = F d
.T h e d i s t a n c e is a lw a y s m e a s u r e d p e r p e n d
i c u l a r l yf r o m t h e l i n e o f a c t i o n o f t h e f o
r c e t o t h e c e n t e ro f r e s i s t a n c e .
A B
~ a / ~ ....................
F i g u r e 4 . A f o r c e a p p l i e d a t t h e b r a c k e
t b l a c k ) w i l lr e s u l t i n b o t h a f o r c e a n d a m
o m e n t a t th e c e n t e r o fr e s i s t a n c e g r a y ) . O
c c l u s a l v i ew A ) , b u c c a l v i e w B ) ,m e s i a l v i
ew C ) . A p p l i e d f o r c e s a n d c o u p l e s a r es h o w
n i n b l a c k t h r o u g h o u t . E q u i v a l e n t f o r c e
s y st e m sf o rc e s a n d m o m e n t s ) a t t h e c e n t e r
o f r e s is t a n c e ar es h o w n i n g r a y .
A ',.mm ~ C 2,g-mmg,~-mm ~ $1 0 0 - - ~ :
1 mm 1 mm' ~ o o g e ~ 2O O g lO O g o
F i g u r e 5. T h e m a g n i t u d e o f a m o m e n t d e p e
n d s o nb o t h t h e m a g n i t u d e o f t h e a p p l i e d f
o r c e a n d i ts p e r -p e n d i c u l a r d i s t a n c e f r o
m t h e c e n t e r o f r e s i st a n ce ,M = F d A ) . T h e m a
g n i t u d e o f t h e m o m e n t i n c r ea s e sa s t h e f o r
c e i n c r e a s e s B ) o r t h e d i s t a n c e i n c r e a s e
sC ) .
ouplesP u r e r o t a t i o n o f a t o o t h w i t h o u t t r
a n s l a t i o n c a nb e a c c o m p l i s h e d b y a p p l y i
n g 2 e q u a l a n d o p p o -s i te f o rc e s t h a t a r e n o
t a c t i n g a l o n g t h e s a m el i n e. T w o e q u a l a n d
o p p o s i t e , n o n c o l i n e a rf o r c e s a r e c a l l e
d a c o u p l e . T h e 2 f o r c e s c a n c e lo u t a n y t e n
d e n c y f o r t h e c e n t e r o f r e s i s ta n c e o ft h e t
o o t h t o m o v e , b u t t h e m o m e n t s c r e a t e d byt h
e 2 f o r c e s d o n o t c a n c e l e a c h o t h e r . T h e t o
o t h ,t h e r e f o r e , r o t a t e s a b o u t i ts c e n t e r
o f r e si s t a n c er e g a r d l e s s o f t h e p o i n t o f a
p p l i c a t i o n o f th ec o u p l e . I f t h e 2 f o r c e s o
f t h e c o u p l e a c t o no p p o s i t e s i de s o f th e c e
n t e r o f r e si s t an c e , t h e i re f f e c t to c r e a t e
a m o m e n t i s a d d i t i v e . I f t h e y a r eo n t h e s a
m e s i d e o f t h e c e n t e r o f r e s is t a n c e , th e ya
r e s u b t r a c t i v e . E i t h e r w a y , n o n e t f o r c e
i s f e l t b yt h e t o o t h , o n l y a t e n d e n c y t o r o
t a t e . I n t h i s w a y ,a c o u p l e a p p l i e s a p u r e
m o m e n t t o a t o o t h .
T h e m a g n i t u d e o f t h e m o m e n t c r e a t e d by
ac o u p l e is d e p e n d e n t o n b o t h f b r c e m a g n i t
u d ea n d d i s t a n c e. T h e m o m e n t o f a c o u p l e i s
r e a l lyt h e s u m o f t h e m o m e n t s c r e a t e d b y e a
c h o f th e2 f o r c e s th a t m a k e u p t h e c o u p l e
:
M c o u p l c = M F o r c c 1 M v o r c e 2 o r M c o u p l c=
Fmd~ + Fzd2 .
F 1 a n d F 2 a r e e q u a l b u t o p p o s i t e . T h e r e
f o r e , t h em a g n i t u d e o f a m o m e n t c r e a t e d b
y a c o u p l e i sF d 1 + d 2 ) o r F ti m e s t h e d i s t a n c
e b e t w e e n t h e2 f o r c e s , M = F d .
A c o u p l e e x e r t s n o n e t f o r c e o n t h e c e n t
e r o fr e s i s ta n c e b e c a u s e t h e 2 f o r c es t h a t
c o m p r i s e i ta r e o p p o s i t e i n d i r e c t i o n a n
d c a n c e l e a c h o t h e r .T h e r e f o r e , a c o u p l e
a l o n e a l w a y s a c t s t o r o t a t e ,t ip , o r t o r q u
e t h e t o o t h a r o u n d t h e c e n t e r o f
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6 StevenJ Lindauer
o f t y p ic a l t o o t h m o v e m e n t , b u t t h e p r i n
c i p l e sd e s c r i b e d a r e r e l e v a n t to o r t h o d o
n t i c t o o t hm o v e m e n t f o r a n y p u r p o s e .
B y v a r y i n g th e r a t i o o f m o m e n t t o f o rc e a
p -p l i e d t o t ee t h , t h e q u a l i t y o f t o o t h m o v
e m e n tc a n b e c h a n g e d a m o n g t i p p in g , c ro w n
m o v e -m e n t , t r a n s l a t i o n , a n d r o o t m o v e m
e n t . 4,7,8W h e n e v e r a f o r c e i s a p p l i e d a t t h
e c r o w n o f at o o t h , a t e n d e n c y f o r t h e t o o t
h t o r o t a te , t i p , o rt o r q u e a m o m e n t ) i s a ls
o c r e a t e d . I n a d d i t i o nt o t h e f o r c e a p p l i
e d , a c o u p l e m a y a l s o b e e n -g a g e d i n t e n t i
o n a l l y t o p a r t ia l l y c o r r e c t , c o m -p l e t e l
y c o r r e c t , o r o v e r c o r r e c t t h is t e n d e n c y
. B yc h a n g i n g t h e r a ti o o f t h e m o m e n t f r o m t
h ea p p l i e d c o u p l e t o t h e f o r c e a p p l i e d , t
h e c e n t e ro f r o t a t i o n o f to o t h m o v e m e n t c a
n b e v a r i e d top r o d u c e t h e t yp e o f t o o t h m o v
e m e n t d e s i re d .
TippingD u r i n g s p a c e c l o s u r e , t h e r e i s n o p
r a c t i c a l, d i -r e c t w a y t o a p p l y a f o r c e d i r
e c t l y t h r o u g h t h ec e n t e r o f re s i s t a n c e o f
a t o o t h t o a c h i e v e p u r et r a n s l a t i o n w i t h
o u t t i p p i n g . T o m o v e a t o o t ht h r o u g h a n e x
t r a c t i o n s p a c e , a f o r c e i s a p p l i e d a tt h e
b r a c k e t t o p u l l th e t o o t h i n t h e d e s i r e dd i
r e c t i o n . T h e f o r c e a l o n e w i l l c a u s e t h e t
o o t h t om o v e i n t h e c o r r e c t d i r e c t i o n , b u
t t h e c r o w n w i l la l so t i p i n t h a t d i r e c t i o n
w h i l e t h e a p e x m o v e s i nt h e o p p o s i t e d i r e
c t i o n . I n t h is ca s e, n o m o m e n th a s b e e n a p p l
i e d t o p r e v e n t t h e t i p p i n g t h a to c c u r s. O n
l y a f o r c e h a s b e e n a p p l i e d F i g 8 A ) .T h e m o
m e n t - t o - f o r c e r a t i o M / F ) a p p l i e d i s 0 /
For 0
W h e n o n l y a f o r c e i s a p p l i e d a t t h e b r a c
k e t t om o v e a t o o t h , t h e e q u i v a l e n t f o r c e
s y s te m a t t h ec e n t e r o f re s i s t a n c e is th e f o
r c e p l u s a m o m e n tt h a t w il l t i p t h e c r o w n i n
t h e d i r e c t i o n o f t h e
B )Figure 8 . T i p p i n g . T h e a p p l i e d f o r c e sy s
t em A ) . T h ee q u i v a l e n t f o r c e sy s t e m a t t h e
c e n t e r o f r e s is t a n c e B ) .T h e p r e d i c t e d t o
o t h m o v e m e n t w i t h a c e n t e r o f ro t a -t i o n , s
h o w n t h r o u g h o u t a s a g r a y d o t , j u s t a p i c a
l to t h ec e n t e r o f r e s i s t a n c e b l a c k d o t s ) C
) .
B C
Figure 9 . C r o w n m o v e m e n t . T h e a p p l i e d f o r
c e s ys -t e m A ) . T h e e q u i v a l e n t f o r c e s y s te
m a t t h e c e n t e r o fr e s is t a n ce B ) . T h e p r e d i
c t e d t o o t h m o v e m e n t w i th ac e n t e r o f r o t a t
i o n a t t h e a p e x C ) .
f o r c e F i g 8 B ) . I f t h e c e n t e r o f r e s i s t a
n c e i s 1 0m m a p i c a l t o th e b r a c k e t , t hi s m o m
e n t is 1 0 m mt i m e s th e m a g n i t u d e o f t h e f o rc e
. F o r e x a m p l e ,a d i s t a l f o r c e o f 1 0 0 g a t t h
e b r a c k e t w i l l c r e a t ea n e q u i v a l e n t o f 1 0
0 g o f d i s ta l f o r c e a t t h ec e n t e r o f r e s i s ta
n c e p l u s a m o m e n t o f 1 ,0 0 0g - m m i n t h e d i r e c
t i o n o f d is t a l c r o w n t ip .
W i t h o u t a m o m e n t t o c o u n t e r a c t t h e t en
-d e n c y o f t h e t o o t h t o t i p i n t h e d i r e c t i o
n o f t h ef o r c e, t h e c e n t e r o f r e s i s t a n c e o f
t h e t o o t h m o v e si n t h e d i r e c t i o n o f t h e f o
r c e , t h e c r o w n m o v e sf u r t h e r t h a n t h e c e n
t e r o f re s i s t a n ce , a n d t h ea p e x a c t u a l ly m o
v e s i n a d i r e c t i o n o p p o s i t e t ot h e f o rc e . T
h e t o o t h a p p e a r s t o ha v e r o t a t e da r o u n d a p
o i n t j u s t a p i c a l t o i t s c e n t e r o f re s is -t a
n c e a s i l l u s t r a t e d in F i g u r e 8 C . T h e r e f o
r e , t h ec e n t e r o f r o t a t i o n w h e n t h e m o m e n
t t o f o r cer a t i o i s 0 i s j u s t a p i c a l t o t h e c e
n t e r o f r e s i s t a n c e .
r o wn M o v e m e n tK n o w i n g t h a t a f o r c e a p p l
i e d a t th e b r a c k e t t om o v e a t o o t h w i ll a l s o
r e s u l t in a t e n d e n c y t o t i p ,a c o u n t e r m o m e
n t i n th e f o r m o f a c o u p l e m a ya l so b e a p p l i e
d i n t e n t i o n a l l y t o c o u n t e r a c t t i p-p i n g .
I f t h e c e n t e r o f r e s i s t a n c e i s 1 0 m m a p i c a
lt o t h e b r a c k e t , a f o r c e a l o n e w i l l p r o d u
c e am o m e n t t h a t i s 1 0 m m t i m e s t h e m a g n i t u
d e o ft h e fo r ce . A p p l y i n g a c o u n t e r m o m e n t
o r c o u p lel e ss t h a n 1 0 m m t i m e s t h e m a g n i t u
d e o f t h ef o r c e w il l r e d u c e t h e t e n d e n c y o f
th e t o o t hc r o w n t o t ip i n t h e d i r e c t i o n o f t
h e f o r c e b u t n o tn e g a t e i t c o m p l e t e l y F i g
9 A ) . A n a p p l i e d M / Fo f a b o u t 7 / 1 w i ll r e s u l
t i n c r o w n m o v e m e n tw h i l e t h e a p e x o f t h e r
o o t r e m a i n s r e l a t iv e l ys t a t i o n a r y . 7
W h e n a M / F o f 7 / 1 i s a p p l i e d a t t h e b r a c k
e t ,t h e e q u i v a l e n t f o r c e s y s t e m a t t h e c e
n t e r o f r e -
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Basics o f Mechanics
s i s t a n c e is a f o r c e t o m o v e t h e t o o t h p l u
s a s m a l ln e t t e n d e n c y f o r t h e c r o w n t o t ip
in t h e d i r e c -t i o n o f t h e f o r c e F i g 9 B ) . F o r
e x a m p l e , i f a f o r c eo f 1 00 g is a p p l i e d t o m o
v e a t o o t h d i s ta l l y , t h e r ei s a t e n d e n c y f r
o m t h a t f o r c e f o r t h e t o o t h c r o w nt o t i p d is
t al l y o f 1 ,0 0 0 g - m m . A c o u n t e r m o m e n to r c o
u p l e o f 1 , 00 0 g - m m i n t h e o p p o s i t e d i r e c -t
i o n a ls o w o u l d n e e d t o b e a p p l i e d i n t e n t i
o n a l l yt o n e g a t e t h e t e n d e n c y t o t ip c o m p l
e t e l y . I f ac o u p l e o f o n l y 7 00 g - m m w e r e a p p
l i e d a t t h eb r a c k e t , t h e t o o t h w o u l d h a v e
a n e t t e n d e n c y f o rt h e c r o w n t o t i p d is t a l l
y o f 3 0 0 g - m m . A t t h ec e n t e r o f re s i s t an c e t
h e r e w o u l d b e a d i st a l f o r c eo f 10 0 g p l u s a m
o m e n t t o ti p t h e c r o w n d i s t a l l yo f 3 0 0 g - m m
. T h e m o m e n t o f t h e c o u p l e ) t of o r c e r a t i o
M / F ) a p p l i e d i s 7 0 0 / 1 0 0 o r 7 / 1 .
W i t h a n a p p l i e d M / F o f 7 / 1 , t h e t o o t h w i
l lm o v e i n t h e d i r e c t i o n o f t h e f o r c e a n d t
h e c r o w nw i ll t i p s o m e w h a t i n t h e s a m e d i r e
c t i o n . W h i l et h e c e n t e r o f r e s i s t a n c e i s
m o v i n g d i s t a l l y a s ar e s u l t o f t h e a p p l i e
d f o r c e , t h e c r o w n i s t i p p i n gd i s t a l ly a t a
g r e a t e r r a t e b u t t h e a p e x i s m o v i n gt o a le s
s e r e x t e n t o r n o t a t al l. T h e t o o t h a p p e a r
st o h a v e r o t a t e d a r o u n d i ts a p e x , a s i l l u s
t r a t e d i nF i g u r e 9 C . T h e r e f o r e , t h e c e n t
e r o f r o t a t i o n w h e nt h e m o m e n t t o f o r c e r a
t i o is 7 / 1 i s a t t h e a p e x o f t h et o o th , a n d o n
l y c r o w n m o v e m e n t o c cu r s.
P u r e T r a n s l a t i o nW h e n a f o r c e t o m o v e a t
o o t h i s a p p l i e d a t ab r a c k e t t h a t is 1 0 m m a w
a y f r o m t h e c e n t e r o fr e s i s t a n c e , a t e n d e
n c y f o r t h e t o o t h t o t i p i sc r e a t e d t h a t is 1
0 m m t i m e s th e m a g n i t u d e o ft h e f o r ce . T o c o
u n t e r a c t t h e t e n d e n c y f o r t ip -p i n g , a c o u
p l e c a n b e a p p l i e d i n t e n t i o n a l l y t op r o d
u c e a m o m e n t o f e q u a l m a g n i t u d e i n t h eo p p
o s i t e d i r e c t i o n F i g 1 0 A ). T h e f o r c e a l o n
e
B C
F i g u r e 1 0 . P u r e t r a n s l a t i o n . T h e a p p l
i e d f o r c e s y s t emA ) . T h e e q u i v a l e n t f o r c e
s y s t e m a t t h e c e n t e r o fr e s is t a nc e B ) . T h e
p r e d i c t e d t o o t h m o v e m e n t C ) . I np u r e t r a
n s l a t i o n , t h e c e n t e r o f r o t a t i o n i s a t i n
f in i t y .
B
F i g u r e 1 1 R o o t m o v e m e n t . T h e a p p l i e d f
o r c e sy s te mA ) . T h e e q u i v a l e n t f o r c e s y s te
m a t t h e c e n t e r o fr e s is t a n ce B ) . T h e p r e d i
c t e d t o o t h m o v e m e n t w i th ac e n t e r o f r o t a t
i o n a t t h e c r o w n C ) .
w o u l d c a u s e t h e t o o t h t o m o v e i n t h e d i r
e c t i o no f t h e f o r c e a n d t h e c r o w n t o t ip i n t
h e s a m ed i r e c t i o n . T h e c o u p l e c o m p l e t e l
y n e g a t e s th i st e n d e n c y t o t ip , b u t t h e t o o
t h s ti ll m o v e s i n t h ed i r e c t i o n o f t h e f o r c
e. T h e M / F a p p l i e d is 1 0 / 1 ,a n d t h e t o o t h t r
a n s l a t e s w i t h o u t t i p p i n g .
W h e n a M / F o f 1 0 / 1 i s a p p l i e d a t t h e b r a c
k e t ,t h e e q u i v a l e n t f o r c e s y s t e m a t t h e c
e n t e r o f r e -s i s t a n c e is a s i n g l e f o r c e w i t
h n o n e t m o m e n t
F i g 1 0 B ). F o r e x a m p l e , i f a f o r c e o f 1 00 g
i sa p p l i e d a t t h e b r a c k e t t o m o v e a t o o t h d
i s t a l ly ,t h e e q u i v a l e n t f o r c e s y s t e m a t
th e b r a c k e t i s1 0 0 g o f d i s ta l f o r c e p l u s a t
e n d e n c y f o r t h ec r o w n t o t i p in t h e d i r e c t i
o n o f t h e f o r c e o f 1 ,0 0 0g - m m . I f a c o u p l e w i
t h a c o u n t e r m o m e n t o f1 ,0 0 0 g - m m t e n d i n g t
o ti p th e c r o w n i n t h ed i r e c t i o n o p p o s i t e t
o t h e f o r c e w e r e a l so a p p l i e di n t e n t i o n a l
l y , t h e n e t e q u i v a l e n t f o r c e s y s t e mw o u l
d b e a s i n g l e d i s t a l f o r c e o f 1 0 0 g t h r o u g
ht h e c e n t e r o f r e s i st a n c e . T h e M / F a p p l i e
d is1 , 0 0 0 / 1 0 0 o r 1 0 / 1 .
W i t h a n a p p l i e d M / F o f 1 0 / 1 , t h e t o o t h w
i llt r a n s l a te i n t h e d i r e c t i o n o f th e f o r c e
w i t h o u tt i p p i n g . T h i s i s o f t e n r e f e r r e d
t o a s p u r e t r a n s -l a t i o n a n d i s s h o w n i n F i
g u r e 1 0 C . I n p u r e t r a n s -l a t io n , t h e c e n t e
r o f r o t a t i o n i s c o n s i d e r e d t o b ea t in f i n i
ty b e c a u s e n o r o t a t i o n o c c u r s.
R o o t M o v e m e n tW h e n t h e c o u n t e r m o m e n t a
p p l i e d i n t e n t io n a l l ya t a b r a c k e t is m o r e
t h a n 1 0 m m t i m e s t h em a g n i t u d e o f t h e f o rc e
a p p l i e d , th e t o o t h m o v e si n t h e d i r e c t i o n
o f th e f o r c e b u t t h e c r o w n t ip s i nt h e o p p o s
i t e d i r e c t i o n F i g l l A ) . T h e f o r c ea l o n e w
o u l d c a u s e t h e t o o t h t o m o v e i n t h ed i r e c t
i o n o f t h e f o r c e a n d t h e c r o w n t o t i p i n t h
es a m e d i r e c t i o n . T h e c o u p l e a p p l i e d m o r
e t h a n
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8 StevenJ Lindauer
n e g a t e s t h i s t e n d e n c y t o t ip , b u t t h e c e
n t e r o fr e s i s t a n c e s t il l m o v e s i n t h e d i r e
c t i o n o f t h ef o r c e w h i l e t h e c r o w n d o e s n o
t . A n a p p l i e d m o -m e n t - t o - f o r c e r a t i o o f
a b o u t 1 3 / 1 w i ll r e s u l t i nr o o t m o v e m e n t w h
i le t h e c r o w n o f t h e t o o t hr e m a i n s r e l a ti v
e l y s t a t i o n a r y . 7
W h e n a M / F o f 1 3 / 1 is a p p l i e d a t th e b r a c k
e t,t h e e q u i v a l e n t f o r c e s y s t e m a t th e c e n
t e r o f re -s i s t a n c e is a f o r c e t o m o v e t h e t o
o t h p l u s a s m a l ln e t t e n d e n c y f o r t h e r o o t
t o ti p in t h e d i r e c t i o no f t h e f o r c e F i g 1 1 B
) . F o r e x a m p l e , i f a f o r c e o f1 0 0 g i s a p p l i
e d t o m o v e a t o o t h d i s t a ll y , t h e r e i sa t e n d
e n c y f r o m t h a t fo r c e f o r th e t o o t h c r o w nt o
ti p d is ta ll y o f 1 ,0 0 0 g - m m . A c o u n t e r m o m e n
to r c o u p l e o f 1 ,0 0 0 g - m m i n t h e o p p o s i t e d i
r e c-t i o n w o u l d a l so n e e d t o b e a p p l i e d t o n
e g a t e t h et e n d e n c y t o t ip c o m p l e t e l y . I f a
c o u p l e o f 1 ,3 0 0g - m m w e r e a p p l ie d , t h e t e n
d e n c y t o ti p w o u l d b em o r e t h a n c o u n t e r a c t
e d a n d t h e r e w o u l d b e an e t t e n d e n c y f o r t h
e c r o w n t o ti p m e s ia l ly o f 3 0 0g - m m . T h e e q u i
v a l e n t f o r c e s y s te m a t t h e c e n t e ro f r e s i s
t a n c e i s a d i s t a l f o r c e o f 1 0 0 g p l u s am o m e
n t t o ti p th e c r o w n m e s i a ll y o f 3 0 0 g - m m .T h e
M / F a p p l i e d a t t h e b r a c k e t is 1 , 3 0 0 / 1 0 0 o
r1 3 / 1 .
W i t h a n a p p l i e d M / F o f 1 3 / 1 , t h e t o o t h w
il lm o v e i n th e d i r e c t i o n o f t h e f o r c e a n d t
h e r o o tw i ll t i p i n t h e s a m e d i r e c t i o n . W h i
l e t h e c e n t e ro f r e s i s t a n c e is m o v i n g d i s
ta l ly a s a r e s u l t o f t h ea p p l i e d f o r c e , t h e
r o o t i s t i p p i n g d i s ta l ly a t ag r e a t e r r a t e
b u t t h e c r o w n is m o v i n g t o a l e s s e re x t e n t o
r n o t a t all . T h e t o o t h a p p e a r s t o h a v er o t a
t e d a r o u n d i ts c r o w n a s i l l u s t ra t e d i n F i g
u r e1 1 C. T h e r e f o r e , t h e c e n t e r o f r o t a t i o
n w h e n t h em o m e n t - t o - f o r c e r a t io i s 1 3 /1 is
a t th e c r o w n o ft h e t o o t h a n d o n l y r o o t m o v e
m e n t o c c u r s.
Pure RotationI f o n l y a c o u p l e , a n d n o n e t f o r c
e , is a p p l i e d t oa to o t h , t h e t o o t h w i ll r o t a
t e a r o u n d its c e n t e r o fr e s i s t a n c e a n d t h e
t o o t h w i ll n o t t r a n s l a t e F i g1 2 A ) . B e c a u s
e t h e a c t i o n o f a c o u p l e d o e s n o td e p e n d o n
i ts p o i n t o f a p p l i c a t io n , a p u r e m o -m e n t a
lw a y s a c t s a t t h e c e n t e r o f r e s i s t a n c e F i
g1 2 B ) . T h e f o r c e s o f t h e c o u p l e c a n c e l o u
t a n yt e n d e n c y f o r t h e c e n t e r o f r e s i st a n c
e o f t h et o o t h t o m o v e , b u t t h e m o m e n t p r o d
u c e d b y t h ec o u p l e c a u s es t h e t o o t h t o ti p as
s h o w n i n F i g u r e1 2 C . T h e m o m e n t - t o - f o r c
e r a t io i s i n f in i te a n dt h e c e n t e r o f r o t a t i
o n i s c o i n c i d e n t w i t h t h e
{ { 6 [ * .
Figure 1 2 . P u r e r o t a t i o n . T h e a p p l i e d f o r
c e s y st e mA ) . T h e e q u i v a l e n t f o r c e s y s t e m
a t t h e c e n t e r o fr e s i st a n c e B ) . T h e p r e d i c
t e d t o o t h m o v e m e n t w i t h ac e n t e r o f r o t a t
i o n a t t h e c e n t e r o f r e s i s ta n c e C ) .c e n t e r
o f re s i s t a n c e . T h i s i s r e f e r r e d t o as p u r
er o t a t i o n .
P r e d i c ti n g t h e F o r c e S y s t e m s C r e a t ed b
yO r t h o d o n t i c p p l i a n c e c t i v a t i o n sT h e o b
j e ct i v e o f a c t iv a t in g a n o r t h o d o n t i c a p p
li -a n c e is t o p r o d u c e f o rc e s a n d m o m e n t s t o
m o v et e e t h in a p r e d i c t a b l e a n d c o n t r o l l e
d m a n n e r .T h e a p p l i a n c e i t s el f o b e y s t h e l
aw s o f p h y s ic s . I ta c t s b y e f f e c t i n g f o r c e
s y s t e m s a t al l p o i n t s o fa t t a c h m e n t . I t is
n o t p o s s i b le t o fu l ly u n d e r s t a n dt h e r e s p o
n s e t o a p p l i a n c e a c t i v a t i o n b y o b s e r v -i
n g th e d i s p l a c e m e n t a t o n e e n d o f a n a p p l i
a n c ew i t h o u t c o n s i d e r i n g t h e e f f ec t s at t
h e o t h e r .
T h e f o r c e sy s te m s p r o d u c e d b y o r t h o d o n
t i ca p p l i a n c e a c t iv a t io n s m u s t b e r e s o l v
e d s e p a r at e l yf r o m t h e a c t u a l f o rc e s a n d m
o m e n t s t h a t i n d i-v i d u a l t e e t h w i ll e x p e r
i e n c e a t t h e i r r e s p e c t i v ec e n t e r s o f r e s
is t a nc e . U n d e r s t a n d i n g t h e p h y si c st h a t d
e t e r m i n e t h e f o r c e s y s te m s g e n e r a t e d b ya
c t iv a t i ng o r t h o d o n t i c a p p l i a n c e s d i sc l
os e s t h ef o r c e s a n d m o m e n t s t h a t a re a p p l i
e d t o te e t h a tt h e i r p o i n t s o f a t t a c h m e n t ,
u s u a ll y a t th e b r a c k -e ts . T o d e d u c e h o w e a c
h t o o t h w i ll b e d i s p l a c e da s a r e s u l t o f t h e
a c t i v a t i o n , t h e e q u i v a l e n t f o r c es y s t e
m s a c t i n g a t t h e c e n t e r s o f r e s i s t a n c e m u
s tb e a s s e s s e d s u b s e q u e n t l y .Sta t i c qu i l i
b r i umW h e n o r t h o d o n t i c a p p l i a n c e s a r e a c
t iv a t e d a n di n s er t ed , t o o t h m o v e m e n t d o e s
n o t o c c u r i m m e -d i a te l y b u t p r o c e e d s s lo w
ly o v e r a n e x t e n d e dp e r i o d o f t i m e . F o r t h i
s re a s o n , t h e p h y s i c a l la w so f s ta ti cs a r e c o
n s i d e r e d a d e q u a t e t o d e s c r ib et h e i n s t a n
t a n e o u s f o r c e s y s te m s p r o d u c e d b yo r t h o d
o n t i c a p p l i a n c e s . T h e s e a r e t h e c o n d i
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Basics of Mechanics
tions under which analysis of orthodontic forcesystems occurs.
The laws of statics can not beused to describe how the force
systems willchange as a tooth moves and an appliance deac-tivates
and alters its configuration.
The physical laws of statics are governed byNewton s third law,
which may be sum mari zed asstating that for every action there
must be anequal and opposite reaction. The orthodonticapplication
is that for every appliance, but notnecessarily for every tooth to
which it is attached,the sum of the forces and the sum of the
mo-ments must be equa l to zero. 9 Tha t is, becausethe appliance
itself does not move instanta-neously once it is placed, the net
force systemproduced by the appliance as a whole mustbe equal to
zero. This condition must be metby every orthodontic appliance
regardless ofthe treatment philosophy used. It is impossibleto
design an appliance that defies the laws ofphysics.E qua l an d O
pposi te F orcesPerhaps the simplest orthodontic appliance
toanalyze is the elastic band. An elastic bandstretc hed bet ween
two points of att ach men t willproduc e a force of some magnit ude
at one endand, by the laws of equilibrium, produce anopposite force
of the same magnitude at theother end.
As far as the elastic ba nd is con cer ned , it is
inequilibrium. The sum of the forces produ ced bythe elastic, equal
and opposite, is zero. The elas-tic itself produces no m oments, so
the sum o fthe moments is also zero and the conditions ofstatic
equilibrium are met.One Couple App l iances Sta t ica l l yD eterm
in ate SystemsA 1-couple orthodontic appliance is insertedinto a
bracket or tube at 1 end and is tied as apoin t c onta ct at the
other, s Because it is notengaged into an orthodontic bracket, the
endthat is tied as a point contact cannot produce acouple but only
a simple force at that site. Theother end, which is engaged in the
bracket slot,can produce both a force and a couple at
thatattachment. The appliance is a 1-couple systembecause a couple
is generated only at the site offull engagement. It is statically
determinate be-cause the magnitudes of the forces and t o o
ments produced can be determined clinicallyafter the appliance
is inserted into the bracket.This can be done by inserting the
appliance intothe bracket and measuring the force required
toactivate the wire to the site where it will be tiedas a point
contact.
An example of a 1-couple orthodontic appli-ance is a long arm or
cantilever activated toextrude a high maxillary buccal canine as
dem-onstrated in Figure 13A. The wire is insertedinto the molar
auxiliary tube and bent to restcoronal to the canine. It is
activated by displac-ing the wire apically and tying it to the
caninebracket to create an extrusive force as in Figure14. In a
1-couple system, the direction of theforces and couple exerted by
the appliance areintuitive and can be deduced in a number ofw a y s
.
The most consistent way to derive the forcesystem produced by
any orthodontic applianceis to remove the activated wire from its
at-tachments and lay it passively over the attach-ment sites. The
angle formed by the wire andthe brackets will show the direc tion
of thecouple pro duc ed at tile site of enga geme ntwhere the angle
between the wire and bracketis largest. 11-~3
Because there is only one site of enga geme nt
t 20 @1 000 g-ram
Figure 13. A long arm or cantilever to extrude a highcanine (A).
The passive wire showing the angle be-tween the molar bracket and
the wire (B). The forcesystem exerted by the wire is in equilibrium
with acrown-mesial moment at the molar, and equal andopposite
extrusive and intrusive forces t the canineand molar, respectively
(C).
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10 Steven J. Lindauer
Figure 14. Clinical example of a wire tied as a pointcontact to
extrude a high canine.in a 1-couple system, the direction of the c
oupleat that site depends only on the angle formedbetween that
bracket and the wire. As shown inFigure 13B, the molar bracket must
rotate coun-terclockwise to engage the wire. The couple pro-duced
by the wire at the molar will therefore bein a direction to tip the
molar crown mesiallyand root distally. The laws of equilibrium
dictatethat the sum of the mo ments exerted by the wirebe equal to
zero. With a counterclockwise mo-ment at the molar, there must be a
clockwisemome nt to maintain equilibrium. That mo men tis produced
by intrusive and extrusive forces ofequal magnitude exerted by the
wire at the mo-lar and canine, respectively, as shown in
Figure13C.
To furt her illustrate the state of equilibr iumof the
appliance, assume that a 50-g extrusiveforce is exerted by the wire
at the canine. Thiscan be measured using a force gauge after
in-serting the wire into the molar tube and beforetying it to the
canine. An intrusive force of 50 gmust, therefore, be present at
the molar to havethe sum of the forces exerted by the
applianceequal to zero. If the mol ar and canine are 20mm apart, a
couple of 50 g times 20 mm or 1,000g-mm is exerted by the appliance
as a whole in aclockwise direction. To have the sum of the mo-ments
equal to zero, a counterclockwise coupleof 1,000 g-ram is exerted
by the appliance at themolar to tip its crown mesially and root
distally,as shown in Figure 13C.Two Couple Appl iancesmStat
icaUyndeterm in ate SystemsA 2-couple appliance is one that is
engaged intoattachments at both ends. A couple, therefore,may be
generated by the wire at either or both
attachment sites. In this case, inserting the wireinto 1
attachment and using a gauge to measurethe force required to
deflect the wire to theother bracket will not necessarily give an
accu-rate assessment of the force produced by theappliance. Because
of the inability to measureforce systems produced by 2-couple
appliancesclinically, they are referred to as being
staticallyindeterminate.
The force systems prod uce d by 2-couple orth-odontic appliances
depend on both the wiregeometry and bracket angulation
relationships.A rough idea of the directions of the forces
andmoments exerted by a 2-couple appliance maybe deduced by the
method described above: thewire is passively placed over both
attachmentsites to determine the angle forme d between thebrack et
slot and the wire at each end. If theangles formed at the 2
attachments are equaland opposite, equal and opposite couples will
begenerated and no force will be produced at ei-the r site, as show
n in Figure 15A. If the an glesare equal and in the same direction,
then bothsites will experience couples in the same direc-tion of
equal magnitude. The forces produc ed atthe attachment sites will
be equal in magnitude
_ _ _
Figure 15. Force systems from a 2-couple appliancedep end on
wire-bracket geometry. Equal and oppo-site bracket-wire angle
relationships result in equaland opposite couples with no forces
A). Equal brack-et-wire angle relationships in the same direction
resultin equal couples in the same direction with largeforces to
maintain appliance equilibrium B). Ifbracket-wire angle
relationships are unequal, the at-tachment with the largest angle
will have the largestcouple and resultant forces will be in a
directionopposite this moment to maintain equilibrium C).
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Basics of M echanics
and opposite in direction to each other. To-gether, the forces
will form a couple on the wireto maintain equilibrium of the
appliance as awhole. That couple formed by the 2 forces willbe
opposite in direction and equal in magnitudeto the sum of those
couples produ ced at theindividual attachment sites, as shown in
Figure15B.
If the angles fo rme d betwee n the passivelyplaced wire and the
2 bracket slots are not equalin magnitude, then the site with the
greatestangle can be determined. This will be the site atwhich the
greatest couple will be generated. Thedirection of the mom ent
produce d at that toothwill be that which will bring the bracket to
thewire. The forces at the 2 atta chme nt sites will beequal in
magnitude and opposite in direction toeach other as always.
Together, those forces willform a couple on the wire to maintain
equilib-rium of the appliance as a whole. The coupleformed by the 2
forces will be opposite in direc-tion to the cou ple exe rted at
the site of attach-ment with the greatest bracket-wire angulationas
shown in Figure 15C. The ma gnit ude of theforces and the direction
of the couple at theother attachment site, however, may not be
ap-parent from clinical observation alone.
The relative momen ts and forces produc ed by2-couple appliance
activations are extre mely sen-sitive to clinical geometry. They
have been de-te rmi ne d for bo th 2-dime nsion al 14-16 an d
3-di-mens iona P 7,1s conditions.
The forces and moments produced by astraight segment of round
wire inserted into 2orthodontic brackets were first described by
Bur-stone and Koenig in 1974.14 Wh en the 2 bracketsare equally but
oppositely angled as shown inFigure 16A, the resultant couples are
mirrorimages of each other their mathematical sum iszero) and no
forces are produced. As the angu-lation of the bracket on the right
is varied whilethe left bracket is held constant, a static
analysiscan be used to determine the relative momentsand
equilibrium forces that result. A nonlinearreduction of the right
moment occurs as theangulation of the right bracket is
decreasedslightly in Figure 16B. As the absolute sum of the2 momen
ts increases because of the cha nge inrelative bracket angulations,
proportional in-creases in the forces are noted that maintain
theconditions of equilibrium Fig 16B). Whe n theangle of the right
bracket is half that of the left,
/ ,
C / ,/ /
B, % / ' , . -- ,,
--, D /E
/ / * / /
TFigure 16. Force systems from a 2-couple appliance:straight
segment of wire in malaligned brackets. Equaland oppositely angled
brackets resuh in equal andopposite couples A). As 1 bracket
angulation de-creases, the magnitude of the couple at that
bracketdecreases and forces result B). When the angnlationof 1
bracket is half that of the other, no couple isexperienced at the
bracket with the smaller angula-tion and forces are consequently
increased C). As thebracket angle continues to decrease at 1 end of
theappliance, the couple at the lesser angled bracket is inthe same
direction as at the greater angled bracketand forces are even
greater D). When both bracketsare equally angled in the same
direction, couples atthe 2 brackets are equal and in the same
directionwith forces at a maximum E).but still opposite in
direction, no couple ispresent at the right bracket, as illustrated
inFigure 16C. Moments at the right bracket in thesame direction as
the left bracket couple in-crease as the right bracket angle
decreases tozero and then increases to mimic the left bracketangul
ation Fig 16D). Finally, when bo th brack-ets are equal in
direction and angulation, cou-ples are also equal their mathemat
ical sum ismaximal), and equilibrium forces are at a max-imu m Fig
16E).
Analogously, the static force systems gener-ated by placing
wires with V- and step-bends intoaligned, coplanar brackets have
been well docu-ment ed. 15 In a 2-dimensional mode l using
2brackets, a symmetrically placed V-bend pro-duces equal and
oppositely directed couples, asshown in Figure 17A. Asymmetri c
bends result invarious combinations of moments and forces.Moving
the be nd slightly off cent er results in anincrease in the couple
generated at the bracketcloser to the bend and a decrease at the
bracket
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1 2 Steven J Lindauer
D
lFigure 1 7 . F o r c e s y s te m s f r o m a 2 - c o u p l e a
p p l i a n c e :b e n t w i r e i n a l i g n e d b r a ck e t s .
W h e n t h e a p e x o f t h e Vb e n d i s h a l f w a y b e t w
e e n t h e b r a c k e t s , t h e w i r e e x e r t se q u a l a
n d o p p o s i t e c o u p l e s a t t h e 2 a t t a c h m e n t s
a n dn o f o r c e s ( A ) . A s t h e V b e n d i s m o v e d o f
f c e n t e r , t h ec o u p l e n e a r e r t h e b e n d i n c r
ea s e s a n d t h e c o u p l ef u r t h e r f r o m t h e b e n d
d e c r e a s e s . F o r c e s r e s u l t t om a i n t a i n a p
p l i a n c e e q u i l i b r i u m ( B ). W h e n t h e Vb e n d i
s a t t h e d i s t a n c e b e t w e e n t h e b r a c k e t s ,
ac o u p l e i s o n l y p r e s e n t a t t h e b r a c k e t n e
a r e r t h e b e n d .N o c o u p l e i s e x p e r i e n c e d a
t t h e b r a c k e t f u r t h e r f r o mt h e b e n d , a n d f
o r c e s r e s u l t t o m a i n t a i n e q u i l i b r i u m( C
) . W h e n t h e V b e n d i s v e r y c l o s e to 1 b r a c k e
t ,c o u p l e s in t h e s a m e d i r e c t i o n a r e e x p e r
i e n c e d a t b o t hb r a c k e t s , w i t h t h e l a r g e r
c o u p l e a t t h e b r a c k e t c l o s e rt o t h e b e n d .
F o r c e s i n c r e a s e a s t h e b e n d i s m o v e df u r t
h e r o f f c e n t e r ( D ) . A s t e p b e n d , r e g a r d l e
s s o fw h e r e i t i s p l a c e d , r e s u l t s i n e q u a l
c o u p l e s i n t h e s a m ed i r e c t i o n a t b o t h a t t
a c h m e n t s . F o r c e s a r e a t a m a x i -m u m ( E )
.
f u r t h e r a w a y , a s s h o w n i n F i g u r e 1 7 B . T
h e s u mo f t h e m o m e n t s f r o m t h e 2 c o u p l e s i s
n o l o n g e rz e r o, s o f o r c e s a r e g e n e r a t e d t h
a t p r o d u c e ac o u n t e r a c t i n g c o u p l e o n t h e
w i r e as a w h o l e t om a i n t a i n e q u i l i b r i u m c o
n d i t i o n s . W h e n t h eb e n d i s l o c a t e d a t 1/3 o
f t h e d i s t a n c e b e t w e e n 2b r a c k e t s , b e c a u
s e o f t h e d e f l e c t i o n a n g l e o f t h ea c t i v a t
e d w i r e , a c o u p l e r e s u l t s a t o n l y t h eb r a c
k e t n e a r e r t h e b e n d ( F i g 1 7 C ). F o r c e s in -c
r e a s e to m a i n t a i n e q u i l i b r i u m . B e n d s c l
o s e r t ot h e b r a c k e t r e s u l t in c o u p l e s a t b o
t h b r a c k e t s i nt h e s a m e d i r e c t io n w i th t h e
g r e a t e r m o m e n tg e n e r a t e d a t t h e b r a c k e t
c l o s e r t o th e b e n d ( F i g1 7 D ) . T h e 2 c o u p l e s
a r e n o w a d d i t iv e , a n d e v e ng r e a t e r f o r c e s
r e s u l t a t th e 2 a t t a c h m e n t s t om a i n t a i n e q
u i l i b r i u m o f t h e a p p l i a n c e . S t epb e n d s a r
e l e ss s e n s i t iv e t h a n V b e n d s t o l o c a t i o na
n d r e s u l t i n m o m e n t s o f e q u a l m a g n i t u d e a
n d
d i r e c t i o n a t b o t h b r a c k e ts i n d e p e n d e n
t o f w h e r et h e y a r e p l a c e d , a s s h o w n i n F i g
u r e 1 7 E. W i t hc o u p l e s o f e q u a l m a g n i t u d e p
r o d u c e d a t b o t ha t t a c h m e n t s it e s i n t h e s a
m e d i r e c t i o n , f o rc e sa r e i n c r e a s e d f u r t h
e r t o m a i n t a i n e q u i l i b r i u mc o n d i t i o n s
.
T h e f o r c e s a n d m o m e n t s s h o w n i n F i g u r e
s 16a n d 1 7 a re c o r r e c t f o r 2 - c o u p l e a p p l i a
n c e s a c ti v ei n a n y 1 d i m e n s i o n . T h e b r a c k e
t s , t h e r e f o r e , a r es h o w n w i t h o u t t e e t h t
o e m p h a s i z e t h a t t h e f o r c es y s t e m s m a y b e
a c t i n g i n t h e o c c l u s a l , l a t e r a l , o rf r o n
t a l p l a n e s . R e l a t i v e s i ze s o f th e a r r o w s a
n dc u r v es r e f l e c t d i f f e r e n c e s i n f o r c e a n
d m o m e n tm a g n i t u d e s , r e s p ec t i ve l y .
P r e d ic t in g H o w T e e t h W i ll M o v e i nR e s p o n
s e t o Or t h o d o n t i c p p l i a n c ec t i v a t i o n sT h
e f i r s t s e c t i o n o f t h i s a r t i c l e , P r e d i c t
i n g H o wT e e t h W i l l M o v e i n R e s p o n s e t o A p p
l i c a t i o n o fO r t h o d o n t i c F o r c e S y st em s , d
e s c r i b e d h o w m o -m e n t s a n d f o r c e s d i s p l a
c e i n d i v i d u a l t e e t h. T h es e c o n d s ec t i o n ,
P r e d i c t i n g t h e F o r c e S y s te m sC r e a t e d b y O
r t h o d o n t i c A p p l i a n c e A c t iv a t i on s ,d i s c
u s s e d th e w a y i n w h i c h m o m e n t s a n d f o r c e sa
r e g e n e r a t e d b y o r t h o d o n t i c a p p l i a n c e s
. T h i sf i n a l s e c t i o n s h o w s h o w t h e s e t w o s
t e p s c a n b eu s e d t o g e t h e r t o p r e d i c t h o w t
e e t h w i ll m o v ew h e n o r t h o d o n t i c a p p l i a n c
e s a re a c ti v a te d . C o n -v e r s el y , t h e 2 s t e p s
c a n b e u s e d t o d e s i g n a na p p l i a n c e t h a t w i
ll m o v e t e e t h i n a p r e d i c t a b l ew a y. I t is i m p
o r t a n t t o d i s t i n g u i s h b e t w e e n t h e 2c o n c
e p t s : t e e t h m o v e i n r e s p o n s e t o f o r c e s a n
dm o m e n t s a p p l i e d at t h e i r b r ac k e t s , a n d o
r t h -o d o n t i c a p p l i a n c e s a r e a c t i v a t e d to
t r a n s f e rf o r c e s a n d m o m e n t s t o th e b r a c k e
t s o f t e e t h.
P r e d i c t i n g t h e R e s u l t o f pp l i ance c t i va t
i onU s i n g t h e l aw s o f s t at i c m e c h a n i c s , t h e
f o r c es y st em s g e n e r a t e d b y a n o r t h o d o n t i
c a p p l i a n c ea r e c a l c u l a t e d t o d i s c e r n t h
e f o r c e s a n d m o -m e n t s t r a n s f e r r e d t o t e e
t h a t t h e i r p o i n t s o fa t t a c h m e n t . S u b s e q
u e n t l y , t h e f o r c e s a n d m o -m e n t s a p p l i e d
a t t h e b r a c k e t s o f t e e t h a r e u s e dt o p r e d i
c t h o w t h e t e e t h w i ll b e d i s p l a c e d i n i-t ia l
ly a n d e v e n t u a l l y m o v e a s a r e s u lt o f b o n er
e m o d e l i n g .
A s a n e x a m p l e o f p r e d i c t i n g h o w t e e t h w
i llm o v e i n r e s p o n s e t o a p p l i a n c e a c t iv a t
i o n, a no r t h o d o n t i c w i r e u s e d t o m o v e a m a x
i l l a r y ca -
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Basics of Mechanics 1 3
n i n e b u c c a l l y , o u t o f l i n g u a l c r o s s b i
t e , i s i l lu s -t r a t e d i n F i g u r e 1 8. T h e a p p l
i a n c e i s a c t i v a t e d bym a k i n g a m e s ia l in - b e
n d n e a r t h e m o l a r a n di n s e r t i n g t h e w ir e i n
to t h e m o l a r t u b e . T h e w i r eis t i e d t o t h e c a
n i n e b u t n o t i n s e r t e d i n t o t h ec a n i n e b r a
c k e t .
B e c a u s e t h e w i r e is e n g a g e d o n l y at t h e m
o l a ra t t a c h m e n t , a c o u p l e is g e n e r a t e d b y
t h e w i reo n l y at t h e m o l a r . T h e c o u p l e is i n a
d i r e c t i o nt h a t w il l r o t a t e t h e m o l a r c r o w
n m e s i o l i n g u a l l y .E q u a l a n d o p p o s i t e l a
t e r a l fo r c e s in a d i r e c t i o nt o c o u n t e r a c t
t h e c o u p l e w i ll b e g e n e r a t e d b yt h e w i re t o
m o v e t h e c a n i n e b u c c a l l y a n d m o l a rl i n g u
a l l y , a s s h o w n i n F i g u r e 1 8 A . T h e b u c c a lf
o r c e a t t h e c a n i n e c a n b e m e a s u r e d u s i n g
af o r c e g a u g e o n c e t h e w i r e is i n s e r t e d i n t
o t h em o l a r t u b e . I f t h e f o r c e m e a s u r e s 5 0
g a n d t h ed i s t a n c e f r o m t h e m o l a r t o t h e c a
n i n e i s 2 0 m m ,t h e m e s i o l i n g u a l c o u p l e a t
t h e m o l a r i s 5 0 g 2 0 m m o r 1 ,0 00 g - m m i n m a g n i
t u d e . T h e l i n g u a lf o r c e a t t h e m o l a r b r a c
k e t i s 5 0 g, a n d t h e b u c c a lf o r c e a t t h e c a n i
n e b r a c k e t i s 5 0 g .
T h e s u m o f t h e f o r c e s e x e r t e d b y t h e w i r
e a n dt h e s u m o f t h e m o m e n t s e x e r t e d b y th e w
i rem u s t b e e q u a l t o z e r o t o s a ti s fy t h e c o n d
i t i o n s o fe q u i l i b r i u m . T h e f o r c e s , 5 0 g l
i n g u a l a t t h e m o -l a r a n d 5 0 g b u c c a l a t t h e
c a n i n e , s u m t o z e ro .T h e 2 f o r c e s c re a t e a c
o u p l e e x e r t e d b y t h e w i r eo f 5 0 g 2 0 m m o r 1 ,
0 00 g - m m i n a c l o c k w i s ed i r e c t i o n , a s s h o w
n i n F i g u r e 1 8 A . T h e c o u p l eg e n e r a t e d a t t
h e m o l a r m u s t b e e q u a l a n d o p -p o s i t e , 1 ,0 0
0 g - r a m c o u n t e r c l o c k w i s e , f o r t h es u m o f
t h e m o m e n t s t o e q u a l z er o . T h e c o u p l ea t t h
e m o l a r i s i n a m e s i o l i n g u a l d i r e c t i o n w i
t h am a g n i t u d e o f 1 ,0 0 0 g - m m .
~ [ ~ ~ 5 0g1,000-mm l0g 20 mm ........ .
B f s
F i g u r e 1 8 . P r e d i c t i n g t h e r e s u l t o f a n
a p p l i a n c e a c t i -v a t i o n . A 1 - c o u p l e a p p l
i a n c e i s u s e d t o m o v e a c a n i n eb u c c a l ly , o u
t o f cr o s s b i te . T h e f o r c e s y s t em e x e r t e d b
yt h e a p p l i a n c e i n t h e o c c l u s a l v i e w A ) . F
o r c e s e x e r t e db y t h e a p p l i a n c e a t t h e b r a
c k e t s b l a c k ) a n d t h e e q u i v -a l e n t t b r c e s
y s te m s a t t h e c e n t e r s o f r e s i s t a n c e g r a y
)f r o m a m e s i a l a s p e c t B ) .
T h e s e c o n d s t e p i n d e t e r m i n i n g h o w t e e
t hw i l l m o v e i s t o d i s c e r n t h e e q u i v a l e n t
f o r c e s ys -t e m s e x e r t e d a t th e c e n t e r s o f r
e s is t a n c e o f th et e e t h i n v o l v e d . T h e s e a r
e s h o w n i n F i g u r e 1 8 B.T h e r e i s a l i n g u a l f o
r c e o f 5 0 g a t th e m o l a rb r a c k e t a n d a m e s i o l
i n g u a l c o u p l e o f 1 ,0 00g - m m . I f t h e c e n t e r
o f re s i s t a n ce o f t h e m o l a r i s1 0 m m a p i c a l t
o t h e b r a c k e t , t h e e q u i v a l e n tf o r c e s y s te
m a t t h e m o l a r c e n t e r o f r e si s t a n c e is5 0 g o
f l i n g u a l f o r c e p l u s a m o m e n t o f 10 m m 5 0 g o
r 5 0 0 g - m m , t e n d i n g t o t i p t h e c r o w nl i n g u
a l l y . I n a d d i t i o n , t h e r e i s a m o m e n t o f 1
,0 0 0g - m m , t e n d i n g t o r o t a te t h e m o l a r m e s
i o l i n -g u a l l y .
A t t h e c a n i n e b r a c k e t t h e r e i s a b u c c a l
f o r c eo f 50 g . I f t h e c a n i n e c e n t e r o f r e s i s
t a n c e i s 1 0m m a p i c a l t o th e c a n i n e b r a c k e t
, t h e e q u i v a l e n tf o r c e s y s t e m a t t h e c a n i
n e c e n t e r o f r e s i s t a n c e is5 0 g o f b u c c a l f o
r c e p l u s a m o m e n t o f 10 m m 5 0 g o r 5 0 0 g - r a m ,
t e n d i n g t o t i p t h e c r o w nb u c c a ll y . T h e m o l
a r w i ll , t h e r e f o r e , t e n d t o m o v el i n g u a l l
y , i t s c r o w n w i ll t ip l i n g u a l l y , a n d t h et o
o t h w i l l r o t a t e m e s i o l i n g u a l ly . T h e c a n
i n e w i llm o v e b u c c a l l y a n d i ts c r o w n w i l l t
i p b u c c a l l y .
D e s i g n i n g a n O r t h o d o n t i c p p l i a n c eT h e
s t e p s o f f o r c e s y s t e m a n a l y s i s a r e r e v e r
s e d i ft h e o b j e c t i v e is d e s i g n i n g a n a p p l i
a n c e t o m o v et e e t h p r e d i c t a b l y . F ir s t , t h
e d e s i r e d f o r c e s y s t em sa t t h e c e n t e r s o f r
e s i s t a n c e a re d e t e r m i n e d .F r o m t h is , th e f
o r c e s y s t e m s r e q u i r e d a t th eb r a c k e t s c a n
b e c a l c u l a t e d a n d a n a p p l i a n c e d e -s i g n e
d t o d e l i v e r t h e d e s i r e d f o r c e s a n d m o -m e
n t s .
A s a n e x a m p l e , a n a p p l i a n c e d e s i g n e d t
oc l o se a p r e m o l a r e x t r a c t i o n s it e b y m o l a
r t r an s -l a t i o n a n d c a n i n e t r a n s l a t i o n i s
i l l u s t ra t e d i nF i g u r e 1 9 . I f t h e m a g n i t u d
e o f th e s p a c e c l o s i n gf o r c e d e s i r e d i s 1 0 0
g , a m e s i a l f o r c e a t t h e m o l a rc e n t e r o f r e
s i s t a n c e o f 1 00 g a n d a d i s t a l f o r c e o f1 0 0 g
a t t h e c a n i n e c e n t e r o f r e s i s t a n c e w i ll b
en e c e s s a r y , a s s h o w n i n F i g u r e 1 9 A . T h i s
w i l l p r o -d u c e m o l a r t r a n s l a t i o n m e s i a l
l y a n d c a n i n et r a n s l a t i o n d i s t a l l y .
T h e f o r c e s y st e m s r e q u i r e d a t t h e b r a c k
e t sF i g 1 9 B ) t h a t w i l l r e s u l t i n t h e s e c e n
t e r - o f - r e si s -
t a n c e e q u i v a l e n ts m u s t t h e n b e d e t e r m i
n e d . I ft h e m o l a r c e n t e r o f r es i s t a n c e is 10
m m a p i c a l t ot i l e b r a c k e t , a m e s i a l f o r c e
o f 1 0 0 g a t t h e m o l a rb r a c k e t w o u l d p r o d u c
e a f o r c e o f 1 00 g a t th e
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Basics of Mechanics 15
it ively. Un wa nte d effects are correc ted by adjust-ments
along the way. The f inal result is achievedby a series of well-p
la nned m echan ical interven-t ions that in it iate and sustain a
controlled bio-logic react ion.
T h e c o m p o n e n t s o f c o m p r e h e n s i v e o r t h
-odon t ic treatment prel iminary al ignm ent over-bite control
space closure root paralle l ing andf inishing rely on a series of
b iom echa nica l pro-cesses. Th e ch oice o f ap p lian ces an d t
ech n iq u e sused by pract it ioners varies radical ly among
in-dividuals but the funda ment al forces and mo-ments they produce
are universal . Applianceswill always act according to the laws of
physics.Un d ers t an d in g t h e b as ic b iomech an ica l p r in
c i -p les in volved in e f f ec tin g con t ro l led t oot h
move-men t mak es ach iev in g su cces s f u l or t h od on t ict
reat men t ou t comes more p red ic t ab le an d con -sistent.
A c k n o w l e d g m e n tThe a uthor thanks Ms. Carol Wilkins
for her h elp i n prepar-ing the illustrations for this
article.
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